A COMPOSITIONAL SIMULATION MODEL FOR
CARBON DIOXIDE FLOODING WITH
IMPROVED FLUID TRAPPING
by
Jeffrey S. Brown
c Copyright by Jeffrey S. Brown, 2014
All Rights Reserved
A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines
in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Petroleum Engineering).
Golden, Colorado
Date
Signed:
Jeffrey S. Brown
Signed:
Dr. Hossein Kazemi
Thesis Advisor
Golden, Colorado
Date
Signed:
Dr. Erdal Ozkan
Professor and Interim Head
Department of Petroleum Engineering
ii
ABSTRACT
A new formulation for fluid trapping using a dual-media approach which includes compositional
trapping and interphase mass transfer was developed, coded, and validated. This formulation does
not exist in notable commercial reservoir simulators. The formulation was incorporated into a
three-dimensional, three-phase, parallel compositional simulator to simulate carbon dioxide (CO2 )
water-alternating-gas (WAG) injection. Fluid phase trapping is both a channeling issue and a porescale issue. Pore-scale phase trapping is strongly related to hysteresis in the relative permeability
and capillary pressure; the simulator incorporates them in a methodology consistent with these
issues. New algorithms were developed to implement the CO2 solubility in water and oil and CO2
phase trapping in a way that preserves the mass balance of the oil, water, and gas phases. The new
simulator was implemented using a parallel infrastructure to facilitate computationally intensive
fine grid systems.
For test examples, we focused on a mixed wet carbonate reservoir in the Middle East. These
tests were used to evaluate the significance of various trapping scenarios. Compositional trapping,
gas relative permeability hysteresis, CO2 solubility in water, and permeability heterogeneity were
found to have significant impacts on oil recovery and timing, as well as CO2 storage and utilization
during waterflood and CO2 WAG processes.
iii
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1
2.2
2.3
Enhanced Oil Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1
Miscible Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2
Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.3
Other Enhanced Oil Recovery Methods . . . . . . . . . . . . . . . . . . . . . 5
CO2 Enhanced Recovery and Sequestration . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1
CO2 Enhanced Oil Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2
CO2 Flood Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3
CO2 WAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.4
CO2 Sequestration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.5
CO2 Simulation with TOUGH . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.6
CO2 Water Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.7
CO2 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.8
Other Articles on CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . 10
Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
iv
2.3.1
Computation Approaches in Reservoir Simulation . . . . . . . . . . . . . . 10
2.3.2
Fractured Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3
Compositional Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . 12
2.3.4
CO2 and Miscible Flood Simulation . . . . . . . . . . . . . . . . . . . . . . 13
2.3.5
Parallel Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.6
Simulation of Trapping and Bypassing . . . . . . . . . . . . . . . . . . . . 15
2.3.7
Simulation of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.8
Additional Simulation Topics
2.3.9
SPE Comparative Solution Projects . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4
Geologic Characterization in Middle East . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5
Relative Permeability and Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . 17
2.6
2.5.1
General Articles on Relative Permeability
. . . . . . . . . . . . . . . . . . 17
2.5.2
General Articles on Capillary Pressure . . . . . . . . . . . . . . . . . . . . 18
2.5.3
Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.4
Three-Phase Relative Permeability . . . . . . . . . . . . . . . . . . . . . . 18
2.5.5
Relative Permeability Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.6
Capillary Pressure Hysteresis
2.5.7
Combined Relative Permeability and Capillary Pressure Hysteresis . . . . 22
2.5.8
Non-zero Relative Permeability Derivative . . . . . . . . . . . . . . . . . . 23
2.5.9
Additional Relative Permeability Effects . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . . . . . 21
Equation of State Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.1
Calculation of Equation of State . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.2
Adjusting Equation of State Parameters . . . . . . . . . . . . . . . . . . . 25
2.6.3
Modifications to Equation of State Model when CO2 is Present . . . . . . 25
v
2.7
2.8
2.9
2.6.4
Phase Behavior Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.5
Other Equation of State References . . . . . . . . . . . . . . . . . . . . . . 26
Pore Scale Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7.1
Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7.2
Micro Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7.3
Additional Pore Scale Simulation Discussion . . . . . . . . . . . . . . . . . 28
Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8.1
Interfacial Tension Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8.2
Interfacial Tension and Relative Permeability
2.8.3
Spreading Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8.4
Interfacial Tension Fit Gas-Oil . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8.5
Water Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.8.6
CO2 -Brine Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . 30
. . . . . . . . . . . . . . . . 29
Liquid-Liquid-Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.10 Asphaltenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
CHAPTER 3 COMPOSITIONAL RESERVOIR SIMULATION OVERVIEW . . . . . . . . 36
3.1
Compositional Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2
Commercial Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3
Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4
Partially Implicit Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1
IMPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2
IMPSEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.3
Fully Implicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.4
Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
vi
3.5
Thermodynamic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6
Typical Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.7
Off-Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.8
Well Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.9
Right Hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.10 Total Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.11 Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.12 Accumulation Pressure Derivatives
. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.13 Accumulation Saturation Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.14 Accumulation Composition Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.15 Pressure Spatial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.16 Fugacity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.17 Computation for Fixed Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.18 Computation for Fixed Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.19 Additional Implicit Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
CHAPTER 4 MATHEMATICAL FORMULATION OVERVIEW . . . . . . . . . . . . . . . 57
4.1
Primary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2
Secondary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1
Calculation of Secondary Variables . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2
Storage of Secondary Variables . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3
List of Secondary Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3
Overview of Simulation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4
Assemble the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.1
Single Medium (No Trapping) . . . . . . . . . . . . . . . . . . . . . . . . . 79
vii
4.4.2
Degenerate Case with Oil and Water Only . . . . . . . . . . . . . . . . . . 79
4.4.3
Degenerate Case with Gas and Water Only . . . . . . . . . . . . . . . . . . 79
4.4.4
Degenerate Case with Gas and Oil Only . . . . . . . . . . . . . . . . . . . 80
4.4.5
Degenerate Case with Water Only . . . . . . . . . . . . . . . . . . . . . . . 80
4.4.6
Degenerate Case with Oil Only . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4.7
Degenerate Case with Gas Only . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.8
Three-Phase Degenerate Case with Fewer Components . . . . . . . . . . . 81
4.5
Rewrite Base Equations for Um Solve . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6
Update Primary Variables at Each Nonlinear Iteration . . . . . . . . . . . . . . . . 85
4.7
Update Primary Variables at Each Nonlinear Iteration: Flash . . . . . . . . . . . . 87
4.8
Update WCO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
CHAPTER 5 TRAPPING FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1
Trapping Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2
Initialize Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3
Update Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.2
Mass at Time n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.3
Transfer Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.4
Update Mole Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.5
Compute the Volumes
5.3.6
Compute the Saturations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4
Single Porosity Irreversible Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5
Dual Porosity as Reversible Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6
Dual Porosity Computation Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
viii
5.7
5.6.1
Implicit Pm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6.2
Explicit Pm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6.3
Implicit τ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6.4
Explicit τ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Computation of the Solution of a Dual Porosity System . . . . . . . . . . . . . . . . 106
CHAPTER 6 TIME DERIVATIVES FORMULATION . . . . . . . . . . . . . . . . . . . . . 108
6.1
Pressure Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2
Saturation Derivatives
6.3
Composition Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
CHAPTER 7 SPACE DERIVATIVES FORMULATION . . . . . . . . . . . . . . . . . . . . 111
7.1
Initial Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2
Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.3
Expand Deltas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.4
Expand Terms on Left-Hand-Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.5
Rearrange Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.6
Combine Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.7
Upstream Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.8
Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
CHAPTER 8 EQUATION OF STATE FORMULATION . . . . . . . . . . . . . . . . . . . . 119
8.0.1
Expand Fugacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.1
Fugacity Equations - Above Bubble Point . . . . . . . . . . . . . . . . . . . . . . . . 120
8.2
Fugacity Equations - Below Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.3
Method for Peng-Robinson Flash Calculation
8.3.1
. . . . . . . . . . . . . . . . . . . . . 122
Peneloux Volume Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . 123
ix
8.3.2
Constants for This Formulation . . . . . . . . . . . . . . . . . . . . . . . . 123
8.3.3
Initial Values, Compute Km , (Full Flash Only)
8.3.4
Flash to Calculate the Vapor Fraction, (Full Flash Only) . . . . . . . . . . 125
8.3.5
Calculate the Mixing Parameters . . . . . . . . . . . . . . . . . . . . . . . 126
8.3.6
Calculate the z̆-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3.7
Calculate the Fugacities f (Not if Only Computing z̆)
8.3.8
Calculate the Tolerance (Full Flash Only) . . . . . . . . . . . . . . . . . . 127
8.3.9
Calculate the Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.3.10
Calculate the saturations (Full Flash Only) . . . . . . . . . . . . . . . . . . 129
. . . . . . . . . . . . . . . 124
. . . . . . . . . . . 127
8.4
Evaluate Fugacity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.5
Evaluate Peng-Robinson Pressure Derivatives
8.6
8.7
. . . . . . . . . . . . . . . . . . . . . 130
8.5.1
Evaluate
∂ξ
∂P
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5.2
Evaluate
∂ z̆
∂P
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5.3
Evaluate Derivatives of f (z̆) . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.5.4
Evaluate Derivatives of A and B . . . . . . . . . . . . . . . . . . . . . . . . 132
Evaluate Peng-Robinson Composition Derivatives . . . . . . . . . . . . . . . . . . . 132
8.6.1
Evaluate
∂ξ
∂Xm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.6.2
Evaluate
∂ z̆
∂Xm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.6.3
Evaluate Derivatives of A and B . . . . . . . . . . . . . . . . . . . . . . . . 133
8.6.4
Evaluate
∂a
∂Xm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.6.5
Evaluate
∂b
∂Xm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Check Fugacity Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.7.2
Fugacity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
x
8.8
8.9
8.7.3
Pressure Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.7.4
Composition Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.7.5
Consistency Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Solving Cubic Equations Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.8.1
Initialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.8.2
Three Distinct Real Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.8.3
One Real Root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.8.4
Three Real Roots, Two or More Coincide . . . . . . . . . . . . . . . . . . . 142
8.8.5
Newton Raphson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Fugacity Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.10 Flash Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.11 Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
CHAPTER 9 FORMULATION OF WELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.1
Well Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2
Flow from Node to Well
9.3
Well Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.4
Properties for Flow in Wellbore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.5
Pressure in Wellbore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.6
Compute the Moody Friction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.7
Computation for Fixed Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9.8
Computation for Fixed Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.9
Wells with Single Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.9.1
Fixed Pressure Producer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.9.2
Fixed Rate Producer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
xi
9.9.3
Fixed Mole Rate Producer . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.9.4
Fixed Pressure Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.9.5
Fixed Rate Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.9.6
Fixed Pressure Producer with Switch to Rate Control . . . . . . . . . . . . 165
9.9.7
Fixed Rate Producer with Switch to Pressure Control . . . . . . . . . . . . 166
9.9.8
Fixed Pressure Injector with Switch to Rate Control . . . . . . . . . . . . 167
9.9.9
Fixed Rate Injector with Switch to Pressure Control . . . . . . . . . . . . 168
CHAPTER 10 MASS BALANCE CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . 170
10.1 Calculate Surface Conditions of Well Fluids Using Separators
. . . . . . . . . . . . 170
10.2 Calculate Surface Conditions of Original Oil in Place . . . . . . . . . . . . . . . . . 173
10.3 Mass Balance Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
CHAPTER 11 RELATIVE PERMEABILITY AND CAPILLARY PRESSURE . . . . . . . . 177
11.1 Three Phase Relative Permeability
. . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.2 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.2.1
Hysteresis Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.2.2
Hysteresis Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.2.3
Combined Three-Phase Relative Permeability and Hysteresis . . . . . . . . 180
11.2.4
Combined Analysis of Algorithms . . . . . . . . . . . . . . . . . . . . . . . 181
11.3 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11.3.1
Composition of Trapped Phase
. . . . . . . . . . . . . . . . . . . . . . . . 183
11.3.2
Simple Trapping Composition . . . . . . . . . . . . . . . . . . . . . . . . . 183
11.3.3
Complex Trapping Composition . . . . . . . . . . . . . . . . . . . . . . . . 184
11.3.4
Composition Trapping Formulation . . . . . . . . . . . . . . . . . . . . . . 184
11.4 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
xii
11.4.1
Interfacial Tension Literature . . . . . . . . . . . . . . . . . . . . . . . . . 185
11.5 Rock Type and Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.5.1
Rock Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.5.2
Wettability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.5.3
Static Wettability Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
11.5.4
Dynamic Wettability Changes . . . . . . . . . . . . . . . . . . . . . . . . . 187
11.6 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.7 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.8 Flow Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.9 Brooks-Corey Properties for Mixed Wet Rock . . . . . . . . . . . . . . . . . . . . . 190
11.9.1
Simplified Three-Phase Relative Permeability . . . . . . . . . . . . . . . . 190
11.9.2
Derivatives of Simplified Three-Phase Relative Permeability . . . . . . . . 192
11.9.3
Two-Phase Relative Permeabilities . . . . . . . . . . . . . . . . . . . . . . 193
11.9.4
Water-Oil Capillary Pressure for Mixed-Wet Systems . . . . . . . . . . . . 194
11.9.5
Gas-Oil Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
11.9.6
Derivatives of Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . 196
11.10 Three Phase Relative Permeability References . . . . . . . . . . . . . . . . . . . . . 196
11.11 Hysteresis References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.12 Combined Three-Phase Relative Permeability and Hysteresis References . . . . . . 198
CHAPTER 12 VISCOSITY FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . 200
12.1 Treatment of Viscosity by Commercial Applications . . . . . . . . . . . . . . . . . . 200
12.2 Other Viscosity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
12.3 Lohrenz-Brae-Clark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
12.3.1
Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
xiii
12.3.2
Time-Dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
12.4 Jossi plus Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
12.5 Corresponding States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.5.1
Methane Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.5.2
Methane Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
12.5.3
Corresponding States Calculations . . . . . . . . . . . . . . . . . . . . . . . 210
12.5.4
Heavy oil adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
12.6 Extended Corresponding States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.6.1
n-Decane Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.6.2
n-Decane Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12.6.3
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12.7 f -Theory Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
12.7.1
Dilute Gas Viscosity and General Properties . . . . . . . . . . . . . . . . . 217
12.7.2
f -Theory Friction Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 217
12.7.3
Mixing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
CHAPTER 13 FORMULATION FOR PROPERTIES OF WATER CONTAINING CO2 . . . 220
13.1 CO2 Solubility in Water
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
13.2 Adjustments to Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
13.3 Other Special Properties of CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
13.4 Properties of Water Containing CO2 , Overview . . . . . . . . . . . . . . . . . . . . 222
13.5 Commercial Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
13.6 Properties of Water Containing CO2 , CMG GEM . . . . . . . . . . . . . . . . . . . 223
13.7 Units of concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
13.8 Selection Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
xiv
13.8.1
Rowe, Brine Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
13.8.2
Garcı́a, CO2 Brine Density and Partial Molar Volume . . . . . . . . . . . . 227
13.8.3
Kestin, Brine Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
13.8.4
Duan, Henry’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
13.9 Correlations for this Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
13.10 Computational Forms of WCO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
13.10.1 Option 0: WCO2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
13.10.2 Option C: Constant WCO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
13.10.3 Option ZW0: Compute ξw using WCO2 = 0 . . . . . . . . . . . . . . . . . . 231
13.10.4 Option ZW1: Compute ξw using WCO2 . . . . . . . . . . . . . . . . . . . . 232
13.10.5 Option KP1: Use a simplified model for WCO2 using YCO2 [Pb ] . . . . . . . 232
13.10.6 Option KP2: Use a simplified model for WCO2 using YCO2 below the
bubble point and YCO2 = 0 above the bubble point . . . . . . . . . . . . . 233
13.10.7 Option KP3: Use a simplified model for WCO2 using YCO2 [Pb ] . . . . . . . 234
n+1
fully implicit . . . . . . . . . . . . . . . . . . . . . . . . . 234
13.10.8 Option 1: WCO
2
n+1
implicit pressure, explicit fugacity coefficient . . . . . . . 235
13.10.9 Option 2: WCO
2
n+1
implicit pressure, explicit fugacity . . . . . . . . . . . . . 235
13.10.10 Option 3: WCO
2
n+1
implicit pressure, fugacity at . . . . . . . . . . . . . . . 236
13.10.11 Option 4: WCO
2
n+1
implicit pressure, explicit fugacity coefficient . . . . . . 236
13.10.12 Option 2Z: WZ,CO
2
n+1
implicit pressure, explicit fugacity . . . . . . . . . . . . 237
13.10.13 Option 3Z: WZ,CO
2
n+1
implicit pressure, fugacity at . . . . . . . . . . . . . . 237
13.10.14 Option 4Z: WZ,CO
2
n+1
partially implicit, function of both Xm and Ym
13.10.15 Option 1XY: WCO
2
. . . 238
n+1
fully implicit . . . . . . . . . . . . . . . . . . . . . . . . 240
13.10.16 Option Y1: WCO
2
n
explicit . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
13.10.17 Option Y5: WCO
2
xv
n+1
13.10.18 Option P1: WCO
[P only] fully implicit . . . . . . . . . . . . . . . . . . . . 241
2
n+1
[YCO2 only], evaluate Y at . . . . . . . . . . . . . . . . 242
13.10.19 Option K1: WCO
2
n+1
[YCO2 only], evaluate Y at . . . . . . . . . . . . . . . . 243
13.10.20 Option K2: WCO
2
13.10.21 Using WCO2 as a Transfer Term . . . . . . . . . . . . . . . . . . . . . . . . 243
13.10.22 Rowe, Brine Density, Eclipse + VIP+CMG, H2 O + NaCl, ρw + Cw . . . . 245
13.10.23 Garcı́a, CMG, Brine Density, H2 O + CO2 + NaCl, ρw + v̄CO2 . . . . . . . . 246
13.10.24 Kestin, Brine Viscosity, Eclipse+VIP, H2 O + NaCl, μw . . . . . . . . . . . 250
13.10.25 Duan, Henry’s Law, H2 O + CO2 + NaCl, WCO2 + HCO2
. . . . . . . . . . 251
13.11 Correlations Used to Evaluate Other Correlations . . . . . . . . . . . . . . . . . . . 253
13.11.1 Zeebe, Henry’s Law for Seawater, H2 O + CO2 + NaCl, H . . . . . . . . . . 254
13.11.2 Duan, Fugacity, H2 O + CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . 254
13.12 Henry’s Law Correlations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
13.12.1 Chang, Mole Fraction, Eclipse + VIP, H2 O + CO2 + NaCl,
WCO2 + Rsw + Bw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
13.12.2 CMG, Henry’s Law, H2 O + CO2 + NaCl, WCO2 + HCO2
. . . . . . . . . . 257
13.12.3 Enick, Henry’s Law, H2 O + CO2 + NaCl, H + Rsw + WCO2 + μw . . . . . 258
13.13 Adjustments to Peng-Robinson Equation of State . . . . . . . . . . . . . . . . . . . 259
13.13.1 Peng-Robinson Equation of State Paramters . . . . . . . . . . . . . . . . . 259
13.13.2 Soreide, EOS, Eclipse, H2 O + CO2 + NaCl, WCO2 + ρaq . . . . . . . . . . . 260
13.13.3 Delshad, EOS and IFT, H2 O + CO2 + NaCl, WCO2 + ρaq + σgw . . . . . . 261
13.13.4 Yan, EOS, H2 O + CO2 + NaCl, WCO2 + ρaq . . . . . . . . . . . . . . . . . 262
13.13.5 Melhem, EOS, H2 O + CO2 , WCO2 + ρaq . . . . . . . . . . . . . . . . . . . 262
13.13.6 Spycher, EOS, Eclipse, H2 O + CO2 + NaCl, WCO2 + ρaq . . . . . . . . . . 263
13.14 Models Considered But Not Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
xvi
CHAPTER 14 COMPUTATION: ASSEMBLY OF JACOBIAN . . . . . . . . . . . . . . . . . 265
14.1 Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
14.2 Diagonal Terms Above the Bubble Point . . . . . . . . . . . . . . . . . . . . . . . . 266
14.3 Diagonal Terms Below the Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14.4 Off-Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14.5 Well Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
14.6 Right Hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
14.7 Total Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
14.8 Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
14.9 Accumulation Derivatives: Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
14.10 Accumulation Derivatives: Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . 271
14.11 Accumulation Derivatives: Composition . . . . . . . . . . . . . . . . . . . . . . . . . 272
14.12 Spatial Derivatives: Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
14.13 Fugacity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
14.14 Fugacity Equations - Above Bubble Point . . . . . . . . . . . . . . . . . . . . . . . . 274
14.15 Fugacity Equations - Below Dew Point . . . . . . . . . . . . . . . . . . . . . . . . . 275
14.16 Computation for Fixed Rate Wells
. . . . . . . . . . . . . . . . . . . . . . . . . . . 275
14.17 Computation for Fixed Pressure Wells
. . . . . . . . . . . . . . . . . . . . . . . . . 276
14.18 Additional Comments on Computation . . . . . . . . . . . . . . . . . . . . . . . . . 277
14.19 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
14.20 Illustration of Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
14.20.1 Illustration of a 5 × 5 × 3 Model, Well Geometry
. . . . . . . . . . . . . . 279
14.20.2 Illustration of a 5 × 5 × 3 Model, Block Values . . . . . . . . . . . . . . . . 280
14.20.3 Illustration of a 5 × 5 × 3 Model, Matrix Assembly . . . . . . . . . . . . . 282
xvii
14.20.4 Illustration of a 5 × 5 × 3 Model, Local LU Decomposition . . . . . . . . . 282
14.20.5 Illustration of a 5 × 5 × 3 Reduced Model
. . . . . . . . . . . . . . . . . . 290
14.20.6 Illustration of a 16 × 16 × 3 Model . . . . . . . . . . . . . . . . . . . . . . 290
CHAPTER 15 COMPUTATION: DESCRIPTION OF LINEAR SOLVERS . . . . . . . . . . 295
15.1 Serial Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
15.1.1
Dense Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 295
15.1.2
Band Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 296
15.1.3
Special Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . 298
15.1.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
15.2 Parallel Solvers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
15.2.1
Direct LU Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
15.2.2
Iterative Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
15.2.3
Parallel Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
CHAPTER 16 COMPUTATATION: PARALLEL COMPUTING . . . . . . . . . . . . . . . . 304
16.1 Computation Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
16.2 Solution Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
16.3 Initialize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
16.4 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
16.4.1
Computation Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
16.4.2
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
CHAPTER 17 VALIDATION CASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
17.1 Validation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
17.2 Description of model 760E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
17.3 Description of model 761E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
xviii
17.4 Description of model 762E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
17.5 Description of model 760F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
17.6 Description of model 761F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
17.7 Description of model 762F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
17.8 Description of model 760G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
17.9 Description of model 761G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
17.10 Description of model 762G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
17.11 Compare CMG Model with my Model 760E and 761E . . . . . . . . . . . . . . . . . 335
17.12 Compare CMG Model with my Model 762E . . . . . . . . . . . . . . . . . . . . . . 339
17.13 Compare CMG Model with my Model 760F, 761F, and 762F . . . . . . . . . . . . . 341
17.14 Compare CMG Model with my Model 760G . . . . . . . . . . . . . . . . . . . . . . 344
17.15 Compare CMG Model with my Model 761G . . . . . . . . . . . . . . . . . . . . . . 346
17.16 Compare CMG Model with my Model 762G . . . . . . . . . . . . . . . . . . . . . . 349
CHAPTER 18 CASE STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
18.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
18.2 Variations in Porosity and Permeability . . . . . . . . . . . . . . . . . . . . . . . . . 356
18.3 Relative Permeability Test Case Literature Review
. . . . . . . . . . . . . . . . . . 359
18.3.1
Water-Oil Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
18.3.2
Gas-Oil Data
18.3.3
Gas-Water Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
18.3.4
Two-Phase Experiments with Different Phases . . . . . . . . . . . . . . . . 365
18.3.5
Three-Phase Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
18.3.6
Relative Permeability Formulations . . . . . . . . . . . . . . . . . . . . . . 366
18.3.7
Relative Permeability Observations . . . . . . . . . . . . . . . . . . . . . . 366
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
xix
18.4 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
18.4.1
Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
18.4.2
Oil/Water Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
18.4.3
Gas/Oil Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
18.4.4
Trapped Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
18.4.5
Trapped Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
18.4.6
Cycle Dependent Residual Oil Saturations . . . . . . . . . . . . . . . . . . 372
18.4.7
Water Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 372
18.4.8
Gas Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
18.4.9
Oil Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
18.5 Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
18.6 Future Test Case Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
CHAPTER 19 DISCUSSION OF RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
19.1 Evaluation of Primary Production Performance . . . . . . . . . . . . . . . . . . . . 395
19.2 Evaluation of Waterflood Performance
. . . . . . . . . . . . . . . . . . . . . . . . . 395
19.3 Evaluation of Continuous CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . . 398
19.4 Evaluation of CO2 WAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
19.5 Evaluation of Compositional Recovery Factor
. . . . . . . . . . . . . . . . . . . . . 409
19.6 Evaluation of CO2 Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
19.7 Evaluation of CO2 Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
CHAPTER 20 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
CHAPTER 21 RECOMMENDED FUTURE WORK
. . . . . . . . . . . . . . . . . . . . . . 423
21.1 Use of This Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
21.2 Formulation and Computation Enhancements . . . . . . . . . . . . . . . . . . . . . 423
xx
21.3 Phase Labeling and Relative Permeability Experiments . . . . . . . . . . . . . . . . 423
CHAPTER 22 NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
APPENDIX - RESULTS FOR SPECIFIC TEST CASES . . . . . . . . . . . . . . . . . . . . 480
A.1
Primary Production Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
A.2
Waterflood Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
A.3
Continuous CO2 Injection Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
A.4
WAG Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
xxi
LIST OF FIGURES
Figure 2.1
Low temperature phase behavior of Wasson crude showing the presence of
two liquid hydrocarbon phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 2.2
Various miscibility regions for a CO2 flood,
Figure 3.1
Block 2: block geometry for the off-block diagonal values with the IMPES
formulation for a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 3.2
Block 4: well terms for the component equations for a NC = 5 problem. . . . . 44
Figure 3.3
Block 6: right-hand-side terms for the component equations for a NC = 5
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 3.4
Block 5: blocks for the well equations for a NC = 5 problem. . . . . . . . . . . 46
Figure 8.1
Illustration of amn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Figure 8.2
Regular flash calculation flow chart.
Figure 8.3
Flash calculation flow chart for thermodynamic minimum miscibility
pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Figure 11.1
Illustration of pore doublet effect in a water-wet rock; green is oil and blue is
water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Figure 11.2
Variation of relative permeability with wettability changes . . . . . . . . . . . 188
Figure 12.1
Compare the density correlation for methane to Pedersen Figure 10.3. . . . . . 207
Figure 12.2
Compare the viscosity correlation for methane to Pedersen Figure 10.4. . . . . 209
Figure 12.3
Compare the Hanley viscosity correlation for methane to Gonzalez Figure 2. . 210
Figure 13.1
Solubility of methane in water . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Figure 13.2
Solubility of CO2 in water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Figure 13.3
Change in z-factor as a function of pressure for CO2 . . . . . . . . . . . . . . . 222
Figure 14.1
Block 1: block geometry for the main block diagonal of a NC = 5 problem. . . 265
Figure 14.2
Block 7: block geometry above the bubble point for the main block diagonal
of a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
xxii
. . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . 149
Figure 14.3
Block 7: block geometry below the dew point for the main block diagonal of
a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Figure 14.4
Block 2: block geometry for the off-block diagonal values with the IMPES
formulation for a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . 268
Figure 14.5
Block 4: well terms for the component equations for a NC = 5 problem. . . . . 268
Figure 14.6
Block 6: right-hand-side terms for the component equations for a NC = 5
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Figure 14.7
Block 5: blocks for the well equations for a NC = 5 problem. . . . . . . . . . . 270
Figure 14.8
Geometry of three horizontal wells for a 5 × 5 × 3 problem.
Figure 14.9
Matrix 0: block banded matrix for a 5 × 5 × 3 problem. . . . . . . . . . . . . . 280
Figure 14.10
Block 1: block geometry for the main block diagonal of a NC = 5 problem. . . 281
Figure 14.11
Block 2: block geometry for the off-block diagonal values with the IMPES
formulation for a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . 281
Figure 14.12
Block 3: block geometry for the off-block diagonal values with the IMPSEC
formulation for a NC = 5 problem. . . . . . . . . . . . . . . . . . . . . . . . . . 281
Figure 14.13
Block 4: well terms for the component equations for a NC = 5 problem. . . . . 281
Figure 14.14
Block 5: blocks for the well equations for a NC = 5 problem. . . . . . . . . . . 282
Figure 14.15
Matrix 1: Spatial derivatives for a 5 × 5 × 3 × 9 problem. . . . . . . . . . . . . 283
Figure 14.16
Matrix 2: Time derivatives for a 5 × 5 × 3 × 9 problem. . . . . . . . . . . . . . 284
Figure 14.17
Matrix 3: Combined matrix for a 5 × 5 × 3 × 9 problem. . . . . . . . . . . . . 285
Figure 14.18
Matrix 4: Well matrix for a 5 × 5 × 3 × 9 problem with three horizontal wells. . 286
Figure 14.19
Matrix 5: Combined matrix with wells for a 5 × 5 × 3 × 9 problem with three
horizontal wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Figure 14.20
or P Matrix 6: Eliminate the qw
well well terms from the component
equations for a 5 × 5 × 3 × 9 problem with three horizontal wells. . . . . . . . 288
Figure 14.21
Matrix 7: Eliminate the off-band well terms from the component equations
for a 5 × 5 × 3 × 9 problem with three horizontal wells. . . . . . . . . . . . . . 289
Figure 14.22
Row 1: An example of a row without well terms for a 5 × 5 × 3 × 9 problem. . 289
xxiii
. . . . . . . . . . 280
Figure 14.23
The non-zero blocks of Row 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Figure 14.24
The non-zero columns of Row 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Figure 14.25
The non-zero columns from Row 1 are stored with the main block diagonal
first. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Figure 14.26
The result of local LU decomposition on Row 1 in the order they are stored. . 290
Figure 14.27
The result of local LU decomposition on Row 1. . . . . . . . . . . . . . . . . . 290
Figure 14.28
Matrix 8: banded matrix for a 5 × 5 × 3 problem without eliminating wells.
Figure 14.29
Matrix 9: banded matrix for a 5 × 5 × 3 problem. . . . . . . . . . . . . . . . . 291
Figure 14.30
Geometry of three horizontal wells for a 16 × 16 × 3 problem. . . . . . . . . . 292
Figure 14.31
Matrix 10: banded matrix for a 16 × 16 × 3 problem without eliminating
wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Figure 14.32
Upper left corner of Matrix 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Figure 14.33
Matrix 11: banded matrix for a 16 × 16 × 3 problem. . . . . . . . . . . . . . . 294
Figure 15.1
Jacobian matrix for a 3 × 1 × 1 system with NC = 5 and Nblock = 9. . . . . . . 295
Figure 15.2
The test case after the first stage of Gaussian elimination. . . . . . . . . . . . 296
Figure 15.3
The test case after the second stage of Gaussian elimination. . . . . . . . . . . 296
Figure 15.4
The banded structure for the test case. . . . . . . . . . . . . . . . . . . . . . . 297
Figure 15.5
The test case after the first stage of Banded Gaussian elimination. . . . . . . . 297
Figure 15.6
The test case after the second stage of Banded Gaussian elimination. . . . . . 298
Figure 15.7
The sparse storage structure of the local LU solvers. . . . . . . . . . . . . . . . 299
Figure 15.8
The first step of the local LU solvers. . . . . . . . . . . . . . . . . . . . . . . . 299
Figure 15.9
The condensed matrix after the local LU decomposition. . . . . . . . . . . . . 300
Figure 15.10
The banded structure of the condensed matrix. . . . . . . . . . . . . . . . . . . 300
Figure 15.11
The condensed matrix after the first step of the band solve. . . . . . . . . . . . 300
Figure 15.12
The condensed matrix after the second step of the band solve. . . . . . . . . . 300
xxiv
. 291
Figure 15.13
Use the values from the condensed matrix solve to perform a back
substitution on each grid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Figure 16.1
Illustration of a group of 9 nodes with 8 processor cores each. . . . . . . . . . 305
Figure 16.2
Illustration of computations with a hybrid MPI/openMP 3 × 3 × 8 processor
grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Figure 16.3
Illustration of computations with an MPI 9 × 8 processor grid. . . . . . . . . . 306
Figure 16.4
Illustration of computations with a linear array of 72 processors. . . . . . . . . 306
Figure 16.5
Parallel boundary computations. . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Figure 16.6
Parallel boundary computations for a 3 × 3 processor grid. . . . . . . . . . . . 307
Figure 16.7
Parallel boundary computations for a 9 × 8 processor grid. . . . . . . . . . . . 307
Figure 16.8
Parallel computations for load balancing. . . . . . . . . . . . . . . . . . . . . . 308
Figure 16.9
Ra bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
Figure 16.10
Efficiency plot for Nx = 80, Ny = 80, and Nz = 15. . . . . . . . . . . . . . . . 315
Figure 16.11
Speedup plot for Nx = 80, Ny = 80, and Nz = 15. . . . . . . . . . . . . . . . . 315
Figure 16.12
Scalability plot for Nx = Ny , Nz = 15, and EP = 0.1. . . . . . . . . . . . . . . 316
Figure 16.13
Memory constrained scalability plot. . . . . . . . . . . . . . . . . . . . . . . . . 317
Figure 17.1
Production rates at reservoir conditions for model 760E. . . . . . . . . . . . . 320
Figure 17.2
Production pressure for model 760E. . . . . . . . . . . . . . . . . . . . . . . . 321
Figure 17.3
Saturation for equivalent one-cell model for model 760E. . . . . . . . . . . . . 321
Figure 17.4
Mole fraction for equivalent one-cell model for model 760E. . . . . . . . . . . . 321
Figure 17.5
Molar recovery factor for model 760E. . . . . . . . . . . . . . . . . . . . . . . . 322
Figure 17.6
Saturation for equivalent one-cell model for model 762E. . . . . . . . . . . . . 323
Figure 17.7
Injection rates at reservoir conditions for model 760F. . . . . . . . . . . . . . . 323
Figure 17.8
Injection pressures for model 760F. . . . . . . . . . . . . . . . . . . . . . . . . 323
Figure 17.9
Production rates at reservoir conditions for model 760F. . . . . . . . . . . . . 324
xxv
Figure 17.10
Production pressure for model 760F.
Figure 17.11
Saturation for equivalent one-cell model for model 760F. . . . . . . . . . . . . 325
Figure 17.12
Mole fraction for equivalent one-cell model for model 760F. . . . . . . . . . . . 325
Figure 17.13
Molar recovery factor for model 760F. . . . . . . . . . . . . . . . . . . . . . . . 325
Figure 17.14
Saturation for equivalent one-cell model for model 762F. . . . . . . . . . . . . 326
Figure 17.15
Injection rates at reservoir conditions for model 760G. . . . . . . . . . . . . . . 327
Figure 17.16
Injection pressures for model 760G. . . . . . . . . . . . . . . . . . . . . . . . . 327
Figure 17.17
Production rates at reservoir conditions for model 760G. . . . . . . . . . . . . 328
Figure 17.18
Production pressure for model 760G. . . . . . . . . . . . . . . . . . . . . . . . 328
Figure 17.19
Saturation for equivalent one-cell model for model 760G. . . . . . . . . . . . . 329
Figure 17.20
Mole fraction for equivalent one-cell model for model 760G. . . . . . . . . . . . 329
Figure 17.21
Molar recovery factor for model 760G. . . . . . . . . . . . . . . . . . . . . . . . 330
Figure 17.22
Injection rates at reservoir conditions for model 761G. . . . . . . . . . . . . . . 330
Figure 17.23
Injection pressures for model 761G. . . . . . . . . . . . . . . . . . . . . . . . . 331
Figure 17.24
Production rates at reservoir conditions for model 761G. . . . . . . . . . . . . 331
Figure 17.25
Production pressure for model 761G. . . . . . . . . . . . . . . . . . . . . . . . 331
Figure 17.26
Saturation for equivalent one-cell model for model 761G. . . . . . . . . . . . . 332
Figure 17.27
Mole fraction for equivalent one-cell model for model 761G. . . . . . . . . . . . 333
Figure 17.28
Molar recovery factor for model 761G. . . . . . . . . . . . . . . . . . . . . . . . 333
Figure 17.29
Injection rates at reservoir conditions for model 762G. . . . . . . . . . . . . . . 333
Figure 17.30
Injection pressures for model 762G. . . . . . . . . . . . . . . . . . . . . . . . . 334
Figure 17.31
Production rates at reservoir conditions for model 762G. . . . . . . . . . . . . 334
Figure 17.32
Production pressure for model 762G. . . . . . . . . . . . . . . . . . . . . . . . 335
Figure 17.33
Saturation for equivalent one-cell model for model 762G. . . . . . . . . . . . . 336
xxvi
. . . . . . . . . . . . . . . . . . . . . . . 324
Figure 17.34
Mole fraction for equivalent one-cell model for model 762G. . . . . . . . . . . . 336
Figure 17.35
Molar recovery factor for model 762G. . . . . . . . . . . . . . . . . . . . . . . . 336
Figure 17.36
Comparison of production rates for model 760E. . . . . . . . . . . . . . . . . . 337
Figure 17.37
Comparison of producer grid cell pressures for model 760E. . . . . . . . . . . . 337
Figure 17.38
Difference of producer grid cell pressures for model 760E. . . . . . . . . . . . . 338
Figure 17.39
Comparison of total molar rates for model 760E. . . . . . . . . . . . . . . . . . 338
Figure 17.40
Comparison of recovery factors for model 760E. . . . . . . . . . . . . . . . . . 338
Figure 17.41
Comparison of recovery factors for model 761E. . . . . . . . . . . . . . . . . . 339
Figure 17.42
Comparison of production rates for model 762E. . . . . . . . . . . . . . . . . . 339
Figure 17.43
Comparison of producer grid cell pressures for model 762E. . . . . . . . . . . . 340
Figure 17.44
Difference of producer grid cell pressures for model 762E. . . . . . . . . . . . . 340
Figure 17.45
Comparison of recovery factors for model 762E. . . . . . . . . . . . . . . . . . 340
Figure 17.46
Comparison of production rates for model 760F. . . . . . . . . . . . . . . . . . 341
Figure 17.47
Comparison of producer grid cell pressures for model 760F. . . . . . . . . . . . 341
Figure 17.48
Comparison of recovery factors for model 760F. . . . . . . . . . . . . . . . . . 342
Figure 17.49
Comparison of water saturation for model 760F at 500 days. . . . . . . . . . . 342
Figure 17.50
Comparison of recovery factors for model 761F. . . . . . . . . . . . . . . . . . 343
Figure 17.51
Comparison of recovery factors for model 762F. . . . . . . . . . . . . . . . . . 343
Figure 17.52
Comparison of production rates for model 760G. . . . . . . . . . . . . . . . . . 344
Figure 17.53
Comparison of producer grid cell pressures for model 760G. . . . . . . . . . . . 344
Figure 17.54
Comparison of recovery factors for model 760G. . . . . . . . . . . . . . . . . . 345
Figure 17.55
Comparison of gas saturation for model 760G at 500 days. . . . . . . . . . . . 345
Figure 17.56
Comparison of water saturation for model 760G at 1000 days. . . . . . . . . . 346
Figure 17.57
Comparison of gas saturation for model 760G at 1000 days. . . . . . . . . . . . 346
xxvii
Figure 17.58
Comparison of production rates for model 761G. . . . . . . . . . . . . . . . . . 347
Figure 17.59
Comparison of producer grid cell pressures for model 761G. . . . . . . . . . . . 347
Figure 17.60
Comparison of recovery factors for model 761G. . . . . . . . . . . . . . . . . . 348
Figure 17.61
Comparison of pressure profiles for model 761G at 1000 days. . . . . . . . . . 348
Figure 17.62
Comparison of water saturation for model 761G at 1000 days. . . . . . . . . . 348
Figure 17.63
Comparison of gas saturation for model 761G at 1000 days. . . . . . . . . . . . 349
Figure 17.64
Comparison of pressure profiles for model 761G at 1500 days. . . . . . . . . . 349
Figure 17.65
Comparison of water saturation for model 761G at 1500 days. . . . . . . . . . 350
Figure 17.66
Comparison of gas saturation for model 761G at 1500 days. . . . . . . . . . . . 350
Figure 17.67
Comparison of recovery factors for model 762G. . . . . . . . . . . . . . . . . . 350
Figure 18.1
Porosity distribution for Facies 5 of Jobe. . . . . . . . . . . . . . . . . . . . . . 357
Figure 18.2
Porosity-permeability correlation for Facies 5 of Jobe. . . . . . . . . . . . . . . 358
Figure 18.3
Porosity and permeability for Geostatistical Realization # 1. . . . . . . . . . . 359
Figure 18.4
Porosity and permeability for Geostatistical Realization # 2. . . . . . . . . . . 360
Figure 18.5
Porosity and permeability for Geostatistical Realization # 3. . . . . . . . . . . 361
Figure 18.6
Porosity and permeability for Geostatistical Realization # 4. . . . . . . . . . . 362
Figure 18.7
Porosity and permeability for Geostatistical Realization # 5. . . . . . . . . . . 363
Figure 18.8
Porosity and permeability for Geostatistical Realization # 6. . . . . . . . . . . 364
Figure 18.9
Oil and water relative permeability curves including the data points. . . . . . 369
Figure 18.10
Gas and oil relative permeability curves including the data points. . . . . . . . 370
Figure 18.11
Trapped gas saturation as a function of maximum achieved gas saturation. . . 371
Figure 18.12
Trapped oil saturation as a function of maximum oil saturation achieved
after the initial oil saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Figure 18.13
Water relative permeability based on a fit to the oil-water data. . . . . . . . . 373
Figure 18.14
Bounding scanning curves for gas relative permeability . . . . . . . . . . . . . 374
xxviii
Figure 18.15
A decreasing gas relative permeability scanning curve. . . . . . . . . . . . . . . 376
Figure 18.16
An increasing gas relative permeability scanning curve. . . . . . . . . . . . . . 378
Figure 18.17
Oil relative permeability based on from the oil/water SCAL . . . . . . . . . . 379
Figure 18.18
Oil relative permeability based on from the gas/oil SCAL . . . . . . . . . . . . 380
Figure 18.19
Compare the krow and the krog . For this data set the curves are very similar. . 380
Figure 18.20
max ≤ S
The krow scanning curves have no hysteresis because Sot
org < Sorw . . . . 381
Figure 18.21
max ≤ S
The krog scanning curves have no hysteresis because Sot
org < Sorw . . . . 381
Figure 18.22
A decreasing oil relative permeability scanning curve. . . . . . . . . . . . . . . 383
Figure 18.23
An increasing oil relative permeability scanning curve. . . . . . . . . . . . . . . 384
Figure 18.24
Oil-water capillary pressure curves including the data points. . . . . . . . . . . 386
Figure 18.25
Capillary pressure bounding curves and interpolated scanning curves. . . . . . 387
Figure 18.26
Decreasing capillary pressure scanning curve. . . . . . . . . . . . . . . . . . . . 388
Figure 18.27
Increasing capillary pressure scanning curve. . . . . . . . . . . . . . . . . . . . 390
Figure A.1
Primary Production Pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
Figure A.2
Primary Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
Figure A.3
Primary Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Figure A.4
Primary nonlinear iteration convergence. . . . . . . . . . . . . . . . . . . . . . 481
Figure A.5
Primary time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Figure A.6
Primary pressure for cells along diagonal between wells. . . . . . . . . . . . . . 482
Figure A.7
Primary total mass of CO2 for cells along diagonal between wells. . . . . . . . 482
Figure A.8
Primary total mass of hydrocarbons (no CO2 ) for cells along diagonal
between wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
Figure A.9
PRIM saturation for equivalent one cell model. . . . . . . . . . . . . . . . . . . 483
Figure A.10
Primary total mole fraction in the reservoir. . . . . . . . . . . . . . . . . . . . 483
Figure A.11
Primary recovery factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
xxix
Figure A.12
Primary compositional recovery factor. . . . . . . . . . . . . . . . . . . . . . . 484
Figure A.13
Distribution of pressures at primary economic limit.
. . . . . . . . . . . . . . 485
Figure A.14
2-D pressure distribution at primary economic limit.
. . . . . . . . . . . . . . 485
Figure A.15
Distribution of oil saturation at primary economic limit. . . . . . . . . . . . . 486
Figure A.16
2-D oil saturation distribution at primary economic limit.
. . . . . . . . . . . 486
Figure A.17
Distribution of gas saturation at primary economic limit.
. . . . . . . . . . . 487
Figure A.18
2-D gas saturation distribution at primary economic limit. . . . . . . . . . . . 487
Figure A.19
Distribution of water saturation at primary economic limit.
Figure A.20
2-D water saturation distribution at primary economic limit.
Figure A.21
Waterflood Injection Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
Figure A.22
Waterflood Injection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
Figure A.23
Waterflood Production Pressures. . . . . . . . . . . . . . . . . . . . . . . . . . 490
Figure A.24
Waterflood Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Figure A.25
Waterflood Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Figure A.26
WF − Primary Oil Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Figure A.27
Waterflood nonlinear iteration convergence. . . . . . . . . . . . . . . . . . . . . 491
Figure A.28
Waterflood time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Figure A.29
Waterflood pressure for cells along diagonal between wells. . . . . . . . . . . . 492
Figure A.30
Waterflood total mass of CO2 for cells along diagonal between wells. . . . . . . 492
Figure A.31
Waterflood total mass of hydrocarbons (no CO2 ) for cells along diagonal
between wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
Figure A.32
WF saturation for equivalent one cell model. . . . . . . . . . . . . . . . . . . . 493
Figure A.33
Waterflood total mole fraction in the reservoir.
Figure A.34
Waterflood recovery factor.
Figure A.35
Waterflood compositional recovery factor.
. . . . . . . . . . 488
. . . . . . . . . 488
. . . . . . . . . . . . . . . . . 493
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
xxx
. . . . . . . . . . . . . . . . . . . . 494
Figure A.36
Distribution of pressures at waterflood economic limit. . . . . . . . . . . . . . 495
Figure A.37
2-D pressure distribution at waterflood economic limit. . . . . . . . . . . . . . 495
Figure A.38
Distribution of oil saturation at waterflood economic limit.
Figure A.39
2-D oil saturation distribution at waterflood economic limit. . . . . . . . . . . 496
Figure A.40
Distribution of gas saturation at waterflood economic limit.
Figure A.41
2-D gas saturation distribution at waterflood economic limit.
Figure A.42
Distribution of water saturation at waterflood economic limit. . . . . . . . . . 498
Figure A.43
2-D water saturation distribution at waterflood economic limit.
Figure A.44
Continuous CO2 Injection Pressure. . . . . . . . . . . . . . . . . . . . . . . . . 499
Figure A.45
Continuous CO2 Injection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 499
Figure A.46
Continuous CO2 Production Pressures. . . . . . . . . . . . . . . . . . . . . . . 500
Figure A.47
Continuous CO2 Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . 500
Figure A.48
Continuous CO2 Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . 500
Figure A.49
GF − WF Oil Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Figure A.50
Continuous CO2 nonlinear iteration convergence.
Figure A.51
Continuous CO2 time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . 502
Figure A.52
Continuous CO2 pressure for cells along diagonal between wells. . . . . . . . . 502
Figure A.53
Continuous CO2 total mass of CO2 for cells along diagonal between wells. . . 503
Figure A.54
Continuous CO2 total mass of hydrocarbons (no CO2 ) for cells along
diagonal between wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Figure A.55
Continuous CO2 saturation for equivalent one cell model. . . . . . . . . . . . . 503
Figure A.56
Continuous CO2 total mole fraction in the reservoir.
Figure A.57
Continuous CO2 recovery factor.
Figure A.58
Continuous CO2 compositional recovery factor.
Figure A.59
Continuous CO2 storage of CO2 .
. . . . . . . . . . 496
. . . . . . . . . . 497
. . . . . . . . . 497
. . . . . . . . 498
. . . . . . . . . . . . . . . . 501
. . . . . . . . . . . . . . 504
. . . . . . . . . . . . . . . . . . . . . . . . . 504
. . . . . . . . . . . . . . . . . 505
. . . . . . . . . . . . . . . . . . . . . . . . . 505
xxxi
Figure A.60
Continuous CO2 utilization of CO2 .
. . . . . . . . . . . . . . . . . . . . . . . 505
Figure A.61
Distribution of pressures at Continuous CO2 economic limit.
. . . . . . . . . 506
Figure A.62
2-D pressure distribution at Continuous CO2 economic limit.
. . . . . . . . . 506
Figure A.63
Distribution of oil saturation at Continuous CO2 economic limit.
Figure A.64
2-D oil saturation distribution at Continuous CO2 economic limit.
Figure A.65
Distribution of gas saturation at Continuous CO2 economic limit. . . . . . . . 508
Figure A.66
2-D gas saturation distribution at Continuous CO2 economic limit. . . . . . . 508
Figure A.67
Distribution of water saturation at Continuous CO2 economic limit.
Figure A.68
2-D water saturation distribution at Continuous CO2 economic limit.
Figure A.69
WAG Injection Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
Figure A.70
WAG Injection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
Figure A.71
WAG Production Pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
Figure A.72
WAG Production Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Figure A.73
WAG Production Ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Figure A.74
WAG − WF Oil Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Figure A.75
WAG nonlinear iteration convergence. . . . . . . . . . . . . . . . . . . . . . . . 512
Figure A.76
WAG time step criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Figure A.77
WAG pressure for cells along diagonal between wells. . . . . . . . . . . . . . . 513
Figure A.78
WAG total mass of CO2 for cells along diagonal between wells. . . . . . . . . . 513
Figure A.79
WAG total mass of hydrocarbons (no CO2 ) for cells along diagonal between
wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
Figure A.80
WAG saturation for equivalent one cell model. . . . . . . . . . . . . . . . . . . 514
Figure A.81
WAG total mole fraction in the reservoir.
Figure A.82
WAG recovery factor.
Figure A.83
WAG compositional recovery factor.
. . . . . . . 507
. . . . . . 507
. . . . . 509
. . . . 509
. . . . . . . . . . . . . . . . . . . . 515
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
xxxii
. . . . . . . . . . . . . . . . . . . . . . . 515
Figure A.84
WAG storage of CO2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Figure A.85
WAG utilization of CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
Figure A.86
Distribution of pressures at WAG economic limit. . . . . . . . . . . . . . . . . 516
Figure A.87
2-D pressure distribution at WAG economic limit.
Figure A.88
Distribution of oil saturation at WAG economic limit.
Figure A.89
2-D oil saturation distribution at WAG economic limit.
Figure A.90
Distribution of gas saturation at WAG economic limit. . . . . . . . . . . . . . 518
Figure A.91
2-D gas saturation distribution at WAG economic limit.
Figure A.92
Distribution of water saturation at WAG economic limit.
Figure A.93
2-D water saturation distribution at WAG economic limit. . . . . . . . . . . . 520
xxxiii
. . . . . . . . . . . . . . . 517
. . . . . . . . . . . . . 517
. . . . . . . . . . . . 518
. . . . . . . . . . . . 519
. . . . . . . . . . . 519
LIST OF TABLES
Table 3.1
Distribution of components in phases for NC = 8 . . . . . . . . . . . . . . . . . 38
Table 4.1
Distribution of components in phases for NC = 5 . . . . . . . . . . . . . . . . . 61
Table 4.2
Primary variables
Table 4.3
Secondary variables which do not vary with time, DA notime . . . . . . . . . . 67
Table 4.4
Secondary variables at n which are not needed for the transmissibility
calculations, DA cell only n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Table 4.5
Secondary variables at n which are needed for the transmissibility
calculations, DA for TRANS n . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Table 4.6
Secondary variables at , DA cell only ell
. . . . . . . . . . . . . . . . . . . . . 70
Table 4.7
Well properties at , stored for each well.
. . . . . . . . . . . . . . . . . . . . . 75
Table 9.1
Superscripts
Table 9.2
Subscripts
Table 9.3
Well variables
Table 12.1
Units for (12.21).
Table 15.1
Computation and memory requirement for 3 different solvers . . . . . . . . . . 299
Table 17.1
Validation cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
Table 19.1
Description of test case scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 394
Table 19.2
Primary production: recovery factor and time to economic limit
Table 19.3
Waterflood time to economic limit . . . . . . . . . . . . . . . . . . . . . . . . . 397
Table 19.4
Waterflood recovery factor
Table 19.5
Continuous CO2 recovery factor
Table 19.6
Continuous CO2 response time . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
. . . . . . . . 396
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
. . . . . . . . . . . . . . . . . . . . . . . . . . 400
xxxiv
Table 19.7
Continuous CO2 response duration . . . . . . . . . . . . . . . . . . . . . . . . . 403
Table 19.8
WAG recovery factor
Table 19.9
WAG recovery factor versus continuous CO2 recovery factor
Table 19.10
WAG response duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
Table 19.11
Compositional recovery factor for waterflood
Table 19.12
Compositional recovery factor for continuous CO2 injection . . . . . . . . . . . 411
Table 19.13
Compositional recovery factor for WAG . . . . . . . . . . . . . . . . . . . . . . 412
Table 19.14
CO2 storage for continuous CO2 injection . . . . . . . . . . . . . . . . . . . . . 413
Table 19.15
CO2 storage for WAG injection . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
Table 19.16
CO2 storage difference for continuous vs WAG CO2 injection . . . . . . . . . . 416
Table 19.17
CO2 utilization for continuous CO2 injection . . . . . . . . . . . . . . . . . . . 417
Table 19.18
CO2 utilization for WAG injection . . . . . . . . . . . . . . . . . . . . . . . . . 419
Table 19.19
CO2 utilization difference for continuous vs WAG CO2 injection
Table 22.1
Subscripts and superscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Table 22.2
Variables used in this document
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
. . . . . . . . . . 407
. . . . . . . . . . . . . . . . . . . 410
. . . . . . . . 420
. . . . . . . . . . . . . . . . . . . . . . . . . . 427
xxxv
ACKNOWLEDGMENTS
I am very thankful to the organizations which have funded portions of this work. These include
Saudi Aramco, The Petroleum Institute of Abu Dhabi, Marathon Center of Excellence for Reservoir
Studies, and Colorado School of Mines. I am very grateful to the members of Marathon Center of
Excellence for Reservoir Studies and the CSM/PI Integrated Carbonate Reservoir Studies groups
for valuable discussions during my time at Mines. I thank the faculty at CSM who have offered a
wonderful integrated learning experience. I would like to thank my committee members, Dr. Mark
Lusk, Dr. Erdal Ozkan, Dr. J. Rick Sarg, and Dr. Yu-Shu Wu, for providing valuable advice
throughout the process. I am indebted to my advisor, Dr. Hossein Kazemi, for his guidance,
support, and the important insight into different formulation possibilities and their importance in
field applications.
xxxvi
This dissertation is dedicated to my wife Ana, my mother Barbara, and my daughter Aiva. I
could not have finished when I did without their support, assistance, and understanding.
xxxvii
CHAPTER 1
INTRODUCTION
Enhanced oil recovery (EOR) is a group of methods designed to increase the production of oil
in addition to waterflooding. These methods are described by Green and Willhite (1998) and Lake
(1989). They include miscible and immiscible gas injection, thermal recovery, mobility control,
and chemical flooding. Based on the 2012 Worldwide EOR Survey conducted by the Oil and Gas
Journal, (Koottungal, 2012), carbon dioxide (CO2 ) enhanced oil recovery is now 351 MBOPD and
thermal recovery is 323 MBOPD with 89 MBOPD for other gas injection and no reported volumes
for chemical methods, carbonated waterflood, or microbial EOR.
Injecting carbon dioxide in oil reservoirs has two advantages: increasing the production of oil
and sequestering CO2 . CO2 may be injected without water or as a water-alternating-gas (WAG)
injection. In the USA, potential enhanced oil recovery from CO2 injection is approximately 80
billion barrels, corresponding to approximately 25 billion metric tons of sequestered CO2 . In the
world, potential enhanced oil recovery from CO2 injection is approximately 880 billion barrels,
corresponding to approximately 260 billion metric tons of sequestered CO2 (Rychel, 2012).
Injecting gas or water into a reservoir helps maintain the pressure and helps displace the oil.
Gas injection lowers the residual oil saturation and enhances gravity drainage. Injecting CO2
causes oil to swell and lowers the residual oil saturation more than methane. Gas injection in the
second or third WAG cycle will continue to decrease the residual oil saturation. CO2 is soluble in
water so it can access trapped oil by traveling through a water block. Mixing CO2 with reservoir
oil changes the viscosity and density of the oil; these changes may make it easier to displace the
oil. If the CO2 is miscible with the oil, it reduces trapping and further decreases the residual oil
saturation. CO2 injection leads to the sequestration of approximately 50% of the injected CO2 .
WAG injection controls the mobility more than continuous gas injection. WAG injection will cause
each CO2 injection cycle to follow different pathways in the reservoir, which leads to increased oil
recovery. WAG is often used for economic reasons because CO2 supply is often more limited than
water supply and CO2 injection is often more expensive than water injection.
1
If two or more phases are present in a pore space, one may become isolated or trapped as one
phase displaces another. Trapped or bypassed fluids consist of isolated pockets of fluids that are
not connected between the injector and producer. In a mixed-wet water-oil system, there may be
connected oil, connected water, connected gas, disconnected oil, disconnected water, and disconnected gas all present at the same time. In core flood experiments, the residual oil saturation is
determined. The residual oil saturation measured is the sum of the residual connected oil and the
residual disconnected oil, although when oil production stops most of the oil is probably disconnected.
Compositional variations of the trapped fluids have a significant impact on the volume of oil recovery; the timing of oil, water, and gas production; and the amount of CO2 storage and utilization.
Disconnected or trapped oil will not automatically equilibrate with mobile oil and gas, especially if
it is isolated by a water phase. Disconnected or trapped gas will not automatically equilibrate with
mobile oil and gas, especially if it is isolated by a water phase. In a water-oil system, an increasing
water saturation, water displacing oil leads to trapped oil. In a water-gas system, an increasing
water saturation leads to trapped gas; this is sometimes called “water blocking”. In a water-oil-gas
system, any of the three phases can be trapped. Trapping can relate to a microscopic effect such as
snap-off or pore doublet trapping. Trapped oil, gas, and water can also be related to a bypassing
effect, where a preferential flow path leaves fluids behind. This effect occurs at scales from the pore
network through inter-well scales.
The goal of this research was to evaluate the effects of variations in the composition of trapped
fluids on CO2 WAG simulation. A three-dimensional, three-phase, parallel compositional simulator
was developed with a specialized formulation to handle compositional trapping and CO2 WAG
injection. This formulation tracks the compositional differences between the trapped oil, gas, and
water and the mobile oil, gas, and water using a dual porosity approach. The mobile oil, gas, and
water (m1 system) are analogous to the fracture system and the disconnected oil, gas, and water
(m2 system) are analogous to the matrix system. The approach differs from Coats, Thomas, and
Pierson (2004a) method for tracking bypassed oil because the gas, oil, and water may all be trapped
with compositional variations. The amount and composition of the trapped fluids changes with
time. Reservoir simulation allows us to predict future performance of CO2 enhanced oil recovery
and sequestration at different scales.
2
Test cases with properties based on mixed wet carbonate reservoirs were used to evaluate the
effects of compositional trapping, gas relative permeability hysteresis, the solubility of CO2 in water,
and other trapping effects on the volume of oil recovery; the timing of oil, water, and gas production;
and the amount of CO2 storage and utilization. Primary production, waterflood, continuous CO2
injection, and CO2 WAG production schemes were evaluated.
3
CHAPTER 2
LITERATURE REVIEW
This chapter presents a literature review of different topics related to compositional simulation
of CO2 enhanced oil recovery. First there is a general discussion of enhanced oil recovery, followed
by papers dealing specifically with CO2 enhanced oil recovery and CO2 sequestration. Next is
a discussion of numerical reservoir simulation; the simulator in this dissertation is an example
of a numerical reservoir simulator. Test cases for this dissertation were based on a field in the
Middle East, and there is a brief review of papers characterizing geology in the Middle East.
Relative permeability is an important property of multiphase fluid flow and is especially important
to understand the effects of trapping. Compositional simulation is based on the calculation of phase
properties from an equation of state model.
2.1
Enhanced Oil Recovery
Enhanced oil recovery (EOR) is a group of methods designed to increase the production of oil
in addition to waterflooding. These methods are described by Green and Willhite (1998) and Lake
(1989). They include miscible and immiscible gas injection, thermal recovery, mobility control,
and chemical flooding. Based on the 2012 Worldwide EOR Survey conducted by the Oil and Gas
Journal, (Koottungal, 2012), carbon dioxide (CO2 ) enhanced oil recovery is now 351 MBOPD and
thermal recovery is 323 MBOPD with 89 MBOPD for other gas injection and no reported volumes
for chemical methods, carbonated waterflood, or microbial EOR.
2.1.1
Miscible Flooding
Katz and Stalkup (1983) discusses some of the limitations of reservoir simulation of miscible
floods. Stalkup (1983) presents an overview of miscible displacement processes. Uleberg and Høier
(2002) describes a method for determining minimum miscibility pressure for a dual porosity system.
2.1.2
Gas Injection
van Vark, Masalmeh, van Dorp, Al Nasr, and Al-Khanbashi (2004) conducted compositional
simulations of an Abu Dhabi reservoir to evaluate different injection mixtures of CH4 , CO2 , and
4
H2 S. H2 S yielded even better miscibility than CO2 . Changes in heterogeneity also had a significant
impact on recovery.
2.1.3
Other Enhanced Oil Recovery Methods
Agbalaka, Dandekar, Patil, Khataniar, and Hemsath (2008) summarizes the conclusion from the
literature that wettability has a significant impact on recovery during water injection, gas injection,
and WAG. Teletzke, Wattenbarger, and Wilkinson (2010) presents an overview of how to set up a
field pilot study for an EOR project.
CO2 Enhanced Recovery and Sequestration
2.2
Injecting carbon dioxide in oil reservoirs has two advantages: increasing the production of oil
and sequestering CO2 . CO2 may be injected without water or as a water-alternating-gas (WAG)
injection. In the USA, potential enhanced oil recovery from CO2 injection is approximately 80
billion barrels, corresponding to approximately 25 billion metric tons of sequestered CO2 . In the
world, potential enhanced oil recovery from CO2 injection is approximately 880 billion barrels,
corresponding to approximately 260 billion metric tons of sequestered CO2 (Rychel, 2012).
2.2.1
CO2 Enhanced Oil Recovery
Holm and Josendal (1974) presents the following summary of the benefits of CO2 . These benefits
are still the primary reasons today for CO2 injection.
• CO2 promotes swelling.
• CO2 reduces oil viscosity.
• CO2 increases oil density.
• CO2 is soluble in water.
• CO2 exerts an acidic effect on rock.
• CO2 can vaporize portions of the oil.
• CO2 can be transported chromatographically through porous rock.
5
Injecting CO2 results in several displacement mechanisms, including solution gas drive, immiscible CO2 , multi-contact miscible CO2 , and miscible CO2 .
Zekri, Shedid, and Almehaideb (2007) conducted core flood experiments related to CO2 EOR
in Abu Dhabi. Rawahi, Hafez, Al-Yafei, Al-Hammadi, Ghori, Putney, Matthews, and Harb (2012)
describes a CO2 EOR pilot design in Abu Dhabi. Yan and Stenby (2009) presents a study incorporating the effects of different CO2 solubilities in water on the oil recovery. Berenblyum, Calderon,
and Surguchev (2009) presents an overview of the mechanisms for CO2 enhanced oil recovery.
Ghedan (2009) presents an overview of laboratory experiments related to CO2 enhanced oil recovery. Al-Abri, Sidiq, and Amin (2009) describes experimental results for enhancing condensate
recovery by injecting CO2 and CH4 . Riazi, Sohrabi, Jamiolahmady, Ireland, and Brown (2009)
describes micromodel experiments for carbonated water injection. Manrique, Thomas, Ravikiran,
Izadi, Lantz, Romero, and Alvarado (2010) presents an overview of enhanced oil recovery projects
based on Oil and Gas Journal reports and additional references. Prieditis, Wolle, and Notz (1991)
describes a CO2 WAG flood in west Texas San Andres formation.
2.2.2
CO2 Flood Simulation
Chase and Todd (1984) describes a compositional reservoir simulator which includes CO2 solubility in brine. Chase and Todd (1984) also use a water blocking function based on Raimondi and
Torcaso (1964)
Stwb =
Sorw
ro
1 + β kkrw
(2.1)
(2.1) uses a parameter β to vary how strong the water blocking effect is; β = 1 would correspond to
a highly water wet sandstone; β = 5 is the much weaker blocking effect in a mixed wet west Texas
San Andres carbonate. Chase and Todd (1984) use a transition parameter α to vary the relative
permeabilities, viscosities, and densities between the oil and gas phases. Jackson, Andrews, and
Claridge (1985) presents simulation analysis of WAG ratio, using (2.1).
LaForce and Jessen (2007) presents an analysis of WAG simulations. Chang, Coats, and Nolen
(1998) describes a compositional reservoir simulator for CO2 flooding, including CO2 solubility in
water. Christensen, Stenby, and Skauge (1998) discusses the results of compositional simulation of
WAG using hysteresis options by Larsen and Skauge (1998). Christensen et al. (1998) concludes that
6
gas relative permeability hysteresis and slug size had very little effect on the results. Oil viscosity,
compositional simulation, and three-phase hysteresis were important for their simulations.
Hustad, Kløv, Lerdahl, Berge, Stensen, and Øren (2002) presents the results of 2D cross-section
simulation models of WAG with hysteresis. Nghiem, Sammon, Grabenstetter, and Ohkuma (2004)
describes modifications to CMG GEM which handle CO2 solubility in the aqueous phase and
aqueous geochemistry for simulation of CO2 sequestration in aquifers. Shtepani (2007) describes
experimental and modeling requirements for CO2 EOR. Shtepani (2007) recommends scaling the
residual oil saturation, gas-oil relative permeability, and gas-oil capillary pressure based on a ratio
of interfacial tensions:
α=
2.2.3
σ − σmin
σmax − σmin
(2.2)
CO2 WAG
Rogers and Grigg (2001) contains a thorough literature review of CO2 WAG processes. Injectivity of CO2 is sometimes higher and sometimes lower than waterflood injectivity. Awan, Teigland,
and Kleppe (2008) presents a review of gas injection and WAG injection projects in the North Sea.
Rogers and Grigg (2001) also contains a discussion of trapping/bypassing; the points summarized here will be discussed with the papers cited by Rogers and Grigg (2001). WAG ratio should
be based on volume, not based on time, and should increase with time for the best results.
Surguchev, Korbol, and Krakstad (1992) discusses calculations of the optimum WAG ratio
which will vary for each field. Based on Surguchev et al. (1992) and Rogers and Grigg (2001),
technical factors include heterogeneity, wettability, fluid properties, miscibility conditions, injection
techniques, WAG parameters, flow geometry, and physical dispersion. Surguchev et al. (1992) uses
a North Sea reservoir example to conclude that optimization of WAG depends on stratification,
hysteresis, and three-phase flow effects.
Gorell (1988) determined the amount of trapped solvent, trapped oil, mobile solvent, mobile
oil, and water as a result of 1-D simulations which simulate WAG as simultaneous gas and water
injection.
Todd, Cobb, and McCarter (1982) presents results for simulation of a field case in west Texas
(Wasson San Andres field). Prieditis and Brugman (1993) presents data at reservoir temperature
7
showing hysteresis in the water relative permeability for West Texas carbonates, showing that the
residual water saturation after waterflood is higher than the conate water saturation. The presence
of a residual miscible oil or CO2 saturation significantly reduces the predicted oil recovery. The
experimental data was simulated using a Todd and Longstaff (1972) approach. Dria, Pope, and
Sephrnoori (1993) presents three-phase relative permeability data for dolomite cores. Schneider
and Owens (1976) present measurements of hysteresis for rich gas injection in oil-wet carbonates
in West Texas.
Rogers and Grigg (2001) based on Wegener and Harpole (1996) states that macroscopic bypassing is related to heterogeneity and mobility differences; this is compounded by effects of varying
trapped gas saturations . Wegener and Harpole (1996) describes composite core flood experiments
of a West Texas carbonate. This study showed hysteresis in water relative permeability, with the
irreducible water saturation 15–20% saturation units higher than the connate water saturation.
Rampersad, Ogbe, Kamath, and Islam (1995) presents a good overview of the performance
effects of oil trapped by water during WAG. Fatemi, Sohrabi, Jamiolahmady, Ireland, and Robertson (2011) presents experimental results for multiple cycles of CO2 WAG in high permeability
water-wet and mixed-wet sandstones.
2.2.4
CO2 Sequestration
Haugen and Eide (1996) discusses CO2 sequestration options; the options have not changed since
1996. Flett, Gurton, and Taggart (2004) concludes that gas-water relative permeability hysteresis
and trapping has a significant effect on the amount of CO2 stored in an aquifer sequestration
case. Bachu and Bennion (2008) measured CO2 -brine relative permeabilities, capillary pressures,
and interfacial tensions for several different reservoirs in Alberta. Flett, Gurton, and Weir (2007)
presents simulation results for CO2 sequestration in aquifers.
Burton and Bryant (2009) presents a method for injecting CO2 dissolved in brine rather than
pure CO2 for CO2 sequestration in aquifers. Noh, Lake, Bryant, and Araque-Martinez (2007)
discusses a fractional flow based analytical model for simulating CO2 sequestration in aquifers.
Thibeau, Nghiem, and Ohkuma (2007) evaluates the long-term effect of geochemical reactions on
CO2 sequestration in aquifers. Bryant (2007) presents an overview of geologic CO2 sequestration.
8
Nattwongasem and Jessen (2009) presents a study of CO2 sequestration using CMG GEM.
Economides and Ehlig-Economides (2009) provides an overview of the volume requirement for a
regional CO2 sequestration subject to injection pressure constraints. Nghiem, Yang, Shrivatava,
Kohse, Hassam, Chen, and Card (2009) optimizes the amount of gas trapped by residual gas
trapping and solubility trapping for an example saline aquifer. Esposito and Benson (2010) presents
a simulation of CO2 leakage from a sequestration site along with possible remediation efforts.
Javaheri and Jessen (2011) measured co-current and counter-current relative permeability curves
and used these to calculate the effect on CO2 sequestration in an aquifer. Altundas, Ramakrishnan,
Chugunov, and de Loubens (2011) presents a simulation study of CO2 trapping caused by capillary
pressure hysteresis.
2.2.5
CO2 Simulation with TOUGH
Battistelli, Calore, and Pruess (1997) describes the EWASG module in TOUGH2 for geothermal
brine plus gas. It includes variations in the salt content and an option for CO2 -brine. Pruess,
Xu, Apps, and Garcia (2003) discusses CO2 injection in aquifers using the TOUGH2 suite. Zhang,
Doughty, Wu, and Pruess (2007) describes a parallel version of the TOUGH2 codes for use in CO2
sequestration studies. Battistelli and Marcolini (2009) presents the TMGAS module in TOUGH2 for
injection of gas into a brine aquifer; the gas may contain CO2 , H2 S, light hydrocarbons, nitrogen,
oxygen, and sulpher dioxide.
2.2.6
CO2 Water Solubility
Enick and Klara (1992) discusses the effect of CO2 solubility in brine on reservoir simulation
models. Do and Pinczewski (1993) discusses how CO2 solubility in water can diffuse through thin
layers of “water blocking” to get to trapped oil. Diffusion equilibrium is reached in approximately
100 hours. Takenouchi and Kennedy (1965) presents experimental work for water containing H2 O,
CO2 , and NaCl.
2.2.7
CO2 Trapping
Dai and Orr Jr. (1987) describes some trapping effects related to CO2 flooding. Dai and Orr Jr.
(1987) categorizes oil into flowing, dendritic, and trapped oil, including the effects of trapped oil
not mixing completely with the dendritic oil. Salter and Mohanty (1982) presents experimental
9
results for tracer floods in a four-foot long Berea sandstone core. Salter and Mohanty (1982) uses a
capacitance model to describe flowing, dendritic, and isolated oil fractions in a strongly water-wet
media, based on Coats and Smith (1964). Coats and Smith (1964) describes dead-end space using
a diffusion model.
2.2.8
Other Articles on CO2 Injection
Patton, Coats, and Spence (1982) describes a CO2 huff-and-puff process for increasing the
production of wells by altering the near-wellbore properties. It also provides data for eventual CO2
EOR operations. Morsi, Leslie, and Macdonald (2004) evaluates different methods for recovering
CO2 from flue gas for use as EOR in Abu Dhabi reservoirs. Seto, Jessen, and Orr (2007) evaluates
CO2 injection in a condensate reservoir. Plug and Bruining (2007) describes an experimental
procedure for measuring capillary pressure for a CO2 -brine system in sand packs. Hassanzadeh,
Pooladi-Darvish, Elsharkawy, Keith, and Leonenko (2008) presents a good review of CO2 properties
in brine, including a diffusion coefficient for gaseous CO2 into brine and a diffusion coefficient for
aqueous CO2 in brine.
2.3
Reservoir Simulation
Numerical reservoir simulation provides a way to understand past performance and predict
future performance of fluid flow in reservoirs. It can also be used to understand the sensitivity of
different parameters. The reservoir simulator created in this dissertation was used to evaluate the
importance of compositional trapping and other trapping related phenomena.
Odeh (1969) provides an overview of reservoir simulation, including of 0-D, 1-D, 2-D, and 3-D
models. Coats (1982) provides an early review of reservoir simulation. Kazemi, Al-Kobaisi, Kurtoglu, Heris, Charoenwongsa, Fakcharoenphol, and Akinboyewa (2012) provides a general discussion
of reservoir simulation in 2012.
Christensen, Larsen, and Nicolaisen (2000) presents a field test case using a WAG flood in a
North Sea reservoir. Masalmeh (2000) discusses oil recovery from transition zones.
2.3.1
Computation Approaches in Reservoir Simulation
Partially implicit methods are not mathematically stable for all combinations of grid cell size
and time step size. Courant, Friedrichs, and Lewy (1967) describes a way to calculate the stability
10
criteria, using a number now called the CFL number. Coats (2003a) describes a way to calculate
the CFL number for a compositional IMPES problem; Coats (2003b) presents related derivations.
In an overview of reservoir simulation methods Coats (1969) specifies that the implicit pressure
explicit saturation (IMPES) approach was first described in Stone and Garder (1961).
Christensen et al. (2000) describes saving a saturation value for use if the saturations oscillate
when calculating hysteresis.
Atan, Al-Matrook, Kazemi, Ozkan, and Gardner (2005) describes a method for reservoir simulation using two different scales of grid cells.
The iterative approach described in Lu, Al-Shaalan, and Wheeler (2007) for a black oil system is
similar to the iterative approach used here, including the possibility of varying the solver tolerances
for pressure and saturation as a function of iteration number. Lu et al. (2007) refers to an older
paper by Dawson, Klı́e, Wheeler, and Woodward (1997) that describes a related method.
Lu and Beckner (2011) describes a methodology for only solving for the grid cells which have
not yet converged.
2.3.2
Fractured Reservoir Simulation
Many hydrocarbon reservoirs are naturally fractured. To simulate naturally fractured reservoirs,
there are several approaches discussed in the literature. Dual porosity and dual permeability
systems partition the reservoir into two media: an interconnected fracture system which has a low
storage capacity but high flow capacity and a matrix system which provides high storage capacity
but low flow capacity. In a dual porosity representation, the matrix system connects to the fracture
system in the same grid cell but does not connect to adjacent fracture or matrix grid cells nor to
the wells. The fracture system connects to the matrix system, to adjacent grid cells, and to the
wells. In a dual permeability representation, the matrix blocks connect to the matrix system in
an adjacent grid cell. Multiple interacting continua (MINC) models provide connections between
several different levels of fracture and/or matrix systems (Wu and Pruess, 1988). Triple porosity
methods provide connections between two fracture systems and one matrix system or two matrix
systems and one fracture system. Fractures may be simulated using a discrete fracture network
where every fracture is represented individually or a network system where a collection of fractures
is represented by an interconnected network of fractures. A network representation often uses a
11
sugar cube model of the reservoir, where the matrix system is inside the sugar cubes and the
fracture system is between the sugar cubes (Warren and Root, 1963).
For this project, trapped fluids are represented using a dual porosity approach. The mobile
fluids are equivalent to the fractures in a dual porosity system: the m1 system is connected to wells,
neighboring grid cells, and to the trapped m2 system within the grid cell. The trapped fluids are
equivalent to the matrix in a dual porosity system: the m2 system is only connected to the m1
system in the same grid cell.
One of the earliest papers discussing naturally fractured reservoir systems is Warren and Root
(1963). Another early paper is Kazemi, Merrill, Porterfield, and Zeman (1976), which describes the
basic formulation used here. Gilman and Kazemi (1988) refines the formulation of Kazemi et al.
(1976), adding additional resolution to the gravity and capillary pressure in the fracture/matrix
transfer. Kazemi, Atan, Al-Matrook, Dreier, and Ozkan (2005) describes simulation of a system
with multiple levels of fractures; it uses an example with three fracture systems and one matrix
system. A slight modification of this approach would apply to a naturally fractured system with
a mobile m1 matrix system and a trapped m2 system. Balogun, Kazemi, Ozkan, Al-Kobaisi, and
Ramirez (2009), Ramirez, Kazemi, Al-Kobaisi, Ozkan, and Atan (2009), and Al-Kobaisi, Kazemi,
Ramirez, Ozkan, and Atan (2009) describe an updated formulation for calculating the water-oil-gas
transfer functions in dual porosity simulation.
Gouth, Moen-Maurel, Jeanjean, Soyeur, and Aziz (2007) describes a triple porosity simulation
in Abu Dhabi. Detwiler, Rajaram, and Glass (2005) describes a method of calculating the fracture
relative permeability using variable aperture fractures. Fung, Middya, and Dogru (2011) presents
results of a triple porosity simulation in Saudi Arabia.
2.3.3
Compositional Reservoir Simulation
The foundation of the compositional simulation formulation used here is described in Kazemi,
Vestal, and Shank (1978). According to Kazemi et al. (1978) the approach of using a separate flash
calculation was first described in Tsutsumi and Dixon (1972). The formulation used here has two
primary differences: the CO2 is soluble in the aqueous phase (WCO2 > 0), and trapping is accounted
for as in a dual porosity system. Coats (1980) describes a similar approach for compositional
simulation, although the approach used by Coats (1980) is fully implicit. Coats (1980) compares
12
their method to several other methods, including the iterative approach of Fussell and Fussell
(1979). Coats (1989) extends the approach of Coats (1980) for a dual porosity compositional
simulator. Acs, Doleschall, and Farkas (1985) describes a slightly different approach using pressure
and the component masses as primary variables. Kendall, Morrell, Peaceman, Silliman, and Watts
(1983) describes the development of the MARS simulator at Exxon. Young and Stephenson (1983)
describes a compositional reservoir simulator. Watts (1986) describes an approach for compositional
simulation; this approach first solves a pressure equation; using the pressure solution it solves for
velocities; using the velocities it solves for implicit saturations and relative permeabilities. This
approach requires the calculation of the derivatives of partial molar volumes. Nghiem and Li (1990)
describes a way to simplify flash calculations for use in a compositional simulator. Nghiem and
Sammon (1997) assumes that the fluids in a grid cell equilibrate based on diffusion rather than
instantaneously being in equilibrium. Haukås, Aavatsmark, Espedal, and Reiso (2007) describes
additional compositional approaches. Wang and Pope (2001) describes the state of the art in 2001
for compositional simulation using an equation of state.
Voskov and Tchelepi (2008) describes performing compositional simulations in using compositional space parameterization rather than simulating based on total number of moles or mole
fractions. Pan and Tchelepi (2011) describes another set of variables for compositional simulation
plus methods for bypassing the stability analysis of the compositional system.
Wong, Firoozabadi, and Aziz (1990) compares several of the previous methods of compositional
simulation, with the conclusion that the methods are more similar than it might appear under
a casual inspection. Coats (2000) compares different compositional formulations and finds them
similar. Nghiem, Fong, and Aziz (1981) describes the earliest version of the CMG methodology for
compositional simulation. It is similar to Kazemi et al. (1978) and discusses convergence issues.
2.3.4
CO2 and Miscible Flood Simulation
Todd and Longstaff (1972) describes a way to calculate miscible flood performance using a four
“component” system consisting of water, oil, gas, and solvent. Todd and Longstaff (1972) also
describes a way to calculate viscosity, density, and relative permeability of a miscible oil and gas
hydrocarbon system.
13
Chase and Todd (1984) presents an early simulation study of CO2 flooding in a San Andres
carbonate reservoir in west Texas). Several features are included that are specific to CO2 floods,
including dropout of heavy components, water blocking, viscous instability and fingering, miscible/immiscible transition, and a non-zero WCO2 . Water blocking was calculated as Sblock =
Sorw
1+β kkro
;
rw
this is used to adjust the saturations accessible to CO2 .
Enick and Klara (1992) discusses the effect of including the CO2 solubility in brine; they conclude that it is frequently necesary for accurate compositional simulation results of CO2 flooding.
Enick and Klara (1992) also provides correlations for calculating WCO2 based on the total dissolved solids present, and a methodology for updating WCO2 in a compitional formulation similar
to Kazemi et al. (1978) and the approach used in this dissertation.
Xiao and Jones (2007) describes a reactive transport model for dolomitization.
Coats, Whitson, and Thomas (2004b) describes modeling of dispersion. Garmeh and Johns
(2010) discusses the importance of mixing within grid cells for reservoir simulation.
2.3.5
Parallel Simulation
Killough and Bhogeswara (1991) describes an early parallel compositional simulator. Domain
decomposition, communication, and load balancing described in this paper are still issues today.
Zhang, Wu, Ding, Pruess, and Elmroth (2001) describes a parallel formulation for TOUGH2. Atan,
Kazemi, and Caldwell (2006) describes a method to use multiscale multimesh reservoir simulation for parallel openMP based computations. Tarman, Wang, Killough, and Sepehrnoori (2011)
describes a method for decomposing a reservoir simulation into rectangular grids for parallel computations.
Dogru, Li, Sunaidi, Habiballah, Fung, Al-Zamil, Shin, McDonald, and Srivastava (1999) describes the initial development of the parallel compositional simulator POWERS (Parallel Oil Water and Gas Reservoir Simulator) at Saudi Aramco. Dogru, Sunaidi, Fung, Habiballah, Al-Zamel,
and Li (2002) describes an update of the work on POWERS. Al-Shaalan, Fung, and Dogru (2003)
describes a dual permeability extension to POWERS using a hybrid MPI/openMP parallelization
scheme. Fung and Dogru (2007) and Fung and Dogru (2008) describes an update to POWERS
using a parallel unstructured solver. Dogru, Fung, Al-Shaalan, Middya, and Pita (2008) describes
the extension of POWERS from mega-cell models to giga-cell models. Al-Shaalan, Klie, Dogru, and
14
Wheeler (2009), Dogru, Fung, Middya, Al-Shaalan, Pita, HemanthKumar, Su, Tan, Hoy, Dreiman,
Hahn, Al-Harbi, Al-Youbi, Al-Zamel, Mezghani, and Al-Mani (2009), and Dogru, Fung, Middya,
Al-Shaalan, Byer, Hoy, Hahn, Al-Zamel, Pita, Hemanthkumar, Mezghani, Al-Mana, Tan, Dreiman,
Fugl, and Al-Baiz (2011) describe extensions to GigaPOWERS.
2.3.6
Simulation of Trapping and Bypassing
Coats et al. (2004a) describes a formulation for accounting for bypassed oil. The formulation
described in Coats et al. (2004a) is the closest to the formulation presented in this dissertation of
all the literature reviewed.
Barker, Prevost, and Pitrat (2005) describes a way to modify the mobile compositions.
2.3.7
Simulation of Diffusion
da Silva and Belery (1989) presents equations for calculating the diffusion coefficients in a
compositional system. Nghiem and Sammon (1997) presents correlations for calculating diffusion
coefficients in a compositional system. Hoteit and Firoozabadi (2006a) describes compositional
simulations of diffusion in naturally fractured reservoirs with gas injection. Bahar and Liu (2008)
measured the diffusion coefficient of gaseous CO2 into brine.
2.3.8
Additional Simulation Topics
Coats (1980) computes oil and gas relative permeabilities weighted by f (σ) =
σ
σ0
1
n1
to scale
krg and krog as the system becomes miscible. It uses Stone (1973) to calculate kro from krog , krow ,
krw , and krg .
Kelly (2006) describes using an equation of state to calculate the density variations within an
injection well as a function of depth; it is important to use multiple depths in the calculation
because CO2 density varies a lot with temperature and pressure. Wu and Bai (2009) describes
a method for simulating low salinity water flooding. Zhang, Yin, Wu, and Winterfeld (2012)
describes a methodology for non-isothermal reactive transport modeling with application to CO2
sequestration using the TOUGH framework. Das, Mirzaei, and Widdows (2006) describes how
microscopic heterogeneities can effect the relative permeability and capillary pressure.
15
2.3.9
SPE Comparative Solution Projects
The SPE Comparative Solution Project is a series of ten articles which compare different reservoir simulators: Odeh (1981), Weinstein, Chappelear, and Nolen (1986), Kenyon and Behie (1987),
Aziz, Ramesh, and Woo (1987), Killough and Kossack (1987), Firoozabadi and Thomas (1990),
Nghiem, Collins, and Sharma (1991), Quandalle (1993), Killough (1995), and Christie and Blunt
(2001a). These articles present test cases which can be used to evaluate new reservoir simulators.
The following provides a brief description of each article:
1. Odeh (1981) presents a 3-D 2-phase black oil problem involving gas injection.
2. Weinstein et al. (1986) present a radial 2-D 3-phase black oil problem involving coning of
both water and gas.
3. Kenyon and Behie (1987) present a 3-D 3-phase compositional problem involving retrograde
gas cycling.
4. Aziz et al. (1987) present a 2-D 3-phase steam injection problem.
5. Killough and Kossack (1987) present a 3-D 3-phase compositional problem involving miscible hydrocarbon gas injection. This could possibly be used as a test case of the simulator
developed here.
6. Firoozabadi and Thomas (1990) present a 2-D 3-phase black oil problem involving a naturally
fractured reservoir.
7. Nghiem et al. (1991) present a 3-D 3-phase black oil problem involving horizontal wells.
8. Quandalle (1993) presents 3-D 3-phase black oil problem which compares different gridding
techniques.
9. Killough (1995) presents a 3-D 3-phase black oil problem with a 9000 grid cell geostatistically
populated grid.
10. Christie and Blunt (2001a), based on (Christie and Blunt, 2001b), present a 3-D 3-phase black
oil problem with a 1.1 million grid cell geostatistically populated grid. The paper focuses on
upscaling techniques.
16
2.4
Geologic Characterization in Middle East
Jobe (2013) presents a detailed review of the geologic characterization of the Abu Dhabi reser-
voirs studied by the CSM/PI Integrated Carbonate Reservoir Studies Group.
Al-Aruri, Ali, Ahmad, and Samad (1998) uses mercury injection capillary pressure data to
help group carbonate facies into petrophysical rock types in Abu Dhabi. Ghedan, Gunningham,
Ehmaid, and Azer (2002) describes a process to upscale a reservoir simulation model in Abu Dhabi.
Bushara, El Tawel, Borougha, Dabbouk, and Qotb (2002) describes a study by Zadco to characterize the fracture permeability. Cantrell and Hagerty (2003) describes a way to characterize the
carbonate rocks in Ghawar. Ottinger, Kompanik, Al Suwaidi, Brantferger, and Edwards (2012)
describes geostatistical mapping of reservoir rock types conducted by Zadco. Yamamoto, Kompanik, Brantferger, Al-Zinati, Ottinger, Al-Ali, Dodge, and Edwards (2012) describes geostatistical
modeling of a dolomitized zone by Zadco.
Ghedan, Thiebot, and Boyd (2004) describes modeling a water-oil transition zone in Abu Dhabi,
including one author from Zadco. Ghedan (2007) uses dynamic reservoir rock types to assign
relative permeability and capillary pressure functions for grid cells in a reservoir simulation model
for Abu Dhabi. It’s important to account for the varying wettability if there is a transition zone
present. Ghedan, Canbaz, Boyd, Mani, and Haggag (2010) describes a new method for measuring
the wettability based on work done with Zadco.
2.5
Relative Permeability and Capillary Pressure
Relative permeability represents the reduced permeability when multiple fluids are present in a
reservoir.
2.5.1
General Articles on Relative Permeability
Mualem (1976) presents a two phase relative permeability model not including hysteresis.
Thomeer (1983) presents a two phase relative permeability model not including hysteresis. Chierici
(1984) presents a two phase relative permeability model not including hysteresis. Kamath, Meyer,
and Nakagawa (2001) presents two-phase oil/water relative permeability data for carbonate rocks.
Bennion and Bachu (2005) and Bennion and Bachu (2008b) present CO2 /brine relative permeability data for carbonate and sandstone cores in Canada. Byrnes and Bhattacharya (2006) presents
17
relative permeability data for carbonate reservoirs. Egermann, Laroche, Manceau, Delamaide, and
Bourbiaux (2007) presents gas/water relative permeability data for vuggy carbonates. Rustad,
Theting, and Held (2008) presents a simulation approach for assessing the uncertainty in relative permeabilities. Gawish and Al-Homadhi (2008) presents relative permeability experiments for
different temperatures, wettabilities, and overburden pressures.
2.5.2
General Articles on Capillary Pressure
Parker, Lenhard, and Kuppusamy (1987) presents a model for capillary pressure. Gray and Hassanizadeh (1991) presents a theoretical discussion of a capillary pressure model. Zhou and Blunt
(1997) presents a discussion of how the three-phase spreading coefficient effects capillary pressures.
Clerke (2009) presents a bimodal capillary pressure distribution. Lamy, Iglauer, Pentland, Blunt,
and Maitland (2010) presents capillary pressure data for carbonate cores. Iglauer, Wülling, Pentland, Al Mansoori, and Blunt (2009) presents a review of capillary trapping in sandstones along
with some new data.
2.5.3
Trapping
Land (1968) provides the trapped gas saturation as a function of the initial gas saturation. This
model is a very commonly used model and the base model for many comparisons. Keelan and Pugh
(1975) presents early experimental data for trapped gas saturations in carbonates. Torquato (1990)
presents a discussion of diffusion controlled trapping. Lin and Huang (1990) presents methods for
calculating trapping in an oil/water system in various wettabilities of Berea cores. Muller and
Lake (1991) presents a model of trapping using diffusion. All trapping amounts are presented as a
function of residence time. Bennion, Thomas, Bietz, and Bennion (1996) presents a discussion of
different trapping mechanisms. Pentland, Al-Mansoori, Iglauer, Bijeljic, and Blunt (2008) presents
measurements of trapping in sand packs.
2.5.4
Three-Phase Relative Permeability
Naar and Wygal (1961) presents an early model of three-phase relative permeability. Stone
(1970) and Stone (1973) present a three-phase relative permeability model. This model is a very
commonly used model and the base model for many comparisons. Dietrich and Bondor (1976)
presents a three-phase relative permeability model. Carlson (1981) presents a three-phase relative
18
permeability model using Killough and Kossack (1987) and Land (1968). This model is a very
commonly used model and the base model for many comparisons. Fayers and Matthews (1984)
analyzes three-phase relative permeability data from various literature sources. Thomas and Coats
(1992) rewrites Stone’s methods in terms of arbitrary permeabilities. Larsen and Skauge (1998)
presents a three-phase relative permeability formulation. Larsen and Skauge (1999) presents an
immiscible WAG simulation using Larsen and Skauge (1998). Blunt (2000) presents a three-phase
relative permeability formulation and a good summary of previous methods. van Dijke, Sorbie, and
McDougall (2000) and van Dijke, Sorbie, and McDougall (2001) present a formulation for threephase relative permeability. Oliveira and Demond (2003) presents a comparison of three-phase
relative permeability models. Juanes and Patzek (2004b) and Juanes and Patzek (2004a) present
a theoretical discussion under what conditions three-phase relative permeability models transition
between hyperbolic and elliptic regions. Yuen, Siu, Shenawi, Bukhamseen, Lyngra, and Al-Turki
(2008) presents a three-phase relative permeability model based on a curve fit to experimental data.
Yuan and Pope (2011) presents a three-phase relative permeability model including a new method
to transition between two phase gas-water and oil-water systems.
Delshad and Pope (1989) presents an analysis of seven different three-phase relative permeability formulations. Baker (1988) presents an analysis of different three-phase relative permeability
formulations. Fayers (1989) presents an analysis of Stone’s methods for three-phase relative permeability formulations. Kokal and Maini (1990) presents analysis of several three-phase relative
permeability experiments and a modification of Stone’s method. Guzman, Giordano, Fayers, Aziz,
and Godi (1994) presents simulation results for WAG injections based on different three-phase
relative permeability models. Pope, Wu, Narayanaswamy, Delshad, Sharma, and Wang (1998)
presents an analysis of three-phase relative permeability data from various sources in terms of
trapping number. Whitson, Fevang, and Saevareid (1999) presents an analysis of three-phase relative permeability data using krg vs krg /kro and the capillary number for Berea sandstone and a
North Sea sandstone. This paper also discusses variations in the relative permeability curves as
a function of miscibility. Kossack (2000) presents a comparison three-phase relative permeability models with hysteresis as implemented in Eclipse. Spiteri and Juanes (2004) and Spiteri and
Juanes (2006) present simulation of WAG injection with different three-phase relative permeability
models. Karkooti, Masoudi, Arif, Darman, and Othman (2011) presents a WAG case study using
19
three-phase relative permeability of a Malaysian field. Shahverdi, Sohrabi, Fatemi, Jamiolahmady,
Irelan, and Robertson (2011) presents a review of three-phase relative permeability formulations
and a simulation of experiments at Heriot-Watt.
Saraf, Batycky, Jackson, and Fisher (1982) presents three-phase relative permeability data for
Berea sandstone. Van Spronsen (1982) presents three-phase relative permeability data collected
using a centrifuge for both Berea and the Weeks Island sandstone. Ehrlich, Tracht, and Kaye (1984)
presents laboratory data for a dolomite reservoir subjected to a lab-based CO2 WAG flood. Oak
(1990) presents the results of very thorough experiments of three-phase relative permeability on
water-wet Berea sandstone. Kalaydjian, Moulu, Vizika, and Munkerud (1997) presents three-phase
relative permeability experiments for Fontainebleau sandstone and Clashach sandstone. Jerauld
(1997) presents three-phase relative permeability data and curve fits for mixed wet Prudhoe Bay
sandstone. Sahni, Burger, and Blunt (1998) presents three-phase relative permeability measurements of packed sands and sandstones. Ebeltoft, Iversen, Vatne, Andersen, and Nordtvedt (1998)
presents three-phase relative permeability data for a chalk reservoir. Moulu, Vizika, Egermann,
and Kalaydjian (1999) presents three-phase relative permeability data for the Vosges sandstone
under different wettabilities. The paper uses a fractal correlation to match the experimental data.
Kralik, Manak, Jerauld, and Spence (2000) presents the results of three-phase relative permeability experiments on an oil-wet sandstone. Egermann, Vizika, Dallet, Requin, and Sonier (2000)
presents simulations of three-phase relative permeability experiments on Estaillades limestone. Element, Masters, Sargent, Jayasekera, and Goodyear (2003) presents WAG experiments that require
a three-phase relative permeability formulation with hysteresis in chalk. Dehghanpour, DiCarlo,
Aminzadeh, and Mirzaei (2010) presents WAG experiments using a water-wet sand pack. Cao and
Siddiqui (2011) presents three-phase relative permeability data for three immiscible fluids (not oil,
gas, and water, but interpreted as similar by the authors) in Berea sandstone. Fatemi and Sohrabi
(2012) presents a review of three-phase relative permeability models and experimental data for multiple WAG cycles. Fatemi, Sohrabi, Jamiolahmady, and Ireland (2012a) presents a history match
of experimental three-phase relative permeability data and a good literature review. Shahverdi and
Sohrabi (2012) presents an analysis of three-phase relative permeability data. Fatemi, Sohrabi,
Jamiolahmady, and Ireland (2012b) presents three-phase relative permeability data for water wet
and mixed wet cores.
20
2.5.5
Relative Permeability Hysteresis
Naar and Henderson (1961) and Naar and Wygal (1961) present an early model of relative
permeability hysteresis; better theories have been presented more recently. Philip (1964) has a
method for calculating hysteresis scanning curves based on the wetting and drying curves. Walsh,
Negahban, and Gupta (1989) uses the difference between the drainage and imbibition curves in
Berea sandstone to calculate the trapped saturation for a CO2 flood. Braun and Holland (1995)
presents experimental oil/water scanning curves for Berea sandstone and an Australian sandstone.
Chang, Mohanty, Huang, and Honarpour (1997) presents experimental measurements of mixed
wet oil/water relative permeability. Lenhard and Oostrom (1998) presents a discussion of twophase oil/water relative permeability with hysteresis. Bennion, Thomas, Jamaluddin, and Ma
(1998) presents a discussion of different kinds of hysteresis. Spiteri, Juanes, Blunt, and Orr (2005)
presents simulation models applied to CO2 injection with relative permeability hysteresis. Zhang,
Falcone, and Teodoriu (2010) presents the effects of relative permeability hysteresis on near-wellbore
pressures. Krause (2012) presents relative permeability data for Berea sandstone showing the 3D
variation in saturations in a core flood. Dernaika, Basioni, Dawoud, Kalam, and Skjaeveland (2012)
presents relative permeability data with hysteresis for various carbonate rocks.
Honarpour, Huang, and Dogru (1996) presents an experimental apparatus to simultaneously
measure relative permeability, capillary pressure, and electrical resistivity during a core flood. Hysteresis data is presented for Berea sandstone. Masalmeh (2001) presents a discussion of hysteresis
in water-wet, oil-wet, and mixed-wet porous media.
Behzadi (2010) presents a simulation of CO2 trapping in the Nugget formation. Altundas et al.
(2011) presents a simulation of CO2 trapping.
2.5.6
Capillary Pressure Hysteresis
Morrow and Harris (1965) provides data for capillary pressure hysteresis measured in a column
packed with glass beads. Morrow (1970) is an early summary of the thermodynamics of capillary
pressure hysteresis. Lenhard, Parker, and Kaluarachchi (1991) presents a two-phase gas/water capillary pressure hysteresis model with experimental data. Kleppe, Delaplace, Lenormand, Hamon,
and Chaput (1997) presents measurements of gas/oil capillary pressure hysteresis and describes a
way to scale the scanning curves.
21
2.5.7
Combined Relative Permeability and Capillary Pressure Hysteresis
Killough and Kossack (1987) provides a method for computing capillary pressure and relative
permeability hysteresis. This model is a very commonly used model and the base model for many
comparisons. Parker and Lenhard (1987) and Lenhard and Parker (1987) present a model for
three-phase capillary pressure and relative permeability hysteresis. Bradford, Abriola, and Leij
(1997) presents a discussion of three-phase relative permeability and capillary pressure models.
Nordtvedt, Ebeltoft, lversen, Sylte, Urkedal, Vatne, and Watson (1997) presents three-phase capillary pressure and relative permeability measurements. The three-phase data was fit using a product
of two splines. Hustad (2002) presents a three-phase capillary pressure and relative permeability
model with hysteresis. Fayers, Foakes, Lin, and Puckett (2000) presents a three-phase capillary
pressure and relative permeability model with hysteresis, including a method for weighting relative
permeabilities as miscibility is developed. Blunt (2000) presents an analysis of three-phase relative
permeability and capillary pressure experiments, including a discussion of trapped oil, spreading oil,
and mobile oil. Hustad et al. (2002) presents WAG simulation results and a description using the
IKU3P model for relative permeability and capillary pressure. Kjosavik, Ringen, and Skjaeveland
(2002) presents a relative permeability and capillary pressure formulation with hysteresis. Delshad,
Lenhard, Oostrom, and Pope (2003) presents a relative permeability and capillary pressure formulation with hysteresis. Hustad and Browning (2009) presents a relative permeability and capillary
pressure formulation with hysteresis.
DiCarlo, Sahni, and Blunt (2000) presents three-phase capillary pressure and relative permeability data for various wettability sandpacks. Masalmeh (2003) presents capillary pressure and
relative permeability data and their variations with wettability. Jackson, Valvatne, and Blunt
(2002) presents relative permeability and capillary pressures calculated from pore network simulations of Berea sandstone. Masalmeh (2002) presents capillary pressure and relative permeability
data for mixed wet and oil wet Middle East carbonates. Masalmeh, Shiekah, and Jing (2007)
presents capillary pressure and relative permeability data and modeling for a carbonate reservoir.
Ghomian, Pope, and Sepehrnoori (2008) presents simulations of CO2 WAG for EOR and sequestration using different three-phase relative permeability and capillary pressure models. Olafuyi,
Cinar, Knackstedt, and Pinczewski (2008) presents experimental capillary pressure and relative
22
permeability data for Berea sandstone, Bentheim sandstone, and Mount Gambier carbonates.
Masalmeh and Wei (2010) presents a study of WAG options using three-phase relative permeability and capillary pressure hysteresis. It uses a linear relative permeability option under miscible
conditions. Bhatti, Kalam, Hafez, and Kralik (2012) presents relative permeability and capillary
pressure data for Abu Dhabi carbonates.
2.5.8
Non-zero Relative Permeability Derivative
Bell, Trangenstein, and Shubin (1986) discusses reasons why the derivative of at least one of
the relative permeabilities with respect to saturation should be non-zero: it leads to a region of
the relative permeability space that has a non-hyperbolic solution; physically we would expect the
solution to be hyperbolic for all saturation values.
lim
Sϕ →Sϕ
∂krϕ [Sϕ ]
= 0
∂S
(2.3)
The following papers present a theoretical discussion under what conditions three-phase relative
permeability models transition between hyperbolic and elliptic regions: van Dijke et al. (2000),
van Dijke et al. (2001), Juanes and Patzek (2004b), and Juanes and Patzek (2004a). According to
Dr. Kazemi, if capillary pressure is appropriately calculated this is no longer required.
2.5.9
Additional Relative Permeability Effects
Wilson (1956) illustrates the effects of overburden pressure on oil/water relative permeability.
Overburden pressure causes Swr to increase and Sowr to decrease. Al-Quraishi and Khairy (2005)
presents experimental results showing changes in oil/water relative permeability as a function of
overburden pressure.
Coats and Smith (1964) describes diffusion-based mass transfer out of trapped pores. Sinnokrot,
Ramey, and Marsden (1971) describes the changes in capillary pressure with temperature. Swr
increases with increasing temperature for sandstone and decreases with increasing temperature for
carbonates.
Torabzadey (1984) and Kumar, Torabzadeh, and Handy (1985) present the experimental variations of water/oil relative permeability in Berea sandstone with temperature and interfacial tension.
The Sorw decreases with increasing temperature, but the change is much smaller for a low inter-
23
facial tension system. The Swr increases with increasing temperature for a low interfacial tension
system but is approximately constant with a high interfacial tension system.
Sorbie and van Dijke (2010) presents an analysis of near-miscible interfacial tension changes. It
also presents the results of some pore-scale and micro-model experiments of near-miscible WAG.
Middya and Dogru (2008) describes a method for calculating well drainage pressure as an average
of multiple grid cells rather than the value of a single cell.
2.6
Equation of State Literature
The phases and compositions are determined using “flash” calculations with an equation of
state. This section describes different methods for calculating an equation of state and methods
for adjusting equation of state parameters to fit experimental data.
2.6.1
Calculation of Equation of State
Rachford and Rice (1952) describes what is now considered the standard method of performing
hydrocarbon flash calculations.
Michelsen (1980) presents a method for calculating phase envelopes. Michelsen (1982a) and
Michelsen (1982b) present a methodology for flash calculations. Michelsen and Mollerup (1986)
specifies derivatives of thermodynamic properties. Whitson and Michelsen (1989) describes a flash
calculation using negative flash. Mollerup and Michelsen (1992) describe computations of thermodynamic derivatives that were used to check the flash calculations used in this dissertation.
Michelsen (1998) describes some ways to speed up flash calculations.
Li and Nghiem (1982) describes several different methods for flash calculations. Nghiem, Aziz,
and Li (1983) describes a flash calculation procedure.
Fussell and Yanosik (1978) presents a flash calculation procedure called Minimum Variable
Newton-Raphson (MVNR). Guehria, Thompson, and Reynolds (1990) describes a flash calculation procedure and ways to calculate derivatives of thermodynamic properties. Nagarajan, Cullick,
and Griewank (1991a) and Nagarajan, Cullick, and Griewank (1991b) describe a method for critical point calculations. Thomas, Bennion, and Bennion (1991) describes a method for calculating
pseudo-ternary diagrams. Firoozabadi and Pan (2002) describes improved stability analysis calculations for compositional modeling. Pan and Firoozabadi (2001) describes improved flash calculations
24
for compositional modeling. Li and Johns (2006) describes improved flash calculations for compositional modeling. Rasmussen, Krejbjerg, Michelsen, and Bjurstrom (2006) describes improved flash
calculations for compositional modeling. Hoteit and Firoozabadi (2006b) provides a good overview
of flash calculations and suggests some improvements in stability testing.
Juanes (2008) describes a method to perform flash calculations by discretizing the tie lines.
Li and Firoozabadi (2010) describes a flash procedure, including liquid-vapor and liquid-liquidvapor systems. Voskov and Tchelepi (2008), Voskov and Tchelepi (2007), and Gasmi, Voskov, and
Tchelepi (2009) describe a flash procedure using a compositional space parameterization. Voskov,
Younis, and Tchelepi (2009) and Voskov (2011) describe several different parameterizations that
can be used for flash calculations and compares each of these methods.
2.6.2
Adjusting Equation of State Parameters
Rowe (1978) presents a methodology for using pseudo components in reservoir simulation.
Whitson (1984) discusses the importance of C7+ properties for EOS predictions. Whitson (1983)
describes methods for splitting the C7+ into pseudocomponents. Pedersen, Thomassen, and Fredenslund (1985) and Pedersen, Thomassen, and Fredenslund (1988) discuss appropriate ways for
fitting EOS parameters. Nishiumi, Arai, and Takeuchi (1988) discuss ways to calculate binary
interaction parameters for fitting an equation of state. Leibovici, Govel, and Piacentino (1993)
describes a method for calculating pseudo-component properties. Gasem, Gao, Pan, and Robinson
(2001) describes some changes for the Peng-Robinson EOS that may improve the fit to experimental data. Jaubert, Vitu, Mutelet, and Corriou (2005) describes modifications to the Peng-Robinson
EOS using experimental information about specific aromatic compounds. Ahmed (2007b) describes
modifications to the Peng-Robinson EOS that make the fits to experimental data better.
2.6.3
Modifications to Equation of State Model when CO2 is Present
The following papers describe modifications to the binary interaction coefficients of CO2 with
hydrocarbons using the Peng-Robinson EOS: Mulliken and Sandler (1980), Kato, Nagahama, and
Hirata (1981), Turek, Metcalfs, Yarborough, and Robinson (1984), Lin (1984), Nishiumi et al.
(1988), Kordas, Tsoutsouras, Stamataki, and Tassios (1994), Vitu, Privat, Jaubert, and Mutelet
(2008).
25
Coutinho, Kontogeorgis, and Stenby (1994) describe modifications to the binary interaction
coefficients of CO2 with hydrocarbons using the Soave-Redlich-Kwong EOS.
Metcalfe and Yarborough (1979) discusses the effects of mixing CO2 with oil on the phase
behavior of the system. Data is presented for a CO2 -CH4 -nC4 -nC10 system, in addition to other
systems. Kuan, Kilpatrick, Sahimi, Scirven, and Davis (1986) presents CO2 -water-hydrocarbon
phase behavior.
Turek et al. (1984) describes some methods for fitting the EOS properties for a system containing
CO2 . Han and McPherson (2008) compares several different CO2 -brine equations of state for CO2
sequestration applications.
2.6.4
Phase Behavior Illustration
Rowe Jr. and Silberberg (1965) describes the phase behavior for an enriched gas injection
process; it has an example of three-dimensional ternary diagrams with pressure along one axis. Rowe
(1967) presents four-dimensional plots of phase behavior. Kalippan and Rowe (1971) illustrates
additional ways to present phase behavior for more than three variables.
2.6.5
Other Equation of State References
Li and Nghiem (1986) describes a way to calculate solubility in the aqueous phase using Henry’s
Law and describes a three-phase oil-water-gas flash procedure. One of the examples is for a CO2 brine system. Broad, Varotsis, and Pasadakis (2001) describes the effect of data quality on the
predictions of an EOS. Nagarajan, Honarpour, and Sampath (2007) describes the sampling processes needed to accurately characterize reservoir fluids.
2.7
Pore Scale Simulation
Pore scale simulation is a specialized category of reservoir simulation devoted to simulating
devoted to simulating microscopic process to help understand macroscopic processes. Some of
these techniques are praising for future study of trapping.
2.7.1
Network Models
Ehrlich and Crane (1969) presents an overview of network models. One approach is to consider
a porous medium as a network of capillary tubes. Hysteresis can be explained by bypassing, Naar
26
and Henderson (1961). A simple pore doublet consists of one small and one large capillary tube.
Naar and Henderson (1961) describes a capillary tube based model for imbibition and drainage
relative permeabilities, including bypassed/blocked/trapped oil.
Blunt, Fenwick, and Zhou (1994) presents a discussion of spreading and non spreading oils, plus
a discussion of how individual pores or pore throats drain.
McDougall and Sorbie (1995) describes a cubic 20 × 20 × 20 network of nodes; each node has
an assigned capillary radius; each node is connected to six neighbors. After creating this network,
McDougall and Sorbie (1995) then conducted waterflood experiments for water wet and mixed wet
systems. The network simulations result in simulated capillary pressure and relative permeabilities.
Blunt (1997) describes network simulations where the contact angle varies between nodes.
Laroche, Vizika, and Kalaydjian (1999) describes another network model.
van Dijke and Sorbie (2003) discusses the results of network model simulations. They observe
that “displacement chains” can occur where an individual fluid is not in contact with the inlet and
outlet but is also not trapped; this is an extension of the “double displacement” concept.
Piri and Blunt (2002) describes a network model that consists of pores connected by pore
throats. Pores and pore throats have triangular, square, or circular cross sections. The network
used in Piri and Blunt (2002) is based on a 27 mm3 core of Berea with 12349 pores and 26146 throats;
condition number varies between 1 and 19, with an average of 4.19. Pores vary from 3.62 μm to
73.54 μm; the throats vary from 0.90 μm to 56.85 μm, with an absolute permeability of 2600 md.
Piri and Blunt (2005a) continue the discussion of how to conduct network simulations. Piri and
Blunt (2005b) discuss the results of mixed-wet network modeling, including relative permeability
predictions and the distribution of fluids in different pore sizes after primary drainage, water flood,
and tertiary gas injection. Oil moves into intermediate sized pores during gas injection as a result
of double displacement. Tertiary gas injection and secondary gas injection predict different relative
permeability curves.
Nguyen, Sheppard, Knackstedt, and Pinczewski (2006) compares the results of a Berea based
network model to the relative permeability measurements of Oak (1990). Suicmez, Piri, and Blunt
(2006) compares the results of a Berea based network model to experimental data of Oak (1990),
Egermann et al. (2000), and Element et al. (2003). Suicmez et al. (2006) hypothesizes that relative
permeability is independent of flow path if mobile saturations are used rather than mobile plus
27
trapped saturations.
Mahmud (2007) uses a cubic 64 × 64 × 64 network model based on Fontainebleau sandstone.
Suicmez, Piri, and Blunt (2008) describes the results of network model simulations, including
trapped oil and gas as a function of initial gas saturation. Suicmez, Piri, and Blunt (2007) presents
results of network simulations of WAG. Pentland, Tanino, Iglauer, and Blunt (2010) compares pore
network models to a new set of coreflood experiments.
Sheng, Thompson, Fredrich, and Salino (2011) compares different methods of network simulation.
2.7.2
Micro Models
Campbell and Orr (1985) used micromodels to visualize CO2 /oil displacements. Their 2D models were 88 mm × 63 mm with etched glass pores that are 750 μm × 140 μm. Sohrabi, Tehrani,
Danesh, and Henderson (2001) uses oil wet and mixed wet micromodels to visualize WAG. Sohrabi,
Tehrani, Danesh, and Henderson (2004) uses high pressure micromodels to visualize WAG processes.
Dong, Foraie, Huang, and Chatzis (2005) uses micromodels to illustrate pore scale effects in immiscible WAG. Chalbaud, Lombard, Martin, Robin, Bertin, and Egermann (2007) presents micromodel
experiments using CO2 and nitrogen. Bondino, McDougall, Ezeuko, and Hamon (2010) presents
the results of a re-pressurization micromodel experiment.
2.7.3
Additional Pore Scale Simulation Discussion
van Dijke, Sorbie, Sohrabi, Tehrani, and Danesh (2002) uses a combination of network simulation and micromodels to help understand WAG processes. Ajo-Franklin (2007) presents an
overview of different techniques for extracting a pore-network model from rocks; these include
2D optical microscope image analysis, micro-CT scans, scanning confocal microscopy, and ablation
combined with 2D imagery. Algive, Bekri, and Vizika (2009) uses a pore network model to evaluate
geochemical changes while injecting CO2 .
Knackstedt, Dance, Kumar, Averdunk, and Paterson (2010) uses QEMscan images to construct
pore network model and compares the network simulation results to drainage and imbibition experiments on the same cores. Youssef, Bauer, Bekri, Rosenberg, and Vizika (2010) uses microCT
scans to measure the in situ saturation during fluid flow experiments.
28
2.8
Interfacial Tension
Interfacial tension (IFT) represents the strength of the interface between two fluids. IFT is
often used to scale relative permeability as miscibility is developed based on a pressure change.
2.8.1
Interfacial Tension Methods
Weinaug and Katz (1943) describes an early approach for calculating the interfacial tension .
Lee and Chien (1984) describes a method for calculating interfacial tension. Danesh, Dandekar,
Todd, and Sarkar (1991) describes an adjustment to the calculation of interfacial tension. Zuo
and Stenby (1998) fits several different methods for calculating interfacial tension to experimental
data; these methods are based on Helmholtz free energies and chemical potential, which can be
calculated from an EOS.
Grigg and Schechter (1998) reviews various interfacial tension methods and concludes that an
exponent of 3.88 is best in the following equation:
σ=
n
n # n
X
−
ξ
P
Y
ξln P#
m m
v m m
3.88
(2.4)
m
Grigg and Schechter (1998) defines the following parachors, consistent with (2.4):
• PCO2 = 82.00
• PCH4 = 74.05
• PnC4 = 193.90
• PnC10 = 440.69
Schechter and Guo (1998) presents different ways to calculate parachors.
2.8.2
Interfacial Tension and Relative Permeability
Bardon and Longeron (1980) conducted gas-oil relative permeability measurements under different interfacial tensions. The krog changed curvature and endpoint saturations significantly over
interfacial tensions from σ = 12.6 × 10−3 N/m to σ = 0.001 × 10−3 N/m. The krg did not change
much above σ = 0.065 × 10−3 N/m, but changes significantly between σ = 0.065 × 10−3 N/m and
σ = 0.001 × 10−3 N/m
29
Harbert (1983) presents water-oil relative permeability data for several formations under interfacial tensions between σ = 0.1 × 10−3 N/m and σ = 2.0 × 10−3 N/m
Shen, Zhu, Li, and Wu (2006) evaluates changes in the krw and krow as the water-oil interfacial
tension changes σow . Al-Wahaibi, Grattoni, and Muggeridge (2006) presents changes in the gas-oil
relative permeability as the interfacial tension σgo changes. Fagerlund, Niemi, and Odén (2006)
scales the relative permeabilities based on interfacial tensions.
2.8.3
Spreading Coefficient
The spreading coefficient is defined as
Sow = σwg − σog − σow
(2.5)
Oren and Pinczewski (1994) uses micromodels to study the effects of the spreading coefficient on
production mechanisms.
2.8.4
Interfacial Tension Fit Gas-Oil
Firoozabadi, Katz, Soroosh, and Sasjjadian (1988) presents interfacial tension fits for various
fluids. Pedersen, Lund, and Fredenslund (1989) presents interfacial tension fits for various fluids.
Rønningesen (1993) presents interfacial tension fits for North Sea fluids.
2.8.5
Water Interfacial Tension
Bahramian, Danesh, Gozalpour, Tohidi, and Todd (2007) presents fits of interfacial tension for a
water-methane-cyclohexane-decane system. Rushing, Newsham, Van Fraassen, Mehta, and Moore
(2008) presents gas-water interfacial tension data for several dry gas systems for temperatures
between 300◦ F and 400◦ F.
2.8.6
CO2 -Brine Interfacial Tension
Bennion and Bachu (2006) presents data for brine-CO2 relative permeability curves, including
how they change with interfacial tension. Chalbaud, Robin, and Egermann (2006) and Chalbaud,
Robin, Lombard, Martin, Egermann, and Bertin (2009) present a correlation for brine-CO2 interfacial tension which includes the effects of different salinities, temperatures, and pressures.
30
g 82 g
N
mol
−3 N
−
ρ
]+
ρbr
σCO2 ,br [ ] = 26×10 [ ]+1.2550mNaCl[
CO
2
m
m
kg
44.01
cm3
cm3
4.7180
T [K]
Tc [K]
1.0243
(2.6)
Bennion and Bachu (2008a) presents correlations for CO2 -brine interfacial tension as a function
of pressure, temperature, and salinity. Delshad, Kong, and Wheeler (2011) presents a formulation
for CO2 -brine interfacial tension, plus some adjustments for relative permeability.
Chun and Wilkinson (1995) presents a correlation for a CO2 -H2 O system; this correlation is
only applicable for this specific system. Ramey (1973) presents a general method for calculating
the oil-water interfacial tension, but the method requires reading one of the values from a graph.
Firoozabadi and Ramey (1988) presents several correlations for gas-water and oil-water interfacial
tensions. Firoozabadi and Ramey (1988) uses the Lee and Chien (1984) parachor for water of
52.0. Zuo and Stenby (1997) describes a way to calculate interfacial tension using gradient theory,
requiring the Helmholtz free energy and chemical potential. There are various adjustments for
pure compounds, including one for H2 O and one for CO2 . Shariat, Moore, Mehta, Van Fraassen,
Newsham, and Rushing (2011) presents a summary of gas-water interfacial tension data.
2.9
Liquid-Liquid-Vapor
At temperatures below 50◦ C (for instance Nghiem and Li (1984)), it is possible for two liquid
hydrocarbon phases to form in addition to a liquid water phase, a gaseous phase, and a solid
asphaltene-rich phase. This is sometimes called the “LLV” region, corresponding to the two liquids
and vapor phase present for the hydrocarbon phases, or the “LLL” region when the aqueous phase
is included. The LLV region is typically relatively narrow in pressure and composition, Figure 2.1.
Jarell, Fox, Stein, and Webb (2002) and Lake (1989) discuss some of these effects. Figure 2.2 shows
the different possible displacement mechanisms for a CO2 flood; type III is the liquid-liquid vapor
region.
Several papers discuss experiments which show liquid-liquid-vapor portions of the phase diagram. Gardner, Orr, and Patel (1981) present experiments using Wasson crude oil and CO2 . Orr,
Yu, and Lien (1981) present experiments with Maljamar crude oil and mixtures of pure components.
Baker, Pierce, and Luks (1982) present experiments using CO2 plus various pure components and
31
Figure 2.1: Low temperature phase behavior of Wasson crude showing the presence of two liquid
hydrocarbon phases (Lake, 1989).
32
Figure 2.2: Various miscibility regions for a CO2 flood, (Klins, 1984). Region I is immiscible. Region
II may develop miscilibility. Region III is a miscible region that may contain two hydrocarbon liquid
phases. Region IV is the first contact miscible region. Region V involves liquid CO2 .
some experiments using CO2 plus Levelland crude oil. Turek et al. (1984) present experiments
using a synthetic oil and CO2 , plus an unnamed reservoir oil. Enick, Holder, and Morsi (1985)
present experimental data for a pure component system of CO2 and tridecane that displays LLV
behavior. Bryant and Monger (1988) present experimental data using Wasson crude oil plus CO2 .
Godbole, Thele, and Reinbold (1995) and Wang and Strycker (2000) present experimental results
for fields in Alaska.
Other papers discuss methods for calculating LLV or LLLV equilibria. Fussell (1979) presents
an early discussion of the Minimum Variable Newton Raphson technique. Risnes and Dalen (1984)
describe a methodology for multi-phase flash. Nghiem and Li (1984) discuss the Quasi Newton
Successive Substitution method used in CMG. Nghiem and Li (1986) continue the discussion of
Nghiem and Li (1984) and illustrate the simulation of a slim tube experiment using Wasson crude
oil. Baker et al. (1982) provide the details of the computation of Gibbs Free Energy for determining
stability. Enick, Holder, and Mohamed (1987) provide a detailed description of the search strategy
for stable phases in a four-phase flash formulation. These are illustrated using Maljamar crude
33
oil mixed with CO2 . Nagarajan et al. (1991a) provide a detailed description of four-phase flash
calculations. Lindeloff and Michelsen (2003) present the methods used by PVTsim and illustrates
these techniques using four different crude oils mixed with CO2 . Li and Firoozabadi (2010) present
a good summary of several different methods for calculating stability analysis and three-phase flash
calculations, illustrated using a Maljamar crude oil mixed with CO2 .
2.10
Asphaltenes
Asphaltenes are heavy hydrocarbon components that are not soluble in pentane, hexane, heptane or CO2 but are soluble in benzene and toluene. Asphaltenes can alter the permeability and
wettability of rocks they are deposited on; this is often a concern for production or pipeline engineers but is of lesser concern to reservoir engineers. Asphaltene literature was reviewed for this
project, but asphaltene deposition and simulation is being evaluated by another Ph.D. student at
Colorado School of Mines, Tadesse Teklu.
Pedersen and Christensen (2007) present a good review of different asphaltene deposition mechanisms. Leontaritis (1989) and Kokal and Sayegh (1995) present a good review of asphaltene literature including a description of different deposition mechanisms. The following additional articles
include good descriptions of specific asphaltene deposition models: Novosad and Costain (1990),
Nghiem, Hassam, Nutakki, and George (1993), Leontaritis, Amaefule, and Charles (1994), Mansoori (1994), Nghiem, Coombe, and Farouq Ali (1998), Nghiem, Kohse, Farouq Ali, and Doan
(2000), Kohse, Nghiem, Maeda, and Ohno (2000), Nghiem, Sammon, and Kohse (2001), Kohse
and Nghiem (2004), and Fazelipour, Pope, and Sepehrnoori (2008).
Leontaritis and Mansoori (1988) and Mohammed, Arisaka, and Kumazaki (1998) provide reviews of asphaltene issues in various fields. Kim, Boudh-Hir, and Mansoori (1990) provide a good
review of the role of asphaltenes in wettability alteration. Collins and Melrose (1983) and Yan,
Plancher, and Morrow (1997) describe experiments designed to measure wettability alteration with
asphaltene deposition. The following additional articles include the results of interesting experiments related to asphaltene deposition in both clastic and carbonate rocks: Hirschberg, deJong,
Schipper, and Meijer (1984), Monger and Fu (1987), Monger and Trujillo (1991), Dubey and Waxman (1991), Minssieux (1997),Srivastava and Huang (1997), Srivastava, Huang, and Dong (1999),
Ali and Islam (1998), Nabzar, Aguilera, and Rajoub (2005), Broad, Al Binbrek, Neilson, and Gib-
34
son (2005), Oskui, Salman, Gholoum, Rashed, Al Matar, Al-Bahar, and Kahali (2006), Loahardjo,
Xie, and Morrow (2008), and Hashmi and Firoozabadi (2010).
35
CHAPTER 3
COMPOSITIONAL RESERVOIR SIMULATION OVERVIEW
The goal of this dissertation is to develop a practical, robust, computationally efficient compositional simulator with improved physics for CO2 flooding. This project consists of the following
three main topics.
1. Compositional simulation formulation
2. Formulation is amenable for parallel computing
3. Science of CO2 water-alternating-gas injection
• Evaluate generalized capillary pressure and relative permeability functional relationships
• Evaluate existing algorithms for three-phase relative permeability
• Evaluate existing and new algorithms for capillary pressure and relative permeability
hysteresis
• Evaluate existing and new algorithms for trapping of various phases, including compositional mixing associated with trapping
• Evaluate relative permeability and capillary pressure changes with wettability and interfacial tension
• Account for CO2 phase behavior
– High solubility of CO2 in water phase
– Adjustments to equation of state
3.1
Compositional Simulation
This project involves the simulation of compositional fluid flow in porous media, as applied
to carbon dioxide (CO2 ) water-alternating-gas (WAG) injection. One of the earliest papers on
compositional simulation in the petroleum industry was Coats (1980). Chase and Todd (1984) was
an early paper on how to simulate CO2 injection. There are three main approaches for compositional
36
simulation formulations, as expressed by Wong et al. (1990), Acs et al. (1985), and Watts (1986).
Some of the details of the formulation used in this project are discussed in lecture notes from
Dr. Kazemi, (Kazemi, 2008a,b, 2009, 2010), and others were derived as part of my dissertation
work.
3.2
Commercial Simulators
There are several commercial reservoir simulators, but the commercial software with the largest
market share and which is most often used as a benchmark for simulation results is Eclipse, (Schlumberger, 2007a). The two primary manuals for Eclipse have a description of all of the available options in Schlumberger (2007a), and a more detailed technical description in Schlumberger (2007b).
Computer Modeling Group provides a suite of reservoir simulators; the CMG simulator applicable
to compositional simulation is “GEM”, CMG (2010). Landmark Graphics Corporation provides a
suite of reservoir simulators, including VIP core and VIP executive, Landmark (2000). In addition
to the commercial simulators, there are a number of proprietary reservoir simulators which have
been developed within large oil companies. Saudi Aramco’s “Gigapowers”, Dogru et al. (2008),
was designed form the beginning as a parallel reservoir simulator, with a specific goal to simulate
reservoirs with a large number of grid cells.
3.3
Mathematical Formulation
This project describes three-phase, compositional fluid flow in porous media, as applicable to
the oil and gas industry. The three phases considered are the oil phase, the gas phase, and the water
phase (also called the aqueous phase). Under normal conditions of pressure and temperature, all
three phases are immiscible with respect to each other, but under some conditions of temperature
and pressure the oil and gas phases become miscible.
The partial differential equations used to solve for compositional fluid flow are second order in
space and first order in time. This formulation uses Po , So , Sg , X1 , . . . , XNC −2 , and Y1 , . . . , YNC −2
as the primary variables1 . NC is defined as the total number of components, including the H2 O
component. This results in 2NC − 1 primary variables. There are NC component equations and
NC − 1 thermodynamic constraints.
1
All variables are defined in Chapter 22.
37
Table 3.1 shows the distribution of the components in the three phases, illustrated with an
8-component system. The formulation used here accounts for the solubility of CO2 in the aqueous phase, but neglects the solubility of H2 O in the oil and gas phases and the solubility of the
hydrocarbon components in the aqueous phase, since they are not expected to have a significant
effect in WAG injection problems. For Table 3.1, the primary variables are Po , So , Sg , and the pure
hydrocarbon components X1 , . . . , XNC −2 , and Y1 , . . . , YNC −2 , or 2NC − 1 = 15 primary variables.
There are NC = 8 component equations, one for each of the composite hydrocarbon components,
one for CO2 , and one for H2 O. There are NC − 1 = 7 thermodynamic constraints, one for each of
the pure hydrocarbons and one for CO2 . The Wm (the solubility of CO2 in water) are evaluated
explicitly for the spatial derivatives.
Table 3.1: Distribution of components in phases for NC = 8
component
C1
CI1
CI2
CH1
CH2
CH3
CO2
H2 O
oil
X1
XI1
XI2
XH1
XH2
XH3
XCO2
0
gas
Y1
YI1
YI2
YH1
YH2
YH3
YCO2
0
aqueous
0
0
0
0
0
0
WCO2
WH2 O
The following set of equations describes the differential equations used to solve for compositional
fluid flow in a porous medium, as used in the oil and gas industry. For each component (total NC ),
the general partial differential equation for the component mass balance is in (3.1).
0.006328∇ · Xm ξo λo k(∇Po − γo ∇D) + 0.006328∇ · Ym ξg λg k(∇Po + ∇Pcgo − γg ∇D) +
0.006328∇ · Wm ξw λw k(∇Po − ∇Pcow − γw ∇D) + Xm ξo q̂o + Ym ξg q̂g + Wm ξw q̂w =
∂
φ(Xm So ξo + Ym Sg ξg + Wm Sw ξw )
(3.1)
∂t
The normalization constraints on each component are represented by (3.2)–(3.3). Because there
is no H2 O in the hydrocarbon liquid or vapor phases, the upper limit of the sums are to NC − 1
not NC .
38
N
C −1
Xm = 1
=⇒
XNC −1 = 1 −
m
N
C −1
Ym = 1
=⇒
YNC −1 = 1 −
N
C −2
m
N
C −2
Xm
(3.2)
Ym
(3.3)
m
m
For the CO2 component, it is useful to recast (3.1) as (3.4). Use (3.2)–(3.3) to reduce the
degrees of freedom of the terms of (3.4).
N
C −2
0.006328∇ · (1 −
Xm )ξo λo k(∇Po − γo ∇D) +
m =1
N
C −2
m =1
0.006328∇ · (1 −
Ym )ξg λg k(∇Po + ∇Pcgo − γg ∇D) +
0.006328∇ · Wm ξw λw k(∇Po − ∇Pcow − γw ∇D) +
(1 −
N
C −2
m =1
Xm )ξo q̂o + (1 −
N
C −2
Ym )ξg q̂g + Wm ξw q̂w =
m =1
N
N
C −2
C −2
∂
φ((1 −
Xm )So ξo + (1 −
Ym )Sg ξg + Wm Sw ξw )
∂t
m =1
(3.4)
m =1
For the H2 O component, (3.1) simplifies to (3.5). It is useful to use WH2 O + WCO2 = 1.
0.006328∇ · (1 − WCO2 ) ξw λw k(∇Po − ∇Pcow − γw ∇D) + (1 − WCO2 ) ξw q̂w =
∂
φ((1 − WCO2 ) Sw ξw )
(3.5)
∂t
3.4
Partially Implicit Formulation
Different primary variables can be evaluated at time n or iteration level . The accumulation
terms are evaluated at time for the new iteration and at n for the previous time step. In the
IMPES formulation, the pressure terms in the spatial derivatives are evaluated at and all other
spatial and well variables are evaluated at n; some IMPES formulations evaluate the pressure in
the well terms at . In the IMPSEC formulation, the pressure and saturation terms in the spatial
derivatives are evaluated at and all other spatial and well variables are evaluated at n. In the
fully implicit formulation, all the primary variables in the spatial derivatives are evaluated at .
39
3.4.1
IMPES
For the Implicit Pressure, Explicit Saturation (IMPES) formulation, the pressure is evaluated
at n + 1 and the saturations and compositions are evaluated at time n. The finite difference form
of (3.1) using the IMPES formulation is as follows, (3.6).
0.006328∇ ·
0.006328∇ ·
0.006328∇ ·
X n ξn
m o n #
kro k (∇Pon+1 − γon ∇D# ) +
μno
Y nξn
m g n #
n
krg k (∇Pon+1 + ∇Pcgo
− γgn ∇D# ) +
n
μg
W n ξn
m w n
n
n
krw k # (∇Pon+1 − ∇Pcow
− γw
∇D# )
n
μw
n n n
n n n
+ Xm
ξo q̂o + Ymn ξgn q̂gn + Wm
ξw q̂w =
1 n+1 n+1 n+1 n+1
n+1 n+1 n+1
φ
(Xm So ξo + Ymn+1 Sgn+1 ξgn+1 + Wm
Sw ξw ) −
Δt
1 n n n n
n n n
φ (Xm So ξo + Ymn Sgn ξgn + Wm
Sw ξw )
Δt
3.4.2
(3.6)
IMPSEC
For the Implicit Pressure, Implicit Saturation, Explicit Composition (IMPSEC) formulation,
the pressure and saturations are evaluated at n + 1 and the compositions are evaluated at time n.
The finite difference form of (3.1) using the IMPSEC formulation is as follows, (3.7).
0.006328∇ ·
0.006328∇ ·
0.006328∇ ·
X n ξn
m o n+1 #
kro k (∇Pon+1 − γon ∇D# ) +
μno
Y nξn
m g n+1 #
n+1
krg k (∇Pon+1 + ∇Pcgo
− γgn ∇D# ) +
n
μg
W n ξn
m w n+1 #
n+1
n
krw k (∇Pon+1 − ∇Pcow
− γw
∇D# ) +
n
μw
n n n
n n n
Xm
ξo q̂o + Ymn ξgn q̂gn + Wm
ξw q̂w =
1 n+1 n+1 n+1 n+1
n+1 n+1 n+1
φ
(Xm So ξo + Ymn+1 Sgn+1 ξgn+1 + Wm
Sw ξw ) −
Δt
1 n n n n
n n n
φ (Xm So ξo + Ymn Sgn ξgn + Wm
Sw ξw )
Δt
3.4.3
(3.7)
Fully Implicit
For the Fully Implicit formulation, everything is evaluated at n + 1. Our expectation is that the
Fully Implicit formulation will not be required for this project. The finite difference form of (3.1)
using the Fully Implicit formulation is as follows, (3.8).
40
0.006328∇ ·
0.006328∇ ·
0.006328∇ ·
X n+1 ξ n+1
o
n+1 #
kro
k (∇Pon+1 − γon+1 ∇D# ) +
μn+1
o
Y n+1 ξ n+1
m
g
n+1 #
n+1
n+1
n+1
#
k
k
(∇P
+
∇P
−
γ
∇D
)
+
rg
o
cgo
g
μn+1
g
W n ξn
m w n+1 #
n+1
n+1
n+1
#
k
k
(∇P
−
∇P
−
γ
∇D
)
+
rw
o
cow
w
μnw
m
n+1 n+1 n+1
n+1 n+1 n+1
=
ξo q̂o + Ymn+1 ξgn+1 q̂gn+1 + Wm
ξw q̂w
Xm
1
n+1 n+1 n+1
n+1 n+1 n+1
φn+1 (Xm
So ξo + Ymn+1 Sgn+1 ξgn+1 + Wm
Sw ξw ) −
Δt
1 n n n n
n n n
φ (Xm So ξo + Ymn Sgn ξgn + Wm
Sw ξw )
Δt
3.4.4
(3.8)
Comparison
The IMPES formulation is computationally very efficient. Using a banded solver with bandwidth β = Nx × Nz and total number of grid cells Nxyz = Nx × Ny × Nz , the computational
order for the IMPES formulation is O β 2 Nxyz . The IMPSEC formulation captures additional
variability in the saturations with the possibility of larger stable timesteps. The banded solver
for an IMPSEC algorithm has computational order O (6β)2 (3Nxyz ) , or O [108 × IMPES]. A fully
implicit algorithm is computationally inefficient. It would only be necessary if the compositional
gradient between grid cells were a significant driver for fluid flow between the grid cells. A fully
implicit algorithm has computational order O (2 · (2NC − 1)β)2 ((2NC − 1)Nxyz ) . For NC = 8
components, this is O (2)2 (15)3 β 2 Nxyz , O [125 × IMPSEC], or O [13500 × IMPES].
3.5
Thermodynamic Constraints
There are NC − 1 thermodynamic constraints evaluated at time n + 1 which represent the
equilibrium conditions between the hydrocarbon liquid and vapor phases. The CO2 in the water
phase is evaluated explicitly using K-values.
n+1
fn+1
om = fgm
(3.9)
For the Peng-Robinson Equation of State (Peng and Robinson, 1976), this is defined as follows:
41
#
bm
z̆ln+1 − 1 − ln z̆ln+1 − Bln+1 +
·P
· exp n+1
bl
√
z̆ln+1 +
2 + 1 Bln+1
An+1
b#
2
l
m
n+1 #
√
− √ n+1 ·
+
Xn amn − n+1 · ln
an+1
bl
2 2Bl
z̆ln+1 −
2 − 1 Bln+1
l
n
#
bm
− Ymn+1 · P n+1 · exp n+1
z̆vn+1 − 1 − ln z̆vn+1 − Bvn+1 +
bv
√
2 + 1 Bvn+1
z̆vn+1 +
An+1
2
b#
v
m
n+1 #
√
− √ n+1 ·
Yn amn − n+1 · ln
an+1
bv
2 2Bv
z̆vn+1 −
2 − 1 Bvn+1
v
n
n+1
Xm
3.6
n+1
(3.10)
Typical Sizes
The formulation above involves three levels of iterations, assuming a direct matrix solve.
• Time loop n: time step sizes range from a few seconds to a maximum of roughly 30 days.
Total time ranges from a few years to about 150 years. Total time steps for a simulation are
typically between 100 and 1000.
• Time loop : the progression to a new time step is an iterative process; typically this involves
between 3 and 15 iterations, with around 5 being typical.
• Flash loop iterations: this typically involves between 3 and 20 iterations, but there are some
cases near phase transitions which may require hundreds of iterations. Typical values are
probably around 5.
Multiplying out these typical values yields an expected value of 300 × 5 × 5 = 7, 500 solutions of
the matrix equation Aδ = R and 7500 · Nxyz flash calculations. A starts out as a [(3NC ) · Nxyz ] ×
[(3NC ) · Nxyz ] matrix. Some of the thermodynamic constraints have been evaluated explicitly in
this formulation, involving the simplification of A to a sparse [(2NC − 1) · Nxyz ] × [(2NC − 1) · Nxyz ]
matrix. This is further simplified into a [3Nxyz ] × [3Nxyz ] matrix for IMPSEC or a Nxyz × Nxyz
matrix for IMPES. The following are some typical values:
• NC , the total number of components, ranges from 4 to about 15, with 7–10 being typical.
Note that this is already simplified down from the hundreds of chemical components typically
present in a hydrocarbon system.
42
• Nxyz represents the total number of simulation grid cells. For a rectangular 3-D matrix,
Nxyz = Nx ∗ Ny ∗ Nz . The problem size is characterized by the following gradational scale for
Nxyz .
– Nxyz : (0, 104 ) most 1-D or 2-D problems and very small 3-D problems
– Nxyz : (104 , 105 ) are considered small problems in industry. Problems through this size
are typically run in serial mode.
– Nxyz : (105 , 106 ) are considered medium problems in industry. Problems through this
size are often run in serial mode, but sometimes run in parallel.
– Nxyz : (106 , 107 ) are considered large or very large problems in industry, depending on
the hardware available. These are almost always run in parallel.
– Nxyz : (107 , 109 +) are considered very large problems. These are always run in parallel,
and only a few companies have simulators that can handle this size model. Saudi Aramco
ran their first billion cell model in the fall of 2008. They are actively developing software
to routinely run these billion cell models.
– Nxyz : 1012 +: it is easy to define mathematically why models of 1012 or more grid cells
would be beneficial. For instance, if we have an oil field that is 10 km × 100 km × 100 m
and we split this into grid cells which are 1 m × 1 m × 0.1 m, this is 1012 grid cells. If we
consider basin modeling for a basin which is 1000 km × 1000 km × 10 km and simulate
this using a 100 m × 100 m × 1 m, this represents 1013 grid cells. Pore scale modeling of
a 10 cm × 10 cm × 10 cm block at a resolution of 1 μm × 1 μm × 1 μm represents 1012
grid cells.
If we use a medium sized problem with 105 grid cells and 8 components, a typical solution might
involve 7, 500 solves of the sparse matrix A of dimensions [(15) · (105 )] × [(15) · (105 )]. If we use a
naive dense matrix solution of O(N 3 ), this represents approximately (1.5 · 106 )3 × 7, 500 = 2.5 · 1022
FLOP. Fortunately, sparse matrix solves have a lower order than O(N 3 ) (more details in final
report).
3.7
Off-Diagonal Terms
Off-diagonal terms have the following form, Figure 3.1.
43
Figure 3.1: Block 2: block geometry for the off-block diagonal values with the IMPES formulation
for a NC = 5 problem. Black represents non-zero values; gray represents zero values.
Po So Sg Xm
Cm
Gm
Ym
X
0
0
0
0
0
0
0
0
0
(3.11)
Off-diagonal bands have the following form, here illustrated for i + 1, j, k.
Po
Cm
DPmn
x
y t,i+1,jk
z
Gm
3.8
0
So
Sg
X m
Ym
0
0
0
0
0
0
0
0
(3.12)
Well Terms
Figure 3.2: Block 4: well terms for the component equations for a NC = 5 problem. Black represents
non-zero values; gray represents zero values.
Well unknowns have the following form, Figure 3.2.
|q Pt,w
t,w
Cm
Gm
(3.13)
X
0
Well unknowns have the following form.
44
Cm
Gm
3.9
|q Pt,w
t,w
WDWmn
ijk
(3.14)
0
Right Hand Side
Figure 3.3: Block 6: right-hand-side terms for the component equations for a NC = 5 problem.
Black represents non-zero values; gray represents zero values.
Right-hand-side, constant terms have the following form, Figure 3.3.
R
Cm
Gm
(3.15)
X
X
Right-hand-side, constant terms without well connections have the following form.
Cm
Gm
m
VR
Δt Accijk
−
R
mn
mn
− DCmn
xt,ijk − DCyt,ijk − DCzt,ijk
m
−fm
o,ijk + fg,ijk
mn
VR
Δt Accijk
(3.16)
Right-hand-side, constant terms with well connections have the following form.
Cm
Gm
m
VR
Δt Accijk
−
mn
VR
Δt Accijk
R
mn
mn
mn
− DCmn
xt,ijk − DCyt,ijk − DCzt,ijk + WCijk
m
−fm
o,ijk + fg,ijk
(3.17)
Right-hand-side, constant terms above the bubble point or below the dew point without well
connections have the following form.
Cm
Gm
GNC −1
m
VR
Δt Accijk
−
R
mn
mn
− DCmn
xt,ijk − DCyt,ijk − DCzt,ijk
m
−fm
o,ijk + fg,ijk
−GNC −1,ijk
mn
VR
Δt Accijk
45
(3.18)
Right-hand-side, constant terms above the bubble point or below the dew point with well
connections have the following form.
R
Cm
Gm
m
VR
Δt Accijk
−
mn
VR
Δt Accijk
−
GNC −1
3.10
mn
DCmn
xt,ijk − DCyt,ijk
m
−fm
o,ijk + fg,ijk
−GNC −1,ijk
mn
− DCmn
zt,ijk + WCijk
(3.19)
Total Rate Equations
Figure 3.4: Block 5: blocks for the well equations for a NC = 5 problem. Black represents non-zero
values; gray represents zero values.
Total rate equations for each well have the following form, Figure 3.4.
Po So Sg Xm
Qw
X
0
0
0
Ym
(3.20)
0
Total rate equations for each well have the following form.
Qw
Po
QDPnijk
So Sg Xm
0
0
0
Ym
(3.21)
0
Diagonal terms for the total rate equations have the following form.
|q Pt,w
t,w
Qw
(3.22)
X
Diagonal terms for the total rate equations have the following form.
Qw
|q Pt,w
t,w
QDWnijk
(3.23)
Right-hand-side, constant terms for the total rate equations have the following form.
R
Qw
(3.24)
X
Right-hand-side, constant terms for the total rate equations have the following form.
46
R
Qw QCn
ijk
3.11
(3.25)
Accumulation
Define the accumulation term
=
φ
ξ
S
X
+
φ
ξ
S
Y
+
φ
ξ
S
W
Accm
i oi oi mi
i gi gi mi
i wi wi mi
i
3.12
(3.26)
Accumulation Pressure Derivatives
For the normal hydrocarbon components,
∂Accmi
∂P ,
for cell i and component m = 1 . . . NC − 2.
∂ξgi
∂Accmi
∂φi
∂φi
∂ξoi
= ξoi
+ ξgi
+ φi Soi
+ φi Sgi
Soi Xmi
Sgi Ymi
Xmi
Ymi
∂P
∂P
∂P
∂P
∂P
For the CO2 component,
∂Accmi
∂P ,
for cell i and component m = NC − 1.
∂Accmi
∂φi
∂φi
∂φi
= ξoi
+ ξgi
+ ξwi
+
Soi Xmi
Sgi Ymi
Swi
Wmi
∂P
∂P
∂P
∂P
∂WCO
∂ξgi
2 ,i
∂ξoi
∂ξwi
+ φi Sgi
+ φi Swi
+ φi ξwi
Xmi
Ymi
Wmi
Swi
φi Soi
∂P
∂P
∂P
∂P
For the H2 O component,
∂Accmi
∂P ,
(3.27)
(3.28)
for cell i and component m = NC .
∂WCO2 ,i
∂Accmi
∂φi
∂ξwi
= ξwi
+ φi Swi
− φi ξwi
Swi
Wmi
Wmi
Swi
∂P
∂P
∂P
∂P
3.13
(3.29)
Accumulation Saturation Derivatives
Evaluate
∂Accmi
∂So .
∂Accmi
= φi ξoi
Xmi
− φi ξwi
Wmi
∂So
Evaluate
(3.30)
∂Accmi
∂Sg .
∂Accmi
= φi ξgi
Ymi
− φi ξwi
Wmi
∂Sg
(3.31)
47
Above the bubble point, Sg = 0 and Sg → Pb becomes a new primary variable and
Below the dew point, So = 0 and So → Pd becomes a new primary variable and
3.14
∂Accmi
∂Pd
∂Accmi
∂Pb
= 0.
=0
Accumulation Composition Derivatives
For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m = 1 . . . NC − 2,
evaluate
∂Accmi
.
∂Xm
∂Accmi
∂ξoi
= φi Soi
Xmi
+ φi ξoi
Soi δm,m
∂Xm
∂Xm
(3.32)
For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m = 1 . . . NC − 2,
evaluate
∂Accmi
.
∂Ym
∂ξgi
∂Accmi
= φi Sgi Ymi
+ φi ξgi
Sgi δm,m
∂Ym
∂Ym
(3.33)
For the CO2 component equation Cm = CNC −1 and m = 1 . . . NC − 2, evaluate
∂Accmi
.
∂Xm
∂AccCO2 ,i
∂ξoi
= φi Soi
XCO
− φi Soi
ξoi
2 ,i
∂Xm
∂Xm
(3.34)
For the CO2 component equation Cm = CNC −1 and m = 1 . . . NC − 2, evaluate
∂Accmi
.
∂Ym
∂ξgi
∂ξ ∂Accmi
∂WCO2
= φi Sgi YCO2
− φi Sgi
ξgi + φi Swi
WCO2 wi + φi Swi
ξwi
∂Ym
∂Ym
∂Ym
∂Ym
For the water component equation Cm = CNC and m = 1 . . . NC − 2, evaluate
(3.35)
∂Accmi
.
∂Xm
∂AccH2 O,i
=0
∂Xm
(3.36)
For the water component equation Cm = CNC and m = 1 . . . NC − 2, evaluate
∂AccH2 O,i
∂ξ ∂WCO2
= φi Swi
WH2 O wi − φi Swi
ξwi
∂Ym
∂Ym
∂Ym
48
∂Accmi
.
∂Ym
(3.37)
3.15
Pressure Spatial Derivatives
The following derivatives are written in terms of x and i ± 1. The same approach applies to y
and j ± 1 and z and k ± 1.
The following are the multiples of δPi±1 . All ± are either positive or negative for this equation.
mn
mn
mn
DPmn
=
T
+
T
+
T
1
1
1
xt,i±1
xo,i±
xg,i±
xw,i±
2
2
(3.38)
2
The following are the multiples of δPi .
mn
mn
=
−
DP
+
DP
DPmn
xt,i
xt,i+1
xt,i−1 =
mn
mn
mn
mn
mn
mn
+
T
+
T
+
T
+
T
+
T
(3.39)
− Txo,i+
1
xo,i− 1
xg,i+ 1
xg,i− 1
xw,i+ 1
xw,i− 1
2
2
2
2
2
2
The following do not multiply deltas. All ± are either positive or negative for this equation.
mn
n
mn
n
n
=
T
·
P
−
γ
D
·
P
−
γ
D
+
P
+
T
DCmn
1
1
1
1
i±1
i±1
xt,i±1
i±1
i±1
cgo,i±1 +
xo,i± 2
o,i± 2
xg,i± 2
g,i± 2
mn
n
n
Pi±1
− γw,i±
(3.40)
Txw,i±
1 ·
1 Di±1 − Pcow,i±1
2
2
The following do not multiply deltas.
mn
n
mn
n
=
−T
·
P
−
γ
D
·
P
−
γ
D
−
T
+
DCmn
xt,i
i
i
xo,i+ 12
o,i+ 12 i
xo,i− 12
o,i− 12 i
mn
n
n
mn
n
n
·
P
−
γ
D
+
P
·
P
−
γ
D
+
P
−
T
− Txg,i+
1
i
cgo,i
i
cgo,i +
g,i+ 12 i
xg,i− 21
g,i− 21 i
2
mn
n
n
mn
n
n
·
P
−
γ
D
−
P
·
P
−
γ
D
−
P
−
T
(3.41)
− Txw,i+
1
i
cow,i
i
cow,i
w,i+ 1 i
xw,i− 1
w,i− 1 i
2
3.16
2
2
2
Fugacity Equations
The fugacities are defined by
m
fm
o = Φo Xm P
Evaluate
∂fm
oi
∂P ,
∂fm
oi
= fm
oi
∂P
m
fm
g = Φ g Ym P
(3.42)
m = 1 . . . NC − 1:
1 ∂Φm
oi
∂P
Φm
oi
+ Φm
oi Xm
(3.43)
49
Evaluate
∂fm
gi
∂P ,
∂fm
gi
= fm
gi
∂P
m = 1 . . . NC − 1:
m
1 ∂Φgi
∂P
Φm
gi
+ Φm
gi Ymi
(3.44)
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂fm
oi
= fm
oi
∂Xm
1 ∂Φm
oi
∂X
Φm
m
oi
1 ∂Φm
gi
∂Y
Φm
m
gi
(3.45)
3.17
1 ∂Φm
oi
∂X
Φm
m
oi
m
1 ∂Φgi
Φm
gi ∂Ym
for m = 1 . . . NC − 2:
+ Φm
gi P δm,m
(3.46)
∂fl
mi
∂Xm
for m = 1 . . . NC − 2:
− Φm
oi P
(3.47)
For the CO2 equations m = 1 . . . NC − 1, evaluate
∂fm
gi
= fm
gi
∂Ym
∂fm
gi
∂P
For the CO2 equations m = NC − 1, evaluate
∂fm
oi
= fm
oi
∂Xm
for m = 1 . . . NC − 2:
+ Φm
oi P δm,m
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂fm
gi
= fm
gi
∂Ym
∂fo
gi
∂Xm
∂fm
gi
∂P
for m = 1 . . . NC − 2:
− Φm
gi P
(3.48)
Computation for Fixed Rate
Each component equation Cw,α,m has a source term. The coefficient of δP is
#
n
n
n
n
n
n
n
n
n
WDPmn
w,α = −WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α
(3.49)
The coefficient of δPw is
#
n
n
n
n
n
n
n
n
n
WDWmn
w,α = WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α
The constant terms associated with the well are
50
(3.50)
#
n
n
n
n
n
n
n
n
n
WCmn
w,α = WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α ·
w ,n
− Pw, + Pw,α
Pw,α
(3.51)
Each well has a total rate equation. This equation has the following form for a fixed rate well.
The coefficient of δP is
QDPnw,α
= − WI#
w,α ·
emax n
qo,w,α
ξo,w,α λno,w,α
w,emax
ξo,w,α
max
+
emax n
qg,w,α
ξg,w,α λng,w,α
w,emax
ξg,w,α
max
+
emax n
qw,w,α
ξw,w,α λnw,w,α
w,emax
ξw,w,α
max
(3.52)
The coefficient of δPw is
QDWnw,α
=
α
max
α =1
WI#
w,α ×
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
(3.53)
w,emax
ξw,w,α
max
The constant terms associated with the constant rate equation
max α
#
= qt,w +
WIw,α ×
Pw,α
RHS
QCn
w,α
3.18
α =1
e
n
n
max
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
w,const @ n
− Pw, + Pw,α
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
×
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(3.54)
Computation for Fixed Pressure
Each component equation Cw,α,m has a source term. This term has the following form for a
fixed pressure well. The coefficient of δP is 0.
WDPmn
w,α = 0
(3.55)
is
The coefficient of δqt,w
WDWmn
w,α =
αmax
α =1
WI#
w,α
n
WI#
w,α λt,w,α
×
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α
Xm,w,α
51
(3.56)
The constant terms associated with the well are
WCmn
w,α = −
αmax
α =1
WI#
w,α
n
WI#
w,α λt,w,α
×
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α
Xm,w,α
(3.57)
Each well has a total rate equation. This equation has the following form for a fixed pressure
well. The coefficient of δP is
QDPnw,α
= WI#
w,α ·
emax n
qo,w,α
ξo,w,α λno,w,α
w,emax
ξo,w,α
max
+
emax n
qg,w,α
ξg,w,α λng,w,α
w,emax
ξg,w,α
max
+
emax n
qw,w,α
ξw,w,α λnw,w,α
(3.58)
w,emax
ξw,w,α
max
is
The coefficient of δqt,w
QDWnw,α = 1
(3.59)
The constant terms associated with the constant rate equation
QCn
w,α
=
,
−qt,w
−
α
max
#
w,n
WIw,α · Pw,α
− P
w,α ·
α =1
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
3.19
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(3.60)
Additional Implicit Decisions
• IMPES Primary variables 1
– implicit: Po is evaluated at time n + 1.
– δPo are computed directly from the matrix equation.
• IMPES Primary variables 2
– mixed: So , Sg are evaluated at time n for the spatial derivatives and at time n + 1 for
the time derivatives.
– mixed: Sw = 1 − So − Sg is evaluated at time n for the spatial derivatives and at time
n + 1 for the time derivatives.
52
– mixed: Xm , Ym are evaluated at time n for the spatial derivatives and at time n + 1 for
the time derivatives.
– mixed: Wm is evaluated at time n for the spatial derivatives and at time n + 1 for the
time derivatives. Evaluate thermodynamics at time .
– Zm = Wm + Xm + Ym
–
m Xm
= 1,
m Ym
= 1,
m Wm
=1
– Xm , Ym , Wm are computed using the local LU decomposition of matrix A
• IMPES Secondary variables 1
– explicit: kro (So , Sw , Sg ), krg (So , Sw , Sg ), krw (So , Sw , Sg ), saturations are evaluated at
time n. Functional dependence on wettability, interfacial tension, trapping, and hysteresis are evaluated less often.
– explicit: Pcgo (So , Sw , Sg ), Pcow (So , Sw , Sg ), Pcgw (So , Sw , Sg ), saturations are evaluated
at time n. Functional dependence on wettability, interfacial tension, trapping, and
hysteresis are evaluated less often. Assume time derivatives of pressure refer to Po .
– mixed: ξo (P, Xm ), ξg (P, Ym ), ξw (P, Wm ) are evaluated at time n for the spatial derivatives and time n + 1 for the time derivatives.
– mixed: Co (P ), Cg (P ) are evaluated at time n for the spatial derivatives and time n + 1
for the time derivatives.
– All kr and all Pc and their derivatives are computed using their own functions
– All ξ, Co , and Cg are computed from the flash computation.
• IMPSEC Primary variables 1
– implicit: Po is evaluated at time n + 1.
– implicit: So , Sg are evaluated at time n + 1.
– implicit: Sw = 1 − So − Sg is evaluated at time n + 1.
– δPo , δSo , and δSg are computed directly from the matrix equation.
• IMPSEC Primary variables 2
53
– mixed: Xm , Ym are evaluated at time n for the spatial derivatives and at time n + 1 for
the time derivatives.
– mixed: Wm is evaluated at time n for the spatial derivatives and at time n + 1 for the
time derivatives. Evaluate thermodynamics at time .
– Zm = Wm + Xm + Ym
–
m Xm
= 1,
m Ym
= 1,
m Wm
=1
– Xm , Ym , Wm are computed using the local LU decomposition of matrix A
• IMPSEC Secondary variables 1
– implicit: kro (So , Sw , Sg ), krg (So , Sw , Sg ), krw (So , Sw , Sg ), saturations are evaluated at
time n + 1. Functional dependence on wettability, interfacial tension, trapping, and
hysteresis are evaluated less often.
– implicit: Pcgo (So , Sw , Sg ), Pcow (So , Sw , Sg ), Pcgw (So , Sw , Sg ), saturations are evaluated
at time n + 1. Functional dependence on wettability, interfacial tension, trapping, and
hysteresis are evaluated less often. Assume time derivatives of pressure refer to Po .
– mixed: ξo (P, Xm ), ξg (P, Ym ), ξw (P, Wm ) are evaluated at time n for the spatial derivatives and time n + 1 for the time derivatives.
– mixed: Co (P ), Cg (P ) are evaluated at time n for the spatial derivatives and time n + 1
for the time derivatives.
– All kr and all Pc and their derivatives are computed using their own functions.
– All ξ, Co , and Cg are computed from the flash computation.
• Secondary variables 2
– explicit, once per time step: μo (P, Xm ), μg (P, Ym ), μw (P, Wm ) are evaluated at time n
since the viscosity does not change rapidly for small pressure changes.
– explicit, once per time step: γo (P, Xm , ξo , MWo ), γg (P, Ym , ξg , MWg ), γw (P, Wm , ξw , MWw )
are evaluated at time n since the specific gravity does not change rapidly for small pressure changes.
54
– explicit, once per time step: Upstream weighting is evaluated at time n. The cell which
is upstream of another cell is used for many of the fluid properties, but the determination
of which cells are upstream is computed at most once per time step for every cell.
– explicit, once per time step: Source and sink terms are evaluated at time n.
– γ and μ are evaluated using their own functions after completion of a flash computation.
– Upstream weighting and source and sink terms are evaluated in their own functions.
• Tertiary variables - constant in this formulation
– constant: k, φ: permeability and porosity may be time dependent at n with asphaltene
deposition, but are constant in this formulation.
– constant: Cw , Cφ are not defined as functions of pressure for this formulation.
– constant: D - gravity does not vary with time
• Other considerations
– explicit, at most once per time step: Swt , Sgt , Sot are evaluated at time n or less
frequently if possible. Changes in the trapped oil, water, and gas are evaluated at most
once per time step for every cell.
– explicit, at most once per time step: k, φ are evaluated at time n. Changes in k and φ
are a result of solid deposition, adsorption, or dissolution. These reactions occur at most
once per time step per cell, but may be less frequent. Note that the compressibility of
the matrix Cφ is handled separately.
– explicit, at most once per time step: Relative permeability hysteresis is evaluated at
time n or less frequently if possible. This means that whether to use the increasing or
decreasing curve is determined at most once per time step for a cell.
– explicit, at most once per time step: Capillary pressure hysteresis is evaluated at time n
or less frequently if possible. This means that whether to use the increasing or decreasing
curve is determined at most once per time step for a cell.
– explicit, at most once per time step: Wettability changes are evaluated at time n or less
frequently if possible.
55
– explicit, at most once per time step: Pressure dependence of relative permeability (related to wettability and miscibility changes) is evaluated at time n or less frequently if
possible. These are treated in a similar way to hysteresis curves.
– explicit, at most once per time step: Pressure dependence of capillary pressure (related
to wettability and miscibility changes) is evaluated at time n or less frequently if possible.
These are treated in a similar way to hysteresis curves.
– explicit, at most once per time step: Adsorption is evaluated at time n or less frequently
if possible.
56
CHAPTER 4
MATHEMATICAL FORMULATION OVERVIEW
The basic equation for each component is:
Cm=1...NC ,m1 : 0.006328 VR ∇ ·
Xn
n
n
mm1 ξom1 krom1
n
μom1
+1
n
km#1 (∇Pom
− γom
∇D# ) +
1
1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
km1 (∇Pom
+ ∇Pcgom
− γgm
∇D# ) +
1
1
1
n
μgm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m
om
cowm
wm
1
1
1
1
μnwm1
0.006328 VR ∇ ·
+1
n
n
n
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 − τmm
=
Xmm
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
1 /m2
VR +1 +1 +1 +1
+1 +1 +1
+1
+1 +1 +1
φm1 Xmm1 Som1 ξom1 + φ+1
m1 Ymm1 Sgm1 ξgm1 + φm1 Wmm1 Swm1 ξwm1 −
Δt
VR n n
n
n
n
(4.1)
φm1 Xmm1 Som
ξ n + φnm1 Ymm
S n ξ n + φnm1 Wmm
S n ξn
1 om1
1 gm1 gm1
1 wm1 wm1
Δt
(5.40) represents the m2 pore system.
+1
Cm=1...NC ,m2 : τmm
=
1 /m2
VR +1 +1 +1 +1
+1 +1 +1
+1
+1 +1 +1
φm2 Xmm2 Som2 ξom2 + φ+1
Y
S
ξ
+
φ
W
S
ξ
m2
mm2 gm2 gm2
m2
mm2 wm2 wm2 −
Δt
VR n n
n
n
n
n
n
n
n
n
n
n
(4.2)
φm2 Xmm2 Som
ξ
+
φ
Y
S
ξ
+
φ
W
S
ξ
m2 mm2 gm2 gm2
m2 mm2 wm2 wm2
2 om2
Δt
The m1 /m2 transfer function is defined by:
+1
+1
+1
×
τmm
= 0.006328 VR σm#1 /m2 km#1 /m2 Pom
− Pom
1
2
1 /m2
up,n
up,n
up,n
up,n
up,n
up,n
up,n
Ymm1 /m2 ξgm1 /m2 krgm
Wmm
ξ up,n k up,n
Xmm1 /m2 ξom1 /m2 krom1 /m2
1 /m2
1 /m2 wm1 /m2 rwm1 /m2
+
+
(4.3)
μup,n
μup,n
μup,n
om1 /m2
gm1 /m2
wm1 /m2
And the thermodynamic constraints are:
+1
Gm=1...NC −1,m1 : f+1
om,m1 − fgm,m1 = 0
(4.4)
+1
Gm=1...NC −1,m2 : f+1
om,m2 − fgm,m2 = 0
(4.5)
57
4.1
Primary Variables
The formulation used here is an isothermal formulation; this means the temperature is constant
for the simulation run. The temperature T is measured in ◦ F and converted as appropriate to
◦ C,
K, or R. The phase behavior, viscosity, density, solubility, capillary pressure, and relative
permeability all change with temperature.
The formulation used here assumes that the salinity remains constant within a simulation
run. The aqueous density and CO2 solubility change with the salinity of the water. Salinity is
represented as an equivalent mole fraction of NaCl, WNaCl , and converted as needed to a mass
fraction, molarity, or molality. Most of the experiments with brines first measure the properties
with a certain salt concentration and then measure the solubility or density changes of an additional
component separately. Both reservoir brines and seawater are dominated by Na and Cl; variations
in the composition of the salts only causes a small change in the solubility. As a result, the salinity
specified is relative to an equivalent system with H2 O and NaCl only.
The pressures (measured in psia) in the oil, gas, and water phases change as a function of time
n
or Pom
and the trapped
and space. After discretization, the mobile oil phase pressure Pom
1 ,ijk
1 ,ijk
n
or Pom
are stored for each grid cell, for the current time step n, and
oil phase pressure Pom
2 ,ijk
2 ,ijk
for the current nonlinear iteration . The gas phase pressure Pg,m1 is expanded using the gas-oil
capillary pressure:
Pgm1 − Pom1 = Pcgo [Sot , Sgt , Swt ]
(4.6)
The gas phase pressure Pg,m2 is expanded using the gas-oil capillary pressure:
Pgm2 − Pom2 = Pcgo [Sot , Sgt , Swt ]
(4.7)
The water phase pressure Pw,m1 is expanded using the oil-water capillary pressure:
Pom1 − Pwm1 = Pcow [Sot , Sgt , Swt ]
(4.8)
The water phase pressure Pw,m2 is expanded using the oil-water capillary pressure:
Pom2 − Pwm2 = Pcow [Sot , Sgt , Swt ]
(4.9)
58
The saturations (measured as a volume fraction) in the oil, gas, and water phases change as a
n
function of time and space. After discretization, the water saturation in the mobile system Swm
1 ,ijk
n
or Swm
and the water saturation in the trapped system Swm
or Swm
are stored for each
1 ,ijk
2 ,ijk
2 ,ijk
grid cell, for the current time step n, and for the current nonlinear iteration . The oil saturation
n
n
or Som
and the oil saturation in the trapped system Som
or
in the mobile system Som
1 ,ijk
1 ,ijk
2 ,ijk
are stored for each grid cell, for the current time step n, and for the current nonlinear
Som
2 ,ijk
iteration . The sum of the mobile saturations is equal to 1:
So,m1 + Sg,m1 + Sw,m1 = 1
(4.10)
The sum of the trapped saturations is equal to 1:
So,m2 + Sg,m2 + Sw,m2 = 1
(4.11)
There are several degenerate cases that need to be considered.
1. All phases are present; store Sw and So and calculate Sg = 1 − Sw − So as needed.
2. The gas saturation Sg = 0; store Sw and calculate So = 1 − Sw as needed.
3. The oil saturation So = 0; store Sw and calculate Sg = 1 − Sw as needed.
4. The water saturation Sw = 1, the oil saturation So = 0, and the gas saturation Sg = 0.
5. The water saturation Sw = 0; store the oil saturation So and calculate the gas saturation
Sg = 1 − So as needed.
6. The water saturation Sw = 0, the oil saturation So = 1, and the gas saturation Sg = 0.
7. The water saturation Sw = 0, the oil saturation So = 0, and the gas saturation Sg = 1.
The mole fractions (measured as a fraction of lbmol) of each component in the oil, gas, and water
phases change as a function of time and space. For this work, only the CO2 and H2 O components
are present in the water phase; the oil and gas phases do not contain any H2 O. For a system with
NC = 5 components, there are three hydrocarbon components, one component for CO2 , and one
component for H2 O; refer to Table 4.1.
59
The mole fractions of each component in the mobile oil phase sum to 1. The mole fractions
X1,m1 , X2,m1 , and X3,m1 are stored and XCO2 ,m1 is calculated when needed.
XCO2 ,m1 = 1 − X1,m1 − X2,m1 − X3,m1
(4.12)
The mole fractions of each component in the trapped oil phase sum to 1. The mole fractions X1,m2 ,
X2,m2 , and X3,m2 are stored and XCO2 ,m2 is calculated when needed.
XCO2 ,m2 = 1 − X1,m2 − X2,m2 − X3,m2
(4.13)
The mole fractions of each component in the mobile gas phase sum to 1. The mole fractions
Y1,m1 , Y2,m1 , and Y3,m1 are stored and YCO2 ,m1 is calculated when needed.
YCO2 ,m1 = 1 − Y1,m1 − Y2,m1 − Y3,m1
(4.14)
The mole fractions of each component in the trapped gas phase sum to 1. The mole fractions Y1,m2 ,
Y2,m2 , and Y3,m2 are stored and YCO2 ,m2 is calculated when needed.
YCO2 ,m2 = 1 − Y1,m2 − Y2,m2 − Y3,m2
(4.15)
The mole fractions of each component in the mobile water phase sum to 1. The mole fraction
WCO2 ,m1 is stored and WH2 O,m1 is calculated when needed.
WH2 O,m1 = 1 − WCO2 ,m1
(4.16)
The mole fractions of each component in the trapped water phase sum to 1. The mole fraction
WCO2 ,m1 is stored and WH2 O,m2 is calculated when needed.
WH2 O,m2 = 1 − WCO2 ,m2
(4.17)
For the formulations used here, WCO2 is calculated as a function of P , T , WNaCl , XCO2 , and YCO2
as needed.
All of the primary variables after simplification are listed in Table 4.2. The two primary variables
which do not depend on the spatial location, T and WNaCl are stored on each processor. The
variables which depend on the spatial location are stored in PetSc distributed arrays, including
both the local values and the “ghost” values for the adjacent grid cells. PetSc automatically handles
60
Table 4.1: Distribution of components in phases for NC = 5
component
C{L}ight or C1 or CH4
C{I}ntermediate or C2 or nC4
C{H}eavy or C3 or nC10
C4 or CO2
C5 or H2 O
oil
X1
X2
X3
X4
0
gas
Y1
Y2
Y3
Y4
0
aqueous
0
0
0
W4
W5
the communication of the ghost properties. There are three arrays of primary variables; one for
time n (DA primary n), one for time (DA primary ell), and one for the best iteration value from
1.. (DA primary best ell) in case the nonlinear iterations fail to converge. Each primary variable
distributed array includes Pom1 , Swm1 , Som1 , WCO2 ,m1 , Xm ,m1 , Ym ,m1 , Pom2 , Swm2 , Som2 , WCO2 ,m2 ,
Xm ,m2 , and Ym ,m2 .
Table 4.2: Primary variables
variable
units
◦F
T#
#
WNaCl
n
Po,m
1 ,ijk
Po,m
1 ,ijk
lbmol/lbmol
psia
psia
n
Po,m
2 ,ijk
Po,m
2 ,ijk
psia
psia
n
Sw,m
1 ,ijk
Sw,m
1 ,ijk
ft3 /ft3
ft3 /ft3
n
Sw,m
2 ,ijk
Sw,m
2 ,ijk
ft3 /ft3
ft3 /ft3
n
So,m
1 ,ijk
So,m
1 ,ijk
ft3 /ft3
ft3 /ft3
n
So,m
2 ,ijk
So,m
2 ,ijk
ft3 /ft3
ft3 /ft3
n
Xm
,m ,ijk
1
lbmol/lbmol
Xm
,m ,ijk
1
lbmol/lbmol
name
Constant temperature.
Constant salinity.
Pressure in mobile oil phase at time n for every grid cell.
Pressure in mobile oil phase at nonlinear iteration for every
grid cell.
Pressure in trapped oil phase at time n for every grid cell.
Pressure in trapped oil phase at nonlinear iteration for every
grid cell.
Saturation in mobile water phase at time n for every grid cell.
Saturation in mobile water phase at nonlinear iteration for
every grid cell.
Saturation in trapped water phase at time n for every grid cell.
Saturation in trapped water phase at nonlinear iteration for
every grid cell.
Saturation in mobile oil phase at time n for every grid cell.
Saturation in mobile oil phase at nonlinear iteration for every
grid cell.
Saturation in trapped oil phase at time n for every grid cell.
Saturation in trapped oil phase at nonlinear iteration for
every grid cell.
Mole fraction in mobile oil phase for component
m = 1 . . . NC − 2 at time n for every grid cell.
Mole fraction in mobile oil phase for component
m = 1 . . . NC − 2 at nonlinear iteration for every grid cell.
Continued.
61
Table 4.2: Continued.
Table 4.2: Primary variables (continued)
variable
n
Xm
,m ,ijk
2
units
lbmol/lbmol
Xm
,m ,ijk
2
lbmol/lbmol
Ymn ,m1 ,ijk
lbmol/lbmol
Ym ,m1 ,ijk
lbmol/lbmol
Ymn ,m2 ,ijk
lbmol/lbmol
Ym ,m2 ,ijk
lbmol/lbmol
n
WCO
2 ,m1 ,ijk
lbmol/lbmol
WCO
2 ,m1 ,ijk
lbmol/lbmol
n
WCO
2 ,m2 ,ijk
lbmol/lbmol
WCO
2 ,m2 ,ijk
lbmol/lbmol
4.2
name
Mole fraction in trapped oil phase for component
m = 1 . . . NC − 2 at time n for every grid cell.
Mole fraction in trapped oil phase for component
m = 1 . . . NC − 2 at nonlinear iteration for every grid cell.
Mole fraction in mobile gas phase for component
m = 1 . . . NC − 2 at time n for every grid cell.
Mole fraction in mobile gas phase for component
m = 1 . . . NC − 2 at nonlinear iteration for every grid cell.
Mole fraction in trapped gas phase for component
m = 1 . . . NC − 2 at time n for every grid cell.
Mole fraction in trapped gas phase for component
m = 1 . . . NC − 2 at nonlinear iteration for every grid cell.
Mole fraction of CO2 in the mobile aqueous phase at time n for
every grid cell.
Mole fraction of CO2 in the mobile aqueous phase at nonlinear
iteration for every grid cell.
Mole fraction of CO2 in the trapped aqueous phase at time n
for every grid cell.
Mole fraction of CO2 in the trapped aqueous phase at
nonlinear iteration for every grid cell.
Secondary Variables
Secondary variables are calculated as a function of the primary variables. The following sec-
ondary variables appear directly in the partial differential equations or the IMPES finite difference
expansion of the partial differential equations.
4.2.1
Calculation of Secondary Variables
WCO2 may be calculated as a primary variable or a secondary variable, but here WCO2 is
calculated as a secondary variable as a function of T , WNaCl , Po,m1 ,ijk , YCO2 , and possibly XCO2 .
The derivatives
∂WCO2 ∂WCO2
∂P , ∂Xm ,
and
∂WCO2
∂Ym
are evaluated analytically from the derivatives of the
correlations. In a three-phase system, the CO2 may partition between the water, oil, and gas
phases. The gas-oil partitioning is handled by a normal two-phase flash calculation. The gas-water
partitioning is handled by a CO2 solubility computation; the gas-water solubility used here is the
model by Duan and Sun (2003).
62
For a water-oil-gas system, several different possibilities are available to use with the Duan and
Sun (2003) correlation; option (4.18) is used here.
WCO2 =
F [P, T, WNaCl , YCO2 ]
(4.18)
WCO2 = αF [P, T, WNaCl , YCO2 ] + (1 − α)F [P, T, WNaCl , XCO2 ]
(4.19)
WCO2 =
(4.20)
WCO2 [P, T, WNaCl , ZCO2 ]
For a two-phase oil-water system, several different possibilities are available to use in the Duan and
Sun (2003) correlation; option (4.21) was found to work best.
WCO2 =
F [Pb , T, WNaCl , YCO2 [Pb ]]
(4.21)
WCO2 =
F [P, T, WNaCl , YCO2 [Pb ]]
(4.22)
WCO2 = αF [P, T, WNaCl , YCO2 [Pb ]] + (1 − α)F [P, T, WNaCl , XCO2 ]
(4.23)
WCO2 =
F [P, T, WNaCl , ZCO2 ]
(4.24)
WCO2 =
F [P, T, WNaCl , YCO2 = 0] =⇒ WCO2 = 0
(4.25)
Unfortunately, for both the oil-water system and the water-oil-gas system insufficient experimental data is available to decide between the different choices of CO2 models. A three-phase flash
calculation based on an equation of state like Peng-Robinson could also be used to represent the
CO2 partitioning in any of these systems, but the three-phase flash would also require additional
experimental data to calibrate the model.
The molar density in the oil and gas phases, ξo and ξg , are calculated as part of the PengRobinson equation of state flash for a gas-oil system. The oil density ξo is a function of P , T , and
Xm . The gas density ξg is a function of P , T , and Ym . The derivatives
∂ξg
∂ξo
∂ξo
∂P , ∂Xm , ∂P ,
and
∂ξg
∂Ym
are evaluated using analytical derivatives of the Peng-Robinson equation of state. The oil specific
gravity γo [psi/ft] is calculated from ξo :
γo =
ξo MWo
144
(4.26)
The gas specific gravity γg [psi/ft] is calculated from ξg :
γg =
ξg MWg
144
(4.27)
63
The molar density in the aqueous phase, ξw , is calculated as a function of P , T , WNaCl , and
WCO2 . If WCO2 is a primary variable, the derivatives
∂ξw
∂P
and
∂ξw
∂WCO2
are evaluated using analytical
derivatives of the correlations. If WCO2 is a secondary variable, the derivatives
∂ξw
∂ξw
∂ξw
∂P , ∂Xm , ∂Ym
are
evaluated using analytical derivatives of the correlations. The water specific gravity γw [psi/ft] is
calculated from ξw :
γw =
ξw MWw
144
(4.28)
The total porosity φ or φt changes as a function of Pom1 . The mobile porosity and trapped
porosity φm1 and φm2 change as the trapping changes. The ratios φm1 /φt and φm2 /φt remain
constant except when the trapping changes. The derivatives
∂φ ∂φm1
∂P , ∂P ,
and
∂φm2
∂P
are evaluated
using analytical derivatives. The total saturations are defined as follows:
Sot =
(Som1 φm1 + Som2 φm2 ) /φt
(4.29)
Sgt =
(Sgm1 φm1 + Sgm2 φm2 ) /φt
(4.30)
Swt = (Swm1 φm1 + Swm2 φm2 ) /φt
(4.31)
The relative permeabilities and capillary pressures in both the mobile and trapped systems are
calculated as a function of the total saturations, Swt , Sot , Sgt , not the mobile or trapped saturation
only. This means that
krw =
krwm1 = krwm2
(4.32)
kro =
krom1 = krom2
(4.33)
krg =
krgm1 = krgm2
(4.34)
Pcgo =
Pcgom1 = Pcgom2
(4.35)
Pcow = Pcowm1 = Pcowm2
(4.36)
The relative permeability and capillary pressures are assumed to be representative of the initial
reservoir pressure and temperature. The water relative permeability krw is a function of Swt , Sot ,
Sgt , and the saturation history of the grid cell. The oil relative permeability kro is a function of Swt ,
Sot , Sgt , and the saturation history of the grid cell. The gas relative permeability krg is a function
of Swt , Sot , Sgt , and the saturation history of the grid cell. The gas-oil relative permeability Pcgo
is a function of Swt , Sot , Sgt , and the saturation history of the grid cell. The oil-water relative
64
permeability Pcow is a function of Swt , Sot , Sgt , and the saturation history of the grid cell.
The fugacity in the oil phase fom and the derivatives of fugacity
∂fom
∂P
and
∂fom
∂Xm
are calculated
using the Peng-Robinson equation of state. The fugacity and fugacity derivatives are functions of
P and Xm . The fugacity in the gas phase fgm and the derivatives of fugacity
∂fgm
∂P
and
∂fgm
∂Ym
are
calculated using the Peng-Robinson equation of state. The fugacity and fugacity derivatives are
functions of P and Ym .
4.2.2
Storage of Secondary Variables
Secondary variables which are different in each grid cell are stored as PetSc distributed arrays.
There are different arrays for values stored at time n, nonlinear iteration , and values which
don’t change with time. Arrays are also divided based on whether they contain “ghost” cells for
neighboring processors and whether they use ghost cells from other arrays. The arrays are also
split based on when the values need to be computed and used. The following list describes the
different distributed arrays in calculation order.
1. Initialization: values which do not change with time
1.1. DA notime, with ghost cells; these properties change with the grid cell but do not change
with time. Includes k, km1 /mtwo , σm1 /m2 , D, cφ , Δx, Δy, Δz, and the constant portion
of the transmissibilities TC. The different hysteresis curves for relative permeability and
capillary pressure are also defined. See Table 4.3.
2. Update at time n
2.1. DA primary n, with ghost cells; the primary variables at n.
2.2. DA before TRANS n, no ghost cells; these values change when when the water, oil, or gas
saturation direction of individual grid cells changes to increasing or decreasing. Includes
the properties needed to calculate the relative permeability and capillary pressure curves
such as the saturation direction, endpoint saturations, maximum historical saturations,
and curvature. These values are calculated based on DA primary n and DA notime.
2.3. DA cell only n, no ghost cells; these properties are calculated at time n for every grid cell.
They are required by the jacobian calculation but are not required by the transmissibility
65
calculation. Includes the mobile and trapped φ, fom , and fgm . These properties are
calculated based on DA primary n and DA notime. See Table 4.4.
2.4. DA for TRANS n, with ghost cells; these values are needed for the transmissibility calculations. Includes the mobile and trapped ξo , ξg , ξw , γo , γg , γw , μo , μg , μw , kro , krg ,
krw , Pcow , and Pcgo . These properties are calculated based on DA primary n, DA notime,
DA cell only n, and DA before TRANS n. After the local properties are calculated, the
ghost values are communicated to the neighboring processors. See Table 4.5.
2.5. DA TRANS n, no ghost cells; the transmissibilities themselves are local but depend on
the local and ghost values of DA primary n, DA notime, and DA for TRANS n. Includes
the upstream potential Ψup , the upstream weighted specific gravities γ up , the inter-grid
transmissibility Tm1 /m1 , and the intra-grid transmissibility Tm1 /m2 .
2.6. DA jacobian n, no ghost cells; these are the jacobian values at time n. The portion of
the jacobian for each grid cell is a two-dimensional array. The 4NC rows of the array
represent each of the component equations and each of the thermodynamic equations for
the mobile system m1 and the trapped system m2 . For a 7-point finite difference stencil
with single completion wells, the 4NC + 7 columns of the array represent the primary
variables, the mobile pressures at the adjacent grid cells, and the right-hand-side of the
jacobian equation corresponding to the grid cell. In degenerate cases or cases without
trapping, a portion of the local jacobian matrix is the identity matrix. DA jacobian n
is calculated based on DA primary n, DA notime, DA for TRANS n, DA TRANS n, and
DA cell only n.
2.7. DA after TRANS n, no ghost cells; the values used in the flash computation for the
primary variables. Includes the mobile and trapped values of Um , α, β, and Z2ph,m .
These properties are calculated based on DA primary n, DA notime, DA for TRANS n,
DA TRANS n, DA cell only n, and DA jacobian n.
3. Update at nonlinear iteration 3.1. DA primary ell, with ghost cells; the primary variables at .
66
3.2. DA cell only ell, no ghost cells; the secondary variables evaluated at . This includes: the
−1 ; the mobile
mobile and trapped water saturation at the previous nonlinear iteration Sw
and trapped values of φ, GCO2 , Gmax , ξo , ξg , ξw , fom , fgm ; and the derivatives of ξo , ξg ,
ξw , WCO2 , fom , fgm , and Gmax . These properties are calculated based on DA primary ell
and DA notime. See Table 4.6.
3.3. DA jacobian ell, no ghost cells; these are the jacobian values at time . The portion of
the jacobian for each grid cell is a two-dimensional array. The 4NC rows of the array
represent each of the component equations and each of the thermodynamic equations for
the mobile system m1 and the trapped system m2 . For a 7-point finite difference stencil
with single completion wells, the 4NC + 7 columns of the array represent the primary
variables, the mobile pressures at the adjacent grid cells, and the right-hand-side of the
jacobian equation corresponding to the grid cell. In degenerate cases or cases without
trapping, a portion of the local jacobian matrix is the identity matrix. DA jacobian ell
is calculated based on DA primary ell, DA notime, DA jacobian n, and DA cell only ell.
3.4. DA solution ell, no ghost cells; the solution vector at . Although the solution vector
contains all the primary variables as a result of LU-decomposition, some of them may
be degenerate and some are calculated by flash. When not degenerate, the pressures
Pom1 and Pom2 , water saturations Swm1 and Swm2 , and primary WCO2 ,m1 and WCO2 ,m2
are calculated from the solution vector. DA solution ell is calculated by the solver based
on DA jacobian ell.
3.5. DA primary best ell, with ghost cells; the value of the primary variables at the best of
the nonlinear iterations 1 . . . .
4.2.3
List of Secondary Variables
Table 4.3: Secondary variables which do not vary with time, DA notime
variable
units
Δtn
day
VR#
ft3
ijk
Continued.
name
Time step size for time n.
Rock volume for each grid cell; does not change with time.
67
Table 4.3: Continued.
Table 4.3: Secondary variables which do not vary with time, DA notime (continued)
variable
#
Dijk
units
ft
#
kijk
#
km
1 ,ijk
md
md
#
km
1 /m2 ,ijk
md
#
kxx,ijk
md
#
kyy,ijk
md
#
kzz,ijk
md
#
σm
1 /m2 ,ijk
1/ft2
WI#
w
ft3 /day
psia/cp
name
Depth to midpoint of each grid cell; does not change with
time.
Permeability of each grid cell; does not change with time.
Permeability of mobile system for each grid cell; does not
change with time.
Permeability of transfer from trapped system to mobile
system for each grid cell; does not change with time.
Permeability in the x or i direction of each grid cell; does
not change with time.
Permeability in the y or j direction of each grid cell; does
not change with time.
Permeability in the z or k direction of each grid cell; does
not change with time.
Shape factor for transfer between mobile and trapped
system for each grid cell; does not change with time.
Well index for each well; does not change with time.
Table 4.4: Secondary variables at n which are not needed for the transmissibility calculations,
DA cell only n
variable
units
3
3
φnt,ijk
ft pore/ft rock
φnm1 ,ijk
ft3 mobile/ft3 rock
φnm2 ,ijk
ft3 trapped/ft3 rock
n
Swt,ijk
ft3 water/ft3 pore
n
Sot,ijk
ft3 oil/ft3 pore
n
Sgt,ijk
ft3 gas/ft3 pore
MWno,m1 ,ijk
lbm/lbmol
MWng,m1 ,ijk
lbm/lbmol
name
Porosity at time n for every grid cell. It is a function of
n
.
Po,m
1 ,ijk
Mobile pore fraction at time n for every grid cell. It is
n
.
a function of Po,m
1 ,ijk
Trapped pore fraction at time n for every grid cell. It is
n
.
a function of Po,m
1 ,ijk
Total water saturation at time n for every grid cell. It
n
, and
is a function of φnt,ijk , φnm1 ,ijk , φnm2 ,ijk , Swm
1 ,ijk
n
Swm2 ,ijk .
Total oil saturation at time n for every grid cell. It is a
n
n
, and Som
.
function of φnt,ijk , φnm1 ,ijk , φnm2 ,ijk , Som
1 ,ijk
2 ,ijk
Total gas saturation at time n for every grid cell. It is a
n
n
, and Sgm
.
function of φnt,ijk , φnm1 ,ijk , φnm2 ,ijk , Sgm
1 ,ijk
2 ,ijk
Molecular weight of mobile oil phase at time n for
n
and MW#
every grid cell. It is a function of Xm,m
m.
1 ,ijk
Molecular weight of mobile oil phase at time n for
n
and MW#
every grid cell. It is a function of Ym,m
m.
1 ,ijk
Continued.
68
Table 4.4: Continued.
Table 4.4: Secondary variables at n which are not needed for the transmissibility calculations,
DA cell only n (continued)
variable
MWnw,m1 ,ijk
units
lbm/lbmol
fnom,m1 ,ijk
psi
fnom,m2 ,ijk
psi
fngm,m1 ,ijk
psi
fngm,m2 ,ijk
psi
name
Molecular weight of mobile oil phase at time n for every
#
n
Wm,m
and MW#
grid cell. It is a function of WNaCl
m.
1 ,ijk
Mobile oil phase fugacity for component m at time n
n
for every grid cell. It is a function of T , Po,m
and
1 ,ijk
n
X1...NC −1,m1 ,ijk .
Trapped oil phase fugacity for component m at time n
n
and
for every grid cell. It is a function of T , Po,m
2 ,ijk
n
X1...NC −1,m2 ,ijk .
Mobile gas phase fugacity for component m at time n
n
for every grid cell. It is a function of T , Po,m
and
1 ,ijk
n
Y1...NC −1,m1 ,ijk .
Trapped gas phase fugacity for component m at time n
n
and
for every grid cell. It is a function of T , Po,m
2 ,ijk
n
Y1...NC −1,m2 ,ijk .
Table 4.5: Secondary variables at n which are needed for the transmissibility calculations, DA for
TRANS n
variable
units
n
ξo,m
1 ,ijk
lbmol/ft3
n
ξo,m
2 ,ijk
lbmol/ft3
n
ξg,m
1 ,ijk
lbmol/ft3
n
ξg,m
2 ,ijk
lbmol/ft3
n
ξw,m
1 ,ijk
lbmol/ft3
n
ξw,m
2 ,ijk
lbmol/ft3
μno,m1 ,ijk
cp
μno,m2 ,ijk
cp
μng,m1 ,ijk
cp
name
Molar density of mobile oil phase at time n for every grid
n
n
and X1...N
.
cell. It is a function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Molar density of trapped oil phase at time n for every grid
n
n
and X1...N
.
cell. It is a function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Molar density of mobile gas phase at time n for every grid
n
n
and Y1...N
.
cell. It is a function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Molar density of trapped gas phase at time n for every grid
n
n
and Y1...N
.
cell. It is a function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Molar density of mobile water phase at time n for every grid
n
n
, and WCO
.
cell. It is a function of T , WNaCl , Po,m
1 ,ijk
2 ,m1 ,ijk
Molar density of trapped water phase at time n for every
n
, and
grid cell. It is a function of T , WNaCl , Po,m
2 ,ijk
n
WCO2 ,m2 ,ijk .
Viscosity of mobile oil phase at time n for every grid cell. It
n
n
and X1...N
.
is a function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Viscosity of trapped oil phase at time n for every grid cell.
n
n
and X1...N
.
It is a function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Viscosity of mobile gas phase at time n for every grid cell.
n
n
and Y1...N
.
It is a function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Continued.
69
Table 4.5: Continued.
Table 4.5: Secondary variables at n which are needed for the transmissibility calculations, DA for
TRANS n (continued)
variable
μng,m2 ,ijk
units
cp
μnw,m1 ,ijk
cp
μnw,m2 ,ijk
cp
n
γo,m
1 ,ijk
psi/ft
n
γg,m
1 ,ijk
psi/ft
n
γw,m
1 ,ijk
psi/ft
n
krw,ijk
md/md
n
kro,ijk
md/md
n
krg,ijk
md/md
n
Pcgo,ijk
psia
n
Pcow,ijk
psia
name
Viscosity of trapped gas phase at time n for every grid cell.
n
n
and Y1...N
.
It is a function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Viscosity of mobile water phase at time n for every grid cell.
n
n
It is a function of T , WNaCl , Po,m
, and WCO
.
1 ,ijk
2 ,m1 ,ijk
Viscosity of trapped water phase at time n for every grid
n
n
, and WCO
.
cell. It is a function of T , WNaCl , Po,m
2 ,ijk
2 ,m2 ,ijk
Specific gravity of mobile oil phase at time n for every grid
n
and MWno,m1 ,ijk .
cell. It is a function of ξo,m
1 ,ijk
Specific gravity of mobile gas phase at time n for every grid
n
cell. It is a function of ξg,m
and MWng,m1 ,ijk .
1 ,ijk
Specific gravity of mobile water phase at time n for every
n
and MWnw,m1 ,ijk .
grid cell. It is a function of ξw,m
1 ,ijk
Relative permeability to water at time n for every grid cell.
n
n
n
, So,t,ijk
, and Sg,t,ijk
.
It is a function of Sw,t,ijk
Relative permeability to oil at time n for every grid cell. It
n
n
n
, So,t,ijk
, and Sg,t,ijk
.
is a function of Sw,t,ijk
Relative permeability to gas at time n for every grid cell. It
n
n
n
, So,t,ijk
, and Sg,t,ijk
.
is a function of Sw,t,ijk
Gas-oil capillary pressure at time n for every grid cell. It is
n
n
n
, So,t,ijk
, and Sg,t,ijk
.
a function of Sw,t,ijk
Oil-water capillary pressure at time n for every grid cell. It
n
n
n
, So,t,ijk
, and Sg,t,ijk
.
is a function of Sw,t,ijk
Table 4.6: Secondary variables at , DA cell only ell
variable
units
3
3
φt,ijk
ft pore/ft rock
φm1 ,ijk
ft3 mobile/ft3 rock
φm2 ,ijk
ft3 trapped/ft3 rock
ξo,m
1 ,ijk
lbmol/ft3
name
Porosity at nonlinear iteration for every grid cell. It is
.
a function of Po,m
1 ,ijk
Mobile pore fraction at nonlinear iteration for every
.
grid cell. It is a function of Po,m
1 ,ijk
Trapped pore fraction at nonlinear iteration for every
.
grid cell. It is a function of Po,m
1 ,ijk
Molar density of mobile oil phase at nonlinear iteration
and
for every grid cell. It is a function of T , Po,m
1 ,ijk
X1...NC −1,m1 ,ijk .
Continued.
70
Table 4.6: Continued.
Table 4.6: Secondary variables at , DA cell only ell (continued)
variable
ξo,m
2 ,ijk
units
lbmol/ft3
ξg,m
1 ,ijk
lbmol/ft3
ξg,m
2 ,ijk
lbmol/ft3
ξw,m
1 ,ijk
lbmol/ft3
ξw,m
2 ,ijk
lbmol/ft3
fom,m1 ,ijk
psi
fom,m2 ,ijk
psi
fgm,m1 ,ijk
psi
fgm,m2 ,ijk
psi
∂φm ,ijk
1
∂Pom1 ,ijk
ft3 /ft3
psia
∂φm ,ijk
2
∂Pom2 ,ijk
ft3 /ft3
psia
∂ξo,m
1 ,ijk
∂Po,m1 ,ijk
lbmol/ft3
psia
name
Molar density of trapped oil phase at nonlinear
iteration for every grid cell. It is a function of T ,
and X1...N
.
Po,m
2 ,ijk
C −1,m2 ,ijk
Molar density of mobile gas phase at nonlinear
iteration for every grid cell. It is a function of T ,
Po,m
and Y1...N
.
1 ,ijk
C −1,m1 ,ijk
Molar density of trapped gas phase at nonlinear
iteration for every grid cell. It is a function of T ,
and Y1...N
.
Po,m
2 ,ijk
C −1,m2 ,ijk
Molar density of mobile water phase at nonlinear
iteration for every grid cell. It is a function of T ,
WNaCl , Po,m
, and WCO
.
1 ,ijk
2 ,m1 ,ijk
Molar density of trapped water phase at nonlinear
iteration for every grid cell. It is a function of T ,
, and WCO
.
WNaCl , Po,m
2 ,ijk
2 ,m2 ,ijk
Mobile oil phase fugacity for component m at nonlinear
iteration for every grid cell. It is a function of T ,
and X1...N
.
Po,m
1 ,ijk
C −1,m1 ,ijk
Trapped oil phase fugacity for component m at
nonlinear iteration for every grid cell. It is a function
and X1...N
.
of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Mobile gas phase fugacity for component m at
nonlinear iteration for every grid cell. It is a function
and Y1...N
.
of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Trapped gas phase fugacity for component m at
nonlinear iteration for every grid cell. It is a function
and Y1...N
.
of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Derivative of mobile pore fraction with respect to
pressure at nonlinear iteration for every grid cell. It is
.
a function of Po,m
1 ,ijk
Derivative of trapped pore fraction with respect to
pressure at nonlinear iteration for every grid cell. It is
.
a function of Po,m
2 ,ijk
Derivative of molar density of mobile oil phase with
respect to pressure at nonlinear iteration for every
and
grid cell. It is a function of T , Po,m
1 ,ijk
X1...NC −1,m1 ,ijk .
Continued.
71
Table 4.6: Continued.
Table 4.6: Secondary variables at , DA cell only ell (continued)
variable
units
∂ξo,m
2 ,ijk
∂Po,m2 ,ijk
lbmol/ft
psia
name
3
∂ξo,m
1 ,ijk
∂Xm ,m ,ijk
lbmol/ft3
lbmol/lbmol
∂ξo,m
2 ,ijk
∂Xm ,m ,ijk
lbmol/ft3
lbmol/lbmol
1
2
∂ξg,m
1 ,ijk
∂Po,m1 ,ijk
lbmol/ft3
psia
∂ξg,m
2 ,ijk
∂Po,m2 ,ijk
lbmol/ft3
psia
Derivative of molar density of trapped oil phase with
respect to pressure at nonlinear iteration for every
and
grid cell. It is a function of T , Po,m
2 ,ijk
X1...NC −1,m2 ,ijk .
Derivative of molar density of mobile oil phase with
respect to each component mole fraction Xm ,m1 at
nonlinear iteration for every grid cell. They are a
and all the X1...N
.
function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Derivative of molar density of trapped oil phase with
respect to each component mole fraction Xm ,m2 at
nonlinear iteration for every grid cell. They are a
and all the X1...N
.
function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Derivative of molar density of mobile gas phase with
respect to pressure at nonlinear iteration for every
and
grid cell. It is a function of T , Po,m
1 ,ijk
Y1...NC −1,m1 ,ijk .
Derivative of molar density of trapped gas phase with
respect to pressure at nonlinear iteration for every
and
grid cell. It is a function of T , Po,m
2 ,ijk
Y1...NC −1,m2 ,ijk .
∂ξg,m
1 ,ijk
∂Ym ,m ,ijk
lbmol/ft3
lbmol/lbmol
∂ξg,m
2 ,ijk
∂Ym ,m ,ijk
lbmol/ft3
lbmol/lbmol
∂WCO
2 ,m1 ,ijk
∂Pom1 ,ijk
lbmol/lbmol
psia
∂WCO
2 ,m2 ,ijk
∂Pom2 ,ijk
Derivative of mobile WCO2 with respect to pressure at
nonlinear iteration for every grid cell. It is a function
, T , WNaCl , XCO
, and YCO
. It is
of Po,m
1 ,ijk
2 ,m1 ,ijk
2 ,m1 ,ijk
evaluated only when WCO2 is a secondary variable.
lbmol/lbmol
psia
Derivative of trapped WCO2 with respect to pressure at
nonlinear iteration for every grid cell. It is a function
, T , WNaCl , XCO
, and YCO
. It is
of Po,m
2 ,ijk
2 ,m2 ,ijk
2 ,m2 ,ijk
evaluated only when WCO2 is a secondary variable.
1
2
Derivative of molar density of mobile gas phase with
respect to each component mole fraction Ym ,m1 at
nonlinear iteration for every grid cell. They are a
and all the Y1...N
.
function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
Derivative of molar density of trapped gas phase with
respect to each component mole fraction Ym ,m2 at
nonlinear iteration for every grid cell. They are a
and all the Y1...N
.
function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
Continued.
72
Table 4.6: Continued.
Table 4.6: Secondary variables at , DA cell only ell (continued)
variable
units
name
∂WCO
2 ,m1 ,ijk
∂XCO2 ,m1 ,ijk
lbmol/lbmol
lbmol/lbmol
∂WCO
2 ,m2 ,ijk
∂XCO2 ,m2 ,ijk
Derivative of mobile WCO2 with respect to XCO2 at
nonlinear iteration for every grid cell. It is a function
, T , WNaCl , XCO
, and YCO
. It is
of Po,m
1 ,ijk
2 ,m1 ,ijk
2 ,m1 ,ijk
evaluated only when WCO2 is a secondary variable.
lbmol/lbmol
lbmol/lbmol
Derivative of trapped WCO2 with respect to XCO2 at
nonlinear iteration for every grid cell. It is a function
, T , WNaCl , XCO
, and YCO
. It is
of Po,m
2 ,ijk
2 ,m2 ,ijk
2 ,m2 ,ijk
evaluated only when WCO2 is a secondary variable.
∂WCO
2 ,m1 ,ijk
∂YCO2 ,m1 ,ijk
lbmol/lbmol
lbmol/lbmol
Derivative of mobile WCO2 with respect to YCO2 at
nonlinear iteration for every grid cell. It is a function
, T , WNaCl , XCO
, and YCO
. It is
of Po,m
1 ,ijk
2 ,m1 ,ijk
2 ,m1 ,ijk
evaluated only when WCO2 is a secondary variable.
∂WCO
2 ,m2 ,ijk
∂YCO2 ,m2 ,ijk
lbmol/lbmol
lbmol/lbmol
Derivative of trapped WCO2 with respect to YCO2 at
nonlinear iteration for every grid cell. It is a function
, T , WNaCl , XCO
, and YCO
. It is
of Po,m
2 ,ijk
2 ,m2 ,ijk
2 ,m2 ,ijk
evaluated only when WCO2 is a secondary variable.
∂ξw,m
1 ,ijk
∂Po,m1 ,ijk
lbmol/ft3
psia
∂ξw,m
2 ,ijk
∂Po,m2 ,ijk
lbmol/ft3
psia
∂ξw,m
1 ,ijk
∂WCO2 ,m1 ,ijk
lbmol/ft3
lbmol/lbmol
∂ξw,m
2 ,ijk
∂WCO2 ,m2 ,ijk
lbmol/ft3
lbmol/lbmol
Derivative of molar density of mobile water phase with
respect to pressure at nonlinear iteration for every
, and
grid cell. It is a function of T , WNaCl , Po,m
1 ,ijk
WCO2 ,m1 ,ijk .
Derivative of molar density of trapped water phase
with respect to pressure at nonlinear iteration for
,
every grid cell. It is a function of T , WNaCl , Po,m
2 ,ijk
and WCO2 ,m2 ,ijk .
Derivative of molar density of mobile water phase with
respect to WCO2 ,m1 at nonlinear iteration for every
, and
grid cell. It is a function of T , WNaCl , Po,m
1 ,ijk
WCO2 ,m1 ,ijk . It is evaluated only when WCO2 is a
primary variable.
Derivative of molar density of trapped water phase
with respect to WCO2 ,m2 at nonlinear iteration for
,
every grid cell. It is a function of T , WNaCl , Po,m
2 ,ijk
and WCO2 ,m2 ,ijk . It is evaluated only when WCO2 is a
primary variable.
Continued.
73
Table 4.6: Continued.
Table 4.6: Secondary variables at , DA cell only ell (continued)
variable
units
name
∂ξw,m
1 ,ijk
∂XCO2 ,m1 ,ijk
lbmol/ft
lbmol/lbmol
∂ξw,m
2 ,ijk
∂XCO2 ,m2 ,ijk
lbmol/ft3
lbmol/lbmol
∂ξw,m
1 ,ijk
∂YCO2 ,m1 ,ijk
lbmol/ft3
lbmol/lbmol
∂ξw,m
2 ,ijk
∂YCO2 ,m2 ,ijk
lbmol/ft3
lbmol/lbmol
3
Derivative of molar density of mobile water phase with
respect to XCO2 ,m1 at nonlinear iteration for every
,
grid cell. It is a function of T , WNaCl , Po,m
1 ,ijk
XCO2 ,m1 ,ijk , and YCO2 ,m1 ,ijk . It is evaluated only when
WCO2 is a secondary variable.
Derivative of molar density of trapped water phase
with respect to XCO2 ,m2 at nonlinear iteration for
,
every grid cell. It is a function of T , WNaCl , Po,m
2 ,ijk
XCO2 ,m2 ,ijk , and YCO2 ,m2 ,ijk . It is evaluated only when
WCO2 is a secondary variable.
Derivative of molar density of mobile water phase with
respect to YCO2 ,m1 at nonlinear iteration for every
,
grid cell. It is a function of T , WNaCl , Po,m
1 ,ijk
XCO2 ,m1 ,ijk , and YCO2 ,m1 ,ijk . It is evaluated only when
WCO2 is a secondary variable.
Derivative of molar density of trapped water phase
with respect to YCO2 ,m2 at nonlinear iteration for
,
every grid cell. It is a function of T , WNaCl , Po,m
2 ,ijk
XCO2 ,m2 ,ijk , and YCO2 ,m2 ,ijk . It is evaluated only when
WCO2 is a secondary variable.
∂fom,m ,ijk
1
∂Po,m1 ,ijk
psia
psia
∂fom,m ,ijk
2
∂Po,m2 ,ijk
Derivative of mobile oil phase fugacity for component
m with respect to pressure at nonlinear iteration for
and
every grid cell. They are a function of T , Po,m
1 ,ijk
X1...NC −1,m1 ,ijk .
psia
psia
∂fgm,m ,ijk
1
∂Po,m1 ,ijk
Derivative of trapped oil phase fugacity for component
m with respect to pressure at nonlinear iteration for
and
every grid cell. They are a function of T , Po,m
2 ,ijk
X1...NC −1,m2 ,ijk .
psia
psia
∂fgm,m ,ijk
2
∂Po,m2 ,ijk
Derivative of mobile gas phase fugacity for component
m with respect to pressure at nonlinear iteration for
and
every grid cell. They are a function of T , Po,m
1 ,ijk
Y1...NC −1,m1 ,ijk .
psia
psia
Derivative of trapped gas phase fugacity for component
m with respect to pressure at nonlinear iteration for
and
every grid cell. They are a function of T , Po,m
2 ,ijk
Y1...NC −1,m2 ,ijk .
Continued.
74
Table 4.6: Continued.
Table 4.6: Secondary variables at , DA cell only ell (continued)
variable
units
name
∂fom,m ,ijk
1
∂Xm ,m ,ijk
psia
lbmol/lbmol
∂fom,m ,ijk
2
∂Xm ,m ,ijk
Derivative of mobile oil phase fugacity for component
m with respect to each component mole fraction Xm ,m1
at nonlinear iteration for every grid cell. They are a
and all the X1...N
.
function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
psia
lbmol/lbmol
∂fgm,m ,ijk
1
∂Ym ,m ,ijk
Derivative of trapped oil phase fugacity for component
m with respect to each component mole fraction Xm ,m2
at nonlinear iteration for every grid cell. They are a
and all the X1...N
.
function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
psia
lbmol/lbmol
∂fgm,m ,ijk
2
∂Ym ,m ,ijk
Derivative of mobile gas phase fugacity for component
m with respect to each component mole fraction Ym ,m1
at nonlinear iteration for every grid cell. They are a
and all the Y1...N
.
function of T , Po,m
1 ,ijk
C −1,m1 ,ijk
psia
lbmol/lbmol
Derivative of trapped gas phase fugacity for component
m with respect to each component mole fraction Ym ,m2
at nonlinear iteration for every grid cell. They are a
and all the Y1...N
.
function of T , Po,m
2 ,ijk
C −1,m2 ,ijk
1
2
1
2
Table 4.7: Well properties at , stored for each well.
variable
4.3
units
qw,w
ft3 /day
qo,w
ft3 /day
qg,w
ft3 /day
name
Water production or injection rate at reservoir conditions
for nonlinear iteration for every well. It is a function of
n
n
n
n
n
WI#
w , Pom1 ,ijk , krw,ijk , kro,ijk , krg,ijk , μw,ijk , μo,ijk , and
n
μg,ijk .
Oil production or injection rate at reservoir conditions for
nonlinear iteration for every well. It is a function of WI#
w,
n
n
n
n
n
n
Pom1 ,ijk , krw,ijk , kro,ijk , krg,ijk , μw,ijk , μo,ijk , and μg,ijk .
Gas production or injection rate at reservoir conditions for
nonlinear iteration for every well. It is a function of WI#
w,
n
n
n
n
n
n
,
k
,
k
,
k
,
μ
,
μ
,
and
μ
.
Pom
rw,ijk
ro,ijk
rg,ijk
w,ijk
o,ijk
g,ijk
1 ,ijk
Overview of Simulation Process
The following steps are involved for a complete simulation run.
1. Initialize
1.1. Load properties for a specific simulation run from standard input and file(s).
75
1.2. Allocate memory: allocate global EOS, allocate local eos, allocate temp EOS, psim initialize
data sizes, psim initialize data 2spot, psim allocate DA variables, psim allocate other variables,
psim allocate init variables.
1.3. Initialize temperature dependent constants: initialize eos X4, initialize temperature constants,
initialize LBC viscosity constants, initialize aqueous constants,
1.4. Initialize grid properties for a specific simulation run: psim initialize 1D 0020, psim init
trap 1D 0020, psim initialize 1D 0001 flash, psim initialize 1D 0001 vector, psim init trap
1D 0001 vector
1.5. Initialize well properties for a specific simulation run: psim initialize 1D 0020 well
1.6. Initialize solver: psim solver init
2. For each time step n: psim solve iterate ell, psim solve n
2.1. Copy final iteration of previous timestep (n − 1, + 1) to new timestep (n, = 0):
psim COPY ell to n
2.2. Calculate interior and ghost cell properties at n; communicate ghost properties needed
for transmissibilities to neighbors at n: psim calculate local n.
2.3. Calculate transmissibilities at n: psim all TRANS nonly
2.4. Calculate and communicate wells at n: psim COPY well n
2.5. Calculate time step size at n: psim local timestep n
2.6. Calculate jacobian and other properties which depend on the transmissibilities at n:
psim calculate all jacobian n, psim calc after TRANS n
2.7. Copy properties for = 0: psim COPY n to ell
3. For each nonlinear iteration , before convergence or before maximum number of iterations
is exceeded: psim solve ell
3.1. When > 0, calculate interior properties at (transmissibilities depend on n not on ,
so no ghost properties are needed): psim calculate local ell.
3.2. Calculate wells at : psim COPY well ell, sim well single completion ell
76
3.3. Calculate and communicate jacobian at : psim calculate all jacobian ell
3.4. Solve matrix equation and communicate solution at : psim solver solve
3.5. Update and communicate primary variables and determine if the solution has converged
at : psim convergence update primary ell
3.6. If not converged and the maximum number of nonlinear iterations has not been exceeded,
return to Step 3.
4. After the final nonlinear iteration of each timestep n: psim solve iterate ell only
4.1. If the maximum number of nonlinear iterations was exceeded, update the primary variables using the best solution at .
4.2. Write out selected grid properties: psim calculate local ell, psim converged local ell
4.3. Write out selected well properties: psim COPY well ell, sim well single completion ell
4.4. If necessary, WAG to gas or WAG to water or transfer additional mass to the trapped
system: psim update trap properties n, psim 1D 0020 well WAG gas, psim 1D 0020 well
WAG water.
4.5. If not the final time step, return to Step 2.
5. Finalize
5.1. Finalize solver
5.2. Deallocate memory
4.4
Assemble the Jacobian
In a three-phase dual medium system with NC = 5 and no degeneracies illustrated in (4.37),
there are 2 × (2 × NC − 1) primary variables: Pm1 , Swm1 , Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 ,
Y3,m1 , Pm2 , Swm2 , Som2 , X1,m2 , X2,m2 , X3,m2 , Y1,m2 , Y2,m2 , and Y3,m2 . The primary variables are
reordered to facilitate the following main steps of the solution.
1. Pm1 is solved for first using a sparse matrix solve of a reduced set of pressure equations,
one per grid cell. The reduced set of equations is obtained by LU decomposition of the full
system of equations for each grid cell. This simplification is possible because only the Pm1
terms appear for off-block-diagonal grid cells.
77
2. Pm2 , Sw,m1 , and Sw,m2 are solved for next by back substitution local to the grid cell.
3. Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 , and Y3,m1 are solved using flash calculations local to
the grid cell.
4. Som2 , X1,m2 , X2,m2 , X3,m2 , Y1,m2 , Y2,m2 , and Y3,m2 are solved using flash calculations local to
the grid cell.
There are also 2 × (2 × NC − 1) equations: C1,m1 , C2,m1 , C3,m1 , C4,m1 4, C5,m1 , G1,m1 , G2,m1 ,
G3,m1 , G4,m1 , C1,m2 , C2,m2 , C3,m2 , C4,m2 4, C5,m2 , G1,m2 , G2,m2 , G3,m2 , and G4,m2 . These are ordered
in the following way to facilitate the LU decomposition:
1. CH2 O,m1 , CH2 O,m2 , CCO2 ,m1 , and CCO2 ,m2 . These component equations come first because they
typically have large non-zero coefficients of Pm1 , Pm2 , Swm1 , and Swm2 .
2. C1...NC −2,m1 and G1...NC −1,m1 ; these are associated with the flash variables Som1 , X1...NC −2,m1 ,
and Y1...NC −2,m1
3. C1...NC −2,m2 and G1...NC −1,m2 ; these are associated with the flash variables Som2 , X1...NC −2,m2 ,
Swm1
Swm2
Som1
Xm=1,m1
Xm=2,m1
Xm=3,m1
Ym=1,m1
Ym=2,m1
Ym=3,m1
Som2
Xm=1,m2
Xm=2,m2
Xm=3,m2
Ym=1,m2
Ym=2,m2
Ym=3,m2
CH2 O,m1
CH2 O,m2
CCO2 ,m1
CCO2 ,m2
Cm=1,m1
Cm=2,m1
Cm=3,m1
#
X
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
#
0
X
0
0
0
0
0
0
0
0
0
0
0
0
0
0
X
X
#
0
X
X
X
X
X
X
X
0
0
0
0
0
0
0
X
X
0
#
0
0
0
0
0
0
0
X
X
X
X
X
X
X
X
X
X
0
#
X
X
X
X
X
X
0
0
0
0
0
0
0
X
X
X
0
X
#
X
X
X
X
X
0
0
0
0
0
0
0
X
X
X
0
X
X
#
X
X
X
X
0
0
0
0
0
0
0
Gg/o,m=1,m1
Gg/o,m=2,m1
Gg/o,m=3,m1
Gg/o,m=4,m1
Cm=1,m2
Cm=2,m2
Cm=3,m2
Gg/o,m=1,m2
Gg/o,m=2,m2
Gg/o,m=3,m2
Gg/o,m=4,m2
X
0
0
0
0
X
X
#
X
X
X
0
0
0
0
0
0
0
X
0
0
0
0
X
X
X
#
X
X
0
0
0
0
0
0
0
X
0
0
0
0
X
X
X
X
#
X
0
0
0
0
0
0
0
X
0
0
0
0
X
X
X
X
X
#
0
0
0
0
0
0
0
X
X
0
X
0
0
0
0
0
0
0
#
X
X
X
X
X
X
X
X
0
X
0
0
0
0
0
0
0
X
#
X
X
X
X
X
X
X
0
X
0
0
0
0
0
0
0
X
X
#
X
X
X
X
0
X
0
0
0
0
0
0
0
0
0
0
X
X
#
X
X
X
0
X
0
0
0
0
0
0
0
0
0
0
X
X
X
#
X
X
0
X
0
0
0
0
0
0
0
0
0
0
X
X
X
X
#
X
0
X
0
0
0
0
0
0
0
0
0
0
X
X
X
X
X
#
Pom1
Pom2
and Y1...NC −2,m2
78
(4.37)
4.4.1
Single Medium (No Trapping)
In a single medium system, all of the m2 variables and equations can be eliminated. These
variables and equations are eliminated in the order of their occurrence from the reordered lists.
The nine remaining variables are the following: Pm1 , Sw,m1 , Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 ,
and Y3,m1 . The nine remaining equations are the following: CH2 O,m1 , CCO2 ,m1 , C1...NC −2,m1 , and
G1...NC −1,m1 .
For degenerate cases where one of the phases or one of the components is not present, the
same process is used. The variables and equations are eliminated in the order of their occurrence
from the reordered lists. Different phases or components can be present in the mobile and trapped
medium, leaving many potential options. To simplify the discussion the single medium case will be
used in the remainder of the discussion.
4.4.2
Degenerate Case with Oil and Water Only
When a three-phase gas-oil-water system degenerates into a two-phase oil-water system, instead
of three saturations Sw , So , Sg = 1−So −Sw , there are now only two saturations, Sw and So = 1−Sw .
Although the gas compositions Y1 , Y2 , Y3 , Y4 can be defined, they are not relevant to a problem
with only oil and water. The five remaining primary variables are the following: Pm1 , Sw,m1 , X1,m1 ,
X2,m1 , and X3,m1 . All of the component equations are still relevant, but note that the terms referring
to gas saturations or gas relative permeabilities are zero. The thermodynamic constraints can be
defined but are not relevant to a problem without both oil and gas. The five remaining equations
are the following: CH2 O,m1 , CCO2 ,m1 , and C1...NC −2,m1 .
4.4.3
Degenerate Case with Gas and Water Only
When a three-phase gas-oil-water system degenerates into a two-phase gas-water system, instead
of three saturations Sw , So , Sg = 1−So −Sw , there are now only two saturations, Sw and Sg = 1−Sw .
Although the oil compositions X1 , X2 , X3 , X4 can be defined, they are not relevant to a problem
with only gas and water. The five remaining primary variables are the following: Pm1 , Sw,m1 , Y1,m1 ,
Y2,m1 , and Y3,m1 . All of the component equations are still relevant, but note that the terms referring
to oil saturations or oil relative permeabilities are zero. The thermodynamic constraints can be
defined but are not relevant to a problem without both oil and gas. The five remaining equations
79
are the following: CH2 O,m1 , CCO2 ,m1 , and C1...NC −2,m1 .
4.4.4
Degenerate Case with Gas and Oil Only
The degenerate case where a three-phase gas-oil-water system degenerates into a two-phase
oil-gas system is unusual, but can occur in several different ways. In a steam injection scenario
(not considered in this dissertation), the water may all be vaporized into gas. For a system with
trapping, the trapped system may not contain any trapped water. A gas-only system or an oil-only
system can also evolve into a two-phase oil-gas system.
Instead of three saturations Sw , So , Sg = 1−So −Sw , there are now only two saturations, So and
Sg = 1 − So . The water component equation does not apply here. The eight remaining variables
are the following: Pm1 , Som1 , X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 , and Y3,m1 . The eight remaining
equations are the following: CCO2 ,m1 , C1...NC −2,m1 , and G1...NC −1,m1 .
4.4.5
Degenerate Case with Water Only
The degenerate case where a three-phase gas-oil-water system degenerates into a single-phase
water system is unusual, but can occur in several different ways. In a steam injection scenario
(not considered in this dissertation), all the steam may condense into liquid water. For a system
with trapping, the trapped system may only contain water. A gas-water system may turn into a
water-only system with water injection or the dissolution of all the gas into the water.
Instead of three saturations Sw , So , Sg = 1 − So − Sw , there is now only one saturation, Sw ;
but because Sw = 1, this saturation can be eliminated too. The X1,m1 , X2,m1 , X3,m1 , Y1,m1 , Y2,m1 ,
and Y3,m1 can be defined but are not applicable. This leaves only one primary variable, Pm1 and
one equation, CH2 O,m1 .
4.4.6
Degenerate Case with Oil Only
The degenerate case where a three-phase gas-oil-water system degenerates into a single-phase
oil system is unusual, but can occur in several different ways. For a system with trapping, the
trapped system may only contain oil. A gas-oil system may turn into an oil-only system based on
a phase transition.
Instead of three saturations Sw , So , Sg = 1 − So − Sw , there is now only one saturation, So ;
but because So = 1, this saturation can be eliminated too. Although the gas compositions Y1 ,
80
Y2 , Y3 , Y4 can be defined, they are not relevant to a problem with oil only. This leaves only four
primary variables: Pm1 , X1,m1 , X2,m1 , and X3,m1 . The water component equation is not applicable
to a system without water. The thermodynamic constraints can be defined but are not relevant
to a problem without both oil and gas. This leaves the following four equations: CCO2 ,m1 and
C1...NC −2,m1 .
4.4.7
Degenerate Case with Gas Only
The degenerate case where a three-phase gas-oil-water system degenerates into a single-phase
gas system is unusual, but can occur in several different ways. For a system with trapping, the
trapped system may only contain gas. A gas-oil system may turn into an oil-only system based on
a phase transition.
Instead of three saturations Sw , So , Sg = 1 − So − Sw , there is now only one saturation, Sg ; but
because Sg = 1, this saturation can be eliminated too. Although the oil compositions X1 , X2 , X3 ,
X4 can be defined, they are not relevant to a problem with gas only. This leaves only four primary
variables: Pm1 , Y1,m1 , Y2,m1 , and Y3,m1 . The water component equation is not applicable to a system
without water. The thermodynamic constraints can be defined but are not relevant to a problem
without both oil and gas. This leaves the following four equations: CCO2 ,m1 and C1...NC −2,m1 .
4.4.8
Three-Phase Degenerate Case with Fewer Components
When one of the components is zero, this eliminates two primary variables (Xm and Ym ) and
two equations (Cm and Gm ). These primary variables and equations are eliminated and the rest
of the order is preserved.
For instance, if Z1,m1 = 0, the seven remaining variables are the following: Pm1 , Sw,m1 , Som1 ,
X2,m1 , X3,m1 , Y2,m1 , and Y3,m1 . The seven remaining equations are the following: CH2 O,m1 , CCO2 ,m1 ,
C2...NC −2,m1 , and G2...NC −1,m1 .
If ZCO2 = 0, then the NC − 2 = 3 component is eliminated from the variables since now
X3,m1 = 1 − X1,m1 − X2,m1 and Y3,m1 = 1 − Y1,m1 − Y2,m1 . The equations for CO2 are still eliminated.
The seven remaining variables are the following: Pm1 , Sw,m1 , Som1 , X1,m1 , X2,m1 , Y1,m1 , and Y2,m1 .
The seven remaining equations are the following: CH2 O,m1 , C1...NC −2,m1 , and G1...NC −2,m1 .
A similar process applies if more than one component is zero.
81
4.5
Rewrite Base Equations for Um Solve
The solution approach used for each nonlinear iteration uses a careful labeling of the terms in
the component equations to solve for So , Sg , Xm , Ym , Wm based on a solution of P and Sw . This
section identifies the new variable definitions of the base equations. Section 4.6 identifies the steps
in the solution.
The component equations for the m1 system, (4.1) are rewritten as follows. The [ + 1] terms
and Pm+1
which have already been calculated during the pressure
are based on the pressure Pm+1
1
2
solve.
+1
Um,m
1
Xn
n
n
ξ
k
[+1]
mm1 om1 rom1 #
n
0.006328 VR ∇ ·
km1 (∇Pom1 − γom
∇D# ) +
1
μnom1
Y n ξ n kn
[+1]
mm1 gm1 rgm1 #
n
n
0.006328 VR ∇ ·
km1 (∇Pom1 + ∇Pcgom
− γgm
∇D# ) +
1
1
n
μgm1
W n ξ n kn
[+1]
mm1 wm1 rwm1 #
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
om
m
cowm
wm
1
1
1
1
μnwm1
=
[+1]
[+1]
[+1]
n
n
n
n
n
n
Xmm1 ξom1 qom1 + Ymm1 ξgm1 qgm1 + Wmm1 ξwm1 qwm1 −
Cm=1...NC ,m1 :
[+1]
+1
Zm,2ph,m
1
[+1]
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2
up,n
ξ up,n k up,n
Xmm
1 /m2 om1 /m2 rom1 /m2
μup,n
om1 /m2
α+1
m1
+
up,n
Ymm
ξ up,n k up,n
1 /m2 gm1 /m2 rgm1 /m2
μup,n
gm1 /m2
n
Zm,2ph,m
1
×
βm+1
1
VR +1 +1 +1
VR
+1 +1
+1
+1 +1 +1
−
φm1 (Som1 ξom1 + Sgm
φ
ξ
)
+W
S
ξ
mm1
1 gm1
Δt
Δt m1 wm1 wm1
+
up,n
Wmm
ξ up,n k up,n
1 /m2 wm1 /m2 rwm1 /m2
μup,n
wm1 /m2
αn
m1
βmn1
VR n
VR n n n n
n
n
n
n
φm1 (Som
φ S
ξ
+
S
ξ
)
+W
ξ
gm1 gm1
mm1
1 om1
Δt
Δt m1 wm1 wm1
(4.38)
The component equations for the m2 system, (4.2) are rewritten as follows. The [ + 1] terms
and Pm+1
which have already been calculated during the pressure
are based on the pressure Pm+1
1
2
solve.
82
+1
Um,m
2
Cm=1...NC ,m2 :
+1
Zm,2ph,m
2
0.006328 VR σm#1 /m2 km#1 /m2
up,n
up,n
k up,n
Xmm1 /m2 ξom
1 /m2 rom1 /m2
μup,n
om1 /m2
α+1
m2
−
[+1]
Pom2
×
up,n
Ymm
ξ up,n k up,n
1 /m2 gm1 /m2 rgm1 /m2
μup,n
gm1 /m2
+
+
up,n
Wmm
ξ up,n k up,n
1 /m2 wm1 /m2 rwm1 /m2
=
μup,n
wm1 /m2
βm+1
2
VR +1 +1 +1
VR
+1 +1
+1
+1 +1 +1
ξ
)
+W
S
ξ
−
φm2 (Som2 ξom2 + Sgm
φ
gm
mm
m
wm
wm
2
2
2
2
2
Δt
Δt 2
[+1]
Pom1
n
Zm,2ph,m
2
αn
m2
βmn2
VR n
VR
n
n
n
n
n
n
n
n
φm2 (Som
φ
ξ
+
S
ξ
)
+W
S
ξ
gm2 gm2
mm2
2 om2
Δt
Δt m2 wm2 wm2
(4.39)
Add the NC equations (4.38) to obtain Ct,m1 . This eliminates Xmm1 , Ymm1 , and Wmm1 because
m Xmm1 = 1,
m Ymm1 = 1, and
m Wmm1 = 1.
+1
Ut,m
1
Ct,m1
ξ n kn
[+1]
n
0.006328 VR ∇ · om1n rom1 km#1 (∇Pom1 − γom
∇D# ) +
1
μom1
ξ n kn
[+1]
gm1 rgm1 #
n
n
0.006328 VR ∇ ·
km1 (∇Pom1 + ∇Pcgom
− γgm
∇D# ) +
1
1
n
μgm1
ξ n kn
[+1]
n
n
#
0.006328 VR ∇ · wm1n rwm1 km#1 (∇Pom1 − ∇Pcowm
−
γ
∇D
)
+
wm
1
1
μwm1
:
=
[+1]
[+1]
[+1]
n
n
n
ξom1 qom1 + ξgm1 qgm1 + ξwm1 qwm1 −
[+1]
up,n
k up,n
ξom
1 /m2 rom1 /m2
μup,n
om1 /m2
α+1
m1
[+1]
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2
+
up,n
ξgm
k up,n
1 /m2 rgm1 /m2
μup,n
gm1 /m2
+
up,n
ξwm
k up,n
1 /m2 rwm1 /m2
×
μup,n
wm1 /m2
βm+1
1
VR
VR
+1 +1
+1 +1
+1 +1 +1
−
φ+1
φ
(S
ξ
+
S
ξ
)
+
S
ξ
om1 om1
gm1 gm1
Δt m1
Δt m1 wm1 wm1
VR
Δt
αn
m1
n
n
φnm1 (Som
ξ n + Sgm
ξn ) +
1 om1
1 gm1
βmn1
VR n n n φ S
ξ
Δt m1 wm1 wm1
(4.40)
Add the NC equations (4.39) to obtain Ct,m2 . This eliminates Xmm2 , Ymm2 , and Wmm2 because
m Xmm2 = 1,
m Ymm2 = 1, and
m Wmm2 = 1.
83
+1
Ut,m
2
Ct,m2 :
[+1]
[+1]
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2 ×
up,n
up,n
up,n
up,n
ξgm
ξwm
k up,n
k up,n
ξom1 /m2 krom
1 /m2
1 /m2 rgm1 /m2
1 /m2 rwm1 /m2
+
+
μup,n
μup,n
μup,n
om1 /m2
gm1 /m2
wm1 /m2
α+1
m2
=
βm+1
2
VR
VR
+1 +1
+1 +1
+1 +1
φ+1
φ+1
m2 (Som2 ξom2 + Sgm2 ξgm2 ) +
m2 Swm2 ξwm2 −
Δt
Δt
VR
Δt
αn
m2
n
φnm2 (Som
ξn
2 om2
βmn2
VR
n
n
n
n
n
φ
+ Sgm
ξ
)
+
S
ξ
2 gm2
Δt m2 wm2 wm2
(4.41)
Write the water equation for m1 .
+1
UWAT,m
1
[+1]
n
n
0.006328 VR ∇ ·
km#1 (∇Pom1 − ∇Pcowm
− γwm
∇D# ) +
1
1
[+1]
n
q
− =
ξ
wm
wm1
:
1
up,n
up,n
ξwm1 /m2 krwm
[+1]
[+1]
1 /m2
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2 ×
μup,n
wm1 /m2
ξn
n
wm1 krwm1
μnwm1
CWAT,m1
βm+1
1
VR
+1 +1
S
ξ
−
φ+1
Δt m1 wm1 wm1
VR
Δt
βmn1
n
ξn
φnm1 Swm
1 wm1
(4.42)
Write the water equation for m2 .
+1
UWAT,m
2
[+1]
[+1]
CWAT,m2 : 0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2
×
up,n
ξwm
k up,n
1 /m2 rwm1 /m2
μup,n
wm1 /m2
βm+1
2
VR
+1 +1
−
φ+1
m2 Swm2 ξwm2
Δt
84
VR
Δt
=
βmn2
n
φnm2 Swm
ξn
2 wm2
(4.43)
4.6
Update Primary Variables at Each Nonlinear Iteration
The process involves the following steps:
1. Assemble the Jacobian using (4.1)–(4.5). The ordering of the primary variables and equations
listed here is for the gas/oil/water system with WCO2 as a secondary variable.
• The primary variables are ordered as follows: Pom1 , Pom2 , Swm1 , Swm2 , Som1 , Xm=1,m1 ,
Xm=2,m1 , Xm=3,m1 (up to Xm=NC −2,m1 ), Ym=1,m1 , Ym=2,m1 , Ym=3,m1 (up to Ym=NC −2,m1 ),
Som2 , Xm=1,m2 , Xm=2,m2 , Xm=3,m2 (up to Xm=NC −2,m2 ), Ym=1,m2 , Ym=2,m2 , Ym=3,m2 (up
to Ym=NC −2,m2 )
• The equations are ordered as follows, and are reordered using partial pivoting. CH2 O,m1 ,
CH2 O,m2 , CCO2 ,m1 , CCO2 ,m2 , Cm=1,m1 , Cm=2,m1 , Cm=3,m1 (up to CNC −2,m1 ), Gm=1,m1 ,
Gm=2,m1 , Gm=3,m1 , Gm=4,m1 (up to GNC −1,m1 ), Cm=1,m2 , Cm=2,m2 , Cm=3,m2 (up to CNC −2,m2 ),
Gm=1,m2 , Gm=2,m2 , Gm=3,m2 , Gm=4,m2 (up to GNC −1,m2 )
2. Set up and solve pressure equation.
2.1. Perform an LU decomposition for each grid cell.
2.2. The upper left corner of each block forms the LU-pressure equation.
+1
using a sparse matrix solver.
2.3. Solve the system of LU-pressure equations for δPom
1
3. In psim convergence update primary ell, calculate primary variables at + 1.
+1
+1
and Pom
3.1. Calculate Pom
1
2
+1
+1
+δP +1 , with additional checks to keep P
3.1.1. Calculate Pom
= Pom
min < Pom1 < Pmax .
om1
1
1
Test for convergence using
+1 max δPom
< P
1 ,ijk
(4.44)
ijk
+1
+1
+1
by back substitution. Calculate Pom
= Pom
+ δPom
, with addi3.1.2. Calculate δPom
2
2
2
2
+1
+1 +1
− Pom
< Pmax .
< Pmax diff and Pmin < Pom
tional checks to ensure Pom
2
1 2
+1
+1
and Swm
3.2. Calculate Swm
1
2
85
+1
3.2.1. If the previous iteration Sw < Swm
< 1−Sw , calculate δSwm
by back substitution.
1
1
+1
+1
+1
= Swm
+ δSwm
, with additional checks to ensure 0 ≤ Swm
≤ 1.
Calculate Swm
1
1
1
1
< Sw or Swm
> 1 − Sw , then
3.2.2. If the previous iteration Swm
1
1
+ 1
3.2.2.1. Calculate an approximate value for φm1 2 .
+ 1
φm1 2 = φm1 +
∂φ
δP
∂P
(4.45)
+ 1
3.2.2.2. Calculate an approximate value for ξw,m21 . Accumulate pressure derivative at
this point.
+ 1
+
ξw,m21 = ξwm
1
∂ξwm1
∂ξwm1
∂ξwm1
δP +
δYm +
δXm
∂P
∂Ym
∂Xm
(4.46)
+1
< 1−Sw , calculate δSwm
by back substitution.
3.2.3. If the previous iteration Sw < Swm
2
2
+1
+1
+1
= Swm
+ δSwm
, with additional checks to ensure 0 ≤ Swm
≤ 1.
Calculate Swm
2
2
2
2
< Sw or Swm
> 1 − Sw , then
3.2.4. If the previous iteration Swm
2
2
+ 1
3.2.4.1. Calculate an approximate value for φm2 2
+ 1
φm2 2 = φm2 +
∂φ
δP
∂P
(4.47)
+ 1
3.2.4.2. Calculate an approximate value for ξw,m22 . Accumulate pressure derivative at
this point.
+ 1
+
ξw,m22 = ξwm
2
∂ξwm2
∂ξwm2
∂ξwm2
δP +
δYm +
δXm
∂P
∂Ym
∂Xm
(4.48)
+1
+1
+1
, qgm
, and qwm
by calling psim COPY well ell and
3.3. Calculate the well properties qom
1
1
1
+1
and properties at n. See
sim well single completion converged ell. as a function of Pom
1
Chapter 9 for a discussion of the calculation of well rates using fixed rate, fixed pressure,
and mixed pressure and rate constraints.
4. In psim convergence update primary ell, call psim new primary from flash ell. Calculate the primary variables at + 1 which depend on flash calculations. See Section 4.7 for more details.
86
• The primary variables calculated by psim new primary from flash ell: Som1 , Som2 , Xm =1...NC−2 ,m1 ,
Ym =1...NC−2 ,m1 , Xm =1...NC−2 ,m2 , Ym =1...NC−2 ,m2
• If the previous iteration Swm
< Sw or Swm
> 1 − Sw , calculate Swm1
1
1
• If the previous iteration Swm
< Sw or Swm
> 1 − Sw , calculate Swm2
2
2
• The secondary variables calculated by psim new primary from flash ell: WCO2 m1 , ξgm1 ,
ξom1 , ξwm1 , WCO2 m2 , ξgm2 , ξom2 , and ξwm2 .
5. In psim convergence update primary ell, calculate mass balance for grid cells in psim converged
local ell.
Mtn =
n
n
n
+ Mgmm
+ Mwmm
Momm
+
1
1
1
m
Mt+1 =
n
n
n
+ Mgmm
+ Mwmm
Momm
2
2
2
(4.49)
m
m
+1
+1
+1
+
M
+
M
Momm
+
gmm
wmm
1
1
1
+1
+1
+1
+1
+1
+1
+
M
+
M
+
q
+
q
Momm
+
q
gmm2
wmm2
om
gm
wm × Δn (4.50)
2
m
m
The residual R+1 is used to determine the best model in case the nonlinear iterations do not
converge.
R+1 =
Mt+1 − Mtn
Mtn
(4.51)
6. In psim convergence update primary ell, print any desired primary variables and any desired
secondary variables at + 1 for all grid cells. If desired, print information on the convergence
process, including the residual and the grid cells with maximum changes in P , S, Xm , Ym ,
and WCO2 .
4.7
Update Primary Variables at Each Nonlinear Iteration: Flash
In psim new primary from flash ell and the subroutines it calls, calculate the primary variables
at + 1 which depend on flash calculations. These include:
• The primary variables calculated by psim new primary from flash ell: Som1 , Som2 , Xm =1...NC−2 ,m1 ,
Ym =1...NC−2 ,m1 , Xm =1...NC−2 ,m2 , Ym =1...NC−2 ,m2
87
• If the previous iteration Swm
< Sw or Swm
> 1 − Sw , calculate Swm1
1
1
• If the previous iteration Swm
< Sw or Swm
> 1 − Sw , calculate Swm2
2
2
• The secondary variables calculated by psim new primary from flash ell: WCO2 m1 , ξgm1 , ξom1 ,
ξwm1 , WCO2 m2 , ξgm2 , ξom2 , and ξwm2 .
1. In psim new primary from flash ell call psim calc after TRANS ell one. Calculate primary variables at + 1.
+1
+1
+1
using (4.38) and (4.52) as a function of Pom
, Pom
, and properties at
1.1. Calculate Umm
1
1
2
n.
0.006328 VR ∇ ·
Xn
n
n
mm1 ξom1 krom1
n
μom1
[+1]
n
km#1 (∇Pom1 − γom
∇D# ) +
1
Y n ξ n kn
[+1]
mm1 gm1 rgm1 #
n
n
#
k
(∇P
+
∇P
−
γ
∇D
)
+
om
m
cgom
gm
1
1
1
1
μngm1
W n ξ n kn
[+1]
mm1 wm1 rwm1 #
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
om
m1
cowm1
wm1
1
μnwm1
[+1]
[+1]
[+1]
n
n
n
ξn q
+ Ymm
ξn q
+ Wmm
ξn q
−
Xmm
1 om1 om1
1 gm1 gm1
1 wm1 wm1
0.006328 VR ∇ ·
+1
Umm
=
1
[+1]
[+1]
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2
up,n
ξ up,n k up,n
Xmm
1 /m2 om1 /m2 rom1 /m2
μup,n
om1 /m2
+
up,n
Ymm
ξ up,n k up,n
1 /m2 gm1 /m2 rgm1 /m2
μup,n
gm1 /m2
+
up,n
Wmm
ξ up,n k up,n
1 /m2 wm1 /m2 rwm1 /m2
μup,n
wm1 /m2
×
(4.52)
+1
+1
+1
1.2. Calculate Umm
using (4.39) and (4.53) as a function of Pom
, Pom
, and properties at
2
1
2
n.
[+1]
+1
Umm
2
[+1]
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2 ×
up,n
up,n
up,n
up,n
up,n
up,n
up,n
up,n
up,n
=
Y
W
k
ξ
k
ξ
k
Xmm1 /m2 ξom
/m
/m
/m
/m
/m
/m
/m
/m
rom
mm
gm
rgm
mm
wm
rwm
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
+
+
μup,n
μup,n
μup,n
om1 /m2
gm1 /m2
wm1 /m2
(4.53)
+1
+1
+1
1.3. Calculate UWATm
using (4.42) and (4.54) as a function of Pom
, Pom
, and properties
1
2
1
at n.
88
[+1]
n
n
km#1 (∇Pom1 − ∇Pcowm
− γwm
∇D# ) +
1
1
[+1]
n
− (4.54)
ξwm
q
wm
1
1
up,n
up,n
ξwm1 /m2 krwm1 /m2
[+1]
[+1]
0.006328 VR σm#1 /m2 km#1 /m2 Pom1 − Pom2 ×
μup,n
wm1 /m2
0.006328 VR ∇ ·
+1
UWATm
=
1
ξn
n
wm1 krwm1
n
μwm1
+1
+1
+1
1.4. Calculate UWATm
using (4.43) and (4.55) as a function of Pom
, Pom
, and properties
1
2
2
at n.
+1
UWATm
2
= 0.006328 VR
σm#1 /m2 km#1 /m2
[+1]
Pom1
−
[+1]
Pom2
×
up,n
k up,n
ξwm
1 /m2 rwm1 /m2
μup,n
wm1 /m2
(4.55)
2. In psim new primary from flash ell, load the previously calculated values of αnm1 , αnm2 , βmn1 ,
βmn2 . These depend only on variables at n, which means they were already calculated in in
psim calc after TRANS n.
αnm1 =
βmn1 =
αnm2
βmn2
=
=
VR n
n
n
n
n
Δt φm1 (Som1 ξom1 + Sgm1 ξgm1 )
VR n
n
n
Δt φm1 Swm1 ξwm1
VR n
n
n
n
n
Δt φm2 (Som2 ξom2 + Sgm2 ξgm2 )
VR n
n
n
Δt φm2 Swm2 ξwm2
(4.56)
(4.57)
(4.58)
(4.59)
3. In psim new primary from flash ell, calculate additional properties at + 1 needed to calculate
+1
+1
and Zm,2ph,m
Zm,2ph,m
1
2
+1
+1
+1
3.1. Calculate Utm
using (4.40) and (4.60) as a function of Pom
, Pom
, and properties at n.
1
1
2
+1
=
Utm
1
+1
Umm
1
(4.60)
m
+1
+1
+1
using (4.41) and (4.61) as a function of Pom
, Pom
, and properties at n.
3.2. Calculate Utm
2
1
2
+1
=
Utm
2
+1
Umm
2
(4.61)
m
=
3.3. Solve (4.42) for βm+1
1
VR +1 +1 +1
Δt φm1 Swm1 ξwm1
+1
= UWAT,m
+ βmn1
1
3.4. Solve (4.43) for βm+1
=
2
VR +1 +1 +1
Δt φm2 Swm2 ξwm2
+1
= UWAT,m
+ βmn2
2
89
3.5. Solve (4.40) for α+1
m1 =
+1 +1
VR +1
Δt φm1 (Som1 ξom1
+1 +1
+1
+ Sgm
ξ ) = Ut,m
+ αnm1 + βmn1 − βm+1
1 gm1
1
1
3.6. Solve (4.41) for α+1
m2 =
+1 +1
VR +1
Δt φm2 (Som2 ξom2
+1 +1
+1
+ Sgm
ξ ) = Ut,m
+ αnm2 + βmn2 − βm+1
2 gm2
2
2
+1
+1
4. In psim new primary from flash ell, calculate Zm,2ph,m
and Zm,2ph,m
1
2
+1
+1
+1
4.1. If α+1
m1 > α and αm2 > α (under normal conditions), calculate Zm,2ph,m1 and Zm,2ph,m2 .
+1
+1
n Zn
=
U
+
α
/α+1
4.1.1. For m = 1 . . . NC − 2, solve (4.38) for Zm,2ph,m
m,m
m1
m
1
m,2ph,m
1
1
1
+1
4.1.2. Calculate ZCO
=1−
2 ,2ph,m1
4.1.3. Ensure that 0 ≤
+1
Zm,2ph,m
1
N
C −2
m =1
+1
Zm
,2ph,m .
1
≤ 1 and that
+1
m Zm,2ph,m1
= 1.
+1
+1
+1
n Zn
=
U
+
α
4.1.4. For m = 1 . . . NC − 2, solve (4.39) for Zm,2ph,m
m,m2
m2 m,2ph,m2 /αm2
2
4.1.5. Calculate
+1
ZCO
2 ,2ph,m2
4.1.6. Ensure that 0 ≤
=1−
N
C −2
m =1
+1
Zm,2ph,m
2
+1
Zm
,2ph,m .
2
≤ 1 and that
+1
m Zm,2ph,m2
= 1.
+1
4.2. If α+1
m1 < α and αm2 < α ,
+1
+1
4.2.1. Set Zm
=CO2 ,2ph,m1 = 0 and Zm=CO2 ,2ph,m1 = 1.
+1
+1
4.2.2. Set Zm
=CO2 ,2ph,m2 = 0 and Zm=CO2 ,2ph,m2 = 1.
+1
+1
4.3. If α+1
m1 > α and αm2 < α , calculate Zm,2ph,m1 .
+1
+1
n Zn
=
U
+
α
/α+1
4.3.1. For m = 1 . . . NC − 2, solve (4.38) for Zm,2ph,m
m,m
m1
m
1
m,2ph,m
1
1
1
+1
4.3.2. Calculate ZCO
=1−
2 ,2ph,m1
4.3.3. Ensure that 0 ≤
+1
Zm,2ph,m
1
N
C −2
m =1
+1
Zm
,2ph,m .
1
≤ 1 and that
+1
m Zm,2ph,m1
= 1.
+1
+1
= Zm,2ph,m
4.3.4. Set Zm,2ph,m
2
1
+1
+1
4.4. If α+1
m1 < α and αm2 > α , calculate Zm,2ph,m2 .
+1
+1
+1
n Zn
=
U
+
α
4.4.1. For m = 1 . . . NC − 2, solve (4.39) for Zm,2ph,m
m,m2
m2 m,2ph,m2 /αm2
2
4.4.2. Calculate
+1
ZCO
2 ,2ph,m2
4.4.3. Ensure that 0 ≤
=1−
+1
Zm,2ph,m
2
N
C −2
m =1
+1
Zm
,2ph,m .
2
≤ 1 and that
+1
m Zm,2ph,m2
= 1.
+1
+1
= Zm,2ph,m
4.4.4. Set Zm,2ph,m
1
2
+1
+1
+1
+1
5. In psim new primary from flash ell, flash Zm,2ph,m
and calculate Xm,m
, Ym,m
, and Swm
.
1
1
1
1
90
+1
+1
+1
+1
+1
+1
+1
5.1. Flash Zm,2ph,m
at Pom
to calculate Xmm
, Ymm
, ξom
, ξgm
, L+1
m1 , Vm1 .
1
1
1
1
1
1
+1
≤ 1 and that
5.2. Ensure that 0 ≤ Xm,m
1
+1
5.3. Ensure that 0 ≤ Ym,m
≤ 1 and that
1
+1
m Xm,m1
+1
m Ym,m1
= 1.
= 1.
+1
< S or Swm
> 1 − S , calculate Swm
here.
5.4. If Swm
1
1
1
+ 1
5.4.1. Calculate an approximate value for ξw,m21 . Accumulate composition derivative at
this point.
+ 1
+
ξw,m21 = ξwm
1
∂ξwm1
∂ξwm1
∂ξwm1
δYm +
δXm
δP +
∂P
∂Ym
∂Xm
+ 1
+ 1
(4.62)
+ 1
5.4.2. Use ξw,m21 calculated above; if ξw,m21 < then set ξw,m21 = ξw [Patm , Tres , WCO2 = 0, WNaCl ].
+ 1
+ 1
+1
= 0.
5.4.3. Use φm1 2 calculated above; if φm1 2 < then Swm
1
+ 1
5.4.4. If φm1 2 > then
+1
=
Swm
1
Δt βm+1
1
VR φ+ 12 ξ + 12
m1
(4.63)
w,m1
+1
≤1
5.4.5. Ensure that 0 ≤ Swm
1
6. In psim new primary from flash ell, call psim primary iterate WCO2 ell. Use an iterative method
+1
+1
and ξw,m
. See Section 4.8 for the details.
to calculate WCO
1
2 ,m1
7. In psim new primary from flash ell, update So , Sg , Xm , and Ym .
+1
≤ 1 and that
7.1. Ensure that 0 ≤ Xm,m
1
+1
≤ 1 and that
7.2. Ensure that 0 ≤ Ym,m
1
+1
=
7.3. Calculate Som
1
+1
7.4. Calculate Sgm
=
1
+1
m Xm,m1
+1
m Ym,m1
= 1.
= 1.
+1
+1
(1 − Sw,m
)L+1
m1 ξg,m1
1
+1
+1 +1
L+1
m1 ξg,m1 + Vm1 ξo,m1
+1
+1
(1 − Sw,m
)Vm+1
ξo,m
1
1
1
+1
+1 +1
L+1
m1 ξg,m1 + Vm1 ξo,m1
+1
+1
+1
+1
8. In psim new primary from flash ell, flash Zm,2ph,m
and calculate Xm,m
, Ym,m
, and Swm
.
2
2
2
2
+1
+1
+1
+1
+1
+1
+1
at Pom
to calculate Xmm
, Ymm
, ξom
, ξgm
, L+1
8.1. Flash Zm,2ph,m
m2 , Vm2 .
2
2
2
2
2
2
+1
≤ 1 and that
8.2. Ensure that 0 ≤ Xm,m
2
+1
≤ 1 and that
8.3. Ensure that 0 ≤ Ym,m
2
+1
m Xm,m2
+1
m Ym,m2
91
= 1.
= 1.
+1
8.4. If Swm
< S or Swm
> 1 − S , calculate Swm
here.
2
2
2
+ 1
8.4.1. Calculate an approximate value for ξw,m22 . Accumulate composition derivative at
this point.
+ 1
+
ξw,m22 = ξwm
2
∂ξwm2
∂ξwm2
∂ξwm2
δYm +
δXm
δP +
∂P
∂Ym
∂Xm
+ 1
+ 1
(4.64)
+ 1
8.4.2. Use ξw,m22 calculated above; if ξw,m22 < then set ξw,m22 = ξw [Patm , Tres , WCO2 = 0, WNaCl ].
+ 1
+ 1
+1
= 0.
8.4.3. Use φm2 2 calculated above; if φm2 2 < then Swm
2
+ 1
8.4.4. If φm2 2 > then
+1
=
Swm
2
Δt βm+1
2
VR φ+ 12 ξ + 12
m2
(4.65)
w,m2
+1
≤1
8.4.5. Ensure that 0 ≤ Swm
2
9. In psim new primary from flash ell, call psim primary iterate WCO2 ell. Use an iterative method
+1
+1
and ξw,m
. See Section 4.8 for the details.
to calculate WCO
2
2 ,m2
10. In psim new primary from flash ell, update So , Sg , Xm , and Ym .
+1
≤ 1 and that
10.1. Ensure that 0 ≤ Xm,m
2
+1
≤ 1 and that
10.2. Ensure that 0 ≤ Ym,m
2
+1
=
10.3. Calculate Som
2
+1
10.4. Calculate Sgm
=
2
+1
m Xm,m2
+1
m Ym,m2
= 1.
= 1.
+1
+1
(1 − Sw,m
)L+1
m2 ξg,m2
2
+1
+1 +1
L+1
m2 ξg,m2 + Vm2 ξo,m2
+1
+1
(1 − Sw,m
)Vm+1
ξo,m
2
2
2
+1
+1 +1
L+1
m2 ξg,m2 + Vm2 ξo,m2
11. In psim new primary from flash ell, update φm1 , φm2 , φt .
+1
as a nonlinear function of Pom
11.1. Calculate φ+1
t
1
#
#
+1
φ+1
=
φ
exp
C
×
P
−
P
φ
t
om1
ref
ref
+1
11.2. Calculate φ+1
m1 and φm2 :
92
(4.66)
4.8
φ+1
= φm1 ×
m1
φ+1
t
φt
(4.67)
φ+1
= φm2 ×
m2
φ+1
t
φt
(4.68)
Update WCO2
+1
+1
In psim primary iterate WCO2 ell, use an iterative method to calculate WCO
and ξw,m
. Up1
2 ,m1
+1 if necessary.
date Sw
+1
+1
The calculation of WCO
and ξw,m
follows the same procedure in
2
2 ,m2
psim primary iterate trap WCO2 ell.
+1
+1
+1
n , U +1 , U +1 , U n
n
1. Calculate P +1 , αn , α+1 , β n , β +1 , Um
t
m
WAT , UWAT , Zm,2ph , Zm,2ph , YCO2 , and
V +1 .
+1,e
+1,e
, YCO
, and V +1,e are updated.
2. Iterative calculation of WCO2 . Each loop, Zm,2ph
2 ,2ph
2.1. If β +1 < S , set WCO2 = 0, exit iterative loop.
+1,e
from the mass balance of the CO2 component equation:
2.2. Calculate WCO
2 ,mbal
+1,e
WCO
2 ,mbal
=
+1 − Z +1,e α+1 + Z n
n
n
n
Um
m,2ph α + WCO2 β
m,2ph
β +1
(4.69)
+1,e
:
2.3. Calculate the CO2 solubility, WCO
2 ,sol
+1,e
WCO
2 ,sol
⎧
+1,e
+1
+1
⎪
⎨ WCO2 [Po , YCO2 = 1] Sw > 1 − S =⇒ Sw ≈ 1
+1,e
=
[Pb ]] V +1,e < V =⇒ Sg ≈ 0
WCO2 [Pb+1 , YCO
2
⎪
⎩ W
+1 , Y +1,e ]
V +1,e ≥ V =⇒ Sg > 0
CO2 [Po
CO2
(4.70)
+1,e
+1,e +1,e
−
W
2.4. If WCO
CO2 ,sol < WCO2 , set WCO2 = WCO2 ,mbal , exit iterative loop.
2 ,mbal
+1,e
:
2.5. For m = 1 . . . NC − 1, compute Mm,2ph,m
1
+1,e
+1,e−1 +1
= Zm,2ph
α
Mm,2ph
(4.71)
+1,e
:
2.6. Calculate ΔMCO
2
+1,e
+1,e
+1,e
+1,e
+1,e
+1
=
M
−
M
=
β
−
W
W
ΔMCO
CO2 ,mbal
CO2 ,sol
CO2 ,mbal
CO2 ,sol
2
93
(4.72)
+1,e
+1,e
+1,e
+1,e
2.7. If WCO
< WCO
and |ΔMCO
| > MCO
2
2 ,mbal
2 ,sol
2 ,2ph
+1,e
+1,e
ΔMCO
= −MCO
2
2 ,2ph
(4.73)
+1,e
2.8. Update MCO
2 ,2ph
+1,e
+1,e
+1,e
MCO
= MCO
+ ΔMCO
2
2 ,2ph
2 ,2ph
2.9. If
(4.74)
+1,e
Mm
,2ph < , all mass is now in the water phase
m
+1 = 1
• Sw
+1,e
=1
• ZCO
2 ,2ph
Otherwise,
+1,e
Mm,2ph
+1,e
= +1,e
Zm,2ph
Mm ,2ph
(4.75)
m
+1,e
≤ 1 and
Ensure 0 ≤ Zm,2ph
+1,e
Zm
,2ph .
m
+1,e
+1,e
at Po+1 to calculate Xm
, Ym+1,e , ξo+1,e , ξg+1,e , L+1,e , V +1,e .
2.10. Flash Zm,2ph
> 1 − and ΔM +1,e > 0
2.11. If Sw
S
CO2
+1
Sw
⎧
+1,e
⎪
ΔMCO
⎪
2
⎪
1
−
, V +1,e < V
⎪
⎨
+1,e
ξo
=
+1,e
⎪
ΔMCO
⎪
⎪
2
⎪
, V +1,e ≥ V
⎩ 1−
ξg+1,e
+1
3. Calculate ξw
94
(4.76)
CHAPTER 5
TRAPPING FORMULATION
This chapter describes different mathematical formulation options for including trapping in a
compositional reservoir simulation model.
5.1
Trapping Variables
• m1 mobile oil
• m2 trapped oil
• m matrix
• f fracture
• Wm[m1 ] , Xm[m1 ] , Ym[m1 ] , Zm[m1 ] mole fractions
m2
m2
m2
m2
• ξo[m1 ] , ξg,[m1 ] , ξw,[m1 ] molar densities
m2
m2
m2
• Som , Sgm , Swm matrix saturations
• Som1 , Sgm1 , Swm1 mobile saturations
• Som2 , Sgm2 , Swm2 immobile, trapped, or bypassed saturations
• Mom,[m1 ] , Mgm,[m1 ] , Mwm,[m1 ] molar mass of each component in each phase
m2
m2
m2
• τomm1 /m2 , τgmm1 /m2 , τwmm1 /m2 transfer from mobile phase to trapped phase
• Vo,[m1 ] , Vg,[m1 ] , Vw,[m1 ] volumes of each phase
m2
m2
m2
• φm matrix porosity
• φm1 mobile matrix porosity
• φm2 trapped matrix porosity
• VR rock volume
95
5.2
Initialize Trapping
Initialize the properties for the total system:
1. Define the initial pressure at a specific depth Pom,ijk .
2. Define the initial water saturation for all grid cells Swm,ijk . The initial water saturation will
vary by rock type and depth.
3. Define the initial total composition Zmm as a constant in all grid cells.
4. Flash Zmm,ijk at Pom,ijk to calculate Xmm,ijk , Ymm,ijk , Wmm,ijk , ξom,ijk , ξgm,ijk , ξwm,ijk .
5. Compute the initial oil and gas saturation Som,ijk and Sgm,ijk based on lm,ijk , vm,ijk , ξom,ijk ,
ξgm,ijk , and Swm,ijk .
Set the properties of the mobile phase m1 to the total properties.
Pom1 ,ijk = Pom,ijk
Pom2 ,ijk = Pom,ijk
(5.1)
φm2 ,ijk = 0
(5.2)
Xmm1 ,ijk = Xmm,ijk
Xmm2 ,ijk = Xmm,ijk
(5.3)
Ymm1 ,ijk = Ymm,ijk
Ymm2 ,ijk = Ymm,ijk
(5.4)
Wmm2 ,ijk = Wmm,ijk
(5.5)
Som1 ,ijk = Som,ijk
Som2 ,ijk = 0
(5.6)
Sgm1 ,ijk = Sgm,ijk
Sgm2 ,ijk = 0
(5.7)
Swm1 ,ijk = Swm,ijk
Swm2 ,ijk = 0
(5.8)
ξom1 ,ijk = ξom,ijk
ξom2 ,ijk = ξom,ijk
(5.9)
ξgm1 ,ijk = ξgm,ijk
ξgm2 ,ijk = ξgm,ijk
(5.10)
ξwm1 ,ijk = ξwm,ijk
ξwm2 ,ijk = ξwm,ijk
(5.11)
φm1 ,ijk = φm,ijk
Wmm1 ,ijk = Wmm,ijk
new trap
new trap
new trap
, Sgm
, Swm
.
Specify the amount of trapping at initial conditions: Som
5.3
Update Trapping
This section describes the procedure for transferring mass between the mobile and trapped
phases. Portions of this procedure were also used to initialize the trapped and mobile saturations.
5.3.1
Input
At the time step level, update the amount of trapping. This may happen at initialization,
at specific transitions like the end of the waterflood or each WAG cycle. Trapping may also be
96
updated when a saturation switches from increasing to decreasing or increasing to decreasing for
any specific grid cell at time n. Trapping could also be updated at each timestep n.
When an incremental amount of trapping occurs, this transfers mass from the mobile m1 phase
to the immobile m2 phase. The amount of mass transfer is based on the newly trapped saturation
new trap
new trap
new trap
, Som
, or Swm
. The densities and mole fractions are based on the upstream
Sgm
properties from m1 .
5.3.2
Mass at Time n
The molar mass for each component and each phase is defined as follows:
n
= X n m1 ξ n m1 S n m1 φnm1 VR
Mom
1
m[m2 ] o[m2 ] o[m2 ] [m2 ]
[m
m2 ]
n
= Y n m1 ξ n m1 S n m1 φnm1 VR
Mgm m1
m[m2 ] g [m2 ] g [m2 ] [m2 ]
[m2 ]
n
Mwm m1 = W n m1 ξ n m1 S n m1 φnm1 VR
m[m2 ] w [m2 ] w [m2 ] [m2 ]
[m2 ]
(5.12)
(5.13)
(5.14)
The transfer from the mobile phases to the trapped phases happens when a saturation switches
from increasing to decreasing.
5.3.3
n
τomm
1 /m2
n
= Xmm
ξ n S new trap φnm VR
1 om1 om
(5.15)
n
τgmm
1 /m2
n
τwmm1 /m2
n ξ n S new trap φn VR
= Ymm
m
1 gm1 gm
new trap n
n
n
= Wmm1 ξwm1 Swm
φm VR
(5.16)
(5.17)
Transfer Mass
If oil is trapped, adjust the oil phase masses in the following way:
new
n
n
= Momm
− τomm
Momm
1
1
1 /m2
new
n
n
Momm
= Momm
+ τomm
2
2
1 /m2
(5.18)
If gas is trapped, adjust the gas phase masses in the following way:
new
n
n
= Mgmm
− τgmm
Mgmm
1
1
1 /m2
new
n
n
Mgmm
= Mgmm
+ τgmm
2
2
1 /m2
(5.19)
If water is trapped, adjust the gas phase masses in the following way:
new
n
n
= Mwmm
− τwmm
Mwmm
1
1
1 /m2
new
n
n
Mwmm
= Mwmm
+ τwmm
2
2
1 /m2
97
(5.20)
5.3.4
Update Mole Fractions
as follows:
If oil or gas is trapped, define Z newm1
m[m2 ],hc
new
Zm=1...N
m1
C −2,[m2 ],hc
M newm1 + M newm1
om[ ]
gm[m2 ]
= N −1 m2
C
new
new
Mom
1 + Mgm m1
[m
[ m2 ]
m2 ]
new
Zm=N
C −1,
= 1−
1
[m
m2 ],hc
N
C −2
m=1
new
Zm
(5.21)
1
[m
m2 ],hc
m=1
new
Flash Z newm1
, ξ new
at P n to calculate X newm1 , Y new
m1 , and ξg m1 .
1
m[m2 ],hc
m[m2 ]
m[m
o
]
[
]
[m2 ]
m2
m2
new
If water is trapped, define W m1 as follows:
m[m2 ]
M new m1
wm[m2 ]
new
Wm
1 =
[m
m2 ]
NC
m=NC −1
(5.22)
new
Mwm
1
[m
m2 ]
−→ ξ new
.
Compute the aqueous density using W new
1
1
m[m
w [m
m2 ]
m2 ]
5.3.5
Compute the Volumes
The mobile and immobile volumes in each phase are calculated as follows:
NC
Vonew
1 =
[m
m2 ]
5.3.6
m=1
new
Mom
1
[m
m2 ]
ξ new
1
o[m
m2 ]
NC
Vgnew
1 =
[m
m2 ]
m=1
new
Mgm
1
[m
m2 ]
ξ new
1
g [m
m2 ]
NC
Vwnew
1 =
[m
m2 ]
m=1
new
Mwm
1
[m
m2 ]
ξ new
1
w [m
m2 ]
(5.23)
Compute the Saturations
Compute the saturations based on the volume fractions:
V new
1
o[m
m2 ]
= new
+ V new
V m1 + V new
1
1
o[m2 ]
g [m
w [m
m2 ]
m2 ]
V new
1
g [m
m2 ]
new
Sg m1 = new
new
[m2 ]
V m1 + V m1 + V new
1
o[m2 ]
g [m2 ]
w [m
m2 ]
V new
1
w [m
m2 ]
new
Sw m1 = new
[m2 ]
+ V new
V m1 + V new
1
1
o[m2 ]
g [m
w [m
m2 ]
m2 ]
Sonew
1
[m
m2 ]
(5.24)
(5.25)
(5.26)
Note that these saturation definitions have the following implications:
98
new
new
Sonew
1 + Sg m 1 + Sw m 1 = 1
[m
[m2 ]
[m2 ]
m2 ]
(5.27)
The total matrix saturations are also computed based on the volume fractions:
new + V new
Vom
om2
1
new + V new + V new + V new + V new + V new
Vom
gm1
wm1
om2
gm2
wm2
1
new + V new
Vgm
gm2
1
= new
new + V new + V new + V new + V new
Vom1 + Vgm
wm1
om2
gm2
wm2
1
new + V new
Vwm
wm
1
2
= new
new + V new + V new + V new + V new
Vom1 + Vgm
wm
om
gm2
wm2
1
1
2
new
=
Som
(5.28)
new
Sgm
(5.29)
new
Swm
(5.30)
Note that these saturation definitions have the following implications:
new
new
new
+ Sgm
+ Swm
=1
Som
(5.31)
The mobile and immobile porosities are calculated based on volume fractions.
new + V new + V new
Vom
gm1
wm1
1
φ
new
new
new
new
new + V new m
Vom1 + Vgm1 + Vwm1 + Vom2 + Vgm
wm2
2
new + V new + V new
Vom
gm2
wm2
2
= new
φ
new + V new + V new + V new + V new m
Vom1 + Vgm
wm1
om2
gm2
wm2
1
=
φnew
m1
(5.32)
φnew
m2
(5.33)
Note that these porosity definitions have the following implications:
φm1 + φm2
5.4
= φm
(5.34)
Som1 φm1 + Som2 φm2
= Som φm
(5.35)
Sgm1 φm1 + Sgm2 φm2
= Sgm φm
(5.36)
Swm1 φm1 + Swm2 φm2 = Swm φm
(5.37)
Single Porosity Irreversible Trapping
A system with irreversible trapping can be handled as a dual porosity system with a mobile
m1 pore system and an immobile m2 pore system. Hysteresis and trapping are handled in between
time steps as a separate calculation, so there is no transfer term in (5.38). Fluids become trapped
if their saturation changes from decreasing or constant to increasing.
99
n
n
mm1 ξom1 krom1 #
+1
n
km1 (∇Pom
− γom
∇D# ) +
1
1
n
μom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m
om
cgom
gm
1
1
1
1
μngm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m
om
cowm
wm
1
1
1
1
μnwm1
n
n
n
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 =
Xmm
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
0.006328 VR ∇ ·
Xn
VR +1 +1 +1 +1
+1 +1 +1
+1
+1 +1 +1
Y
S
ξ
+
φ
W
S
ξ
φm1 Xmm1 Som1 ξom1 + φ+1
m1
mm1 gm1 gm1
m1
mm1 wm1 wm1 −
Δt
VR n n
n
n
n
n
n
n
n
n
n
n
ξ
+
φ
Y
S
ξ
+
φ
W
S
ξ
φm1 Xmm1 Som
m1 mm1 gm1 gm1
m1 mm1 wm1 wm1
1 om1
Δt
(5.38)
• m1 mobile oil: unless otherwise specified, all m1 properties are updated using (5.38) and the
trapping update procedure described in Section 5.3.
• m2 trapped oil: unless otherwise specified, all m2 properties are updated only using the
trapping update procedure described in Section 5.3.
• Som , Sgm , Swm : matrix saturations can be calculated using (5.35)–(5.37).
• λom1 = krom1 /μom1 ; λgm1 = krgm1 /μgm1 ; λwm1 = krwm1 /μwm1
• krom1 , krgm1 , krwm1 , Pcowm1 , Pcgom1 are all calculated using the total matrix saturations Som ,
Sgm , and Swm . This assumes that the trapping is representative of effects that are smaller than
the core-scale so that core measurements yield krom rather than krom1 . This could work with
commercial simulators only if the endpoints are adjusted based on the trapped saturations.
• φm , φm1 , φm2 : this formulation ignores the compressibility of the m2 portion of the porosity.
If we add the φm2 Xmm2 Som2 ξom2 and similar terms to the right hand side of (5.38), then we
have too many unknowns for the number of equations. If we change the right hand side terms
to φm Xmm Som ξom then the formulation has an inconsistent mass balance.
5.5
Dual Porosity as Reversible Trapping
A system with reversible trapping can be handled as a dual porosity system with a mobile m1
pore system and an immobile m2 pore system. Hysteresis and trapping are handled in between
100
time steps as a separate calculation and also as a transfer function. Fluids become trapped if their
saturation changes from decreasing or constant to increasing. Fluids can also move from m1 to m2
if the potential Ψm1 > Ψm2 . Fluids can move from m2 to m1 if the potential Ψm2 > Ψm1 . (5.39)
represents the m1 pore system.
n
n
mm1 ξom1 krom1 #
+1
n
#
k
(∇P
−
γ
∇D
)
+
m
om
om
1
1
1
μnom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m1
om1
cgom1
gm1
μngm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m1
om1
cowm1
wm1
μnwm1
+1
n
n
n
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 − τmm
=
Xmm
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
1 /m2
0.006328 VR ∇ ·
Xn
VR +1 +1 +1 +1
+1 +1 +1
+1
+1 +1 +1
φm1 Xmm1 Som1 ξom1 + φ+1
m1 Ymm1 Sgm1 ξgm1 + φm1 Wmm1 Swm1 ξwm1 −
Δt
VR n n
n
n
n
n
n
n
n
n
n
n
φm1 Xmm1 Som
ξ
+
φ
Y
S
ξ
+
φ
W
S
ξ
m1 mm1 gm1 gm1
m1 mm1 wm1 wm1
1 om1
Δt
(5.39)
(5.40) represents the m2 pore system.
+1
τmm
=
1 /m2
VR +1 +1 +1 +1
+1 +1 +1
+1
+1 +1 +1
φm2 Xmm2 Som2 ξom2 + φ+1
m2 Ymm2 Sgm2 ξgm2 + φm2 Wmm2 Swm2 ξwm2 −
Δt
VR n n
n
n
n
n
n
n
n
n
n
n
φm2 Xmm2 Som
ξ
+
φ
Y
S
ξ
+
φ
W
S
ξ
om
m
mm
gm
gm
m
mm
wm
wm
2
2
2
2
2
2
2
2
2
2
Δt
(5.40)
Evaluate Pcgom1 , Pcgom2 , Pcowm1 , and Pcowm2 using the total saturations. This means there
is no capillary pressure difference between the trapped phase and the mobile phase. Given this
assumption, the transfer function is defined by:
+1
+1
+1
×
τmm
= 0.006328 VR σm#1 /m2 km#1 /m2 Pom
− Pom
1
2
1 /m2
up,n
up,n
up,n
up,n
k up,n
ξ up,n k up,n
ξ up,n k up,n
Xmm1 /m2 ξom
Ymm
Wmm
1 /m2 rom1 /m2
1 /m2 gm 1 /m2 rgm1 /m2
1 /m2 wm1 /m2 rwm1 /m2
+
+
μup,n
μup,n
μup,n
om1 /m2
gm1 /m2
wm1 /m2
(5.41)
• m1 mobile oil: unless otherwise specified, all m1 properties are updated using (5.39)–(5.41)
and the trapping update procedure described in Section 5.3.
101
• m2 trapped oil: unless otherwise specified, all m2 properties are updated using (5.39)–(5.41)
and the trapping update procedure described in Section 5.3.
• Som , Sgm , Swm : matrix saturations can be calculated using (5.35)–(5.37).
• λom1 = krom1 /μom1 ; λgm1 = krgm1 /μgm1 ; λwm1 = krwm1 /μwm1
• kro[m1 ] , krg[m1 ] , krw[m1 ] , Pcow[m1 ] , Pcgo[m1 ] : there are four options:
m2
m2
m2
m2
m2
– Option 1, use total matrix saturations Som , Sgm , and Swm . This assumes that the
trapping is representative of effects that are smaller than the core-scale so that core
measurements yield krom , krgm , and krwm . If this option is used, note that Pcowm1 =
Pcowm2 , Pcgom1 = Pcgom2 , krom1 = krom2 , krgm1 = krgm2 , and krwm1 = krwm2 .
– Option 2, use Som1 , Sgm1 , and Swm1 to calculate krom1 , krgm1 , krwm1 , Pcowm1 , and Pcgom1 .
Use Som2 , Sgm2 , and Swm2 to calculate krom2 , krgm2 , krwm2 , Pcowm2 , and Pcgom2 . This
assumes that the trapping, or bypassing, is representative of effects that are between the
core-scale and the reservoir grid scale. This means that core measurements represent
krom1 , krgm1 , and krwm1 and there is no direct measurement of m2 .
– Option 3, reset all endpoints and then use Som1 , Sgm1 , and Swm1 to calculate krom1 , krgm1 ,
krwm1 , Pcowm1 , and Pcgom1 . Use Som2 , Sgm2 , and Swm2 to calculate krom2 , krgm2 , krwm2 ,
Pcowm2 , and Pcgom2 . The difficulty with this method is determining how to adjust the
endpoints. For the m2 system, one approach is to assume all the endpoints are 0.
– Option 4, assume Pcgom2 = 0 and Pcowm2 = 0.
– Option 2, 3, and 4 are possible in commercial simulators, although they ignore the effects
of Section 5.3.
• φm , φm1 , φm2 : this formulation considers the compressibility of the m2 portion of the porosity.
• km1 /m2 ; calculate equivalent k to a diffusion system.
−3 kg/(ms)
D 10−3 cm2 /s μo cp · 10 cp
md
≈ 5 · 10−5 md (5.42)
k [md] =
−12 cm2
6.894·106 kg/(ms2 )
9.869
·
10
kro P psia
psia
102
• Define σm1 /m2 as follows:
σm1 /m2 = 4
1
1
1
+
+
2
2
min(DX/2, 5)
min(DY/2, 5)
min(DZ/2, 1)2
=4
1
1
1
+ 2+ 2
2
5
5
1
(5.43)
• Upstream weighting for all properties in the transfer function (5.41).
– Pom2 > Pom1 then m2 else m1
– Pom2 − Pcgom2 > Pom1 − Pcgom1 then m2 else m1
– Pom2 + Pcowm2 > Pom1 − Pcowm1 then m2 else m1
5.6
Dual Porosity Computation Options
This section describes the computation of the Um and Jacobian matrix for the dual porosity
option. Several definitions will help simplify the notation.
Tmn1 /m2 ,mo
Tmn1 /m2 ,mg
Tmn1 /m2 ,mw
5.6.1
= 0.006328 VR
#
σm
k#
1 /m2 m1 /m2
= 0.006328 VR
#
σm
k#
1 /m2 m1 /m2
= 0.006328 VR
#
σm
k#
1 /m2 m1 /m2
×
×
×
up,n
Xmm
ξ up,n kup,n
1 /m2 om1 /m2 rom1 /m2
μup,n
om1 /m2
up,n
Ymm
ξ up,n kup,n
1 /m2 gm1 /m2 rgm1 /m2
μup,n
gm1 /m2
up,n
Wmm
ξ up,n kup,n
1 /m2 wm1 /m2 rwm1 /m2
μup,n
wm1 /m2
(5.44)
(5.45)
(5.46)
Accm1 =
VR φm1 Xmm1 Som1 ξom1 + φm1 Ymm1 Sgm1 ξgm1 + φm1 Wmm1 Swm1 ξwm1
Δt
(5.47)
Accm2 =
VR φm2 Xmm2 Som2 ξom2 + φm2 Ymm2 Sgm2 ξgm2 + φm2 Wmm2 Swm2 ξwm2
Δt
(5.48)
Implicit Pm2
For the implicit calculation of Pm2 , both Pm1 and Pm2 are evaluated at + 1. The m1 and m2
equations are fully coupled; Pm1 and Pm2 appear in both the m1 and the m2 equations.
Component equations for m1 system.
103
n
n
mm1 ξom1 krom1 #
+1
n
km1 (∇Pom
− γom
∇D# ) +
1
1
n
μom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m
om
cgom
gm
1
1
1
1
μngm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m
om
cowm
wm
1
1
1
1
μnwm1
n
n
n
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 +
Xmm
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
0.006328 VR ∇ ·
Xn
+1
τmm
/m
1
2
− Tmn1 /m2 ,mo + Tmn1 /m2 ,mg + Tmn1 /m2 ,mw Pm 1 + δPm1 − Pm 2 − δPm2 =
n
Acc+1
m1 − Accm1
(5.49)
Component equations for the m2 system.
5.6.2
+1
τmm
/m
1
Tmn1 /m2 ,mo
+
Tmn1 /m2 ,mg
2
+
Tmn1 /m2 ,mw
n
Pm 1 + δPm1 − Pm 2 − δPm2 = Acc+1
m2 − Accm2
(5.50)
Explicit Pm2
For the explicit calculation of Pm2 , Pm1 is evaluated at + 1 and Pm2 is evaluated at n. This
decouples the m1 and m2 equations; the m1 equation still requires a global matrix solve, but the m2
equations are now a local matrix solve.
Component equations for m1 system.
n
n
mm1 ξom1 krom1 #
+1
n
km1 (∇Pom
− γom
∇D# ) +
1
1
n
μom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m
om
cgom
gm
1
1
1
1
μngm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m
om
cowm
wm
1
1
1
1
μnwm1
n
n
n
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 +
Xmm
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
0.006328 VR ∇ ·
Xn
+1
τmm
/m
1
2
− Tmn1 /m2 ,mo + Tmn1 /m2 ,mg + Tmn1 /m2 ,mw Pm 1 + δPm1 − Pmn2 =
n
Acc+1
m1 − Accm1
104
(5.51)
Component equations for the m2 system.
5.6.3
+1
τmm
/m
1
Tmn1 /m2 ,mo
+
Tmn1 /m2 ,mg
+
2
Tmn1 /m2 ,mw
n
Pm 1 + δPm1 − Pmn2 = Acc+1
m2 − Accm2
(5.52)
Implicit τ = 0
For the implicit calculation of τ = 0, Pm1 is evaluated at + 1 and Pm2 is not used as a primary
variable. This means that Pm1 = Pm2 . The accumulation term evaluated at is used in the m1
equations to account for the changes in compressibility of the system. The accumulation term is
evaluated at instead of + 1 because it is difficult to calculate the derivatives
∂Accm2 ∂Accm2
∂Som1 , ∂Xm m
1
, and
∂Accm2
∂Ym m
1
∂Accm2 ∂Accm2
∂Pm1 , ∂Swm1 ,
.
Component equations for m1 system.
n
n
mm1 ξom1 krom1 #
+1
n
km1 (∇Pom
− γom
∇D# ) +
1
1
n
μom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m
om
cgom
gm
1
1
1
1
μngm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m1
om1
cowm1
wm1
μnwm1
n
n
n
Xmm
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 =
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
n
Acc+1
m1 − Accm1
0.006328 VR ∇ ·
5.6.4
Xn
+ Accm2 − Accnm2
(5.53)
Explicit τ = 0
For the explicit calculation of τ = 0, Pm1 is evaluated at + 1 and Pm2 is not used as a primary
variable. This means that Pm1 = Pm2 . The m2 accumulation term is ignored for the m1 equations,
which means the compressibility of the m2 system is ignored.
Component equations for m1 system.
105
n
n
mm1 ξom1 krom1 #
+1
n
km1 (∇Pom
− γom
∇D# ) +
1
1
n
μom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m
om
cgom
gm
1
1
1
1
μngm1
W n ξ n kn
mm1 wm1 rwm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
−
∇P
−
γ
∇D
)
+
m
om
cowm
wm
1
1
1
1
μnwm1
n
n
n
ξ n q +1 + Ymm
ξ n q +1 + Wmm
ξ n q +1 =
Xmm
1 om1 om1
1 gm 1 gm1
1 wm1 wm1
0.006328 VR ∇ ·
Xn
n
Acc+1
m1 − Accm1
5.7
(5.54)
Computation of the Solution of a Dual Porosity System
The way of solving a dual porosity compositional system is similar to the way of solving a single
porosity system described in Chapter 4, with the following differences. Use (5.39)–(5.41) instead
of (3.1). This leads to twice the number of equations and primary variables as the single porosity
system. The order of the equations in the LU decomposition is slightly different. At each step,
both the m1 and the m2 properties are calculated, rather than only the m1 properties.
The process involves the following steps:
1. Assemble the Jacobian. The ordering of the primary variables and equations listed here is
for the gas/oil/water system with WCO2 as a primary variable. The simplified systems for
gas/oil, gas/water, water only, both with and without WCO2 as a primary variable can be
created by deleting the rows which aren’t applicable.
• The primary variables are ordered as follows: Pom1 , Pom2 , Swm1 , Swm2 , WCO2 m1 , WCO2 m2 ,
Som1 , Xm=1,m1 , Xm=2,m1 , Xm=3,m1 (up to Xm=NC −2,m1 ), Ym=1,m1 , Ym=2,m1 , Ym=3,m1
(up to Ym=NC −2,m1 ), Som2 , Xm=1,m2 , Xm=2,m2 , Xm=3,m2 (up to Xm=NC −2,m2 ), Ym=1,m2 ,
Ym=2,m2 , Ym=3,m2 (up to Ym=NC −2,m2 )
• The equations are ordered as follows: CH2 O,m1 , CH2 O,m2 , CCO2 ,m1 , CCO2 ,m2 , Gg/w,CO2 ,m1 ,
Gg/w,CO2 ,m2 , Cm=1,m1 , Cm=2,m1 , Cm=3,m1 (up to CNC −2,m1 ), Gm=1,m1 , Gm=2,m1 , Gm=3,m1 ,
Gm=4,m1 (up to GNC −1,m1 ), Cm=1,m2 , Cm=2,m2 , Cm=3,m2 (up to CNC −2,m2 ), Gm=1,m2 ,
Gm=2,m2 , Gm=3,m2 , Gm=4,m2 (up to GNC −1,m2 )
n , U n , αn , αn , β n , β n .
2. Calculate Umm
mm2
m1
m2
m1
m2
1
106
[+1]
3. Solve the matrix equation for Pom1 .
[+1]
[+1]
[+1]
[+1]
[+1]
4. Back substitute to calculate Pom2 , Swm1 , Swm2 , WCO2 m1 , WCO2 m2
[+1]
[+1]
[+1]
[+1]
[+1]
5. Compute Umm1 , Umm2 , Utm1 , and Utm2 as a function of Pom1
[+1]
6. Compute φm1
[+1]
and φm2
[+1]
[+1]
and Pom2 .
[+1]
as a function of Pom1 and Pom2
[+1]
[+1]
[+1]
7. If WCO2 is a secondary variable, compute WCO2 m1 and WCO2 m2 as a function of Pom1
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
and
[+1]
Pom2 . Compute ξwm1 and ξwm2 as a function of Pom1 , Pom2 , WCO2 m1 , and WCO2 m2 .
[+1]
and βm2
[+1]
and αm2
8. Compute βm1
9. Compute αm1
[+1]
as a function of φm1 , Swm1 , ξwm1 , φm2 , Swm2 , and ξwm2 .
[+1]
as a function of Utm1 , βm1 , Utm2 , and βm2 .
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
10. Compute Z2ph,m,m1 and Z2ph,m,m2 as a function of Umm1 , αm1 , Umm2 , and αm2 .
[+1]
[+1]
11. Flash Z2ph,m,m1 at Pom1
[+1]
Pom2
[+1]
[+1]
[+1]
[+1]
[+1]
[+1]
to calculate Xmm2 , Ymm2 , ξom2 , and ξgm2 .
[+1]
[+1]
[+1]
[+1]
[+1]
to calculate Xmm1 , Ymm1 , ξom1 , and ξgm1 . Flash Z2ph,m,m2 at
[+1]
[+1]
12. Calculate Som1 , Sgm1 , Som2 , Sgm2 .
107
CHAPTER 6
TIME DERIVATIVES FORMULATION
All the accumulation terms are local to a specific cell. The notation in this section uses i to
represent this cell. This applies equally well to a 1D, 2D, or 3D cell.
The accumulation term
∂
∂Accmi
=
φi ξoi Soi Xmi + φi ξgi Sgi Ymi + φi ξwi Swi Wmi
∂t
∂t
(6.1)
Evaluate the Taylor series expansion of (6.1).
n
Accn+1
1 ∂Accmi
+1
n
mi − Accmi
=
Acc
≈
−
Acc
mi
mi
∂t
tn+1 − tn
Δt
(6.2)
Expand Acc+1
mi for NC components; all terms are evaluated for cell i and component m.
Acc+1
mi
=
N
C −2
∂Accmi
∂Accmi
∂Accmi
Accmi +
δPi +
δSoi +
δSgi +
∂Pi
∂Soi
∂Sgi
m =1
∂Accmi
∂Accmi
δXm i +
δYm i
∂Xm i
∂Ym i
(6.3)
6.1
Pressure Derivatives
For the normal hydrocarbon components,
∂Accmi
∂P ,
for cell i and component m = 1 . . . NC − 2.
∂ξgi
∂Accmi
∂φi
∂φi
∂ξoi
= ξoi
+ ξgi
+ φi Soi
+ φi Sgi
Soi Xmi
Sgi Ymi
Xmi
Ymi
∂P
∂P
∂P
∂P
∂P
For the CO2 component,
∂Accmi
∂P ,
for cell i and component m = NC − 1.
∂Accmi
∂φi
∂φi
∂φi
= ξoi
+ ξgi
+ ξwi
+
Soi Xmi
Sgi Ymi
Swi
Wmi
∂P
∂P
∂P
∂P
∂WCO
∂ξgi
2 ,i
∂ξoi
∂ξwi
+ φi Sgi Ymi
+ φi Swi Wmi
+ φi ξwi Swi
φi Soi Xmi
∂P
∂P
∂P
∂P
For the H2 O component,
∂Accmi
∂P ,
(6.4)
for cell i and component m = NC .
108
(6.5)
∂WCO2 ,i
∂Accmi
∂φi
∂ξwi
= ξwi
+ φi Swi
− φi ξwi
Swi
Wmi
Wmi
Swi
∂P
∂P
∂P
∂P
(6.6)
The porosity increases with depth at constant overburden stress.
φ[P ] = φref · exp [Cφ · (P − Pref )] ≈ φref · (1 + Cφ · (P − Pref ))
(6.7)
Use the definition of Cφ .
Cφ =
6.2
1 ∂φ
φ ∂P
∂φ
= φCφ
∂P
(6.8)
Saturation Derivatives
Evaluate
∂Accmi
∂So .
∂Accmi
= φi ξoi
Xmi
− φi ξgi
Ymi
∂So
Evaluate
(6.9)
∂Accmi
∂Sw .
∂Accmi
= φi ξwi
Wmi
− φi ξgi
Ymi
∂Sw
(6.10)
Above the bubble point, Sg = 0 and Sg → Pb becomes a new primary variable and
Below the dew point, So = 0 and So → Pd becomes a new primary variable and
6.3
∂Accmi
∂Pd
∂Accmi
∂Pb
= 0.
=0
Composition Derivatives
For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m = 1 . . . NC − 2,
evaluate
∂Accmi
.
∂Xm
∂Accmi
∂ξoi
= φi Soi
Xmi
+ φi ξoi
Soi δm,m
∂Xm
∂Xm
(6.11)
For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m = 1 . . . NC − 2,
evaluate
∂Accmi
.
∂Ym
∂ξgi
∂Accmi
= φi Sgi Ymi
+ φi ξgi
Sgi δm,m
∂Ym
∂Ym
(6.12)
109
For the CO2 component equation Cm = CNC −1 and m = 1 . . . NC − 2, evaluate
∂Accmi
.
∂Xm
∂AccCO2 ,i
∂ξoi
∂ξwi
∂WCO2
= φi Soi
XCO
−
φ
S
ξ
+
φ
S
W
+ φi Swi
ξwi
CO
,i
i
oi
oi
i
wi
2
2
∂Xm
∂Xm
∂Xm
∂Xm
For the CO2 component equation Cm = CNC −1 and m = 1 . . . NC − 2, evaluate
∂Accmi
.
∂Ym
∂ξgi
∂Accmi
∂ξwi
∂WCO2
= φi Sgi
YCO
−
φ
S
ξ
+
φ
S
W
+ φi Swi
ξwi
i gi gi
i wi CO2
2
∂Ym
∂Ym
∂Ym
∂Ym
For the water component equation Cm = CNC and m = 1 . . . NC − 2, evaluate
(6.14)
∂Accmi
.
∂Xm
∂AccH2 O,i
∂ξ ∂WCO2
= φi Swi
WH2 O wi − φi Swi
ξwi
∂Xm
∂Xm
∂Xm
For the water component equation Cm = CNC and m = 1 . . . NC − 2, evaluate
∂AccH2 O,i
∂ξ ∂WCO2
= φi Swi
WH2 O wi − φi Swi
ξwi
∂Ym
∂Ym
∂Ym
110
(6.13)
(6.15)
∂Accmi
.
∂Ym
(6.16)
CHAPTER 7
SPACE DERIVATIVES FORMULATION
This chapter describes the mathematical expansion of the spatial derivatives in the finite difference expansion of the partial differential equations.
7.1
Initial Expansion
This section assumes implicit pressure, explicit saturation, and explicit composition. Start with
(3.1); multiply through by the rock volume VRi = Δxi Δyi Δzi .
0.006328 · VRi ∇ · Xm ξo λo k(∇Po − γo ∇D) +
0.006328 · VRi ∇ · Ym ξg λg k(∇Po + ∇Pcgo − γg ∇D) +
0.006328 · VRi ∇ · Wm ξw λw k(∇Po − ∇Pcow − γw ∇D) + Xm ξo qo + Ym ξg qg + Wm ξw qw =
∂
φ(Xm So ξo + Ym Sg ξg + Wm Sw ξw )
(7.1)
VRi
∂t
Write the finite difference expansion of Um . Each of the terms are labeled.
LHS1
n
Um
(P n+1 )
= (Xm ξo qo + Ym ξg qg + Wm ξw qw )n
i
LHS2
(Xm ξo λo kxx )n
i+ 1
n+1
· Pi+1
− Pin+1
− Δyi Δzi
2
+ Δyi Δzi
Δxi+ 1
LHS3
(Xm ξo λo kxx )n
i− 1
2
Δxi− 1
2
LHS4
(Ym ξg λg kxx )n
i+ 1
Δxi+ 1
n+1
· Pi+1
− Pin+1
− Δyi Δzi
(Ym ξg λg kxx )n
i− 1
2
Δxi− 1
(Wm ξw λw kxx )n
i+ 1
Δxi+ 1
·
n+1
Pi+1
−
Pin+1
− Δyi Δzi
− Δyi Δzi
2
Δxi− 1
(Xm ξo λo kxx )n
i+ 1
2
2
n+1
· Pin+1 − Pi−1
2
LHS8
Δxi+ 1
LHS7
(Wm ξw λw kxx )n
i− 1
2
n+1
· Pin+1 − Pi−1
2
LHS6
2
+ Δyi Δzi
· Pin+1 − Pi−1
LHS5
2
n+1
2
2
+ Δyi Δzi
n
γo,i+
− Δyi Δzi
1 · (Di+1 − Di )
2
LHS9
(Xm ξo λo kxx )n
i− 1
2
Δxi− 1
2
n
γo,i−
1 · (Di − Di−1 )
2
+ · · · Continued in next equation · · ·
111
(7.2)
n
Um
(P n+1 ) = · · · Continued from previous equation · · · +
− Δyi Δzi
LHS10
(Ym ξg λg kxx )n
i+ 1
2
Δxi+ 1
− Δyi Δzi
(Wm ξw λw kxx )n
i+ 1
2
+ Δyi Δzi
(Ym ξg λg kxx )n
i− 1
2
Δxi− 1
n
γw,i+
− Δyi Δzi
1 · (Di+1 − Di )
(Wm ξw λw kxx )n
i− 1
2
Δxi− 1
n
n
· Pcgo,i+1
− Pcgo,i
− Δyi Δzi
− Δyi Δzi
n
γw,i−
1 · (Di − Di−1 )
2
(Ym ξg λg kxx )n
i− 1
2
Δxi− 1
n
n
· Pcgo,i
− Pcgo,i−1
2
LHS16
(Wm ξw λw kxx )n
i+ 1
2
Δxi+ 1
LHS15
2
2
2
LHS14
Δxi+ 1
n
γg,i−
1 · (Di − Di−1 )
LHS13
2
(Ym ξg λg kxx )n
i+ 1
2
2
LHS12
Δxi+ 1
LHS11
2
2
n
γg,i+
− Δyi Δzi
1 · (Di+1 − Di )
2
·
n
Pcow,i+1
−
n
Pcow,i
LHS17
(Wm ξw λw kxx )n
i− 1
2
+ Δyi Δzi
Δxi− 1
2
n
n
· Pcow,i
− Pcow,i−1
2
(7.3)
7.2
Transmissibility
There are several transmissibilities used in this formulation. The following is an example in the
x-direction, where X = X|Y |W , ξo = ξo |ξg |ξw , μo = μo |μg |μw , and ± is either positive or negative
everywhere.
mn
Txo,i±
1
2
= 0.006328 · Δyi Δzi
n
n
Xm,i±
kn
k#
1ξ
o,i± 1 ro,i± 1 xx,i± 1
2
2
2
2
(7.4)
μno,i± 1 Δx#
i± 21
2
This is an example in the y-direction:
mn
Tyo,j±
1 = 0.006328 · Δxj Δzj
n
n
Xm,j±
kn
k#
1ξ
o,j± 1 ro,j± 1 yy,j± 1
2
2
2
2
(7.5)
#
μno,j± 1 Δyj±
1
2
2
2
This is an example in the z-direction:
mn
Tzo,k±
1
2
7.3
= 0.006328 · Δxk Δyk
n
n
Xm,k±
kn
k#
1ξ
o,k± 1 ro,k± 1 zz,k± 1
2
2
2
#
μno,k± 1 Δzk±
1
2
2
Expand Deltas
Approximate P n+1 .
112
2
(7.6)
P n+1 ≈ P +1
δP = P +1 − P (7.7)
P n+1 ≈ P + δP
7.4
Expand Terms on Left-Hand-Side
Substitute (7.7) into (7.2), LHS2 .
mn
Txo,i+
1
2
LHS2 =
· Pi+1
+ δPi+1 − Pi − δPi
(7.8)
Substitute (7.7) into (7.2), LHS3 .
mn
Txo,i−
1
2
LHS3 =
· Pi + δPi − Pi−1
− δPi−1
(7.9)
Substitute (7.7) into (7.2), LHS4 .
LHS4 =
mn
Txg,i+
·
Pi+1
+ δPi+1 − Pi − δPi
1
2
(7.10)
Substitute (7.7) into (7.2), LHS5 .
mn
Txg,i−
1
2
LHS5 =
· Pi + δPi − Pi−1
− δPi−1
(7.11)
Substitute (7.7) into (7.2), LHS6 .
mn
Txw,i+
1
2
LHS6 =
· Pi+1
+ δPi+1 − Pi − δPi
(7.12)
Substitute (7.7) into (7.2), LHS7 .
LHS7 =
mn
Txw,i−
1
2
· Pi + δPi − Pi−1
− δPi−1
(7.13)
Substitute (7.7) into (7.2), LHS8 .
LHS8 =
mn
n
Txo,i+
Di+1 − Di
1γ
1 ·
o,i+
2
(7.14)
2
113
Substitute (7.7) into (7.2), LHS9 .
mn
n
LHS9 = Txo,i− 1 γo,i− 1 · Di − Di−1
2
(7.15)
2
Substitute (7.7) into (7.2), LHS10 .
LHS10
mn
n
= Txg,i+ 1 γg,i+ 1 · Di+1 − Di
2
(7.16)
2
Substitute (7.7) into (7.2), LHS11 .
LHS11
mn
n
= Txg,i− 1 γg,i− 1 · Di − Di−1
2
(7.17)
2
Substitute (7.7) into (7.2), LHS12 .
LHS12 =
mn
n
Txw,i+
Di+1 − Di
1γ
1 ·
w,i+
2
(7.18)
2
Substitute (7.7) into (7.2), LHS13 .
LHS13 =
mn
n
Txw,i−
Di − Di−1
1γ
1 ·
w,i−
2
(7.19)
2
Substitute (7.7) into (7.2), LHS14 .
LHS14 =
mn
n
n
Txg,i+
·
Pcgo,i+1
− Pcgo,i
1
(7.20)
2
Substitute (7.7) into (7.2), LHS15 .
LHS15 =
mn
n
n
Txg,i−
·
Pcgo,i
− Pcgo,i−1
1
(7.21)
2
Substitute (7.7) into (7.2), LHS16 .
114
LHS16
mn
n
n
= Txw,i+ 1 · Pcow,i+1
− Pcow,i
(7.22)
2
Substitute (7.7) into (7.2), LHS17 .
LHS17
7.5
mn
n
n
= Txw,i− 1 · Pcow,i
− Pcow,i−1
(7.23)
2
Rearrange Terms
There are 24 + 20 terms in equations (7.8)–(7.23). These terms need to be rearranged in the
following way:
• Multiples of δP at i and i ± 1; these will end up in A in the matrix equation.
• Terms which do not multiply a δ; these will end up in R in the matrix equation.
The following are the multiples of δPi±1 . All ± are either positive or negative for this equation.
mn
mn
mn
=
T
+
T
+
T
DPmn
xt,i±1
xo,i± 1
xg,i± 1
xw,i± 1
2
2
(7.24)
2
The following are the multiples of δPi .
mn
mn
DPmn
xt,i = − DPxt,i+1 + DPxt,i−1 =
mn
mn
mn
mn
mn
mn
(7.25)
− Txo,i+
1 + T
1 + T
1 + T
1 + T
1 + T
1
xo,i−
xg,i+
xg,i−
xw,i+
xw,i−
2
2
2
2
2
2
The following do not multiply deltas. All ± are either positive or negative for this equation.
mn
n
mn
n
n
=
−T
·
P
−
γ
D
·
P
−
γ
D
+
P
−
T
DCmn
1
1
1
1
xt,i±1
i±1
i±1
cgo,i±1 +
xo,i± 2
o,i± 2 i±1
xg,i± 2
g,i± 2 i±1
mn
n
n
Pi±1
− γw,i±
(7.26)
− Txw,i±
1 ·
1 Di±1 − Pcow,i±1
2
The following do not multiply deltas.
115
2
mn
n
mn
n
DCmn
=
T
·
P
−
γ
D
·
P
−
γ
D
+
T
+
1
1
1
1
xt,i
i
i
xo,i+ 2
o,i+ 2 i
xo,i− 2
o,i− 2 i
mn
n
n
mn
n
n
Pi − γg,i+
Pi − γg,i−
+ Txg,i−
Txg,i+
1 ·
1 Di + Pcgo,i
1 ·
1 Di + Pcgo,i +
2
2
2
2
mn
n
n
mn
n
n
(7.27)
Txw,i+ 1 · Pi − γw,i+ 1 Di − Pcow,i + Txw,i− 1 · Pi − γw,i−
1 Di − Pcow,i
2
7.6
2
2
2
Combine Terms
The final form of Umx is:
mn
Uxi
=
mn
mn
DPmn
xt,i+1,jk δPi+1,jk + DPxt,ijk δPijk + DPxt,i−1,jk δPi−1,jk +
mn
mn
mn
−DCxt,i+1,jk − DCxt,ijk − DCxt,i−1,jk
(7.28)
The final form of Umy is:
mn
Uyi
=
mn
mn
DPmn
yt,ij+1,k δPij+1,k + DPyt,ijk δPijk + DPyt,ij−1,k δPij−1,k +
mn
mn
mn
−DCyt,ij+1,k − DCyt,ijk − DCyt,ij−1,k
(7.29)
The final form of Umz is:
mn
=
Uzi
mn
mn
DPmn
zt,ijk+1 δPijk+1 + DPzt,ijk δPijk + DPzt,ijk−1 δPijk−1 +
mn
mn
−DCmn
zt,ijk+1 − DCzt,ijk − DCzt,ijk−1
mn
n
n
n
= Umxi
+ Umyi
+ Umzi
+ (Xm ξo qo + Ym ξg qg + Wm ξw qw )ni
Uti
7.7
(7.30)
(7.31)
Upstream Weighting
At time n, it is necessary to evaluate which cells are upstream of other cells in order to calculate
the appropriate i ±
1
2
terms. This computation involves the following equations
#
n
n
n #
− γoi±1
Di±1
) − (Pin − γoi
Di )
Ψnoi± 1 = (Pi±1
2
#
n
n
n
n
n #
n
− γgi±1
Di±1
+ Pcgo,i±1
) − (Pin − γgi
Di + Pcgo,i
)
Ψgi± 1 = (Pi±1
2
#
n
n
n
n
n
n
− γwi±1
Di±1
− Pcow,i±1
) − (Pin − γwi
Di# − Pcow,i
)
Ψwi± 1 = (Pi±1
2
(7.32)
(7.33)
(7.34)
For normal evaluation, evaluate all the fluid properties at the upstream node. This applies to
the following properties.
n
n
n
n
, ξϕ,i±
• γϕ,i±
1, μ
1, k
ϕ,i± 1
rϕ,i± 1
2
2
2
2
116
n
n
n
• Xm,i±
, Wm,i±
1, Y
1
m,i± 1
2
2
2
The upstream weighting for all of these properties is defined by (7.35), using a generic variable
χ.
χnϕ,i± 1
2
=
χnϕ,i , χnϕ,i±1 U
χnϕ,i±1 Ψnϕ,i± 1 > 0
=
2
(7.35)
Ψnϕ,i± 1 ≤ 0
χnϕ,i
2
Evaluate the permeability as a weighted harmonic average:
kxx,i± 1
2
Δxi± 1
=
2
kxx,i kxx,i±1
,
Δxi Δxi±1
=
H
2
Δxi
kxx,i
+
(7.36)
Δxi±1
kxx,i±1
Using a combination of (7.35) and (7.36),
mn
Txo,i±
1
2
7.8
= 0.006328·Δyi Δzi ·
n ξ n kn
Xm
o ro
μno
,
i
n ξ n kn
Xm
o ro
μno
·
i±1
U
kxx
Δx
,
i
kxx
Δx
(7.37)
i±1
H
Time Steps
The maximum time step size is determined by the “CFL” constraint, based on the original
paper, Courant et al. (1967). The basic CFL constraint is defined for IMPES models by:
uΔt
≤1
Δx
(7.38)
For practical reasons, it is often better to use:
uΔt
≤ 0.1
Δx
(7.39)
The CFL constraint is defined for each phase across each boundary.
Δtn+1
x g
y
z
o
w
i± 12
≤
0.1φi VRi (ξoi Soi + ξgi Sgi + ξwi Swi )
mn
n T x g
Ψ x g
1
1
y o i±
y o i±
m
z
w
2
z
w
(7.40)
2
Another constraint is that the volumes moving through a grid cell at any time should also be
less than 10% cell pore volumes in a time step. For a fixed rate injector:
117
Δtn+1
well,i ≤
0.1φi VRi
|qt,w |
(7.41)
For a fixed pressure injector:
Δtn+1
well,i ≤
0.1φi VRi
n )|
| − WIi λnt,i (Pin − Pw,i
(7.42)
For a fixed rate producer:
Δtn+1
well,i ≤
0.1φi VRi
n |
|qt,i
(7.43)
For a fixed pressure producer:
Δtn+1
well,i ≤
0.1φi VRi
n ))|
|(−WIi λnt,i (Pin − Pw,i
(7.44)
The eventual time step sized used is based on the minimum across all the wells and all the
interfaces:
Δt
n+1
= MIN MIN[Δtn+1
well,w ],
∀w
, Δtn+1
, Δtn+1
, Δtn+1
, Δtn+1
, Δtn+1
],
MIN[Δtn+1
xg,i+ 21 ,j,k
xg,i− 12 ,j,k
xo,i+ 12 ,j,k
xo,i− 12 ,j,k
xw,i+ 21 ,j,k
xw,i− 21 ,j,k
∀ijk
, Δtn+1
, Δtn+1
, Δtn+1
, Δtn+1
, Δtn+1
],
MIN[Δtn+1
yg,i,j+ 1 ,k
yg,i,j− 1 ,k
yo,i,j+ 1 ,k
yo,i,j− 1 ,k
yw,i,j+ 1 ,k
yw,i,j− 1 ,k
∀ijk
2
2
2
2
2
2
, Δtn+1
, Δtn+1
, Δtn+1
, Δtn+1
, Δtn+1
]
MIN[Δtn+1
zg,i,j,k+ 21
zg,i,j,k− 21
zo,i,j,k+ 21
zo,i,j,k− 21
zw,i,j,k+ 21
zw,i,j,k− 21
∀ijk
118
(7.45)
CHAPTER 8
EQUATION OF STATE FORMULATION
All the fugacity terms are local to a specific cell. The notation in this section uses i to represent
this cell. This applies equally well to a 1D, 2D, or 3D cell.
The Gm = 1 . . . NC − 1 fugacity equations are:
n+1
fn+1
omi − fgmi = 0
(8.1)
Evaluate the Taylor series expansion for fn+1
omi :
fn+1
omi
≈
f+1
omi
=
fomi
N
C −2 ∂fomi
∂fomi
+
δPi +
δXm i
∂Pi
∂Xm i
(8.2)
m =1
Evaluate the Taylor series expansion for fn+1
gmi :
+1
fn+1
gmi ≈ fgmi
N
C −2
∂fgmi
= fgmi +
δPi +
∂Pi
m =1
8.0.1
∂fgmi
δYm i
∂Ym i
(8.3)
Expand Fugacities
The fugacities are defined by (8.4).
flm = Φlm Xm P
Evaluate
∂fl
mi
∂P ,
fvm = Φvm Ym P
(8.4)
m = 1 . . . NC − 1:
∂Φl
∂fl
mi
l
= Xmi
Pi mi + Φl
mi Xm = fmi
∂P
∂P
Evaluate
∂fv
mi
∂P ,
1 ∂Φl
mi
∂P
Φl
mi
+ Φl
mi Xm
(8.5)
+ Φv
mi Ymi
(8.6)
m = 1 . . . NC − 1:
∂Φv
∂fv
mi
v
= Ymi
Pi mi + Φv
mi Ymi = fmi
∂P
∂P
1 ∂Φv
mi
∂P
Φv
mi
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
119
∂fl
mi
∂Xm
for m = 1 . . . NC − 2:
∂Φl
∂fl
mi
l
mi
= Xmi
P
+ Φl
mi Pi δm,m = fmi
∂Xm
∂Xm
1 ∂Φl
mi
l
∂X
Φmi
m
+ Φl
mi P δm,m
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂Φv
∂fv
mi
v
= Ymi
Pi mi + Φv
mi Pi δm,m = fmi
∂Ym
∂Ym
For the CO2 equations m = NC − 1, evaluate
∂Φl
∂fl
mi
l
mi
= Xmi
Pi
− Φl
mi Pi = fmi
∂Xm
∂Xm
1 ∂Φv
mi
∂Y
Φv
m
mi
∂fl
mi
∂Xm
∂Φv
∂fv
mi
v
= Ymi
Pi mi − Φv
mi Pi = fmi
∂Ym
∂Ym
8.1
for m = 1 . . . NC − 2:
+ Φv
mi P δm,m
(8.8)
for m = 1 . . . NC − 2:
1 ∂Φl
mi
∂X
Φl
m
mi
For the CO2 equations m = 1 . . . NC − 1, evaluate
∂fv
mi
∂P
(8.7)
− Φl
mi P
∂fv
mi
∂P
1 ∂Φv
mi
∂Y
Φv
m
mi
(8.9)
for m = 1 . . . NC − 2:
− Φv
mi P
(8.10)
Fugacity Equations - Above Bubble Point
Above the bubble point, Sg = 0. Sg is replaced by a new variable, the bubble point pressure
Pb . One of the fugacity equations (8.1) is replaced by (8.11).
GNC −1 :
Pbn+1
−
N
C −1
m=1
N
C −1 n+1
n+1 P n+1
fom [Pb , X]Y
fn+1
m
om [Pb , X]
n+1
b
=
P
=0
−
b
n+1
Φn+1
[P
,
Y
]
f
[P
,
Y
]
gm
gm
b
b
(8.11)
m=1
The other fugacity equations are evaluated at Pb for m from 1 to NC − 2.
n+1
G1...NC −2 : fn+1
om [Pb , X] − fgm [Pb , Y ] = 0
(8.12)
Evaluate the Taylor series expansion for GNC −1 :
Gn+1
NC −1,i
≈
G+1
NC −1,i
=
GNC −1,i
+
∂GNC −1,i
δPbi +
∂Pbi
N
C −2
∂GN −1,i
C
m =1
∂Xm i
120
δXm i
+
N
C −2
m =1
∂GNC −1,i
∂Ym i
δYm i
(8.13)
The derivatives,
∂fom ∂fgm ∂fom
∂Pb , ∂Pb , ∂Xm ,
and
∂fgm
∂Ym
are evaluated using (8.5)–(8.8) with P → Pb . In
order to calculate both GNC −1 and the derivatives of GNC −1 , define:
Gm =
P
fom [Pb , X]Y
m b
fgm [Pb , Y ]
Evaluate the derivative
(8.14)
∂GN −1
C
∂Pb :
N
C −1
1 ∂fom
1 ∂fgm
∂GNC −1
1
=1−
Gm
+
−
∂Pb
Pb fom ∂Pb
fgm ∂Pb
(8.15)
m=1
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
∂GN −1
C
∂Xm :
N
C −1
Gm ∂fom
∂GNC −1
=−
∂Xm
fom ∂Xm
(8.16)
m=1
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
N
C −1 ∂fgm
∂GNC −1
=−
fom Pb δm,m − Gm
∂Ym
∂Ym
∂GN −1
C
∂Ym :
/fgm
(8.17)
m=1
8.2
Fugacity Equations - Below Dew Point
Below the dew point, So = 0. So is replaced by a new variable, the dew point pressure Pd . One
of the fugacity equations (8.1) is replaced by (8.18).
GNC −1 :
Pdn+1
−
N
C −1
m=1
N
C −1 n+1
n+1 P n+1
]Xm
fn+1
fgm [Pd , Y
gm [Pd , Y ]
d
n+1
= Pd −
=0
n+1
n+1
Φom [Pd , X]
fom [Pd , X]
(8.18)
m=1
The other fugacity equations are evaluated at Pd for m from 1 to NC − 2.
n+1
G1...NC −2 : fn+1
om [Pd , X] − fgm [Pd , Y ] = 0
(8.19)
Evaluate the Taylor series expansion for GNC −1 :
121
Gn+1
NC −1,i
≈
G+1
NC −1,i
=
GNC −1,i
+
∂GNC −1,i
δPdi +
∂Pdi
N
C −2
∂GN −1,i
C
∂Xm i
m =1
The derivatives,
∂fom ∂fgm ∂fom
∂Pd , ∂Pd , ∂Xm ,
and
∂fgm
∂Ym
δXm i
+
N
C −2
m =1
∂GNC −1,i
∂Ym i
δYm i
(8.20)
are evaluated using (8.5)–(8.8) with P → Pd . In
order to calculate both GNC −1 and the derivatives of GNC −1 , define:
Gm =
P
]Xm
fgm [Pd , Y
d
fom [Pd , X]
Evaluate the derivative
(8.21)
∂GN −1
C
∂Pd :
N
C −1
1 ∂fgm
1 ∂fom
1
∂GNC −1
=1−
Gm
+
−
∂Pd
P
f
∂P
f
gm
om ∂Pd
d
d
m=1
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
N
C −1 ∂fom
∂GNC −1
=−
fgm Pd δm,m − Gm
∂Xm
∂Xm
m=1
∂GN −1
C
∂Xm :
/fom
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
(8.23)
∂GN −1
C
∂Ym :
N
C −1
Gm ∂fgm
∂GNC −1
=−
∂Ym
f ∂Ym
m=1 gm
8.3
(8.22)
(8.24)
Method for Peng-Robinson Flash Calculation
The equation of state is a mechanism to calculate the Xm , Ym , ξo , and ξg .
The non-ideal gas law is defined by (8.25).
P =
z̆RT
v
(8.25)
The single component Peng-Robinson equation of state is defined by (8.26).
P =
a
RT
−
v − b v(v + b) + b(v − b)
(8.26)
122
8.3.1
Peneloux Volume Adjustment
This section is based on Pedersen and Christensen (2007) and Ahmed (2007a). The PengRobinson Equation of State with the Peneloux volume correction (Péneloux and Rauzy, 1982) is
defined by (8.27).
P =
a
RT
−
v − b (v + c)(v + 2c + b) + (b + c)(v − b)
(8.27)
This results in an adjustment to the specific volumes and the densities, but does not adjust the
phase splitting.
vnew = vEOS −
Xm cm
(8.28)
In some cases (for instance the Eclipse SSHIFT parameter sm ) , the volume shift is defined as a
multiplier to the bm :
cm = bm sm
(8.29)
The fugacities are adjusted as follows:
fm,new = fm,EOS exp[−cm
P
]
RT
(8.30)
In practice, this is accomplished by adjusting the fugacity coefficient:
ln Φm,new = ln Φm,EOS − cm
P
RT
(8.31)
There are several correlations in the literature for initial values of the volume shift parameter.
In practice, they are typically used as fitting parameters for tuning the equation of state.
8.3.2
Constants for This Formulation
For this formulation, the temperature is constant it the initial value T # . It may be necessary
to vary the temperature with depth in the future.
Compute κm , am , and bm , constant for a specific T and P . The Ωa (Eclipse OMEGAA) and Ωb
(Eclipse OMEGAB) are defined by (8.32).
123
Ωa = 0.4572355289
Ωb = 0.0777960739
(8.32)
The κ is defined by (8.33), where ω (Eclipse ACF) is the acentric factor.
2
κm = 0.37464 + 1.54226ωm − 0.26992ωm
(8.33)
In 1978, Peng and Robinson defined a new κ as follows:
2
0.37464 + 1.54226ωm − 0.26992ωm
ωm ≤ 0.491
2 + 0.016666ω 3 , ω > 0.491
0.379642 + 1.48503ωm − 0.164423ωm
m
m
κm =
(8.34)
The a is defined by (8.35). Tcm is the critical temperature (Eclipse TCRIT) and Pcm is the critical
pressure (Eclipse PCRIT).
am =
2
R2 Tcm
Ωa
Pcm
1 + κm
!
1−
T
Tcm
2
(8.35)
The b is defined by (8.36).
bm = Ω b
RTcm
Pcm
(8.36)
The amn is defined by (8.37). The binary interaction coefficient is δ̆mn (Eclipse BIC). In PR78, δ̆mn
is a function of temperature.
1/2
amn = (1 − δ̆mn )a1/2
m an
8.3.3
(8.37)
Initial Values, Compute Km , (Full Flash Only)
e1 ,
If there is a pre-existing value of Km
e1 +1
e1
= Km
Km
flm
fvm
(8.38)
If there is no pre-existing value of Km , compute the initial estimate of Km based on (8.39),
Wilson’s equation.
e1
Km
Pcm
Tcm
= exp 5.3727(1 + ωm ) 1 −
P
T
124
(8.39)
8.3.4
Flash to Calculate the Vapor Fraction, (Full Flash Only)
The flash computation is defined by (8.40).
f (V
e2
)=
m
e1 − 1)Z (Km
m
e1
(Km
− 1)V e2 + 1
(8.40)
The flash derivative is
e1 − 1)2 Z ∂f
(Km
m
=−
2
∂V
e1
e
m
(Km − 1)V 2 + 1
(8.41)
e1 and Y e1
If f (0) ≤ 0 and f (1) ≤ 0, then V e2 = 0 and the components are all liquid. Define Xm
m
using (8.42).
V e2 = 0
e1
Xm
= Zm
e1 Yme1 = Km
Zm
(8.42)
e1 and Y e1
If f (1) ≥ 0 and f (0) ≥ 0, then V e2 = 1 and the components are all vapor. Define Xm
m
using (8.43).
V e2 = 1
Yme1 = Zm
e1
Xm
=
Zm
e1
Km
(8.43)
Calculate V e2 using Newton Raphson iteration with a starting value of either V e2 =0 = 0.5 or the
previous estimate of V e2 . Solve for V e2 when f (V e2 ) = 0.
V e2 +1 = V e2 − f (V e2 )
∂f (V e2 )
∂V
(8.44)
Convergence is defined by
e +1
V 2 − V e2 ≤ V
V e2
(8.45)
e1 and Y e1 based on (8.46).
Calculate Xm
m
e1
=
Xm
V
Zm
− 1) + 1
e1
(Km
e1
e1
Yme1 = Km
· Xm
125
(8.46)
If a temporary step for (8.46) evaluates V e2 ≤ 0, compute the next iteration based on f (0) and
f (0). If a temporary step for (8.46) evaluates V e2 ≥ 1, compute the next iteration based on f (1)
and f (1).
Normalize
e1
m Xm
= 1 and
e1
m Ym
= 1 to avoid any numerical errors. Calculate L.
Le 2 = 1 − V e 2
8.3.5
(8.47)
Calculate the Mixing Parameters
The multi-component Peng-Robinson equation of state is defined by a set of mixing rules.
Compute the mixed a using (8.48).
ale1 =
e1 e1
amn Xm
Xn
ave1 =
m,n
amn Yme1 Yne1
(8.48)
m,n
The mixing rule for b is defined by (8.49).
ble1 =
e1
bm Xm
bve1 =
m
bm Yme1
(8.49)
m
Define A by (8.50).
Ale1 =
ale1 P R2 T 2
Ave1 =
ave1 P R2 T 2
(8.50)
B ve1 =
bve1 P RT
(8.51)
Define B by (8.51).
B le1 =
8.3.6
ble1 P RT
Calculate the z̆-factor
Calculate z̆ le3 , the smallest positive real root in (8.52), using Ale1 and B le1 . Calculate z̆ ve3 ,
the largest real root in (8.52), using Ave1 and B ve1 .
f (z̆) = z̆ 3 + (B − 1)z̆ 2 + (A − 3B 2 − 2B)z̆ + (B 3 − AB + B 2 ) = 0
(8.52)
Define the following coefficients of the terms of f (z̆).
a0 = (B 3 − AB + B 2 )
a1 = (A − 3B 2 − 2B)
126
a2 = (B − 1)
(8.53)
Solve the cubic equation using the methods of Section 8.8.
When A and B are constants,
∂f
∂ z̆
evaluates to (8.54). This is used for the Newton-Raphson
flash calculation.
∂f = 3z̆ 2 + 2(B − 1)z̆ + (A − 3B 2 − 2B)
∂ z̆ A,B
8.3.7
(8.54)
Calculate the Fugacities f (Not if Only Computing z̆)
1
Compute the liquid fugacity coefficients, Φle
m using (8.55).
bm le3
(z̆ − 1) − ln (z̆ le3 − B le1 )+
ble1
√
2 + 1 B le1
z̆ le3 +
2
bm
cm P
Ale1
e1
√
Xn amn − le1 · ln
−
− √
le1
le
le
le
1
3
1
a
b
RT
2 2B
z̆ −
2−1 B
n
1
ln Φle
m =
(8.55)
1
Compute the vapor fugacity coefficients, Φve
m using (8.56).
bm ve3
(z̆
− 1) − ln (z̆ ve3 − B ve1 )+
bve1 √
2 + 1 B ve1
z̆ ve3 +
2
bm
cm P
Ave1
e1
√
Y
a
−
·
ln
−
− √
mn
n
ve
ve
b 1
RT
2 2B ve1 a 1
z̆ ve3 −
2 − 1 B ve1
n
1
ln Φve
m =
(8.56)
The fugacities are defined by (8.57).
le1 e1 1
fle
m = Φm Xm P
8.3.8
ve1 e1 1
fve
m = Φ m Ym P
(8.57)
Calculate the Tolerance (Full Flash Only)
e1 using (8.58).
Compute Rm
e1
=
Rm
1
fle
m
ve1
fm
(8.58)
Convergence is defined by
e1
− 1| ≤ f
|Rm
∀m
(8.59)
e1 − 1| ≥ for any m, compute K e1 +1 = Re1 K e1 and return to Section 8.3.4.
If |Rm
f
m m
m
127
Eclipse uses (8.60), but it has several problems. First, it requires values at lots of nodes, whereas
the convergence in (8.59) is local to one grid cell. Second, it does not identify which grid cells are
having problems with the flash computation.
fle1
m
8.3.9
m
1
fve
m
2
−1
< f
(8.60)
Calculate the Densities
Once the fugacities converge in Step 8, calculate the ξ using (8.61).
ξo =
P
ξg =
z̆ l RT
P
(8.61)
z̆ v RT
Calculate vt .
vtl =
1
ξo
1
ξg
vtv =
(8.62)
If there is a Peneloux volume adjustment factor cm , then first calculate the specific volume vt
using (8.63) and then calculate ξ using (8.64).
vtl =
z̆ l RT −
Xm cm
P
m
ξo =
1
vtl
vtv =
z̆ v RT −
Ym cm
P
m
(8.63)
1
vtv
ξg =
(8.64)
Compute the molecular weight for the liquid and gas from (8.65).
MWo =
MWm Xm
MWg =
m
MWm Ym
(8.65)
m
Compute the densities using (8.66).
ρo = ξo MWo
ρg = ξg MWg
Compute the specific gravities, in
γo =
0.433 ·
ρo
ρw,sc
=
(8.66)
psi
ft
0.433
ρo
62.4
based on ρ
=
1
ρo
144
128
lbm
ft3
.
γg =
1
ρg
144
(8.67)
8.3.10
Calculate the saturations (Full Flash Only)
Calculate the saturations for two phases only:
L=
ξo So
ξo So + ξg Sg
n
=
Soi
8.4
V =
n Ln
ξgi
i
n V n + ξ n Ln
ξoi
i
gi i
ξg Sg
ξo So + ξg Sg
n
Sgi
=
(8.68)
nV n
ξoi
i
n V n + ξ n Ln
ξoi
i
gi i
(8.69)
Evaluate Fugacity Derivatives
1
Compute the liquid fugacity coefficients, Φle
m using (8.70).
bm le3
(z̆ − 1) − ln (z̆ le3 − B le1 )+
ble1
√
le3 +
le1
2
+
1
B
z̆
2
b
cm P
Ale1
m
√
Xne1 amn − le
· ln
−
− √
le
1
1
le
le
le
1
3
1
a
b
RT
2 2B
z̆ −
2−1 B
n
1
ln Φle
m =
(8.70)
1
Compute the vapor fugacity coefficients, Φve
m using (8.71).
bm ve3
(z̆
− 1) − ln (z̆ ve3 − B ve1 )+
bve1 √
2 + 1 B ve1
z̆ ve3 +
2
bm
cm P
Ave1
e1
√
Yn amn − ve1 · ln
−
− √
ve1
ve
ve
ve
1
3
1
a
b
RT
2 2B
z̆ −
2−1 B
n
1
ln Φve
m =
Evaluate
∂Φl
m
∂P .
Note that
∂A/B
∂P
(8.71)
= 0.
l
bm ∂ z̆ l
1
∂ z̆
∂B l
1 ∂Φl
m
=
−
−
+
l
l
l
l
Φm ∂P
b ∂P
(z̆ − B ) ∂P
∂P
l
√
√
l
l
NC −1
∂ z̆
∂ z̆ l
b
2 + 1 ∂B
2 − 1 ∂B
cm
2 Al
m
∂P +
∂P
∂P −
∂P
√
√
Xn amn − l ·
−
−
·
− √
l
l
l
l
l
l
a
b
RT
2 2B
z̆ +
2+1 B
z̆ −
2−1 B
n=1
(8.72)
Evaluate
∂Φv
m
∂P .
v
bm ∂ z̆ v
1
∂ z̆
∂B v
1 ∂Φv
m
=
−
−
+
v
v
v
v
Φm ∂P
b ∂P
(z̆ − B ) ∂P
∂P
v
√
√
v
v
NC −1
∂ z̆
∂ z̆ v
b
2 + 1 ∂B
2 − 1 ∂B
cm
2 Av
m
∂P +
∂P
∂P −
∂P
√
√
Yn amn − v ·
−
−
· v
− √
v +
v
v −
v
a
b
RT
2 2B v
z̆
2
+
1
B
z̆
2
−
1
B
n=1
(8.73)
129
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂Φl
m
∂Xm
for m = 1 . . . NC − 2.
Since amm = 0, the derivatives are the same for m = NC − 1.
1 ∂bl
1 ∂Φl
∂ z̆ l
bm
∂B l
∂ z̆ l
1
m
l
=
−
1)
−
−(z̆
+
−
+
l
l
l
l
l
Φm ∂Xm
b
b ∂Xm
∂Xm
(z̆ − B ) ∂Xm
∂Xm
√
NC −1
b
z̆ l +
2 + 1 B l
1 ∂Al
Av
1 ∂B l
2 m
√
√
−
− l
X amn − l · ln
+
·
·
Al ∂Xm
B ∂Xm
al n=1 n
b
2 2B v
z̆ l −
2 − 1 B l
NC −1
1 ∂al
Av
1 ∂bl
2 2(am,m − am,M ) bm
√
−
X amn
+ l
·
+
·
− l
al n=1 n
a ∂Xm
al
b
bl ∂Xm
2 2B v
√
z̆ l +
2 + 1 B l
√
+
ln
z̆ l −
2 − 1 B l
⎞
√
√
⎛ ∂ z̆l
∂B l
∂ z̆ l
∂B l
NC −1
b
+
2 + 1 ∂X
−
2 − 1 ∂X
Al
2 ∂X
∂X
m
m
m
m ⎠
√
√
√
−
X amn − l · ⎝ m
−
·
al n=1 n
b
2 2B l
z̆ l +
2 + 1 B l
z̆ l −
2 − 1 B l
(8.74)
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂Φv
m
∂Ym
for m = 1 . . . NC − 2.
Since amm = 0, the derivatives are the same for m = NC − 1.
v
1 ∂bv
∂ z̆
1 ∂Φv
bm
∂B v
∂ z̆ v
1
m
v
=
−
1)
−
−(z̆
+
−
+
v
v
v
v
v
Φm ∂Ym
b
b ∂Ym
∂Ym
(z̆ − B ) ∂Ym
∂Ym
√
NC −1
b
z̆ v +
2 + 1 B v
1 ∂Av
Av
1 ∂B v
2 m
√
√
−
− v
Y amn − v · ln
+
·
·
Av ∂Ym
B ∂Ym
av n=1 n
b
2 2B v
z̆ v −
2 − 1 B v
NC −1
1 ∂av
Av
1 ∂bv
2 2(am,m − am,M ) bm
√
−
Y amn
+ v
·
+
·
− v
av n=1 n
a ∂Ym
al
b
bv ∂Ym
2 2B v
√
2 + 1 B v
z̆ v +
√
+
ln
z̆ v −
2 − 1 B v
⎞
√
√
⎛ ∂ z̆v
∂B v
∂ z̆ v
∂B v
NC −1
b
+
2 + 1 ∂Y
−
2 − 1 ∂Y
Av
2 ∂Y
∂Y
m
m
m ⎠
√
√
√
−
Y amn − v · ⎝ m
− m
·
av n=1 n
b
2 2B v
z̆ v +
2 + 1 B v
z̆ v −
2 − 1 B v
(8.75)
8.5
Evaluate Peng-Robinson Pressure Derivatives
Required derivatives:
∂ z̆
∂P
130
∂ξ
∂P
∂A
∂P
∂B
∂P
(8.76)
All of these derivatives are required for both the oil phase and the gas phase.
8.5.1
∂ξ
∂P
Evaluate
Use the definition of ξ =
vPR =
z̆RT
P
ξ=
1
v
in the equation of state; solve for ξ.
P
1
=
z̆RT
vPR
(8.77)
Derivative of ξ:
∂ξ
=ξ
∂P
1
1 ∂ z̆
−
P
z̆ ∂P
(8.78)
Where there is a Peneloux volume adjustment, evaluate
ξ=
vPR −
(8.79)
m Xm cm
Evaluate
1 ∂ z̆
1
−
P
z̆ ∂P
(8.80)
∂ z̆
∂P
Evaluate cubic equation, solve for
∂ z̆
∂P
. Evaluate the derivative of both sides of (8.52).
∂f ∂ z̆
∂f ∂A
∂f ∂B
∂f (z̆, A, B)
=
+
+
=0
∂P
∂ z̆ ∂P
∂A ∂P
∂B ∂P
Solve (8.81) for
∂ z̆
=−
∂P
8.5.3
as follows:
1
∂ξ
= ξ 2 · vPR ·
∂P
8.5.2
∂ξ
∂P
(8.81)
∂ z̆
∂P
∂f ∂A
∂A ∂P
+
∂f
∂ z̆
∂f ∂B
∂B ∂P
(8.82)
Evaluate Derivatives of f (z̆)
Evaluate
∂f
∂ z̆
∂f
= 3z̆ 2 + 2(B − 1)z̆ + (A − 3B 2 − 2B)
∂ z̆
Evaluate
(8.83)
∂f
∂A
131
∂f
= z̆ − B
∂A
Evaluate
(8.84)
∂f
∂B
∂f
= z̆ 2 − 6B z̆ − 2z̆ + 3B 2 − A + 2B
∂B
8.5.4
(8.85)
Evaluate Derivatives of A and B
A is defined by (8.86):
Al =
Thus
∂A
∂P
al P R2 T 2
(8.86)
is:
al
∂Al
= 2 2
∂P
R T
1
1 ∂Al
=
A ∂P
P
(8.87)
B is defined by (8.88):
Bl =
Thus
∂B
∂P
bl P RT
is:
bl
∂B l
=
∂P
RT
8.6
(8.88)
1 ∂B l
1
=
B ∂P
P
(8.89)
Evaluate Peng-Robinson Composition Derivatives
Required derivatives:
∂ z̆
∂Xm
∂ξ
∂Xm
∂A
∂Xm
∂B
∂Xm
∂a
∂Xm
All of these derivatives are required for both the oil phase and the gas phase.
132
∂b
∂Xm
(8.90)
8.6.1
∂ξ
∂Xm
Evaluate
Where there is no Peneloux volume adjustment, evaluate
∂ξ
∂P
using the definition of ξ =
1
v
in
the equation of state; solve for ξ.
vPR =
z̆RT
P
ξ=
P
1
=
z̆RT
vPR
(8.91)
Derivative of ξ:
∂ξ
= −ξ
∂Xm
1 ∂ z̆
z̆ ∂Xm
(8.92)
Where there is a Peneloux volume adjustment, evaluate
ξ=
vPR −
1
(8.94)
∂ z̆
∂Xm
Evaluate
Evaluate the cubic equation; solve for
∂ z̆
∂Xm .
Evaluate the derivative of both sides of (8.52).
∂f ∂ z̆
∂f ∂A
∂f ∂B
∂f (z̆, A, B)
=
+
+
=0
∂Xm
∂ z̆ ∂Xm
∂A ∂Xm
∂B ∂Xm
Solve (8.95) for
∂ z̆
=−
∂Xm
8.6.3
as follows:
(8.93)
m Xm cm
1 ∂ z̆
∂ξ
2
= −ξ vPR
− (cm − cM )
∂Xm
z̆ ∂Xm
8.6.2
∂ξ
∂Xm
(8.95)
∂ z̆
∂Xm :
∂f ∂A
∂A ∂Xm
+
∂f
∂ z̆
∂f ∂B
∂B ∂Xm
(8.96)
Evaluate Derivatives of A and B
A is defined by:
Thus
Al =
al P R2 T 2
∂A
∂Xm
is
(8.97)
133
P ∂al
∂Al
= 2 2
∂Xm
R T ∂Xm
1 ∂al
1 ∂Al
=
A ∂Xm
a ∂Xm
(8.98)
B is defined by:
bl P RT
Bl =
Thus
∂B
∂P
(8.99)
is
P ∂bl
∂B l
=
∂Xm
RT ∂Xm
8.6.4
1 ∂bl
1 ∂B l
=
B ∂Xm
b ∂Xm
(8.100)
∂a
∂Xm
Evaluate
For this section, M = NC −1. Figure 8.1 illustrates how the amn is split into pieces. Figure 8.1(a)
shows the pieces which contain Xk , using k = m = 1 for this illustration. Figure 8.1(b) shows
the pieces for m = M or n = M (ie contains XM ). Figure 8.1(c) shows the overlap between
Figure 8.1(a) and Figure 8.1(b).
і
m=k
m:1..M
і
m=k
ї
m=M
ј n=k
ј n=k
ј n=k
n:1..M
ї
m=M
n:1..M
m:1..M
n:1..M
і
m=k
љ
љ n=M
(a) Values with Xk .
љ
n=M
(b) Values with XM .
m:1..M
ї
m=M
n=M
(c) Values with both Xk and XM .
Figure 8.1: Illustration of amn .
Recall the definition of a:
al =
M M
1/2
(1 − δ̆mn )a1/2
m an Xm Xn
(8.101)
m=1 n=1
Expand (8.101) into the terms that contain Xk . Use Figure 8.1 for reference. Although akk = 0
and aM M = 0, the derivation is actually simpler if we ignore this.
134
no Xk , no XM
one Xk , no XM
no Xk , one XM
M
−1
M
−1
M
−1
M
−1
a=
amn Xm Xn + 2 ·
amk Xm Xk + 2 ·
amM Xm XM +
m=k n=k
m=k
m=k
two Xk , no XM
no Xk , two XM
one Xk , one XM
akk Xk Xk + aMM XM XM + 2 · akM Xk XM (8.102)
∂XM
∂Xk
The derivative
is defined by
∂
∂XM
=
(1 − X1 − X2 − · · · − Xk − · · · − XM −2 − XM −1 ) = −1
∂Xk
∂Xk
Calculate the derivatives
∂a
∂Xk ,
(8.103)
for k = m = 1 . . . NC − 2, using (8.102).
M
−1
M
−1
∂a
= 2·
amk Xm +−2·
amM Xm +2·akk Xk +−2·aMM XM +2·akM XM −2·akM Xk (8.104)
∂Xk
m=k
m=k
(8.104) can be simplified by using
M
amk Xm =
m=1
M
−1
amk Xm + akk Xk + akM XM
(8.105)
m=k
The final result is the derivative
∂a
∂Xk :
∂a
=2·
Xm (amk − amM )
∂Xk
M
(8.106)
m=1
8.6.5
Evaluate
∂b
∂Xm
For this section, M = NC − 1. Recall the definition of b:
b=
M
bm Xm
(8.107)
m=1
Expand (8.107) into (8.108).
b = b1 X1 + · · · + bk Xk + · · · + bM (1 − X1 − · · · − XM −1 )
Evaluate
∂b
∂Xk ,
for k = m = 1 . . . NC − 2.
135
(8.108)
∂b
= bk − bM = bm − bCO2
∂Xk
8.7
(8.109)
Check Fugacity Derivatives
This section provides some internal consistency checks for fugacity derivatives based on Michelsen
and Mollerup (2007) and Mollerup and Michelsen (1992). Mollerup and Michelsen (1992) provides
derivatives based on the Redlich-Kwong equation of state; this section provides the derivations for
the Peng-Robinson equation of state.
8.7.1
Introduction
The mixing rules for both equations of state are:
b=
Xi bi
B = nb =
i
a=
ni bi
(8.110)
i
i
Xi Xj aij
D = n2 a =
j
i
ni nj aij
(8.111)
j
The general form of the equation of state is
P =
a
RT
−
v − b (v + δ1 b)(v + δ2 b)
(8.112)
For Redlich Kwong, δ1 = 1 and δ2 = 0. For Peng-Robinson, δ1 = 1 +
√
2 and δ2 = 1 −
√
2.
The derivations are based on the reduced residual Helmholtz energy.
D
V + δ1 B
V
−
ln
F = n ln
V −B
B(δ1 + δ2 )RT
V + δ2 B
8.7.2
(8.113)
Fugacity Coefficient
The fugacity coefficient is defined as
ln Φi =
∂F
∂ni
This is based on
− ln z̆
(8.114)
T,V
∂F
∂ni
136
∂F
∂ni
Evaluate
= Fn + FB
T,V
∂B
∂D
+ FD
∂ni
∂ni
(8.115)
∂F
∂n
∂F
V
Fn =
= ln
∂n
V −B
Evaluate
(8.116)
∂F
∂B
n
D
∂F
=
−
FB =
∂B
V − B B(δ1 + δ2 )RT
Evaluate
1
V + δ1 B
δ1
δ2
− ln
+
−
B
V + δ2 B
V + δ1 B V + δ2 B
∂F
∂D
1
V + δ1 B
∂F
=−
ln
FD =
∂D
B(δ1 + δ2 )RT
V + δ2 B
Evaluate
(8.117)
(8.118)
∂D
∂ni
∂D
=2·
nj aij
∂ni
(8.119)
j
Evaluate
∂B
∂ni
∂B
= bi
∂ni
8.7.3
(8.120)
Pressure Derivative
Evaluate
∂ ln Φi
∂P
as
v̄i
1
∂ ln Φi
=
−
∂P
RT
P
(8.121)
Evaluate v̄i
v̄i =
∂V
∂ni
=−
T,P
∂P
∂ni
T,V
∂P
∂V
n,T
(8.122)
137
∂P
∂V
Evaluate
∂P
∂V
= −RT
T,n
∂P
∂ni
−
T,n
nRT
V2
(8.123)
∂P
∂ni
Evaluate
∂2F
∂V2
= −RT
T,V,nj=i
∂2F
∂V ∂ni
+
T,nj
RT
V
(8.124)
Evaluate FV V
FV V =
∂2F
∂V2
=−
T,n
4Dδ1 V
RT (δ1 + δ2 ) V 2 −
2
B 2 δ12
+
D
+
RT (V − Bδ1 )2 (Bδ2 + V )
Bn(B − 2V )
D
− 2
RT (V − Bδ1 )(Bδ2 + V )2
V (B − V )2
(8.125)
Evaluate FV,ni
FV,ni =
∂2F
∂V ∂ni
= FnV + FBV
T,nj
∂B
∂D
+ FDV
∂ni
∂ni
(8.126)
Evaluate FnV
FnV =
B
∂2F
=
∂n∂V
BV − V 2
(8.127)
Evaluate FBV
FBV =
4DBδ13
∂2F
=
∂B∂V
RT (δ1 + δ2 ) V 2 − B 2 δ12
2
+
DV − 2DBδ1
+
BRT (Bδ1 − V )2 (Bδ2 + V )
n
DV
−
2
BRT (Bδ1 − V )(Bδ2 + V )
(B − V )2
(8.128)
Evaluate FDV
FDV =
8.7.4
δ1 − δ2
∂2F
=
∂D∂V
RT (δ1 + δ2 )(Bδ1 + V )(Bδ2 + V )
Composition Derivative
Evaluate
∂ ln Φi
∂nj
as
138
(8.129)
∂ ln Φi
=
∂nj
∂ ln Φi
∂nj
=
T,P
∂2F
∂ni ∂nj
+ RT
∂P
∂ni
T,V
T,V
∂P
∂V
∂P
∂nj
T,V
+
1
n
(8.130)
T,n
Evaluate Fni ,nj
Fni ,nj =
∂2F
∂ni ∂nj
= FnB
T,V
∂B
∂B
+
∂ni ∂nj
+ FBB
∂2B
FB
+ FBD
∂ni ∂nj
∂B ∂B
+
∂ni ∂nj
∂B ∂D
∂B ∂D
+
∂ni ∂nj
∂nj ∂ni
+ FD
∂2D
∂ni ∂nj
(8.131)
Evaluate FnB
FnB =
1
∂2F
=
∂n∂B
V −B
(8.132)
Evaluate FBB
2D log
Bδ1 +V
Bδ2 +V
D(δ1 V − δ2 V )
+
∂
+ δ2 )
+ δ2 )(Bδ1 + V )(Bδ2 + V )
Dδ1 V (δ1 − δ2 )
DV (δ1 − δ2 )
+
+
2
B RT (δ1 + δ2 )(Bδ1 + V )(Bδ2 + V ) BRT (δ1 + δ2 )(Bδ1 + V )2 (Bδ2 + V )
n
Dδ2 V (δ1 − δ2 )
+
BRT (δ1 + δ2 )(Bδ1 + V )(Bδ2 + V )2
(B − V )2
FBB =
∂2F
B2
=−
B 3 RT (δ1
+
B 2 RT (δ1
(8.133)
Evaluate FBD
FBD =
∂2F
∂B∂D
log
=
Bδ1 +V
Bδ2 +V
B 2 RT (δ1
+ δ2 )
−
V (δ1 − δ2 )
BRT (δ1 + δ2 )(Bδ1 + V )(Bδ2 + V )
(8.134)
Evaluate Dni ,nj
Dni ,nj =
∂2D
= 2 · aij
∂ni ∂nj
(8.135)
Evaluate Bni ,nj
Bni ,nj =
∂2B
=0
∂ni ∂nj
(8.136)
139
8.7.5
Consistency Check
Internal consistency of the implementation of
ni
i
∂ ln Φi
∂P
can be evaluated as follows:
∂ ln Φi
(z̆ − 1)n
z̆ − 1
∂ ln Φi
Xi
=
=⇒
=
∂P
P
∂P
P
(8.137)
i
Mollerup and Michelsen (1992) is based on total number of moles. Two additional derivatives
are needed to create the derivatives with respect to mole fractions. Use
∂
∂Xj
=
∂nj
∂nj
nj
n1 + n2 + · · · + nj + · · · + nM
Internal consistency of the implementation of
∂ ln Φi
∂nj
=
∂nj
∂ni
= 0, and also:
1
(1 − Xj )
N
can be evaluated as follows:
∂ ln Φj
∂ ln Φi
∂ ln Φj
∂ ln Φi
=
=⇒ (1 − Xj )
= (1 − Xi )
∂nj
∂ni
∂Xj
∂Xi
ni
i
8.8
(8.138)
∂ ln Φi
∂ ln Φi
= 0 =⇒
Xi (1 − Xj )
=0
∂nj
∂Xj
(8.139)
(8.140)
i
Solving Cubic Equations Numerically
There is a closed form solution for a cubic equation that involves complex variables. This section
describes how to computationally implement this closed form solution.
8.8.1
Initialize
For this section, the following variables are used with arbitrary units: x, a0 , a1 , a2 , Q, R, θ, A,
B.
The following solution procedure is based on Press, Flannery, Tukolsky, and Vetterling (1992)
and Wang (2006) with a derivation in Weisstein (2006). A general cubic equation has the form:
x3 + a2 x2 + a1 x + a0 = 0
(8.141)
Define Q and R as follows. Note that if a2 , a1 , and a0 are real then Q and R are real.
Q=
a22 − 3a1
9
R=
2a32 − 9a2 a1 + 27a0
54
140
(8.142)
8.8.2
Three Distinct Real Roots
If R2 < Q3 , then there are three distinct real roots. Note that since R2 ≥ 0, Q3 > 0 and
&
&
√
therefore Q > 0 and thus Q3 and Q exist and Q3 > 0. Define the angle θ. Arccos represents
the principle arc-cosine.
R
θ = Arccos
Q3/2
(8.143)
The real roots are
x1
x2
x3
8.8.3
&
a2
θ
−
= −2 Q cos
3
3
&
θ + 2π
−
= −2 Q cos
3
&
θ − 2π
−
= −2 Q cos
3
(8.144)
a2
3
a2
3
(8.145)
(8.146)
One Real Root
If R2 > Q3 , then there is one real root and two complex roots. For real Q and R, compute A
as follows:
1/3
&
A = −sgn[R] |R| + R2 − Q3
(8.147)
If A = 0, then there are three identical roots
x1 = x2 = x3 = −
a2
3
(8.148)
If A = 0, then the real root is x1 and the complex conjugate roots are x2 and x3 :
Q a2
x1 = A + −
A
3
Q
1
A+
x2 = −
2
A
Q
1
A+
x3 = −
2
A
√ a2
3
−
+i
A−
3
2
√ a2
3
−
−i
A−
3
2
(8.149)
Q
A
(8.150)
Q
A
(8.151)
141
8.8.4
Three Real Roots, Two or More Coincide
If R2 = Q3 then there are three real roots and two or more coincide. The approach for R2 > Q3
simplifies to the following. For this case, A = B.
A = B = −sgn[R] (|R|)1/3
(8.152)
If A = 0 then there are three identical real roots,
x1 = x2 = x3 = −
a2
3
(8.153)
If A = 0,
a2
3
a2
= −A −
3
x1 = 2A −
x2 = x3
8.8.5
(8.154)
(8.155)
Newton Raphson
The roots of a polynomial equation are subject to significant roundoff errors. This can easily
be illustrated with a polynomial such as
(x − 109 )(x − 2)(x − 1) = x3 − (109 + 3)x2 + (3 · 109 + 2)x − (2 · 109 )
(8.156)
As a result of these numerical inaccuracies, it is common practice to “polish” numerical roots using
Newton Raphson. Newton Raphson is an excellent tool for this since it has quadratic convergence
close to a root. The down side of Newton Raphson is that it is sensitive to the initial estimate.
This is overcome by using an algebraic solution as a starting value.
xe3 +1 = xe3 − f (x)
∂f (x)
∂x
(8.157)
Convergence is defined by
e +1
x 3 − xe 3 ≤ x
xe 3
(8.158)
142
8.9
Fugacity Computations
The fugacity, fugacity coefficients, and fugacity derivatives are computed in the same subroutine
by first defining some temporary variables. These variables are highlighted in (8.159), (8.160), and
(8.161): variables a1 , . . . , a12 do not depend on m; variables b1m , b2m , b3m , c1m , c2m , c4m depend
1
on m; variable c3mm depends on both m and m . Compute the liquid fugacity coefficients, Φle
m
using (8.159).
b1m
c1m
a4
a1 c4m
a3
a
2 1
l
l
l
−
ln Φl
m = bm l z̆ − 1 − ln z̆ − B
b
b3m
⎛
⎞
⎛
a9
1/a11
⎞
b2m
4m
⎟ b
⎜
a7
⎟
⎜
a5
⎜ a6
b1m ⎟
a
14
√
⎟ ⎜
⎜
⎟
⎟
⎜
l
l
C −1
N
b ⎟
2+1 B ⎟
⎜ z̆ +
Al ⎜
⎜ 2
m ⎟
⎟ + − 1 cm P
⎜
√
×
−
ln
X
a
⎟
⎜
mn
n
⎟
⎜
l
l
l
1/a
12
a
b ⎟
RT
2 2B ⎜
⎜
n=1
⎟
⎟
⎜
⎟
⎜
⎠
⎝
a8
⎟
⎜
⎠
⎝
√
l
z̆ −
2 − 1 B l
Evaluate
∂Φl
m
∂P
using (8.160).
c2m
b1m
l
bm ∂ z̆ l
1 ∂Φm
= l
−
Φl
b
∂P
m ∂P
−
(8.159)
1
a3
z̆ l − B l
∂ z̆ l
∂B l
−
∂P
∂P
+
b3m
⎞
a13
NC −1
b ⎟
⎜ 1 Al
m
√
Xn amn − l ⎟
×⎜
2 l
⎠×
⎝
a
b
2 2B l
n=1
a5
⎛
a10
⎞
b4m
a8
a
l ⎟
14
√
⎜ l
⎟ ⎜ ∂ z̆
∂ z̆ l
2 − 1 ∂B
⎜ ∂P
∂P −
∂P ⎟
⎟ + − 1 cm
⎜
−
⎟
⎜
1/a11
1/a12
RT
⎟
⎜ √ √ ⎠
⎝
z̆ l +
z̆ l −
2 + 1 B l
2 − 1 B l
⎛
a7
√
l
+
2 + 1 ∂B
∂P
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂Φl
m
∂Xm
using (8.161). Since amm = 0, the derivatives are the same for m = NC − 1.
143
(8.160)
for m = 1 . . . NC − 2
c3mm
b1m ⎛
a2
l
1 ∂bl
bm ⎝
1 ∂Φm
l
−
(z̆
=
−
1)
Φl
bl
bl ∂Xm
m ∂Xm
⎞
a10
∂ z̆ l
∂B l
1
∂ z̆ ⎠
− l
−
+
∂Xm
(z̆ − B l ) ∂Xm
∂Xm
l
b3m
+
a
9
a5
√
N
−1
C
l
l
l
l
v
z̆
+
2
+
1
B
A
1 ∂A
bm
1 ∂B
2
√
√
+
−
− l
X amn − l · ln
·
·
Al ∂Xm
B ∂Xm
al n=1 n
b
2 2B v
z̆ l −
2 − 1 B l
⎛
⎞
b2m
a5
b1m
a6
⎜
⎟
C −1
1 ∂al
⎜ 2 N
2
1 ∂bl ⎟
bm
Av
⎜
⎟·
√
−
Xn amn
− l
· ⎜ l
+ l (am,m − am,M ) + l
l ∂X ⎟
a
∂X
a
b
b
2 2B v
m
m
⎝a
⎠
n=1
ln
a9
√
z̆ l +
2 + 1 B l
√
+
z̆ l −
2 − 1 B l
⎛
a7
√
∂B l
+
2 + 1 ∂X
b3m
⎜
⎜ ∂ z̆l
NC −1
b
⎜ ∂Xm
2 Al
m
m
Xn amn − l · ⎜
− √
·
⎜
l
l
1/a11
a
b
2 2B
⎜
n=1
√ ⎝ l
z̆ +
2 + 1 B l
a5
⎞
a8
⎟
√
∂ z̆ l
∂B l ⎟
2 − 1 ∂X
⎟
∂Xm −
m
⎟
−
⎟
1/a12
⎟
√ ⎠
z̆ l −
2 − 1 B l
(8.161)
8.10
Flash Calculations
The flash calculations described here are based on Rachford and Rice (1952). Michelsen and
Mollerup (2007, pg 252) has a similar “successive substitution” algorithm. This algorithm has more
detailed checks for degenerate cases and division by zero.
1. Calculate initial values of Km values; use Wilson. (Commentary: at one point, I tried using
the previous Km values; most of the time this works; some of the time it caused convergence
failures or convergence to the wrong solution. If the previous solution had some problem, this
compounds that problem. Now I always use Wilson. Also note, storing all the Km for all the
grid cells in a big grid requires a lot of memory. )
e1
Km
Pcm
Tcm
= exp 5.3727(1 + ωm ) 1 −
P
T
2. Calculate a weighted critical temperature.
144
(8.162)
Tc,mix
m MWm Zm Tc,m
= m MWm Zm
(8.163)
3. itK = 0; convergedK = TRUE
4. Loop K: WHILE((itK < itK,max )OR(convergedK))
5. itK = itK + 1
6. Calculate f [V = 0], f [V = 1]
7. Initial estimate of V . Make single phase computation consistent. (Commentary: before doing
this, some single phase cases were giving inconsistent results. Single component cases were
not consistent with the rest of the phase diagram.)
IF((f [V = 0] >= 0)OR(f [V = 1] <= 0))THEN
IF(T >= Tc,mix ) THEN V = 1 ELSE V = 0 ENDIF
ELSE
V = 0.5
ENDIF
(8.164)
8. Loop f (V )
9. IF (V >= 1 − V ); current estimate is 100% gas.
10. Fill in later
11. ELSEIF (V <= V ); current estimate is 100% liquid.
12. Fill in later
13. ELSE (V < V < 1 − V ); current estimate is two-phase gas-liquid.
14. Initialize Vmin , Vmax , and V , current version. (Commentary: current code uses this version,
but version below is fine for negative flash; I don’t remember if there was a reason for changing
this.)
145
Vmin =
0
(8.165)
Vmax =
1
(8.166)
V = 0.5
(8.167)
15. Initialize Vmin , Vmax , and V , negative flash
Vmin =
Vmax =
V =
−1
(8.168)
Km,max − 1
−1
Km,min − 1
Vmin + Vmax
2
(8.169)
(8.170)
16. Loop V :
17. Calculate f (V )
18. IF (f (V ) > 0), update Vmin = max(V, Vmin )
19. IF (f (V ) <= 0), update Vmax = min(V, Vmax )
20. Vold = V
21. IF |f (V )| < small , avoid dividing by zero; set f (V ) = small .
22. Update V using Newton Raphson
V = Vold −
f (V )
f (V )
(8.171)
23. Check if V out of range; update using binary search if necessary.
IF ((V <= Vmin ) OR (V >= Vmax ))THEN V =
Vmin + Vmax
2
24. Calculate convergence criteria and avoid dividing by zero:
146
(8.172)
IF (|Vold | < V ) THEN
convV = |V − Vold |
ELSE
V − Vold convV = Vold (8.173)
25. End of loop V
26. Loop m #1: (update Xm and Ym and avoid dividing by zero)
27. Calculate temporary variable a1 to avoid dividing by zero.
a1 = (Km − 1) ∗ V + 1
(8.174)
28. Update Xm and Ym
IF (|a1 | < small ) THEN
Xm = 1
Ym = 0
ELSE
Zm
Xm =
a1
Zm
Ym = Km ∗
a1
(8.175)
29. End of loop m #1
30. Loop m #2: (make sure all Xm and Ym are in range)
31. IF (Xm < Zm ) THEN Xm = 0
32. IF (Xm > 1) THEN Xm = 1
33. IF (Ym < Zm) THEN Ym = 0
34. IF (Ym > 1) THEN Ym = 1
35. End of loop m #2
36. Renormalize Xm and Ym
147
IF (
Xm < small ) THEN
m
Xm = Zm
ELSE
Xm
Xm = Xm
(8.176)
m
IF (
Ym < small ) THEN
m
Ym = Zm
ELSE
Ym
Ym = Ym
(8.177)
m
37. End of loop f (V )
38. End of loop K
8.11
Flowchart
This section presents flow charts for the flash calculations, Figure 8.2 and Figure 8.3.
148
Flash Calculation Flowchart
•
•
•
INPUT P, T , Z m
OUTPUT X m , Ym , Km , Pb , Pd ,V
Initial Values
–
–
–
–
–
3
1
Wilson : K m
Previous Km or Wilson
§
Previous V or 0.5; V : ¨¨ ( K 1 1) , ( K
©
Previous Pb or P
Previous Pd or P
Be careful of dividing by 0
m ,max
4
liquid : f (0) 0
2
V
0
V
4.1
·
¸¸
1)
m ,min
¹
1
5
Ym
Ym
Km Zm
Xm
z
L /V
,)
L /V
m
L /V
m
,f
Pb , X m , Ym
V e1 V e V e1
Ve
z
,)
V e1 V e d 106
Zm
5.4
Zm / Km
Xm
L /V
m
Zm
( K m 1)V 1
Normalize : ¦ X m
5.5
m
5.6
Calculate :
L /V
f (V e )
f '(V e )
5.2
m
4.3
Calculate :
two phases : f (0) ! 0 and f (1) 0
5.1
Normalize : ¦ X m 1
4.2
( K m 1) Z m
m 1)V 1
1
Zm
m
3.3
M
¦ (K
Calculate : f (V )
vapor : f (1) ! 0
Xm
Normalize : ¦ Ym 1
3.2
ª
§
1
1 ·º
exp «5.3727(1 Zm ) ¨1 ¸»
Prm
«¬
© Trm ¹ »¼
m 1
5.3
3.1
Start
L /V
m
,f
6
¦Y
m
1
m
z L /V , ) mL /V , fmL /V P, X m , Ym
Calculate : Pb
fmL
¦)
m
7
Rm
V
m
Pd
fmV
¦)
m
L
m
fmL / fmV
8
9
Rm 1 d 106
Figure 8.2: Regular flash calculation flow chart.
149
1
Km X m
Calculate :
Pd , X m , Ym
End
Ym
Kme1
Rme Kme
Extended Flash
Calculation Flowchart
Start
1
•
•
•
•
Wilson : K m
Used for thermodynamic MMP
INPUT P, T , Z
OUTPUT X , Y , K ,V
Initial Values
ª
§
1
1 ·º
exp «5.3727(1 Zm ) ¨1 ¸»
Prm
«¬
© Trm ¹ »¼
m
m
m
2
M
– Previous Km or Wilson
§
– Previous V or 0.5 V : ¨¨ (K
©
1
m ,max
m 1
·
1
,
¸
1) ( K m,min 1) ¸¹
5.1
V e1 V e – Be careful of dividing by 0
f (V e )
f '(V e )
5.2
• vaporizing gas drive
– Solve Zminitial oil
– MMP where
( K m 1) Z m
m 1)V 1
¦ (K
Calculate : f (V )
m
5.3
Ve
V e1
V e1 V e d 106
(E ( Km 1) 1) X m
E
10
5.4
• condensing gas drive
– Solve ZmINJ (E ( Km 1) 1) X m
– MMP where E 10
Xm
Zm
( K m 1)V 1
Normalize : ¦ X m
5.5
m
5.6
Ym
1
Km X m
¦Y
m
1
m
Calculate :
z L /V , ) mL /V , fmL /V P, X m , Ym
7
Rm
fmL / fmV
8
End
9
Rm 1 d 106
Kme1
Rme Kme
Figure 8.3: Flash calculation flow chart for thermodynamic minimum miscibility pressure.
150
CHAPTER 9
FORMULATION OF WELLS
This chapter is based on Kazemi et al. (1978) and course notes from Dr. Kazemi’s classes
(Kazemi, 2008a, 2009, 2010). All of the equations were re-derived as part of this dissertation.
9.1
Well Notation
The following notations are specific to wells.
Table 9.1: Superscripts
variable
units
name
#
n
script
index
n+1
index
index
e
index
emax
w
w
index
script
script
script
represents a constant term, one that does not vary with time
temporal index representing full time step; represents variables
evaluated explicitly at n
temporal index representing full time step; represents variables
evaluated implicitly at n + 1
temporal index representing nonlinear iteration level between
n = ( = 0) and n + 1 = ( + 1).
represents iterative solution for well connections; this iteration is done
for each step
represents converged solution for well connections
indicates that a variable is within the wellbore, not the reservoir
represents the terms of the well equation that do not depend on the
primary variables at time represents total properties for well w
Table 9.2: Subscripts
variable
units
name
w
α
index
index
α
index
indicates this property is for well number 1, 2, . . .
index for completions in a well; starts at the toe of the well and
increases towards the heel of the well. For a fully penetrating vertical
well, α = kmax − k + 1.
index for completions in a well, used in summations
Table 22.2 identifies the variables used in this document. The units given are typical units. The
units for empirical correlations are listed in a particular section are listed within each section that
contains correlations.
151
Table 9.3: Well variables
variable
units
Cαw
Dα
dw
α
γw
γo
γg
hf,α+ 1
day/ft3
ft
ft
psi/ft
psi/ft
psi/ft
ft
well bore storage coefficient
total vertical depth of completion α in a well
well inside diameter
specific gravity of aqueous phase
specific gravity of oil phase
specific gravity of gas phase
friction adjustment based on the length of the well segment
k
λw
λo
λg
μw
μo
μg
NRe
Q
q
rw,α
md
1/cp
1/cp
1/cp
cp
cp
cp
unitless
lbmol/day
ft3 /day
ft
ρ
sα
Vαw
w
vϕα
WI#
α
lbmol/ft3
unitless
ft3
3
ft /day
3
(ft /day)(cp/psi)
permeability
mobility of water phase
mobility of oil phase
mobility of gas phase
viscosity of water phase
viscosity of oil phase
viscosity of gas phase
Reynold’s number
molar flux rate
volumetric rate
effective wellbore radius for flow between the reservoir and
the well
density
skin factor for well
volume within wellbore
velocity of phase ϕ in wellbore
well index
2
9.2
name
Flow from Node to Well
Since fluid flow in the reservoir is based on average pressures represented at the center of the
grid cell, P̄w,α = Pw,α .
The flow rate from each perforated cell for producing wells is defined by:
w
qo,w,α = −WI#
w,α λo,w,α (P̄w,α − Pw,α )
w
qg,w,α = −WI#
w,α λg,w,α (P̄w,α − Pw,α )
(9.1)
w
qw,w,α = −WI#
w,α λw,w,α (P̄w,α − Pw,α )
The flow rate from each perforated cell for injection wells is defined by:
w
qt,w,α = −WI#
w,α λt,w,α (P̄w,α − Pw,α )
(9.2)
152
9.3
Well Index
The well index for a vertical well. If it’s a fractured well, k → kf,eff = kf φf + (1 − φf )km .
WI#
w,α
&
&
0.006328( kx,w,α )( ky,w,α )(2π)(Δzw,α )
√
=
√
( Δxw,α )( Δyw,α )
√
ln
− 12 + sw,α
( π)rw,w,α
(9.3)
For a horizontal well in the x-direction.
WI#
w,α
9.4
&
&
0.006328( ky,w,α )( kz,w,α )(2π)(Δxw,α )
√
=
√
( Δyw,α )( Δzw,α )
√
ln
− 12 + sw,α
( π)rw,w,α
(9.4)
Properties for Flow in Wellbore
Define the mass flow rates for producers as:
Qo,w,α = qo,w,α ξo,w,α
Qg,w,α = qg,w,α ξg,w,α
Qw,w,α = qw,w,α ξw,w,α
(9.5)
w
Qw,w,α = qw,w,α ξw,w,α
(9.6)
Define the mass flow rates for injectors as:
w
Qo,w,α = qo,w,α ξo,w,α
w
Qg,w,α = qg,w,α ξg,w,α
For both injectors and producers:
Qw
w,w,α =
α
Qw,w,α
Qw,w = Qw
w,w,αmax
(9.7)
α =1
For both injectors and producers:
Qw
hc,w,α
=
α
Qo,w,α + Qg,w,α
(9.8)
α =1
Define the total mole fraction for each component for producers as follows. If the denominator
w
w
= Zm,w,α−1
.
is approximately zero, then set Zm,w,α
w
Zm,w,α
α
=
α =1
Xmw,α Qo,w,α + Ymw,α Qg,w,α
α
α =1 Qo,w,α + Qg,w,α
153
(9.9)
Define the total mole fraction for each component for injectors as follows. If the denominator
w
w
= Zm,w,α+1
.
is approximately zero, then set Zm,w,α
αmax w
=
Zm,w,α
w
w
α =α Xmw,α Qo,w,α + Ymw,α Qg,w,α
αmax
α =α Qo,w,α + Qg,w,α
(9.10)
Compute the well mole fractions and densities using flash:
w
w
w
w
w
w
w
w
w
w
, Tw,α
, Zmw,α
} −−−→ {lw
{Pw,α
w,α , vw,α , Xm,w,α , Ym,w,α , Wm,w,α , ξo,w,α , ξg,w,α , ξw,w,α ,
flash
w
w
w
w
w
w
w
w
w
w
w
, γg,w,α
, γw,w,α
, ρw
γo,w,α
o,w,α , ρg,w,α , ρw,w,α , μo,w,α , μg,w,α , μw,w,α , Co,w,α , Cg,w,α , Cw,w,α } (9.11)
w
w
Qw
o,w,α = lw,α Qhc,w,α
Qo,w = Qw
o,w,αmax
w
w
w
Qw
t,w,α = Qo,w,α + Qg,w,α + Qw,w,α
w
w
Qw
g,w,α = vw,α Qhc,w,α
Qg,w = Qw
g,w,αmax (9.12)
Qt,w = Qw
t,w,αmax
(9.13)
Qw
w,w,α
w
ξw,w,α
(9.14)
Define the flow rates
w
=
qo,w,α
Qw
o,w,α
w
ξo,w,α
Qw
g,w,α
w
ξg,w,α
w
qg,w,α
=
w
w
w
w
= qo,w,α
+ qg,w,α
+ qw,w,α
qt,w,α
w
=
vo,w,α
w
qo,w,α
π w
2
4 (dw,α )
w
qw,w,α
=
w
qt,w
= qt,w,α
max
w
qg,w,α
π w
2
4 (dw,α )
w
vg,w,α
=
(9.15)
w
vw,w,α
=
w
qw,w,α
π w
2
4 (dw,α )
(9.16)
w
w
w
w
= vo,w,α
+ vg,w,α
+ vw,w,α
vt,w,α
9.5
(9.17)
Pressure in Wellbore
The well pressures are defined relative to the reference well pressure, Pw , which is defined at
the heel of the well, or α = αmax for well w.
WBS→0
w
Pw,α
= Pw +
α V w 1 · Cw 1
w,α +
w,α +
2
α =1
2
Δt
·
w,n+1
w,n
Pw,α
− Pw,α
gravity
+
154
αmax
−1
α =α
w
γt,w,α
·
Dw,α − Dw,α +1 − hf,w,α + 1
2
(9.18)
w
Cw,α+
1
2
=
w
γt,w,α+
1
2
Co,w,α+ 1 Qw
+ Cg,w,α+ 1 Qw
+ Cw,w,α+ 1 Qw
o,w,α+ 1
g,w,α+ 1
w,w,α+ 1
2
=
2
2
2
2
Qw
t,w,α+ 12
2
(9.19)
w
w
w
w
w
w
γo,w,α+
+ γg,w,α+
+ γw,w,α+
1Q
1Q
1Q
o,w,α+ 1
g,w,α+ 1
w,w,α+ 1
2
2
2
2
Qw
t,w,α+ 1
2
2
(9.20)
2
π
1
2
w
2
(Lw,α − Lw,α +1 ) ·
(dw
w,α ) + (dw,α +1 )
4
2
w
Vw,α
+ 1 =
2
vw
2
1
t,w,α+ 2
hf,w,α+ 1 = sign[q] · f[NRe,w,α+ 1 ] ·
2
2
(9.21)
86,400
(dw
) · (32.2)
w,α+ 1
· (Lw,α − Lw,α+1 )
(9.22)
2
⎛
·⎝
NRe,w,α+ 1 = 0.017224·dw
w,α+ 1
2
ρw
|v w
|
o,w,α+ 1 o,w,α+ 1
2
2
μw
o,w,α+ 1
2
2
+
ρw
|v w
|
g,w,α+ 1 g,w,α+ 1
2
μw
g,w,α+ 1
2
+
ρw
|v w
|
w,w,α+ 1 w,w,α+ 1
2
2
μw
w,w,α+ 1
2
⎞
⎠
2
(9.23)
Compute the following α + 12 terms using the following types of weighting: upstream weighting
for an injection well is α + 1, upstream weighting for an production well is α.
• Co : upstream weighted
• γ o : upstream weighted
g ,w,α+ 12
w
g ,w,α+ 12
w
• Qw
o
g ,w,α+ 12
w
: upstream weighted
: upstream weighted
• Qw
t,w,α+ 1
2
• vwo g ,w,α+ 12
w
: upstream weighted
w
• vt,w,α+
1 : upstream weighted
2
• ρ o g ,w,α+ 12
w
: upstream weighted
155
• μo g ,w,α+ 12
w
: upstream weighted
: arithmetic average
• dw
w,α+ 1
2
9.6
Compute the Moody Friction Factor
Calculate the Moody friction factor as follows:
f[NRe < 2000] : f1 =
64
NRe
(9.24)
For large Reynold’s numbers, compute f using Newton Raphson iteration of the following:
f[NRe
2εRe
18.574
1
√
= 1.7384 − 2 · log10
+
< 4000] : √
d
( f2 )
(NRe )( f2 )
f[2000 < NRe < 4000] : f = f1 [2000] + (f2 [4000] − f1 [2000]) ·
9.7
NRe − 2000
4000 − 2000
(9.25)
(9.26)
Computation for Fixed Rate
is fixed, calculate the flow rates, pressures, and other properties in the
When the rate qw
following order. This requires an iterative approach because the friction term in the pressure
calculation depends on the flow rate in a nonlinear way.
e
using (9.27) for producers and (9.28) for injectors.
1. Initialize qw,α
qeo g ,w,α
w
= α
max
1
#
n
α =1 WIw,α λt,w,α
n
WI#
w,α λ o
g ,w,α
w
qw
n
WI#
w,α λt,w,α
e
= α
qw
qt,w,α
#
max
n
WI
λ
α =1
w,α t,w,α
(9.27)
(9.28)
w,e
using (9.2).
2. Calculate Pw,α
w,e
=
Pw,α
n+1
e
n
qt,w,α
+ WI#
w,α λt,w,α Pw,α
(9.29)
n
WI#
w,α λt,w,α
w,e
w,e
using (9.9); flash to calculate γt,w,α
using (9.11).
3. Calculate Zmw,α
156
w,e+1
4. Calculate Pw,α
from bottom to top, (9.18).
w ,e
Pw,α
w,e+1
Pw,α
Pw
−1 αmax
w,e
w,e
= Pw,α
+
γt,w,α
· Dw,α − Dw,α +1 − hw,e
max
f,w,α + 1
(9.30)
2
α =α
5. Calculate qw,α from bottom to top using (9.2).
w,e+1
e+1
n
n+1
= −WI#
qt,w,α
w,α λt,w,α (Pw,α − Pw,α )
(9.31)
.
6. Calculate qw
e+1
=
qt,w
α
max
α =1
e+1
qt,w,α
(9.32)
,e+1 − qt,w
7. Repeat steps 2–6 until qt,w
< qw , using
e+2
qt,w,α
qt,w
=
e+1
qt,w
e+1
· qt,w,α
(9.33)
w ,e
.
Define Pw,α
w ,e
Pw,α
=
αmax
−1 α =α
w,e
γt,w,α
· Dw,α − Dw,α +1 − hw,e
f,w,α + 1
(9.34)
2
Each component equation Cw,α,m has a source term. This term has the following form for a
fixed rate well:
w,emax
n
n
n
n
n
n
n
n
n
n+1
−WI#
w,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α · Pw,α − Pw,α
(9.35)
In terms of δP and δPw :
n
n
n
n
n
n
n
n
n
− WI#
w,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α ·
RHS
RHS
well
RHS
diagonal
,
w ,emax
Pw,α + δPw,α − Pw + δPw + Pw,α
157
(9.36)
The coefficient of δP is
#
n
n
n
n
n
n
n
n
n
WDPmn
w,α = −WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α
(9.37)
The coefficient of δPw is
#
n
n
n
n
n
n
n
n
n
WDWmn
w,α = WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α
(9.38)
The constant terms associated with the well are
#
n
n
n
n
n
n
n
n
n
WCmn
w,α = WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α ·
w ,emax
− Pw, + Pw,α
Pw,α
(9.39)
Each well has a total rate equation. This equation has the following form for a fixed rate well:
=−
qt,w
α
max α =1
w,emax
n+1
−
P
WI#
·
P
·
w,α
w,α
w,α
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
(9.40)
w,emax
ξw,w,α
max
In terms of δP and δPw :
RHS
RHS
RHS
RHS
well
α
max allα #
,
w ,emax
qt,w = −
WIw,α ·
Pw,α + δPw,α − Pw + δPw + Pw,α
α =1
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
·
emax
n
n
qw,w,α
ξw,w,α λw,w,α
(9.41)
w,emax
ξw,w,α
max
The coefficient of δP is
QDPnw,α
= − WI#
w,α ·
emax n
ξo,w,α λno,w,α
qo,w,α
w,emax
ξo,w,α
max
+
emax n
qg,w,α
ξg,w,α λng,w,α
The coefficient of δPw is
158
w,emax
ξg,w,α
max
+
emax n
qw,w,α
ξw,w,α λnw,w,α
w,emax
ξw,w,α
max
(9.42)
QDWnw,α
=
α
max
α =1
#
WIw,α ×
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
(9.43)
w,emax
ξw,w,α
max
The constant terms associated with the constant rate equation
QCn
w,α
9.8
=
qt,w
+
α
max
#
WIw,α ×
Pw,α
α =1
e
n
n
max
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
w ,emax
− Pw, + Pw,α
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
×
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(9.44)
Computation for Fixed Pressure
When the pressure Pw is fixed, calculate the flow rates, pressures, and other properties in the
following order for the first timestep only.
1. Initialize properties for grid cell w, αmax .
w,e
= Pw .
1.1. Pw,α
max
1.2. Initialize qw,αmax using (9.2).
w,e
e
n
n+1
= −WI#
qt,w,α
w,αmax λt,w,αmax (Pw,αmax − Pw,αmax )
max
(9.45)
w,e
w,e
using (9.9); flash to calculate γt,w,α
using (9.11).
1.3. Calculate Zmw,α
max
max
2. Initialize from top to bottom:
w,e
assuming hf = 0 using (9.18).
2.1. Pw,α
w,e
Pw,α
=
Pw
+
αmax
−1
w,e
γt,w,α
max
· Dw,α
α =α
→0
− Dw,α +1 − hw,e
f,w,α + 1
(9.46)
2
e
using (9.2).
2.2. Calculate qt,w,α
w,e
e
n
n+1
= −WI#
qt,w,α
w,α λt,w,α (Pw,α − Pw,α )
159
(9.47)
w,e+1
w,e+1
3. Calculate Zmw,α
using (9.9); flash to calculate γt,w,α
using (9.11).
w from bottom to top using (9.18).
4. Calculate Pw,α
w,e+1
Pw,α
=
Pw
+
αmax
−1 w,e+1
γt,w,α
α =α
· Dw,α − Dw,α +1 − hw,e
f,w,α + 1
(9.48)
2
5. Calculate qw,α from bottom to top using (9.2).
w,e+1
e+1
n
n+1
= −WI#
qt,w,α
w,α λt,w,α (Pw,α − Pw,α )
(9.49)
w,e+1
w,e w
− Pw,α
6. Repeat 3–5 until max Pw,α
< Pw,α
When the pressure Pw is fixed, calculate the flow rates, pressures, and other properties in the
following order for all timesteps after the first.
1. Initialize properties from the previous timestep.
w,e
w,n−1
= γtw,α
1.1. Initialize γtw,α
w,n−1
1.2. Initialize hw,e
f,w,α = hf,w,α
2. Initialize from top to bottom
w,e
using (9.18).
2.1. Pw,α
w,e
Pw,α
=
Pw
+
αmax
−1
w,e
γt,w,α
α =α
· Dw,α − Dw,α +1 − hw,e
f,w,α + 1
(9.50)
2
e
using (9.2).
2.2. Calculate qt,w,α
w,e
e
n
n+1
= −WI#
qt,w,α
w,α λt,w,α (Pw,α − Pw,α )
(9.51)
w,e+1
w,e+1
using (9.9); flash to calculate γtw,α
using (9.11).
3. Calculate Zmw,α
w from bottom to top using (9.18).
4. Calculate Pw,α
w,e+1
Pw,α
=
Pw
+
αmax
−1 w,e+1
γt,w,α
α =α
· Dw,α − Dw,α +1 − hw,e
f,w,α + 1
2
160
(9.52)
e+1
5. Calculate qt,w,α
from bottom to top using (9.2).
w,e+1
e+1
n
n+1
= −WI#
qt,w,α
w,α λt,w,α (Pw,α − Pw,α )
(9.53)
w,e+1
w,e w
− Pw,α
6. Repeat 3–5 until max Pw,α
< Pw,α
Each component equation Cw,α,m has a source term. This term has the following form for a
fixed pressure well:
αmax
WI#
w,α
#
n
α =1 WIw,α λt,w,α
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α × qt,w
× Xm,w,α
(9.54)
:
In terms of δqw
αmax
α =1
WI#
w,α
×
n
WI#
w,α λt,w,α
n
n
ξo,w,α
λno,w,α
Xm,w,α
+
n
n
Ym,w,α
ξg,w,α
λng,w,α
+
n
n
Wm,w,α
ξw,w,α
λnw,w,α
RHS
well
,
× qt,w + δqt,w
(9.55)
The coefficient of δP is 0.
WDPmn
w,α = 0
(9.56)
is
The coefficient of δqt,w
WDWmn
w,α
αmax
=
α =1
WI#
w,α
×
n
WI#
w,α λt,w,α
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α
Xm,w,α
(9.57)
The constant terms associated with the well are
WCmn
w,α = −
αmax
α =1
WI#
w,α
n
WI#
w,α λt,w,α
×
,
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α × qt,w
(9.58)
Xm,w,α
161
Each well has a total rate equation. This equation has the following form for a fixed pressure
well:
=−
qt,w
α
max α =1
w,n+1
n+1
−
P
WI#
·
P
×
w,α
w,α
w,α
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
(9.59)
w,emax
ξw,w,α
max
:
In terms of δP and δqw
RHS
RHS
RHS
well
α
max allα ,
w,emax
δP
+ δqt,w
=−
+
WI#
×
P
P
qt,w
−
w,α
w,α
w,α
w,α
α =1
emax n
qo,w,α ξo,w,α λno,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
×
emax
n
n
qw,w,α
ξw,w,α λw,w,α
(9.60)
w,emax
ξw,w,α
max
The coefficient of δP is
QDPnw,α
= WI#
w,α ×
emax n
ξo,w,α λno,w,α
qo,w,α
w,emax
ξo,w,α
max
+
emax n
qg,w,α
ξg,w,α λng,w,α
w,emax
ξg,w,α
max
+
emax n
qw,w,α
ξw,w,α λnw,w,α
w,emax
ξw,w,α
max
(9.61)
is
The coefficient of δqt,w
QDWnw,α = 1
(9.62)
The constant terms associated with the constant rate equation
QCn
w,α
=
,
−qt,w
−
α
max
#
w,emax
WIw,α × Pw,α
×
− P
w,α
α =1
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
9.9
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(9.63)
Wells with Single Completions
For single well completions, the approach described in (9.27)–(9.63) can be simplified. The
source term for each component can be solved for directly, so the total well equations are not
necessary. The values of WDP and WC can be solved without iterations. There is no need for a well
variable, so the coefficient WDW = 0 for all wells. Because there are no total-well equations and
162
no well-specific primary variables, the Jacobian matrix will have a strictly block-banded structure.
9.9.1
Fixed Pressure Producer
For a fixed pressure producer with a single completion, the bottom hole producing pressure Pw
is specified at the elevation of the completion. There is no total well equation and no well variables,
so QDP = 0, QDW = 0, QC = 0, and WDW = 0.
WDPmn
ijk =
−WI#
w,ijk
WCm
ijk =
WI#
w,ijk
n
n
n
ξo,ijk
kro,ijk
Xm,ijk
μno,ijk
n
n
n
ξo,ijk
kro,ijk
Xm,ijk
μno,ijk
+
+
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
μng,ijk
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
μng,ijk
+
+
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
μnw,ijk
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
μnw,ijk
(9.64)
− Pw
Pijk
(9.65)
9.9.2
Fixed Rate Producer
<0
For a fixed rate producer with a single completion, the total bottom hole production rate qw
is specified. There is no total well equation and no well variables, so QDP = 0, QDW = 0, QC = 0,
and WDW = 0. The total rate is multiplied by various variables at n, so there is no dependence on
n+1
, and WDP = 0.
Pijk
λnt,ijk
=
λno,ijk
+
λng,ijk
+
λnw,ijk
=
n
kro,ijk
μno,ijk
+
n
krg,ijk
μng,ijk
+
n
krw,ijk
(9.66)
μnw,ijk
WDPm = 0
WCmn
ijk = −
9.9.3
(9.67)
qw
λnt,ijk
×
n
n
n
Xm,ijk
ξo,ijk
kro,ijk
μno,ijk
+
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
μng,ijk
+
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
μnw,ijk
(9.68)
Fixed Mole Rate Producer
For a fixed total molar rate producer with a single completion, the total bottom hole production
rate Qw,lbmol [lbmol/day] < 0 is specified. There is no total well equation and no well variables, so
QDP = 0, QDW = 0, QC = 0, and WDW = 0. The total rate is multiplied by various variables at
n+1
, and WDP = 0.
n, so there is no dependence on Pijk
163
λnt,ijk = λno,ijk + λng,ijk + λnw,ijk =
n
kro,ijk
μno,ijk
+
n
krg,ijk
μng,ijk
+
n
krw,ijk
(9.69)
μnw,ijk
WDPm = 0
(9.70)
WCmn
ijk = −Qw,lbmol
(9.71)
The following relates the molar rate to the volumetric rate.
qw
9.9.4
=
Qw,lbmol × λnt,ijk
(9.72)
n
n
n
ξo,ijk
λno,ijk + ξg,ijk
λng,ijk + ξw,ijk
λnw,ijk
Fixed Pressure Injector
For a fixed pressure injector with a single completion, the bottom hole injection pressure Pw
is specified at the elevation of the completion. As is typical for injectors, it is based on the total
w,n
n to
is flashed at Pijk
mobility of the grid cell injected into. The total specified composition Zm
w,n
w,n
w,n
, Ymw,n , Wm
, ξow,n , ξgw,n , and ξw
. There is no total well equation and no well
determine Xm
variables, so QDP = 0, QDW = 0, QC = 0, and WDW = 0.
λnt,ijk
=
λno,ijk
+
λng,ijk
+
λnw,ijk
=
n
kro,ijk
μno,ijk
+
n
krg,ijk
μng,ijk
+
n
krw,ijk
(9.73)
μnw,ijk
#
n
w,n w,n
w,n w,n
+ Ymw,n ξgw,n + Wm
ξw
WDPmn
ijk = −WIw,ijk λt,ijk Xm ξo
#
n
w,n w,n
w,n w,n
+ Ymw,n ξgw,n + Wm
ξw
WCm
ijk = WIw,ijk λt,ijk Xm ξo
9.9.5
− Pw
Pijk
(9.74)
(9.75)
Fixed Rate Injector
>0
For a fixed rate producer with a single completion, the total bottom hole production rate qw
is specified. There is no total well equation and no well variables, so QDP = 0, QDW = 0, QC = 0,
and WDW = 0. The total rate is multiplied by various variables at n, so there is no dependence on
n+1
, and WDP = 0.
Pijk
164
WDPm = 0
(9.76)
w = Z w . The aqueous density ξ w,n is determined based on W w ,
For a water injector, Wm
m
CO2
w,ijk
w , the reservoir temperature T # , and the reservoir pressure P n .
WNaCl
ijk
w w,n
WCmn
ijk = −qw Wm ξw
(9.77)
w,n
n to determine X w,n ,
is flashed at Pijk
For a gas injector, the total specified composition Zm
m
w,n
.
Ymw,n , ξow,n , ξgw,n , Lw,n
o , and Vg
w,n w,n w,n
WCmn
+ Ymw,n ξgw,n Vgw,n
ijk = −qw × Xm ξo Lo
9.9.6
(9.78)
Fixed Pressure Producer with Switch to Rate Control
Calculate the total mobility
λt =
kro krg
krw
+
+
μo
μg
μw
(9.79)
Calculate the total flow rate, which should be less than zero.
qcheck = −WI · λt (P +1 − P )
(9.80)
If the calculated total flow rate is greater than a maximum flow rate, |qcheck | > |qprodmax |, then
set the flow rate to qprodmax and calculate the well properties using rate control.
WDPm = 0
WCm
qprodmax
=−
λt
(9.81)
n
n
n
ξo,ijk
kro,ijk
Xm,ijk
μno,ijk
+
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
μng,ijk
+
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
μnw,ijk
(9.82)
If the calculated total flow rate qcheck ≥ 0, then set the flow rate to 0 and calculate the well
properties using rate control.
WDPm = 0
(9.83)
165
WCm = 0
(9.84)
If the calculated total flow rate is between 0 and the maximum flow rate qprodmax , −|qprodmax | ≤
qcheck ≤ 0, calculate the well properties using pressure control.
WDPmn
ijk
=
−WI#
w,ijk
n
n
n
ξo,ijk
kro,ijk
Xm,ijk
μno,ijk
+
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
μng,ijk
+
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
(9.85)
μnw,ijk
− Pw
WCm = −WDPm Pijk
9.9.7
(9.86)
Fixed Rate Producer with Switch to Pressure Control
Calculate the total mobility
λt =
kro krg
krw
+
+
μo
μg
μw
(9.87)
Calculate the bottom hole producing pressure
Pcheck =
q
+ P +1
WI · λt
(9.88)
If the calculated bottom hole producing pressure Pcheck < PBHPmin , then set the producing
pressure to PBHPmin and calculate the well properties using pressure control.
#
WDPmn
ijk = −WIw,ijk
n
n
n
ξo,ijk
kro,ijk
Xm,ijk
μno,ijk
+
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
− PBHPmin
WCm = −WDPm Pijk
μng,ijk
+
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
μnw,ijk
(9.89)
(9.90)
If the calculated bottom hole producing pressure Pcheck > PBHPmin , then calculate the well
properties using rate control.
WDPm = 0
(9.91)
166
WCmn
ijk = −
9.9.8
qw
λnt,ijk
×
n
n
n
Xm,ijk
ξo,ijk
kro,ijk
μno,ijk
+
n
n
n
Ym,ijk
ξg,ijk
krg,ijk
μng,ijk
+
n
n
n
Wm,ijk
ξw,ijk
krw,ijk
μnw,ijk
(9.92)
Fixed Pressure Injector with Switch to Rate Control
Calculate the total mobility
λt =
kro krg
krw
+
+
μo
μg
μw
(9.93)
Calculate the total flow rate, which should be greater than zero.
qcheck = −WI · λt (P +1 − P )
(9.94)
If the calculated total flow rate is greater than a maximum flow rate, |qcheck | > qinjmax , then set
the flow rate to qinjmax and calculate the well properties using rate control.
WDPm = 0
(9.95)
w = Z w . The aqueous density ξ w,n is determined based on W w ,
For a water injector, Wm
m
CO2
w,ijk
w , the reservoir temperature T # , and the reservoir pressure P n .
WNaCl
ijk
w w,n
WCmn
ijk = −qinjmax Wm ξw
(9.96)
w,n
n to determine X w,n ,
is flashed at Pijk
For a gas injector, the total specified composition Zm
m
w,n
.
Ymw,n , ξow,n , ξgw,n , Lw,n
o , and Vg
w,n w,n w,n
+ Ymw,n ξgw,n Vgw,n
WCmn
ijk = −qinjmax · Xm ξo Lo
(9.97)
If the calculated total flow rate qcheck ≤ 0, then set the flow rate to 0 and calculate the well
properties using rate control.
WDPm = 0
(9.98)
WCm = 0
(9.99)
167
If the calculated total flow rate is between 0 and the maximum flow rate qinjmax , 0 ≤ qcheck <
qinjmax , calculate the well properties using pressure control.
#
n
w,n w,n
w,n w,n
WDPmn
+ Ymw,n ξgw,n + Wm
ξw
ijk = −WIw,ijk λt,ijk Xm ξo
WCm = −WDPm
9.9.9
Pijk
−
Pw
(9.100)
(9.101)
Fixed Rate Injector with Switch to Pressure Control
Calculate the total mobility
λt =
kro krg
krw
+
+
μo
μg
μw
(9.102)
Calculate the bottom hole injection pressure
Pcheck =
q
+ P +1
WI · λt
(9.103)
If the calculated bottom hole injection pressure Pcheck > Pinjmin , then set the injection pressure
to Pinjmin and calculate the well properties using pressure control.
#
n
w,n w,n
w,n w,n
+ Ymw,n ξgw,n + Wm
ξw
WDPmn
ijk = −WIw,ijk λt,ijk Xm ξo
− Pinjmin
WCm = −WDPm Pijk
(9.104)
(9.105)
If the calculated bottom hole injection pressure Pcheck < Pinjmin , then calculate the well properties using rate control.
WDPm = 0
(9.106)
w = Z w . The aqueous density ξ w,n is determined based on W w ,
For a water injector, Wm
m
CO2
w,ijk
w , the reservoir temperature T # , and the reservoir pressure P n .
WNaCl
ijk
w w,n
WCmn
ijk = −qw Wm ξw
(9.107)
168
w,n
n to determine X w,n ,
For a gas injector, the total specified composition Zm
is flashed at Pijk
m
w,n
.
Ymw,n , ξow,n , ξgw,n , Lw,n
o , and Vg
w,n w,n w,n
WCmn
+ Ymw,n ξgw,n Vgw,n
ijk = −qw · Xm ξo Lo
169
(9.108)
CHAPTER 10
MASS BALANCE CALCULATIONS
This chapter describes methods to calculate the properties of well fluids at surface conditions
using separators, the calculation of original oil in place at surface conditions, and the mass balance
calculations used to evaluate the computational success of each nonlinear iteration and each time
step.
10.1
Calculate Surface Conditions of Well Fluids Using Separators
For this formulation, the WCO2 at surface conditions is assumed to be zero. All CO2 dissolved
in water at reservoir conditions is assumed to go into the gas line at surface conditions.
Define the mass flux at each well in lbmol/day.
Qw
om =
Qw
gm
w
Qwm
Qw
hc,m
Qw
o
Qw
g
Qw
w
Qw
hc
w ξw qw
Xm
o o
(10.1)
=
Ymw ξgw qgw
(10.2)
=
wξw qw
Wm
w w
w
Qom + Qw
gm
ξow qow
ξgw qgw
w qw
ξw
w
Qw
+
Qw
o
g
(10.3)
=
=
=
=
=
(10.4)
(10.5)
(10.6)
(10.7)
(10.8)
Define the hydrocarbon mole fractions:
w
Zhc,m
=
Qw
hc,m
(10.9)
Qw
hc
w
at the conditions of separator 1.
Flash Zhc,m
P sep1 ,T sep1
w
sep1
−−−−−−→ Xm
, Ymsep1 , Lsep1 , V sep1 , ξosep1 , ξgsep1
Flash Zhc,m
Define the new mass flux rates after separator 1.
170
(10.10)
Qsep1
o
Qsep1
g
Qsep1
om
Qsep1
gm
= Lsep1 Qw
hc
(10.11)
= V sep1 Qw
hc
sep1 sep1
= Xm
Qo
= Ymsep1 Qsep1
g
(10.12)
(10.13)
(10.14)
The liquid output of separator 1 goes to separator 2. The gas output of separator 1 goes into
sep1
at the conditions of separator 2.
the gas line. Flash Xm
P sep2 ,T sep2
sep1
sep2
−−−−−−→ Xm
, Ymsep2 , Lsep2 , V sep2 , ξosep2 , ξgsep2
Flash Xm
(10.15)
Define the new mass flux rates after separator 2.
Qsep2
o
= Lsep2 Qsep1
o
(10.16)
Qsep2
g
Qsep2
om
Qsep2
gm
=V
sep2 Qsep1
o
(10.17)
=
=
sep2 sep2
Xm
Qo
sep2 sep2
Y m Qg
(10.18)
(10.19)
The liquid output of separator 2 goes to separator 3. The gas output of separator 2 goes into
sep2
at the conditions of separator 3.
the gas line. Flash Xm
P sep3 ,T sep3
sep2
sep3
−−−−−−→ Xm
, Ymsep3 , Lsep3 , V sep3 , ξosep3 , ξgsep3
Flash Xm
(10.20)
Define the new mass flux rates after separator 3.
Qsep3
o
= Lsep3 Qsep2
o
(10.21)
Qsep3
g
Qsep3
om
Qsep3
gm
=V
sep3 Qsep2
o
(10.22)
=
=
sep3 sep3
Xm
Qo
sep3 sep3
Y m Qg
(10.23)
(10.24)
Compute the oil rate and molar oil rate at standard conditions.
sep3
/ξosep3 = Lsep3 Lsep2 Lsep1 Qw
qosc [SCF/day] = Qsep3
o
hc /ξo
Qsc
om [lbmol/day]
=
Qsep3
om
=
sep3 sep3 sep2 sep1 w
Xm
L L L Qhc
171
(10.25)
(10.26)
Compute the gas line molar rates:
line
= Qsep1
+ Qsep2
+ Qsep3
+ Qw
Qgas
g
g
g
g
w,CO2 =
sep2 sep1 w
w
L Qhc + V sep3 Lsep2 Lsep1 Qw
V sep1 Qw
hc + V
hc + Qw,CO2
(10.27)
Compute the gas line molar rates for each component (m = 1 . . . NC − 1):
line
sep2
sep3
w
= Qsep1
Qgas
gm
gm + Qgm + Qgm + Qw,m =
sep2 sep2 sep1 w
w
L Qhc + Ymsep3 V sep3 Lsep2 Lsep1 Qw
Ymsep1 V sep1 Qw
hc + Ym V
hc + Qw,m (10.28)
Calculate the mole fractions in the gas line:
gas line
Zm
=
line
Qgas
gm
(10.29)
line
Qgas
g
The non-ideal gas law is
P V = z̆nRT
(10.30)
To calculate the volume at standard conditions as a function of the number of moles, z̆ = 1. In
imperial units (P [psia], V [ft3 ], n[lbmol], and T [R]), use the following:
V [ft3 ] = nz̆RT /P =
R
T
P
3
ft psia
◦
◦
× (60 F + 459.67 F) / (14.7 psia) =
n[lbmol] × (1) × 10.731592
R lbmol
3 ft
(10.31)
n[lbmol] × 379.38
lbmol
z̆
Compute the gas rate at standard conditions.
line
× 379.38
qgsc [SCF/day] = Qgas
g
Qsc
gm [lbmol/day] =
SCF
lbmol
line
Qgas
gm
(10.32)
(10.33)
The water density at standard conditions is calculated from
sc
= ξw [P sep3 , T sep3 , WCO2 = 0, WNaCl ]
ξw
(10.34)
172
Compute the water rate at standard conditions
sc
sc
[SCF/day] = Qw
qw
w,H2 O /ξw
Qsc
w,CO2 [lbmol/day]
sc
Qw,H2 O [lbmol/day]
10.2
(10.35)
=
0
(10.36)
=
Qw
w,H2 O
(10.37)
Calculate Surface Conditions of Original Oil in Place
For this formulation, the WCO2 at surface conditions is assumed to be zero. All CO2 dissolved
in water at reservoir conditions is assumed to go into the gas line at surface conditions.
Define the initial fluids in the reservoir in lbmol.
init
Mom
init ξ init
= VRφinit Soinit Xm
o
(10.38)
init
Mgm
init
Mwm
init
Mhc,m
Moinit
Mginit
Mwinit
init
Mhc
= VRφinit Sginit Yminit ξginit
(10.39)
init W init ξ init
VRφinit Sw
m
w
init + M init
= Mom
gm
init
init
= VRφ So ξoinit
= VRφinit Sginit ξginit
init ξ init
= VRφinit Sw
w
init
init
= Mo + Mg
(10.40)
=
(10.41)
(10.42)
(10.43)
(10.44)
(10.45)
Define the hydrocarbon mole fractions:
init
=
Zhc,m
init
Mhc,m
(10.46)
init
Mhc
init at the conditions of separator 1.
Flash Zhc,m
P sep1 ,T sep1
init
sep1
−−−−−−→ Xm
, Ymsep1 , Lsep1 , V sep1 , ξosep1 , ξgsep1
Flash Zhc,m
(10.47)
Define the new mass flux rates after separator 1.
Mosep1
Mgsep1
sep1
Mom
sep1
Mgm
init
= Lsep1 Mhc
(10.48)
init
= V sep1 Mhc
sep1
= Xm
Mosep1
= Ymsep1 Mgsep1
(10.49)
(10.50)
(10.51)
173
The liquid output of separator 1 goes to separator 2. The gas output of separator 1 goes into
sep1
at the conditions of separator 2.
the gas line. Flash Xm
P sep2 ,T sep2
sep1
sep2
Flash Xm
−−−−−−→ Xm
, Ymsep2 , Lsep2 , V sep2 , ξosep2 , ξgsep2
(10.52)
Define the new mass flux rates after separator 2.
Mosep2
= Lsep2 Mosep1
(10.53)
Mgsep2
sep2
Mom
sep2
Mgm
=V
sep2 M sep1
o
(10.54)
=
=
sep2
Xm
Mosep2
Ymsep2 Mgsep2
(10.55)
(10.56)
The liquid output of separator 2 goes to separator 3. The gas output of separator 2 goes into
sep2
at the conditions of separator 3.
the gas line. Flash Xm
P sep3 ,T sep3
sep2
sep3
−−−−−−→ Xm
, Ymsep3 , Lsep3 , V sep3 , ξosep3 , ξgsep3
Flash Xm
(10.57)
Define the new mass flux rates after separator 3.
Mosep3
= Lsep3 Mosep2
(10.58)
Mgsep3
sep3
Mom
sep3
Mgm
=V
sep3 M sep2
o
(10.59)
=
=
sep3
Xm
Mosep3
Ymsep3 Mgsep3
(10.60)
(10.61)
Compute the initial volume of oil at standard conditions.
init /ξ sep3
OOIP[SCF] = Mosep3 /ξosep3 = Lsep3 Lsep2 Lsep1 Mhc
o
sc
[lbmol]
Mom
=
sep3
Mom
=
sep3 sep3 sep2 sep1
init
Xm
L L L Mhc
(10.62)
(10.63)
Compute the gas line molar volumes:
init
=
Mggas line = Mgsep1 + Mgsep2 + Mgsep3 + Mw,CO
2
init
init
init
init
+ V sep2 Lsep1 Mhc
+ V sep3 Lsep2 Lsep1 Mhc
+ Mw,CO
V sep1 Mhc
2
174
(10.64)
gas line
sep1
sep2
sep3
init
Mgm
= Mgm
+ Mgm
+ Mgm
+ Mw,m
=
init
init
init
init
+ Ymsep2 V sep2 Lsep1 Mhc
+ Ymsep3 V sep3 Lsep2 Lsep1 Mhc
+ Mw,m
(10.65)
Ymsep1 V sep1 Mhc
Calculate the mole fractions in the gas line:
gas line
=
Zm
gas line
Mgm
(10.66)
Mggas line
Compute the initial volume of free and associated gas at standard conditions.
OGIP[SCF] = Mggas line × 379.38
sc
Mgm
[lbmol]
SCF
lbmol
(10.67)
gas line
Mgm
=
(10.68)
The water density at standard conditions is calculated from
sc
= ξw [P sep3 , T sep3 , WCO2 = 0, WNaCl ]
ξw
(10.69)
Compute the water rate at standard conditions
w
sc
/ξw
Mwsc [SCF] = Mw,H
2O
sc
[lbmol]
Mw,CO
2
sc
Mw,H2 O [lbmol]
(10.70)
=
0
(10.71)
=
w
Mw,H
2O
(10.72)
Calculate the recovery factor:
RF =
10.3
cumulative qosc
OOIP
(10.73)
Mass Balance Calculations
Define the mass in lbmol of each phase:
Motm1 ,ijk
= Som1 ,ijk ξom1 ,ijk φm1 ,ijk VR
(10.74)
Mgtm1 ,ijk
= Sgm1 ,ijk ξgm1 ,ijk φm1 ,ijk VR
(10.75)
Mwtm1 ,ijk = Swm1 ,ijk ξwm1 ,ijk φm1 ,ijk VR
(10.76)
= Som2 ,ijk ξom2 ,ijk φm2 ,ijk VR
(10.77)
Motm2 ,ijk
175
Mgtm2 ,ijk
= Sgm2 ,ijk ξgm2 ,ijk φm2 ,ijk VR
(10.78)
Mwtm2 ,ijk = Swm2 ,ijk ξwm2 ,ijk φm2 ,ijk VR
(10.79)
Define the total system mass as
= Motm
+ Mgtm
+ Mwtm
+ Motm
+ Mgtm
+ Mwtm
Mijk
1 ,ijk
1 ,ijk
1 ,ijk
2 ,ijk
2 ,ijk
2 ,ijk
M =
Mijk
(10.80)
(10.81)
ijk
Define the injection and production rates in lbmol of each phase:
Qotm1 ,ijk
= qo,ijk ξom1 ,ijk Δt
(10.82)
Qgtm1 ,ijk
= qg,ijk ξgm1 ,ijk Δt
(10.83)
Qwtm1 ,ijk = qw,ijk ξwm1 ,ijk Δt
(10.84)
Define the total flux in lbmol.
Qijk = Qotm1 ,ijk + Qgtm1 ,ijk + Qwtm1 ,ijk
Q =
(10.85)
Qijk
(10.86)
ijk
Use (10.87) to determine the best solution, especially if there was no convergence.
+1
massbal = M +1 − M n − Q+1
inj + |Qprod |
(10.87)
Use (10.88) for the incremental mass balance.
massbalincr = 1 −
M +1 − M n
+1
Q+1
inj − |Qprod |
(10.88)
Use (10.89) for the cumulative mass balance.
massbalcum
n
n
0
n =0 M − M
= 1 − n
n
n
n =0 Qinj − |Qprod |
(10.89)
176
CHAPTER 11
RELATIVE PERMEABILITY AND CAPILLARY PRESSURE
There are two principal ways to specify the capillary pressure; capillary pressure as a function of
saturation or J-function as a function of saturation. Relative permeability is specified as a function
of saturation. For a more generalized formulation, capillary pressure and relative permeability are
functions of the following:
• Saturation of oil, water, and gas
• Hysteresis: direction of change of oil, water, and gas
• Trapping: trapped oil, water, and gas
• Interfacial tension and miscibility: explicit functional dependence on interfacial tension for
J-function specification of capillary pressure; need to specify relationship for relative permeability
• Rock Type, including porosity, permeability (explicit functional dependence for J-function
specification of capillary pressure), single porosity / dual porosity, and anisotropy. Rock type
may change with fluid-rock chemical interactions.
• Wettability: explicit functional dependence on contact angle for J-function specification of
capillary pressure; need to specify relationship for relative permeability
• Temperature
• Composition of fluids
• Fluid flow rate dependence
Traditionally, gas relative permeability is assumed to be a function of gas saturation only and
water relative permeability is assumed to be a function of water saturation only. Oil relative
permeability is usually assumed to be a function of all three phase saturations. For some oil wet
reservoirs, oil may be approximately a function of the oil saturation only. For mixed wet reservoirs
177
and for CO2 WAG scenarios in all kinds of reservoirs, the gas, oil, and water relative permeabilities
may be a function of all three phase saturations.
11.1
Three Phase Relative Permeability
Three-phase relative permeability models describe a way to use sets of two-phase relative permeabilities to calculate the three-phase relative permeability. These models almost always apply
to calculation of kro ; some may also be applied to krg and krw . The following articles discuss
three-phase relative permeability models.
• Land (1968), SPE #1942
• Stone (1970), SPE #2116: “Stone 1”
• Stone (1973): “Stone 2”
• Fayers and Matthews (1984), SPE #11277: Analysis of 3-phase relative permeability experiments. Modification of Stone 1.
• Baker (1988), SPE #17369: Analysis of 3-phase relative permeability experiments. Uses
saturation weighted relative permeabilities.
• Delshad and Pope (1989): Comparison of 7 different 3-phase relative permeability models2 .
Presents a different model for calculating kro .
• Fayers (1989), SPE #16965: Describes alternate ways to calculate Sorm for Stone 1 algorithm.
• Larsen and Skauge (1998), SPE #38456
• Pope et al. (1998), SPE #49266: Gas condensate relative permeability model
• Paterson, Painter, Zhang, and Pinczewski (1998), SPE #50938: Gas condensate relative
permeability model
• van Dijke et al. (2000), SPE #59310
2
All variables are defined in Chapter 22.
178
There are some limitations of these algorithms which are discussed in later literature. Many
articles mention difficulties with Stone 1 (Stone, 1970) and Stone 2 (Stone, 1973), including: Kleppe
et al. (1997), Larsen and Skauge (1998), Blunt (2000), Element et al. (2003), and Spiteri and Juanes
(2004). Drawbacks of Land (1968) are mentioned in Jerauld (1997), Blunt (2000), and Element
et al. (2003).
11.2
Hysteresis
Hysteresis means that a function depends not only on the current state but on some of the
past history. The term was originally coined in approximately 1800 to describe the lag in response
to magnetic forces. It was based on the Greek word “hystérēsis”, which means “deficiency”, or
“the state of being behind or late” (Gove, 1986; Wikipedia, 2010a). The relative permeability
and capillary pressure functions may be different depending on the increasing (I), decreasing (D),
or constant (C) state of each of the phases. This leads to the following twelve legal states for a
two-phase or three-phase system, listed as a 3-tuple in the order water-oil-gas:
• IID, IDI, DII
• IDC, ICD, DIC, DCI, CID, CDI
• IDD, DID, DDI
The initial state of the system is determined by the geologic history of the formation. The state
of individual grid cells are dynamic effects which change with the reservoir simulation.
11.2.1
Hysteresis Applications
Hysteresis is important when coning of water or gas is present, during immiscible gas injection,
during miscible gas injection, and during water-alternating-gas (WAG) injection. Hysteresis effects
change the relative permeability and capillary pressure based on changes in the increasing or decreasing state of water, oil, and gas phases. Hysteresis causes changes in recovery and different
timing of breakthrough or coning.
11.2.2
Hysteresis Literature
The following articles discuss hysteresis algorithms.
179
• Killough (1976), SPE #5106: Hysteresis algorithm
• Carlson (1981), SPE #10157: Non-wetting phase hysteresis
• Delshad et al. (2003), SPE #86916: Mixed wet model with hysteresis
• Spiteri et al. (2005), SPE #96448: New hysteresis model
There are some limitations of these algorithms which are discussed in later literature. Articles
which discuss limitations of the other algorithms include: Kleppe et al. (1997), Element et al.
(2003), and Spiteri and Juanes (2004).
11.2.3
Combined Three-Phase Relative Permeability and Hysteresis
Many more recent articles discuss both both three-phase relative permeability and hysteresis,
including:
• Jerauld (1997), SPE #36178: Correlations for 3-phase relative permeability and hysteresis to
fit an extensive mixed wet data set in Alaska.
• Blunt (2000), SPE #67950: 3-phase relative permeability and hysteresis model; comparison
of different models
• Egermann et al. (2000), SPE #65127: 3-phase relative permeability model with hysteresis
• Hustad et al. (2002), SPE #75138: Presentation of what Eclipse calls the “IKU” method;
comparison of different models
• Hustad (2002), SPE #74705: Presentation of what Eclipse calls the “ODD3P” method;
comparison of different models.
• Element et al. (2003), SPE #84903: Evaluation of different relative permeability and hysteresis models
• Spiteri and Juanes (2004), SPE #89921: Evaluation of different relative permeability and
hysteresis models
180
11.2.4
Combined Analysis of Algorithms
The following articles include a discussion of many of the previous algorithms.
• Blunt (2000)
– Review of previous algorithms, plus presentation of new model
– multiple problems indicated with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973), and
LandLand (1968)
– discussion of the other models: Vizika and Lombard (1996), Larsen and Skauge (1998),
Jerauld (1997)
– Element et al. (2003)
– Review of previous algorithms, using new data set
– Larsen and Skauge (1998): Problems because uses Stone 1; no trapping of water; no
variation of Land constant with cycle
– Blunt (2000): No wetting phase hysteresis; hysteresis is in a closed loop and shouldnt
be; no variation of Land constant with cycle
– Egermann et al. (2000): No variation of Land constant with cycle
• Spiteri and Juanes (2004)
– Review of previous algorithms, using Oaks data (Oak, 1990)
– Stone 1, Stone 2, Baker (1988): All are bad fits for kro
– Larsen and Skauge (1998): Not suitable for krg
– Other models: Killough (1976), Carlson (1981), Lenhard and Oostrom (1998), Jerauld
(1997), Blunt (2000)
11.3
Trapping
Trapping refers to the process of making a portion of one of the phases immobile. In two
dimensions trapping can be illustrated using capillary tubes of different sizes, Figure 11.1. In
three dimensions, variations in the pore size distribution, variation in the possible paths in three
181
dimensions, and the differing wettability along different mineral grains lead to many additional
ways for trapping to occur. Different histories of increasing and decreasing of each of the phase
saturations can lead to each of the following six possibilities, several of which may be present in
any grid cell.
• Gas trapped by oil
• Gas trapped by water
• Oil trapped by gas
• Oil trapped by water
• Water trapped by gas
• Water trapped by oil
Trapping typically occurs when one phase switches from increasing to decreasing saturation,
Figure 11.1(a) and Figure 11.1(b). Additional fluids may be trapped by later cycles. Trapped volumes will only decrease by diffusive processes or compositional effects. Trapped fluid compositions
change only by diffusive processes. Trapping changes are dynamic effects which change with the
reservoir simulation.
(a) Early displacement of oil by water; water
moves faster through the smaller pore throats.
(b) Later displacement of oil by water, showing
trapped oil.
Figure 11.1: Illustration of pore doublet effect in a water-wet rock; green is oil and blue is water.
Algorithms to calculate trapping are typically combined with three-phase relative permeability
and hysteresis in the literature.
182
11.3.1
Composition of Trapped Phase
One of the fundamental assumptions of reservoir simulation is that the phases and components
are in equilibrium with each other within each grid cell for each time step. This assumption is not
very good when it comes to the interaction of components in a trapped phase with components in
the mobile phases.
When a phase is trapped, its starting composition is the same as the mobile phase. Later on, the
composition of the trapped phase may change through diffusive processes or may remain constant.
None of the published articles on compositional simulation include the different compositions of
the trapped phase from the mobile phase.
If the compositions of the trapped phases are not tracked separately (base case), then the
following are required for every grid cell:
• Gas, Sg , Ym
• Oil, So , Xm
• Water, Sw , Wm
11.3.2
Simple Trapping Composition
The easiest composition trapping option to implement is for the trapped compositions to be
locked in place. The memory requirements are smaller than the option including diffusion, and it
is computationally much closer to the base case. If additional oil is trapped in a cell that already
has trapped oil, then the trapped composition Xmt is a weighted average of the existing Xmt and
the added mobile oil Xm . This option only adds additional explicit computations when a phase
changes from increasing to decreasing, triggering trapping. This option requires the following in
addition to the base case:
• Trapped gas, Sgt , Ymt ; only interaction Ym → Ymt
• Trapped oil, Sot , Xmt ; only interaction Xm → Xmt
• Trapped water, Swt, Wmt ; only interaction Wm → Wmt
183
To track hysteresis effects, the trapped saturations may also be required. These can be stored
individually for every grid cell, or a flag can track the presence of hysteresis in every grid cell and
then the trapped saturations are only stored for those cells where they are needed.
11.3.3
Complex Trapping Composition
An early reference for diffusion constrained trapping is Coats and Smith (1964). A more complicated composition tracking option involves the diffusive mixing of the trapped compositions.
Here it is necessary to track both the trapped phases and also which phase is doing the trapping,
because the kinetics and relevant equations are different. In addition, it also adds another explicit
primary equation every time step for each diffusion option that is used. This requires the following
in addition to the base case:
• Gas trapped by oil, Sgto , Ymto ; initial Ym → Ymto ; later interaction Ymto ↔ Xm
• Gas trapped by water, Sgtw , Ymtw ; initial Ym → Ymtw ; later interaction Ymtw ↔ Wm ; here
applies only to CO2 component
• Oil trapped by gas, Sotg , Xmtg ; initial Xm → Xmtg ; later interaction Xmtg ↔ Ym
• Oil trapped by water, Sotw , Xmtw ; initial Xm → Xmtw ; later interaction Xmtw ↔ Wm ; here
applies only to CO2 component
• Water trapped by gas, Swtg , Wmtg ; initial Wm → Wmtg ; later interaction Wmtg ↔ Ym ; here
applies only to CO2 component
• Water trapped by oil, Swto , Wmto ; initial Wm → Wmto ; later interaction Wmto ↔ Xm ; here
applies only to CO2 component
11.3.4
Composition Trapping Formulation
When trapping is added this yields (11.1).
0.006328∇ · Xm ξo λo k(∇Po − γo ∇D) + 0.006328∇ · Ym ξg λg k(∇Po + ∇Pcgo − γg ∇D) +
0.006328∇ · Wm ξw λw k(∇Po − ∇Pcow − γw ∇D) + Xm ξo q̂o + Ym ξg q̂g + Wm ξw q̂w − τmtt =
∂
φ(Xm So ξo + Ym Sg ξg + Wm Sw ξw )
(11.1)
∂t
184
The total transfer function for the trapped phase is (11.2).
τmtt = τmotw + τmotg + τmgtw + τmgto + τmwto + τmwtg
(11.2)
(11.3)–(11.8) represents the transfer functions for each phase trapped by every other.
τmgto = Kmgto σ (Xm So ξo ) − (Ymto Sgto ξgto ) =
τmgtw = Kmgtw σ (Wm Sw ξw ) − (Ymtw Sgtw ξgtw ) =
τmotg = Kmotg σ (Ym Sg ξg ) − (Xmtg Sotg ξotg ) =
τmotw = Kmotw σ (Wm Sw ξw ) − (Xmtw Sotw ξotw ) =
∂
φYmto Sgto ξgto )
∂t
∂
φYmtw Sgtw ξgtw )
∂t
∂
φXmtg Sotg ξotg )
∂t
∂
φXmtw Sotw ξotw )
∂t
(11.3)
(11.4)
(11.5)
(11.6)
τmwtg = Kmwtg σ (Ym Sg ξg ) − (Wmtg Swtg ξwtg ) =
∂
φWmtg Swtg ξwtg )
∂t
(11.7)
τmwto = Kmwto σ (Xm So ξo ) − (Wmto Swto ξwto ) =
∂
φWmto Swto ξwto )
∂t
(11.8)
11.4
Interfacial Tension
As oil and gas phases transition from immiscible to miscible or from miscible to immiscible, the
relative permeability and capillary pressure can change significantly. Computationally, the miscible
hydrocarbon phase may be labeled as either “oil” or “gas”. The hydrocarbon and water relative
permeabilities need to be the same regardless of how this phase is labeled. Capillary pressure using
the J-function formulation explicitly accounts for the contact angle between the phases. Miscibility
changes are dynamic effects which change with the reservoir simulation.
11.4.1
Interfacial Tension Literature
For any simulation model that includes miscibility, it is necessary to consider the changes of
relative permeability with interfacial tension. Several authors have used the capillary number
185
(Nc =
uμ
σ )
to scale the relative permeability, (Gibson, 2006; Stegemeier, 1977). Several authors
have used the ratio of the interfacial tension to a reference interfacial tension, σ/σ0 , (Coats, 1980;
Hustad, 2002; Karimaie and Torsæter, 2008). Several authors have used a density weighting function
fh =
ξh −ξg
ξo −ξg
in addition to the capillary number, (Blunt, 2000; Jerauld, 1997). Chase and Todd
(1984) use a “miscibility weighting function” α, which ranges between 0 for immiscible and 1 for
miscible. Schlumberger (2007a,b) use a ratio of the temperature to the critical temperature, T /Tc .
There does not seem to be any paper in the literature which compares these different techniques.
11.5
Rock Type and Wettability
This section describes the effects of rock type and wettability on relative permeability and
capillary pressure.
11.5.1
Rock Type
Different rock types are often used to initialize a simulation with different static relative permeability and capillary pressure curves. Different curves are also specified for the fracture system
and the matrix system in dual porosity or dual permeability regions. Different end points may
also be associated with individual grid cells to add additional variability to the properties. Rock
type changes are typically used to lump the effects of different mineralogy, permeability, porosity,
and pore size distribution. If the simulation grid is the result of upscaling a finer scaled geologic
distribution of properties, then it may be necessary to have the relative permeability and capillary
pressure functions differ in different directions.
11.5.2
Wettability Definitions
A good definition of wettability is “the tendency of one fluid to spread on or adhere to a solid
surface in the presence of other immiscible fluids” (Anderson, 1986a). There are also several other
definitions in 13933, including definitions based on contact angle, Amott index, and USBM index,
as well as some rules of thumb for wettability based on relative permeability, capillary pressure,
and residual saturations. The following six articles are a series of articles published in 1986-1987.
They review the wettability literature of the time and focus on different effects of wettability.
• Anderson (1986a), SPE #13932: General overview of wettability. Reference on wettability
definitions.
186
• Anderson (1986b), SPE #13933: Discussion of wettability measurement techniques. Good
discussion of different wettability indices.
• Anderson (1986c), SPE #13934: Discussion of how wettability affects the electrical properties.
Related to other work of the CSM/PI team.
• Anderson (1987a), SPE #15271: Discussion of how wettability affects capillary pressure.
• Anderson (1987b), SPE #16323: Discussion of how wettability effects relative permeability,
but does not define a functional relationship. As the wettability changes, the endpoints and
curvature both change.
• Anderson (1987c), SPE #16471: Discussion of how wettability affects waterflooding. This
has an impact on the total recovery from reservoirs as well as the remaining oil available for
enhanced oil recovery.
There are several illustrations of the variations of relative permeability with wettability, including several figures in Anderson (1987b), for instance, Figure 11.2. None of these articles give a
formula for this change as a function of contact angle.
11.5.3
Static Wettability Changes
Rocks with different wettabilities have different relative permeability and capillary pressure
functions. If the wettability is constant within a given rock type then the different relative permeability and capillary pressure functions may be specified with static properties based on the
rock types. In some reservoirs, there are variations in wettability within the same rock type; this
may be a result of the deposition of different amounts of asphaltenes over geologic time based on
compositional gradients and variations in the oil water contact elevation. Capillary pressure using
the J-function methodology incorporates the changes in wettability explicitly through variations in
the contact angle. It is possible to specify the variation between relative permeability in a variety
of ways, but a comparison of these methods is not discussed in the literature.
11.5.4
Dynamic Wettability Changes
Wettability may also change dynamically with the deposition of asphaltenes, with temperature
changes, and with fluid-rock chemical interactions. If any of these dynamic effects are simulated,
187
Figure 11.2: Variation of relative permeability with wettability changes (Anderson, 1987b).
188
then it is also necessary to specify how the relative permeability curves change dynamically with
the changes in wettability. The same approach used for wettability gradients should also work for
dynamic changes in wettability.
The permeability and porosity will also change with the deposition of asphaltenes. Because
asphaltenes will not be deposited uniformly in all pores and pore throats, this could also cause a
dynamic variation in the relative permeability and capillary pressure functions. There is insufficient
experimental data to understand exactly how the functional form of the capillary pressure and
relative permeability changes with this deposition. If asphaltene deposition is simulated, then the
dynamic impacts of this deposition will be accounted for using dynamic wettability changes and
an additional scaling of the end points.
11.6
Temperature
The relative permeability and capillary pressure functions change with temperature variations.
There is insufficient data in the literature to document whether these changes are solely a result
of the changes in the interfacial tension and wettability with changes in temperature. If there are
additional variations, then a method to change capillary pressure and relative permeability functions
would need to be created. Under static conditions, most reservoirs experience a temperature
gradient. For many simulation models, these temperature changes may be ignored, but they are
significant for some reservoirs. There also are some large reservoirs which have lateral variations in
initial reservoir temperature. Temperature may also change dynamically by the injection of cold
water into a initially hot reservoir, or by the injection of steam or other hot fluid into a reservoir. For
this dissertation, initial temperature variations may be simulated but time dependent temperature
changes will not be.
11.7
Composition
Relative permeability and capillary pressure are a function of different fluid compositions. This
is why most experimental relative permeability and capillary pressure functions are determined
using oil, gas, and brine typical of the producing formation. In addition to these effects, there also
seem to be different relative permeabilities for a gaseous CO2 :water system and a gaseous H2 S:water
system, (Bennion and Bachu, 2008b). This indicates that there are probably dynamic variations
189
in the relative permeability and capillary pressure functions with the changes in the composition
of the reservoir fluids, but insufficient experimental data exists to create a general compositional
model of these variations.
11.8
Flow Rates
Vary rapid flow rates will generate non-Darcy effects. It is assumed that these non-Darcy effects
are simulated using a different technique than variations in the relative permeability. Rapid flow
rates or geomechanical effects may also change the porosity, permeability, and structure of the
rock. These changes would also cause changes in the capillary pressure and relative permeability,
but these effects are beyond the scope of this work.
11.9
Brooks-Corey Properties for Mixed Wet Rock
The definitions of the saturations, porosities, and other terms used to describe the single porosity, dual porosity, and dual permeability systems with various amounts of trapping are described
in Chapter 5.
11.9.1
Simplified Three-Phase Relative Permeability
Relative permeability varies with the rock type; the wettability; the interfacial tension; the
previous maxima and minima of each saturation; and the increasing, decreasing, or constant status
of each phase. If all of these changes can be expressed as adjustments to the saturation endpoints
or the relative permeability value at the maximum saturation, then the following equations can be
used for all relative permeability calculations.
The total water relative permeability is defined by (11.9). For many systems, the water relative
permeability curve does not have significant hysteresis, so even when other forms are used to
calculate the three-phase relative permeability for the oil system, (11.9) may still be used for the
water system.
krw = krw
Sw − Sw,min
Sw,max − Sw,min
nw
= krw
1 − Sorm1
Sw − Swrm1 − Swm2
− Som2 − Sgrm1 − Sgm2 − Swrm1 − Swm2
nw
(11.9)
krw [Sw ≤ Sw,min ] = 0
krw [Sw ≥ Sw,max ] =
krw
190
(11.10)
The total oil relative permeability may be defined by (11.11). For many systems, it is defined
instead as a mixture of krow and krog .
kro =
kro
So − So,min
So,max − So,min
kro [So ≤ So,min] = 0
no
=
kro
1 − Sorm1
So − Sorm1 − Som2
− Som2 − Sgrm1 − Sgm2 − Swrm1 − Swm2
kro [So ≥ So,max ] = kro
no
(11.11)
(11.12)
The total gas relative permeability may be defined by (11.13). The form of this equation is
different from (11.9) and (11.11) in order to ensure that the derivative
∂kr
∂S
as S → Smin is non-zero
for at least one of the phases. Having one non-zero derivative and two zero derivatives stabilizes
the mathematics of the three-phase relative permeability calculations. Although the trapped and
residual properties change with the system, Sg,min stays the same for some systems. In this case,
krg is often specified using (11.13) rather than as a mixture of krgo and krgw .
krg =
krg1
Sg − Sg,min
Sg,max − Sg,min
krg [Sg ≤ Sg,min ] = 0
+
(krg
−
krg1
)
Sg − Sg,min
Sg,max − Sg,min
ng
(11.13)
krg [Sg ≥ Sg,max ] = krg
(11.14)
For a spreading oil, it is logical to rewrite kro using a non-zero derivative and write krg with a
zero derivative. In this case, (11.11) is rewritten as (11.15) and (11.13) is rewritten as (11.16).
kro = kro1
krg =
krg
So − So,min
So,max − So,min
Sg − Sg,min
Sg,max − Sg,min
− kro1
)
+ (kro
So − So,min
So,max − So,min
no
(11.15)
ng
(11.16)
Some cases with hysteresis require S-shaped scanning curves, for instance an increasing scanning
curve where the Smin endpoint was not reached, (11.17). This format is flexible; if krL [Smax1 ] =
krR [Smax1 ],
∂krL
∂S [Smax1 ]
=
∂krR
∂S [Smax1 ],
krR [Smax2 ] is specified, and n2 is negative, then this gener-
ates an S-shaped curve.
⎧
n1
S−Smin
⎨ krL = kr
,
S < Smax1
Smax1 −S
min
n2
S−Smin
⎩ krR = k + k
, S ≥ Smax1
r2
r3 Smax2 −Smin
191
(11.17)
Another approach used to define three-phase relative permeabilities is to specify individual
two-phase properties and then mix them in some way to obtain three-phase relative permeabilities.
11.9.2
Derivatives of Simplified Three-Phase Relative Permeability
For the IMPSEC formulation, it is necessary to calculate the derivatives of the relative permeability with respect to saturation. The following are the derivatives of the equations in Section 11.9.1. All adjustments to the Smin , Smax , and kr are made at time n, so these terms are
constants for purposes of the derivative calculations.
The derivatives of krw with respect to saturation from (11.9) are defined by:
nw
Sw − Sw,min
krw =
Sw,max − Sw,min
n
Sw − Sw,min
krw
∂krw
w
=
∂Sw
(Sw,max − Sw,min ) Sw,max − Sw,min
∂krw
[Sw → Sw,min ] = 0
∂Sw
∂krw
∂krw
=−
∂So
∂Sw
∂krw
∂krw
=−
∂Sg
∂Sw
krw
(11.18)
nw −1
=
nw
krw
(Sw − Sw,min)
(11.19)
(11.20)
(11.21)
(11.22)
The derivatives of kro with respect to saturation from (11.11) for a non-spreading oil are defined
by:
no
So − So,min
So,max − So,min
n
So − So,min
kro
∂kro
o
=
∂So
(So,max − So,min ) So,max − So,min
∂kro
[So → So,min ] = 0
∂So
∂krw
∂kro
=−
∂Sw
∂So
∂krw
∂kro
=−
∂Sg
∂So
kro =
kro
(11.23)
no −1
=
no
kro
(So − So,min )
(11.24)
(11.25)
(11.26)
(11.27)
The derivatives of krg with respect to saturation from (11.13) for the gas associated with a
non-spreading oil are defined by:
192
krg = krg1 + krg2
Sg − Sg,min
krg1 = krg1
Sg,max − Sg,min
ng
Sg − Sg,min
krg2 = (krg − krg1 )
Sg,max − Sg,min
krg1
∂krg1
=
∂Sg
(Sg,max − Sg,min )
− k )n
ng −1
(krg
Sg − Sg,min
ng
∂krg2
rg1 g
· krg2
=
=
∂Sg
(Sg,max − Sg,min ) Sg,max − Sg,min
(Sg − Sg,min )
krg1
ng
∂krg
+
· krg2
=
∂Sg
(Sg,max − Sg,min ) (Sg − Sg,min )
krg1
∂krg
[Sg → Sg,min ] =
∂Sg
(Sg,max − Sg,min )
∂krw
∂krg
=−
∂So
∂Sg
∂krw
∂krg
=−
∂Sw
∂Sg
(11.28)
(11.29)
(11.30)
(11.31)
(11.32)
(11.33)
(11.34)
(11.35)
(11.36)
The derivative of the S-shaped relative permeabilities are:
⎧
n1
S−Smin
⎨ krL = kr
,
Smax1 −Smin
n2
S < Smax1
S−Smin
⎩ krR = k + krR3 = k + k
, S ≥ Smax1
r2
r2
r3 Smax2 −Smin
n−1
n
krL
S − Smin
n
∂krL
=
krL
=
∂S
(Smax − Smin ) Smax − Smin
(S − Smin )
∂krL
[S → Smin ] = 0
∂S
n2 −1
n
kr3
S − Smin
n2
∂krR
2
=
krR3
=
∂S
(Smax2 − Smin ) Smax2 − Smin
(S − Smin )
11.9.3
(11.37)
(11.38)
(11.39)
(11.40)
Two-Phase Relative Permeabilities
The two-phase oil-water relative permeability is defined by (11.41). Note that this is a subset
of (11.11), but (11.11) explicitly accounts for the effects of trapped phases and the presence of gas.
krow =
∗
krow
So − Sowr
1 − Swr − Sowr
now
(11.41)
The two-phase water-oil relative permeability is defined by (11.42). Note that this is a subset
of (11.9), but (11.9) explicitly accounts for the effects of trapped phases and the presence of gas.
193
∗
krw
krw =
Sw − Swr
1 − Swr − Sowr
nw
(11.42)
The two-phase oil-gas relative permeability is defined by (11.43). Note that this is a subset of
(11.11), but (11.11) explicitly accounts for the effects of trapped phases and the presence of residual
phases of each type. (11.11) is also written in terms of So as a primary variable.
krog =
∗
krog
1 − Sg − Swr − Sogr
1 − Swr − Sogr
nog
(11.43)
The two-phase gas-liquid relative permeability is defined by (11.44). Note that this is a subset
of (11.13), but (11.13) explicitly accounts for the effects of trapped phases and the presence of
residual phases of each type.
∗
krg = krg
Sg
1 − Swr − Sogr
ng
(11.44)
There are several ways of combining separate two-phase relative permeabilities into three phase
relative permeabilities. One of these approaches is based on Stone’s method.
kro,mix =
11.9.4
Sg krog + (Sw − Swr )krow
Sg + Sw − Swr
(11.45)
Water-Oil Capillary Pressure for Mixed-Wet Systems
As usual, the following definitions are used for Sw,min and Sw,max :
Sw,min = Sw,m1 ,r + Sw,m2
(11.46)
Sw,max = 1 − So,m1 ,r − So,m2 − Sg,m1 ,r − Sg,m2
(11.47)
The oil-water capillary pressure for a mixed wet system is defined by both of these equations:
Sw − Sw,min
Swx − Sw,min
Sw,max − Sw
= Pcow,thr + αow (Sw,max − Swx ) ln
Sw,max − Swx
Pcow1 = Pcow,thr − αow (Swx − Sw,min ) ln
(11.48)
Pcow2
(11.49)
194
⎧
Pcow,max ,
Sw ≤ Sw,min
⎪
⎪
⎪
⎪
P
,
S
< Sw ≤ Sw,min,clip
⎪
cow,max
w,min
⎪
⎨
Pcow1 ,
Sw,min,clip < Sw ≤ Swx
=
Pcow2 ,
Swx < Sw ≤ Sw,max,clip
⎪
⎪
⎪
⎪
⎪
P
,
S
w,max,clip < Sw ≤ Sw,max
⎪
⎩ cow,min
Pcow,min ,
Sw ≥ Sw,max
Pcow
(11.50)
An alternate definition uses ow ≈ 10−4 to avoid calculating ln[0].
Sw − Sw,min + ow
= Pc,thr − αow (Swx − Sw,min + ow ) ln
Swx − Sw,min + ow
Sw,max − Sw + ow
= Pc,thr + αow (Sw,max − Swx + ow ) ln
Sw,max − Swx + ow
Pcow1
Pcow2
⎧
Pcow,max ,
Sw ≤ Sw,min
⎪
⎪
⎨
Pcow1 ,
Sw,min < Sw ≤ Swx
=
P
,
S
⎪
cow2
wx < Sw ≤ Sw,max
⎪
⎩
Pcow,min ,
Sw ≥ Sw,max
Pcow
(11.51)
(11.52)
(11.53)
Note that for both of these equations, Pcow1 [Swx ] = Pcow2 [Swx ] = 0.
11.9.5
Gas-Oil Capillary Pressure
As usual, the following definitions are used for Sg,min and Sg,max :
Sg,min = Sg,m1 ,r + Sg,m2
(11.54)
Sg,max = 1 − So,m1 ,r − So,m2 − Sw,m1 ,r − Sw,m2
(11.55)
The gas-oil capillary pressure for the primary drainage cycle is defined using Pcgo1 with an extra
threshhold term.
Pcgo1
Sg − Sg,min
= −αgo (Sg,max − Sg,min ) ln
Sgmax − Sg,min
(11.56)
⎧
⎪
⎪
⎨
Pcgo
Pcgo,max ,
Sg ≤ Sg,min
Pcgo,max ,
Sg,min < Sg ≤ Sg,min,clip
=
P
+ Pcgo1 , Sg,min,clip ≤ Sg ≤ Sg,max
⎪
⎪
⎩ cgoth
0,
Sg ≥ Sg,max
195
(11.57)
11.9.6
Derivatives of Capillary Pressure
The derivatives of the mixed wet oil-water capillary pressure are
∂Pcow1
∂Sw
∂Pcow1
∂So
∂Pcow1
∂Sg
∂Pcow2
∂Sw
∂Pcow2
∂So
∂Pcow2
∂Sg
= −αow
Swx − Sw,min
Sw − Sw,min
(11.58)
∂Pcow1
∂Sw
∂Pcow1
=−
∂Sw
Sw,max − Swx
= −αow
Sw,max − Sw
∂Pcow2
=−
∂Sw
∂Pcow2
=−
∂Sw
=−
(11.59)
(11.60)
(11.61)
(11.62)
(11.63)
∂Pcow2
∂Pcow1
[Swx ] =
[Swx ] = −αow .
∂Sw
∂Sw
The derivatives of the gas-oil capillary pressure are
Note that the derivatives
Sg,max − Sg,min
∂Pcgo1
= −αgo
∂Sg
Sg − Sg,min
∂Pcgo1
∂Pcgo1
=−
∂So
∂Sg
∂Pcgo1
∂Pcgo1
=−
∂Sw
∂Sg
11.10
(11.64)
(11.65)
(11.66)
Three Phase Relative Permeability References
The following articles discuss three-phase relative permeability models.
• Land (1968), SPE #1942: 3-phase relative permeability model
• Stone (1970), SPE #2116: “Stone 1”, 3-phase relative permeability model
• Stone (1973): “Stone 2”, 3-phase relative permeability model
• Fayers and Matthews (1984), SPE #11277: Analysis of 3-phase relative permeability experiments. Modification of Stone 1.
• Baker (1988), SPE #17369: Analysis of 3-phase relative permeability experiments. Uses
saturation weighted relative permeabilities.
196
• Delshad and Pope (1989): Comparison of 7 different 3-phase relative permeability models.
Presents a different model for calculating kro .
• Fayers (1989), SPE #16965: Describes alternate ways to calculate Sorm for Stone 1 algorithm.
• Pope et al. (1998), SPE #49266: Gas condensate relative permeability model
• Paterson et al. (1998), SPE #50938: Gas condensate relative permeability model
• van Dijke et al. (2000), SPE #59310: 3-phase relative permeability model
• Egermann et al. (2000), SPE #65127: 3-phase relative permeability model with hysteresis
The following articles discuss problems with some of the previous algorithms: Selected literature
indicating problems with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973)
• Kleppe et al. (1997)
• Larsen and Skauge (1998)
• Blunt (2000)
• Element et al. (2003)
• Spiteri and Juanes (2004)
11.11
Hysteresis References
Hysteresis models:
• Killough (1976), SPE #5106: Hysteresis calculation
• Carlson (1981), SPE #10157: Non-wetting phase hysteresis
• Blunt (2000), SPE #67950: 3-phase relative permeability and hysteresis model; comparison
of different models
• Delshad et al. (2003), SPE #86916: Mixed wet model with hysteresis
• Spiteri et al. (2005), SPE #96448: New hysteresis model
197
11.12
Combined Three-Phase Relative Permeability and Hysteresis References
Models for both three-phase relative permeability and hysteresis:
• Jerauld (1997), SPE #36178: Correlations for 3-phase relative permeability and hysteresis to
fit an extensive mixed wet data set.
• Larsen and Skauge (1998), SPE #38456: 3-phase relative permeability model
• Blunt (2000), SPE #67950: 3-phase relative permeability and hysteresis model; comparison
of different models
• Egermann et al. (2000), SPE #65127: 3-phase relative permeability model with hysteresis
• Hustad et al. (2002), SPE #75138: Presentation of the “IKU” method (named by Eclipse);
comparison of different models
• Hustad (2002), SPE #74705: 3-phase relative permeability and capillary pressure model with
hysteresis; called the “ODD3P” model by Eclipse.
• Element et al. (2003), SPE #84903: Evaluation of different relative permeability and hysteresis models
• Spiteri and Juanes (2004), SPE #89921: Evaluation of different relative permeability and
hysteresis models
Many of the algorithms presented in the literature have some limitations or drawbacks. The
following articles discuss problems with some of the previous algorithms:
• Selected literature indicating problems with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973)
– Kleppe et al. (1997)
– Larsen and Skauge (1998)
– Blunt (2000)
– Element et al. (2003)
– Spiteri and Juanes (2004)
198
• Jerauld (1997): Problems with Land (1968)
• Kleppe et al. (1997): Problems with Killough (1976)
• Blunt (2000): Review of previous algorithms, plus presentation of new model; multiple problems indicated with Stone 1 (Stone, 1970), Stone 2 (Stone, 1973), LandLand (1968); discussion
of the other models: Vizika and Lombard (1996), Larsen and Skauge (1998), Jerauld (1997)
• Element et al. (2003)
– Review of previous algorithms, using new data set
– Larsen and Skauge (1998): Problems because uses Stone 1; no trapping of water; no
variation of Land constant with cycle
– Blunt (2000): No wetting phase hysteresis; hysteresis is in a closed loop and shouldnt
be; no variation of Land constant with cycle
– Egermann et al. (2000): No variation of Land constant with cycle
• Spiteri and Juanes (2004)
– Review of previous algorithms, using Oaks data (Oak, 1990)
– Stone 1, Stone 2, Baker (1988): All are bad fits for kro
– Larsen and Skauge (1998): Not suitable for krg
– Other models: Killough (1976), Carlson (1981), Lenhard and Oostrom (1998), Jerauld
(1997), Blunt (2000)
199
CHAPTER 12
VISCOSITY FORMULATION
This chapter describes several different mathematical formulations to calculate the oil and gas
viscosity.
12.1
Treatment of Viscosity by Commercial Applications
There are two primary models for calculating the viscosity for a compositional model. The LBC
model, Lohrenz, Bray, and Clark (1964), has five regular tuning parameters plus the critical volumes
for each parameter. It is a reasonably good model if tuned, but is frequently off by 50% if it is not
tuned. The Pedersen model, Pedersen and Fredenslund (1987), is based on a corresponding states
model which maps the viscosity of a mixture to the viscosity of methane. It is tuned in different
ways by different programs. The Aasberg model, Aasberg-Petersen, Knudsen, and Fredenslund
(1991), is based on a corresponding states model which maps the viscosity of a mixture to the
viscosity of methane and the viscosity of n-decane. None of the commercial software packages
implement the Aasberg model.
Schlumberger Eclipse implements the Pedersen and the LBC models. It is possible to tune the
parameters for the LBC model by adjusting the 5 values in the polynomial correlation based on
the reduced density or by adjusting the vc values. No tuning is allowed for the Pedersen model.
Haliburton Landmark VIP implements the Pedersen and the LBC models. It is possible to tune
the parameters for the LBC model by adjusting the 5 values in the polynomial correlation based
on the reduced density or by adjusting the vc values. It is possible to tune the parameters for the
Pedersen model by adjusting the parameter values in μCH4 ,1 and μCH4 ,2 . VIP also allows binary
interaction parameters for the viscosity equation.
Computer Modeling Group GEM and Winprop implement the Pedersen model but does not
implement the LBC model. It is possible to tune the parameters for the Pedersen model by adjusting
the parameter values in the MWmix and the parameters in calculating α.
Calsep PVTsim implements the Pedersen and the LBC models. It is possible to tune the
parameters for the LBC model by adjusting the 5 values in the polynomial correlation based on
200
the reduced density or by adjusting the vc values. The Pedersen model can be tuned by adjusting
the parameters for calculating the MWmix and the parameters in μCH4 ,2 .
Powers implements the LBC model, but does not implement the Pedersen model. It is possible
to tune the parameters for the LBC model by adjusting the 5 values in the polynomial correlation
based on the reduced density or by adjusting the vc values.
12.2
Other Viscosity Models
Several other viscosity models that may have predictive capabilities were discovered in the
literature search.
The f -theory model, Quiñones-Cisneros, Zéberg-Mikkelsen, and Stenby (2001a), defines viscosity correlations based on friction theory using a dilute gas viscosity correlation and correlations based on the attractive and repulsive forces of the Peng-Robinson model. There are sixteen
parameters which have been determined based on fitting viscosity data for various hydrocarbon
components. There are another eleven parameters which have been determined for the dilute gas
viscosity correlation. There is also the option to tune the data using a single linear parameter, plus
the possibility of adjusting the critical volume and/or critical viscosity of various components. In
its simplest form, the model requires the Peng-Robinson correlation (including the critical temperature, critical pressure, acentric factor, molecular weight), plus the critical volume and the critical
viscosity. If the critical volume and critical viscosity are not known for a component, then they can
be calculated from the provided self-consistent correlations.
12.3
Lohrenz-Brae-Clark Model
This section discusses how viscosity changes with composition, based on correlations of Lohrenz,
Brae, and Clark (Lohrenz et al., 1964). The correlations in this section use T [R], P [psi], MW[lb/lbmol],
and μ[cp], ξ[lbmol/ft3 ].
12.3.1
Constants
The Lohrenz-Brae-Clark model requires the molecular weight, MWm , the critical pressure Pcm ,
the critical temperature Tcm , and the critical volume vcm for each component. Given a phase density
ξϕ , the mole fractions Xm , and a temperature T , the method will then calculate the viscosity of
201
the phase μϕ . Typically, ξϕ [P, T, X] is obtained from the equation of state, requiring ωm , δ̆mn , Ωa ,
Ωb , and cm in addition to Pcm and Tcm .
Define λm , with T in R and P in psia.
# 2/3
# 1/2 Pcm
MW
m
14.7
1
=
# 1/6
#
λm
Tcm
(12.1)
1.8
Define the relative temperature
#
=
Trm
T#
(12.2)
#
Tcm
Calculate the component viscosity at low pressure.
⎧
0.94
#
⎪
⎪
T
⎪
rm
⎪
−5
⎪
⎪
⎨34 · 10 ·
λ#
m
[cp]
=
μ#
5/8
m
⎪
⎪
⎪
17.78 · 10−5 · 4.58 · Tr# − 1.67
⎪
⎪
⎪
⎩
λ#
m
12.3.2
#
Trm
≤ 1.5
(12.3)
#
Trm
> 1.5
Time-Dependent
For the oil phase, the pseudo-reduced properties of mixtures are defined as follows:
pPcn =
n #
Xm
Pcm
pTcn =
pξcn = n #
Xm
Tcm
1
n #
vcm
Xm
(12.4)
For the gas phase, the pseudo-reduced properties of mixtures are defined as follows:
pPcn =
#
Ymn Pcm
pTcn =
#
Ymn Tcm
pξcn = 1
#
Ymn vcm
(12.5)
For the oil phase, the total properties are defined as follows
MWnt =
n
Xm
MW#
m
(12.6)
m
202
1/2
n #
Xm
μm MW#
m
1/2
n
Xm
MW#
m
μnt,low =
(12.7)
For the gas phase, the total properties are defined as follows
MWnt =
Ymn MW#
m
(12.8)
m
1/2
#
MW
Ymn μ#
m
m
1/2
Ymn MW#
m
μnt,low =
(12.9)
The λt uses the same equation for both the oil and gas phases:
λnt
=
pTcn
1.8
1/6
(MWnt )−1/2
pPcn
14.7
−2/3
(12.10)
Calculate the relative density, using ϕ = o, g:
n
ξϕr
ξϕn
= n
pξc
(12.11)
Calculate the viscosity. The Lohrenz, Brae, and Clark model uses the correlation presented by
Jossi, Stiel, and Thodos (1962).
(μnϕ [cp] − μnt,low ) · λnt + 10−4
1/4
=
2
3
4
+ −0.040758 · ξϕr
+ 0.0093324 · ξϕr
(12.12)
0.10230 + 0.023364 · ξϕr + 0.058533 · ξϕr
12.4
Jossi plus Lee
CMG does not provide the Lohrenz, Brae, and Clark (Lohrenz et al., 1964) model. The Jossi
model (Jossi et al., 1962) with low pressure viscosity calculated based on Lee and Eakin (1964) is
a model that can be implemented relatively easily to compare with CMG models. The correlations
in this section use T [R], P [psi], MW[lb/lbmol], and μ[cp], ξ[lbmol/ft3 ]; CMG uses T [K], P [atm],
MW[g/gmol], μ[cp], and ξ[kmol/m3 ]
For the oil phase, the pseudo-reduced properties of mixtures are defined as follows:
203
pPcn =
n #
Xm
Pcm
pTcn =
pξcn = n #
Xm
Tcm
1
n #
Xm
vcm
(12.13)
For the gas phase, the pseudo-reduced properties of mixtures are defined as follows:
pPcn =
#
Ymn Pcm
pTcn =
pξcn = #
Ymn Tcm
1
#
Ymn vcm
(12.14)
For the oil phase, the total properties are defined as follows
MWnt =
n
Xm
MW#
m
(12.15)
m
For the gas phase, the total properties are defined as follows
MWnt =
Ymn MW#
m
(12.16)
m
Based on Lee and Eakin (1964), the low pressure viscosity is calculated based on
3
μt,low =
10−4 (7.43 + 0.0133MWt ) (T [R]) 2
T [R] + 75.4 + 13.9MWt
(12.17)
The λt uses the same equation for both the oil and gas phases:
λnt
=
pTcn
1.8
1/6
(MWnt )−1/2
pPcn
14.7
−2/3
(12.18)
Calculate the relative density, using ϕ = o, g:
n
=
ξϕr
ξϕn
pξcn
(12.19)
Calculate the viscosity.
(μnϕ [cp] − μnt,low ) · λnt + 10−4
1/4
=
2
3
4
+ −0.040758 · ξϕr
+ 0.0093324 · ξϕr
(12.20)
0.10230 + 0.023364 · ξϕr + 0.058533 · ξϕr
204
12.5
Corresponding States
The corresponding states model, as presented by Pedersen and Christensen (2007), relates the
viscosity of an oil or gas mixture to the viscosity of methane. The viscosity of methane is calculated
from a very detailed correlation of methane density and viscosity as a function of pressure and
temperature, with some additional adjustments below the freezing point of methane. The units in
this section use T [K], P [atm], ρ[g/cm3 ], ξ[gmol/L], and μ[cp].
Table 12.1: Units for (12.21).
variable
units
P
P
ξ
atm
bar
gmol/L
T
μ
K
cP
name
pressure P [atm] = P [psi]/14.6959488
pressure P [bar] = P [atm] ∗ 1.01325
molar density ξ[gmol/L] = 16.0184634 · ξ[lbmol/ft3 ]
gmol = lbmol/453.59237
liter = ft3 /28.31684659
temperature T [K] = 273.15 + (T [F ] − 32) · 59
viscosity
The corresponding states model requires the critical pressure Pcm , the critical temperature
Tcm , the molecular weight MWm , and a detailed model of the viscosity and density as a function
of temperature and pressure for methane. Given the pressure P , the temperature T , and the mole
fractions of each component Xm the method will then calculate the viscosity of the phase μϕ .
12.5.1
Methane Density
The following correlation for pressure as a function of methane density is based on Pedersen
and Christensen (2007) and McCarty (1974). This correlation is in the form of P [ξ, T ] but what we
need is ξ[P, T ]; this is calculated using Newton-Raphson iterations. The solution seems to converge
to the correct value for a wide range of pressure and temperatures if ξ = 25gmol/L is used as an
initial estimate. It is important not to use the value estimated from the Peng-Robinson equation
of state, since this correlation is more accurate for methane than the more general Peng-Robinson
EOS module.
205
F ξCH4 [ξ, P, T ] :
0 = −P + 0.08205616 · ξ · T +
ξ 2 · (−0.018439486666 · T + 1.0510162064 · T 1/2 +
− 16.057820303 + 848.44027562 · T −1 + −4.2738409106 · 104 · T −2 )+
ξ 3 · (7.6565285254 · 10−4 · T + −0.48360724197 + 85.195473835 · T −1 + −1.6607434721 · 104 · T −2 )+
ξ 4 · (−3.7521074532 · 10−5 · T + 2.8616309259 · 10−2 + −2.868528973 · T −1 )+
ξ 5 (1.1906973942 · 10−4 )+
ξ 6 · (−8.5315715699 · 10−3 · T −1 + 3.8365063841 · T −2 )+
ξ 7 · (2.4986828379 · 10−5 · T −1 )+
ξ 8 · (5.7974531455 · 10−6 · T −1 + −7.1648329297 · 10−3 · T −2 )+
ξ 9 · (1.2577853784 · 10−4 · T −2 )+
exp[−0.0096 · ξ 2 ] · ξ 3 · (2.2240102466 · 104 · T −2 + −1.4800512328 · 106 · T −3 )+
ξ 5 · (50.498054887 · T −2 + 1.6428375992 · 106 · T −4 )+
ξ 7 · (0.21325387196 · T −2 + 37.791273422 · T −3 )+
ξ 9 · (−1.1857016815 · 10−5 · T −2 + −31.630780767 · T −4 )+
ξ 11 · (−4.1006782941 · 10−6 · T −2 + 1.4870043284 · 10−3 · T −3 )+
ξ 13 · (3.1512261532 · 10−9 · T −2 + −2.1670774745 · 10−6 · T −3 + 2.4000551079 · 10−5 · T −4 )
(12.21)
The Newton-Raphson evaluation of ξ is as follows
ξ
+1
∂F ξCH4 [ξ , P, T ]
= ξ − F ξCH4 [ξ , P, T ] /
∂ξ
(12.22)
With convergence criteria
+1
ξ
− ξ < ξ
ξ
(12.23)
Figure 12.1 shows that this correlation for density has been properly coded using the correct
units.
12.5.2
Methane Viscosity
The critical properties for methane, from Hanley, McCarty, and Haynes (1975) are the following.
It is important to use these values exactly to reproduce the correlation accurately.
• molecular weight MW = 16.043g/gmol.
206
Compare to Pedersen Figure 10.3
1000
800
P bar
600
400
200
0
24
26
28
30
32
34
CH4 density molL
Figure 12.1: Compare the density correlation for methane to Pedersen Figure 10.3. Each set of
dots represents steps of 30bar for a specific temperature from the code.
207
• freezing temperature TF = 91K
• critical temperature Tc = 190.55K
• critical pressure Pc = 45.387atm
• critical molar density ξc = 10.15gmol/L
• critical density ρc = 0.162836g/cm3
There is extensive data on the viscosity of methane. This correlation is based on Pedersen
and Christensen (2007) and Hanley et al. (1975), with a low temperature adjustment based on
Pedersen and Fredenslund (1987). The tanh terms are adjusted to fit Pedersen and Christensen
(2007), Figure 10.4. The coefficient of 1.0ΔT does not reproduce Figure 10.4.
μCH4 [ρ[g/cm3 ], T [K]] = 10−4 · μCH4 ,0 [T ] + μCH4 ,1 [T ]+
1 − tanh[0.1(T − TF,CH4 )]
1 + tanh[0.1(T − TF,CH4 )]
μCH4 ,2 [ρ, T ] +
μCH4 ,3 [ρ, T ]
2
2
(12.24)
μCH4 ,0 [T [K]] =
− 2.090975 · 105 · T −1 + 2.647269 · 105 · T −2/3 + −1.472818 · 105 · T −1/3 +
4.716740 · 104 + −9.491872 · 103 · T 1/3 + 1.219979 · 103 · T 2/3 +
− 9.627993 · 101 · T + 4.274152 · T 4/3 + −8.141531 · 10−2 · T 5/3 (12.25)
T
μCH4 ,1 [T [K]] = 1.696985927 + −0.133372346 · 1.4 − ln
168.0
2
(12.26)
μCH4 ,2 [ρ[g/cm3 ], T [K]] = exp −10.35060586 + 188.73011594 · T −1 ×
exp ρ0.1 · (17.571599671 + −3019.3918656 · T −3/2 )+
ρ − ρc √
· ρ · (0.042903609488 + 145.29023444 · T −1 + 6127.6818706 · T −2 ) − 1.0
ρc
208
(12.27)
μCH4 ,3 [ρ[g/cm3 ], T [K]] = exp −9.74602 + 44.6055 · T −1 ×
exp ρ0.1 · (18.0834 + −4126.66 · T −3/2 )+
ρ − ρc √
· ρ · (0.976544 + 81.8134 · T −1 + 15649.9 · T −2 ) − 1.0
ρc
(12.28)
Figure 12.2 shows that this correlation for viscosity has been properly coded using the correct
units. The Pedersen and Fredenslund (1987) model is very good for 100bar and 800bar. The
coefficient of the tanh term was adjusted to get as good a fit as possible at 2000bar. All fits were
very good for the Hanley et al. (1975) model, provided that the identical numerical values for the
critical pressure, temperature, and density values are used.
Compare to Pedersen Figure 10.4
Μcp
70
1.6
80
90
100
110
120
1.6
1.2
1.2
0.8
0.8
0.4
0.4
0.
70
80
90
100
110
0.
120
TK
Figure 12.2: Compare the viscosity correlation for methane to Pedersen Figure 10.4. The dots are
temperature steps of 1K. The red dots are for the Pedersen 1987 model. The green dots are for
the Hanley 1975 model.
Figure 12.3 shows that the Hanley et al. (1975) correlation is a also good predictor of the
experimental data at higher temperatures and pressures specified in Gonzalez, Bukacek, and Lee
(1967). Note that the Hanley et al. (1975) correlation falls within the range of the experimental
209
data, although at higher pressures it is further from the Gonzalez et al. (1967) correlation.
Compare to Gonzalez Figure 2
Μ104 cp
40
390
80
120
160
200
240
280
320
360
400
440
390
370
370
350
350
330
330
310
310
290
290
270
270
250
250
230
230
210
210
190
190
170
170
150
40
80
120
160
200
240
280
320
360
400
150
440
T°F
Figure 12.3: Compare the Hanley viscosity correlation for methane to Gonzalez Figure 2. The dots
are temperature steps of 5◦ F.
12.5.3
Corresponding States Calculations
The mixed critical temperature is defined as
# 1/3 ⎞3 )
# 1/3
Tcj
T
#
ci
⎠
Xin Xjn ⎝
+
Tci# · Tcj
#
#
Pci
Pcj
j
⎛
# 1/3 ⎞3
# 1/3
Tcj
Tci
⎠
Xin Xjn ⎝
+
#
#
P
P
ci
cj
i
j
n
[Xın ] =
Tc,mix
i
⎛
The mixed critical pressure is defined as
210
(12.29)
i
Pc,mix [Xın ] = 8 ·
⎛
⎛
Xin Xjn ⎝
j
Tci#
Pci#
1/3
+
#
Tcj
1/3 ⎞3 )
⎠
#
Pcj
#
Tci# · Tcj
⎞
⎛
1/3 # 1/3 ⎞3 2
#
Tcj
⎜ n n ⎝ Tci
⎠ ⎟
Xi Xj
+
⎝
⎠
#
#
Pci
Pcj
i
j
(12.30)
The following properties are not time-dependent and can be pre-calculated for Tc,mix and Pc,mix
⎛
TcPc#
ij
=⎝
Tci#
Pci#
1/3
+
i
#
Tcj
#
Pcj
1/3 ⎞3
⎠
i
=
TcPc#
ij
Xin Xjn TcPc#
ij
)
#
· Tci# · Tcj
Xin Xjn TcTc#
ij
j
n
[Xın ] = Tc,mix
TcTc#
ij
n
Pc,mix
[Xın ] = 8 · ⎛
⎝
j
i
i
Xin Xjn TcTc#
ij
j
(12.31)
⎞2
(12.32)
⎠
Xin Xjn TcPc#
ij
j
The reduced density is defined by
*
ξCH4
ξrn [P n , T # , Xın ] =
#
#
T # · Tc,CH
P n · Pc,CH
4
4
,
n
n
Pc,mix
Tc,mix
+
(12.33)
#
ξc,CH
4
The molecular weight of the mixture is defined by
MWnmix [Xın ] = 1.304 · 10−4 · ( MWnw )2.303 − ( MWnn )2.303 + MWnn
(12.34)
Using the weight averaged and number averaged molecular weights:
MWnw [Xın ] =
i
2
Xin MW#
i
Xin MW#
i
MWnn [Xın ] =
Xin MW#
i
(12.35)
i
i
The adjustment factor αmix is defined by:
αnmix [P n , T # , Xın ] = 1.00 + 7.378 · 10−3 · (ξrn )1.847 (MWnmix )0.5173
211
(12.36)
The methane adjustment factor αCH4 is defined by:
αnCH4 [P n , T # ] = 1.00 + 7.378 · 10−3 ·
ξCH4 P n , T #
1.847 #
ξc,CH
4
MW#
CH4
0.5173
(12.37)
The adjusted pressure and temperature are:
P0n [P n , T # , Xın ]
=
#
P n · Pc,CH
· αnCH4
4
T0n [P n , T # , Xın ]
n
Pc,mix
· αnmix
=
#
T # · Tc,CH
· αnCH4
4
n
Tc,mix
· αnmix
(12.38)
The viscosity of the mix is defined by:
μnmix,L [P n , T # , Xın ] =
12.5.4
n
Tc,mix
−1/6 #
Tc,CH
4
·
n
Pc,mix
2/3 ·
#
Pc,CH
4
MWnmix
MW#
CH4
1/2 αnmix
·
αnCH4
·μCH4 [P0n , T0n ] (12.39)
Heavy oil adjustment
For heavy oils, the corresponding states model based on methane is not accurate. The following
adjusted viscosity for heavy oils is based on Pedersen and Christensen (2007) and Rønningesen
(1993).
μmix,H [P n , T # , Xın ] = 10
−1
−0.07995−0.1101M n −371.8(T # )
−1
+6.215M n (T # )
exp[0.008 · (P n − 1.000)]
(12.40)
Here, M is defined by the following
⎧
MWnw
⎪
⎪
⎪
⎨ MWn ≤ 1.5,
n
n
n
M [Xı ] =
n
MW
⎪
w
⎪
⎪
⎩ MWn ≥ 1.5,
n
MWnn
MWnn ·
,
MWnw
1
·
1.5 MWnn
(12.41)
Define the viscosity of the mixture in the following way, using (12.38) to define the mix temperature.
• If T0 ≥ 75K, μmix = μmix,L based on (12.39).
• If T0 ≤ 65K, μmix = μmix,H based on (12.40).
• If 65K < T0 < 75K, μmix = (μmix,H + μmix,L )/2.
212
Since the Rønningesen (1993) model is an adjustment to the corresponding states model, it
requires the same things as the corresponding states model: the critical pressure Pcm , the critical
temperature Tcm , the molecular weight MWm , and a detailed model of the viscosity and density
as a function of temperature and pressure for methane. Given the pressure P , the temperature T ,
and the mole fractions of each component Xm the method will then calculate the viscosity of the
phase μϕ .
12.6
Extended Corresponding States
The extended corresponding states model, as presented by Aasberg-Petersen et al. (1991),
relates the viscosity of an oil or gas mixture to the viscosity of methane and n-decane. It is based
on Pedersen and Fredenslund (1987). As presented here, the methane correlations from Pedersen
and Fredenslund (1987) are used directly. For n-decane, the viscosity correlation from AasbergPetersen et al. (1991) is used, and compared graphically to Lee and Ellington (1965). The density
correlation in Aasberg-Petersen et al. (1991) does not provide the full details of the method, and
references other works that are not available or not cited in their bibliography. As a result, ndecane density correlations from Audonnet and Pádua (2004), Cibulka and Hnědkovský (1996),
and Assael, Dymond, and Exadaktilou (1994). These are compared graphically to density values
from Sage and Lacey (1950).
The corresponding states model requires the critical pressure Pcm , the critical temperature
Tcm , the molecular weight MWm , and a detailed model of the viscosity and density as a function
of temperature and pressure for methane. Given the pressure P , the temperature T , and the mole
fractions of each component Xm the method will then calculate the viscosity of the phase μϕ .
12.6.1
n-Decane Density
Aasberg-Petersen et al. (1991) defines a correlation for n-decane, but it references a dissertation
by Jensen at the Technical University of Denmark which was not available and an article by Chueh
and Prausnitz which is not listed in their bibliography. As a result, there is insufficient information
to be able to implement the Aasberg-Petersen et al. (1991) n-decane density correlation.
Audonnet and Pádua (2004) uses T [K], P [MPa], ρ[kg/m3 ]. Audonnet and Pádua (2004) uses
the Tait equation, Dymond and Malhotra (1988), to define the pressure and temperature variation
213
of density.
B+P
ρ[T, P ] − ρ0 [T, P0 ]
= C log10
ρ[T, P ]
B + P0
(12.42)
For the n-decane correlation by Audonnet and Pádua (2004), the parameter values are:
P0 = 25MPa
(12.43)
C = 0.2252
(12.44)
B = 436.929 − 2.17957T + 0.00272365T 2
(12.45)
ρ0 = 1133.36 − 2.39023T + 0.00225011T
(12.46)
2
Cibulka and Hnědkovský (1996) collected a lot of density data for various n-paraffins, including
n-decane; in addition, the article provides a correlation for n-decane density based on the Tait
equation. Cibulka and Hnědkovský (1996) uses T [K], P [MPa], ρ[kg/m3 ].
ρ[T, P ] =
ρ0 [T, P0 ]
B+P
1 − C ln B+P
0
(12.47)
For n-decane correlation by Cibulka and Hnědkovský (1996), the parameter values are:
P0 = 0.101325MPa
(12.48)
T0 = 294.35K
C = 0.087992 − 0.000816
B = 83.5746 − 61.9418
− 6.4316
T − T0
100
(12.49)
T − T0
100
T − T0
100
(12.50)
+ 21.8935
3
+ 1.0545
T − T0
100
T − T0
100
2
+
(12.51)
4
(12.52)
Cibulka and Hnědkovský (1996) uses Assael et al. (1994) as a correlation for ρ0 . Cibulka and
Hnědkovský (1996) uses T [K], P [MPa], ρ[kg/m3 ].
T
ρ0 [T, P0 ] = 239 1 + 0.329139 + 7.364340 1 −
617.65
T
− 9.985096 1 −
617.65
214
1
3
+
2
3
T
5.283608 1 −
617.65
(12.53)
12.6.2
n-Decane Viscosity
The viscosity model of Aasberg-Petersen et al. (1991) uses T [K], P [atm], ρ[g/cm3 ], and μ[cP].
• molecular weight MW = 142.284g/gmol.
• freezing temperature TF = 243.5K
• critical temperature Tc = 617.40K
• critical pressure Pc = 20.18atm
• critical density ρc = 0.2269g/cm3
−4
3
μnC10 [ρ[g/cm ], T [K]] = 10
· μnC10 ,0 [T ] + ρ · μnC10 ,1 [T ] + μnC10 ,2 [ρ, T ]
(12.54)
μnC10 ,0 [T [K]] = 0.2640 · T −1 + 0.9487 · T −2/3 + 71.0 · T −1/3
(12.55)
T
μnC10 ,1 [T [K]] = 2.48 · 10−4 + 81.35 · 5.9583 − ln
490
(12.56)
μnC10 ,2 [ρ[g/cm ], T [K]] = exp −11.739 − 811.3 · T
3
−1
2
· exp ρ0.1 · (16.092 + −18464 · T −3/2 )+
ρ − ρc √
· ρ · (1.9745 + 898.45 · T −1 + 119620 · T −2 ) − 1.0
ρc
12.6.3
(12.57)
Calculations
The extended corresponding states model of Aasberg-Petersen et al. (1991) is a specific example
of the generalized corresponding states model described in Teja and Rice (1981).
The mixed critical temperature is defined as
# 1/3 ⎞3 )
# 1/3
Tcj
T
#
ci
⎠
Xin Xjn ⎝
+
Tci# · Tcj
#
#
Pci
Pcj
j
⎛
# 1/3 ⎞3
# 1/3
Tcj
T
ci
⎠
Xin Xjn ⎝
+
#
#
P
P
ci
cj
i
j
n
[Xın ] =
Tcx
i
⎛
215
(12.58)
The mixed critical pressure is defined as
i
Pcx [Xın ] = 8 ·
⎛
Xin Xjn ⎝
j
Tci#
Pci#
1/3
+
#
Tcj
#
Pcj
1/3 ⎞3 )
⎠
#
Tci# · Tcj
⎛
⎞
⎛
1/3 # 1/3 ⎞3 2
#
Tcj
⎜ n n ⎝ Tci
⎠ ⎟
Xi Xj
+
⎝
⎠
#
#
Pci
Pcj
i
j
(12.59)
The molecular weight of the mixture is defined by
MWnx [Xın ] = 0.00867358 · ( MWnw )1.56079 − ( MWnn )1.56079 + MWnw
(12.60)
Using the weight averaged and number averaged molecular weights:
i
MWnw [Xın ] =
2
Xin MW#
i
MWnn [Xın ] =
Xin MW#
i
Xin MW#
i
(12.61)
i
i
Define the mixture viscosity using the reference viscosity of methane and n-decane using
μmix
μcx · μCH4 [T1 , P1 ]
=
·
μc1
μnC10 [T2 , P2 ] · μc1
μCH4 [T1 , P1 ] · μc2
MWx −MWCH
4
MWnC −MWCH
10
4
(12.62)
Define the reference temperatures and pressures as follows:
T1 =
T · Tc,CH4
Tcx
T2 =
T · Tc,nC10
Tcx
P1 =
P · Pc,CH4
Pcx
P2 =
P · Pc,nC10
Pcx
(12.63)
Define the reference mixture properties
1
2
1
μcx = (MWx ) 2 (Pcx ) 3 (Tcx )− 6
1
2
2
3
(12.64)
− 16
μc1 = (MWCH4 ) (Pc,CH4 ) (Tc,CH4 )
1
2
2
3
(12.65)
− 16
μc2 = (MWnC10 ) (Pc,nC10 ) (Tc,nC10 )
12.7
(12.66)
f -Theory Model
The f -theory model is described by Quiñones-Cisneros et al. (2001a). There are additional
derivations and comparisons to experimental data in Quiñones-Cisneros, Zéberg-Mikkelsen, and
216
Stenby (2000) and Quiñones-Cisneros, Zéberg-Mikkelsen, and Stenby (2001b). For this model, the
viscosity is in μP, the specific volume is in cm3 /mol, the temperature is in K, and the pressure is
in bar.
The viscosity is split into components depending on the dilute gas viscosity μ0 and the frictionbased viscosity μf . The friction-based viscosity is split into a portion that multiplies the PengRobinson repulsive pressure, the Peng-Robinson repulsive pressure squared, and the Peng-Robinson
attractive pressure.
μ = μ0 + μf = μ0 + κa PPRa + κr PPRr + κrr (PPRr )2
12.7.1
(12.67)
Dilute Gas Viscosity and General Properties
The dilute gas viscosity is defined based on the work of Chung.
μ0 = 40.785
(MW · T )1/2 Fc
(12.68)
(vc )2/3 Ω
Where Fc and Ω are defined using
Fc = 1 − 0.2756ω
T = 1.2593
T
Tc
(12.69)
Ω = 1.16145 (T )−0.14874 + 0.52487 exp [−0.7732 · T ] + 2.16178 exp [−2.43787 · T ]
− 6.435 · 10−4 (T )0.14874 · sin 18.0323 · (T )−0.76830 − 7.27371
(12.70)
The critical density may be calculated from the following correlation if it has not been defined from
another correlation:
vc
12.7.2
1
cm3
=
Pc
mol
0.000235751 + 3.42770 RT
c
(12.71)
f -Theory Friction Properties
The critical viscosity is defined using the following correlation based on Uyehara or the tabulated
values.
217
μc = 7.9483 (MW)1/2 (Pc )2/3 (Tc )−1/6
(12.72)
• N2 , μc [μP] = 174.179
• CO2 , μc [μP] = 376.872
• CH4 , μc [μP] = 152.930
• C2 , μc [μP] = 217.562
• C3 , μc [μP] = 249.734
• iC4 , μc [μP] = 271.155
• nC4 , μc [μP] = 257.682
• iC5 , μc [μP] = 275.073
• nC5 , μc [μP] = 258.651
• C6 , μc [μP] = 257.841
The Peneloux adjusted Peng-Robinson terms are defined by:
PPRa =
−a
(v + c)(v + 2c + b) + (b + c)(v − b)
PPRr =
RT
v−b
(12.73)
The following definitions use the reduced variables Γ and Ψ:
Γ=
Tc
T
Ψ=
RTc
Pc
(12.74)
The attractive term is defined by the following equation, using seven pre-fit coefficients.
Pc
= −0.140464 + −4.89197 · 10−2 (Γ − 1) +
μc
0.270572 + −1.10473 · 10−4 Ψ · (exp [Γ − 1] − 1) +
κa ·
−4.48111 · 10−2 + 4.08972 · 10−5 Ψ + −5.79765 · 10−9 Ψ2 · (exp [2Γ − 2] − 1) (12.75)
The repulsive term is defined by the following equation, using seven pre-fit coefficients.
218
Pc
= 1.19902 · 10−2 + −0.357875 (Γ − 1) +
μc
0.637572 + −6.02128 · 10−5 Ψ · (exp [Γ − 1] − 1) +
κr ·
−7.9024 · 10−2 + 3.72408 · 10−5 Ψ + −5.561 · 10−9 Ψ2 · (exp [2Γ − 2] − 1) (12.76)
The squared repulsive term is defined by the following equation, using two pre-fit coefficients.
κrr ·
12.7.3
(Pc )2
= 8.55115 · 10−4 + 1.37290 · 10−8 · Ψ · (exp [2Γ] − 1) · (Γ − 1)2
μc
(12.77)
Mixing Rules
The following mixing rules apply to the f -theory model.
*
μ0,mx = exp
+
Xi ln[μ0,i ]
(12.78)
i
Define X by weighting the mole fractions with the molecular weight raised to a power, here
−0.30.
⎛
Xi = Xi (MWi )−0.3 / ⎝
⎞
Xj (MWj )−0.3 ⎠
(12.79)
j
For κa , κr , and κrr , the mixture properties are defined as:
κmx =
κi Xi
(12.80)
i
219
CHAPTER 13
FORMULATION FOR PROPERTIES OF WATER CONTAINING CO2
There are several ways CO2 differs from the hydrocarbon components of oil and natural gas.
CO2 is much more soluble in water than hydrocarbon components, so for simulation of CO2 injection
it is necessary to include this solubility effect. There are some adjustments to the Peng-Robinson
equation of state which make the EOS more accurate in the presence of CO2 . Asphaltene deposition
may be significant when the CO2 composition of the oil phase is between certain thresholds. When
mixed with certain oils at temperatures below 150◦ F, CO2 can cause the formation of two liquid
hydrocarbon phases, plus a gas phase, plus an aqueous phase. The critical point for CO2 is within
the normal operating conditions; as a result CO2 injection is normally as a supercritical fluid that
has some properties of a liquid and some properties of a gas.
13.1
CO2 Solubility in Water
Figure 13.1 shows the solubility of methane in water. Figure 13.2 shows the solubility of CO2
in water. Note that the solubility of CO2 in water is about ten times the solubility of methane in
water. For a CO2 flood, it is necessary to consider the CO2 solubility in water, but is not necessary
to consider the solubility of methane. There are additional properties defined in Klins (1984) which
define the variation in the water viscosity based on dissolved CO2 , water compressibility, water
formation volume factor, water density, and adjustments to solution gas-water ratio with salinity.
13.2
Adjustments to Equation of State
There are some improvements to the Peng-Robinson equation of state discussed in the literature.
Two of these references with extra details include Ahmed (1989) and Ahmed (2007b). There are
also some versions of Peng-Robinson that handle the water phase (Whitson and Brulé, 2000).
13.3
Other Special Properties of CO2
CO2 has some unusual properties. Lake (1989) discusses some of these effects. Rogers and
Grigg (2001) provides a literature survey of the variation in injection of CO2 . Because the criti-
220
Figure 13.1: Solubility of methane in water (Klins, 1984).
Figure 13.2: Solubility of CO2 in water (Klins, 1984).
221
cal temperature for CO2 is 87.91◦ F, the z-factor for CO2 dips very steeply at low temperatures,
Figure 13.3.
Figure 7-4 Compressibility chart for carbon dioxide (CO2) (from Gibbs, 1971)
Figure 13.3: Change in z-factor as a function of pressure for CO2 (Lake, 1989).
13.4
Properties of Water Containing CO2 , Overview
There are several approaches to calculate the solubility of non-water components in the aqueous
phase. The approaches described here include Henry’s Law correlations and adjustments to the
Peng-Robinson or other equation of state for systems containing H2 O. Solubility may be based on
three different approaches: a pure H2 O aqueous phase; only CO2 is soluble in the aqueous phase;
and multiple components are soluble in the aqueous phase. For this work, only CO2 is soluble in
the aqueous phase. If H2 S were present, it would also need to be soluble in the aqueous phase.
There are also two options for the vapor phase: H2 O may be present or absent in the vapor phase.
For this work, H2 O is assumed to be absent from the vapor phase.
The equation of state based models are more general than the Henry’s Law correlations, but
they are also more time consuming and harder to validate for the case where only CO2 is soluble
in the aqueous phase and H2 O is not present in the oleic or vapor phases. In the case where the
water content in the vapor phase is neglected and only CO2 and H2 O is present in the aqueous
phase, Henry’s Law correlations seem to yield sufficiently accurate predictions.
222
13.5
Commercial Simulators
Schlumberger (2007b) describes several methods for calculating the properties of CO2 in the
aqueous phase. The CO2SOL option uses the Henry’s law based model of Chang et al. (1998). It
is described by the Eclipse manual as the most applicable method for enhanced oil recovery. This
method is also used by VIP (Landmark, 2000). The CO2STORE option is designed for two phases,
a CO2 -rich phase and a H2 O-rich phase. It uses the equation of state procedure by Spycher and
Pruess (2005) to calculate the mole fraction in the aqueous phase. It uses the method by Kell and
Whalley (1975) to calculate the pure water density and then the method of Ezrokhi described in
Zaytsev and Aseyev (1992) to adjust for the salt content. The viscosity is calculated using Vesovic,
Wakeham, Olchowy, Sengers, Watson, and Millat (1990) and Fenghour, Wakeham, and Vesovic
(1998). The GASWAT option is most applicable to CO2 storage in an aquifer or a depleted gas
reservoir. It accounts for the presence of H2 O in the gas phase. It uses the Søreide and Whitson
(1992) modifications to the Peng-Robinson EOS.
13.6
Properties of Water Containing CO2 , CMG GEM
In CMG GEM, CMG (2010), the aqueous viscosity may be specified as a simple function
of pressure or calculated by Kestin, Khalifa, and Correia (1981). The mole fractions of CO2
are calculated from Henry’s Law, using Li and Nghiem (1986) or Harvey (1996). The Harvey
(1996) calculations also require several additional correlations: the saturation pressure for water
is calculated using Saul and Wagner (1987); the partial molar volume for CO2 is calculated using
Garcı́a (2001); the salinity adjustment is calculated using Bakker (2003). The fugacity of saturated
water is calculated using Canjar and Manning (1967). The molar volume of water is calculated
using Rowe and Chou (1970).
13.7
Units of concentration
The following concentration units are used, with appropriate conversions:
• mi represents the molality, moli /masssolvent . For this chapter, the solvent is H2 O. Note that
the denominator is the mass of the solvent, not the total mass of the solution. Typically
measured in m = mol/kg or mol/lbm. For the following conversions, mi is in units of mol/kg.
223
• ci represents the molarity, moli /volt . Typically measured in mol/m3 , mol/L, mol/cm3 , or
mol/ft3 . For the following conversions, ci is in units of mol/L.
• Xi , or WCO2 represents the mole fraction, moli /molt . For the following conversions, Wi is in
units of mol/mol.
• wi represents the mass fraction, massi /masst . Typically measured in m3 /m3 , ft3 /ft3 , or
ppmw. For the following conversions, wi is in units of kg/kg.
• Rsw represents the gas solubility in scf/stb.
For the following conversions, MWi is in units of g/mol, and ρ is in units of kg/m3 = g/L. Most
of the correlations use the equivalent concentration of NaCl rather than the specific composition of
the salts. Based on Duan and Sun (2003), this is a good assumption for most cations and anions
except for SO−−
4 . The molecular weight of H2 O is 18.0153, the molecular weight of CO2 is 44.0096,
and the molecular weight of NaCl is 58.4430. The molality of an aqueous phase containing only
H2 O is 55.5084 mol/kg.
To convert from weight fraction wi into mole fraction Xi
αi =
wi MWj
MWi
j
αi
Xi = j αj
(13.1)
To convert from weight fraction wi into molal units mi
mi = 1000
wi
MWi
(13.2)
/wsolvent
To convert from weight fraction wi into molar units ci
ci =
wi ρt
MWi
(13.3)
To convert from mole fraction Xi into weight fraction wi
Xi MWi
wi = j Xj MWj
(13.4)
To convert from mole fraction Xi into molal units mi , first convert Xi into wi , then convert wi
into mi .
224
Xi MWi
wi = j Xj MWj
mi = 1000
wi
MWi
(13.5)
/wsolvent
To convert from mole fraction Xi into molar units ci
ci = Xi
ρt
X
j j MWj
(13.6)
To convert from molal units mi into weight fraction wi
mi MWi
wi = j mj MWj
(13.7)
To convert from molal units mi into mole fraction Xi
mi
Xi = j mj
(13.8)
To convert from molal units mi into molar units ci
ci =
mi
j mj MWj
(13.9)
ρt
To convert from molar units ci into weight fraction wi
wi =
ci MWi
ρt
(13.10)
To convert from molar units ci into weight fraction xi
ci
Xi = (13.11)
j cj
To convert from molar units ci into molal units mi , first convert from molar units ci into mole
fraction Xi , then convert from mole fraction Xi into weight fraction wi , then convert from weight
fraction wi into molal units mi .
ci
Xi = j cj
Xi MWi
wi = j Xj MWj
mi = 1000
225
wi
MWi
/wsolvent
(13.12)
To convert from weight fraction wi in kg/kg into the gas solubility in Rsw,i in scf/stb uses the
following equation. The conversion constant 2130.3 is based on 379.423 scf/mol from Klins (1984).
Not all of the correlations are clear about which “standard conditions” are used for temperature
and pressure.
Rsw,i =
2130.3ρt wi
MWi
(13.13)
To convert from gas solubility in Rsw,i in scf/stb into weight fraction wi in kg/kg
wi =
13.8
MWi Rsw,i
2130.3ρt
(13.14)
Selection Process
This section describes how the algorithms for calculating brine density, brine viscosity, and CO2
solubility in water were selected.
13.8.1
Rowe, Brine Density
Rowe and Chou (1970) is used by all of the commercial simulators to calculate brine density as
a function of H2 O and NaCl content. It is also the preferred method by many other authors which
need a method for calculating brine density, including Kestin, Khalifa, Abe, Grimes, Sookiazian,
and Wakeham (1978), Kestin et al. (1981), Kestin and Shankland (1984), Chang et al. (1998), Enick
and Klara (1992), and Li and Nghiem (1986). Rowe and Chou (1970) reports that their correlation
is within 3% of the experimental data for both density and the derivatives of density with respect
to pressure and temperature for a range of temperatures from 0◦ C to 150◦ C, 0 to 25% weight
percent NaCl, and pressures from 1 to 350 kg/cm2 = 4978 psia. To check the implementation, the
correlations were validated using all of the figures and tables in Rowe and Chou (1970). It compares
favorably to figures 3.42 and 3.43A in Klins (1984). The values of the density and compressibility
were also compared favorably with several web sources, including:
• http://en.wikipedia.org/wiki/Properties_of_water
• http://www.engineeringtoolbox.com/fluid-density-temperature-pressure-d_309.html
• http://www.searchanddiscovery.com/documents/2006/06015powley/images/a03.htm
It also compares favorably to the data in ASME (1935).
226
13.8.2
Garcı́a, CO2 Brine Density and Partial Molar Volume
Garcı́a (2001) provides a way to calculate the density of a brine containing NaCl, H2 O, and
CO2 . It uses Rowe and Chou (1970) to calculate the NaCl plus H2 O brine density. This method
is referred to in many more recent articles as a way to calculate the partial molar volume of CO2 .
Garcı́a (2001) reports that their correlation is valid for temperatures between 0◦ C and 300◦ C,
and from 0 to 0.05 mole fraction CO2 . The authors compare their correlations to four previous
correlations. To check the implementation, the correlations were validated using all of the figures
and tables in Garcı́a (2001).
13.8.3
Kestin, Brine Viscosity
Kestin et al. (1978) describes a correlation for the viscosity of NaCl brine solutions for 20–150◦ C
and pressures of 0.1–35 MPa, and 0–5.4 molal. Kestin et al. (1978) report that their correlation has
a maximum deviation of 1.4% with a standard deviation of 0.5%. Sayegh and Najman (1987) shows
that CO2 has a negligible impact on the viscosity of the H2 O+NaCl system. The Kestin correlations
are used by Eclipse and VIP. To check the implementation, the correlations were validated using all
of the figures and tables in Kestin et al. (1978), Kestin et al. (1981), Kestin and Shankland (1984),
and figures 3.44 and 3.45 from Klins (1984).
13.8.4
Duan, Henry’s Law
There are a lot of different methods for solubility calculations. The methods of the commercial
simulators Eclipse (Schlumberger, 2007b), VIP (Landmark, 2000), and CMG CMG (2010) were
reviewed, plus all the articles they cite related to CO2 solubility. An independent literature search
was also conducted. Methods were selected for further study which have CO2 solubility as a function
of temperature, pressure, and NaCl salinity.
Several methods are based on adjustments of an equation of state. These include Søreide and
Whitson (1992), Delshad et al. (2011), Yan and Stenby (2009), Melham and Little (1989), Spycher
and Pruess (2005), and Li and Nghiem (1986). Because it is more difficult to validate these methods
and because the presence of H2 O in the vapor phase is neglected, these methods were described
but not implemented.
227
Four methods based on Henry’s Law calculations were also selected for evaluation. These
includes the methods of Duan and Sun (2003); Chang et al. (1998); Enick and Klara (1990); and
CMG; including the methods of Harvey (1996), Saul and Wagner (1987), Garcı́a (2001), and Bakker
(2003). All four of these methods were implemented, and Duan, Møller, and Weare (1992) was
used to calculate fugacity (accurate to within 5%) for comparison purposes. Each of the methods
were validated using the figures and tables presented in the articles that describe the correlations.
The method of Duan and Sun (2003) was selected based on the best fit with data from a variety of
sources. First, Duan and Sun (2003) was validated using plots from Duan and Sun (2003). Duan
and Sun (2003), Spycher, Pruess, and Ennis-King (2003), and Spycher and Pruess (2005) have
detailed figures including solubility data from a large number of sources. The correlations from
Duan and Sun (2003) were compared to figures and tables from those sources as well as Harvey
(1996), Klins (1984), Obeida, Kalam, Al-Sahn, Gibson, Masaleeh, and Zhang (2009), Rumpf,
Nicolaisen, Öcal, and Maurer (1994), Li and Nghiem (1986), Søreide and Whitson (1992), Yan and
Stenby (2009), and Zeebe and Wolf-Gladrow (2001). Duan and Sun (2003) is valid for 273 K to
533 K, 0 to 2000 bar = 29007 psia, and 0 to 4 mol/kg = 0.189 kg/kg NaCl, and with a correlation
and experimental accuracy of 7% CO2 solubility.
13.9
Correlations for this Project
The following is a short summary of the procedure for calculating the viscosity, solubility, and
density of the aqueous phase containing CO2 , H2 O, and NaCl. These correlations involve conversion
between different units for pressure, temperature, and concentration. The mole fraction of NaCl,
n
n
, in a NaCl-H2 O system is used as the concentration input. WNaCl
is calculated explicitly at
WNaCl
each time step n after all other properties have been calculated at n. The weight fractions for a
NaCl-H2 O system are defined with a superscript:
,n
=
wNaCl
,n
WNaCl
MWNaCl
,n
,n
WNaCl MWNaCl + (1 − WNaCl
)MWH2 O
(13.15)
,n
=
wH
2O
,n
(1 − WNaCl
)MWH2 O
,n
,n
WNaCl MWNaCl + (1 − WNaCl
)MWH2 O
(13.16)
The molalities for a NaCl-H2 O system are defined with a superscript:
228
m,n
NaCl
,n
1000 wNaCl
= ,n
wH2 O MWNaCl
(13.17)
m,n
H2 O
=
,n
1000 wH2 O
,n
MWH2 O
wH
2O
(13.18)
The viscosity is calculated using Kestin et al. (1981) and Kestin et al. (1978). For the IMPES
formualtion, it is only computed at the time step level n and no derivatives are required.
n
μnw = μnw [P n , T # , m,n
NaCl [WNaCl ]]
(13.19)
The solubility WCO2 in a CO2 + NaCl + H2 O system is calculated using the following procedure
from Duan and Sun (2003).
,n
,n
], mH2 O [WNaCl
], fCO2 ] =
WCO2 = WCO2 [P, T # , mNaCl [WNaCl
mCO2 [P, T # , fCO2 , mNaCl ]
,n
mCO2 [P, T # , fCO2 , mNaCl ] + m,n
NaCl + mH2 O
(13.20)
The mnCO2 is calculated as follows:
mnCO2 = mCO2 [P, T, fCO2 , mNaCl [WNaCl ]] =
Y n P n ΦnCO2 [P n , T # , Ymn ]
fCO2
= CO2n
HCO2
HCO2 [P n , T # , m,n
NaCl ]
(13.21)
The molar density of the aqueous phase ξw in a CO2 + NaCl + H2 O system is calculated based
on correlations by Rowe and Chou (1970) and Garcı́a (2001), using the WNaCl and WCO2 . ρbrine
represents the density of a NaCl + H2 O system.
n
= ξw [P, T, WCO2 , wNaCl
[WNaCl ], MWw,t [WCO2 , WNaCl ]] =
ξw
,n
]/MWnw,t
ρnbrine [P n , T # , wNaCl
1+
n
WCO
2
MWn
w,t
,n
]v̄CO2 [T # ]10−3
−MWCO2 + ρnbrine [P n , T # , wNaCl
(13.22)
n+1 is evaluated as follows:
The aqueous density ξw
n+1 , T # , w ,n ]/MWn+1
ρn+1
w,t
NaCl
brine [P
n+1
=
ξw
1+
n+1
WCO
2
MWn+1
w,t
n+1 , T # , w ,n ]v̄
#
−3
−MWCO2 + ρn+1
NaCl CO2 [T ]10
brine [P
229
(13.23)
The total molecular weight is defined in the following way because the experiments were first
conducted to measure the properties of the NaCl + H2 O system, with an adjustment added for the
NaCl + H2 O + CO2 system.
MWNaCl +(1−WNaCl
)MWH2 O )
MWw,t = MWw,t [WCO2 , WNaCl ] = WCO2 MWCO2 +(1−WCO2 )(WNaCl
(13.24)
13.10
Computational Forms of WCO2
Recall that Duan and Sun (2003) uses the following version of Henry’s Law for mCO2 :
mCO2 =
fCO2
HCO2
(13.25)
To calculate the derivative of WCO2 with respect to pressure,
∂WCO2
∂P ,
use the conversion from
molality to mole fraction.
mi
Wi = j mj
After solving
∂WCO2
∂P
(13.26)
and
∂mH2 O
∂P
simultaneously, this yields
1 ∂mCO2
∂WCO2
=
∂P
∂P
j mj
(13.27)
The derivative of WCO2 with respect to Ym ,
∂WCO2
∂Ym
is
1 ∂mCO2
∂WCO2
=
∂Ym
j mj ∂Ym
(13.28)
The derivative of mCO2 with respect to pressure,
1
∂mCO2
=
∂P
HCO2
∂HCO2
∂fCO2
− mCO2
∂P
∂P
∂mCO2
∂P
is
(13.29)
The derivative of mCO2 with respect to Ym is
1 ∂fCO2
∂mCO2
=
∂Ym
HCO2 ∂Ym
(13.30)
230
13.10.1
Option 0: WCO2 = 0
This option is the simplest to implement and compute. It completely neglects the solubility of
CO2 in the aqueous phase.
WCO2 = 0
∂WCO2
=0
∂P
∂WCO2
=0
∂Ym
(13.31)
This option has a cumulative mass balance error less than 10−4 for all components but does
not accurately represent the physics of CO2 solubility.
13.10.2
Option C: Constant WCO2
This option is the simplest to implement and compute. The CO2 solubility is not a function of
pressure or composition, but is non-zero.
WCO2
∂WCO2
∂P
∂WCO2
∂Xm
∂WCO2
∂Ym
= constant
(13.32)
=
0
(13.33)
=
0
(13.34)
=
0
(13.35)
For the flash calculation, use
+1
=
βw
VR +1 +1 +1 +1
#
φ Sw ξw [P , WCO
]
2
Δt
(13.36)
This option has a low cumulative mass balance error and the appropriate behavior for the water
saturation, but does not accurately represent the physics of CO2 solubility.
13.10.3
Option ZW0: Compute ξw using WCO2 = 0
This computation uses Rowe and Chou (1970) rather than Rowe and Chou (1970) plus Garcı́a
(2001). The following pressure derivatives are the following; all other derivatives are zero.
∂ρbrine
= ρbrine Cw
∂P
1
∂ρbrine
∂ξw
=
= ξ w Cw
∂P
MWbrine ∂P
(13.37)
(13.38)
231
13.10.4
Option ZW1: Compute ξw using WCO2
ρw,t =
ρbrine
WCO2
1+
· −MWCO2 + ρbrine · v̄CO2 · 10−3
MWw,t
MWw,t = MWCO2 WCO2 + MWH2 O (1 − WCO2 )
(13.39)
(13.40)
The molar density ξw is defined by
ξw =
ρw,t
MWw,t
(13.41)
The derivative of ξw with respect to pressure
1
∂ξw
=
∂P
MWw,t
∂ξw
∂P
∂MWw,t
∂ρw,t
− ξw
∂P
∂P
(13.42)
The derivative of ξw with respect to mole fraction
1
∂ξw
=
∂Ym
MWw,t
13.10.5
∂ξw
∂Ym
is
∂MWw,t
∂ρw,t
− ξw
∂Ym
∂Ym
(13.43)
The derivative of ξw with respect to mole fraction
1
∂ξw
=
∂Xm
MWw,t
is
∂ξw
∂Xm
is
∂MWw,t
∂ρw,t
− ξw
∂Xm
∂Xm
(13.44)
Option KP1: Use a simplified model for WCO2 using YCO2 [Pb ]
If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 [Pb ]. Use the
brine density rather than the total aqueous density. Define WCO2 as follows:
Rsw =
α=
WCO2 =
200
379
(1 − exp[−0.001386P ])
(13.45)
Rsw + 5.6146ξbrine
Rsw
YCO2
Rsw + 5.6146ξbrine
(13.46)
(13.47)
The derivatives are calculated as follows:
232
∂Rsw
=
∂P
∂WCO2
=
∂P
∂WCO2
=
∂Ym
200
379
YCO2
α
· 0.001386 · exp[−0.001386P ]
∂Rsw
∂P
−
−
Rsw ∂Rsw
α
∂P
− 5.6146 Rαsw
∂ξbrine
∂P
(13.48)
(13.49)
Rsw
Rsw + 5.6146ξbrine
(13.50)
(13.51)
This option has a low cumulative mass balance error and the appropriate behavior for the water
saturation. Option KP1 is computationally stable with both option ZW0 and ZW1. This approach
is good if the salinity and temperature are constant, but the correlation needs to be updated for a
specific salinity and temperature.
13.10.6
Option KP2: Use a simplified model for WCO2 using YCO2 below the bubble
point and YCO2 = 0 above the bubble point
If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 = 0 =⇒
WCO2 = 0. Use the brine density rather than the total aqueous density. Define WCO2 as follows:
Rsw =
α=
WCO2 =
200
379
(1 − exp[−0.001386P ])
(13.52)
Rsw + 5.6146ξbrine
Rsw
YCO2
Rsw + 5.6146ξbrine
(13.53)
(13.54)
The derivatives are calculated as follows:
∂Rsw
=
∂P
∂WCO2
=
∂P
∂WCO2
=
∂Ym
200
379
YCO2
α
· 0.001386 · exp[−0.001386P ]
∂Rsw
∂P
−
−
Rsw ∂Rsw
α
∂P
− 5.6146 Rαsw
Rsw
Rsw + 5.6146ξbrine
∂ξbrine
∂P
(13.55)
(13.56)
(13.57)
(13.58)
This option has a low cumulative mass balance error and the appropriate behavior for the water
saturation. Option KP2 is computationally stable with both option ZW0 and ZW1. This approach
is good if the salinity and temperature are constant, but the correlation needs to be updated for a
specific salinity and temperature.
233
13.10.7
Option KP3: Use a simplified model for WCO2 using YCO2 [Pb ]
If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 [Pb ]. Use the
brine density rather than the total aqueous density. This model of WCO2 is based on a fit of Duan
and Sun (2003) using T = 200◦ F and ws = 0.225, for pressures from P = 100 psia to P = 5000 psia
in steps of 100 psia, and for YCO2 ranging from 0 to 1 in steps of 0.05.
Rsw = 0.195028 (1 − exp[−0.000571152P ])
(13.59)
Rsw + 5.6146ξbrine
Rsw
YCO2
Rsw + 5.6146ξbrine
α=
WCO2 =
(13.60)
(13.61)
The derivatives are calculated as follows:
∂Rsw
= 0.195028 · 0.000571152 · exp[−0.000571152P ]
∂P
∂WCO2
YCO2 ∂Rsw
Rsw ∂Rsw
Rsw ∂ξbrine
=
−
−
5.6146
α
∂P
α
∂P
α
∂P
∂P
Rsw
∂WCO2
=
−
∂Ym
Rsw + 5.6146ξbrine
(13.62)
(13.63)
(13.64)
(13.65)
This option has a low cumulative mass balance error and the appropriate behavior for the water
saturation. Option KP3 is computationally stable with both option ZW0 and ZW1. This approach
is better than KP1 because it is based on a specific temperature and salinity relevant for offshore
Abu Dhabi.
13.10.8
n+1
fully implicit
Option 1: WCO
2
The mn+1
CO2 is calculated as follows, using a implicit approach:
mn+1
CO2 =
n+1
∂WCO
∂ξw
2
∂P , ∂P ,
fn+1
CO2
n+1
HCO
2
=
n+1 n+1 n+1
YCO
P
ΦCO2 [P n+1 , T # , Ymn+1
]
2
(13.66)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
are computed using
n+1
∂mn+1
∂WCO
CO2
∂ξw
2
.
∂P
∂Ym , ∂Ym ,
234
are computed using
∂mn+1
CO2
∂Ym .
∂mn+1
CO2
=
∂P
1
n+1
HCO
2
∂mn+1
CO2
=
∂Ym
∂fn+1
CO2
∂P
− mn+1
CO2
1
n+1
HCO
2
n+1
∂HCO
2
∂P
(13.67)
∂fn+1
CO2
∂Ym
(13.68)
This option causes a steady decrease in the water saturation, even when no water is injected
or produced. This inconsistency arises because the CO2 solubility experiments used only a pure
supercritical or vapor CO2 phase and a water phase to measure the water solubility.
13.10.9
n+1
Option 2: WCO
implicit pressure, explicit fugacity coefficient
2
The mn+1
CO2 is calculated as follows, using an implicit pressure explicit fugacity coefficient approach:
mn+1
CO2
=
n P n+1 Φn
n
#
n
YCO
CO2 [P , T , Ym ]
2
Using this approach,
∂mn+1
CO2
∂P
(13.69)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
n+1
∂WCO
2
∂Ym
= 0 and
n+1
∂ξw
∂Ym
= 0, and
n+1
∂WCO
2
∂P
and
∂ξw
∂P
are computed using
⎞
⎛
same
different
same
⎜ n n+1 ⎟
∂HCO
1 ⎜ fCO2
n+1
2⎟
=
−
m
CO2 ∂P ⎟
n+1 ⎜
n
P
⎠
⎝
HCO2
∂mn+1
CO2
=
∂Ym
∂mn+1
CO2
∂P .
(13.70)
0
(13.71)
This option causes a steady decrease in the water saturation, even when no water is injected
or produced. This inconsistency arises because the CO2 solubility experiments used only a pure
supercritical or vapor CO2 phase and a water phase to measure the water solubility.
13.10.10
n+1
implicit pressure, explicit fugacity
Option 3: WCO
2
The mn+1
CO2 is calculated as follows, using an implicit pressure explicit fugacity approach:
mn+1
CO2
=
fnCO2
n+1
HCO
2
Using this approach,
=
n P n Φn
n
#
n
YCO
CO2 [P , T , Ym ]
2
(13.72)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
n+1
∂WCO
2
∂Ym
= 0 and
n+1
∂ξw
∂Ym
= 0, and
235
n+1
∂WCO
2
∂P
and
∂ξw
∂P
are computed using
∂mn+1
CO2
∂P .
∂mn+1
CO2
∂P
⎞
⎛
same
same
⎜different n+1 ⎟
∂HCO
1 ⎜ n+1
2⎟
=
0 − mCO2
⎟
n+1 ⎜
∂P ⎠
HCO2 ⎝
∂mn+1
CO2
=
∂Ym
(13.73)
0
(13.74)
This option causes a steady decrease in the water saturation, even when no water is injected
or produced. This inconsistency arises because the CO2 solubility experiments used only a pure
supercritical or vapor CO2 phase and a water phase to measure the water solubility.
13.10.11
n+1
implicit pressure, fugacity at Option 4: WCO
2
The mn+1
CO2 is calculated as follows, using an implicit pressure explicit fugacity approach:
mn+1
CO2 =
fCO2
n+1
HCO
2
Using this approach,
∂mn+1
CO2
∂P
=
P Φ
#
YCO
CO2 [P , T , Ym ]
2
(13.75)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
n+1
∂WCO
2
∂Ym
= 0 and
n+1
∂ξw
∂Ym
= 0, and
n+1
∂WCO
2
∂P
and
∂ξw
∂P
are computed using
⎞
⎛
same
same
⎜different n+1 ⎟
∂HCO
1 ⎜ n+1
2⎟
=
0 − mCO2
⎟
n+1 ⎜
∂P
⎠
⎝
HCO2
∂mn+1
CO2
=
∂Ym
0
∂mn+1
CO2
∂P .
(13.76)
(13.77)
This option causes a steady decrease in the water saturation, even when no water is injected
or produced. This inconsistency arises because the CO2 solubility experiments used only a pure
supercritical or vapor CO2 phase and a water phase to measure the water solubility.
13.10.12
n+1
implicit pressure, explicit fugacity coefficient
Option 2Z: WZ,CO
2
The mn+1
Z,CO2 is calculated as follows, using an implicit pressure explicit fugacity coefficient
approach:
mn+1
Z,CO2 =
n P n+1 Φn
n
#
n
ZCO
Z,CO2 [P , T , Zm ]
2
(13.78)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
236
Using this approach,
n+1
∂WZ,CO
2
∂Ym
∂P
n+1
∂ξw
∂Ym
= 0, and
n+1
∂WZ,CO
2
∂P
and
∂ξw
∂P
are computed using
⎞
same
different
⎜ n
n+1 ⎟
∂HCO
1 ⎜ fZ,CO2
n+1
2⎟
=
−
m
⎟
⎜
Z,CO2
n+1 ⎝ P n
∂P ⎠
HCO
same
∂mn+1
Z,CO2
= 0 and
∂mn+1
Z,CO2
.
∂P
⎛
(13.79)
2
∂mn+1
Z,CO2
∂Ym
=
0
(13.80)
This option has a mass balance error of 10−2 in the CO2 component.
13.10.13
n+1
implicit pressure, explicit fugacity
Option 3Z: WZ,CO
2
The mn+1
Z,CO2 is calculated as follows, using an implicit pressure explicit fugacity approach:
mn+1
Z,CO2 =
fnZ,CO2
n+1
HCO
2
Using this approach,
=
∂P
(13.81)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
n+1
∂WZ,CO
2
∂Ym
= 0 and
n+1
∂ξw
∂Ym
= 0, and
n+1
∂WZ,CO
2
∂P
and
∂ξw
∂P
are computed using
⎞
⎜different
n+1 ⎟
∂HCO
1 ⎜ n+1
2⎟
=
0
−
m
⎟
⎜
Z,CO2
n+1 ⎝
∂P ⎠
HCO
same
∂mn+1
Z,CO2
n P n Φn
n
#
n
ZCO
Z,CO2 [P , T , Zm ]
2
⎛
∂mn+1
Z,CO2
.
∂P
same
(13.82)
2
∂mn+1
Z,CO2
∂Ym
=
0
(13.83)
This option has a mass balance error of 10−2 in the CO2 component.
13.10.14
n+1
implicit pressure, fugacity at Option 4Z: WZ,CO
2
The mn+1
CO2 is calculated as follows, using an implicit pressure explicit fugacity approach:
mn+1
Z,CO2
=
fZ,CO2
n+1
HCO
2
Using this approach,
=
]
ZCO
P ΦCO2 [P , T # , Zm
2
(13.84)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
n+1
∂WZ,CO
2
∂Ym
= 0 and
n+1
∂ξw
∂Ym
= 0, and
237
n+1
∂WZ,CO
2
∂P
and
∂ξw
∂P
are computed using
∂mn+1
Z,CO2
.
∂P
∂mn+1
Z,CO2
∂P
∂mn+1
Z,CO2
∂Ym
⎞
⎛
same
same
⎜different n+1 ⎟
∂HCO
1 ⎜ n+1
2⎟
=
0 − mZ,CO2
⎟
n+1 ⎜
∂P ⎠
HCO2 ⎝
=
0
(13.85)
(13.86)
This option has a mass balance error of 10−2 in the CO2 component.
13.10.15
n+1
partially implicit, function of both Xm and Ym
Option 1XY: WCO
2
Here, we assume that WCO2 is a linear combination of the WCO2 [P, T, Xm ] and WCO2 [P, T, Ym ].
This means the derivatives of WCO2 are also linear combinations of the derivatives of WCO2 [P, T, Xm ]
and WCO2 [P, T, Ym ].
WCO2 ,v
WCO2
WCO2 ,l
= V · WCO2 [P, T, Ym ] +L · WCO2 [P, T, Xm ]
(13.87)
Expand this in the following way, ignoring the derivatives of V and L.
n+1
WCO
2
]
]
∂W
∂W
[P,
T,
Y
[P,
T,
Y
m
m
CO2
CO2
δP +
= V WCO
[P, T, Ym ] +
δYm +
2
∂P
∂Ym
]
]
∂W
∂W
[P,
T,
X
[P,
T,
X
m
m
CO2
CO2
δP +
[P, T, Xm ] +
δXm
(13.88)
L WCO
2
∂P
∂Xm
is calculated as
The WCO
2
= V · WCO
+ L · WCO
WCO
2
2 ,v
2 ,l
(13.89)
mCO2 ,v
=
WCO
2 ,v
j mj,v
(13.90)
mCO2 ,l
WCO
=
2 ,l
j mj,l
mCO2 ,v is calculated as follows:
mCO2 ,v [P, T, Ym ] =
fCO2 ,v
HCO
2
=
P Φ
#
YCO
CO2 ,v [P , T , Ym ]
2
HCO
[P , T # , m,n
NaCl ]
2
238
(13.91)
mCO2 ,l is calculated as follows:
mCO2 ,l [P, T, Xm ] =
The
∂WCO
2
∂P
fCO2 ,l
=
HCO
2
]
XCO
P ΦCO2 ,l [P , T # , Xm
2
HCO
[P , T # , m,n
NaCl ]
2
is calculated as:
∂WCO
∂WCO
∂WCO
2 ,v
2 ,l
2
=V·
+ L ·
∂P
∂P
∂P
The
∂WCO
2
∂P
are calculated using
(13.93)
∂mCO
2
∂P :
1 ∂mCO2
∂WCO2
= ∂P
∂P
j mj
The
∂mCO
2
∂P
∂mCO2 ,v
∂P
∂mCO2 ,l
∂P
The
∂WCO
2
∂Ym
(13.94)
are calculated as:
=
=
1
HCO
2
1
HCO
∂fCO ,v
2
∂P
2
∂fCO ,l
2
∂P
− mCO2 ,v
∂HCO
2
∂P
(13.95)
− mCO2 ,l
∂HCO
2
∂P
(13.96)
is calculated as:
∂WCO
∂WCO
2 ,v
2
=V ·
∂Ym
∂Ym
The
∂WCO
2 ,v
∂Ym
are calculated using
(13.97)
∂mCO ,v
2
∂Ym :
∂mCO2 ,v
∂WCO2 ,v
1
= ∂Ym
j mj,v ∂Ym
∂mCO ,v
2
∂Ym
(13.98)
is calculated as:
∂mCO2 ,v
∂Ym
The
(13.92)
∂WCO
2
∂Xm
=
1
∂fCO2 ,v
(13.99)
∂Ym
HCO
2
is calculated as:
239
∂WCO
∂WCO
2 ,l
2
= L ·
∂Xm
∂Xm
The
∂WCO
2 ,l
∂Xm
(13.100)
are calculated using
∂mCO ,l
2
∂Xm :
∂mCO2 ,l
∂WCO2 ,l
1
= ∂Xm
j mj,l ∂Xm
∂mCO ,l
2
∂Xm
(13.101)
is calculated as:
∂mCO2 ,l
∂Xm
13.10.16
=
∂fCO2 ,l
1
(13.102)
∂Xm
HCO
2
n+1
Option Y1: WCO
fully implicit
2
When there is no gas in the system, WCO2 = 0. When there is gas in the system, use a fully
+1 =
, P n+1 ]. For the flash calculation, use βw
implicit calculation of WCO2 [Ymn+1
VR +1 +1 Sw ζw .
Δt φ
The mn+1
CO2 is calculated as follows, using a implicit approach:
mn+1
CO2
=
n+1
∂WCO
∂ξw
2
∂P , ∂P ,
fn+1
CO2
n+1
HCO
2
=
n+1 n+1 n+1
YCO
P
ΦCO2 [P n+1 , T # , Ymn+1
]
2
are computed using
∂mn+1
CO2
=
∂P
∂mn+1
CO2
=
∂Ym
1
n+1
HCO
2
(13.103)
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
∂fn+1
CO2
∂P
n+1
∂mn+1
∂WCO
CO2
∂ξw
2
.
∂P
∂Ym , ∂Ym ,
− mn+1
CO2
1
n+1
HCO
2
n+1
∂HCO
2
∂P
are computed using
∂mn+1
CO2
∂Ym .
(13.104)
∂fn+1
CO2
∂Ym
(13.105)
This option causes a non-physical change in the water saturation, even when no water is injected
or produced.
13.10.17
n
explicit
Option Y5: WCO
2
When there is no gas in the system, WCO2 = 0. When there is gas in the system, use an explicit
calculation of WCO2 [Ymn , P n ].
240
n
WCO
= WCO
2
2
n
∂WCO
2
∂P
n
∂WCO
2
∂Xm
n
∂WCO
2
∂Ym
= WCO2 [P n , T # , Ymn ]
(13.106)
=
0
(13.107)
=
0
(13.108)
=
0
(13.109)
Because the composition derivatives of WCO2 are zero,
∂ξw
= 0
∂Xm
∂ξw
= 0
∂Ym
(13.110)
(13.111)
occurs, use
Everywhere ξw
n
= ξw [P , WCO
]
ξw
2
(13.112)
For the flash calculation, use
+1
=
βw
VR +1 +1 +1 +1
n
φ Sw ξw [P , WCO
]
2
Δt
(13.113)
This option causes a non-physical change in the water saturation, even when no water is injected
or produced. There is also a large mass balance error in the CO2 .
13.10.18
n+1
[P only] fully implicit
Option P1: WCO
2
#
= 1.
This option defines WCO2 [P n+1 , T # , Y # ]. Typically use the constant YCO
2
WCO2
∂WCO2
∂P
∂WCO2
∂Xm
∂WCO2
∂Ym
#
= WCO2 [P, YCO
]
2
=
(13.114)
∂mCO2
1
mj ∂P
(13.115)
j
=
0
(13.116)
=
0
(13.117)
The mn+1
CO2 is calculated as follows, using a implicit approach:
241
mn+1
CO2 =
fn+1
CO2
n+1
HCO
2
=
#
n+1 , T # , Y # ]
YCO
P n+1 Φn+1
CO2 [P
m
2
n+1
HCO
[P n+1 , T # , m,n
NaCl ]
2
(13.118)
The pressure derivative of mCO2 is defined as:
∂mn+1
1
CO2
= n+1
∂P
HCO2
n+1
∂HCO
∂fn+1
CO2
n+1
2
− mCO2
∂P
∂P
(13.119)
For the flash calculation, use
+1
=
βw
VR +1 +1 +1 +1
#
+1
φ Sw ξw [P , WCO
[P +1 , YCO
]]
2
2
Δt
(13.120)
This option causes a non-physical change in the water saturation, even when no water is injected
or produced. There is also a large mass balance error in the CO2 .
13.10.19
n+1
[YCO2 only], evaluate Y at Option K1: WCO
2
#
= 0.005.
This option defines WCO2 [YCO2 only]. Typically use the constant KCO
2
WCO2
∂WCO2
∂P
∂WCO2
∂Xm
∂WCO2
∂Ym
= KCO2 YCO2
(13.121)
=
0
(13.122)
=
0
(13.123)
=
−KCO2
(13.124)
If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 [Pb ].
For the flash calculation, use the following:
+1
=
βw
VR +1 +1 +1 +1
+1
φ Sw ξw [P , WCO
[YCO
]]
2
2
Δt
(13.125)
This option has a low cumulative mass balance error and the appropriate behavior for the water
saturation, but does not represent the pressure dependence of CO2 solubility.
At a temperature of 200◦ F a salinity of 0.225 and an average reservoir pressure of 2000 psia,
K = 0.007.
242
13.10.20
n+1
Option K2: WCO
[YCO2 only], evaluate Y at 2
#
= 0.005.
This option defines WCO2 [YCO2 only]. Typically use the constant KCO
2
WCO2
∂WCO2
∂P
∂WCO2
∂Xm
∂WCO2
∂Ym
= KCO2 YCO2
(13.126)
=
0
(13.127)
=
0
(13.128)
=
−KCO2
(13.129)
If there is gas in the system, use YCO2 . If there is no gas in the system, use YCO2 = 0.
For the flash calculation, use the following:
+1
=
βw
VR +1 +1 +1 +1
+1
φ Sw ξw [P , WCO
[YCO
]]
2
2
Δt
(13.130)
The water saturation behavior for this model is good. There are significant mass balance errors
introduced when a two phase oil-water system transitions to a three-phase oil-water-gas system.
13.10.21
Using WCO2 as a Transfer Term
(13.131) represents the m1 pore system for the gas and oil phases:
n
n
mm1 ξom1 krom1 #
+1
n
#
k
(∇P
−
γ
∇D
)
+
m1
om1
om1
μnom1
Y n ξ n kn
mm1 gm1 rgm1 #
+1
n
n
#
0.006328 VR ∇ ·
k
(∇P
+
∇P
−
γ
∇D
)
+
m1
om1
cgom1
gm1
μngm1
+1
n
n
Xmm
ξ n q +1 + Ymm
ξ n q +1 − τmm
− τmm1 ,hc/w =
1 om1 om1
1 gm 1 gm1
1 /m2
0.006328 VR ∇ ·
Xn
VR +1 +1 +1 +1
+1 +1 +1
φm1 Xmm1 Som1 ξom1 + φ+1
m1 Ymm1 Sgm1 ξgm1 −
Δt
VR n n
n
n
n
n
n
n
φm1 Xmm1 Som
ξ
+
φ
Y
S
ξ
m1 mm1 gm1 gm1
1 om1
Δt
(13.131)
The water equation uses a water component that contains NaCl, H2 O, and CO2 .
0.006328 VR ∇ ·
ξn
n
wm1 krwm1
μnwm1
+1
n
n
#
km#1 (∇Pom
−
∇P
−
γ
∇D
)
+
cowm
wm
1
1
1
+1
n
q +1 − τmm
+ τmm1 ,hc/w =
ξwm
1 wm1
1 /m2
VR +1 +1 +1 VR n n n ξ
ξ
−
φ S
φ S
Δt m1 wm1 wm1
Δt m1 wm1 wm1
243
(13.132)
The WCO2 equation
0.006328 VR ∇ ·
Wn
n
ξ n q +1 −
Wmm
1 wm1 wm1
n
n
mm1 ξwm1 krwm1 #
+1
km1 (∇Pom
−
1
μnwm1
+1
τW,mm
+ τW,mm1 ,hc/w =
1 /m2
n
n
#
∇Pcowm
−
γ
∇D
)
+
wm1
1
VR +1 +1 +1 +1 VR n
n
n
n
φm1 Wmm1 Swm1 ξwm1 −
φm1 Wmm
S
ξ
wm
wm
1
1
1
Δt
Δt
(13.133)
(13.134) represents the m2 pore system for the hydrocarbon components
+1
τmm
− τmm2 ,hc/w =
1 /m2
VR +1 +1 +1 +1
+1 +1 +1
φm2 Xmm2 Som2 ξom2 + φ+1
m2 Ymm2 Sgm2 ξgm2 −
Δt
VR n n
n
n
φm2 Xmm2 Som
ξ n + φnm2 Ymm
S n ξn
2 om2
2 gm2 gm 2
Δt
(13.134)
(13.135) represents the m2 pore system for the aqueous component.
+1
τmm
+ τmm2 ,hc/w =
1 /m2
VR +1 +1 +1 VR n n n −
φ S
φ S
ξ
ξ
Δt m2 wm2 wm2
Δt m2 wm2 wm2
(13.135)
(13.136) represents the m2 pore system for WCO2 .
+1
τW,mm
+ τW,mm2 ,hc/w =
1 /m2
VR +1 +1 +1 +1 VR n
n
φm2 Wmm2 Swm2 ξwm2 −
φm2 Wmm
S n ξn
2 wm2 wm2
Δt
Δt
(13.136)
(13.137) represents the m1 /m2 transfer for the hydrocarbon components.
+1
+1
+1
×
τmm
= 0.006328 VR σm#1 /m2 km#1 /m2 Pom
− Pom
1
2
1 /m2
up,n
up,n
k up,n
Xmm1 /m2 ξom
1 /m2 rom1 /m2
μup,n
om1 /m2
+
up,n
Ymm
ξ up,n k up,n
1 /m2 gm 1 /m2 rgm1 /m2
μup,n
gm1 /m2
(13.137)
(13.138) represents the m1 /m2 transfer for the aqueous component.
+1
τmm
1 /m2
= 0.006328 VR
σm#1 /m2 km#1 /m2
+1
Pom
1
−
+1
Pom
2
244
×
up,n
ξwm
k up,n
1 /m2 rwm1 /m2
μup,n
wm1 /m2
(13.138)
(13.139) represents the m1 /m2 transfer for the aqueous component.
+1
τW,mm
1 /m2
13.10.22
= 0.006328 VR
σm#1 /m2 km#1 /m2
+1
Pom
1
−
+1
Pom
2
×
up,n
ξ up,n k up,n
Wmm
1 /m2 wm1 /m2 rwm1 /m2
μup,n
wm1 /m2
(13.139)
Rowe, Brine Density, Eclipse + VIP+CMG, H2 O + NaCl, ρw + Cw
Rowe and Chou (1970) defines a correlation for water density using specific volume v[cm3 /g],
P [kg/cm2 ], T [◦ K], ws [wt fraction], and Cw [cm2 /kg] . This correlation is used by Eclipse, VIP, and
CMG. It is based on experimental data from T = 0◦ C to T = 180◦ C, pressures up to 400 kg/cm2 ,
and salt concentrations up to 25 wt%. The ws used here is based on a system with only H2 O and
NaCl. Since this is also the basis of the measurement of salinity, use wH2 O = 1 − wNaCl .
1
v[P, T, ws ] = A[T ]−P ·B[T ]−P 2 ·C[T ]+ws ·D[T ]+ws2 ·E[T ]−ws ·P ·F [T ]−ws2 ·P ·G[T ]− ws ·P 2 ·H[T ]
2
(13.140)
A[T ] = 5.916365 − 0.01035794T + 0.9270048 · 10−5 T 2 − 1127.522T −1 + 100674.1T −2
B[T ] =
(13.141)
0.5204914 · 10−2 − 0.10482101 · 10−4 T +
0.8328532 · 10−8 T 2 − 1.1702939T −1 + 102.2783T −2
(13.142)
C[T ] = 0.118547 · 10−7 − 0.6599143 · 10−10 T
(13.143)
D[T ] = −2.5166 + 0.0111766T − 0.170552 · 10−4 T 2
(13.144)
E[T ] = 2.84851 − 0.0154305T + 0.223982 · 10−4 T 2
(13.145)
F [T ] = −0.0014814 + 0.82969 · 10−5 T − 0.12469 · 10−7 T 2
(13.146)
245
G[T ] = 0.0027141 − 0.15391 · 10−4 T + 0.22655 · 10−7 T 2
(13.147)
H[T ] = 0.62158 · 10−6 − 0.40075 · 10−8 T + 0.65972 · 10−11 T 2
(13.148)
The compressibility is defined as
1 ∂v
=
Cw [P, T, ws , v] = −
v ∂P T,ws
1
−B[T ] − 2 · P · C[T ] − ws · F [T ] − ws2 · G[T ] − ws · P · H[T ]
−
v
(13.149)
The density is defined as
ρbrine = v −1
(13.150)
The derivative of the density with respect to pressure,
∂ρbrine
∂P
is defined as
∂1/v
Cw
∂v
∂ρbrine
=
= −v −2
=
= ρbrine Cw
∂P
∂P
∂P
v
The derivative of the density with respect to mole fraction,
(13.151)
∂ρbrine
∂Ym
is defined as
∂ρbrine
=0
∂Ym
13.10.23
(13.152)
Garcı́a, CMG, Brine Density, H2 O + CO2 + NaCl, ρw + v̄CO2
Garcı́a (2001) defines the density and partial molar volume of H2 O plus CO2 . The units are
T [◦ C], ρ[kg/m3 ], v̄CO2 [cm3 /mol], c[mol/L], MW[g/mol]. The partial molar volume is calculated
based on temperatures from 0◦ C to 300◦ C and from 0 to 5% molarity.
v̄CO2 [T ] = 37.51 − 9.585 · 10−2 T + 8.740 · 10−4 T 2 − 5.044 · 10−7 T 3
(13.153)
Garcı́a (2001) also presents the following equation for calculating the aqueous density.
ρaq = ρH2 O + MWCO2 · cCO2 − cCO2 · ρH2 O · v̄CO2 · 10−3
246
(13.154)
If we can assume that the H2 O-CO2 system can be decoupled from the H2 O-NaCl, then we can
use the following, where ρbrine comes from Rowe and Chou (1970). Since the molar concentration
cCO2 is a function of the total density ρw,t , this is an iterative calculation.
ρw,t = ρbrine + MWCO2 · cCO2 − cCO2 · ρbrine · v̄CO2 · 10−3
(13.155)
Rewrite cCO2 in terms of WCO2 :
cCO2 = WCO2
WCO2 ρw,t
ρw,t =
MWw,t
j Wj MWj
(13.156)
Substitute for cCO2 in (13.155).
ρw,t =
ρbrine
WCO2
1+
· −MWCO2 + ρbrine · v̄CO2 · 10−3
MWw,t
(13.157)
The total molecular weight for the aqueous phase, MWw,t is defined as follows:
MWw,t = MWCO2 WCO2 + MWbrine (1 − WCO2 )
(13.158)
Where the brine molecular weight is defined by:
MWbrine = MWNaCl WNaCl + MWH2 O (1 − WNaCl )
(13.159)
The derivative of the total molecular weight with respect to WCO2 is
∂MWw,t
= MWCO2 − MWbrine
∂WCO2
(13.160)
g
kg
mol
], ρ[ m
Garcia uses units of T [◦ C], MW[ mol
3 ], and cCO2 [ L ]. (13.157) converts the units to field
lbm
], ρ[ lbm
], and WCO2 [ mol
units, MW[ lbmol
mol ]. A represents converting from
ft3
units of
lbm
.
ft3
ρw,t =
kg
m3
to
lbm
,
ft3
so ρ is in
Two terms in (13.157) are defined locally as α and β to simplify the derivatives.
ρbrine
α
β
WCO2 1/
· −MWCO2 + ρbrine · v̄Garcia
1+
MWw,t
247
(13.161)
The derivative of ρw,t with respect to WCO2 is:
ρw,t β
∂ρw,t
=−
∂WCO2
α MWwt
WCO2 ∂MWwt
1−
MWwt ∂WCO2
(13.162)
The derivative of ρw,t with respect to ρbrine is:
∂ρw,t
= ρwt
∂ρbrine
1/
1
ρbrine
WCO2 v̄Garcia
−
MWwt α
(13.163)
The derivative of ρw,t with respect to pressure
∂ρw,t
∂P
is defined by
∂ρw,t ∂ρbrine
∂ρw,t ∂WCO2
∂ρw,t
= +
∂P
∂ρbrine ∂P
∂WCO2 ∂P
(13.164)
The derivative of ρw,t with respect to mole fraction
∂ρw,t
∂Ym
is defined by
∂ρw,t ∂WCO2
∂ρw,t
=
∂Ym
∂WCO2 ∂Ym
(13.165)
The derivative of ρw,t with respect to mole fraction
∂ρw,t
∂Xm
∂ρw,t ∂WCO2
∂ρw,t
=
∂Xm
∂WCO2 ∂Xm
is defined by
(13.166)
The molar density ξw is defined by
ξw =
ρw,t
MWw,t
(13.167)
The derivative of ξw with respect to WCO2 is
1
∂ξw
=
∂WCO2
MWw,t
∂ρw,t
∂MWw,t
− ξw
∂WCO2
∂WCO2
The derivative of ξw with respect to pressure
1
∂ξw
=
∂P
MWw,t
∂ξw
∂P
(13.168)
is
∂ρw,t
∂MWw,t ∂WCO2
− ξw
∂P
∂WCO2 ∂P
The derivative of ξw with respect to mole fraction
248
(13.169)
∂ξw
∂Ym
is
1
∂ξw
=
∂Ym
MWw,t
∂ρw,t
∂MWw,t ∂WCO2
− ξw
∂Ym
∂WCO2 ∂Ym
The derivative of ξw with respect to mole fraction
1
∂ξw
=
∂Xm
MWw,t
1
=
MWw,t
∂ξw
∂Xm
∂ρw,t
∂MWw,t ∂WCO2
− ξw
∂Xm
∂WCO2 ∂Xm
∂ρw,t
∂MWw,t
− ξw
∂WCO2
∂WCO2
∂WCO2
∂Ym
(13.170)
is
1
=
MWw,t
∂ρw,t
∂MWw,t
− ξw
∂WCO2
∂WCO2
∂WCO2
∂Xm
(13.171)
∂ρw,t
∂P
is defined
When WCO2 is a primary variable, the derivative of ρw,t with respect to pressure
by
∂ρw,t ∂ρbrine
∂ρw,t
= ∂P
∂ρbrine ∂P
(13.172)
When WCO2 is a primary variable, the derivative of ρw,t with respect to mole fraction
∂ρw,t
∂Ym
is
defined by
∂ρw,t
=0
∂Ym
(13.173)
When WCO2 is a primary variable, the derivative of ρw,t with respect to mole fraction
∂ρw,t
∂Xm
is
defined by
∂ρw,t
=0
∂Xm
(13.174)
When WCO2 is a primary variable, the derivative of ξw with respect to pressure
1
∂ξw
=
∂P
MWw,t
∂ξw
∂P
is
∂ρw,t
∂P
(13.175)
When WCO2 is a primary variable, the derivative of ξw with respect to mole fraction
∂ξw
∂Ym
∂ξw
=0
∂Ym
is
(13.176)
When WCO2 is a primary variable, the derivative of ξw with respect to mole fraction
249
∂ξw
∂Xm
is
∂ξw
=0
∂Xm
13.10.24
(13.177)
Kestin, Brine Viscosity, Eclipse+VIP, H2 O + NaCl, μw
Kestin et al. (1981) and Kestin et al. (1978) describe the viscosity of NaCl brine solutions for 20–
150◦ C and pressures of 0.1–35 MPa, and 0–5.4 molal. These correlations are based on Kestin et al.
(1978) and Rowe and Chou (1970). Sayegh and Najman (1987) shows that CO2 has a negligible
impact on the viscosity of the H2 O+NaCl system. These correlations are defined using the following
units: ms [molNaCl /kgH2 O ], μ[μPa · s], P [MPa], T [◦ C], β[1/GPa] The Kestin correlations are based
on a system with H2 O and NaCl only. As a result, the mNaCl and mH2 O are calculated using WNaCl
and WH2 O = 1 − WNaCl .
μ[P, T, ms ] = μ0 [T, ms ] · 1 + β[T, ms ] · 10−3 · P
(13.178)
μ0 [T, ms ] = μ0w [T ] · μ0r [T, ms ]
(13.179)
μ0w [20◦ C] = 1002.0
(13.180)
0
0
μ [T ]
μw [T ]
= log10
=
log10 0 w ◦
μw [20 C]
1002.0
1.2378(20 − T ) − 1.303 · 10−3 (20 − T )2 + 3.06 · 10−6 (20 − T )3 + 2.55 · 10−8 (20 − T )4
96 + T
(13.181)
log10 μ0r [T, ms ] = 3.324 · 10−2 ms + 3.624 · 10−3 m2s − 1.879 · 10−4 m3s +
0
μw [T ]
−2
−2 2
−4 3
(13.182)
−3.96 · 10 ms + 1.02 · 10 ms − 7.02 · 10 ms ∗ log10 0 ◦
μw [20 C]
β[T, ms ] = βsE [T ]β [T, ms ] + βw [T ]
(13.183)
250
βw [T ] = −1.297 + 5.74 · 10−2 T − 6.97 · 10−4 T 2 + 4.47 · 10−6 T 3 − 1.05 · 10−8 T 4
(13.184)
βsE [T ] = 0.545 + 2.8 · 10−3 T − βw [T ]
(13.185)
ms [T ] = 6.044 + 2.8 · 10−3 T + 3.6 · 10−5 T 2
(13.186)
β [T, ms ] = 2.5
13.10.25
ms
ms [T ]
− 2.0
ms
ms [T ]
2
+ 0.5
ms
ms [T ]
3
(13.187)
Duan, Henry’s Law, H2 O + CO2 + NaCl, WCO2 + HCO2
The Duan and Sun (2003) model presents correlations for calculating the Henry’s Law constants
for aqueous solutions containing CO2 , NaCl, plus additional salts. The following units are used in
bar·L
this section: P [bar], T [K], m[mol/kg], v[L/mol], fCO2 [bar mol
mol ], and R[ mol·K ] = 0.08314467.
The Duan correlation is based on experimental data. These data were collected by first creating
a H2 O and NaCl mixture and then measuring the solubility after equilibrium was reached with a
CO2 vapor system. Because of this experimental process and for consistency with the Kestin and
Rowe models, mNaCl and mH2 O are calculated based on WNaCl and WH2 O = 1−WNaCl . Next, mCO2
is calculated using the Duan and Sun (2003) correlation, and then WCO2 is calculated using mNaCl ,
mCO2 , and mH2 O . The value of WNaCl is not changed. The value of WH2 O = 1 − WNaCl − WCO2 is
used for further calculations.
YCO2 P ΦCO2
ln
mCO2
fCO2
= ln
mCO2
l(0)
= ln HCO2
μ
= CO2 + 2mNa λCO2 ,Na + mNa mCl ζCO2 ,Na,Cl
RT
= A[P, T ] + 2mNaCl B[P, T ] + m2NaCl C[P, T ] (13.188)
251
l(0)
μCO2
= A[P, T ] =
RT
28.9447706 + −0.0354581768T + −4770.67077T −1 + 1.02782768 · 10−5 T 2 +
33.8126098
+ 9.04037140 · 10−3 P + −1.14934031 · 10−3 P ln[T ]+
630 − T
−0.0907301486P
9.32713393 · 10−4 P 2
−0.307405726P
+
+
T
630 − T
(630 − T )2
∂A
= 9.04037140 · 10−3 + −1.14934031 · 10−3 ln[T ]+
∂P
−0.307405726
−0.0907301486
9.32713393 · 10−4 P
+
+2·
T
630 − T
(630 − T )2
λCO2 ,Na = B[P, T ] =
(13.190)
−0.411370585 + 6.07632013 · 10−4 T + 97.5347708T −1 +
0.0170656236P
−0.0237622469P
+
+ 1.41335834 · 10−5 T ln[P ]
T
630 − T
−0.0237622469 0.0170656236
T
∂B
=
+
+ 1.41335834 · 10−5
∂P
T
630 − T
P
ζCO2 ,Na,Cl = C[P, T ] =
(13.189)
(13.191)
(13.192)
3.36389723 · 10−4 + −1.98298980 · 10−5 T +
−5.24873303 · 10−3 P
2.12220830 · 10−3 P
+
T
630 − T
2.12220830 · 10−3 −5.24873303 · 10−3
∂C
=
+
∂P
T
630 − T
(13.193)
(13.194)
Use the following definition of mCO2 :
mCO2 =
fCO2
HCO2
(13.195)
The derivative of mCO2 with respect to pressure,
1
∂mCO2
=
∂P
HCO2
∂HCO2
∂fCO2
− mCO2
∂P
∂P
∂mCO2
∂P
is
(13.196)
252
The derivative of HCO2 with respect to pressure,
1
HCO2
∂HCO2
∂P
is
∂A
∂HCO2
∂B
∂C
=
+ 2mNaCl
+ m2NaCl
∂P
∂P
∂P
∂P
(13.197)
To calculate the derivative of WCO2 with respect to pressure,
∂WCO2
∂P ,
use the conversion from
molality to mole fraction.
mi
Wi = j mj
After solving
∂WCO2
∂P
(13.198)
and
∂mH2 O
∂P
simultaneously, this yields
∂WCO2
1 ∂mCO2
=
∂P
∂P
j mj
(13.199)
The derivative of mCO2 with respect to Ym ,
∂mCO2
∂Ym
is
1 ∂fCO2
∂mCO2
=
∂Ym
HCO2 ∂Ym
(13.200)
The derivative of WCO2 with respect to Ym ,
∂WCO2
∂Ym
is
1 ∂mCO2
∂WCO2
=
∂Ym
j mj ∂Ym
(13.201)
Duan and Sun (2003) also tested extending the model to solutions containing Ca2+ , K+ , Mg2+ ,
2−
−
SO2−
4 , CO3 , and HCO3 . They were able to approximate all monovalent cations with mNa+ and
all divalent cations with 2 ∗ mNa+ . Only SO2−
4 required an adjustment factor. (13.188) becomes:
YCO2 P ΦCO2
ln
mCO2
fCO2
= ln
mCO2
= ln H =
l(0)
μCO2
+ 2 (mNa + mK + 2 ∗ mMg + 2 ∗ mCa ) λCO2 ,Na +
RT
(mNa + mK + mMg + mCa ) mCl ζCO2 ,Na,Cl − 0.07mSO4
13.11
Correlations Used to Evaluate Other Correlations
The correlations described in this section are used to evaluate the other correlations.
253
(13.202)
13.11.1
Zeebe, Henry’s Law for Seawater, H2 O + CO2 + NaCl, H
Zeebe and Wolf-Gladrow (2001) describes properties of seawater, as well as a Henry’s Law
correlation for CO2 in seawater. The units of the following correlation are m[molCO2 /kgH2 O ],
fCO2 [atm], H −1 [(molCO2 /kgH2 O )/atm], ws [wt fraction] (assumes salinity S[pptw]), T [◦ C]. This
correlation is the one recommended by Zeebe and Wolf-Gladrow (2001) based on Weiss (1974).
fCO2 = H · mCO2
ln H
−1
(13.203)
T
+
= −60.2409 + 9345.17/T + 23.3585 ln
100
ws · 1000 ·
13.11.2
0.023517 − 0.00023656 · T + 0.0047036
T
100
2
(13.204)
Duan, Fugacity, H2 O + CO2
To compare Henry’s Law computations with direct computations of CO2 solubility, Duan and
Sun (2003) presents correlations for the fugacity of CO2 based on Duan et al. (1992). These
fugacities are only used to check the other correlations. Their accuracy for the CO2 -H2 O system
is reported as within 5%. For simulation with the three-phase system, the fugacity of CO2 is
calculated using the Peng-Robinson equation of state for the gas-oil system. The correlation is
valid in the range 36 to 1000◦ C and 0 to 8000 bar.
fCO2 = YCO2 P ΦCO2
YCO2 =
(13.205)
P − PHs 2 O
P
(13.206)
The water saturation pressure is calculated using Pc,H2 O = 220.85 bar and Tc,H2 O = 647.29 K.
PHs 2 O
Pc T
T − Tc
T − Tc 1.9
=
+ 5.8948420
1 + −38.640844 −
+
Tc
Tc
Tc
T − Tc 3
T − Tc
T − Tc 2
+ 26.654627
+ 10.637097
59.876516
Tc
Tc
Tc
254
4
(13.207)
The fugacity coefficient for the CO2 -H2 O system is calculated using Tr = T /Tc , Pr = P/Pc ,
vr =
vPc
RTc ,
Tc,CO2 = 304.2 K, and Pc,CO2 = 73.825 bar.
1
1
1
ln ΦCO2 = z̆CO2 − 1 − ln z̆CO2 + A1 vr−1 + A2 vr−2 + A3 vr−4 + A4 vr−5 +
2
4
5
a15
a15
1 a13
a14 + 1 − a14 + 1 + 2 exp − 2
2 Tr3 a15
vr
vr
z̆CO2
Pr vr
a13
=
= 1 + A1 vr−1 + A2 vr−2 + A3 vr−4 + A4 vr−5 + 3 2
Tr
Tr vr
a15
a14 + 2
vr
a15
exp − 2
vr
(13.208)
(13.209)
A1 = 8.99288497 · 10−2 + −4.94783127 · 10−1 Tr−2 + 4.77922245 · 10−2 Tr−3
(13.210)
A2 = 1.0380883 · 10−2 + −2.8516861 · 10−2 Tr−2 + 9.49887563 · 10−2 Tr−3
(13.211)
A3 = 5.20600880 · 10−4 + −2.93540971 · 10−4 Tr−2 + −1.77265112 · 10−3 Tr−3
(13.212)
A4 = −2.51101973 · 10−5 + 8.93353441 · 10−5 Tr−2 + 7.88998563 · 10−5 Tr−3
(13.213)
a13 = −1.66727022 · 10−2
(13.214)
13.12
a15 = 2.96 · 10−2
a14 = 1.398
Henry’s Law Correlations
The three methods in this section use Henry’s Law to calculate the solubility of CO2 in the
aqueous phase.
13.12.1
Chang, Mole Fraction, Eclipse + VIP, H2 O + CO2 + NaCl, WCO2 + Rsw + Bw
Chang et al. (1998) defines a correlation for the solubility of CO2 in H2 O. This method is used
by VIP and Eclipse. The units of these correlations use Rsw [SCFCO2 /STBH2 O ], T [◦ F], P [psia],
Bw [RB/STB], ρ[lbm/ft3 ], ws [wt fraction], and Cw [psi−1 ]. This model uses local constants a, b, c,
d, and P 0 .
255
Rsw [P, T ] =
P < P 0 : a · P · 1 − b · sin π2 ·
P ≥ P 0 : a · P 0 (1 − b3 ) + d(P
c·P
c·P +1
− P 0)
(13.215)
The brine solubility is defined by
log10
Rsb [P, T, ws ]
= −2.8037 · ws · T −0.12039
Rsw
(13.216)
The parameters in (13.215) are defined by the following, using
a[T ] = 1.16306+−16.6304·10−3 T +111.07305·10−6 T 2 +−376.85925·10−9 T 3 +524.88916·10−12 T 4
(13.217)
b[T ] = 0.96509+−0.27255·10−3 T +0.09234·10−6 T 2 +−0.10083·10−9 T 3 +0.09979·10−12 T 4 (13.218)
c[T ] = 1.28030 · 10−3 + −10.75660 · 10−6 T + 52.69622 · 10−9 T 2 +
− 222.39488 · 10−12 T 3 + 462.67255 · 10−15 T 4 (13.219)
P 0 [T ] =
sin−1 [b2 ]
2
·
π c 1 − π2 · sin−1 [b2 ]
(13.220)
cP 0
cP 0
cP 0
π
π
π
·
·
·
cos
+
d[T ] = a − ab sin
2 1 + cP 0
2 (1 + cP 0 )2
2 1 + cP 0
(13.221)
The formation volume factor Bw is defined as follows, with ρw,sc and ρw,atm defined by Rowe
and Chou (1970).
Bw [P, T, ws ] =
ρw,sc[P = 14.7 psia, T = 60◦ F, ws ] + 0.02066 · Rsb
ρw,atm [P = 14.7 psia, T, ws ] + 0.0058 · Rsb
(13.222)
The water compressibility Cw is defined based on Rowe and Chou (1970) and the following:
P > 5000 :
1
1
=
+ 7.033(P − 5000)
Cw [P, T, ws ]
Cw,5000 [P = 5000 psia, T, ws ]
Chang et al. (1998) suggests using Kestin et al. (1978) for the viscosity correlation.
256
(13.223)
13.12.2
CMG, Henry’s Law, H2 O + CO2 + NaCl, WCO2 + HCO2
The GEM User’s Manual, CMG (2010), specifies several correlations for calculating the mole
fraction of CO2 in the aqueous phase using Henry’s Law correlations. There are also correlations
listed for N2 , H2 S, and CH4 . These correlations use His [MPa], P [MPa], T [K], v̄[cm3 /mol], and
Cs [mol/kgH2 O ]. The critical properties of water are Tc,H2 O = 647.14 K and Pc,H2 O = 22.064 MPa
The following correlation for the Henry’s Law is based on Harvey (1996), based on the H2 O saturation pressure.
s
[T ] = ln PHs 2 O + −9.4234
ln HCO
2
−1
T
Tc,H2 O
+ 4.0087 1 −
10.3199 exp 1 −
T
T
Tc,H2 O
Tc,H2 O
0.355 −1
T
Tc,H2 O
T
+
−0.41
(13.224)
Tc,H2 O
The saturation pressure is defined by Saul and Wagner (1987). Saul and Wagner (1987) also defines
the density, specific enthalpy, and specific entropy of water at the saturation pressure, but these
properties are not needed here.
PHs 2 O [T ]
T
Tc,H2 O
T
−7.85823 1 −
=
+ 1.83991 1 −
Pc,H2 O
T
Tc,H2 O
Tc,H2 O
3
3.5
T
T
+ 22.6705 1 −
+
− 11.7811 1 −
Tc,H2 O
Tc,H2 O
4
T
T
+ 1.77516 1 −
− 15.9393 1 −
Tc,H2 O
Tc,H2 O
ln
1.5
+
7.5
(13.225)
The Henry’s Law constant is defined by
s
[T ] +
ln HCO2 [P, T ] = ln HCO
2
1
RT
.
P
s
PH
v̄CO2 [T ]dP
2O
1
s
s
v̄
[T
]
+
[T
]
P
−
P
[T
]
= ln HCO
CO
O
H
2
2
2
RT
(13.226)
The partial molar volume is calculated using Garcı́a (2001). The salinity effects are calculated
using a “salting out” coefficient ksalt,CO2 .
ln
Hsalt,CO2
HCO2
= ksalt,CO2 Cs
(13.227)
257
The salting out coefficient is defined based on Bakker (2003), for temperatures from 0◦ C to 300◦ C.
ksalt,CO2 [T ] = 0.11572−6.0293·10−4 (T −273.15)+3.5817·10−6 (T −273.15)2 −3.7772·10−9 (T −273.15)3
(13.228)
The fugacity of the water phase is defined by
faq,CO2 = WCO2 P ΦCO2 = WCO2 HCO2
13.12.3
(13.229)
Enick, Henry’s Law, H2 O + CO2 + NaCl, H + Rsw + WCO2 + μw
Enick and Klara (1992) and Enick and Klara (1990) describe ways to calculate the solubility of
CO2 .
The following correlations are from Enick and Klara (1990). This uses T [K], P [MPa], v[cm3 /mol],
ws [wt fraction], wCO2 [wt fraction], ws [wt fraction], WCO2 [mol fraction]. Cs is the total dissolved
solids in weight percent excluding dissolved gases. Several correlations are defined for Hi∗ and vi∞ .
These were evaluated by Enick and Klara (1990) for temperatures between 298 K–523 K and a
pressure range from 3.4 MPa–85 MPa.
fCO2
ln
WCO2
∗
+
= ln H = ln HCO
2
v̄ ∞ P
A
(WH2 2 O − 1) + CO2
RT
RT
∗
= −5076.29 + 31.9877T − 0.057691T 2 + 3.18012 · 10−5 T 3
HCO
2
(13.230)
(13.231)
A = −2.08184 · 106 + 2.13034 · 104 T − 79.8190T 2 + 0.129991T 3 − 7.76471 · 10−5 T 4 (13.232)
∞
[T ] = 1799.36 − 17.8218T + 0.0659297T 2 − 1.05786 · 10−4 T 3 + 6.200275 · 10−8 T 4 (13.233)
v̄CO
2
The following equation is for weight fractions wCO2 .
wCO2 ,b = wCO2 ,w 1 − 4.893414 · 10−2 (100ws )+
0.1302838 · 10−2 (100ws )2 − 0.1871199 · 10−4 (100ws )3
258
(13.234)
Enick and Klara (1992) uses the model by Enick and Klara (1990).
WCO2 ,b =
wCO2 ,b
44
wCO2 ,b
44
(1−wCO2 ,b )
+ MW
b
(13.235)
1051.2
58.4 − 0.404CTDS
MWb =
(13.236)
ρb = ξb MWb
(13.237)
μb = μw 1 + 1.892 · 10−2 Cs + 1.215 · 10−4 (Cs )2 + 1.941 · 10−5 (Cs )3
(13.238)
Cs [wt%] =
13.13
58.4WNaCl
18WH2 O + 58.4WNaCl
(13.239)
Adjustments to Peng-Robinson Equation of State
The methods in this section use modifications to the equation of state to calculate the CO2
solubility in the aqueous phase.
13.13.1
Peng-Robinson Equation of State Paramters
The Peng-Robinson Equation of State Peng and Robinson (1976) with the Peneloux volume
correction (Péneloux and Rauzy, 1982) is defined by:
P =
a
RT
−
v − b (v + c)(v + 2c + b) + (b + c)(v − b)
(13.240)
The am are defined by:
am =
2
R2 Tcm
Ωa
Pcm
1 + κm [ωm ] 1 −
!
T
Tcm
2
(13.241)
The bm are defined by:
bm = Ω b
RTcm
Pcm
(13.242)
259
This results in an adjustment to the specific volumes and the densities, but does not adjust the
phase splitting.
vnew = vEOS −
Xm cm
(13.243)
In some cases (for instance the Eclipse SSHIFT parameter sm ) , the volume shift is defined as a
multiplier to the bm :
cm = bm sm
(13.244)
The amn is defined by (13.245). The binary interaction coefficient is δ̆mn (Eclipse BIC). In the
Peng-Robinson 1978 version, δ̆mn is a function of temperature. δ̆mn is often labeled kmn , but the
symbol δ̆mn is used here to avoid confusion with permeability.
1/2
amn = (1 − δ̆mn )a1/2
m an
(13.245)
Compute the mixed a using:
ale1 =
e1 e1
amn Xm
Xn
ave1 =
m,n
amn Yme1 Yne1
(13.246)
m,n
Compute the mixed b using:
ble1 =
e1
bm Xm
bve1 =
m
bm Yme1
(13.247)
cm Yme1
(13.248)
m
Compute the mixed c using:
cle1 =
m
13.13.2
e1
cm Xm
cve1 =
m
Soreide, EOS, Eclipse, H2 O + CO2 + NaCl, WCO2 + ρaq
Eclipse describes modifications of the Peng-Robinson equation of state based on Søreide and
Whitson (1992). Redefine the am for the water component, using the following units: P [bar],
Cs [molCO2 /kgH2 O ], T [◦ C] for temperatures from 0◦ C to 325◦ C and salt concentrations from 0 to
5 mol/kg.
260
am =
2
R2 Tcm
Ωa
Pcm
T 1.1
1 + 0.4530 1 − 1 − 0.0103(Cs )
+
Tcm
T
0.0034
Tcm
−3
2
−1
(13.249)
The unit conversions for molality are:
molality = 1000 ∗
molsalt
masswater
Csmolal =
Csppmw
58440.0 − 0.05844 ∗ Csppmw
(13.250)
The binary interaction coefficients in the aqueous phase for water are defined with a temperature
and salinity dependence, based on Søreide and Whitson (1992).
aq
−0.1
=
1.112
−
1.7369ω
(1 + 0.017407Cs ) +
δ̆j,H
j
2O
T
+
(1.1001 + 0.836ωj ) (1 + 0.033516Cs )
Tc,H2 O
(−0.15742 − 1.0988ωj ) (1 + 0.011478Cs )
T
2
(13.251)
Tc,H2 O
The binary interaction coefficients in the aqueous phase for CO2 are defined with a temperature
and salinity dependence, based on Søreide and Whitson (1992).
aq
0.7505
=
−0.31092
1
+
0.15587
(C
)
+
δ̆CO
s
2 ,H2 O
T
+
0.23580 1 + 0.17837 (Cs )0.979
Tc,CO2
− 21.2566 exp −6.7222
13.13.3
T
Tc,CO2
− Cs
(13.252)
Delshad, EOS and IFT, H2 O + CO2 + NaCl, WCO2 + ρaq + σgw
Delshad et al. (2011) describes adjustments to the EOS and calculation of calculation of the
interfacial tension. These equations use T [◦ F] and total dissolved solids Cs [ppm].
δ̆H2 O,CO2 = −0.093625 + 4.861 · 10−4 (T − 113) + 2.29 · 10−7 Cs
It uses a volume shift defined by
261
(13.253)
cH2 O = 0.179 + 2.2222 · 10−4 (T − 113) + 4.9867 · 10−7 Cs
(13.254)
The gas-water interfacial tension is calculated using the following correlations. This correlation
uses T [◦ C], P [MPa], salinity Cs [wt%], and σ[mN/m]. It reproduces the trend in reduced interfacial
tension, but the absolute magnitudes do not fit the experimental data of Bennion and Bachu (2008b)
very well (see Delshad et al. (2011), figure 1–2).
σwg = 71.69243P 0.432629 + 0.210558T 0.900261 + 0.075859Cs1.457937
13.13.4
(13.255)
Yan, EOS, H2 O + CO2 + NaCl, WCO2 + ρaq
Yan and Stenby (2009) uses the Søreide and Whitson (1992) model with some adjustments. For
a system where the CO2 is soluble in water but water is not present in the hydrocarbon phase, the
binary interaction term is adjusted using the following equation. Yan and Stenby (2009) makes the
comment that assuming no H2 O in the vapor phase leads to large inaccuracies in the CO2 fugacity
at temperatures above 150◦ C and/or low pressures. The units are Cs [molal] and T [◦ C].
δ̆H2 O,CO2 = −0.00739470Cs − 0.443752 + 4.55173 · 10−5 Cs + 0.00111209 T
13.13.5
(13.256)
Melhem, EOS, H2 O + CO2 , WCO2 + ρaq
Melham and Little (1989) defines some modifications to the Peng-Robinson equation of state.
Redefine the am for water and CO2 , using T [K], P [atm]. Tc,CO2 [K] = 304.2, Pc,CO2 [atm] = 72.8,
Tc,H2 O [K] = 647.3, and Pc,H2 O [atm] = 217.6.
aCO2 =
Ωa
aH 2 O =
Ωa
2
R2 Tc,CO
2
Pc,H2 O
exp 0.6877 1 −
Pc,CO2
2
R2 Tc,H
2O
*
*
exp 0.8893 1 −
T
Tc,CO2
T
Tc,H2 O
262
,
T
+ 0.3813 1 −
+ 0.0151 1 −
2 +
(13.257)
Tc,CO2
,
T
Tc,H2 O
2 +
(13.258)
13.13.6
Spycher, EOS, Eclipse, H2 O + CO2 + NaCl, WCO2 + ρaq
The mole fraction of H2 O in the gas phase and CO2 in the aqueous phase in the presence of NaCl
is defined by Spycher and Pruess (2005). The correlations require values of the thermodynamic
0
0
,KCO
) , the
equilibrium constant at temperature T and reference pressure P 0 = 1 bar (KH
2O
2
fugacity coefficient of each species in the gas phase (ΦH2 O , ΦCO2 ), the average partial molar volume
over the pressure range P − P 0 (V̄H2 O , V̄CO2 ) from Spycher et al. (2003).
The activity coefficient of CO2 in a mixture containing various salts can be calculated using
several different techniques from the literature. The following two methods give the best results
according to Spycher and Pruess (2005). Duan and Sun (2003) defines the activity coefficient in
terms of pressure, temperature, molality of various salts, and the molality in a pure CO2 -H2 O
mixture. Rumpf et al. (1994) defines the activity coefficient in terms of temperature and the
molality of various salts, and the molality in a pure CO2 -H2 O mixture.
Together, Spycher and Pruess (2005) and Spycher et al. (2003) describe a Redlich-Kwong equation of state model for the H2 O+CO2 +NaCl system. Because the model is based on Redlich-Kwong
rather than Peng-Robinson, another option is preferred if possible.
13.14
Models Considered But Not Used
Li and Nghiem (1986) defines correlations for Henry’s Law constants, for H2 O + CO2 + NaCl.
CMG uses this model to calculate three-phase equilibria. The model is more complicated than the
other models in Section 13.12. It requires parameters from scaled particle theory which may not
be commonly available. Because of the additional data requirements, added complexity, and focus
on three-phase equilibrium calculations, this technique was not selected for this project.
The methods of Kell and Whalley (1975) and Zaytsev and Aseyev (1992) define methods for
calculating the density as a function of the detailed salt composition. The method is used by the
CO2STORE option of Eclipse, but is overly complicated for this work. Kell and Whalley (1975)
defines a correlation for the pure water density as a function of temperature and pressure for 0–
1000 bar and 0–150◦ C. Zaytsev and Aseyev (1992) describes a method originally based on Erzokhi
to adjust the density of water based on the concentrations of various salts. These specify different
correlation for each salt to adjust the density.
263
The methods of Vesovic et al. (1990) and Fenghour et al. (1998) define correlations for the
viscosity of pure CO2 . These methods are used by the CO2STORE option of Eclipse, but is too
specialized for this work. Vesovic et al. (1990) describes a correlation for the viscosity of CO2 in
terms of μ0 (which is defined as a correlation in terms of temperature), a complex correlation for
near-critical behavior in terms of density and temperature, and a correlation in terms of density
and temperature. Fenghour et al. (1998) provides a simpler correlation for μ0 and Δμ, but uses
the same correlation near the critical region.
Duan, Hu, Li, and Mao (2008) presents a detailed review of the experimental data for the
H2 O + CO2 + NaCl system. The data review is good, but the correlations are not well presented.
Majer, Sedlbauer, and Bergin (2008) presents a detailed model for calculating the Henry’s Law
constant for aqueous H2 O + CO2 , plus various other components. It is a complicated model that
does not include the effect of salinity, so it will not be used here.
Fernández-Prini, Alvarez, and Harvey (2003) provides an updated correlation for Henry’s Law
constants for aqueous H2 O+CO2 , plus various other components. Since the model does not account
for salinity and requires an additional saturation pressure correlation, it is not used here.
Rumpf et al. (1994) conducted experiments of the solubility in the H2 O + CO2 + NaCl system.
They present a complex model to calculate this solubility. Their data is used by several of the more
recent articles. Other articles provide a simpler approach which is more applicable to this project.
264
CHAPTER 14
COMPUTATION: ASSEMBLY OF JACOBIAN
0.006328∇ ·
0.006328∇ ·
0.006328∇ ·
X n ξn
m o n #
kro k (∇Pon+1 − γon ∇D# ) +
μno
Y nξn
m g n #
n
krg k (∇Pon+1 + ∇Pcgo
− γgn ∇D# ) +
n
μg
W n ξn
m w n
n
n
krw k # (∇Pon+1 − ∇Pcow
− γw
∇D# )
n
μw
n n n
n n n
+ Xm
ξo q̂o + Ymn ξgn q̂gn + Wm
ξw q̂w =
1 n+1 n+1 n+1 n+1
n+1 n+1 n+1
φ
(Xm So ξo + Ymn+1 Sgn+1 ξgn+1 + Wm
Sw ξw ) −
Δt
1 n n n n
n n n
φ (Xm So ξo + Ymn Sgn ξgn + Wm
Sw ξw )
Δt
14.1
(14.1)
Diagonal Terms
Figure 14.1: Block 1: block geometry for the main block diagonal of a NC = 5 problem. Black
represents non-zero values; gray represents zero values.
Diagonal terms have the following form, Figure 14.1.
Po So Sg Xm
Cm
Gm
Ym
X
X
X
X
X
X
0
0
X
X
(14.2)
Diagonal terms have the following form. If there are no well connections to cell ijk, then
WDP = 0.
265
Po
m
VR ∂Accijk
− Δt ∂P +
mn
mn
DPmn
xt,ijk + DPyt,ijk + DPzt,ijk +
WDPmn
ijk
Cm
∂fm
o,ijk
∂P
Gm
−
So
m
VR ∂Accijk
− Δt ∂So
Sg
m
VR ∂Accijk
− Δt ∂Sg
0
0
∂fm
g,ijk
∂P
X m
m
VR ∂Accijk
− Δt ∂X m
∂fm
o,ijk
∂Xm
−
∂fm
g,ijk
∂Xm
Ym
m
VR ∂Accijk
− Δt ∂Y m
∂fm
o,ijk
∂Ym
−
∂fm
g,ijk
∂Ym
(14.3)
14.2
Diagonal Terms Above the Bubble Point
Figure 14.2: Block 7: block geometry above the bubble point for the main block diagonal of a
NC = 5 problem. Black represents non-zero values; gray represents zero values.
Diagonal terms have the following form, Figure 14.2.
Po So Pb Xm
Cm
Gm
Ym
X
X
0
X
X
0
0
X
X
X
(14.4)
Diagonal terms above the bubble point (Sg = 0) have the following form. If there are no well
connections to cell ijk, then WDP = 0.
Po
m
VR ∂Accijk
− Δt ∂P +
mn
mn
DPmn
xt,ijk + DPyt,ijk + DPzt,ijk +
WDPmn
ijk
So
m
VR ∂Accijk
− Δt ∂So
Gm
0
0
GNC −1
0
0
Cm
Pb
X m
m
VR ∂Accijk
− Δt ∂X m
0
∂fm
o,ijk
∂Pb
−
∂fm
g,ijk
∂Pb
∂GN −1
C
∂Pb
∂fm
o,ijk
∂Xm
−
∂fm
g,ijk
∂Xm
∂GN −1,ijk
C
∂Xm
Ym
m
VR ∂Accijk
− Δt ∂Y m
∂fm
o,ijk
∂Ym
−
∂fm
g,ijk
∂Ym
∂GN −1,ijk
C
∂Ym
(14.5)
266
14.3
Diagonal Terms Below the Dew Point
Figure 14.3: Block 7: block geometry below the dew point for the main block diagonal of a NC = 5
problem. Black represents non-zero values; gray represents zero values.
Diagonal terms have the following form, Figure 14.3.
Po Pd Sg Xm
Cm X 0
X
Gm 0
X
0
Ym
X
X
X
X
(14.6)
Diagonal terms below the dew point (So = 0) have the following form. If there are no well
connections to cell ijk, then WDP = 0.
Cm
Po
m
VR ∂Accijk
− Δt ∂P +
mn
mn
DPmn
xt,ijk + DPyt,ijk + DPzt,ijk +
WDPmn
ijk
Gm
0
GNC −1
0
Pd
Sg
m
VR ∂Accijk
− Δt ∂Sg
0
∂fm
o,ijk
∂Pd
−
∂fm
g,ijk
∂Pd
∂GN −1,ijk
C
∂Pd
0
0
X m
m
VR ∂Accijk
− Δt ∂X m
∂fm
o,ijk
∂Xm
−
∂fm
g,ijk
∂Xm
∂GN −1,ijk
C
∂Xm
Ym
m
VR ∂Accijk
− Δt ∂Y m
∂fm
o,ijk
∂Ym
−
∂fm
g,ijk
∂Ym
∂GN −1,ijk
C
∂Ym
(14.7)
14.4
Off-Diagonal Terms
Off-diagonal terms have the following form, Figure 14.4.
Po So Sg Xm
Cm
Gm
Ym
X
0
0
0
0
0
0
0
0
0
(14.8)
267
Figure 14.4: Block 2: block geometry for the off-block diagonal values with the IMPES formulation
for a NC = 5 problem. Black represents non-zero values; gray represents zero values.
Off-diagonal bands have the following form, here illustrated for i + 1, j, k.
Po
14.5
Cm
DPmn
x
y t,i+1,jk
z
Gm
0
So
Sg
X m
Ym
0
0
0
0
0
0
0
0
(14.9)
Well Terms
Figure 14.5: Block 4: well terms for the component equations for a NC = 5 problem. Black
represents non-zero values; gray represents zero values.
Well unknowns have the following form, Figure 14.5.
|q Pt,w
t,w
Cm
Gm
(14.10)
X
0
Well unknowns have the following form.
Cm
Gm
|q Pt,w
t,w
WDWmn
ijk
(14.11)
0
268
14.6
Right Hand Side
Figure 14.6: Block 6: right-hand-side terms for the component equations for a NC = 5 problem.
Black represents non-zero values; gray represents zero values.
Right-hand-side, constant terms have the following form, Figure 14.6.
R
Cm
Gm
(14.12)
X
X
Right-hand-side, constant terms without well connections have the following form.
Cm
Gm
m
VR
Δt Accijk
−
R
mn
mn
+ DCmn
xt,ijk + DCyt,ijk + DCzt,ijk
m
m
−fo,ijk + fg,ijk
mn
VR
Δt Accijk
(14.13)
Right-hand-side, constant terms with well connections have the following form.
R
Cm
Gm
m
VR
Δt Accijk
−
mn
VR
Δt Accijk
+
mn
DCmn
xt,ijk + DCyt,ijk
m
m
−fo,ijk + fg,ijk
mn
+ DCmn
zt,ijk + WCijk
(14.14)
Right-hand-side, constant terms above the bubble point or below the dew point without well
connections have the following form.
Cm
Gm
GNC −1
m
VR
Δt Accijk
−
R
mn
mn
+ DCmn
xt,ijk + DCyt,ijk + DCzt,ijk
m
−fm
o,ijk + fg,ijk
−GNC −1,ijk
mn
VR
Δt Accijk
(14.15)
Right-hand-side, constant terms above the bubble point or below the dew point with well
connections have the following form.
269
Cm
Gm
m
VR
Δt Accijk
−
mn
VR
Δt Accijk
GNC −1
14.7
R
mn
mn
mn
+ DCmn
xt,ijk + DCyt,ijk + DCzt,ijk + WCijk
m
−fm
o,ijk + fg,ijk
−GNC −1,ijk
(14.16)
Total Rate Equations
Figure 14.7: Block 5: blocks for the well equations for a NC = 5 problem. Black represents non-zero
values; gray represents zero values.
Total rate equations for each well have the following form, Figure 14.7.
Po So Sg Xm
Qw
X
0
0
0
Ym
(14.17)
0
Total rate equations for each well have the following form.
Qw
Po
QDPnijk
So Sg Xm
0
0
0
Ym
(14.18)
0
Diagonal terms for the total rate equations have the following form.
|q Pt,w
t,w
Qw
(14.19)
X
Diagonal terms for the total rate equations have the following form.
Qw
|q Pt,w
t,w
QDWnijk
(14.20)
Right-hand-side, constant terms for the total rate equations have the following form.
R
Qw
(14.21)
X
Right-hand-side, constant terms for the total rate equations have the following form.
R
Qw QCn
ijk
(14.22)
270
14.8
Accumulation
Define the accumulation term
=
φ
ξ
S
X
+
φ
ξ
S
Y
+
φ
ξ
S
W
Accm
i
oi
oi
mi
i
gi
gi
mi
i
wi
wi
mi
i
14.9
(14.23)
Accumulation Derivatives: Pressure
For the normal hydrocarbon components,
∂Accmi
∂P ,
for cell i and component m = 1 . . . NC − 2.
∂ξgi
∂Accmi
∂φi
∂φi
∂ξoi
= ξoi
+ ξgi
+ φi Soi
+ φi Sgi
Soi Xmi
Sgi Ymi
Xmi
Ymi
∂P
∂P
∂P
∂P
∂P
For the CO2 component,
∂Accmi
∂P ,
(14.24)
for cell i and component m = NC − 1.
∂Accmi
∂φi
∂φi
∂φi
= ξoi
+ ξgi
+ ξwi
+
Soi Xmi
Sgi Ymi
Swi
Wmi
∂P
∂P
∂P
∂P
∂WCO
∂ξgi
2 ,i
∂ξoi
∂ξwi
+ φi Sgi
+ φi Swi
+ φi ξwi
Xmi
Ymi
Wmi
Swi
φi Soi
∂P
∂P
∂P
∂P
For the H2 O component,
∂Accmi
∂P ,
(14.25)
for cell i and component m = NC .
∂WCO2 ,i
∂Accmi
∂φi
∂ξwi
= ξwi
+ φi Swi
− φi ξwi
Swi
Wmi
Wmi
Swi
∂P
∂P
∂P
∂P
14.10
(14.26)
Accumulation Derivatives: Saturation
Evaluate
∂Accmi
∂So .
∂Accmi
= φi ξoi
Xmi
− φi ξwi
Wmi
∂So
Evaluate
(14.27)
∂Accmi
∂Sg .
∂Accmi
= φi ξgi
Ymi
− φi ξwi
Wmi
∂Sg
(14.28)
Above the bubble point, Sg = 0 and Sg → Pb becomes a new primary variable and
Below the dew point, So = 0 and So → Pd becomes a new primary variable and
271
∂Accmi
∂Pd
∂Accmi
∂Pb
=0
= 0.
14.11
Accumulation Derivatives: Composition
For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m = 1 . . . NC − 2,
evaluate
∂Accmi
.
∂Xm
∂Accmi
∂ξoi
= φi Soi
Xmi
+ φi ξoi
Soi δm,m
∂Xm
∂Xm
(14.29)
For the normal hydrocarbon component equations Cm = 1 . . . NC − 2 and m = 1 . . . NC − 2,
evaluate
∂Accmi
.
∂Ym
∂ξgi
∂Accmi
= φi Sgi
Ymi
+ φi ξgi
Sgi δm,m
∂Ym
∂Ym
(14.30)
For the CO2 component equation Cm = CNC −1 and m = 1 . . . NC − 2, evaluate
∂Accmi
.
∂Xm
∂AccCO2 ,i
∂ξoi
= φi Soi
XCO
− φi Soi
ξoi
2 ,i
∂Xm
∂Xm
(14.31)
For the CO2 component equation Cm = CNC −1 and m = 1 . . . NC − 2, evaluate
∂Accmi
.
∂Ym
∂ξgi
∂ξ ∂Accmi
∂WCO2
= φi Sgi
YCO
− φi Sgi
ξgi + φi Swi
WCO2 wi + φi Swi
ξwi
2
∂Ym
∂Ym
∂Ym
∂Ym
For the water component equation Cm = CNC and m = 1 . . . NC − 2, evaluate
(14.32)
∂Accmi
.
∂Xm
∂AccH2 O,i
=0
∂Xm
(14.33)
For the water component equation Cm = CNC and m = 1 . . . NC − 2, evaluate
∂AccH2 O,i
∂ξ ∂WCO2
= φi Swi
WH2 O wi − φi Swi
ξwi
∂Ym
∂Ym
∂Ym
14.12
∂Accmi
.
∂Ym
(14.34)
Spatial Derivatives: Pressure
The following derivatives are written in terms of x and i ± 1. The same approach applies to y
and j ± 1 and z and k ± 1.
The following are the multiples of δPi±1 . All ± are either positive or negative for this equation.
272
mn
mn
mn
DPmn
=
T
+
T
+
T
xt,i±1
xo,i± 1
xg,i± 1
xw,i± 1
2
2
(14.35)
2
The following are the multiples of δPi .
mn
mn
=
−
DP
+
DP
DPmn
xt,i
xt,i+1
xt,i−1 =
mn
mn
mn
mn
mn
mn
− Txo,i+
(14.36)
1 + T
1 + T
1 + T
1 + T
1 + T
1
xo,i−
xg,i+
xg,i−
xw,i+
xw,i−
2
2
2
2
2
2
The following do not multiply deltas. All ± are either positive or negative for this equation.
mn
n
mn
n
n
=
T
·
P
−
γ
D
·
P
−
γ
D
+
P
+
T
DCmn
xt,i±1
i±1
i±1
cgo,i±1 +
xo,i± 12
o,i± 12 i±1
xg,i± 21
g,i± 21 i±1
mn
n
n
Txw,i±
·
P
−
γ
D
−
P
(14.37)
1
i±1
cow,i±1
w,i± 1 i±1
2
2
The following do not multiply deltas.
mn
n
mn
n
DCmn
xt,i = −Txo,i+ 1 · Pi − γo,i+ 1 Di − Txo,i− 1 · Pi − γo,i− 1 Di +
2
2
2
2
mn
n
n
mn
n
n
+
− Txg,i+ 1 · Pi − γg,i+ 1 Di + Pcgo,i − Txg,i− 1 · Pi − γg,i− 1 Di + Pcgo,i
2
2
2
2
mn
n
n
mn
n
n
Pi − γw,i+
Pi − γw,i−
− Txw,i−
(14.38)
− Txw,i+
1 ·
1 Di − Pcow,i
1 ·
1 Di − Pcow,i
2
14.13
2
2
2
Fugacity Equations
The fugacities are defined by
m
fm
o = Φo Xm P
Evaluate
∂fm
oi
∂P ,
∂fm
oi
= fm
oi
∂P
Evaluate
∂fm
gi
= fm
gi
∂P
(14.39)
m = 1 . . . NC − 1:
∂fm
gi
∂P ,
m
fm
g = Φ g Ym P
1 ∂Φm
oi
∂P
Φm
oi
+ Φm
oi Xm
(14.40)
m = 1 . . . NC − 1:
1 ∂Φm
gi
m
Φgi ∂P
+ Φm
gi Ymi
(14.41)
273
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂fm
oi
= fm
oi
∂Xm
1 ∂Φm
oi
∂X
Φm
m
oi
1 ∂Φm
gi
∂Y
Φm
m
gi
(14.42)
14.14
1 ∂Φm
oi
Φm
oi ∂Xm
1 ∂Φm
gi
∂Y
Φm
m
gi
for m = 1 . . . NC − 2:
+ Φm
gi P δm,m
(14.43)
∂fl
mi
∂Xm
for m = 1 . . . NC − 2:
− Φm
oi P
(14.44)
For the CO2 equations m = 1 . . . NC − 1, evaluate
∂fm
gi
= fm
gi
∂Ym
∂fm
gi
∂P
For the CO2 equations m = NC − 1, evaluate
∂fm
oi
= fm
oi
∂Xm
for m = 1 . . . NC − 2:
+ Φm
oi P δm,m
For the normal hydrocarbon equations m = 1 . . . NC − 2, evaluate
∂fm
gi
= fm
gi
∂Ym
∂fo
gi
∂Xm
∂fm
gi
∂P
for m = 1 . . . NC − 2:
− Φm
gi P
(14.45)
Fugacity Equations - Above Bubble Point
Above the bubble point, Sg = 0. Sg is replaced by a new variable, the bubble point pressure
Pb .
GNC −1 = Pb −
N
C −1
m=1
fom [Pb , X]
]
Φgm [Pb , Y
Evaluate the derivative
(14.46)
∂GNC −1
∂Pb :
N
C −1
1
∂GNC −1
=1−
v
∂Pb
m=1 Φm [Pb , Y ]
∂Φv [Pb , Y
]
∂flm [Pb , X]
fl [Pb , X]
m
− m v
∂Pb
Φm
∂Pb
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
N
C −1
1
∂GNC −1
=−
v
∂Xm
m=1 Φm [Pb , Y ]
∂flm [Pb , X]
∂Xm
(14.47)
∂GNC −1
∂Xm :
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
274
(14.48)
∂GNC −1
∂Ym :
N
C −1
∂GNC −1
=
∂Ym
m=1
14.15
∂Φvm [Pb , Y
]
flm [Pb , X]
2
∂Ym
(Φvm )
(14.49)
Fugacity Equations - Below Dew Point
Above the bubble point, So = 0. So is replaced by a new variable, the dew point pressure Pd .
GNC −1 = Pd −
N
C −1
m=1
]
fgm [Pd , Y
Φom [Pd , X]
Evaluate the derivative
(14.50)
∂GNC −1
∂Pb :
N
C −1
1
∂GNC −1
=1−
l
∂Pd
m=1 Φm [Pd , X]
] fv [Pd , Y
] ∂Φl [Pd , X]
∂fvm [Pd , Y
m
− m l
∂Pd
Φm
∂Pd
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
N
C −1
∂GNC −1
=
∂Xm
m=1
fvm [Pd , Y ] ∂Φlm [Pd , X]
∂Xm
(Φlm )2
14.16
(14.51)
∂GNC −1
∂Xm :
(14.52)
Evaluate the derivative for m = 1 . . . NC − 2, evaluate
N
C −1
1
∂GNC −1
=−
l
∂Ym
m=1 Φm [Pd , X]
]
∂fvm [Pd , Y
∂Ym
∂GNC −1
∂Ym :
(14.53)
Computation for Fixed Rate Wells
Each component equation Cw,α,m has a source term. The coefficient of δP is
#
n
n
n
n
n
n
n
n
n
WDPmn
w,α = −WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α
(14.54)
The coefficient of δPw is
#
n
n
n
n
n
n
n
n
n
WDWmn
w,α = WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α
The constant terms associated with the well are
275
(14.55)
#
n
n
n
n
n
n
n
n
n
WCmn
w,α = WIw,α · Xm,w,α ξo,w,α λo,w,α + Ym,w,α ξg,w,α λg,w,α + Wm,w,α ξw,w,α λw,w,α ·
w ,n
− Pw, + Pw,α
Pw,α
(14.56)
Each well has a total rate equation. This equation has the following form for a fixed rate well.
The coefficient of δP is
QDPnw,α
= − WI#
w,α ·
emax n
qo,w,α
ξo,w,α λno,w,α
w,emax
ξo,w,α
max
+
emax n
qg,w,α
ξg,w,α λng,w,α
w,emax
ξg,w,α
max
+
emax n
qw,w,α
ξw,w,α λnw,w,α
w,emax
ξw,w,α
max
(14.57)
The coefficient of δPw is
QDWnw,α =
α
max
α =1
WI#
w,α ·
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(14.58)
The constant terms associated with the constant rate equation
max α
#
= qt,w +
WIw,α ·
Pw,α
RHS
QCn
w,α
α =1
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
14.17
+
w,const @ n
− Pw, + Pw,α
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
·
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(14.59)
Computation for Fixed Pressure Wells
Each component equation Cw,α,m has a source term. This term has the following form for a
fixed pressure well. The coefficient of δP is 0.
WDPmn
w,α = 0
(14.60)
is
The coefficient of δqt,w
WDWmn
w,α
=
αmax
α =1
WI#
w,α
n
WI#
w,α λt,w,α
×
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α
Xm,w,α
276
(14.61)
The constant terms associated with the well are
WCmn
w,α = −
WI#
w,α
αmax
n
WI#
w,α λt,w,α
α =1
×
n
n
n
n
n
n
ξo,w,α
λno,w,α + Ym,w,α
ξg,w,α
λng,w,α + Wm,w,α
ξw,w,α
λnw,w,α
Xm,w,α
(14.62)
Each well has a total rate equation. This equation has the following form for a fixed pressure
well. The coefficient of δP is
QDPnw,α = WI#
w,α ×
emax n
qo,w,α
ξo,w,α λno,w,α
w,emax
ξo,w,α
max
+
emax n
qg,w,α
ξg,w,α λng,w,α
w,emax
ξg,w,α
max
+
emax n
qw,w,α
ξw,w,α λnw,w,α
w,emax
ξw,w,α
max
(14.63)
is
The coefficient of δqt,w
QDWnw,α = 1
(14.64)
The constant terms associated with the constant rate equation
QCn
w,α
=
,
−qt,w
−
14.18
α
max
#
w,n
WIw,α × Pw,α
− P
w,α ×
α =1
emax n
n
qo,w,α
ξo,w,α λo,w,α
w,emax
ξo,w,α
max
+
emax n
n
qg,w,α
ξg,w,α λg,w,α
w,emax
ξg,w,α
max
+
emax
n
n
qw,w,α
ξw,w,α λw,w,α
w,emax
ξw,w,α
max
(14.65)
Additional Comments on Computation
This chapter contains additional information and illustrations written since the April 6, 2011
report. The first section describes various efforts we have already made to make the program
computationally efficient. There is more work to be done in several of these categories. The
second section shows graphical illustrations of the process of solving the linear system of equations
←
→ A δ = b. This linear system of equations is solved at each nonlinear iteration , where δ represents
the differences between nonlinear iteration + 1 and nonlinear iteration for the primary variables.
Hopefully the new figures will provide a better understanding of the solution process.
14.19
Computational Efficiency
Several steps were taken to ensure efficient computations.
277
1. The appropriate physics were selected: the model assumes a constant temperature; the compositional formulation assumes that the mole fraction of the components other than CO2 and
H2 O are not present in the aqueous phase; it is also assumed that H2 O is not present in the
oleic phase or the vapor phase.
2. Each calculation was written in a computationally efficient way.
3. Nonlinear iterations are used in order to linearize the partial differential equations.
4. The matrix equations are written in terms of nonlinear differences rather than the variables
directly, δP = P +1 − P rather than P +1 . This normalizes the units of the different primary
variables and acts as a pre-conditioner for the linear solver.
5. The well terms are eliminated to reduce the bandwidth of the matrix and to regularize the
sparsity structure of the matrix.
6. Local LU decomposition is conducted on the equations for each grid cell. This serves to
extract the largest eigenvalues of the sparse matrix and greatly reduce the size of the linear
system, from (2NC − 1) ∗ Nxyz to between 1 ∗ Nxyz and 3 ∗ Nxyz depending on the formulation.
It also acts as another pre-conditioner.
7. For small models, where small is determined by memory requirements and computation time,
use a direct sparse parallel solver.
8. For larger models, use an iterative solver with a pre-conditioner. For example, GMRES with an
ILU(0) pre-conditioner or BICGSTAB with an ILU(0) pre-conditioner. Either of these solution
approaches may be faster depending on the model size and the number of iterations required.
14.20
Illustration of Solution Procedure
Two models are used for illustrations. A small model with Nx = 5, Ny = 5, and Nz = 3 is
used for most of the illustrations. A larger, but still very small model with Nx = 16, Ny = 16, and
Nz = 3 is used for some illustrations.
There are several other typical problem sizes, but they are too large to illustrate the full structure
of the problem at the resolution of the images. One typical problem has Nx = 80, Ny = 80, and
278
Nz = 15. For this system, Nxyz = 96000 and half bandwidth β = Ny ∗ Nz + Nz + 1 = 1216. This
problem can easily be scaled up or down by keeping Nz fixed and varying Nx = Ny . Nz can also
be increased to 30, 45, or 60 layers. The typical number of components ranges from NC = 5 to
NC = 15, with NC = 8 components as the most common. This leads to block sizes of Nb = 2·NC −1
from 9 to 29 with 15 most common. Some formulations for natural fracture systems can double
the block size, from 18 to 58 with 30 most common. If the model uses horizontal wells, there are
typically Nw = 3 horizontal wells aligned with the y-axis, with Nwc = Ny = 80 well completions
for each well. If the model uses vertical wells, there are typically Nw = 5 vertical wells arranged
in a 5-spot pattern aligned with the z-axis, with Nwc = Nz = 15 well completions for each well.
Another problem that has the same aspect ratio as this one has Nx = 16, Ny = 16, and Nz = 3.
This is used for some of the illustrations.
Another typical problem has Nx = 320, Ny = 320, and Nz = 15. For this system, Nxyz =
1563000 and half bandwidth β = Ny ∗ Nz + Nz + 1 = 4816. This problem can easily be scaled up or
down by keeping Nz fixed and varying Nx = Ny . Nz can also be increased to 30 or 45 or 60 layers.
The number of equations per grid cell vary in the same way as described above for the 80 × 80 × 15
model. If the model uses horizontal wells, there are typically Nw = 36 horizontal wells aligned
with the y-axis, with Nwc = Ny = 80 well completions for each well. If the model uses vertical
wells, there are typically Nw = 81 vertical wells aligned in 16 5-spot patterns with the z-axis, with
Nwc = Nz = 15 well completions for each well.
14.20.1
Illustration of a 5 × 5 × 3 Model, Well Geometry
This section illustrates the solution procedure for a small problem. The problem dimensions
were selected so that it is still possible to see the structure at the resolution of the images. The
system illustrated here has Nx = 5, Ny = 5, and Nz = 3. For this system, Nxyz = 75 and half
bandwidth β = Ny ∗Nz +Nz +1 = 19. There are Nw = 3 horizontal wells aligned with the y-axis, see
Figure 14.8, with Nwc = 5 well completions for each well. There are NC = 5 components, leading to
a block size of Nb = 2 · NC − 1 = 9. Unless otherwise specified, the formulation is IMPES (implicit
pressure, explicit saturation and composition), based on an 11-point finite difference scheme (9
points in the xy plane and 2 points in ±z).
279
Figure 14.8: Geometry of three horizontal wells for a 5 × 5 × 3 problem.
14.20.2
Illustration of a 5 × 5 × 3 Model, Block Values
Figure 14.9 shows the block banded matrix structure which is sent to the solver. The rest of this
section describes how this matrix is created. Each block on the main diagonal (red in Figure 14.9)
has the structure illustrated in Figure 14.10 for a NC = 5, Nb = 2NC − 1 = 9 problem. Each block
on the off-diagonal (blue in Figure 14.9) has the structure illustrated in Figure 14.11 for a NC = 5
problem. For the IMPSEC formulation, the off-diagonal blocks have the structure illustrated in
Figure 14.12 for a NC = 5 problem. The well terms for the component equations are represented
by Figure 14.13. The well equations are represented by Figure 14.14.
Figure 14.9: Matrix 0: block banded matrix for a 5 × 5 × 3 problem. Red cells are on the main
block diagonal; blue cells are non-zero values off of the main block diagonal.
280
Figure 14.10: Block 1: block geometry for the main block diagonal of a NC = 5 problem. Black
represents non-zero values; gray represents zero values.
Figure 14.11: Block 2: block geometry for the off-block diagonal values with the IMPES formulation
for a NC = 5 problem. Black represents non-zero values; gray represents zero values.
Figure 14.12: Block 3: block geometry for the off-block diagonal values with the IMPSEC formulation for a NC = 5 problem. Black represents non-zero values; gray represents zero values.
Figure 14.13: Block 4: well terms for the component equations for a NC = 5 problem. Black
represents non-zero values; gray represents zero values.
281
Figure 14.14: Block 5: blocks for the well equations for a NC = 5 problem. Black represents
non-zero values; gray represents zero values.
14.20.3
Illustration of a 5 × 5 × 3 Model, Matrix Assembly
The spatial derivatives are illustrated in Matrix 1, Figure 14.15. The time derivatives of the
accumulation term are illustrated in Matrix 2, Figure 14.16. When Matrix 1 and Matrix 2 are added
together, they yield Matrix 3, Figure 14.17. Matrix 4 shows the well coefficients, Figure 14.18.
Matrix 5 shows Matrix 3 combined with Matrix 4, Figure 14.19. Matrix 6 shows the results of
or P eliminating the qw
well well terms from the component equations, Figure 14.20. This generates
some terms which are not on the block-banded structure of Matrix 3. Matrix 7 shows the results
of eliminating the off-band well terms from the component equations, Figure 14.21. The off-band
terms are eliminated using δP −1 .
14.20.4
Illustration of a 5 × 5 × 3 Model, Local LU Decomposition
One way to simplify the matrix solve for the system described in Figure 14.20 is to perform a
local LU decomposition on the equations for each grid cell and then extract the upper left corner of
each block. If LU with partial pivoting is used, this corresponds to extracting the largest eigenvalue
for each grid cell. This local LU decomposition operates on Matrix 7 one block-row at a time.
Figure 14.22 shows one row extracted from Matrix 7, Figure 14.21, called Row 1 for this discussion. Figure 14.23 shows the results of removing the zero blocks from Figure 14.22. Figure 14.24
shows the results of removing the zero columns from Figure 14.23. Figure 14.25 shows the way Row
1 is stored for the local LU decomposition, with the main block diagonal values on the left followed
by the off-diagonal terms. Figure 14.26 shows the result of applying local LU decomposition to Row
1 as they are stored in memory. Figure 14.27 shows the result of applying local LU decomposition
to Row 1, showing only the non-zero blocks in the same format as Figure 14.23. The red and blue
values in Figure 14.26 and Figure 14.27 are used for the global matrix solution.
282
Figure 14.15: Matrix 1: Spatial derivatives for a 5 × 5 × 3 × 9 problem. Black represents non-zero
values; gray represents zero values within each block; white represents zero values.
283
Figure 14.16: Matrix 2: Time derivatives for a 5 × 5 × 3 × 9 problem. Black represents non-zero
values; gray represents zero values within each block; white represents zero values.
284
Figure 14.17: Matrix 3: Combined matrix for a 5 × 5 × 3 × 9 problem. Black represents non-zero
values; gray represents zero values within each block; white represents zero values.
285
Figure 14.18: Matrix 4: Well matrix for a 5 × 5 × 3 × 9 problem with three horizontal wells. Black
represents non-zero values; gray represents zero values within each block; white represents zero
values.
286
Figure 14.19: Matrix 5: Combined matrix with wells for a 5 × 5 × 3 × 9 problem with three
horizontal wells. Black represents non-zero values; gray represents zero values within each block;
white represents zero values.
287
or P Figure 14.20: Matrix 6: Eliminate the qw
well well terms from the component equations for a
5 × 5 × 3 × 9 problem with three horizontal wells. Black represents non-zero values; gray represents
zero values within each block; white represents zero values.
288
Figure 14.21: Matrix 7: Eliminate the off-band well terms from the component equations for a
5 × 5 × 3 × 9 problem with three horizontal wells. Black represents non-zero values; gray represents
zero values within each block; white represents zero values.
Figure 14.22: Row 1: An example of a row without well terms for a 5 × 5 × 3 × 9 problem. Black
represents non-zero values; gray represents zero values within each block; white represents zero
values.
Figure 14.23: The non-zero blocks of Row 1. Black represents non-zero values; gray represents zero
values within each block.
Figure 14.24: The non-zero columns of Row 1. Black represents non-zero values; gray represents
zero values within each block.
289
Figure 14.25: The non-zero columns from Row 1 are stored with the main block diagonal first.
Black represents non-zero values; gray represents zero values within each block.
Figure 14.26: The result of local LU decomposition on Row 1 in the order they are stored. Black
represents non-zero values; gray represents zero values within each block; red values are extracted
from the main block diagonal; blue values are extracted from the off-diagonal.
14.20.5
Illustration of a 5 × 5 × 3 Reduced Model
After the local LU decomposition, Matrix 7, Figure 14.21 becomes Matrix 8, Figure 14.28. The
direct solvers do not require any additional steps, but for the iterative solvers it may be a good idea
to eliminate the well values, green in Figure 14.28. For vertical wells, this will often be worthwhile.
For horizontal wells, it may be worthwhile or it may be better to evaluate wells at rather than at
n + 1. After elimination, this results in Matrix 9, Figure 14.29.
14.20.6
Illustration of a 16 × 16 × 3 Model
This illustration has Nx = 16, Ny = 16, and Nz = 3. For this system, Nxyz = 768 and half
bandwidth β = Ny ∗ Nz + Nz + 1 = 52. This problem was selected because it has the same aspect
ratio as a typical problem with Nx = 80, Ny = 80, and Nz = 15. This model has Nw = 3 horizontal
wells aligned with the y-axis, with Nwc = Ny = 16 well completions for each well, Figure 14.30.
Matrix 10, Figure 14.31 shows the banded matrix for this system with the well terms. Figure 14.32
zooms in on the upper left corner of Matrix 10, since it is hard to see the well terms in Figure 14.31.
Figure 14.27: The result of local LU decomposition on Row 1. Black represents non-zero values;
gray represents zero values within each block; red values are extracted from the main block diagonal;
blue values are extracted from the off-diagonal.
290
Figure 14.28: Matrix 8: banded matrix for a 5 × 5 × 3 problem without eliminating wells. Red cells
are on the main block diagonal; blue cells are non-zero values off of the main block diagonal; green
cells are the result of wells.
Figure 14.29: Matrix 9: banded matrix for a 5 × 5 × 3 problem. Red cells are on the main block
diagonal; blue cells are non-zero values off of the main block diagonal.
291
Matrix 11, Figure 14.33 shows the banded matrix for this system without the well terms. This is
the form that is typically solved.
Figure 14.30: Geometry of three horizontal wells for a 16 × 16 × 3 problem.
292
Figure 14.31: Matrix 10: banded matrix for a 16 × 16 × 3 problem without eliminating wells. Red
cells are on the main block diagonal; blue cells are non-zero values off of the main block diagonal;
green cells are the result of wells.
Figure 14.32: Upper left corner of Matrix 10. Red cells are on the main block diagonal; blue cells
are non-zero values off of the main block diagonal; green cells are the result of wells.
293
Figure 14.33: Matrix 11: banded matrix for a 16 × 16 × 3 problem. Red cells are on the main block
diagonal; blue cells are non-zero values off of the main block diagonal.
294
CHAPTER 15
COMPUTATION: DESCRIPTION OF LINEAR SOLVERS
Because in our simulator the linear solver lies within the inner most loop, the quality of the
simulator depends heavily on the quality of the linear solvers. This report analyzes various linear
solvers and evaluates their computation complexity, accuracy, robustness, memory requirement
and scalability in parallel environment. An extremely small test case with N X = 3, N Y = 1, and
N Z = 1 is used to illustrate these solvers.
Figure 15.1: Jacobian matrix for a 3 × 1 × 1 system with NC = 5 and Nblock = 9. Black represents
non-zero values; gray represents zero values within the block diagonals; white represents other zero
values.
15.1
Serial Solvers
Three serial solvers are currently proposed and implemented: dense Gaussian elimination, band
Gauss elimination and a special LU solver.
15.1.1
Dense Gaussian Elimination
This is the simple LU with partial pivoting based linear solver, with the matrix in a traditional
2-D array dense form. It corresponds to the DGESV subroutine of the high quality LAPACK
package. This solver is stable and has been used for decades in scientific computing. The error
source in this case is only roundoff error, which is nicely bounded most of the time in practice.
However, since the matrix we are dealing with is highly structured and sparse, it’s a huge waste of
computation and storage to store a such sparse matrix in a dense form.
295
The computation complexity of dense Gaussian elimination solver is around 486 · n3xyz . The
memory requirement is 81 · n2xyz .
The solution procedure starts with the Jacobian matrix in Figure 15.1. The first stage of
Gaussian elimination creates zeroes above the diagonal, Figure 15.2. The second stage of Gaussian
elimination creates zeroes below the diagonal, Figure 15.3, resulting in a solution for each unknown.
Figure 15.2: The test case after the first stage of Gaussian elimination. The above diagonal values
are eliminated first by column and then by row, moving from lower left to upper right. Black
represents non-zero values; gray represents zero values within the block diagonals; white represents
other zero values; cyan represents zero values that have been created.
Figure 15.3: The test case after the second stage of Gaussian elimination. The below diagonal
values are eliminated by back substitution, first by column and then by row, moving from upper
left to lower right. Black represents non-zero values; gray represents zero values within the block
diagonals; white represents other zero values; cyan represents zero values that have been created.
15.1.2
Band Gaussian Elimination
A slightly better format to store the matrix band storage format, which then can be solved by
LAPACK DGBSV subroutine. Since in our matrix the furthest subdiagonal or superdiagonal is
9·nz ·ny away from main diagonal, it’s meaningful to arrange the matrix into a band format(despite
296
that it’s still quite sparse in band format) which ignores all left-bottom and right-top zeros. The
algorithm and accuracy behavior is similar to dense Gaussian elimination but the computation and
storage requirements are greatly reduced.
The computation complexity of band Gaussian elimination solver is around 486 · n2y · n2z · nxyz .
The memory requirement is 27 · ny · nz · nxyz .
The solution procedure starts with the Jacobian matrix in Figure 15.1. The matrix is stored
based on diagonals as shown in Figure 15.4. The first stage of Banded Gaussian elimination creates
zeroes above the diagonal, Figure 15.5. The second stage of Banded Gaussian elimination creates
zeroes below the diagonal, Figure 15.6, resulting in a solution for each unknown.
Figure 15.4: The banded structure for the test case. Everything not in purple is stored in memory
by diagonal and manipulated by the band solver. Black represents non-zero values; gray represents
zero values within the block diagonals; white represents other zero values; purple represents zero
values outside of the bandwidth.
Figure 15.5: The test case after the first stage of Banded Gaussian elimination. The above diagonal
values are eliminated first by column and then by row, moving from lower left to upper right. Black
represents non-zero values; gray represents zero values within the block diagonals; white represents
other zero values; purple represents zero values outside of the bandwidth; cyan represents zero
values that have been created.
297
Figure 15.6: The test case after the second stage of Banded Gaussian elimination. The below
diagonal values are eliminated by back substitution, first by column and then by row, moving from
upper left to lower right. Black represents non-zero values; gray represents zero values within the
block diagonals; white represents other zero values; purple represents zero values outside of the
bandwidth; cyan represents zero values that have been created.
15.1.3
Special Gaussian Elimination
For our particular matrix structure there’s a trick which essentially LU decomposes each block
and generate a substantially more compact linear system. After solving the compact system a
cheap back substitution will solve the whole system.
Suppose we are using the band solver for the compact system and standard back substitution,
the computation complexity is (1440 + 2ny · nz ) · nxyz , and the memory storage requirement is
(153 + 2ny · nz ) · nxyz .
Another nice property is that the compact system has a diagonal dominant matrix which means
to solve it the LU decomposition process does not need any pivoting - a great potential for performance improvement since pivoting involves communication which degrades performance considerably, especially for parallel computations.
The solution procedure starts with the Jacobian matrix in Figure 15.1, but because this solution
procedure utilizes the block structure of the matrix, it is stored as Figure 15.7.
First, conduct a local LU decomposition on each grid cell, reducing the system from Nblock = 9
to Nblock = 1; Figure 15.8. Next, extract the upper left corner of each block and assemble a
new, condensed matrix, Figure 15.9. For a condensed banded solve, store the matrix diagonals,
Figure 15.10. The first stage of Banded Gaussian elimination creates zeroes above the diagonal,
Figure 15.11. The second stage of Banded Gaussian elimination creates zeroes below the diagonal,
Figure 15.12, resulting in a solution for the pressures. Using the results of the condensed matrix
298
Figure 15.7: The sparse storage structure of the local LU solvers. Everything in purple are zero
values that are not stored explicitly. Black represents non-zero values; gray represents zero values
within the block diagonals; white represents other zero values.
solution for pressures, locally back substitute in each grid cell to obtain solutions for the other
primary variables, Figure 15.13.
Figure 15.8: The first step of the local LU solvers. Everything in purple are zero values that are
not stored explicitly. Black represents non-zero values; gray represents zero values within the block
diagonals; white represents other zero values; cyan represents zero values that have been created;
red represents values on the main diagonal to be extracted; blue represents values off of the main
diagonal to be extracted.
15.1.4
Summary
From the above discussion we can see that the special Gaussian elimination has much less
computation complexity and memory requirements, as shown in Table 15.1. There is a large piece
of memory associated with the Jacobian calculations for each grid cell.
Table 15.1: Computation and memory requirement for 3 different solvers
Jacobian
Computation
Memory
608 · nxyz
Dense Gauss
486 · n3xyz
81 · n2xyz
Band Gauss
486 · n2y · n2z · nxyz
27 · ny · nz · nxyz
299
Special Gauss
(1440 + 2ny · nz ) · nxyz
(153 + 2ny · nz ) · nxyz
Figure 15.9: The condensed matrix after the local LU decomposition. Red represents values on
the main diagonal that were extracted from the full matrix; blue represents values off of the main
diagonal that were extracted from the full matrix; white represents other zero values.
Figure 15.10: The banded structure of the condensed matrix. Everything not in purple is stored
in memory by diagonal and manipulated by the band solver. Red represents values on the main
diagonal that were extracted from the full matrix; blue represents values off of the main diagonal
that were extracted from the full matrix; purple represents zero values outside of the bandwidth.
Figure 15.11: The condensed matrix after the first step of the band solve. The above diagonal
values are eliminated first by column and then by row, moving from lower left to upper right. Red
represents values on the main diagonal that were extracted from the full matrix; blue represents
values off of the main diagonal that were extracted from the full matrix; purple represents zero
values outside of the bandwidth; cyan represents zero values that have been created.
Figure 15.12: The condensed matrix after the second step of the band solve. The below diagonal
values are eliminated by back substitution, first by column and then by row, moving from upper
left to lower right. Red represents values on the main diagonal that were extracted from the full
matrix; blue represents values off of the main diagonal that were extracted from the full matrix;
purple represents zero values outside of the bandwidth; cyan represents zero values that have been
created.
300
Figure 15.13: Use the values from the condensed matrix solve to perform a back substitution on
each grid cell. Everything in purple are zero values that are not stored explicitly. Black represents
non-zero values; gray represents zero values within the block diagonals; white represents other
zero values; cyan represents zero values that have been created; red represents values on the main
diagonal that were extracted and solved in the condensed version; blue represents values off of the
main diagonal that were extracted and solved in the condensed version.
When the problem scales(nx , ny , nz gets bigger) we can see that the special Gaussian elimination
approach is far more favorable and thus should be used in practice. It’s also worth noting that
these 3 solvers generally solves the linear system with almost the same accuracy subject to roundoff
errors.
15.2
Parallel Solvers
When going into parallel the choices of solvers become much more subtle, mainly because the
computation and memory requirements can not be as easily determined as in serial case. When we
are in the parallel realm we must be solving a much bigger problem which means a dense or band
matrix format are simply too expensive; we need a sparse matrix format. This not only changes the
algorithms and storage scheme, it also induces some uncertainty as to computation and memory
requirements. Even a direct solver on sparse format matrix(like LU solver) will have different
computation and memory behavior depending on the input matrix and the specific implementation
of the algorithm. More oftentimes, an iterative solver is desired because a direct solver may not scale
well. Iterative solvers are especially hard to predict its execution time and memory consumption
because the number of steps it requires to converge is not known before actually executing it.
From the discussion of the serial solver section we are convinced that the special Gaussian
elimination should be used. We thus only focus on solving the compact system(which can yet be
very large in bigger problems) since this part is the dominant computation task in that approach.
301
15.2.1
Direct LU Solver
Since the compact matrix is highly structured and sparse and diagonal dominant, there’s a good
chance a direct sparse LU solver(like SuperLU) can perform quite well in this case. However, it
still remains to be seen whether direct solvers are appropriate for our problem.
A direct solver is generally very robust and stable like the serial solvers discussed earlier; but its
performance depends heavily on the matrix and the implementation. We propose to use SuperLU
and UMFPACK to test the feasibility to use a direct solver.
15.2.2
Iterative Solvers
Many iterative methods exist for solving large sparse systems. Typically iterative methods enjoy
a relatively low memory footprint and are more scalable than direct sparse solvers. The trick is
how to find the most efficient method for the problem at hand. Unfortunately no universal method
works well for all problems, thus insights into the problem and iterative methods are required.
Another issue is that iterative methods often depends on preconditioner to be effective. The choice
of preconditioners, again, is subtle and no universal scheme proves to work for all problems. All
this requires deeper understanding of the problem and algorithms.
For a starting point we propose to use BICGSTAB or GMRES with ILU or block Jacobi
preconditioner.
15.2.3
Parallel Framework
PETSc is the planed framework that we are going to work with developing the parallel simulator. PETSc includes an expanding suite of parallel linear, nonlinear equation solvers and time
integrators that may be used in application codes written in Fortran, C, C++, Python, and MATLAB (sequential). PETSc provides many of the mechanisms needed within parallel application
codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically,
enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for
users.
The PETSc package provides a infrastructure for our parallel programming, and it provides all
our proposed solvers built-in which makes it an ideal platform for developing and testing various
302
approaches.
303
CHAPTER 16
COMPUTATATION: PARALLEL COMPUTING
The following is an outline of the overall parallel solution approach. For each of the steps,
the amount of expected parallelism is described, including a reference size and an estimate of
the memory, computation time, and communication time if applicable. These sizes have various
constant multipliers, some of which can be quite large. There are also lower order polynomial terms
not represented by the O notation. The combination of all of all these steps is memory limited.
The problem has been demonstrated by Saudi Aramco (Dogru et al., 2008) to be scalable for sizes
up to Nxyz = 109 . At Saudi Aramco they have spent approximately 150 man-years developing
Powers, so I would not expect my results in a few months to be as good as theirs3 .
The total number of grid cells is Nxyz = Nx × Ny × Nz . The bandwidth β is used the banded
solver algorithms; for a system with Nx = Ny > Nz , β = Nx ×Nz . The total number of components,
NC , includes the hydrocarbon components, CO2 component, and H2 O component. The number of
processing nodes, Nn , represents the number of different machines involved in the computations.
The number of processing cores on each node is Np . The total number of cores on all nodes is Nnp .
Message Passing Interface (Gabriel, Fagg, Bosilca, Angskun, Dongarra, Squyres, Sahay, Kambadur, Barrett, Lumsdaine, Castain, Daniel, Graham, and Woodall, 2004; Wikipedia, 2010b,c,
MPI, ), is a language independent communications protocol for parallel computers, including both
shared and distributed memory computers. Open Multi Processing (OpenMP, 2008; Wikipedia,
2010d, OpenMP, ), is a different language independent communications protocol applicable only
to shared memory computers. The two can be combined in a hybrid MPI/OpenMP framework,
using the MPI interface to communicate between nodes and the OpenMP interface to communicate between processors on the same node. For an MPI parallel implementation, all computations,
communication, and memory are divided among Nnp nodes. For a hybrid MPI/OpenMP implementation, memory and communication are among Nn nodes. Computations are divided among
Nn nodes and then further divided among Np processing cores.
3
All variables are defined in Chapter 22.
304
16.1
Computation Grid
The matrix coefficients are defined using a grid of parallel processors. The following figures use
Nx = Ny = 100, Nn = 9, Np = 8, and Nnp = 72. Figure 16.1 shows a grouping of 9 nodes with
8 cores. Figure 16.2 shows the hybrid approach, with a 3 × 3 × 8 grid of processors. Figure 16.3
shows the pure MPI approach, with a 9 × 8 grid of processors. The matrix solver requires a linear
processor array of all 72 processors, Figure 16.4.
Figure 16.1: Illustration of a group of 9 nodes with 8 processor cores each.
Figure 16.2: Illustration of computations with a hybrid MPI/openMP 3 × 3 × 8 processor grid.
The normal boundary computations are illustrated in Figure 16.5. Figure 16.6 shows how this
applies to a 3 × 3 processor grid, and Figure 16.7 shows how this applies to a 9 × 8 processor grid.
Because there are normally a lot of computations for each grid cell for compositional simulation, it
is normally better to compute them on one processor and then send them to the adjacent processor
rather than to make the computations twice.
When the number of grid cells on each processor are not identical, it may be beneficial to
subcontract the grid cells to another processor. Figure 16.8 shows how these computations may be
305
Figure 16.3: Illustration of computations with an MPI 9 × 8 processor grid.
Figure 16.4: Illustration of computations with a linear array of 72 processors.
centralarea
centralarea
innerborder
outerborder
innerborder
outerborder
Figure 16.5: Parallel boundary computations.
306
Figure 16.6: Parallel boundary computations for a 3 × 3 processor grid.
Figure 16.7: Parallel boundary computations for a 9 × 8 processor grid.
307
subcontracted. Using Nx = Ny = 100 and Nn = 9, the processor grid is a 3 × 3 array, Figure 16.2.
If the load on each processor were completely balanced, then there would be 1111.11 2-D grid cells
in on each processor. Using the most efficient grid of processors, there will be one 34 × 34 = 1156,
four 34 × 33 = 1122, and four 33 × 33 = 1089. Using Nx = Ny = 100 and Nnp = 72, the processor
grid is a 9× 8 array, Figure 16.3. If the load on each processor were completely balanced, then there
would be 138.89 grid cells on each processor. Using the most efficient grid of processors, there will
be four 13 × 12 = 156, four 12 × 12 = 144, thirty-two 13 × 11 = 143, and thirty-two 12 × 11 = 132.
Subcontracting for load balancing means that the computations for some of the grid cells on the
centralarea
centralarea
innerborder
outerborder
subcontract
nodes with more cells are sent to the nodes with fewer processors.
innerborder
outerborder
Figure 16.8: Parallel computations for load balancing.
16.2
Solution Steps
The following steps are involved in the solution procedure.
1. Initialization - calculate the grid geometry and distribute the data among the processors
• Reference size α = Nxyz
• Computation: O [α]; computation time: some parts O [α/Nnp ], other parts O [α] depending on details of implementation
• Memory: O [α]; local memory: O [α/Nn ]
• Communication (one-to-many): O [α]; communication time: O [α log2 Nnp ] for the MPI
approach and O [α log2 Nn ] for the hybrid approach
2. Start of time step n or nonlinear iteration 308
3. Calculating the coefficients of the matrix equation that depend only on the local grid cell
(block-diagonal terms)
• Reference size α = 7 · (2NC − 1)2 · Nxyz , using a 7-point finite difference stencil
• Computation: (big constant) ∗ O [α]; computation time: (big constant) ∗ O [α/Nnp ]
• Memory: O [α]; local memory: O [α/Nn ]
• Communication is only required if subcontracting is required for load balancing. Assume
5% of the cells require subcontracting for balancing.
• Communication (one-to-one): O [0.05 · α]; communication time: O [0.05 · α/Nnp ] for the
MPI approach and O [0.05 · α/Nn ] for the hybrid approach
4. Perform local LU decomposition to transform the fully implicit matrix into an IMPES or
IMPSEC matrix. (Press, Teukolsky, Vetterling, and Flannery, 2007)
• Reference size α = 7 · (2NC − 1)2 · Nxyz , using a 7-point finite difference stencil
• Computation: O [(2NC − 1)α]; computation time: O [(2NC − 1)α/Nnp ]
• Memory: O [α]; local memory: O [α/Nn ]
5. Calculating the off block-diagonal coefficients of the matrix equation
• Reference size: α = 4β
&
√
Nnp ; hybrid reference size: α = 4β Nn
• Computation: O [α]; computation time: O [α/Nnp ]
• Memory: O [α]; local memory: O [α/Nn ]
• Communication (one-to-one): O [α]; communication time: O [α/Nnp ] for the MPI approach and O [α/Nn ] for the hybrid approach
6. Set up the matrix solver
• Reference size: IMPES α = 7Nxyz ; IMPSEC: α = 7 · 32 · Nxyz
• Memory: O [2α]; local memory: O [2α/Nn ]
• Communication (many-to-many): O [α]; communication time: O [α log2 Nnp ] for the
MPI approach and O [α log2 Nn ] for the hybrid approach
309
7. Perform matrix solve; (Gauss: Press et al. (2007), Banded: Arbenz, Cleary, Dongarra, and
Hegland (2001); Cleary and Dongarra (1997))
• Reference size, IMPES α = Nxyz ; IMPSEC α = 3 · Nxyz ; banded IMPES β = Nx · Nz ;
banded IMPSEC β = 2 · 3 · Nx · Nz
• Gauss computation: O α3 ; computation time: O α3 /Nnp ; IMPES = 27 · IMPSEC
• Gauss memory: O α2 ; local memory: O α2 /Nnp ; IMPES = 9 · IMPSEC
• Gauss communication: O α2 ; communication time: O α2 log2 Nnp ; IMPES = 9 ·
IMPSEC
• Banded computation: O β 2 α ; computation time: O β 2 α/Nnp ; IMPES = 108·IMPSEC
• Banded memory: O [βα]; local memory: O [βα/Nnp ]; IMPES = 18 · IMPSEC
• Banded communication: O [βα]; communication time: O [βα log2 Nnp ]; IMPES = 18 ·
IMPSEC
8. Transfer results of matrix solve back to grid cells
• Reference size: IMPES α = Nxyz ; IMPSEC: α = 3 · Nxyz
• Memory: O [2α]; local memory: O [2α/Nn ]
• Communication (many-to-many): O [α]; communication time: O [α log2 Nnp ] for the
MPI approach and O [α log2 Nn ] for the hybrid approach
9. local LU back substitution
• Timing already accounted for in Item 4
10. For each new time step or nonlinear iteration, go back to Item 2
16.3
Initialize
Initialization - calculate the grid geometry and distribute the data among the processors
• Reference size α = Nxyz
• Computation: O [α]; computation time: some parts O [α/Nnp ], other parts O [α] depending
on details of implementation
310
• Memory: O [α]; local memory: O [α/Nn ]
• Communication (one-to-many): O [α]; communication time: O [α log2 Nnp ] for the MPI approach and O [α log2 Nn ] for the hybrid approach
Some of the properties need to be distributed to all nodes. Others will be stored only on the
appropriate node. There are some simple initialization parameters that have to be distributed
to all the processors, such as Nx , Ny , Nz , and NC . If grid-based properties, such as porosity φ
and permeability k, are constants or vary only with z, then these need to be distributed to all
processors as well. This type of distribution is best handled by MPI_BROADCAST, which has total
communication Nnp log2 Nnp or communication timing log2 Nnp . Fortunately, this initialization
only needs to occur once at the beginning of the run.
If grid-based properties vary in 3-D, then only the portion of the grid that lives on each processor
needs to be distributed or it needs to be read locally from a file on that node. Wells only need to
live the specific processor that contains the well. If all the data of a particular kind is loaded on
one processor, then this communication is best handled by MPI_SPLIT.
Information relating to variable time step size needs to be distributed to all processors or
calculated locally on each processor. This needs to happen with every time step. Determination of
convergence also needs to happen across all processors.
16.4
Scalability
Scalability evaluations start with the evaluation of the computation time, communication time,
and memory demands for an algorithm on various numbers of processors for various sizes of problems. The following variables are used in this description:
• O [x]: computational order of x.
• Coff [x]: off-node communication time for Ra for data size x.
• Nx : number of grid cells in x-direction.
• Ny : number of grid cells in y-direction.
• Nz : number of grid cells in z-direction.
311
• Nxyz : total number of grid cells.
• β: bandwidth for banded solver
• P : number of processing cores
• n: number of time steps
• : average number of nonlinear iterations for each time step
• COMPT1 : the computation time for a single processor
• COMPT1/P :
T1
P ;
used because it is part of the efficiency calculations and the total TP .
• COMPTP : the computation time for multiple processors
• COMMTP : the communication time for multiple processors
• MP : the memory requirement for each processor for a model
16.4.1
Computation Magnitude
Two clusters at Colorado School of Mines were used as part of this dissertation: RA and MIO.
The following problems are illustrated using Nx = Ny , and β = Nx × Nz . It’s based on properties of
RA thin nodes: 16GB per 8-core node. Values of the coefficients are estimated based on theoretical
calculations and some timing estimates on MIO. Additional calculations are necessary on MIO and
RA. The computation order for a single processor, written as T1/P =
T1
P :
Nxyz
Nxyz
β 2 Nxyz
Nxyz
+O n
+ O n
+ O n
COMPT1/P = O [Nxyz ] + O
P
P
P
P
β 2 Nxyz
Nxyz
Nxyz
Nxyz
4
3
3
3
+15∗10 ·O n
+5∗10 ·O n
+8·O n
≈ 10 ·O [Nxyz ]+10 ·O
P
P
P
P
(16.1)
Rewriting (16.1) using some estimated coefficients, n = 200, and = 5, for NC = 8 components,
(16.1) becomes (16.2).
2
Nxyz
β Nxyz
3
+ 8 ∗ 10 · O
= 10 · O [Nxyz ] + 8 ∗ 10 · O
P
P
COMPT1/P
4
6
The computation order for P processors is
312
(16.2)
β
β
COMPTP = COMPT1/P + O n √
+ O n √
+ O nβ 3 log2 [P − 1]
P
P
β
β
+ 100 · O n √
+ 10 · O nβ 3 log2 [P − 1]
≈ COMPT1/P + 2000 · O n √
P
P
(16.3)
Using estimated constants in (16.3):
β
+ 104 · O β 3 log2 [P − 1]
+ 5 ∗ 10 · O √
P
COMPTP = COMPT1/P
5
(16.4)
The communication order for P processors, using the communication time for each transmission
of N double precision numbers as COMM = ts + N tp .
√
Nxyz
β
∗ P ∗ n+
∗P +O √
COMMTP = O [log2 [P − 1]] + O
P
P
2
√
β
Nxyz
β
∗ P ∗ n + O
∗ P ∗ n + O
∗ P log2 [P − 1] ∗ n
O √
P
P
P
√
β
3
2 Nxyz
∗ P ∗ n+
∗ P + 4 ∗ Coff 15 √
≈ Coff 10 ∗ log2 [P − 1] + Coff 10
P
P
2
√
Nxyz
Nxyz
β
β
∗P ∗n+Coff
∗P ∗n+4∗Coff
∗P log2 [P −1]∗n
4∗Coff √ ∗ P ∗n+Coff 8
P
P
P
P
(16.5)
Using estimated constants in (16.5), using the bandwidth computations from Figure 16.9.
2 Nxyz
∗ P+
COMMTP = Coff 10 ∗ log2 [P − 1] + Coff 10
P
√
√
Nxyz
β
β
∗ P + 4000 · Coff √
∗ P + 1000 · Coff 8
∗ P+
800 · Coff 15 √
P
P
P
2
Nxyz
β
∗ P + 4000 · Coff
∗ P log2 [P − 1] (16.6)
1000 · Coff
P
P
3
The total memory required for each processor in a system using P processors is defined by:
313
Figure 16.9: Ra bandwidth.
2
Nxyz
βNxyz
β
β
Nxyz
+O
+O
+O √
+O
MP = O
P
P
P
P
P
2
β
βNxyz
β
Nxyz
+4·O
+O
+ 120 · O √
(16.7)
≈ 1500 · O
P
P
P
P
Using estimated constants in (16.7) yields the following in gigabytes.
−9
MP = 8 · 10
16.4.2
2
β
βNxyz
β
Nxyz
+4·O
+O
+ 120 · O √
· 10 + 1500 · O
P
P
P
P
4
(16.8)
Analysis
The efficiency EP is defined by (16.9). Figure 16.10 shows the efficiency versus the number of
processors for a model with 80 × 80 × 15 grid cells, using (16.9) with the constants in (16.2), (16.4),
and (16.6).
EP =
T1/P
COMPT1/P
T1
=
=
P · TP
TP
COMPT1/P + COMPTP + COMMTP
314
(16.9)
Efficiency for 80x80x15 Model
1.0
0.9
0.8
Ep
0.7
0.6
0.5
0.4
0.3
2
5
10
20
P
Figure 16.10: Efficiency plot for Nx = 80, Ny = 80, and Nz = 15.
The speedup SP is defined by (16.10). Figure 16.11 shows the speedup versus the number of
processors for a model with 80 × 80 × 15 grid cells, using (16.9) with the constants in (16.2), (16.4),
and (16.6). A typical rule of thumb is to use the number of processors at the inflection point. For
this case, that would be between 100 and 400 cores, or 10 to 50 nodes on Ra.
SP =
T1/P · P
T1
=
= EP · P
TP
TP
(16.10)
Speedup for 80x80x15 Model
9
8
Sp
7
6
5
4
3
2
2
5
10
20
P
Figure 16.11: Speedup plot for Nx = 80, Ny = 80, and Nz = 15.
The number of processors for efficiency EP = 0.1 as a function of the number of grid cells Nxyz
is shown in Figure 16.12. This uses a model with Nx × Nx × 15 grid cells, with the constants in
(16.2), (16.4), and (16.6). Since there exists a number of processors P [Nxyz ] for all numbers of cells
315
Nxyz , this illustrates that the algorithms are scalable. The plot has a similar shape for all values
of EP .
Number of Processors for E p 0.1
105
P
104
1000
100
10
104
105
106
107
108
109
1010
Nxyz
Figure 16.12: Scalability plot for Nx = Ny , Nz = 15, and EP = 0.1.
Figure 16.13 shows the memory constrained scalability for a model with Nx × Nx × 15 grid cells
with the constants in (16.2), (16.4), and (16.6).
316
Memory Constrained Scaling for E p 0.1
1000
500
Nodes
100
50
10
5
1
104
105
106
107
108
Nxyz
Figure 16.13: Memory constrained scalability plot. The red line shows the upper limit of applicability of the banded solver. The purple line shows the minimum number of processors required for
the memory needs. The green line shows the maximum number of processors for EP ≥ 0.1. The
dashed black line shows the maximum number of processors for EP ≥ 0.01. The solid black line
shows the maximum number of processors for EP ≥ 0.50. The blue shading shows the valid region
for EP ≥ 0.1.
317
CHAPTER 17
VALIDATION CASES
This chapter describes a series of comparison cases with GEM, a commercial compositional
simulator by Computer Modeling Group. Roughly a hundred models comparing CMG and my
code were run, including 1-D homogeneous, 2-D homogeneous, 2-D heterogeneous 5-spot models
based on a Middle East field, and 2-D heterogeneous 5-spot models of a fluvial system. Nine models
were selected to present here.
Before running the cases described here, the initialization of the simulations and the status after
a one-day time step were compared. The following items matched exactly between the two models
without any adjustments:
• The initial cell volumes, pore volumes, porosities, pressures, mole fractions, and saturations
were the same.
• The base relative permeability curves without trapping or hysteresis were the same.
• The base capillary pressure curves without trapping or hysteresis were the same.
After evaluating the initialization and the simulations after a one-day time step, several modifications were made to make the simulations more comparable.
• EOS was modified so my model had identical properties for CO2 , CH4 , nC4 , and nC10 as
the default pure-component properties in GEM. Turn off volume shift since the two codes
calculate it differently.
• Oil and gas viscosity model: implemented GEM’s viscosity model in my code since GEM does
not support LBC.
• Water viscosity and density model: implemented GEM’s water viscosity and density model
in my code.
• Well index: calculated well index in my code and then assigned GEM’s well index to this
calculated value.
318
• After adjusting the EOS, the initial moles in the system are off by less than 0.05%.
• After adjusting the EOS, the fugacities of each component are off by less than 0.1%.
Even after making the above adjustments to make the two simulators as comparable as possible,
there were still the following differences.
• Well constraints are the same but GEM’s algorithm to enforce mixed pressure and rate
boundary conditions are different than mine.
• GEM requires well grid cells to be fully implicit even if the rest of the model is IMPES. Over
time, GEM will add additional fully implicit grid cells. There also seem to be differences in
how the pressure is calculated for the production grid cells.
• GEM’s time stepping algorithm is different from mine.
• If my model fails to converge, it takes the value of the best nonlinear iteration and then
continues. If GEM fails to converge, it tries to reduce the time step. If after several reductions
in time step size it still hasn’t converged then the model stops completely.
• GEM calculates hysteresis differently than my code.
17.1
Validation Cases
All of the validation cases described here, Table 17.1, are 1-D homogeneous models with 101 grid
cells in the x-direction. Each model was 1000 ft×100 ft×44 ft, with each grid cell 10 ft×100 ft×44 ft.
Initial reservoir pressure was 3850 psia with a reservoir temperature of 210◦ F. The system has four
components, CO2 , CH4 , nC4 , and nC10 . System permeability is 200 md, system porosity is 17.2%.
CO2 solubility in water was set to zero to simplify the comparisons.
17.2
Description of model 760E
Model 760E is a 1-D model with primary production and no trapping or hysteresis. Since this
is a primary production case, the injection rate is 0. The production well is constrained initially
by a maximum rate of 100 RB/day, Figure 17.1. At about 1100 days the well switches from rate
control to bottom hole producing pressure control, Figure 17.2. The system is above the bubble
319
Table 17.1: Validation cases
Name
760E
761E
762E
760F
761F
762F
760G
761G
762G
Production Scenario
primary production
primary production
primary production
waterflood
waterflood
waterflood
primary production then waterflood
primary production then waterflood
primary production then waterflood
Hysteresis and Trapping
no hysteresis, no trapping
gas hysteresis, no trapping
gas hysteresis, compositional trapping
no hysteresis, no trapping
gas hysteresis, no trapping
gas hysteresis, compositional trapping
no hysteresis, no trapping
gas hysteresis, no trapping
gas hysteresis, compositional trapping
point until about 100 RB/day; there is a large bend in the pressure curve (Figure 17.2) as gas
production starts (Figure 17.1).
Production Rate RBPD
100
q RBPD
80
qTOT
60
qo qg
40
qw
20
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.1: Production rates at reservoir conditions for model 760E. Black is total production in
RB/day; green is oil production; red is gas production; blue is water production.
The average saturations in the reservoir are shown in Figure 17.3. For this model, the water
saturation stays approximately constant. The gas saturation increases as the pressure drops below
the bubble point and stabilizes when the well switches to pressure control.
As shown in Figure 17.4, the average mole fraction of methane decreases with time, the CO2
stays approximately constant, the mole fraction of nC10 increases, and the mole fraction of nC4
increases slightly. Figure 17.5 shows the recovery factor for each of the hydrocarbon components.
Methane has the highest recovery, followed by nC4 and nC10 .
320
Production Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.2: Production pressure for model 760E. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model. At this scale, the blue and orange curve
visually overlay each other.
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
SgM1
0.0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.3: Saturation for equivalent one-cell model for model 760E. For each time step, green is
the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells; blue
is water and red is gas.
Mole Fraction in Reservoir
1.0
CH4 , nC4 , nC10 , CO2
0.8
CH4total moles
0.6
nC4total moles
nC10total moles
0.4
CO2total moles
0.2
0.0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.4: Mole fraction for equivalent one-cell model for model 760E. For each time step, red is
the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;
CO2 is in orange.
321
Produced Fraction by Component
1.0
CH4 , nC4 , nC10
0.8
RF CH4
0.6
RF nC4
0.4
RF nC10
0.2
0.0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.5: Molar recovery factor for model 760E. For each time step, red is the ratio of the
cumulative produced moles of methane to the original number of moles of methane in the reservoir;
nC4 is in green; nC10 is in cyan.
17.3
Description of model 761E
Model 761E is a 1-D model with primary production and gas hysteresis. Model 761E gives
almost identical results to model 760E; because the gas saturation is always increasing there is no
liquid phase to induce the trapping of gas. Different values of the critical gas saturation would
affect these cases.
17.4
Description of model 762E
Model 762E is a 1-D model with primary production with compositional trapping and gas
hysteresis.
The injection rates and pressures for model 762E are visually the same as model 760E. The
production rates and production pressures for model 762E are visually the same as model 760E as
shown in Figure 17.1 and Figure 17.2.
For model 762E, the total model water saturation stays approximately constant. The gas
saturation increases as the pressure drops below the bubble point and stabilizes when the well
switches to pressure control, Figure 17.6. Figure 17.6 is very similar to Figure 17.3
17.5
Description of model 760F
Model 760F is a 1-D waterflood model with no trapping or hysteresis.
The injector has both a maximum bottom hole injection pressure of 3850 psia and a rate
constraint of 100 RB/day, Figure 17.7 and Figure 17.8.
322
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
SgM1
0.0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.6: Saturation for equivalent one-cell model for model 762E. For each time step, green
is the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;
blue is water and red is gas. Purple is the trapped water, cyan is the trapped oil, and yellow is the
trapped gas.
Injection Rate RBPD
100
80
qw or qg bbl
qTOT,inj
60
qo,inj
qg,inj
40
qw,inj
20
0
0
500
1000
1500
2000
tday
Figure 17.7: Injection rates at reservoir conditions for model 760F. Black is total injection in
RB/day; red is gas injection; blue is water injection.
Injection Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
500
1000
1500
2000
tday
Figure 17.8: Injection pressures for model 760F. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
323
The production well has a maximum total production rate of 100 RB/day, Figure 17.9. There
is also a minimum bottom hole producing pressure of 500 psia, but for this case it does not control
the produciton well, Figure 17.10. The system is above the bubble point for the entire simulation.
Production Rate RBPD
100
q RBPD
80
qTOT
60
qo qg
40
qw
20
0
0
500
1000
1500
2000
tday
Figure 17.9: Production rates at reservoir conditions for model 760F. Black is total production in
RB/day; green is oil production; red is gas production; blue is water production.
Production Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
500
1000
1500
2000
tday
Figure 17.10: Production pressure for model 760F. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
The average saturations in the reservoir are shown in Figure 17.3. For this model, the water
saturation progressively increases as the oil saturation decreases. After water breakthrough there
is only a little additional recovery of oil.
As shown in Figure 17.12, the mole fractions of each component remain nearly constant through
the simulation. The compositional recovery factors of each component are also nearly the same,
Figure 17.13.
324
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
SgM1
0.0
0
500
1000
1500
2000
tday
Figure 17.11: Saturation for equivalent one-cell model for model 760F. For each time step, green is
the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells; blue
is water and red is gas.
Mole Fraction in Reservoir
1.0
CH4 , nC4 , nC10 , CO2
0.8
CH4total moles
0.6
nC4total moles
nC10total moles
0.4
CO2total moles
0.2
0.0
0
500
1000
1500
2000
tday
Figure 17.12: Mole fraction for equivalent one-cell model for model 760F. For each time step, red
is the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;
CO2 is in orange.
Produced Fraction by Component
1.0
CH4 , nC4 , nC10
0.8
RF CH4
0.6
RF nC4
0.4
RF nC10
0.2
0.0
0
500
1000
1500
2000
tday
Figure 17.13: Molar recovery factor for model 760F. For each time step, red is the ratio of the
cumulative produced moles of methane to the original number of moles of methane in the reservoir;
nC4 is in green; nC10 is in cyan. In this figure, all three compositional recovery factors visually
overlay each other.
325
17.6
Description of model 761F
Model 761F is a 1-D waterflood model with gas hysteresis. Model 761F gives almost identical
results to model 760F; because the system is always above the bubble point the gas saturation is
always 0, so the option for gas hysteresis is not relevant.
17.7
Description of model 762F
Model 762F is a 1-D waterflood model with compositional trapping and gas hysteresis. Model
762F gives almost identical results to model 760F; because the system is always above the bubble
point the gas saturation is always 0, so compositional trapping is not relevant.
For model 762F, the water saturation progressively increases as the oil saturation decreases,
Figure 17.14. Because the pressure remains above the bubble point, splitting the water and oil into
trapped and mobile fractions has no visual impact on the results (Figure 17.11).
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
0.0
SgM1
0
500
1000
1500
2000
tday
Figure 17.14: Saturation for equivalent one-cell model for model 762F. For each time step, green
is the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;
blue is water and red is gas. Purple is the trapped water, cyan is the trapped oil, and yellow is the
trapped gas.
The results as a function of time and a function of space at a fixed time are visually the same
between model 762F and model 760F.
17.8
Description of model 760G
Model 760G is a 1-D model with primary production followed by a waterflood with no trapping
or hysteresis.
326
The injector has both a maximum bottom hole injection pressure of 3850 psia and a rate
constraint of 100 RB/day, Figure 17.15 and Figure 17.16. For this case only the rate constraint is
needed.
Injection Rate RBPD
100
80
qw or qg bbl
qTOT,inj
60
qo,inj
qg,inj
40
qw,inj
20
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.15: Injection rates at reservoir conditions for model 760G. Black is total injection in
RB/day; red is gas injection; blue is water injection.
Injection Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.16: Injection pressures for model 760G. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
The production well has a maximum total production rate of 100 RB/day, Figure 17.17. There
is also a minimum bottom hole producing pressure of 500 psia, but for this case it does not control
the produciton well, Figure 17.18.
The average saturations in the reservoir are shown in Figure 17.3. For this model, the oil
saturation decreases through the entire simulation. The gas saturation increases initially and then
decreases, going to zero at about the time of water breakthrough. The water saturation is initially
constant and then increases during the waterflood.
327
Production Rate RBPD
100
q RBPD
80
qTOT
60
qo qg
40
qw
20
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.17: Production rates at reservoir conditions for model 760G. Black is total production in
RB/day; green is oil production; red is gas production; blue is water production.
Production Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.18: Production pressure for model 760G. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
328
As shown in Figure 17.20, the mole fraction of CH4 decreases with time. The nC4 and nC10
mole fractions increase with time, with the increase in nC10 bigger than for nC4 . The CO2 is
approximately constant. The compositional recovery factors of CH4 is greater than the recovery
factor for nC4 which is greater than the recovery factor for nC10 , Figure 17.21.
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
SgM1
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.19: Saturation for equivalent one-cell model for model 760G. For each time step, green
is the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;
blue is water and red is gas.
Mole Fraction in Reservoir
1.0
CH4 , nC4 , nC10 , CO2
0.8
CH4total moles
0.6
nC4total moles
nC10total moles
0.4
CO2total moles
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.20: Mole fraction for equivalent one-cell model for model 760G. For each time step, red
is the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;
CO2 is in orange.
17.9
Description of model 761G
Model 761G is a 1-D model with primary production followed by a waterflood with gas hysteresis.
The injector has both a maximum bottom hole injection pressure of 3850 psia and a rate
constraint of 100 RB/day, Figure 17.22 and Figure 17.23. For this case only the rate constraint
329
Produced Fraction by Component
1.0
CH4 , nC4 , nC10
0.8
RF CH4
0.6
RF nC4
0.4
RF nC10
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.21: Molar recovery factor for model 760G. For each time step, red is the ratio of the
cumulative produced moles of methane to the original number of moles of methane in the reservoir;
nC4 is in green; nC10 is in cyan.
is needed. The injection profile is the same as the injection profile for model 760G, Figure 17.15.
The injection pressure profiles are different right after the start of water injection, Figure 17.16
Injection Rate RBPD
100
80
qw or qg bbl
qTOT,inj
60
qo,inj
qg,inj
40
qw,inj
20
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.22: Injection rates at reservoir conditions for model 761G. Black is total injection in
RB/day; red is gas injection; blue is water injection.
The production well has a maximum total production rate of 100 RB/day, Figure 17.24. The
oil response to the waterflood is earlier for model 761G (Figure 17.24) than for model 760G (Figure 17.17). There is also a minimum bottom hole producing pressure of 500 psia; this controls
production shortly after water breakthrough, Figure 17.25. This is different from the producer in
model 760G, Figure 17.18.
The average saturations in the reservoir are shown in Figure 17.3. For this model, the oil
saturation decreases through the entire simulation. The gas saturation increases initially and then
330
Injection Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.23: Injection pressures for model 761G. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
Production Rate RBPD
100
q RBPD
80
qTOT
60
qo qg
40
qw
20
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.24: Production rates at reservoir conditions for model 761G. Black is total production in
RB/day; green is oil production; red is gas production; blue is water production.
Production Pressure psia
4000
PBHP,Pores
3000
PBHP
2000
Pcell
1000
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.25: Production pressure for model 761G. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
331
decreases, going to zero at about the time of water breakthrough. The water saturation is initially
constant and then increases during the waterflood. The gas saturation profile is different between
500 days and 1000 days between model 761G (Figure 17.26) and 760G (Figure 17.19).
As shown in Figure 17.27, the mole fraction of CH4 decreases and then increases again. This
is different than model 760G (Figure 17.20) where it just decreases. This shows the importance of
gas hysteresis. The nC10 mole fraction increases and then decreases with time, also different from
model 760G. The nC4 and CO2 mole fractions are both approximately constant.
The compositional recovery factors of CH4 is greater than the recovery factor for nC4 which is
greater than the recovery factor for nC10 , Figure 17.28. The difference between the CH4 recovery
and the nC10 recovery is much less for model 761G than for model 760G (Figure 17.21). This shows
that a moderate amount of methane is trapped based on the gas hysteresis effects.
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
0.0
SgM1
0
500
1000
1500
2000
2500
3000
tday
Figure 17.26: Saturation for equivalent one-cell model for model 761G. For each time step, green
is the volume of oil in all the grid cells divided by the total volume of fluids in all the grid cells;
blue is water and red is gas.
17.10
Description of model 762G
Model 762G is a 1-D model with primary production followed by a waterflood with compositional trapping and gas hysteresis.
The injector has both a maximum bottom hole injection pressure of 3850 psia and a rate
constraint of 100 RB/day, Figure 17.29 and Figure 17.30. For this case only the rate constraint
is needed. The injection profile is the same as the injection profile for model 761G, Figure 17.22.
The injection pressure profiles are quite different during water injection, Figure 17.23.
332
Mole Fraction in Reservoir
1.0
CH4 , nC4 , nC10 , CO2
0.8
CH4total moles
0.6
nC4total moles
nC10total moles
0.4
CO2total moles
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.27: Mole fraction for equivalent one-cell model for model 761G. For each time step, red
is the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;
CO2 is in orange.
Produced Fraction by Component
1.0
CH4 , nC4 , nC10
0.8
RF CH4
0.6
RF nC4
0.4
RF nC10
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.28: Molar recovery factor for model 761G. For each time step, red is the ratio of the
cumulative produced moles of methane to the original number of moles of methane in the reservoir;
nC4 is in green; nC10 is in cyan.
Injection Rate RBPD
100
80
qw or qg bbl
qTOT,inj
60
qo,inj
qg,inj
40
qw,inj
20
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.29: Injection rates at reservoir conditions for model 762G. Black is total injection in
RB/day; red is gas injection; blue is water injection.
333
Injection Pressure psia
3500
PBHP,Pores
3000
2500
PBHP
Pcell
2000
1500
1000
0
500
1000
1500
2000
2500
3000
tday
Figure 17.30: Injection pressures for model 762G. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
The production well has a maximum total production rate of 100 RB/day, Figure 17.31. The oil
response to the waterflood is earlier and higher for model 762G (Figure 17.31) than for model 761G
(Figure 17.24). There is also a minimum bottom hole producing pressure of 500psia; this controls
production shortly after water breakthrough, Figure 17.32. This is similar to the producer in model
761G, Figure 17.25 but the duration is much shorter. Production pressures are much higher in the
model with compositional trapping than in model 761G without compositional trapping.
Production Rate RBPD
100
q RBPD
80
qTOT
60
qo qg
40
qw
20
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.31: Production rates at reservoir conditions for model 762G. Black is total production in
RB/day; green is oil production; red is gas production; blue is water production.
The average saturations in the reservoir are shown in Figure 17.3. For this model, the oil
saturation decreases through the entire simulation. The portion of the oil that is trapped steadily
increases after the start of water injection. The gas saturation increases initially and then decreases.
After the start of water injection most of the gas is trapped. The water saturation is initially
334
Production Pressure psia
3500
PBHP,Pores
3000
2500
PBHP
2000
Pcell
1500
1000
500
0
500
1000
1500
2000
2500
3000
tday
Figure 17.32: Production pressure for model 762G. Blue is bottom hole injection pressure from my
model; orange is grid cell injection pressure for my model.
constant and then increases during the waterflood. The saturation profile is similar to model 761G
(Figure 17.26), although the shape of the water saturation profile is smoother before and after
water breakthrough.
As shown in Figure 17.34, the mole fraction of CH4 decreases and then increases again. In
model 762G it actually increases above the initial mole fraction. This is because the gas which
becomes trapped gas has a high CH4 content; after it is trapped it can only get produced through
a slow transfer back to the mobile system. The nC10 and nC4 mole fraction increase and then
decrease with time to a value lower than the initial mole fraction; this is different than model
761G, Figure 17.27. The CO2 mole fraction increases slightly with time.
The compositional recovery factors of nC4 and nC10 (Figure 17.35) is visually similar to model
761G (Figure 17.28). The CH4 recovery is much lower for model 762G than for model 761G; the
CH4 recovery in model 761G is lower than model 760G. Both gas hysteresis and compositional
trapping increase the amount of methane that remains in the reservoir.
17.11
Compare CMG Model with my Model 760E and 761E
The production wells have a mixed pressure and rate constraint; they start out controlled by
the production rate, at around 1100 days they switch to bottom hole producing pressure control.
Figure 17.36 shows the bottom hole production rate for my model and the GEM model. Figure 17.37
shows the grid cell pressure for the production cell for my model and the GEM model. There is
a big change in the shape of the pressure profile as the system drops below the bubble point at
335
Saturation for Equivalent OneCell Model
1.0
SwM2
0.8
Total S
SwM1
0.6
SoM2
SoM1
0.4
SgM2
0.2
SgM1
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.33: Saturation for equivalent one-cell model for model 762G. For each time step, green
is the volume of mobile oil in all the grid cells divided by the total volume of fluids in all the grid
cells; cyan is the volume of trapped oil; red is the mobile gas; yellow is the trapped gas; blue is the
mobile water; purple is the trapped water.
Mole Fraction in Reservoir
1.0
CH4 , nC4 , nC10 , CO2
0.8
CH4total moles
0.6
nC4total moles
nC10total moles
0.4
CO2total moles
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.34: Mole fraction for equivalent one-cell model for model 762G. For each time step, red
is the ratio of the moles of methane to the total number of moles; nC4 is in green; nC10 is in cyan;
CO2 is in orange.
Produced Fraction by Component
1.0
CH4 , nC4 , nC10
0.8
RF CH4
0.6
RF nC4
0.4
RF nC10
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.35: Molar recovery factor for model 762G. For each time step, red is the ratio of the
cumulative produced moles of methane to the original number of moles of methane in the reservoir;
nC4 is in green; nC10 is in cyan.
336
around 100 days.
The pressures in Figure 17.37 are very similar; Figure 17.38 illustrates the difference between the
two models is consistently less than 15 psia with a moderately constant offset after the well drops
below the bubble point pressure. These slight differences in pressure lead to different times when the
well transitions from rate control to pressure control (Figure 17.36). The different pressures leads
to different flash conditions, which leads to the variations in molar rate shown in Figure 17.39. The
difference in flash conditions also leads to a different amount of produced oil compared to produced
gas, which leads to a different oil recovery factor, Figure 17.40.
Producer Rate RBD
100
80
60
q
JSB
CMG
40
20
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.36: Comparison of production rates for model 760E. Green is from my model; purple is
from the GEM model.
Producer Cell Pressure psia
4000
P
3000
JSB
2000
CMG
1000
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.37: Comparison of producer grid cell pressures for model 760E. Green is from my model;
purple is from the GEM model.
Model 761E gives almost identical results to model 760E; because the gas saturation is always
increasing there is no liquid phase to induce the trapping of gas. Different values of the critical gas
337
Producer Cell Pressure psia
10
P
5
0
5
10
0
200
400
600
800
1000
1200
1400
tday
Figure 17.38: Difference of producer grid cell pressures for model 760E.
Molar Rate
250
HC lbmolday
200
150
JSB
CMG
100
50
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.39: Comparison of total molar rates for model 760E. Green is from my model; purple is
from the GEM model.
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.40: Comparison of recovery factors for model 760E. Green is from my model; purple is
from the GEM model.
338
saturation would affect these cases. The difference in flash conditions leads to a different amount of
produced oil compared to produced gas, which leads to a different oil recovery factor, Figure 17.41.
Figure 17.41: Comparison of recovery factors for model 761E. Green is from my model; purple is
from the GEM model.
17.12
Compare CMG Model with my Model 762E
Figure 17.42 shows the bottom hole production rate for my model and the GEM model. There
is more pressure difference between 762E and the GEM model, Figure 17.43, than there is between
760E and GEM, Figure 17.43. This difference is even more obvious in Figure 17.44. The recovery
factors are shown in Figure 17.45.
Producer Rate RBD
100
80
60
q
JSB
CMG
40
20
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.42: Comparison of production rates for model 762E. Green is from my model; purple is
from the GEM model.
339
Producer Cell Pressure psia
4000
3000
JSB
P
2000
CMG
1000
0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.43: Comparison of producer grid cell pressures for model 762E. Green is from my model;
purple is from the GEM model.
Producer Cell Pressure psia
40
P
20
0
20
40
0
200
400
600
800
1000
1200
1400
tday
Figure 17.44: Difference of producer grid cell pressures for model 762E.
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
200
400
600
800
1000
1200
1400
tday
Figure 17.45: Comparison of recovery factors for model 762E. Green is from my model; purple is
from the GEM model 761E.
340
17.13
Compare CMG Model with my Model 760F, 761F, and 762F
Figure 17.46 shows the bottom hole production rate for my model and the GEM model; note
that GEM handles mixed pressure and rate constraints differently than my model; it has trouble
finding a solution at the time of water breakthrough. Figure 17.47 shows the grid cell pressure for
the production cell for my model and the GEM model. There is a difference of around 100 psia
between the two models after the initial time steps. This pressure difference may be a result of the
convergence failure in the first few time steps. The recovery factor (Figure 17.48) are very similar
between the two models because the system always stays above the bubble point.
Producer Rate RBD
q
100.05
JSB
100.00
CMG
99.95
0
500
1000
1500
2000
tday
Figure 17.46: Comparison of production rates for model 760F. Green is from my model; purple is
from the GEM model.
Producer Cell Pressure psia
4000
P
3000
JSB
2000
CMG
1000
0
0
500
1000
1500
2000
tday
Figure 17.47: Comparison of producer grid cell pressures for model 760F. Green is from my model;
purple is from the GEM model.
The water saturation profile is very similar, with small differences in the shape of the front;
these differences may be a result of GEM using a mixed fully implicit and IMPES scheme while
341
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
500
1000
1500
2000
tday
Figure 17.48: Comparison of recovery factors for model 760F. Green is from my model; purple is
from the GEM model.
my code uses an IMPES scheme, Figure 17.49.
Sw t500
0.7
0.6
Sw
0.5
0.4
JSB
0.3
CMG
0.2
0.1
0.0
0
200
400
600
800
1000
xft
Figure 17.49: Comparison of water saturation for model 760F at 500 days. Green is from my model;
purple is from the GEM model.
Model 761F gives almost identical results to model 760F; because the system is always above the
bubble point the gas saturation is always 0, so gas hysteresis is not relevant. The recovery factors
(Figure 17.50) are very similar between the two models because the system always stays above
the bubble point. Model 762F gives almost identical results to model 760F; because the system
is always above the bubble point the gas saturation is always 0, so compositional trapping is not
relevant. The recovery factors (Figure 17.51) are very similar between the two models because the
system always stays above the bubble point.
342
Figure 17.50: Comparison of recovery factors for model 761F. Green is from my model; purple is
from the GEM model.
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
500
1000
1500
2000
tday
Figure 17.51: Comparison of recovery factors for model 762F. Green is from my model; purple is
from the GEM model.
343
17.14
Compare CMG Model with my Model 760G
Figure 17.52 shows the bottom hole production rate for my model and the GEM model; note
that GEM handles mixed pressure and rate contraints differently than my model. Figure 17.53
shows the grid cell pressure for the production cell for my model and the GEM model. The
recovery factors (Figure 17.54) are similar between the two models. The differences are likely tied
to the pressure differences in Figure 17.53, which lead to different flash conditions.
Producer Rate RBD
1000
q
800
JSB
600
CMG
400
200
0
500
1000
1500
2000
2500
3000
3500
tday
Figure 17.52: Comparison of production rates for model 760G. Green is from my model; purple is
from the GEM model.
Producer Cell Pressure psia
4000
P
3000
JSB
2000
CMG
1000
0
0
500
1000
1500
2000
2500
3000
3500
tday
Figure 17.53: Comparison of producer grid cell pressures for model 760G. Green is from my model;
purple is from the GEM model.
The waterflood starts at 500 days. The gas saturation profile before the start of water injection
is different, probably because of the difference in flash pressures, Figure 17.55.
The waterflood starts at 500 days. At 1000 days is after some water injection but before
water breakthrough. The water saturation profiles are very similar between GEM and my model,
344
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
500
1000
1500
2000
2500
3000
3500
tday
Figure 17.54: Comparison of recovery factors for model 760G. Green is from my model; purple is
from the GEM model.
Sg t500
0.20
0.15
Sg
JSB
CMG
0.10
0.05
0.00
0
200
400
600
800
1000
xft
Figure 17.55: Comparison of gas saturation for model 760G at 500 days. Green is from my model;
purple is from the GEM model.
345
Figure 17.56. The gas saturations are different due to differences in flash pressures, Figure 17.57.
Sw t1000
0.7
0.6
Sw
0.5
0.4
JSB
0.3
CMG
0.2
0.1
0.0
0
200
400
600
800
1000
xft
Figure 17.56: Comparison of water saturation for model 760G at 1000 days. Green is from my
model; purple is from the GEM model.
Sg t1000
0.10
Sg
0.08
JSB
0.06
CMG
0.04
0.02
0.00
0
200
400
600
800
1000
xft
Figure 17.57: Comparison of gas saturation for model 760G at 1000 days. Green is from my model;
purple is from the GEM model.
17.15
Compare CMG Model with my Model 761G
Figure 17.58 shows the bottom hole production rate for my model and the GEM model; note
that GEM handles mixed pressure and rate constraints differently than my model; this leads to
large differences in the production rates between 1200 days and 1400 days when water breakthrough
occurs. Figure 17.59 shows the grid cell pressure for the production cell for my model and the GEM
model. There are some differences during the waterflood and larger differences during and after
water breakthrough. The recovery factor (Figure 17.60) are similar between the two models. The
differences are likely tied to the pressure differences in Figure 17.59, which lead to different flash
346
conditions.
Producer Rate RBD
800
700
600
q
500
JSB
400
CMG
300
200
100
0
500
1000
1500
2000
2500
3000
tday
Figure 17.58: Comparison of production rates for model 761G. Green is from my model; purple is
from the GEM model.
Producer Cell Pressure psia
4000
P
3000
JSB
2000
CMG
1000
0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.59: Comparison of producer grid cell pressures for model 761G. Green is from my model;
purple is from the GEM model.
The waterflood starts at 500 days. At 1000 days is after some water injection but before water breakthrough. During waterflood, the pressure differences are bigger after the water front has
passed, Figure 17.61. The water saturation profiles are similar between GEM and my model, Figure 17.62. My model is smoother than the GEM model, probably a result of the difference between
IMPES and the mixture of IMPES and fully implicit that GEM uses. The gas saturations are
different due to differences in flash pressures, Figure 17.63. There may also be other computational
differences; again my model has a much more smooth distribution than the GEM model. The GEM
model has spikes which do not make physical sense in a homogenous model.
347
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.60: Comparison of recovery factors for model 761G. Green is from my model; purple is
from the GEM model.
Grid Cell Pressure t1000
4000
Ppsia
3000
JSB
2000
CMG
1000
0
0
200
400
600
800
1000
xft
Figure 17.61: Comparison of pressure profiles for model 761G at 1000 days. Green is from my
model; purple is from the GEM model.
Sw t1000
0.7
0.6
Sw
0.5
0.4
JSB
0.3
CMG
0.2
0.1
0.0
0
200
400
600
800
1000
xft
Figure 17.62: Comparison of water saturation for model 761G at 1000 days. Green is from my
model; purple is from the GEM model.
348
Sg t1000
0.20
Sg
0.15
JSB
0.10
CMG
0.05
0.00
0
200
400
600
800
1000
xft
Figure 17.63: Comparison of gas saturation for model 761G at 1000 days. Green is from my model;
purple is from the GEM model.
1500 days is after water breakthrough. There are still pressure differences between GEM and
my model, Figure 17.64, but they are smaller than at 1000 days. The water saturation profiles
are similar between GEM and my model, Figure 17.65. After water breakthrough my model shows
some weird variations in water saturations. This may be related to the gas saturations still present
in my model but absent from the GEM model, Figure 17.66.
Grid Cell Pressure t1500
4000
Ppsia
3000
JSB
2000
CMG
1000
0
0
200
400
600
800
1000
xft
Figure 17.64: Comparison of pressure profiles for model 761G at 1500 days. Green is from my
model; purple is from the GEM model.
17.16
Compare CMG Model with my Model 762G
The recovery factor (Figure 17.67) are different between the two models as a result of the
compositional trapping changing the mobile saturations at different times; compare Figure 17.33
to Figure 17.26.
349
Sw t1500
0.7
0.6
Sw
0.5
0.4
JSB
0.3
CMG
0.2
0.1
0.0
0
200
400
600
800
1000
xft
Figure 17.65: Comparison of water saturation for model 761G at 1500 days. Green is from my
model; purple is from the GEM model.
Sg t1500
0.025
0.020
JSB
Sg
0.015
CMG
0.010
0.005
0.000
0
200
400
600
800
1000
xft
Figure 17.66: Comparison of gas saturation for model 761G at 1500 days. Green is from my model;
purple is from the GEM model.
Recovery Factor, CMG vs JSB
1.0
PV RCF
RF
produced oil RCF
0.8
0.6
JSB
CMG
0.4
0.2
0.0
0
500
1000
1500
2000
2500
3000
tday
Figure 17.67: Comparison of recovery factors for model 762G. Green is from my model; purple is
from the GEM model 761G.
350
CHAPTER 18
CASE STUDIES
The test cases used in this thesis are based on a low permeability carbonate reservoir in Abu
Dhabi. The work for this study was conducted in collaboration with the CSM/PI Integrated Carbonate Reservoir Research Group, and some of the information comes from discussion with members
of this research group. Portions of proprietary reservoir studies conducted by past operators of the
field were used for some properties of the reservoir simulation. Several publications were especially
valuable for data used here, including Alameri (2010), Jobe (2013), and Shibasaki, Edwards, Qotb,
and Akatsuka (2006).
18.1
Initial Conditions
The following initial conditions are specified.
• Reservoir depth 7550 ft, based on reservoir study information and Alameri (2010).
• Based on the reservoir studies, Jobe (2013), and Shibasaki et al. (2006) the expected dip is
less than 1◦ ; 0◦ is used here.
init = 3842 psia, based on reservoir study information.
• Initial reservoir pressure Pom
1
• Reservoir temperature T = 210◦ F, based on reservoir study information.
• WNaCl = 0.082142, from 200, 000 ppm, based on reservoir study information.
0 = {CH , nC , nC , CO } = {0.25, 0.25, 0.45, 0.05}, based on reservoir study information
• Zm
4
4
10
2
and converted from a 8-hydrocarbon component EOS to a 4-hydrocarbon component EOS.
init = S
• Initial water saturation Sw
wr = 0.059.
The following well constraints are specified. Wells in 1-D simulations and wells in the corner of
a quarter five-spot or a five-spot are 1/4 of these rates.
• Fracture pressure Pfrac = 5662 psia, corresponding to a fracture gradient of 0.75 psia/ft based
on Alameri (2010).
351
• Maximum injection pressure PBHIP = 5000 psia, based on Alameri (2010).
• Injection rate qinj = 400 RB/day = 2245.84 RCF/day, based on Alameri (2010).
• Bottom hole producing pressure PBHPP = 500 psia, based on Alameri (2010).
• Production rate qprod = 400 RB/day = 2245.84 RCF/day, based on Alameri (2010).
• Well radius rw = 0.5 ft.
• Wellbore skin s = 0.
The following are the grid properties for a 2-D 5-spot pattern:
• Δx = 100 m = 328 ft, NX = 21, based on the current 2 km × 2 km development pattern in
Shibasaki et al. (2006).
• Δy = 100 m = 328 ft, NY = 21, based on the current 2 km × 2 km development pattern in
Shibasaki et al. (2006).
• Δz = 44 ft, NZ = 1 based on Jobe (2013) and Shibasaki et al. (2006).
The following are the grid properties for a 2-D 1/4 5-spot pattern:
• Δx = 100 m = 328 ft, NX = 11, based on the current 2 km × 2 km development pattern in
Shibasaki et al. (2006).
• Δy = 100 m = 328 ft, NY = 11, based on the current 2 km × 2 km development pattern in
Shibasaki et al. (2006).
• Δz = 44 ft, NZ = 1 based on Jobe (2013) and Shibasaki et al. (2006).
The following are the grid properties for a 1-D pattern:
• Δx = 141.4 m = 464 ft, NX = 11, based on the 1414 m diagonal of the 1/4 5-spot.
• Δy = 141.4 m = 464 ft, NY = 1, based on the 1414 m diagonal of the 1/4 5-spot.
• Δz = 44 ft, NZ = 1 based on the total thickness of the reservoir in Jobe (2013) and Shibasaki
et al. (2006).
352
Summary of rock properties; variations in permeability and porosity are described in Section 18.2.
• kxx = kyy = kzz = 5.6 md, based on average values for facies “L5A” from Shibasaki et al.
(2006) which corresponds to facies “F5” Jobe (2013). Shibasaki et al. (2006) indicates that
effective permeability from well tests may be up to 5×kcore . The maximum listed permeability
in “L5A” from Shibasaki et al. (2006) is 63.6 md.
• φ = 0.19 based on average values for facies “L5A” from Shibasaki et al. (2006).
• Cφ = 4 · 10−6 psi−1
Trapping properties are defined as follows
• km2 = 5×10−4 md; this is computed to match a liquid diffusion coefficient of D = 10−5 cm2 /s.
D
k[md] =
μo
−5
2
(10 cm /s) × ( 1cp )
ΔP
k
ro
( 0.3 ) × (1000 psia)
×
1 md
1 psia
10−3 kg/(ms)
×
(18.1)
×
−12
2
9.9869 × 10
cm
1 cp
6894 kg/(ms2 )
• σm1 /m2 = 4.32 ft−2 is the shape factor. This is calculated in the same way as a fracture-matrix
shape factor:
σm1 /m2 = 4
1
1
1
+
+
2
2
(5 ft)
(5 ft)
(1 ft)2
Summary of relative permeability; refer to Section 18.4 for a full description.
• Swr = 0.059
• Sorw = 0.231
• Sorg = 0.15
• Sgr = 0.00
• nw = 4.49
• now = 3.76
353
(18.2)
• nog = 4.18
• ng = 2.3147
= 0.093
• krw
= 0.3
• kro
= 0.3
• krg
The gas-oil capillary pressure is assumed to be 0, based on the assumptions of the reservoir
studies and lack of data. See Section 18.5 for a full description of the water-oil capillary pressures;
the parameters are as follows.
• Swr = 0.059
• Sorw = 0.231
I = 0.28
• Swx
D = 0.33
• Swx
• αIow = 5
• αD
ow = 6.5
I
= −3.7
• Pc,offset
D
= −1.7
• Pc,offset
• Pcow,min = −20
• Pcow,max = 20
The following Peng-Robinson Equation of State properties are used:
• MWm = {16.043, 58.124, 142.285, 44.010}
• Pcm = {667.2, 551.1, 305.68, 1069.87}
• Tcm = {343.08, 765.36, 1111.68, 547.56}
354
• ωm = {0.008, 0.193, 0.49, 0.225}
• Pm = {77.3, 191.7, 431.0, 78.0}
• sm = {−0.19404, −0.08625, 0.08563, −0.06155} (cm = bm sm )
• vm = {1.59, 4.08, 9.66, 1.51}
⎛
⎞
0
0
0.0422 0.12
⎜
⎟
⎜
⎟
⎜
0
0
0.0078 0.12 ⎟
⎜
⎟
• δ̆mn = ⎜
⎟
⎜
⎟
0
0.1141 ⎟
⎜ 0.0422 0.0078
⎝
⎠
0.12
0.12 0.1141
0
0.
Initial conditions based on flash of Zm
• So = 0.941
• Sg = 0
• V=0
• Estimated bubble point Pb = 1268.
0 = {0, 0, 0, 0.000525, 0.999475}
• Wm
0 = {0.25, 0.25, 0.45, 0.05, 0.0}
• Xm
• ξw = 3.311
• ξo = 0.461
• ρw = 70.4281 lbmol/ft3
• ρo = 39.09 lbmol/ft3
• γw = 0. lbmol/ft3
• γo = 0.271 lbmol/ft3
• μw = 0.517
• μo = 0.233
355
• kro = 0.3
• krg = 0
• krw = 0
• Pcow = 20
• Pcgo = 0
Initial injection conditions based on flash of {0, 0, 0, 1, 0}.
• ξg = 0.175
18.2
Variations in Porosity and Permeability
Jobe (2013) described 12 cores from different parts of the field of interest. Routine porosity and
permeability analysis was previously conducted on these cores, but it was discovered later that the
CMS300 used for the analysis had not been calibrated for 20 years.
One of the facies from Jobe (2013) was selected for the 2-D studies in this dissertation. Facies 5
of Jobe (2013) corresponds to facies “L5A” and “L5B” of Shibasaki et al. (2006) and lithotype “23”
of the previous reservoir studies. Based on Jobe (2013), Facies 5 is a Lithocodium-Bacinella Wackestone with abundant oncoidal Lithocodium-Bacinella, common echinoderm, coral, bivalve skeletal
debris, and benthic forams including Miliolida, Textularia, and Orbitolina. It is a heterogenous
bioclastic boundstone with both micro and macro porosity.
The porosity distribution for Facies 5 is shown in Figure 18.1. After the outliers beyond three
standard deviations were excluded from the analysis, the mean porosity was 22.3% with a standard
deviation of 3.957. The distribution is symmetric and approximately normal. Based on the known
calibration errors in the porosity measurements, the porosity distribution was shifted based on the
mean values of Shibasaki et al. (2006). The new distribution had a mean of 19% and a standard
deviation of 3.957.
φF 5 [%] = Normal[μ = 19.0%, σ = 3.957%]
(18.3)
356
Porosity Distribution, Facies 5
0.10
frequency
0.08
0.06
0.04
0.02
0.00
0
10
20
30
40
porosity Figure 18.1: Porosity distribution for Facies 5 of Jobe (2013).
Within Facies 5, the permeability is weakly correlated with the porosity, Figure 18.2. The outliers beyond three standard deviations in the original fit were eliminated (cyan dots in Figure 18.2).
A new log-normal fit was created, shown in blue in Figure 18.2. This fit is not exact, so the additional variability is represented by a normal distribution with mean 0 and standard deviation
0.166.
%
kF 5 [md] = 10(−0.498+0.0427×φF 5 +Normal[0,0.166])
(18.4)
Heterogeneity is expected to have a significant effect on reservoir performance. To understand
the effect of heterogeneity, geostatistical analysis was conducted and geostatistical realizations were
created for a typical 5-spot pattern.
The spatial variability of the porosity and permeability was simulated using geostatistics. We
did not have access to enough data to conduct variogram analysis, so a semivariogram was created
that yielded distributions of porosity and permeability that looked reasonable. This semivariagrom
is based on a spherical variogram with a longer range in the NW-SE direction and a lower range
in the NE-SW direction. The distances are all represented in units of m here.
• h = lag; distance between two points.
357
PorosityPermeability Distribution, Facies 5
permeability md
10.0
5.0
2.0
1.0
0.5
0
5
10
15
20
25
30
porosity Figure 18.2: Porosity-permeability correlation for Facies 5, Jobe (2013). The blue line is a lognormal fit to the permeability-porosity trend. The red lines represent one and two standard deviations away from this primary trend. The blue dots are the core plug measurements. The cyan dots
were more than three standard deviations away from the original trend.
• a = range; beyond this distance points are not correlated.
– aNW = 2000 m
– aNE = 1000 m
• c = 0.7 = sill; constant value beyond range.
⎧
3
⎪
h
h
⎪
−
0.5
, h ≤ aNW , primary direction, NW-SE
0.3
+
0.7
×
1.5
⎪
aNW
aNW
⎪
⎪
⎪
⎨
1.0,
h > aNW , primary direction, NW-SE
γ[h] =
3
⎪
h
h
⎪
− 0.5 aNE
, h ≤ aNE , secondary direction, NE-SW
0.3 + 0.7 × 1.5 aNE
⎪
⎪
⎪
⎪
⎩
1.0,
h > aNE , secondary direction, NE-SW
(18.5)
Based on the variograms, a series of 100 Sequential Gaussian Simulations were conducted using
GSLIB (Deutsch and Journel, 1992). These simulations were based on a normal distribution with
mean 0 and standard deviation 1. The control data assumed that all the wells of the 5-spot pattern
have average properties (mean 0). Although the realizations were simulated using Normal[0, 1], the
sample mean for a particular realization will not be equal to 0. The realizations were first shifted to
have a mean of 0, and then transformed into the Normal[19, 3.957] distribution of the porosity for
F5. Six of the one hundred realizations were selected for further use. For each of the six realizations,
358
10 sequences of uncorrelated random numbers were used to generate the permeability using (18.4).
One of the 10 permeability distributions was selected for further use. Figure 18.3–Figure 18.8 show
the selected porosity and permeability spatial distributions as well as the porosity and permeability
histograms.
Permeability md; Facies 5; ID 14
Porosity Distribution; Facies 5; ID 14
1
1.5
2
2.5 3
4
5
7
10
15
20
25 30
0.12
0.15
0.10
count
freq
0.08
0.06
0.10
0.04
0.05
0.02
0.00
0
10
20
30
0.00
40
0.0
porosity 0.2
0.4
0.6
0.8
1.0
1.2
1.4
log10 permeability md
(a) Porosity distribution
(b) Permeability distribution
Porosity; Facies 5; ID 14
log10 Permeability; Facies 5; ID 14
Φ log10 k
30
1.50
1.25
25
1.00
20
0.75
15
0.50
10
0.25
0
5
(d) Permeability map
(c) Porosity map
Figure 18.3: Porosity and permeability for Geostatistical Realization # 1.
18.3
Relative Permeability Test Case Literature Review
Previously, a literature search was conducted for “three-phase relative permeability”, “relative
permeability hysteresis”, “relative permeability in carbonates”, “mixed wettability”, and “Abu
Dhabi fields”. These papers were reviewed again looking for data that might add additional constraints on the relative permeability curves for this test case.
359
Permeability md; Facies 5; ID 26
Porosity Distribution; Facies 5; ID 26
1
0.12
1.5
2
2.5 3
4
5
7
10
15
20
25 30
0.10
0.15
0.06
count
freq
0.08
0.10
0.04
0.05
0.02
0.00
0
10
20
30
0.00
40
0.0
porosity 0.2
0.4
0.6
0.8
1.0
1.2
1.4
log10 permeability md
(a) Porosity distribution
(b) Permeability distribution
Porosity; Facies 5; ID 26
log10 Permeability; Facies 5; ID 26
Φ log10 k
30
1.50
1.25
25
1.00
20
0.75
15
0.50
10
0.25
0
5
(d) Permeability map
(c) Porosity map
Figure 18.4: Porosity and permeability for Geostatistical Realization # 2.
360
Permeability md; Facies 5; ID 39
Porosity Distribution; Facies 5; ID 39
1
1.5
2
2.5 3
4
5
7
10
15
20
25 30
0.10
0.15
0.06
count
freq
0.08
0.10
0.04
0.05
0.02
0.00
0
10
20
30
0.00
40
0.0
porosity 0.2
0.4
0.6
0.8
1.0
1.2
1.4
log10 permeability md
(a) Porosity distribution
(b) Permeability distribution
Porosity; Facies 5; ID 39
log10 Permeability; Facies 5; ID 39
Φ log10 k
30
1.50
1.25
25
1.00
20
0.75
15
0.50
10
0.25
0
5
(d) Permeability map
(c) Porosity map
Figure 18.5: Porosity and permeability for Geostatistical Realization # 3.
361
Permeability md; Facies 5; ID 42
Porosity Distribution; Facies 5; ID 42
1
0.12
0.10
1.5
2
2.5 3
4
5
7
10
15
20
25 30
0.15
0.06
count
freq
0.08
0.10
0.04
0.05
0.02
0.00
0
10
20
30
0.00
40
0.0
porosity 0.2
0.4
0.6
0.8
1.0
1.2
1.4
log10 permeability md
(a) Porosity distribution
(b) Permeability distribution
Porosity; Facies 5; ID 42
log10 Permeability; Facies 5; ID 42
Φ log10 k
30
1.50
1.25
25
1.00
20
0.75
15
0.50
10
0.25
0
5
(d) Permeability map
(c) Porosity map
Figure 18.6: Porosity and permeability for Geostatistical Realization # 4.
362
Permeability md; Facies 5; ID 92
Porosity Distribution; Facies 5; ID 92
1
1.5
2
2.5 3
4
5
7
10
15
20
25 30
0.20
0.12
0.10
0.15
count
freq
0.08
0.06
0.10
0.04
0.05
0.02
0.00
0
10
20
30
0.00
40
0.0
porosity 0.2
0.4
0.6
0.8
1.0
1.2
1.4
log10 permeability md
(a) Porosity distribution
(b) Permeability distribution
Porosity; Facies 5; ID 92
log10 Permeability; Facies 5; ID 92
Φ log10 k
30
1.50
1.25
25
1.00
20
0.75
15
0.50
10
0.25
0
5
(d) Permeability map
(c) Porosity map
Figure 18.7: Porosity and permeability for Geostatistical Realization # 5.
363
Permeability md; Facies 5; ID 99
Porosity Distribution; Facies 5; ID 99
1
1.5
2
2.5 3
4
5
7
10
15
20
25 30
0.12
0.15
0.10
count
freq
0.08
0.06
0.10
0.04
0.05
0.02
0.00
0
10
20
30
0.00
40
0.0
porosity 0.2
0.4
0.6
0.8
1.0
1.2
1.4
log10 permeability md
(a) Porosity distribution
(b) Permeability distribution
Porosity; Facies 5; ID 99
log10 Permeability; Facies 5; ID 99
Φ log10 k
30
1.50
1.25
25
1.00
20
0.75
15
0.50
10
0.25
0
5
(d) Permeability map
(c) Porosity map
Figure 18.8: Porosity and permeability for Geostatistical Realization # 6.
364
18.3.1
Water-Oil Data
Spiteri et al. (2005) present simulation models applied to CO2 injection with relative permeability hysteresis. Masalmeh (2002) presents capillary pressure and relative permeability data for
mixed wet and oil wet Middle East carbonates. Masalmeh (2003) presents capillary pressure and
relative permeability data and their variations with wettability. Lamy et al. (2010) presents capillary pressure data for carbonate cores. Honarpour et al. (1996) presents an experimental apparatus
to simultaneously measure relative permeability, capillary pressure, and electrical resistivity during
a core flood. Hysteresis data is presented for Berea sandstone. Jerauld (1997) presents three-phase
relative permeability data and curve fits for mixed wet Prudhoe Bay sandstone. Kralik et al. (2000)
presents the results of three-phase relative permeability experiments on an oil-wet sandstone. Dernaika et al. (2012) presents relative permeability data with hysteresis for various carbonate rocks.
18.3.2
Gas-Oil Data
Fatemi et al. (2012a) presents a history match of experimental three-phase relative permeability
data and a good literature review. Fatemi et al. (2012b) presents three-phase relative permeability
data for water wet and mixed wet cores.
18.3.3
Gas-Water Data
Levine (2011) presents CO2 /brine relative permeability in sandstone and constructed cores.
Bennion and Bachu (2005) and Bennion and Bachu (2008b) present CO2 /brine relative permeability
data for carbonate and sandstone cores in Canada.
18.3.4
Two-Phase Experiments with Different Phases
Aljarwan, Belhaj, Haroun, and Ghedan (2012) present oil/water and gas/oil data for an Abu
Dhabi reservoir. Ehrlich et al. (1984) presents laboratory data for a dolomite reservoir subjected
to a lab-based CO2 WAG flood. Bhatti et al. (2012) presents relative permeability and capillary
pressure data for Abu Dhabi carbonates.
18.3.5
Three-Phase Experiments
Spiteri and Juanes (2004) and Spiteri and Juanes (2006) present simulation of WAG injection
with different three-phase relative permeability models. Al-Dhahli, Geiger, and van Dijke (2012),
365
and Fatemi and Sohrabi (2012), and Element et al. (2003) present experimental results which
show cycle dependent residual oil saturations. Oak (1990) presents the results of very thorough
experiments of three-phase relative permeability on water-wet Berea sandstone.
18.3.6
Relative Permeability Formulations
For the test cases used here, krow is very close to krog for all values of So . For this specific
application oil relative permeability was calculated as a function of the oil saturation only. The
following articles are a selected group of three-phase relative permeability and kro calculation
methods. One of these methods would most likely be selected to calculate oil relative permeability
if the SCAL data did not support the specialized simplification.
Spiteri and Juanes (2004), presents an evaluation of different relative permeability and hysteresis models, including the presentation of a new method for three-phase relative permeability
hysteresis. Hustad et al. (2002) presents the results of 2D cross-section simulation models of WAG
with hysteresis. Hustad (2002) presents a three-phase capillary pressure and relative permeability
model with hysteresis. Larsen and Skauge (1998) presents a three-phase relative permeability formulation. Dietrich and Bondor (1976) presents a three-phase relative permeability model. Coats
and Smith (1964) describes dead-end space using a diffusion model. Hustad and Browning (2009)
presents a relative permeability and capillary pressure formulation with hysteresis. Blunt (2000)
presents an analysis of three-phase relative permeability and capillary pressure experiments, including a discussion of trapped oil, spreading oil, and mobile oil. Baker (1988) presents an analysis
of different three-phase relative permeability formulations. Fayers and Matthews (1984) analyzes
three-phase relative permeability data from various literature sources. Kossack (2000) presents a
comparison three-phase relative permeability models with hysteresis as implemented in Eclipse.
Kokal and Maini (1990) presents analysis of several three-phase relative permeability experiments
and a modified Stone’s method. Delshad and Pope (1989) presents an analysis of seven different
three-phase relative permeability formulations. Killough (1976), present a hysteresis algorithm.
18.3.7
Relative Permeability Observations
The following articles are related to mixed wet and/or carbonate reservoir relative permeability
and capillary pressure: Masalmeh (2001), Byrnes and Bhattacharya (2006), Syed, Ghedan, Al-
366
Hage, and Tariq (2012), Dabbouk, Liaqat, Williams, and Beattie (2002), Keelan and Pugh (1975),
Wegener and Harpole (1996) After evaluating these articles and the articles listed above, trends
from these articles were used but the data was not directly used.
Ghomian et al. (2008) presents simulations of CO2 WAG for EOR and sequestration using
different three-phase relative permeability and capillary pressure models. Iglauer et al. (2009)
presents a review of capillary trapping in sandstones along with some new data. Shahverdi and
Sohrabi (2012) presents an analysis of three-phase relative permeability data. Masalmeh and Wei
(2010) presents a study of WAG options using three-phase relative permeability and capillary
pressure hysteresis. Ahmed Elfeel, Al-Dhahli, Geiger, and van Dijke (2013) uses tables of threephase relative permeability from pore-network models to simulate WAG.
18.4
Relative Permeability
This section describes the relative permeability curves used in the test cases.
18.4.1
Experimental Data
All the SCAL data we had access to from previous reservoir studies was reviewed. Lithotype
23, 6.49 md corresponds to facies F5 of Jobe (2013) and facies ”L5a” and ”L5b” of Shibasaki et al.
= krog
= 0.3. Both the
(2006) . Based on data from Tadesse Teklu and Waleed Al-Ameri, krow
oil-water and gas-oil kr values are multiplied by 0.3 before curve fitting.
We don’t have measurements of the interfacial tensions to allow for the calculation of the
spreading coefficient. The values of the relative permeability to oil are very small (kro [So = 0.265] =
1.09×10−5 for the water oil F5 experiment). This very small kro makes it difficult to justify a linear
layer flow model for the oil. As a result, kro is fit using a Corey model without an additional linear
flow component. Based on recommendations of Dr. Kazemi, the presence of capillary pressure
makes it unnecessary to have
lim
S→Smin
∂kr
=0
∂S
(18.6)
for any of the saturations. As a result, Corey curves were also used for krg and krw .
van Dijke et al. (2000) and van Dijke et al. (2001) present a formulation for three-phase relative
permeability. Juanes and Patzek (2004b) and Juanes and Patzek (2004a) present a theoretical
367
discussion under what conditions three-phase relative permeability models transition between hyperbolic and elliptic regions.
18.4.2
Oil/Water Experiment
D fit based on data for the water-oil system SCAL (WOG=IDC), scaled to k The krow
row = 0.3.
The data and the fits are shown in Figure 18.9. A standard Corey curve fits the krow .
⎧
⎨ 0,
krow =
⎩ krow
So − Sorw
1 − Swr − Sorw
now
So ≤ Sorw
, So > Sorw
(18.7)
• Swr = 0.059
• Sorw = 0.231
= 0.3
• krow
• now = 3.76
I fit based on data for the water-oil system SCAL (WOG=IDC), scaled to k The krw
row = 0.3 :
The data and the fits are shown in Figure 18.9. A standard Corey curve fits the krw .
krw
⎧
⎨ 0, =
⎩ krw
Sw − Swr
1 − Swr − Sorw
nw
Sw ≤ Swr
, Sw > Swr
(18.8)
• Swr = 0.059
• Sorw = 0.231
= 0.093
• krw
• nw = 4.49
18.4.3
Gas/Oil Experiment
D fit based on data for the gas-oil system SCAL (WOG=IDC), scaled to k = 0.3. The
The krog
rog
data and the fits are shown in Figure 18.10. A standard Corey curve fits the krog .
368
OilWater Relative Permeability
0.5
0.4
Swr ,krow
kr
0.3
0.2
Swr
0.1
0.0
0.0
Sorw
1Sowr ,krwo
krow
0.2
0.4
0.6
0.8
krw
1.0
Sw
Figure 18.9: Oil and water relative permeability curves including the data points. The green curve
and data points are krow . The blue curve and data points are krw .
⎧
⎨ 0,
krog =
⎩ krog
So − Sorg
1 − Swr − Sorg
nog
So ≤ Sorg
, So > Sorg
(18.9)
• Swr = 0.059
• Sorg = 0.15
= 0.3
• krog
• nog = 4.18
I fit based on data for the gas-oil system SCAL (WOG=IDC), scaled to k = 0.3. The
The krg
rog
I .
data and the fits are shown in Figure 18.10. A standard Corey curve fits the krg
I
krg
⎧
⎨ 0, =
⎩ krg
Sg
1 − Swr − Sorg
ng
Sg ≤ 0
, Sg > 0
• Swr = 0.059
• Sorg = 0.15
= 0.3
• krg
• ng = 2.3147
369
(18.10)
GasOil Relative Permeability
0.6
0.5
kr
0.4
Sgr ,krog
0.3
1Sorg Swr ,krgo
Sorg
0.2
krog
0.1
0.0
0.0
Swr
krg
0.2
0.4
0.6
0.8
1.0
Sg
Figure 18.10: Gas and oil relative permeability curves including the data points. The green curve
and data points are krog . The red curve and data points are krg .
18.4.4
Trapped Gas
We did not have access to any trapped gas or gas hysteresis measurements for the field test cased
used in this thesis. Based on a literature review of mixed wet sandstones and carbonates, typical
maximum trapped gas saturation is between 0.2 and 0.3. The shape of this curve for carbonates
and mixed wet sandstones is better fit by Jerauld (1997) than by Land (1968). Specify the trapping
function based on Jerauld (1997), illustrated in Figure 18.11.
Sg,trap [Sg ] =
1+
Sg
1
Sgt,max
S
− 1 × Sg
gt,max
1+b 1−S
(18.11)
gt,max
• b = 1, indicates 0 slope at Sg = 1.
• Sgt,max = 0.25; the maximum amount of trapped gas.
18.4.5
Trapped Oil
The values of trapped oil saturations vary significantly in the literature. Mixed wet sandstones
and carbonates have relatively low trapped oil saturations, with approximately 0.10 − 0.15 typical.
The literature often does not report the trapped oil or kro hysteresis for mixed wet reservoirs. The
trapping function based on Jerauld (1997) is illustrated in Figure 18.12.
370
Trapped Gas
0.30
0.25
Sgtrap
0.20
0.15
0.10
0.05
0.00
0.0
0.2
0.4
0.6
0.8
1.0
Sg
Figure 18.11: Trapped gas saturation as a function of maximum achieved gas saturation.
So
So,trap [So ] =
1+
1
Sot,max
(18.12)
S
− 1 × So
ot,max
1+b 1−S
ot,max
• b = 1, indicates 0 slope at So = 1.
• Sot,max = 0.10; the maximum amount of trapped oil.
Trapped Oil
0.20
Sotrap
0.15
0.10
0.05
0.00
0.0
0.2
0.4
0.6
0.8
1.0
So
Figure 18.12: Trapped oil saturation as a function of maximum oil saturation achieved after the
initial oil saturation.
371
18.4.6
Cycle Dependent Residual Oil Saturations
Al-Dhahli et al. (2012), and Fatemi and Sohrabi (2012), and Element et al. (2003) present
experimental results which show cycle dependent residual oil saturations. The following values
were selected based on the trends in these articles.
0
1
2
= Sorw
= Sorw
= 0.231 based on a fit to the
• Saturation path 1, water injection: Sorw = Sorw
oil/water SCAL data for rock type F5.
0
1
2 = 0.15 based on Dr. Kazemi’s
= Sorg
= Sorg
• Saturation path 2, gas injection: Sorg = Sorg
experience that the gas/oil Sorg from the SCAL data of 0.05 was too low. Individual cores
often have a very low Sorg , but this is not representative of a reservoir scale simulation.
3
= 0.13
• Saturation path 3, water injection: Sorw
4 = 0.12
• Saturation path 4, gas injection: Sorg
5
= 0.11
• Saturation path 5, water injection: Sorw
6 = 0.10
• Saturation path 6, gas injection: Sorg
18.4.7
Water Relative Permeability
The following assumptions were selected for the water relative permeability.
• krw is a function of Sw only. This is valid for all water wet reservoirs and seems valid for
mixed wet reservoirs also. It is not a good assumption for strongly oil wet reservoirs.
• There is no water trapping by oil or gas.
• No physical/chemical process considered here reduces Swr .
• There is no water hysteresis.
• The krwg = krwo = krw and is based on krw calculated from the WOG=IDC waterflood
process for a water-oil system.
• When Sor changes from Sorw to Sorg , the krw follows the krwo curve to a higher endpoint
.
saturation if necessary. In this case, krw [1 − Sorg ] > krw
372
• The krw is illustrated in Figure 18.13.
0 for the water relative permeability based on the fit to the
Specify the reference function krw
water/oil data, Figure 18.13. It is only necessary to specify one reference function because the
water relative permeability curve does not change even if the residual saturations change.
0
• Sw,min
= Swr = 0.059
0
= 1 − Sorw = 1 − 0.231 = 0.791
• Sw,max
0
= 0.093
= krw
• krw,max
• n0w = nw = 4.49
0
[Sw ] =
krw
⎧
⎪
⎨ 0,
0
⎪
⎩ krw,max ×
0
Sw − Sw,min
0
Sw < Sw,min
n w
(18.13)
0
, Sw ≥ Sw,min
0
0
Sw,max
− Sw,min
Water Relative Permeability
0.30
0.25
krw
0.20
Swr
Sorw
0.15
1Sorg ,krw 1Sorg 0.10
1Sorw ,krwo
0.05
0.00
0.0
Swr ,0
0.2
0.4
0.6
0.8
1.0
Sw
Figure 18.13: Water relative permeability based on a fit to the oil-water data in Figure 18.9.
18.4.8
Gas Relative Permeability
The following assumptions were selected for the gas relative permeability.
• krg is a function of Sg only. This is valid for all the reviewed experiments in the literature.
• Gas is trapped using a formulation by Jerauld (1997); this fits the observed data better for
mixed wet carbonate reservoirs than the formulation by Land (1968).
373
• The trapped gas specified by Land (1968) and Jerauld (1997) refer to the total saturations.
To convert between total saturations and the saturations in the m2 systems, use the following
equation.
Sgt = Sg,m2
φm2
φt
(18.14)
• The hysteresis in the gas relative permeability is related to the trapping of gas. Additional
gas is trapped when switching from an increasing scanning curve to a decreasing scanning
curve.
• The krg is based on data for a WOG=CDI gas-oil experiment.
• When Sor changes from Sorw to Sorg , the krg curve extends to a higher endpoint saturation
.
(1 − Sor − Swr ) if necessary. In this case, krg [1 − Sor − Swr ] > krg
The gas relative permeability bounding curves are shown in Figure 18.14. The bounding curves
are the increasing relative permeability at zero initial gas saturation and the decreasing relative
permeability curve at maximum trapped gas saturation.
Gas Relative Permeability
0.6
0.5
krg
0.4
G1
SgG1 ,krg
0.3
max
Sgt
Sorg
0.2
Swr
0.1
G0
SgG0 ,krg
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Sg
Figure 18.14: Bounding scanning curves for gas relative permeability based on Figure 18.10.
The following steps are associated with gas relative permeability hysteresis. The case described
first has an initial water flood with the initial Sg = 0, followed by alternating WAG cycles of gas
374
and water. It is assumed for this description that no gas comes out of solution during the initial
water flood (WOG=IDC).
0 for the gas relative permeability based on the fit to
First, specify the reference function krg
the gas/oil data, Figure 18.14. It is only necessary to specify one reference function because the
drainage and imbibition bounding curves and all the scanning curves have the same curvature
0
0
0
, Sg,max
, krg,max
, and ng do not change even if the residual saturations
ng . The parameters Sg,min
change. If the initial gas saturation is 0, then use the reference curve during the initial waterflood.
If the gas saturation is still 0 at the end of the initial waterflood, also use the reference curve for
the first gas injection cycle.
0
= Sgr = 0
• Sg,min
0
0 = 1 − 0.059 − 0.15 = 0.791
• Sg,max
= 1 − Swr − Sogr
0
= 0.3
= krg
• krg,max
• n0g = ng = 2.3147
0
[Sg ] =
krg
⎧
⎪
⎨ 0,
0
⎪
⎩ krg,max ×
0
Sg − Sg,min
0
Sg,max
−
0
Sg < Sg,min
n g
0
, Sg ≥ Sg,min
0
Sg,min
(18.15)
During water injection, the gas saturation decreases (WOG=IDD). Figure 18.15 illustrates a
decreasing scanning curve in green. Start by calculating the trapped gas based on Jerauld (1997),
Figure 18.11. Specify the minimum gas saturation for the scanning curve based on the trapped gas.
The maximum gas saturation and relative permeability for the scanning curve are the values at
the end of the previous increasing cycle. The two known points on the scanning curve are (Sgm2 , 0)
A ), points and in Figure 18.15.
and (SgA , krg
3
2
• SgA is the gas saturation at the end of the previous cycle, point 2 in Figure 18.15.
A is the gas relative permeability at the end of the previous cycle, point in Figure 18.15.
• krg
2
A =S
A
• Sgt
g,trap Sg .
• Calculate the trapped gas
375
n
Sgt
= max
φnm2 A
n
Sg,m2 n , Sgt
φt
(18.16)
n may have decreased from S prev max based on flash changes and transfer between
Note that Sgt
gt
the trapped and mobile systems.
D
= Sgm2 , point • Sg,min
3 in Figure 18.15.
D
= SgA , point • Sg,max
2 in Figure 18.15.
D
A , point in Figure 18.15.
= krg
• krg,max
2
• nD
g = ng
D
krg
[Sg ] =
⎧
⎪
⎨ 0,
D
⎪
⎩ krg,max ×
D
Sg − Sg,min
D
Sg,max
−
D
Sg < Sg,min
n g
D
Sg,min
(18.17)
D
, Sg ≥ Sg,min
Gas Relative Permeability
0.4
4: Sgmax ,krgmax krg
0.3
max
Sgt
Sorg
0.2
Swr
2: SgA ,krgA 0.1
1
0.0
0.0
14: Sgtmax ,0
3: SgM2 ,0
0.2
0.4
0.6
0.8
1.0
Sg
decr
Figure 18.15: A decreasing gas relative permeability scanning curve shown in green, 2 −−→ .
3
A
A
This assumes that the previous values increased to 2 with values of (Sg , krg ). Bounding curves
incr
decr
14
from Figure 18.14 are shown in red, 4 and 4 −−→ .
1 −−→ During gas injection, the gas saturation increases (WOG=DDI). Figure 18.16 illustrates an
I
I
and krg,max
for this scanning
increasing scanning curve in cyan. Start by specifying the Sg,max
376
0
0
I
curve. If Sorg is constant, this is (Sg,max
, krg,max
). Calculate Sgmin
so that the new scanning curve
A ), or (0, 0) if this is
passes through the values at the end of the previous decreasing cycle (SgA , krg
the first gas injection cycle and there was no initial free gas. The two known points on the scanning
A ) and (S I
I
curve are (SgA , krg
3 and 4 in Figure 18.16.
g,max , krg,max ), points • SgA is the gas saturation at the end of the previous cycle, point 3 in Figure 18.16.
A is the gas relative permeability at the end of the previous cycle, point in Figure 18.16.
• krg
4
I
n , point in Figure 18.16. Note that S n may change with the WAG
= 1 − Swr − Sorg
• Sg,max
4
org
cycle.
I
0 [S I
= krg
• krg,max
4 in Figure 18.16.
gmax ], point I [S A ] = k A to obtain S I
13 in Figure 18.16.
• Use the constraint that krg
g
rg
gmin , point α=
I
=
Sgmin
A
krg
1
ng
(18.18)
I
krg,max
I
α · Sgmax
− SgA
α−1
(18.19)
• nIg = ng
I
krg
[Sg ] =
18.4.9
⎧
⎪
⎨ 0,
I
⎪
⎩ krg,max ×
I
Sg − Sg,min
I
Sg,max
−
I
Sg,min
I
Sg < Sg,min
n g
I
, Sg ≥ Sg,min
(18.20)
Oil Relative Permeability
The following assumptions were selected for the oil relative permeability.
• krow and krog are very close to each other (Figure 18.17,Figure 18.18,Figure 18.19). This
means that kro is a function of So if the Sor is adjusted appropriately.
• Oil is trapped using a formulation by Jerauld (1997); this fits the observed data better for
mixed wet carbonate reservoirs than the formulation by Land (1968).
377
Gas Relative Permeability
0.4
4: Sgmax ,krgmax krg
0.3
max
Sgt
Sorg
0.2
Swr
2
0.1
1
0.0
0.0
A
12:SgM2
,0
I 14: S
13:Smin ,0 gtmax ,0
0.2
3:
SgA ,krgA 0.4
0.6
0.8
1.0
Sg
decr
Figure 18.16: An increasing gas relative permeability scanning curve shown in cyan, 3 −−→ .
4
A
A
This assumes that the previous values decreased to 3 with values of (Sg , krg ). The previous
decr
decr
A ).
, but the saturation did not drop below (SgA , krg
decreasing scanning curve was 3 −−→ 12
2 −−→ incr
decr
14
Bounding curves from Figure 18.14 are shown in red, 4 and 4 −−→ .
1 −−→ • The trapped oil specified by Land (1968) and Jerauld (1997) refer to the total saturations.
To convert between total saturations and the saturations in the m2 systems, use the following
equation.
Sot = So,m2
φm2
φt
(18.21)
• The maximum trapped oil Sot,max is less than the residual oil saturation Sorg or Sorw , even
when Sor is cycle-dependent. This means there is no hysteresis in the kro .
• The krog is based on data for a WOG=CDI gas-oil experiment.
• The krow is based on data for a WOG=IDC water-oil experiment.
• When Sor changes from Sorw to Sorg , and may also continue to decrease with the WAG cycle.
The oil relative permeability krow from the oil-water SCAL data is illustrated in Figure 18.17.
The oil relative permeability krog from the gas-oil SCAL data is illustrated in Figure 18.18. The
krow and krog curves are very similar for this data, as illustrated in Figure 18.19. The properties
of the krow and krog reference curves are as follows:
378
0
W 2 = 0.13 and S W 3 = 0.11.
• Sorw
= 0.231; if Sorw is cycle-dependent, then Sorw
orw
0 = 0.15; if S
G2
G3
• Sorg
org is cycle-dependent, then Sorg = 0.12 and Sorg = 0.10.
0
• So,max
= 1 − Swr = 1 − 0.059 = 0.941
0
= k
= krog
• kro,max
row = 0.3
• n0ow = now = 3.76
• n0og = now = 4.17
0
[So ]
krow
⎧
⎨ 0,
=
0
[So ] =
krog
0
×
⎩ kro,max
⎧
⎪
⎨ 0,
0
⎪
⎩ kro,max ×
0
So − Sorg
0
0
So,max
− Sorg
0
So < Sorw
now
0
So − Sorw
0
0
So,max
− Sorw
(18.22)
0
, So ≥ Sorw
0
So < Sorg
nog
(18.23)
0
, So ≥ Sorg
Oil Relative Permeability
0.4
1Swr ,krow
krow
0.3
0.2
Swr
Sorw
Sorg
0.1
Sorw ,0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
So
Figure 18.17: Oil relative permeability based on from the oil/water SCAL, Figure 18.9.
max < S
The maximum amount of trapped oil, Sot
org < Sorw , Figure 18.20 and Figure 18.21. Using
the approach for scanning curves described in Section 18.4.8 would mean that the krog and krow
would follow the same scanning curves, illustrated in Figure 18.20 and Figure 18.21. Because the
residual oil saturation Sorw or Sorg changes it is still necessary to calculate increasing and decreasing
scanning curves. The trapped oil is still calculated at the end of oil-increasing saturation paths;
379
Oil Relative Permeability
0.4
1Swr ,krow
krog
0.3
0.2
Swr
Sorw
Sorg
0.1
Sorg ,0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
So
Figure 18.18: Oil relative permeability based on from the gas/oil SCAL, Figure 18.10.
Oil Relative Permeability
0.4
krow,krog
0.3
0.2
Swr
Sorw
Sorg
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
So
Figure 18.19: Compare the krow in cyan, Figure 18.17, and the krog in purple, Figure 18.18. For
this data set the curves are very similar.
380
this trapped oil saturation effects the composition of Som1 and Som2 even though it does not effect
the kro . The trapped oil, Sot renormalized as Som2 , only interacts with the mobile system through
a transfer function. Mobile oil Sot < S < Sor is in thermodynamic equilibrium with the additional
mobile oil, gas, and water; the compositions flash between Som1 , Sgm1 , and Swm1 at each nonlinear
iteration.
Oil Relative Permeability
0.4
1Swr ,krow
krow
0.3
0.2
0.1
Swr
Sorw
Sorg
max
Sot
max
Sot
,0
0.0
0.0
Sorw ,0
0.2
0.4
0.6
0.8
1.0
So
max ≤ S
Figure 18.20: The krow scanning curves have no hysteresis because Sot
org < Sorw .
Oil Relative Permeability
0.4
1Swr ,krow
krog
0.3
0.2
0.1
Swr
Sorw
Sorg
max
Sot
maxSorg ,0
Sot
,0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
So
max ≤ S
Figure 18.21: The krog scanning curves have no hysteresis because Sot
org < Sorw .
Although the trapped oil does not effect the oil relative permeability hysteresis, the oil relative
permeability curves still change based on changes in the residual oil saturation.
Figure 18.22 illustrates a decreasing scanning curve in black. The two known points on the
A ), points and in Figure 18.15.
scanning curve are (Sor , 0) and (SoA , kro
5
3
• SoA is the oil saturation at the end of the previous cycle, point 3 in Figure 18.22.
A is the oil relative permeability at the end of the previous cycle, point in Figure 18.22.
• kro
3
381
A =S
A
• Sot
o,trap So .
• Calculate the trapped oil
φnm2 A
n
n
= max So,m
,
S
Sot
2
φnt ot
(18.24)
n may have decreased from S prev max based on flash changes and transfer between
Note that Sot
ot
the trapped and mobile systems.
cycle
D
D
W 1 . For the
= Sor
, point = Sorw
• So,min
5 in Figure 18.22. For the first water injection, So,min
D
G1 .
= Sorg
first gas injection, So,min
D
= 1 − Swr , point • So,max
2 in Figure 18.22.
D
• If the current cycle is gas injection, nD
o = nog . If the current cycle is water injection, no = now .
D
D [S A ] = k A .
, point • Solve for kro,max
2 in Figure 18.22, using the constraint that kro
o
ro
D
A
= kro
×
kro,max
D
[So ] =
kro
⎧
0,
⎪
⎨
⎪
⎩
D
kro,max
D
SoA − So,min
(18.25)
D
D
So,max
− So,min
×
−nD
o
D
So − So,min
D
So,max
−
D
So < So,min
n D
o
D
, So ≥ So,min
D
So,min
(18.26)
I
Figure 18.23 illustrates an increasing scanning curve in black. Start by specifying the So,max
I
I
and kro,max
for this scanning curve. Calculate Somin
so that the new scanning curve passes through
A ). The two known points on the scanning curve
the values at the end of the previous cycle (SoA , kro
A ) and (S I
I
are (SoA , kro
3 and 2 in Figure 18.23.
o,max , kro,max ), points • SoA is the oil saturation at the end of the previous cycle, point 3 in Figure 18.23.
A is the oil relative permeability at the end of the previous cycle, point in Figure 18.23.
• kro
3
I
= 1 − Swr , point • So,max
2 in Figure 18.23.
382
Oil Relative Permeability
0.4
1:1Swr ,krow
krow
0.3
D
D
2:Somax
,kromax
Swr
Sorw
0.2
3:SoA ,kroA Sorg
0.1
4:Sorw ,0
D
5:Somin
,0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
So
decr
Figure 18.22: A decreasing oil relative permeability scanning curve is shown in black, 3 −−→ .
5
A
A ). The
This assumes the values at the end of the previous cycle, ,
3 achieved values of (So , kro
decr
decr
reference curve is shown in green, 3 −−→ .
4
1 −−→ I
0 [S I
= kro
• kro,max
2 in Figure 18.23.
omax ] = kro , point • If the current cycle is gas injection, nIo = nog . If the current cycle is water injection, nIo = now .
I [S A ] = k A to obtain S I
• Use the constraint that kro
4 in Figure 18.23.
g
ro
omin , point α=
I
=
Somin
I
[So ] =
kro
18.5
1
nI
o
A
kro
(18.27)
I
kro,max
I
α · Somax
− SoA
α−1
⎧
0,
⎪
⎨
I
⎪
×
⎩ kro,max
(18.28)
I
So − So,min
I
So,max
−
I
So,min
I
So < So,min
nIo
I
, So ≥ So,min
(18.29)
Capillary Pressure
The capillary pressure curves are represented by equations of the following form.
Pc1
S − Smin
= Pc,offset − α(Sx − Smin ) × log
Sx − Smin
383
(18.30)
Oil Relative Permeability
0.4
1:1Swr ,krow
I
I
2:Somax ,kromax krow
0.3
0.2
Swr
Sorw
Sorg
0.1
3:SoA ,kroA 4:Sorw ,0
I
5:Somin
,0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
So
incr
Figure 18.23: An increasing oil relative permeability scanning curve is shown in black, 3 −−→ .
2
A
A ). The
This assumes the values at the end of the previous cycle, ,
3 achieved values of (So , kro
decr
decr
reference curve is shown in green, 3 −−→ .
4
1 −−→ Pc2 = Pc,offset + α(Smax − Sx ) × log
Pcow
⎧
Pcow,max ,
⎪
⎪
⎪
⎪
P
⎪
cow,max ,
⎪
⎨
Pc1 ,
=
Pc2 ,
⎪
⎪
⎪
⎪
⎪
P
,
⎪
⎩ cow,min
Pcow,min ,
Smax − S
Smax − Sx
S ≤ Smin
Pc1 > Pcow,max
Smin < S < Sx
Sx < S < Smax
Pc2 < Pcow,min
S ≥ Smax
(18.31)
(18.32)
The increasing water (imbibition) oil-water capillary pressure data is shifted to correspond to
the Swr measured for the relative permeability data. After this shift, a capillary pressure curve
I0 of the following form is fit to the data. Figure 18.24 illustrates both the increasing capillary
Pcow
pressure curve and the data points.
• Smin = Swr = 0.059
• Sorw = 0.231; Smax = 1 − Sorw
I = 0.28
• Swx
• αIow = 5
384
I
• Pc,offset
= −3.7
• Pcow,min = −20
• Pcow,max = 20
D0 was then estimated so that the capillary
A bounding decreasing capillary pressure curve Pcow
pressure hysteresis is consistent with the literature. Figure 18.24 illustrates the decreasing capillary
pressure curve.
• Smin = Swr = 0.059
• Sorw = 0.231; Smax = 1 − Sorw
D = 0.33
• Swx
• αD
ow = 6.5
D
= −1.7
• Pc,offset
• Pcow,min = −20
• Pcow,max = 20
Scanning curves are calculated based on interpolating between the bounding curves; see Figure 18.25. When switching from increasing to decreasing or decreasing to increasing scanning
curves, the last achieved saturation SA is used in the following interpolation.
S
[S, SA ]
Pcow
=
I0
Pcow
[S]
+
D0
(Pcow
[S]
I0
− Pcow
[S])
×
0
SA − Smin
0
0
Smax
− Smin
(18.33)
Decreasing capillary pressure scanning curves are calculated as illustrated in Figure 18.26. The
following procedure is used.
A based
1. The previous achieved value of SA corresponds to the previous capillary pressure Pcow
on the previous scanning curve, the black star in Figure 18.26.
S [S, S ], the green curve in Figure 18.26.
2. Calculate the new scanning curve Pcow
A
385
OilWater Capillary Pressure
20
Pcow
10
0
10
20
0.0
0.2
0.4
0.6
0.8
1.0
Sw
Figure 18.24: Oil-water capillary pressure curves including the data points. The blue curve and
data points represent increasing water saturation, decreasing oil saturation (imbibition). The red
curve represents decreasing water saturation, increasing oil saturation (drainage).
386
OilWater Capillary Pressure
20
Pcow
10
0
10
20
0.0
0.2
0.4
0.6
0.8
1.0
Sw
Figure 18.25: Capillary pressure bounding curves and interpolated scanning curves. The blue curve
is the bounding increasing water curve. The red curve is the bounding decreasing water curve. The
green curves are interpolated scanning curves.
387
S [S , S ] = P A .
3. Calculate the saturation SB where Pcow
B
A
cow
A ). Define the new
4. Shift the scanning curve to match the points (Smin , Pcow,max ) and (SA , Pcow
increasing scanning curve, the black curve in Figure 18.26, as follows.
D
Pcow
=
S
Pcow
0
Smin
+ (S −
0
Smin
)×
0
SB − Smin
0
SA − Smin
, SA
(18.34)
OilWater Capillary Pressure
20
Swr ,Pmax
cow Pcow
10
0
B
SB ,Pcow
A
S A ,Pcow
10
20
0.0
0.2
0.4
0.6
0.8
1.0
Sw
Figure 18.26: Decreasing capillary pressure scanning curve in black from the blue star to the black
star and back towards the blue star. The blue curve is the bounding increasing water curve. The
red curve is the bounding decreasing water curve. The green curve is the interpolated decreasing
0
and SB is mapped onto the
scanning curve corresponding to SA . The green curve between Smin
0
black curve between Smin to SA .
Increasing capillary pressure scanning curves are calculated as illustrated in Figure 18.27. The
following procedure is used.
388
A based
1. The previous achieved value of SA corresponds to the previous capillary pressure Pcow
on the previous scanning curve, the black star in Figure 18.27.
S [S, S ], the green curve in Figure 18.27.
2. Calculate the new scanning curve Pcow
A
S [S , S ] = P A .
3. Calculate the saturation SB where Pcow
B
A
cow
A ) and (S
4. Shift the scanning curve to match the points (SA , Pcow
max , Pcow,min ). Define the new
decreasing scanning curve, the black curve in Figure 18.27, as follows.
I
Pcow
18.6
=
S
Pcow
SB + (S − SA ) ×
0
− SB
Smax
0
Smax − SA
, SA
(18.35)
Future Test Case Scenarios
Various additional test cases could be run to understand the sensitivities.
1. Easiest to evaluate; no code changes required
1.1. Review additional cases with heterogeneity.
1.2. An alternate reservoir depth of for instance 5000 ft could be used to simulate a reservoir
with similar properties at a shallower depth. The reservoir pressure is based on the
same gradient of 0.508 psia/ft would remain the same, leading to a reservoir pressure
of 2550 psia. The fracture gradient of 0.75 psia/ft would remain the same, leading to a
fracture pressure of 3750 psia. The temperature gradient of 0.019◦ F/ft would remain the
same, leading to a reservoir temperature of 162◦ F. If there is an initial gas saturation,
may need to adjust the krg curves.
1.3. Alternate horizontal grid spacing; for instance 21 × 21 → 42 × 42. Very easy if either
homogeneous or integer multiple of previous model
1.4. Alternate injection rates, production rates, and production schemes: initial waterflood,
initial gas flood, initial WAG.
1.5. Vary WAG ratio and length.
1.6. Evaluate different criteria to switch from primary production to waterflood to gas flood
to WAG.
389
OilWater Capillary Pressure
20
10
Pcow
B
SB ,Pcow
A
S A ,Pcow
0
10
1Sorw ,Pmin
cow 20
0.0
0.2
0.4
0.6
0.8
1.0
Sw
Figure 18.27: Increasing capillary pressure scanning curve in black from the red star to the black
star and back towards the red star. The blue curve is the bounding increasing water curve. The
red curve is the bounding decreasing water curve. The green curve is the interpolated decreasing
0
is mapped onto the
scanning curve corresponding to SA . The green curve between SB and Smax
0
black curve between SA and Smax
390
1.7. Add horizontal anisotropy to the permeability.
2. Moderately easy to evaluate; some small code changes
2.1. Default: change kro , krg , Pcow , Sorw and Sorg everywhere when switch between water
and gas injection. Another option is to switch when the gas/water starts increasing in
a cell.
2.2. For the base case, the krow and krog are very close, so kro [So only]. A different three-phase
relative permeability curve could be used, such as a renormalized Stone II Stone (1973),
a saturation weighted approach such as Baker (1988), or a new published approach.
2.3. An alternate reservoir dip of 1◦ . Need to initialize P as a function of depth.
2.4. Evaluate case with 2 km × 2 km development; for initial tests it was difficult to get
realistic well performance without 10× or more increase in effective permeability
3. More difficult to evaluate; more code changes and testing
3.1. 2-D cross-section or 3-D model. Need to initialize P as a function of depth. May need a
Pcgo . Need a different set of kr and Pc curves for each layer. Adjust the thicknesses of
each layer to match Shibasaki et al. (2006). Need permeability and porosity distribution
for each layer. Vary the vertical permeability based on the presence of stylolites. Use
Zm as constant throughout and temperature as constant.
3.2. Add simulation of a natural fracture system.
3.3. Add horizontal wells.
3.4. Add hydraulically fractured horizontal wells
3.5. Use a tracer to identify when injected gas arrives in a cell (as opposed to solution gas).
Use this to switch relative permeability curves.
391
CHAPTER 19
DISCUSSION OF RESULTS
Different scenarios were created and simulated with varying formulation options expected to
have an impact on the trapped fluids:
• Trapping: model with no compositional trapping (single media system m1 ) versus a model
with compositional trapping (dual media system m1 and m2 ).
• Heterogeneity: homogeneous or heterogeneous.
• Geometry: 2-D 1/4 5-spot pattern or 2-D injector centered 5-spot pattern.
• Mass Transfer: Vary km2 to represent a slower or faster transfer rate between the m1 and m2
systems.
• WCO2 : No aqueous CO2 solubility (WCO2 = 0), or with the most stable and accurate formulation for WCO2 > 0.
• Gas relative permeability hysteresis: No gas relative permeability hysteresis and no gas trapI0 ); with gas relative permeability hysteresis but without the compositional
ping (krg = krg
variations of the trapped gas; or with gas relative permeability hysteresis and compositional
trapped gas.
• Trapped oil: Trapped oil based only on Jerauld (1997), or trapped oil based on Jerauld (1997)
plus an additional 0.10 saturation units immediately after the waterflood.
• Vary Sor : With or without cycle-dependent residual oil saturations. With either option the
Sorw and Sorg are different, but with cycle-dependent residual saturations the Sorw and Sorg
decrease during the first three WAG cycles.
Table 19.1 provides a description of the specific test cases selected with comments on the
purpose of each one. Models with names starting with W are homogeneous 2-D models without
compositional trapping. Models with names starting with X are homogeneous 2-D models with
392
compositional trapping. Models with names starting with Y are heterogeneous 2-D models without
compositional trapping. Models with names starting with Z are heterogeneous 2-D models with
compositional trapping.
Each scenario is run under four different production schemes.
• Primary production (40 acre): Primary production to an economic limit of 10 RBOPD.
• Waterflood (20 acre): Primary production followed by initiation of water injection approximately 180 days after the economic limit for primary production is reached. Simulate the
waterflood until the rate drops again to an economic limit of 10 RBOPD.
• CO2 injection (20 acre): Primary production followed by waterflood followed by initiation of
CO2 injection approximately 360 days after the economic limit for the waterflood is reached.
Simulate the CO2 injection until the rate drops again to an economic limit of 10 RBOPD,
which typically occurs at a minimum point in the CO2 utitlization curve. Production also
typically stops when gas represents 100% of the production. Although this may occur simply
due to the single production phase getting labeled as “gas” rather than “oil”, the producing
compositions confirm that the production is almost all CO2 at this point.
• CO2 WAG (20 acre): Primary production followed by waterflood followed by CO2 injection
followed by initiation of WAG when the oil rate increases again above 10 RBOPD. Simulate
the CO2 WAG injection until the rate drops again to an economic limit of 10 RBOPD.
The scenarios were evaluated based on several different criteria. The most important criterion
is the recovery factor at the economic limit for each production scheme, which was evaluated
at reservoir conditions, but may be flashed to surface conditions in a separate calculation. The
following criteria were evaluated at the economic limit of each production scheme.
• Recovery factor (RB/RB)
• Time to economic limit.
• CO2 storage (lbmol/lbmol)
CO2 storage =
cumulative CO2 injected − cumulative CO2 produced
cumulative CO2 injected
393
(19.1)
Heterogeneity
Transfer: km2
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
yes
no
yes
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
yes
yes
yes
yes
NA
NA
NA
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−7
5 · 10−9
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
5 · 10−5
NA
5 · 10−5
0
>0
0
0
>0
>0
0
0
>0
>0
>0
>0
0
>0
>0
>0
>0
0
0
>0
0
>0
Cycle dependent Sor
Compositional trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
5-spot
5-spot
Extra trapped oil after WF
Pattern
551
561
562
563
564
550
552
553
554
571
572
573
574
575
576
577
578
579
560
570
580
581
Hysteresis of krg
Model #
W
W
W
W
W
X
X
X
X
X
X
X
X
X
X
X
X
X
Y
Z
Y
Z
WCO2 option
Model type
Table 19.1: Description of test case scenarios
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
NA
NA
NA
NA
NA
yes
no
yes
no
yes
yes
yes
yes
yes
yes
no
yes
no
NA
yes
NA
yes
no
no
no
yes
yes
yes
no
yes
no
yes
yes
yes
yes
yes
yes
yes
no
no
no
yes
no
yes
394
Purpose
Least base: single-media + least trapping
Least base + WCO2
Least base + gas hysteresis
Least base + cycle
Single-media: most trapping
Most base: dual-media + with most trapping
Dual-media + mostly gas trap
Dual-media + mostly oil trap
Dual-media + mostly CO2 trap
Most base + lower km2
Most base + much lower km2
Most base + higher km2
Most base: WCO2 = 0
Most base: hysteresis + no trap gas
Most base: no trap gas + no hysteresis
Most base: no extra oil trap after WF
Most base + no cycle Sor
Dual-media: least trapping
Least base + heterogeneity
Most base + heterogeneity
Least base + heterogeneity
Most base + heterogeneity
• CO2 utilization (MCF/RB)
CO2 storage =
cumulative CO2 injected
oil
− cumulative
produced with waterflood
cumulative oil
produced with CO2
(19.2)
• Compositional recovery factor (lbmol/lbmol) for CH4 , nC4 , and nC10
19.1
Evaluation of Primary Production Performance
Table 19.2 presents the recovery factor (RF) and time to economic limit (EL) for primary
production for all of the scenarios. The economic limit for primary production is the same for all
of the homogeneous models without trapping. The economic limit for primary production is the
same for all of the homogeneous models with compositional trapping. The models with trapping
include a 0.01 SU of water and gas during the primary production to stabilize the computations.
This small amount of trapping leads to 90 days of additional time before the economic limit is
reached. For heterogeneous cases, the time to the economic limit varies because of variations in the
porosity, permeability, transmissibility, and flow paths.
The primary recovery factor for the homogeneous models without trapping is the lowest. The
presence of a little bit of trapped water and oil causes more time to pass before the economic limit
is reached and this also corresponds to a larger recovery factor than without trapping.
19.2
Evaluation of Waterflood Performance
Table 19.3 presents the time to economic limit (EL) for primary and waterflood (WF) and the
recovery factor (RF) for the primary and waterflood, plus the incremental time and incremental
recovery. Table 19.3 is ordered based on the incremental time between the economic limit of
primary production and the economic limit of waterflood production. The timing for the end of
primary production was similar for all the models, so the ranking of the economic limit at the end
of the waterflood and the additional days of production between the end of primary production and
the end of the waterflood are the same. The economic limit for the waterflood is reached earliest
for the homogeneous cases without trapping. For the cases with trapping, the economic limit for
the waterflood is reached earliest for the cases with no trapped gas or gas relative permeability
hysteresis. Although the times to the waterflood economic limit varies with the value of km2 , the
changes in the times do not follow an obvious pattern.
395
Heterogeneity
Transfer: km2
no
yes
no
yes
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
no
NA
5 · 10−5
NA
5 · 10−5
NA
NA
NA
NA
NA
5 · 10−9
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−7
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
RF Primary (RCF)
Compositional Trapping
1/4
1/4
5-spot
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
EL Primary (10 days)
Pattern
560
570
580
581
551
561
562
563
564
572
550
552
553
554
571
573
574
575
576
577
578
579
Hysteresis of krg
Model #
Y
Z
Y
Z
W
W
W
W
W
X
X
X
X
X
X
X
X
X
X
X
X
X
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
662
671
680
689
720
720
720
720
720
722
729
729
729
729
729
729
729
729
729
729
729
729
0.192441
0.203522
0.179983
0.190667
0.187443
0.191192
0.191192
0.191192
0.191192
0.199968
0.202195
0.202195
0.202195
0.202195
0.202042
0.202195
0.202195
0.202195
0.202195
0.202195
0.202195
0.202195
WCO2 Option
Model Type
Table 19.2: Primary production: recovery factor and time to economic limit
0
>0
0
>0
0
>0
0
0
>0
>0
>0
0
0
>0
>0
>0
0
>0
>0
>0
>0
0
396
RF WF (RCF)
RF WF − Primary (RCF)
NA
NA
NA
NA
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−9
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
5 · 10−7
5 · 10−5
RF Primary (RCF)
Transfer: km2
yes
no
no
no
no
no
no
no
no
no
no
no
yes
no
no
no
no
no
no
yes
no
yes
Time WF − Primary (10 days)
Heterogeneity
no
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
yes
yes
EL WF (10 days)
Compositional Trapping
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
EL Primary (10 days)
Pattern
580
561
551
563
562
564
576
575
553
579
552
574
581
572
573
550
554
577
578
560
571
570
Hysteresis of krg
Model #
Y
W
W
W
W
W
X
X
X
X
X
X
Z
X
X
X
X
X
X
Y
X
Z
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
hysteresis + no trap
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
680
720
720
720
720
720
729
729
729
729
729
729
689
722
729
729
729
729
729
662
729
671
1489
1539
1564
1564
1572
1573
1606
1646
1668
1668
1896
1896
1915
1999
2002
2054
2054
2054
2054
2073
2215
2194
789
789
814
814
822
823
856
896
918
918
1146
1146
1215
1249
1252
1304
1304
1304
1304
1383
1465
1504
0.179983
0.191192
0.187443
0.191192
0.191192
0.191192
0.202195
0.202195
0.202195
0.202195
0.202195
0.202195
0.190667
0.199968
0.202195
0.202195
0.202195
0.202195
0.202195
0.192441
0.202042
0.203522
0.562401
0.638989
0.603017
0.615077
0.620464
0.643650
0.588463
0.660872
0.570173
0.570173
0.678063
0.678063
0.646166
0.733150
0.650930
0.683429
0.683429
0.683429
0.683429
0.625756
0.726529
0.673626
0.382418
0.447797
0.415574
0.423885
0.429272
0.452458
0.386268
0.458677
0.367978
0.367978
0.475868
0.475868
0.455499
0.533182
0.448735
0.481234
0.481234
0.481234
0.481234
0.433315
0.524487
0.470104
WCO2 Option
Model Type
Table 19.3: Waterflood time to economic limit
0
>0
0
0
0
>0
>0
>0
0
0
0
0
>0
>0
>0
>0
>0
>0
>0
0
>0
>0
no
no
no
no
397
For both the cases with and without compositional trapping, the cases are also ordered from
earliest time to economic limit to latest time to economic limit as follows:
1. WCO2 > 0, no gas relative permeability hysteresis
2. WCO2 > 0, no compositional gas trapping but gas relative permeability hysteresis
3. WCO2 = 0, no gas relative permeability hysteresis
4. WCO2 = 0, with gas trapping and hysteresis
5. WCO2 > 0, with gas trapping and hysteresis
If two cases with compositional trapping are compared, the change in producing time is bigger
than if two cases without compositional trapping are compared. The presence of CO2 solubility in
water makes the difference in the calculation of the gas relative permeability hysteresis much more
significant, especially when compositional trapping effects are considered.
Table 19.4 presents the time to economic limit (EL) for primary and waterflood (WF) and the
recovery factor (RF) for the primary and waterflood, plus the incremental time and incremental
recovery. Table 19.4 is ordered based on the incremental recovery between the economic limit of
primary production and the economic limit of waterflood production. The lowest recovery factor
at the economic limit of the waterflood and also the lowest incremental waterflood recovery over
primary production occurs for the cases with compositional trapping but with no gas hysteresis.
Next are the cases with no compositional trapping. The compositional trapping cases with different
km2 increase their waterflood recovery as the km2 decreases. The cases with WCO2 = 0 have lower
recovery than the cases with WCO2 > 0. The cases with gas relative permeability hysteresis have
higher recoveries at the end of waterflood than cases with no gas relative permeability hysteresis.
For the compositional trapping cases, the gas relative permeability hysteresis has a larger impact
than the WCO2 . For the system without compositional trapping, the WCO2 is more important than
the hysteresis in the gas relative permeability.
19.3
Evaluation of Continuous CO2 Injection
Table 19.5 presents the start time of the waterflood, the start time of the continuous CO2
injection, the time of the increased oil production corresponsding to CO2 response, and the time
398
RF WF (RCF)
RF WF − Primary (RCF)
5 · 10−5
5 · 10−5
NA
5 · 10−5
NA
NA
NA
NA
NA
5 · 10−3
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−7
5 · 10−9
RF Primary (RCF)
Transfer: km2
no
no
yes
no
no
no
no
yes
no
no
no
yes
no
yes
no
no
no
no
no
no
no
no
Time WF − Primary (10 days)
Heterogeneity
yes
yes
no
yes
no
no
no
no
no
yes
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
EL WF (10 days)
Compositional Trapping
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
EL Primary (10 days)
Pattern
553
579
580
576
551
563
562
560
561
573
564
581
575
570
552
574
550
554
577
578
571
572
Hysteresis of krg
Model #
X
X
Y
X
W
W
W
Y
W
X
W
Z
X
Z
X
X
X
X
X
X
X
X
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
729
729
680
729
720
720
720
662
720
729
720
689
729
671
729
729
729
729
729
729
729
722
1668
1668
1489
1606
1564
1564
1572
2073
1539
2002
1573
1915
1646
2194
1896
1896
2054
2054
2054
2054
2215
1999
918
918
789
856
814
814
822
1383
789
1252
823
1215
896
1504
1146
1146
1304
1304
1304
1304
1465
1249
0.202195
0.202195
0.179983
0.202195
0.187443
0.191192
0.191192
0.192441
0.191192
0.202195
0.191192
0.190667
0.202195
0.203522
0.202195
0.202195
0.202195
0.202195
0.202195
0.202195
0.202042
0.199968
0.570173
0.570173
0.562401
0.588463
0.603017
0.615077
0.620464
0.625756
0.638989
0.650930
0.643650
0.646166
0.660872
0.673626
0.678063
0.678063
0.683429
0.683429
0.683429
0.683429
0.726529
0.733150
0.367978
0.367978
0.382418
0.386268
0.415574
0.423885
0.429272
0.433315
0.447797
0.448735
0.452458
0.455499
0.458677
0.470104
0.475868
0.475868
0.481234
0.481234
0.481234
0.481234
0.524487
0.533182
WCO2 Option
Model Type
Table 19.4: Waterflood recovery factor
0
0
0
>0
0
0
0
0
>0
>0
>0
>0
>0
>0
0
0
>0
>0
>0
>0
>0
>0
no
no
no
no
no
no
399
Model Type
Model #
Pattern
Compositional Trapping
Heterogeneity
Transfer: km2
WCO2 Option
Hysteresis of krg
Extra Trapped Oil After WF
Start WF (10 days)
Start GF (10 days)
Oil Response GF (10 days)
Economic Limit GF (10 days)
Time GF − WF (10 days)
Time GF − GF response (10 days)
RF WF (RCF)
RF GF (RCF)
RF GF − WF (RCF)
Table 19.5: Continuous CO2 recovery factor
X
X
X
X
X
X
Z
X
X
X
X
W
W
Y
W
W
552
574
573
578
550
553
570
575
576
577
579
563
551
560
564
561
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
no
no
no
no
no
no
no
no
yes
no
no
no
no
no
no
yes
no
no
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
0
0
>0
>0
>0
0
>0
>0
>0
>0
0
0
0
0
>0
>0
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no
yes
yes
yes
yes
yes
yes
yes
yes
no
no
NA
NA
NA
NA
NA
750
750
750
750
750
750
690
750
750
750
750
750
750
690
750
750
1930
1930
2040
2090
2070
1700
2230
1680
1640
2090
1700
1600
1580
2110
1610
1570
2774
2314
2468
2524
2497
2272
2672
2299
2211
2505
2241
2172
2145
2629
2105
2055
2785
2352
2525
2580
2552
2382
2848
2464
2381
2628
2545
2574
2545
3182
2609
2555
844
384
428
434
427
572
442
619
571
415
541
572
565
519
495
485
11
38
57
56
55
110
176
165
170
123
304
402
400
553
504
500
0.678063
0.678063
0.650930
0.683429
0.683429
0.570173
0.673626
0.660872
0.588463
0.683429
0.570173
0.615077
0.603017
0.625756
0.643650
0.638989
0.691281
0.698792
0.680371
0.713990
0.714275
0.619315
0.724226
0.732055
0.660004
0.757512
0.749044
0.846343
0.847951
0.883803
0.907230
0.906359
0.013218
0.020729
0.029441
0.030561
0.030846
0.049142
0.050600
0.071183
0.071541
0.074083
0.178871
0.231266
0.244934
0.258047
0.263580
0.267370
400
to the economic limit (EL) of the CO2 flood. Table 19.5 also presents the recovery factor (RF) for
the waterflood (WF), continuous CO2 injection gas flood (GF), and the incremental recovery due
to the CO2 flood. Table 19.5 is ordered based on the incremental recovery between the waterflood
and the gas flood.
The system without compositional trapping has significantly higher recoveries at the economic
limit of CO2 injection than the cases which account for trapping. This is true both for the incremental recovery above the waterflood and for the total recovery from the start of the simulation.
For both the cases with and without compositional trapping, accounting for the CO2 solubility in
water increases the continuous CO2 recovery factor. For both the cases with and without compositional trapping, gas relative permeability hysteresis decreases the continuous CO2 recovery factor.
Gas relative permeability hysteresis is more significant than the WCO2 for a system with compositional trapping, but WCO2 is more significant than krg hysteresis for systems without compositional
trapping.
For the system with dual-media compositional trapping, the incremental recovery from CO2
injection over the waterflood case varies between 0.01 and 0.18. The presence or absence of additional trapped oil after the waterflood causes a large amount of variability in the recovery factor
but does not follow a trend.
Table 19.6 presents the start time of the waterflood, the start time of the continuous CO2
injection, the time of the increased oil production corresponsding to CO2 response, and the time
to the economic limit (EL) of the CO2 flood. Table 19.6 also presents the recovery factor (RF) for
the waterflood (WF), continuous CO2 injection gas flood (GF) and the incremental recovery due
to the CO2 flood. Table 19.6 is ordered based on the time between the start of CO2 injection and
the time of the CO2 response.
The time from the end of waterflood to the increase in oil production corresponding to CO2
response varies between 3840 days and 8440 days. The results for models with compositional
dual-media trapping and without trapping are intermingled in CO2 response time. Cases with
gas relative permeability hysteresis have faster response times than cases without. Cases with
no compositional gas trapping but with gas relative permeability hysteresis seem to have longer
response times than any of the other cases, but since only one case was run with this option it is
difficult to evaluate.
401
Model Type
Model #
Pattern
Compositional Trapping
Heterogeneity
Transfer: km2
WCO2 Option
Hysteresis of krg
Extra Trapped Oil After WF
Start WF (10 days)
Start GF (10 days)
Oil Response GF (10 days)
Economic Limit GF (10 days)
Time GF − WF (10 days)
Time GF − GF response (10 days)
RF WF (RCF)
RF GF (RCF)
RF GF − WF (RCF)
Table 19.6: Continuous CO2 response time
X
X
X
X
X
Z
W
W
W
Y
X
W
X
W
X
X
X
574
577
550
573
578
570
561
562
564
560
579
551
576
563
553
575
552
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
yes
yes
yes
yes
yes
yes
no
no
no
no
yes
no
yes
no
yes
yes
yes
no
no
no
no
no
yes
no
no
no
yes
no
no
no
no
no
no
no
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
NA
NA
NA
NA
5 · 10−5
NA
5 · 10−5
NA
5 · 10−5
5 · 10−5
5 · 10−5
0
>0
>0
>0
>0
>0
>0
0
>0
0
0
0
>0
0
0
>0
0
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
hysteresis + no trap
trap + hysteresis
yes
no
yes
yes
yes
yes
NA
NA
NA
NA
no
NA
yes
NA
yes
yes
no
750
750
750
750
750
690
750
750
750
690
750
750
750
750
750
750
750
1930
2090
2070
2040
2090
2230
1570
1610
1610
2110
1700
1580
1640
1600
1700
1680
1930
2314
2505
2497
2468
2524
2672
2055
2100
2105
2629
2241
2145
2211
2172
2272
2299
2774
2352
2628
2552
2525
2580
2848
2555
NA
2609
3182
2545
2545
2381
2574
2382
2464
2785
384
415
427
428
434
442
485
490
495
519
541
565
571
572
572
619
844
38
123
55
57
56
176
500
NA
504
553
304
400
170
402
110
165
11
0.678063
0.683429
0.683429
0.650930
0.683429
0.673626
0.638989
0.620464
0.643650
0.625756
0.570173
0.603017
0.588463
0.615077
0.570173
0.660872
0.678063
0.698792
0.757512
0.714275
0.680371
0.713990
0.724226
0.906359
0.626867
0.907230
0.883803
0.749044
0.847951
0.660004
0.846343
0.619315
0.732055
0.691281
0.020729
0.074083
0.030846
0.029441
0.030561
0.050600
0.267370
0.006403
0.263580
0.258047
0.178871
0.244934
0.071541
0.231266
0.049142
0.071183
0.013218
402
Model Type
Model #
Pattern
Compositional Trapping
Heterogeneity
Transfer: km2
WCO2 Option
Hysteresis of krg
Extra Trapped Oil After WF
Start WF (10 days)
Start GF (10 days)
Oil Response GF (10 days)
Economic Limit GF (10 days)
Time GF − WF (10 days)
Time GF − GF response (10 days)
RF WF (RCF)
RF GF (RCF)
RF GF − WF (RCF)
Table 19.7: Continuous CO2 response duration
X
X
X
X
X
X
X
X
X
Z
X
W
W
W
W
Y
552
574
550
578
573
553
577
575
576
570
579
551
563
561
564
560
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
no
no
no
no
no
no
no
no
no
no
no
yes
no
no
no
no
no
yes
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
0
0
>0
>0
>0
0
>0
>0
>0
>0
0
0
0
>0
>0
0
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
NA
NA
NA
NA
NA
750
750
750
750
750
750
750
750
750
690
750
750
750
750
750
690
1930
1930
2070
2090
2040
1700
2090
1680
1640
2230
1700
1580
1600
1570
1610
2110
2774
2314
2497
2524
2468
2272
2505
2299
2211
2672
2241
2145
2172
2055
2105
2629
2785
2352
2552
2580
2525
2382
2628
2464
2381
2848
2545
2545
2574
2555
2609
3182
844
384
427
434
428
572
415
619
571
442
541
565
572
485
495
519
11
38
55
56
57
110
123
165
170
176
304
400
402
500
504
553
0.678063
0.678063
0.683429
0.683429
0.650930
0.570173
0.683429
0.660872
0.588463
0.673626
0.570173
0.603017
0.615077
0.638989
0.643650
0.625756
0.691281
0.698792
0.714275
0.713990
0.680371
0.619315
0.757512
0.732055
0.660004
0.724226
0.749044
0.847951
0.846343
0.906359
0.907230
0.883803
0.013218
0.020729
0.030846
0.030561
0.029441
0.049142
0.074083
0.071183
0.071541
0.050600
0.178871
0.244934
0.231266
0.267370
0.263580
0.258047
403
Table 19.7 presents the start time of the waterflood, the start time of the continuous CO2
injection, the time of the increased oil production corresponsding to CO2 response, and the time
to the economic limit (EL) of the CO2 flood. Table 19.7 also presents the recovery factor (RF) for
the waterflood (WF), continuous CO2 injection gas flood (GF), and the incremental recovery due
to the CO2 flood. Table 19.7 is ordered based on the time between the time of the CO2 response
and the time when the economic limit is reached.
The time from the continuous CO2 response until the economic limit of 10 RBOPD varies
between 380 days and 5040 days. For the cases with no compositional trapping the response lasts
significantly longer than when dual-media compositional trapping is considered. For cases with
compositional trapping, accounting for WCO2 increases the response time; some of the injected
CO2 is stored in the water phase. For cases with no compositional trapping, accounting for WCO2
decreases response time. For cases with compositional trapping, gas relative permeability hysteresis
decreases the response time. For cases with no compositional trapping, gas relative permeability
hysteresis increases response time.
19.4
Evaluation of CO2 WAG
Water-alternating-gas injection is used for several different reasons in field development. The
cost of water is often cheaper than the cost of CO2 ; as a result if the recovery is similar between gas
injection and WAG injection it is often cheaper to operate a WAG flood. WAG also helps to lower
the amount of produced gas, which decreases the cost for processing the gas in order to extract the
CO2 for re-injection. WAG helps control the mobility of the fluids; this causes the CO2 to follow a
different path through the reservoir. Mobility control helps both areal and vertical sweep efficiency.
With more heterogeneity, mobility control becomes more important. Mobility control is also very
important in thicker reservoirs where gas override is a larger problem. If there is cycle-dependent
Sor , then WAG also performs better than continuous CO2 injection. Vertical 2D cross sections or
3D cases would emphasize the differences between WAG and continuous CO2 injection, but these
cases were not simulated here.
The CO2 WAG cases start with primary production to the economic limit, followed by waterflood to the economic limit, followed by continuous CO2 injection until response is observed, followed by water-alternating-gas injection. In the cases described here, water is injected for 200 days
404
followed by gas for 200 days, followed by repeating cycles of the same length. It is typical to start
industry WAG at a 1 : 1 ratio, either by time or volume. Changes may then be made based on
observations of the wells and varying CO2 or water supply.
Table 19.8 presents the recovery factors (RF) for the waterflood (WF), continuous CO2 gas
flood (GF), and CO2 WAG plus the incremental recovery factors of gas flood versus waterflood,
WAG versus waterflood, and WAG versus continuous CO2 injection. Table 19.8 is ordered based
on the incremental recovery of the WAG flood versus the waterflood.
The cases without compositional trapping have consistently higher recovery factors after the
waterflood as a result of combined CO2 injection and WAG. Gas relative permeability hysteresis
also significantly decreases the effectiveness of CO2 and WAG injection for the cases with dualmedia compositional trapping. More trapped oil after the waterflood decreases the effectiveness of
CO2 and WAG injection. Although CO2 solubility in water causes variations in the response to
CO2 and WAG injection, the models do not follow an observable trend.
Table 19.9 presents the recovery factors (RF) for the waterflood (WF), continuous CO2 gas
flood (GF), and CO2 WAG plus the incremental recovery factors of gas flood versus waterflood,
WAG versus waterflood, and WAG versus continuous CO2 injection. Table 19.9 is ordered based
on the difference between the WAG flood recovery and the continuous CO2 flood recovery.
Comparing the incremental recovery after waterflood, some of the models have more incremental recovery from continuous CO2 injection and some have more incremental recovery from CO2
injection followed by CO2 WAG. The effects likely to most significantly effect WAG versus continuous CO2 injection include heterogeneity, 3D gravity effects, economics, and operational flexibility.
Although not many heterogeneous simulations were conducted, the largest observed incremental
oil recovery from WAG is for case Y560.
Table 19.10 presents the start times of the waterflood (WF), continuous CO2 gas flood (GF),
and WAG with the economic limits (EL) for each production phase. Table 19.10 is ordered based
on the difference between the WAG flood recovery and the continuous CO2 flood recovery.
The amount of time between the start of WAG and reaching the economic limit varies significantly between 120 days and 5610 days. There are some models that have no incremental production from WAG and some models that have more than 6000 days of incremental production from
WAG, but there were computational difficulties with the models at both extremes that may mask
405
RF GF − WF (RCF)
RF WAG − WF (RCF)
RF WAG − GF (RCF)
>0
0
0
>0
>0
>0
>0
>0
0
>0
>0
0
>0
0
0
>0
0
RF WAG (RCF)
5 · 10−5
NA
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
RF GF (RCF)
Transfer: km2
no
no
no
no
no
no
yes
no
no
no
no
no
no
no
no
no
yes
RF WF (RCF)
Heterogeneity
yes
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
no
no
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
554
562
574
573
550
578
570
577
553
575
576
579
564
563
551
561
560
Hysteresis of krg
Model #
X
W
X
X
X
X
Z
X
X
X
X
X
W
W
W
W
Y
WCO2 Option
Model Type
Table 19.8: WAG recovery factor
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
hysteresis + no trap
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no
NA
yes
yes
yes
yes
yes
no
yes
yes
yes
no
NA
NA
NA
NA
NA
no
no
yes
yes
yes
no
yes
yes
yes
yes
yes
no
yes
yes
no
no
no
0.683429
0.620464
0.678063
0.650930
0.683429
0.683429
0.673626
0.683429
0.570173
0.660872
0.588463
0.570173
0.643650
0.615077
0.603017
0.638989
0.625756
0.684152
0.626867
0.698792
0.680371
0.714275
0.713990
0.724226
0.757512
0.619315
0.732055
0.660004
0.749044
0.907230
0.846343
0.847951
0.906359
0.883803
0.684133
0.630089
0.690718
0.676915
0.709605
0.713955
0.724035
0.735339
0.624546
0.736300
0.664551
0.750321
0.863659
0.847772
0.853358
0.914345
0.911696
0.000723
0.006403
0.020729
0.029441
0.030846
0.030561
0.050600
0.074083
0.049142
0.071183
0.071541
0.178871
0.263580
0.231266
0.244934
0.267370
0.258047
0.000704
0.009625
0.012655
0.025985
0.026176
0.030526
0.050409
0.051910
0.054373
0.075428
0.076088
0.180148
0.220009
0.232695
0.250341
0.275356
0.285940
-0.000019
0.003222
-0.008074
-0.003456
-0.004670
-0.000035
-0.000191
-0.022173
0.005231
0.004245
0.004547
0.001277
-0.043571
0.001429
0.005407
0.007986
0.027893
406
RF GF − WF (RCF)
RF WAG − WF (RCF)
RF WAG − GF (RCF)
>0
>0
0
>0
>0
>0
>0
>0
0
0
0
>0
>0
0
0
>0
0
RF WAG (RCF)
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
RF GF (RCF)
Transfer: km2
no
no
no
no
no
yes
no
no
no
no
no
no
no
no
no
no
yes
RF WF (RCF)
Heterogeneity
no
yes
yes
yes
yes
yes
yes
yes
yes
no
no
yes
yes
yes
no
no
no
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
564
577
574
550
573
570
578
554
579
563
562
575
576
553
551
561
560
Hysteresis of krg
Model #
W
X
X
X
X
Z
X
X
X
W
W
X
X
X
W
W
Y
WCO2 Option
Model Type
Table 19.9: WAG recovery factor versus continuous CO2 recovery factor
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
NA
no
yes
yes
yes
yes
yes
no
no
NA
NA
yes
yes
yes
NA
NA
NA
yes
yes
yes
yes
yes
yes
no
no
no
yes
no
yes
yes
yes
no
no
no
0.643650
0.683429
0.678063
0.683429
0.650930
0.673626
0.683429
0.683429
0.570173
0.615077
0.620464
0.660872
0.588463
0.570173
0.603017
0.638989
0.625756
0.907230
0.757512
0.698792
0.714275
0.680371
0.724226
0.713990
0.684152
0.749044
0.846343
0.626867
0.732055
0.660004
0.619315
0.847951
0.906359
0.883803
0.863659
0.735339
0.690718
0.709605
0.676915
0.724035
0.713955
0.684133
0.750321
0.847772
0.630089
0.736300
0.664551
0.624546
0.853358
0.914345
0.911696
0.263580
0.074083
0.020729
0.030846
0.029441
0.050600
0.030561
0.000723
0.178871
0.231266
0.006403
0.071183
0.071541
0.049142
0.244934
0.267370
0.258047
0.220009
0.051910
0.012655
0.026176
0.025985
0.050409
0.030526
0.000704
0.180148
0.232695
0.009625
0.075428
0.076088
0.054373
0.250341
0.275356
0.285940
-0.043571
-0.022173
-0.008074
-0.004670
-0.003456
-0.000191
-0.000035
-0.000019
0.001277
0.001429
0.003222
0.004245
0.004547
0.005231
0.005407
0.007986
0.027893
407
Economic Limit GF (10 days)
Economic Limit WAG (10 days)
Time WAG − GF (10 days)
>0
0
>0
>0
>0
0
>0
>0
>0
>0
0
0
>0
0
0
>0
Oil Response GF (10 days)
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
Start GF (10 days)
Transfer: km2
no
no
no
no
no
no
no
no
no
yes
no
no
no
no
yes
no
Start WF (10 days)
Heterogeneity
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
no
no
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
554
574
578
573
550
553
577
576
575
570
579
551
561
563
560
564
Hysteresis of krg
Model #
X
X
X
X
X
X
X
X
X
Z
X
W
W
W
Y
W
WCO2 Option
Model Type
Table 19.10: WAG response duration
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
hysteresis + no trap
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
NA
NA
NA
NA
NA
no
yes
no
yes
yes
yes
yes
yes
yes
yes
no
no
no
yes
no
yes
750
750
750
750
750
750
750
750
750
690
750
750
750
750
690
750
2090
1930
2090
2040
2070
1700
2090
1640
1680
2230
1700
1580
1570
1600
2110
1610
2058
2314
2524
2468
2497
2272
2505
2211
2299
2672
2241
2145
2055
2172
2629
2105
2070
2352
2580
2525
2552
2382
2628
2381
2464
2848
2545
2545
2555
2574
3182
2609
2070
2355
2600
2588
2619
2410
2705
2418
2509
2946
2550
2530
2537
2667
3190
2705
12
41
76
120
122
138
200
207
210
274
309
385
482
495
561
600
408
the actual production performance. For the successful models, the models without compositional
trapping have consistently longer incremental production from WAG. Although the calculation of
WCO2 , the presence or absence of gas relative permeability hysteresis, the presence or absence of
additional trapped oil after the waterflood, and cycle-dependent residual oil saturations change the
amount of time of incremental production, there is no trend in how these change for the test cases
simulated here.
19.5
Evaluation of Compositional Recovery Factor
Table 19.11, Table 19.12, and Table 19.13 present the compositional recovery factors for CH4 ,
nC4 , and nC10 and the difference between nC10 and CH4 for the waterflood (WF), continuous CO2
gas flood (GF), and WAG. The trends for waterflood, continuous CO2 injection, and WAG are all
the same; Table 19.11, Table 19.12, and Table 19.13 would be combined in one table if it could
fit on one page. Table 19.11 is ordered based on the difference between the nC10 and CH4 for
the waterflood. Table 19.12 is ordered based on the difference between the nC10 and CH4 for the
continuous CO2 gas flood. Table 19.13 is ordered based on the difference between the nC10 and
CH4 for WAG.
The compositional recovery factors represent the number of moles of methane, butane, or decane
produced as a fraction of the original number of moles in the reservoir. The compositional variation
follows the same trend for the waterflood, CO2 flood, and WAG flood. All of the models without
compositional trapping and none of the models with compositional trapping have approximately the
same recovery for methane, butane, and decane. All the models with compositional trapping that
include gas relative permeability hysteresis have more decane production than methane production.
All the models with compositional trapping but with no gas relative permeability hysteresis have
more methane production than decane production. Decreasing the km2 causes an increase in the
difference between decane production and methane production because it is more difficult for the
methane in the trapped gas to move back into the mobile fluid.
19.6
Evaluation of CO2 Storage
Table 19.14 and Table 19.15 present the CO2 storage and CO2 utilization for continuous CO2
injection and WAG. Table 19.14 is ordered based on the amount of CO2 storage at the economic
409
RF nC10 − CH4 WAG
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
RF nC10 − CH4 GF
no
no
no
no
no
no
no
no
RF nC10 − CH4 WF
0
0
>0
0
0
>0
0
0
0
>0
>0
>0
>0
>0
0
0
>0
>0
>0
>0
>0
>0
RF WF (lbmol nC10 )
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
NA
NA
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−9
5 · 10−7
RF WF (lbmol nC4 )
Transfer: km2
no
no
no
no
no
no
yes
yes
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
RF WF (lbmol CH4 )
Heterogeneity
yes
yes
yes
no
no
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
553
579
576
551
563
561
560
580
562
564
573
575
570
581
552
574
550
554
577
578
572
571
Hysteresis of krg
Model #
X
X
X
W
W
W
Y
Y
W
W
X
X
Z
Z
X
X
X
X
X
X
X
X
WCO2 Option
Model Type
Table 19.11: Compositional recovery factor for waterflood
yes
no
yes
NA
NA
NA
NA
NA
NA
NA
yes
yes
yes
yes
no
yes
yes
no
no
yes
yes
yes
yes
no
yes
no
yes
no
no
no
no
yes
yes
yes
yes
yes
no
yes
yes
no
yes
no
yes
yes
0.674197
0.674197
0.668508
0.648968
0.648968
0.657580
0.636077
0.641361
0.641383
0.650195
0.611407
0.596050
0.540548
0.630521
0.546910
0.546910
0.564804
0.564804
0.564804
0.564804
0.464729
0.481982
0.644331
0.644331
0.659455
0.643662
0.643662
0.656146
0.634787
0.643041
0.644134
0.656060
0.674177
0.677093
0.632271
0.717750
0.641899
0.641899
0.660843
0.660843
0.660843
0.660843
0.641281
0.665055
0.637915
0.637915
0.658303
0.642873
0.642873
0.655923
0.634527
0.643449
0.644470
0.656853
0.683346
0.690110
0.635686
0.731673
0.655798
0.655798
0.674870
0.674870
0.674870
0.674870
0.667150
0.691852
-0.036282
-0.036282
-0.010205
-0.006095
-0.006095
-0.001657
-0.001550
0.002088
0.003087
0.006658
0.071939
0.094060
0.095138
0.101152
0.108888
0.108888
0.110066
0.110066
0.110066
0.110066
0.202421
0.209870
-0.037973
-0.031835
-0.012460
-0.003265
-0.003333
-0.000086
-0.001264
-0.004327
-0.004739
0.003705
0.059675
0.072808
0.110104
0.062464
0.109196
0.113115
0.115095
0.109401
0.120713
0.115175
0.230762
0.229581
-0.037967
-0.032329
-0.012478
-0.003228
-0.003329
-0.000069
-0.001180
NA
0.001640
0.004633
0.068333
0.072551
0.136710
NA
0.171066
0.111579
0.107757
0.110748
0.097278
0.113153
0.230764
0.230495
410
RF nC10 − CH4 WF
RF nC10 − CH4 GF
RF nC10 − CH4 WAG
0
0
>0
0
0
0
0
0
>0
>0
>0
>0
>0
0
>0
>0
0
>0
>0
>0
>0
>0
RF WF (lbmol nC10 )
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
NA
NA
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−7
5 · 10−9
RF WF (lbmol nC4 )
Transfer: km2
no
no
no
no
yes
no
no
yes
no
no
no
yes
no
no
no
yes
no
no
no
no
no
no
RF WF (lbmol CH4 )
Heterogeneity
yes
yes
yes
no
no
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
5-spot
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
553
579
576
562
580
563
551
560
561
564
573
581
575
552
554
570
574
550
578
577
571
572
Hysteresis of krg
Model #
X
X
X
W
Y
W
W
Y
W
W
X
Z
X
X
X
Z
X
X
X
X
X
X
WCO2 Option
Model Type
Table 19.12: Compositional recovery factor for continuous CO2 injection
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
yes
no
yes
NA
NA
NA
NA
NA
NA
NA
yes
yes
yes
no
no
yes
yes
yes
yes
no
yes
yes
yes
no
yes
no
no
yes
no
no
no
yes
yes
yes
yes
no
no
yes
yes
yes
no
yes
yes
yes
0.705464
0.803976
0.713648
0.656507
0.653966
0.829678
0.832960
0.866301
0.865994
0.864346
0.674242
0.749390
0.656086
0.558095
0.567815
0.564331
0.552456
0.568922
0.568981
0.598775
0.441645
0.409479
0.674068
0.777490
0.702625
0.652391
0.650930
0.826777
0.830118
0.865216
0.865923
0.867607
0.726323
0.804886
0.718585
0.654891
0.663246
0.660318
0.651136
0.669351
0.669479
0.704106
0.641932
0.610758
0.667491
0.772141
0.701188
0.651768
0.649639
0.826345
0.829695
0.865037
0.865908
0.868051
0.733917
0.811854
0.728894
0.667291
0.677216
0.674435
0.665571
0.684017
0.684156
0.719488
0.671226
0.640241
-0.036282
-0.036282
-0.010205
0.003087
0.002088
-0.006095
-0.006095
-0.001550
-0.001657
0.006658
0.071939
0.101152
0.094060
0.108888
0.110066
0.095138
0.108888
0.110066
0.110066
0.110066
0.209870
0.202421
-0.037973
-0.031835
-0.012460
-0.004739
-0.004327
-0.003333
-0.003265
-0.001264
-0.000086
0.003705
0.059675
0.062464
0.072808
0.109196
0.109401
0.110104
0.113115
0.115095
0.115175
0.120713
0.229581
0.230762
-0.037967
-0.032329
-0.012478
0.001640
NA
-0.003329
-0.003228
-0.001180
-0.000069
0.004633
0.068333
NA
0.072551
0.171066
0.110748
0.136710
0.111579
0.107757
0.113153
0.097278
0.230495
0.230764
411
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
RF nC10 − CH4 WAG
no
no
no
no
no
no
no
RF nC10 − CH4 GF
0
0
>0
0
0
0
>0
0
>0
>0
>0
>0
>0
>0
0
>0
>0
0
>0
>0
RF nC10 − CH4 WF
WCO2 Option
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
NA
NA
NA
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−7
5 · 10−9
RF WF (lbmol nC10 )
Transfer: km2
no
no
no
no
no
yes
no
no
no
no
no
no
no
no
no
no
yes
no
no
no
RF WF (lbmol nC4 )
Heterogeneity
yes
yes
yes
no
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
RF WF (lbmol CH4 )
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Cycle dependent Sor
Pattern
553
579
576
563
551
560
561
562
564
573
575
577
550
554
574
578
570
552
571
572
Extra Trapped Oil After WF
Model #
X
X
X
W
W
Y
W
W
W
X
X
X
X
X
X
X
Z
X
X
X
Hysteresis of krg
Model Type
Table 19.13: Compositional recovery factor for WAG
yes
no
yes
NA
NA
NA
NA
NA
NA
yes
yes
no
yes
no
yes
yes
yes
no
yes
yes
yes
no
yes
yes
no
no
no
no
yes
yes
yes
yes
yes
no
yes
no
yes
no
yes
yes
0.706385
0.802514
0.713444
0.829989
0.834826
0.890282
0.869894
0.654133
0.845257
0.675334
0.657157
0.618137
0.610048
0.563680
0.554238
0.577063
0.569071
0.521725
0.438057
0.409475
0.674992
0.775724
0.702405
0.827096
0.832016
0.889393
0.869838
0.655605
0.849335
0.734113
0.719434
0.709026
0.706101
0.660314
0.651575
0.675877
0.683153
0.669230
0.639127
0.610755
0.668418
0.770185
0.700966
0.826660
0.831598
0.889102
0.869825
0.655773
0.849890
0.743667
0.729708
0.715415
0.717805
0.674428
0.665817
0.690216
0.705781
0.692791
0.668552
0.640239
-0.036282
-0.036282
-0.010205
-0.006095
-0.006095
-0.001550
-0.001657
0.003087
0.006658
0.071939
0.094060
0.110066
0.110066
0.110066
0.108888
0.110066
0.095138
0.108888
0.209870
0.202421
-0.037973
-0.031835
-0.012460
-0.003333
-0.003265
-0.001264
-0.000086
-0.004739
0.003705
0.059675
0.072808
0.120713
0.115095
0.109401
0.113115
0.115175
0.110104
0.109196
0.229581
0.230762
-0.037967
-0.032329
-0.012478
-0.003329
-0.003228
-0.001180
-0.000069
0.001640
0.004633
0.068333
0.072551
0.097278
0.107757
0.110748
0.111579
0.113153
0.136710
0.171066
0.230495
0.230764
412
Utilization GF (MCF/RB)
Utilization WAG (MCF/RB)
Utilization WAG − GF (MCF/RB)
0
>0
>0
0
>0
>0
0
0
>0
0
>0
>0
>0
>0
>0
0
0
Storage WAG − GF (lbmol)
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
5 · 10−5
NA
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
Storage WAG (lbmol)
Transfer: km2
yes
no
yes
no
no
no
no
no
no
no
no
no
no
no
no
no
no
Storage GF (lbmol)
Heterogeneity
no
yes
yes
yes
yes
no
yes
no
no
no
yes
yes
yes
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
560
554
570
552
577
561
579
551
564
563
578
550
576
575
573
553
574
Hysteresis of krg
Model #
Y
X
Z
X
X
W
X
W
W
W
X
X
X
X
X
X
X
WCO2 Option
Model Type
Table 19.14: CO2 storage for continuous CO2 injection
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
hysteresis + no trap
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
NA
no
yes
no
no
NA
no
NA
NA
NA
yes
yes
yes
yes
yes
yes
yes
no
no
yes
no
yes
no
no
no
yes
yes
no
yes
yes
yes
yes
yes
yes
0.479
0.492
0.702
0.805
0.864
0.870
0.872
0.877
0.877
0.879
0.890
0.894
0.899
0.906
0.914
0.916
0.918
0.398
NA
0.804
0.805
0.894
0.829
0.851
0.844
0.893
0.845
0.883
0.891
0.884
0.894
0.921
0.903
0.931
-0.081
NA
0.102
0.000
0.030
-0.040
-0.021
-0.033
0.015
-0.033
-0.006
-0.003
-0.016
-0.012
0.007
-0.014
0.013
7.06
684.12
16.93
289.64
9.09
4.38
5.68
4.89
4.48
5.07
24.22
24.51
11.98
13.57
27.61
17.79
43.60
4.04
150.71
11.27
NA
11.43
3.25
4.60
3.75
4.15
3.93
22.48
30.63
10.16
11.58
31.78
14.63
115.99
-3.02
-533.41
-5.66
NA
2.34
-1.13
-1.08
-1.14
-0.33
-1.14
-1.74
6.12
-1.83
-1.99
4.17
-3.15
72.39
413
Utilization GF (MCF/RB)
Utilization WAG (MCF/RB)
Utilization WAG − GF (MCF/RB)
0
>0
0
>0
0
0
0
>0
>0
>0
>0
>0
>0
0
>0
0
Storage WAG − GF (lbmol)
NA
5 · 10−5
5 · 10−5
NA
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
Storage WAG (lbmol)
Transfer: km2
yes
yes
no
no
no
no
no
no
no
no
no
no
no
no
no
no
Storage GF (lbmol)
Heterogeneity
no
yes
yes
no
no
no
yes
yes
yes
yes
no
yes
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
560
570
552
561
551
563
579
578
576
550
564
577
575
553
573
574
Hysteresis of krg
Model #
Y
Z
X
W
W
W
X
X
X
X
W
X
X
X
X
X
WCO2 Option
Model Type
Table 19.15: CO2 storage for WAG injection
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
NA
yes
no
NA
NA
NA
no
yes
yes
yes
NA
no
yes
yes
yes
yes
no
yes
no
no
no
yes
no
no
yes
yes
yes
yes
yes
yes
yes
yes
0.479
0.702
0.805
0.870
0.877
0.879
0.872
0.890
0.899
0.894
0.877
0.864
0.906
0.916
0.914
0.918
0.398
0.804
0.805
0.829
0.844
0.845
0.851
0.883
0.884
0.891
0.893
0.894
0.894
0.903
0.921
0.931
-0.081
0.102
0.000
-0.040
-0.033
-0.033
-0.021
-0.006
-0.016
-0.003
0.015
0.030
-0.012
-0.014
0.007
0.013
7.06
16.93
289.64
4.38
4.89
5.07
5.68
24.22
11.98
24.51
4.48
9.09
13.57
17.79
27.61
43.60
4.04
11.27
NA
3.25
3.75
3.93
4.60
22.48
10.16
30.63
4.15
11.43
11.58
14.63
31.78
115.99
-3.02
-5.66
NA
-1.13
-1.14
-1.14
-1.08
-1.74
-1.83
6.12
-0.33
2.34
-1.99
-3.15
4.17
72.39
414
limit of continuous CO2 injection. Table 19.15 is ordered based on the amount of CO2 storage at
the economic limit of WAG.
CO2 storage is a measure of how much injected CO2 remains in the reservoir when an economic
limit is reached. For the homogeneous cases, the amount of CO2 storage is typically 80%–93%.
Heterogeneity can cause a significantly reduced amount of CO2 storage.
For the CO2 storage after continuous CO2 injection or WAG injection, there are variations
based on the WCO2 , gas relative permeability hysteresis, and km2 , but no obvious trends. There is a
small increase in the CO2 storage when the trapped oil increases and for cycle-dependent residual
oil saturations.
Table 19.16 presents the CO2 storage and CO2 utilization for continuous CO2 injection and
WAG. Table 19.16 is ordered based on the difference between CO2 storage at the economic limit
of WAG and at the economic limit of continuous CO2 injection.
For the homogeneous cases, CO2 storage varies between slightly less and slightly more storage
with the WAG flood than with pure CO2 injection. For the heterogeneous cases, the CO2 is utilized
better and less is stored during WAG than with continuous CO2 injection. For case Y560, the CO2
storage is much lower than in the homogeneous cases.
Cases with gas relative permeability hysteresis have more storage during WAG relative to continuous CO2 injection. If CO2 is soluble in water it causes a slight increase in the WAG storage
relative to the continuous CO2 storage. Cycle dependent residual oil saturation causes a slight
increase in the WAG storage relative to the continuous CO2 storage.
19.7
Evaluation of CO2 Utilization
CO2 utilization is a measure of how much CO2 injection it takes to produce an incremental
amount of oil. The lower the CO2 utilization, the better the performance. CO2 utilization values
of 10 MCF/STB are typically economical in the USA.
CO2 utilization is lower (better) for the cases without compositional trapping than for the cases
with compositional trapping. This is the case for both continuous CO2 utilization and CO2 WAG
utilization. The priority of the other options are different for continuous CO2 utilization and CO2
WAG utilization.
415
Utilization GF (MCF/RB)
Utilization WAG (MCF/RB)
Utilization WAG − GF (MCF/RB)
0
>0
0
0
0
>0
0
>0
>0
>0
0
>0
0
>0
>0
>0
>0
Storage WAG − GF (lbmol)
NA
NA
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
NA
5 · 10−5
5 · 10−5
5 · 10−5
Storage WAG (lbmol)
Transfer: km2
yes
no
no
no
no
no
no
no
no
no
no
no
no
no
no
yes
no
Storage GF (lbmol)
Heterogeneity
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
560
561
563
551
579
576
553
575
578
550
552
573
574
564
577
570
554
Hysteresis of krg
Model #
Y
W
W
W
X
X
X
X
X
X
X
X
X
W
X
Z
X
WCO2 Option
Model Type
Table 19.16: CO2 storage difference for continuous vs WAG CO2 injection
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
hysteresis + no trap
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
NA
NA
NA
NA
no
yes
yes
yes
yes
yes
no
yes
yes
NA
no
yes
no
no
no
yes
no
no
yes
yes
yes
no
yes
no
yes
yes
yes
yes
yes
no
0.479
0.870
0.879
0.877
0.872
0.899
0.916
0.906
0.890
0.894
0.805
0.914
0.918
0.877
0.864
0.702
0.492
0.398
0.829
0.845
0.844
0.851
0.884
0.903
0.894
0.883
0.891
0.805
0.921
0.931
0.893
0.894
0.804
NA
-0.081
-0.040
-0.033
-0.033
-0.021
-0.016
-0.014
-0.012
-0.006
-0.003
0.000
0.007
0.013
0.015
0.030
0.102
NA
7.06
4.38
5.07
4.89
5.68
11.98
17.79
13.57
24.22
24.51
289.64
27.61
43.60
4.48
9.09
16.93
684.12
4.04
3.25
3.93
3.75
4.60
10.16
14.63
11.58
22.48
30.63
NA
31.78
115.99
4.15
11.43
11.27
150.71
-3.02
-1.13
-1.14
-1.14
-1.08
-1.83
-3.15
-1.99
-1.74
6.12
NA
4.17
72.39
-0.33
2.34
-5.66
-533.41
416
Utilization GF (MCF/RB)
Utilization WAG (MCF/RB)
Utilization WAG − GF (MCF/RB)
>0
>0
0
0
0
0
>0
>0
>0
>0
0
>0
>0
>0
0
0
>0
Storage WAG − GF (lbmol)
NA
NA
NA
NA
5 · 10−5
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
5 · 10−5
Storage WAG (lbmol)
Transfer: km2
no
no
no
no
no
yes
no
no
no
yes
no
no
no
no
no
no
no
Storage GF (lbmol)
Heterogeneity
no
no
no
no
yes
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
561
564
551
563
579
560
577
576
575
570
553
578
550
573
574
552
554
Hysteresis of krg
Model #
W
W
W
W
X
Y
X
X
X
Z
X
X
X
X
X
X
X
WCO2 Option
Model Type
Table 19.17: CO2 utilization for continuous CO2 injection
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
hysteresis + no trap
trap + hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
NA
NA
NA
NA
no
NA
no
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
yes
no
yes
no
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
no
0.870
0.877
0.877
0.879
0.872
0.479
0.864
0.899
0.906
0.702
0.916
0.890
0.894
0.914
0.918
0.805
0.492
0.829
0.893
0.844
0.845
0.851
0.398
0.894
0.884
0.894
0.804
0.903
0.883
0.891
0.921
0.931
0.805
NA
-0.040
0.015
-0.033
-0.033
-0.021
-0.081
0.030
-0.016
-0.012
0.102
-0.014
-0.006
-0.003
0.007
0.013
0.000
NA
4.38
4.48
4.89
5.07
5.68
7.06
9.09
11.98
13.57
16.93
17.79
24.22
24.51
27.61
43.60
289.64
684.12
3.25
4.15
3.75
3.93
4.60
4.04
11.43
10.16
11.58
11.27
14.63
22.48
30.63
31.78
115.99
NA
150.71
-1.13
-0.33
-1.14
-1.14
-1.08
-3.02
2.34
-1.83
-1.99
-5.66
-3.15
-1.74
6.12
4.17
72.39
NA
-533.41
417
Table 19.17 presents the CO2 storage and CO2 utilization for continuous CO2 injection and
WAG. Table 19.17 is ordered based on the CO2 utilization at the economic limit of continuous CO2
injection.
Continuous CO2 utilization is lower (better) with CO2 solubility in water than with WCO2 = 0.
This effect is much bigger for cases with compositional trapping than for cases without compositional trapping. For the dual-media compositional trapping cases, additional trapped oil leads to
increased (worse) continuous CO2 utilization. Gas relative permeability hysteresis leads to much
worse CO2 utilization for the cases with dual-media compositional trapping.
Table 19.18 presents the CO2 storage and CO2 utilization for continuous CO2 injection and
WAG. Table 19.18 is ordered based on the CO2 utilization at the economic limit of WAG.
WAG CO2 utilization is lower (better) with CO2 solubility in water than with WCO2 = 0. This
effect is much bigger for cases with compositional trapping than for cases without compositional
trapping. WAG CO2 utilization is higher (worse) with gas relative permeability hysteresis. This
effect is much bigger for cases with compositional trapping than for cases without compositional
trapping. Changing the trapped oil after waterflood or adding cycle-dependent residual oil saturation causes variations in the WAG CO2 utilization but no trend was observed in these test
cases.
Table 19.19 presents the CO2 storage and CO2 utilization for continuous CO2 injection and
WAG. Table 19.19 is ordered based on the difference between CO2 utilization at the economic limit
of WAG and at the economic limit of continuous CO2 injection.
WAG CO2 utilization is lower (better) than continuous CO2 utilization in some cases and higher
(worse) in others. For the cases without compositional trapping and the heterogeneous cases, the
WAG CO2 utilization is lower (better) than the continuous CO2 utilization. The other properties
cause variations in the CO2 utilization between WAG and continuous CO2 utilization but no trend
is observed.
418
Storage WAG − GF (lbmol)
Utilization GF (MCF/RB)
Utilization WAG (MCF/RB)
Utilization WAG − GF (MCF/RB)
NA
NA
NA
NA
NA
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
Storage WAG (lbmol)
Transfer: km2
no
no
no
yes
no
no
no
yes
no
no
no
no
no
no
no
no
Storage GF (lbmol)
Heterogeneity
no
no
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
561
551
563
560
564
579
576
570
577
575
553
578
550
573
574
554
Hysteresis of krg
Model #
W
W
W
Y
W
X
X
Z
X
X
X
X
X
X
X
X
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
NA
NA
NA
NA
NA
no
yes
yes
no
yes
yes
yes
yes
yes
yes
no
no
no
yes
no
yes
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
0.870
0.877
0.879
0.479
0.877
0.872
0.899
0.702
0.864
0.906
0.916
0.890
0.894
0.914
0.918
0.492
0.829
0.844
0.845
0.398
0.893
0.851
0.884
0.804
0.894
0.894
0.903
0.883
0.891
0.921
0.931
NA
-0.040
-0.033
-0.033
-0.081
0.015
-0.021
-0.016
0.102
0.030
-0.012
-0.014
-0.006
-0.003
0.007
0.013
NA
4.38
4.89
5.07
7.06
4.48
5.68
11.98
16.93
9.09
13.57
17.79
24.22
24.51
27.61
43.60
684.12
3.25
3.75
3.93
4.04
4.15
4.60
10.16
11.27
11.43
11.58
14.63
22.48
30.63
31.78
115.99
150.71
-1.13
-1.14
-1.14
-3.02
-0.33
-1.08
-1.83
-5.66
2.34
-1.99
-3.15
-1.74
6.12
4.17
72.39
-533.41
WCO2 Option
Model Type
Table 19.18: CO2 utilization for WAG injection
>0
0
0
0
>0
0
>0
>0
>0
>0
0
>0
>0
>0
0
>0
no
no
no
no
419
Utilization GF (MCF/RB)
Utilization WAG (MCF/RB)
Utilization WAG − GF (MCF/RB)
>0
>0
0
0
>0
>0
>0
0
0
>0
0
>0
>0
>0
>0
0
Storage WAG − GF (lbmol)
5 · 10−5
5 · 10−5
5 · 10−5
NA
5 · 10−5
5 · 10−5
5 · 10−5
NA
NA
NA
5 · 10−5
NA
5 · 10−5
5 · 10−3
5 · 10−5
5 · 10−5
Storage WAG (lbmol)
Transfer: km2
no
yes
no
yes
no
no
no
no
no
no
no
no
no
no
no
no
Storage GF (lbmol)
Heterogeneity
yes
yes
yes
no
yes
yes
yes
no
no
no
yes
no
yes
yes
yes
yes
Cycle dependent Sor
Compositional Trapping
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
1/4
Extra Trapped Oil After WF
Pattern
554
570
553
560
575
576
578
551
563
561
579
564
577
573
550
574
Hysteresis of krg
Model #
X
Z
X
Y
X
X
X
W
W
W
X
W
X
X
X
X
WCO2 Option
Model Type
Table 19.19: CO2 utilization difference for continuous vs WAG CO2 injection
trap + hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
hysteresis + no trap
no trap + no hysteresis
trap + hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
no trap + no hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
trap + hysteresis
no
yes
yes
NA
yes
yes
yes
NA
NA
NA
no
NA
no
yes
yes
yes
no
yes
yes
no
yes
yes
no
no
yes
no
no
yes
yes
yes
yes
yes
0.492
0.702
0.916
0.479
0.906
0.899
0.890
0.877
0.879
0.870
0.872
0.877
0.864
0.914
0.894
0.918
NA
0.804
0.903
0.398
0.894
0.884
0.883
0.844
0.845
0.829
0.851
0.893
0.894
0.921
0.891
0.931
NA
0.102
-0.014
-0.081
-0.012
-0.016
-0.006
-0.033
-0.033
-0.040
-0.021
0.015
0.030
0.007
-0.003
0.013
684.12
16.93
17.79
7.06
13.57
11.98
24.22
4.89
5.07
4.38
5.68
4.48
9.09
27.61
24.51
43.60
150.71
11.27
14.63
4.04
11.58
10.16
22.48
3.75
3.93
3.25
4.60
4.15
11.43
31.78
30.63
115.99
-533.41
-5.66
-3.15
-3.02
-1.99
-1.83
-1.74
-1.14
-1.14
-1.13
-1.08
-0.33
2.34
4.17
6.12
72.39
420
CHAPTER 20
CONCLUSIONS
The three-phase compositional reservoir simulator developed here was used to evaluate the
effects of compositional trapping, gas relative permeability hysteresis, the solubility of CO2 in water,
and areal heterogeneity. Other options evaluated include cycle-dependent residual oil saturations,
mass transfer between the trapped and mobile systems, and additional mechanisms for trapped oil.
Compositional recovery factors are different if and only if compositional trapping is used. Compositional trapping is the most significant option for differences in waterflood duration (more trapping is better), gas flood recovery factor (less trapping is better), CO2 response duration (less
trapping is better), WAG recovery factor (less trapping is better), WAG duration (more trapping
is better), and CO2 utilization for WAG and continuous CO2 injection (WAG is better with more
trapping). Compositional trapping has a secondary effect on waterflood recovery (more trapping
is better).
(1) My results indicate that compositional trapping, gas relative permeability hysteresis, and
the solubility of CO2 in water, have a significant impact on the volume of oil produced, the timing
of oil, water, and gas production, and the amount of CO2 stored and CO2 utilized. Primary production, waterflood, continuous CO2 injection, and CO2 WAG production schemes were evaluated.
Permeability and porosity heterogeneity are important to the timing, recovery, CO2 storage, and
CO2 utilization; the effects of heterogeneity need to be evaluated more thoroughly in future work.
(2) Gas relative permeability hysteresis is the most significant parameter in waterflood recovery
(more trapped gas is better) and WAG recovery (with compositional trapping, more trapped gas
is better). Gas relative permeability hysteresis has a secondary effect on the waterflood timing
(more trapped gas is better), gas flood recovery (more trapped gas is worse), duration of gas flood
response (with compositional trapping, more trapped gas is better), compositional recovery factor,
and CO2 utilization for WAG and continuous CO2 injection (more trapped gas is worse). Gas
relative permeability hysteresis was more important than expected.
(3) Solubility of CO2 in water is not the most important option for any of the evaluation
criteria, but it is of secondary importance for waterflood duration (more WCO2 is better), gas flood
421
recovery (more WCO2 is better), gas flood response duration (with compositional trapping, more
WCO2 is better), CO2 storage, and CO2 utilization (more WCO2 is worse). Solubility of CO2 was
less important than expected.
(4) The cycle-dependent residual oil saturations, mass transfer between the trapped and mobile
systems, and additional mechanisms for trapped oil caused small variations in the observations but
were never as significant as compositional trapping, gas relative permeability hysteresis, solubility
of CO2 in water, or heterogeneity. This was less impact than expected.
422
CHAPTER 21
RECOMMENDED FUTURE WORK
The recommended future work includes the following categories.
• Use of this model
• Formulation enhancements
• Computation enhancements
• Laboratory experiments
21.1
Use of This Model
The three-phase parallel compositional simulator developed here can be used to evaluate additional fields or projects. For the test cases described here, more detailed evaluation of the heterogeneity and how it interacts with compositional trapping would be beneficial.
21.2
Formulation and Computation Enhancements
Adding the dual-porosity simulation of naturally or hydraulically fractured reservoirs would add
flexibility to the evaluation of CO2 WAG cases in carbonate reservoirs. Running additional simulations at different scales between the pore-scale and field-scale would be valuable in characterizing
the importance of measurements at different scales.
The simulator developed here is built on a parallel framework. Additional work to improve the
performance of the simulator would benefit future users of the simulator.
21.3
Phase Labeling and Relative Permeability Experiments
When miscibility develops, a two-phase oil-gas system becomes a single hydrocarbon phase.
This can present a problem in calculating relative permeability. If the single phase is labeled as a
gas, then kr,hc = krg whereas if the single phase is labeled as oil then kr,hc = kro . Often these are
simulated by weighting the krg and kro curves using an interfacial tension.
423
If miscibility develops gradually and the transition occurs from two-phase to one-phase, then the
mixture relative permeability could be estimated using interfacial tension. Unfortunately interfacial
tension is not a reliable measure of the change in relative permeability. There are some difficulties
in the correlations for interfacial tension at low interfacial tension values. It is also difficult to
experimentally measure very low interfacial tensions.
Transitions can also occur from single phase “oil” to single phase “gas” (or vice versa) in
the supercritical region of the fluids. These are actually gradual changes with no phase change
present, but depending on how the phases are labeled it may lead to inconsistencies in the relative
permeabilities. Part of the problem is that relative permeability is usually measured for a oil-water
system and a gas-oil system or for a gas-water system, but is not measured for a “supercritical
fluid”-water system. Supercritical fluids are common, especially when dealing with CO2 injection.
Adjusting the relative permeability based on interfacial tension won’t detect this change because
it was a single phase before and after the “transition”.
Experiments by Bennion and Bachu (2005) and Bennion and Bachu (2008b) illustrate the
differences between CO2 -water and H2 S-water relative permeability. It is likely that other kinds of
gas-water and gas-oil systems will have differing relative permeability based on the composition of
the gas.
For the life cycle of a reservoir undergoing a CO2 flood, several different gas relative permeabilities are needed. During primary production, the gas is a hydrocarbon gas in equilibrium with the
oil; this is either part of an initial gas cap or solution gas that forms as the pressure drops near the
producer. The gas is increasing during this stage. During water injection it is necessary to have a
decreasing relative permeability to gas. The gas is still a hydrocarbon gas in equilibrium with the
oil.
If CO2 is injected, then it would be nice to have measurements of the CO2 -oil-water relative
permeability. As the CO2 mixes with the oil, it will vaporize some of the components of the oil.
It would also be nice to have a (CO2 + hydrocarbon)-oil-water relative permeability. Three phase
relative permeability measurements would be helpful. It would also be useful to have measurements
of the CO2 -water relative permeability and CO2 -water-residual oil relative permeability. The residual oil changes depending on the saturation history, so several different CO2 -water-residual oil
experiments would be necessary.
424
During CO2 WAG operations, the gas increases and decreases in different parts of the reservoir.
It would be nice to have measurements of the hysteresis of the relative permeability and capillary
pressure as well as measurements of how the composition varies. As miscibility develops, it would
be nice to have measurements of the relative permeability for the hydrocarbon-water system.
There is limited three-phase or compositional two-phase relative permeability data available,
especially for mixed-wet carbonate rocks. Over the years there have been many proposed formulations for three-phase relative permeability and hysteresis, but without experimental data it is
difficult to evaluate the different methods or propose a new high-quality physically-based method.
425
CHAPTER 22
NOMENCLATURE
This chapter identifies all variables used in this document. Table 22.1 identifies all subscripts
and superscripts.
Table 22.1: Subscripts and superscripts
variable
units
name
#
script
i
j
k
n
index
index
index
index
index
m
m
index
index
α
w
index
index
represents a constant term, one that does not vary
with time
spatial index in x-direction
spatial index in y-direction
spatial index in z-direction
temporal index representing full time step
temporal index representing nonlinear iteration level
between n = ( = 0) and n + 1 = ( + 1).
component index, typically runs from 1..NC − 1
primary variable component index, typically runs
from 1..NC − 2
index for completions in a well
indicates that a variable is within the wellbore, not
the reservoir
generic phase; may be o, w, g, or t.
orientation of directional permeability
represents properties specific to x-direction
represents properties specific to y-direction
represents properties specific to z-direction
water phase
oil phase
gas phase
asphaltene phase
trapped phase
gas trapped by oil
gas trapped by water
oil trapped by gas
oil trapped by water
water trapped by gas
water trapped by oil
ϕ
index
θ
index
x
script
y
script
z
script
w
script
o
script
g
script
a
script
t
script
gto
script
gtw
script
otg
script
otw
script
wtg
script
wto
script
Continued.
426
Table 22.1: Continued.
Table 22.1: Subscripts and superscripts (continued)
variable
t
units
script
m
m1
m2
f
f/m
m/f
l
v
CH4
C1
CI1
CI2
CH1
CH2
CH3
CO2
H2 O
script
script
script
script
script
script
script
script
script
script
script
script
script
script
script
script
script
name
total, refers to sum of phases or sum of m1 and m2
systems
matrix properties
interconnected matrix properties
trapped matrix properties
fracture properties
transfer from fracture into matrix
transfer from matrix into fracture
liquid phase
vapor phase
methane component
methane component
intermediate hydrocarbon pseudo-component 1
intermediate hydrocarbon pseudo-component 2
heavy hydrocarbon pseudo-component 1
heavy hydrocarbon pseudo-component 2
heavy hydrocarbon pseudo-component 3
carbon dioxide component
water component
Table 22.2 identifies the variables used in this document. The units given are typical units. The
units for empirical correlations are listed in a particular section are listed within each section that
contains correlations.
Table 22.2: Variables used in this document
variable
Accnmi
A
Al
Av
amn
al
av
α
αow
Bl
Bv
bm
units
name
3
lbmol/ft
varies
3
ft /lbmol
ft3 /lbmol
psi · ft3 /lbmol
psi · ft3 /lbmol
psi · ft3 /lbmol
varies
unitless
ft3 /lbmol
ft3 /lbmol
ft3 /lbmol
Continued.
accumulation term
general matrix
Peng-Robinson parameter
Peng-Robinson parameter
Peng-Robinson coefficient
Peng-Robinson parameter
Peng-Robinson parameter
General parameter
capillary pressure coefficient
Peng-Robinson parameter
Peng-Robinson parameter
Peng-Robinson coefficient
427
Table 22.2: Continued.
Table 22.2: Variables used in this document (continued)
variable
bl
bv
β
Cw
Co
Cg
Cφ
Cwi
units
ft /lbmol
ft3 /lbmol
unitless
1/psi
1/psi
1/psi
1/psi
fraction
Cs
fraction
Cαw
Coff [f ]
day/ft3
time
Con [f ]
time
3
Cm
cm
DPmn
xt
DCmn
xt
DSOmn
xt
DSGmn
xt
Dmol
Ddisp
l/l
Dowo,mm1 /m2
lbmol/day
ft3 /lbmol
(lbmol/day)/psi
(lbmol/day)
(lbmol/day)
(lbmol/day)
ft2 /day
ft2 /day
ft2 /day
Dgwg,mm1 /m2
l/v
ft2 /day
Dwow,mm1 /m2
l/l
ft2 /day
D
Dα
δ
δ̂mn
δPo
ft
ft
varies
unitless
psi
δSo
unitless
name
Peng-Robinson parameter
Peng-Robinson parameter
bandwidth
water compressibility
oil compressibility
gas compressibility
formation compressibility
concentration, used to track mixing between injected
brine and reservoir brine
salt concentration; units include weight fraction,
mole fraction, volume fraction, mass/volume, and
mole/volume
well bore storage coefficient
communication time between cores on different
nodes for f
communication time between cores on the same
node for f
component equation
Peneloux volume adjustment
coefficient of δP
coefficient which does not multiply δ
coefficient which multiplies δSo
coefficient which multiplies δSg
molecular diffusion coefficient
dispersivity coefficient
liquid-liquid molecular diffusion for
om2 → wm1 → om1
gas-liquid molecular diffusion for gm2 → wm1 → gm1
liquid-liquid molecular diffusion for
wm2 → om1 → wm1
depth
total vertical depth of completion α in a well
general solution vector
binary interaction coefficient
primary variable; change in oil pressure over a
nonlinear iteration
primary variable; change in oil saturation over a
nonlinear iteration
Continued.
428
Table 22.2: Continued.
Table 22.2: Variables used in this document (continued)
variable
δSg
units
unitless
δXm
unitless
δYm
unitless
δm,m
unitless
dw
α
εRe
fom
fgm
f
Gm
γw
γo
γg
hf,α+ 1
ft
unitless
varies
psi
psi
unitless
psi
psi/ft
psi/ft
psi/ft
ft
hϕ,f
hϕ,m1
K
ft
ft
2
ft /day
2
e1
Km
k
kθ
kxx
kyy
kzz
kr
krφ
krw
kro
krg
krow
krwo
md
md
md
md
md
unitless
md
unitless
unitless
unitless
unitless
unitless
Continued.
name
primary variable; change in gas saturation over a
nonlinear iteration
primary variable; change in liquid mole fraction of
component m over a nonlinear iteration
primary variable; change in gas mole fraction of
component m over a nonlinear iteration
Krönecker delta function, evaluates to 1 if m = m
and 0 otherwise
well inside diameter
pipe roughness for Reynold’s number calculation
tolerance or threshhold
oil phase fugacity
gas phase fugacity
Moody friction factor
thermodynamic constraint equation
specific gravity of aqueous phase
specific gravity of oil phase
specific gravity of gas phase
friction adjustment based on the length of the well
segment
fluid height for phase ϕ in the fracture
fluid height for phase ϕ in the matrix
diffusion coefficient for mass transfer between
trapped and mixable phases
multiplier in solution of Peng-Robinson equation of
state
permeability
permeability in direction θ
permeability in x-direction
permeability in y-direction
permeability in z-direction
relative permeability
value of relative permeability at maximum
saturation for phase ϕ
total relative permeability to water
total relative permeability to oil
total relative permeability to gas
relative permeability to oil in presence of water
relative permeability to water in presence of oil
429
Table 22.2: Continued.
Table 22.2: Variables used in this document (continued)
variable
krog
krgo
krwg
krgw
κm
LHS
Lα
l
lw
α
units
unitless
unitless
unitless
unitless
unitless
varies
ft
fraction
fraction
λw
λo
λg
MW
μw
μo
μg
Nx
Ny
Nz
Nxyz
NC
Nc
Nb
Nn
Np
Nnp
NRe
nϕ
O [f ]
P
Pb
Pd
Pcm
Pw
Po
Pg
Pc
Pcow
1/cp
1/cp
1/cp
lbm/mol
cp
cp
cp
unitless
unitless
unitless
unitless
unitless
unitless
unitless
unitless
unitless
unitless
unitless
unitless
time
psi
psi
psi
psi
psi
psi
psi
psi
psi
Continued.
name
relative permeability to oil in presence of gas
relative permeability to gas in presence of oil
relative permeability to water in presence of gas
relative permeability to gas in presence of water
Peng-Robinson parameter
term on left hand side of equation
measured depth along wellbore
mole fraction of liquid phase after flash
mole fraction of liquid phase after flash of
cumulative fluid in wellbore
mobility of water phase
mobility of oil phase
mobility of gas phase
molecular weight
viscosity of water phase
viscosity of oil phase
viscosity of gas phase
number of grid cells in x direction
number of grid cells in y direction
number of grid cells in z direction
total number of grid cells
number of components, including H2 O
capillary number
bond number
number of processing nodes
number of processing cores per node
total number of processing cores on all nodes
Reynold’s number
relative permeability exponent
computational order of f
pressure; if no phase subscript, measured in oil phase
bubble point pressure
dew point pressure
critical pressure
pressure measured in water phase
pressure measured in oil phase
pressure measured in gas phase
capillary pressure
water-oil capillary pressure
430
Table 22.2: Continued.
Table 22.2: Variables used in this document (continued)
variable
Pcgo
Pαw
P
Φm
φ
Ψϕ
Q
Qt
Qo,α
units
psi
psi
psi
unitless
fraction
psi
lbmol/day
lbmol/day
lbmol/day
Qw
o,α
lbmol/day
q
n
q̂wi
n
q̂oi
n
q̂gi
qw
qo
qg
qt
q
RHS
R
R
e1
Rm
Rsw
rw,α
ft3 /day
1/day
1/day
1/day
ft3 /day
ft3 /day
ft3 /day
ft3 /day
ft3 /day
varies
3
psi · ft /lbmol · ◦ F
varies
unitless
SCF/STB
ft
ρ
S
Sw
So
Sg
Sϕ∗
Swt
Sot
Sgt
Swot
Swgt
lbmol/ft3
fraction
fraction
fraction
fraction
fraction
fraction
fraction
fraction
fraction
fraction
Continued.
name
gas-oil capillary pressure
pressure in wellbore
reference producing pressure at the heel of the well
fugacity coefficient
porosity
potential of phase ϕ
molar flux rate
molar flux rate at heel of well
molar flux rate from the well into the reservoir at
completion α
cumulative molar flux rate in the wellbore at
completion α
volumetric rate
water source volumetric rate per reservoir volume
oil source volumetric rate per reservoir volume
gas source volumetric rate per reservoir volume
water rate
oil rate
gas rate
total rate
total volumetric flow rate at heel of well
term on right hand side of equation
Ideal gas law coefficient
general right-hand-side of matrix equation
convergence criteria for flash
solubility of gas in water
effective wellbore radius for flow between the
reservoir and the well
density
short notation for So , Sg , Sw
water saturation
oil saturation
gas saturation
normalized saturation for phase ϕ
total trapped water saturation
total trapped oil saturation
total trapped gas saturation
water trapped by oil phase
water trapped by gas phase
431
Table 22.2: Continued.
Table 22.2: Variables used in this document (continued)
variable
Sowt
Sogt
Sgwt
Sgot
sα
sm
σ
σ
T
Tcm
mn
Txw,i±
1
units
fraction
fraction
fraction
fraction
unitless
unitless
dyne/cm
1/ft2
R
R
(lbmol/ft2 )psi−1
name
oil trapped by water phase
oil trapped by gas phase
gas trapped by water phase
gas trapped by oil phase
skin factor for well
Peneloux volume shift
interfacial tension
shape factor
temperature
critical temperature
transmissibility term for water phase in x direction
mn
Txo,i±
1
(lbmol/ft2 )psi−1
transmissibility term for oil phase in x direction
mn
Txg,i±
1
(lbmol/ft2 )psi−1
transmissibility term for gas phase in x direction
2
2
2
mn
Tyw,i±
1
2
mn
Tyo,i± 1
2
mn
Tyg,i±
1
2
mn
Tzw,i± 1
2
mn
Tzo,i±
1
2
mn
Tzg,i±
1
2
2
−1
(lbmol/ft )psi
transmissibility term for water phase in y direction
(lbmol/ft2 )psi−1
transmissibility term for oil phase in y direction
(lbmol/ft2 )psi−1
transmissibility term for gas phase in y direction
2
−1
(lbmol/ft )psi
transmissibility term for water phase in z direction
(lbmol/ft2 )psi−1
transmissibility term for oil phase in z direction
(lbmol/ft2 )psi−1
transmissibility term for gas phase in z direction
t
ts
Δt
τ
τt,m1 /m2
days
days
days
lbmol/day
lbmol/day
τt,f/m1
lbmol/day
τmgto
lbmol/day
τmgtw
lbmol/day
τmotg
lbmol/day
τmotw
lbmol/day
τmwtg
lbmol/day
time
time step size
time step size
transfer function
transfer function
matrix phases
transfer function
matrix phases
transfer function
by oil
transfer function
by water
transfer function
gas
transfer function
water
transfer function
by gas
Continued.
432
between trapped and mixable
between fracture and mixable
for component m for gas trapped
for component m for gas trapped
for component m for oil trapped by
for component m for oil trapped by
for component m for water trapped
Table 22.2: Continued.
Table 22.2: Variables used in this document (continued)
variable
τmwto
units
lbmol/day
n
Umi
n
Umx,i
n
Umy,i
n
Umz,i
VR
Vαw
vw
α
v
vEOS
(lbmol/ft3 )/day
(lbmol/ft3 )/day
(lbmol/ft3 )/day
(lbmol/ft3 )/day
ft3
ft3
fraction
ft3 /lbmol
ft3 /lbmol
w
vϕα
WI#
α
Wm
Ωa
Ωb
ωm
X
Xm
Δx
χ
ξw
ξo
ξg
Y
Ym
Δy
Zm
Δz
z̆l
z̆v
ft3 /day
(ft /day)(cp/psi)
unitless
lbmol/ft3
unitless
unitless
fraction
fraction
ft
varies
lbmol/ft3
lbmol/ft3
lbmol/ft3
fraction
fraction
ft
fraction
ft
unitless
unitless
3
name
transfer function for component m for water trapped
by oil
total of spatial terms and source terms
total of spatial terms in x direction
total of spatial terms in y direction
total of spatial terms in z direction
rock volume
volume within wellbore
mole fraction of vapor phase after flash
specific volume
specific volume calculated by Peng-Robinson
equation of state before Peneloux volume adjustment
velocity of phase ϕ in wellbore
well index
mole fraction in aqueous phase
Peng-Robinson constant
Peng-Robinson constant
Peng-Robinson acentric factor
short notation for X1 , X2 , . . . , XNC −1 , XNC
mole fraction in oil phase
grid cell size in x direction
general variable
molar density of aqueous phase
molar density of oil phase
molar density of gas phase
short notation for Y1 , Y2 , . . . , YNC −1 , YNC
mole fraction in gas phase
grid cell size in y direction
total mole fraction
grid cell size in z direction
Peng-Robinson z-factor
Peng-Robinson z-factor
433
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APPENDIX - RESULTS FOR SPECIFIC TEST CASES
A.1
Primary Production Results
Figure A.1 illustrates the primary production pressures for W551 and X550.
Production Pressure psia
Production Pressure psia
2500
PBHP ,Pores
PBHP ,Pores
2500
2000
1500
2000
1500
1000
1000
0
2000
4000
6000
8000
10 000
0
2000
4000
tday
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.1: Primary Production Pressures.
Figure A.2 illustrates the primary production rates for W551 and X550.
Production Rate RBPD
200
150
qTOT ,qo ,qg ,qw
150
qTOT ,qo ,qg ,qw
Production Rate RBPD
200
100
50
100
50
0
0
0
2000
4000
6000
8000
10 000
0
tday
2000
4000
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.2: Primary Production Rates.
Figure A.3 illustrates the primary production ratios for W551 and X550.
Figure A.4 illustrates the primary nonlinear iteration convergence for W551 and X550.
Figure A.5 illustrates the primary CFL criteria (Courant et al., 1967) on time step size for
W551 and X550.
480
Production Ratio
1.0
0.8
0.8
qo qT ,qg qT ,qwqT
qo qT ,qg qT ,qwqT
Production Ratio
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
2000
4000
6000
8000
0.0
10 000
0
2000
4000
tday
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.3: Primary Production Ratios.
Average it 2.574
6
2.5
5
2.0
4
nonlinear it
nonlinear it
Average it 1.046
3.0
1.5
3
1.0
2
0.5
1
0.0
0
2000
4000
6000
8000
0
10 000
0
2000
4000
tday
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.4: Primary nonlinear iteration convergence.
Maximum ts from CFL
20
20
15
max ts size day
max ts size day
15
10
5
5
0
10
0
0
2000
4000
6000
8000
10 000
0
2000
4000
6000
8000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.5: Primary time step criteria.
481
10 000
Figure A.6 illustrates the primary pressure for cells along the diagonal between wells for W551
and X550.
Pressure Across All Cells
3500
3500
3000
3000
Ppsi
Ppsi
Pressure Across All Cells
2500
2500
2000
2000
1500
1500
1000
1000
0
2000
4000
6000
8000
10 000
0
2000
4000
tday
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.6: Primary pressure for cells along diagonal between wells.
Figure A.7 illustrates the primary total mass of CO2 for cells along diagonal between wells for
W551 and X550.
Total CO2 Across All Cells
700
600
600
500
500
lbmol CO2
lbmol CO2
Total CO2 Across All Cells
700
400
300
400
300
200
200
100
100
0
0
2000
4000
6000
8000
10 000
0
0
tday
2000
4000
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.7: Primary total mass of CO2 for cells along diagonal between wells.
Figure A.8 illustrates the primary total mass of hydrocarbons (no CO2 ) for cells along diagonal
between wells for W551 and X550.
Figure A.9 illustrates the primary saturation for equivalent one cell model for W551 and X550.
Figure A.10 illustrates the primary total mole fraction in the reservoir for W551 and X550.
Figure A.11 illustrates the primary recovery factor for W551 and X550.
Figure A.12 illustrates the primary compositional recovery factor for W551 and X550.
482
Total HC no CO2 Across Diagonal Cells
Total HC no CO2 Across Diagonal Cells
14 000
12 000
12 000
lbmol HC no CO2 trap mobile
14 000
lbmol HC
10 000
8000
6000
4000
2000
0
10 000
8000
6000
4000
2000
0
2000
4000
6000
8000
0
10 000
0
2000
4000
tday
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.8: Primary total mass of hydrocarbons (no CO2 ) for cells along diagonal between wells.
Saturation for Equivalent OneCell Model
1.0
0.8
0.8
0.6
0.6
Total S
Total S
Saturation for Equivalent OneCell Model
1.0
0.4
0.4
0.2
0.2
0.0
0
5000
10 000
15 000
0.0
20 000
0
5000
10 000
15 000
20 000
25 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.9: Primary saturation for equivalent one cell model. Purple is trapped water, blue is
mobile water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile gas.
Mole Fraction in Reservoir
1.0
0.8
0.8
CH4 , nC4 , nC10 , CO2
CH4 , nC4 , nC10 , CO2
Mole Fraction in Reservoir
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
2000
4000
6000
8000
10 000
0.0
0
tday
2000
4000
6000
8000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.10: Primary total mole fraction in the reservoir.
483
10 000
0.8
0.6
0.6
0.4
RF
0.4
PV RCF
0.8
produced oil RCF
PV RCF
Recovery Factor Econ Limit 0.202195
1.0
RF
produced oil RCF
Recovery Factor Econ Limit 0.191192
1.0
0.2
0.0
0.2
0
2000
4000
6000
8000
0.0
10 000
0
2000
4000
tday
6000
8000
10 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.11: Primary recovery factor.
Produced Fraction by Component
1.0
0.8
0.8
CH4 , nC4 , nC10
CH4 , nC4 , nC10
Produced Fraction by Component
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
2000
4000
6000
8000
10 000
0.0
0
tday
2000
4000
6000
8000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.12: Primary compositional recovery factor.
484
10 000
Figure A.13 illustrates the distribution of pressures at the economic limit of primary for W551
and X550.
Presure Distribution time 7290.
0.010
0.008
0.008
0.006
0.006
freq
freq
Presure Distribution time 7200.
0.010
0.004
0.004
0.002
0.002
0.000
0.000
1000
2000
3000
4000
5000
1000
P psia
2000
3000
4000
5000
P psia
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.13: Distribution of pressures at primary economic limit.
Figure A.14 illustrates the 2-D pressure distribution at the economic limit of primary for W551
and X550.
Presure time 7200.
Presure time 7290.
P psia
P psia
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.14: 2-D pressure distribution at primary economic limit.
Figure A.15 illustrates the distribution of oil saturations at the economic limit of primary for
W551 and X550.
Figure A.16 illustrates the 2-D oil saturation distribution at the economic limit of primary for
W551 and X550.
485
Total Oil Saturation Distribution time 7290.
100
50
80
40
60
30
freq
freq
Oil Saturation Distribution time 7200.
40
20
20
10
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
So
0.2
0.4
0.6
0.8
1.0
SoT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.15: Distribution of oil saturation at primary economic limit.
Oil Saturation time 7200.
Total Oil Saturation time 7290.
So
SoT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.16: 2-D oil saturation distribution at primary economic limit.
486
Figure A.17 illustrates the distribution of gas saturations at the economic limit of primary for
W551 and X550.
Total Gas Saturation Distribution time 7290.
50
80
40
60
30
freq
freq
Gas Saturation Distribution time 7200.
100
40
20
20
10
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Sg
0.2
0.4
0.6
0.8
1.0
SgT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.17: Distribution of gas saturation at primary economic limit.
Figure A.18 illustrates the 2-D gas saturation distribution at the economic limit of primary for
W551 and X550.
Gas Saturation time 7200.
Total Gas Saturation time 7290.
Sg
SgT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.18: 2-D gas saturation distribution at primary economic limit.
Figure A.19 illustrates the distribution of water saturations at the economic limit of primary
for W551 and X550.
Figure A.20 illustrates the 2-D water saturation distribution at the economic limit of primary
for W551 and X550.
487
Total Water Saturation Distribution time 7290.
100
50
80
40
60
30
freq
freq
Water Saturation Distribution time 7200.
40
20
20
10
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Sw
0.2
0.4
0.6
0.8
1.0
SwT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.19: Distribution of water saturation at primary economic limit.
Water Saturation time 7200.
Total Water Saturation time 7290.
Sw
SwT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.20: 2-D water saturation distribution at primary economic limit.
488
A.2
Waterflood Results
Figure A.21 illustrates the waterflood injection pressures for W551 and X550.
Injection Pressure psia
3500
3500
3000
3000
PBHP ,Pores
PBHP ,Pores
Injection Pressure psia
2500
2500
2000
2000
1500
1500
1000
1000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.21: Waterflood Injection Pressure.
Figure A.22 illustrates the waterflood injection rates for W551 and X550.
Injection Rate RBPD
200
150
150
qw or qg bbl
qw or qg bbl
Injection Rate RBPD
200
100
50
100
50
0
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
tday
5000
10 000
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.22: Waterflood Injection Rates.
Figure A.23 illustrates the waterflood production pressures for W551 and X550.
Figure A.24 illustrates the waterflood production rates for W551 and X550.
Figure A.25 illustrates the waterflood production ratios for W551 and X550.
Figure A.26 illustrates the waterflood oil production rate minus the primary production rate
for W551 and X550.
Figure A.27 illustrates the waterflood nonlinear iteration convergence for W551 and X550.
Figure A.28 illustrates the waterflood CFL criteria (Courant et al., 1967) on time step size for
W551 and X550.
489
Production Pressure psia
Production Pressure psia
2500
PBHP ,Pores
PBHP ,Pores
2500
2000
1500
2000
1500
1000
1000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.23: Waterflood Production Pressures.
Production Rate RBPD
200
150
qTOT ,qo ,qg ,qw
150
qTOT ,qo ,qg ,qw
Production Rate RBPD
200
100
50
100
50
0
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.24: Waterflood Production Rates.
Production Ratio
1.0
0.8
0.8
qo qT ,qg qT ,qwqT
qo qT ,qg qT ,qwqT
Production Ratio
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.25: Waterflood Production Ratios.
490
30 000
NewOld Production Rate RBPD
NewOld Production Rate RBPD
30
40
25
20
qo RB
qo RB
30
20
15
10
10
5
0
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.26: WF − Primary Oil Rate .
Average it 1.28686
Average it 27.1783
3.0
50
2.5
40
nonlinear it
nonlinear it
2.0
1.5
20
1.0
10
0.5
0.0
30
0
5000
10 000
15 000
20 000
25 000
30 000
0
35 000
0
5000
10 000
15 000
20 000
25 000
30 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.27: Waterflood nonlinear iteration convergence.
Maximum ts from CFL
20
20
15
max ts size day
max ts size day
15
10
5
5
0
10
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
15 000
20 000
25 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.28: Waterflood time step criteria.
491
30 000
Figure A.29 illustrates the waterflood pressure for cells along the diagonal between wells for
W551 and X550.
Pressure Across All Cells
3500
3500
3000
3000
Ppsi
Ppsi
Pressure Across All Cells
2500
2500
2000
2000
1500
1500
1000
1000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.29: Waterflood pressure for cells along diagonal between wells.
Figure A.30 illustrates the waterflood total mass of CO2 for cells along diagonal between wells
for W551 and X550.
Total CO2 Across All Cells
Total CO2 Across All Cells
700
3000
600
2500
lbmol CO2
lbmol CO2
500
400
300
1500
1000
200
500
100
0
2000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
0
5000
10 000
15 000
20 000
25 000
30 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.30: Waterflood total mass of CO2 for cells along diagonal between wells.
Figure A.31 illustrates the waterflood total mass of hydrocarbons (no CO2 ) for cells along
diagonal between wells for W551 and X550.
Figure A.32 illustrates the waterflood saturation for equivalent one cell model for W551 and
X550.
Figure A.33 illustrates the waterflood total mole fraction in the reservoir for W551 and X550.
Figure A.34 illustrates the waterflood recovery factor for W551 and X550.
Figure A.35 illustrates the waterflood compositional recovery factor for W551 and X550.
492
Total HC no CO2 Across Diagonal Cells
Total HC no CO2 Across Diagonal Cells
14 000
12 000
12 000
lbmol HC no CO2 trap mobile
14 000
lbmol HC
10 000
8000
6000
4000
2000
0
10 000
8000
6000
4000
2000
0
5000
10 000
15 000
20 000
25 000
30 000
0
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.31: Waterflood total mass of hydrocarbons (no CO2 ) for cells along diagonal between
wells.
Saturation for Equivalent OneCell Model
1.0
0.8
0.8
0.6
0.6
Total S
Total S
Saturation for Equivalent OneCell Model
1.0
0.4
0.4
0.2
0.2
0.0
0
5000
10 000
15 000
20 000
25 000
30 000
0.0
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.32: Waterflood saturation for equivalent one cell model. Purple is trapped water, blue is
mobile water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile gas.
Mole Fraction in Reservoir
1.0
0.8
0.8
CH4 , nC4 , nC10 , CO2
CH4 , nC4 , nC10 , CO2
Mole Fraction in Reservoir
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.33: Waterflood total mole fraction in the reservoir.
493
30 000
0.8
0.6
0.6
0.4
RF
0.4
PV RCF
0.8
produced oil RCF
PV RCF
Recovery Factor Econ Limit 0.683429
1.0
RF
produced oil RCF
Recovery Factor Econ Limit 0.615077
1.0
0.2
0.0
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
0.0
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.34: Waterflood recovery factor.
Produced Fraction by Component
1.0
0.8
0.8
CH4 , nC4 , nC10
CH4 , nC4 , nC10
Produced Fraction by Component
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.35: Waterflood compositional recovery factor.
494
30 000
Figure A.36 illustrates the distribution of pressures at the economic limit of waterflood for W551
and X550.
Presure Distribution time 20540.
0.0025
0.0020
0.0020
0.0015
0.0015
freq
freq
Presure Distribution time 15640.
0.0025
0.0010
0.0010
0.0005
0.0005
0.0000
0.0000
1000
2000
3000
4000
5000
1000
P psia
2000
3000
4000
5000
P psia
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.36: Distribution of pressures at waterflood economic limit.
Figure A.37 illustrates the 2-D pressure distribution at the economic limit of waterflood for
W551 and X550.
Presure time 15640.
Presure time 20540.
P psia
P psia
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.37: 2-D pressure distribution at waterflood economic limit.
Figure A.38 illustrates the distribution of oil saturations at the economic limit of waterflood for
W551 and X550.
Figure A.39 illustrates the 2-D oil saturation distribution at the economic limit of waterflood
for W551 and X550.
495
Oil Saturation Distribution time 15640.
Total Oil Saturation Distribution time 20540.
10
12
8
10
6
freq
freq
8
6
4
4
2
2
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
So
0.2
0.4
0.6
0.8
1.0
SoT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.38: Distribution of oil saturation at waterflood economic limit.
Oil Saturation time 15640.
Total Oil Saturation time 20540.
So
SoT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.39: 2-D oil saturation distribution at waterflood economic limit.
496
Figure A.40 illustrates the distribution of gas saturations at the economic limit of waterflood
for W551 and X550.
Total Gas Saturation Distribution time 20540.
Gas Saturation Distribution time 15640.
100
40
80
freq
freq
30
60
20
40
10
20
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Sg
0.2
0.4
0.6
0.8
1.0
SgT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.40: Distribution of gas saturation at waterflood economic limit.
Figure A.41 illustrates the 2-D gas saturation distribution at the economic limit of waterflood
for W551 and X550.
Gas Saturation time 15640.
Total Gas Saturation time 20540.
Sg
SgT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.41: 2-D gas saturation distribution at waterflood economic limit.
Figure A.42 illustrates the distribution of water saturations at the economic limit of waterflood
for W551 and X550.
Figure A.43 illustrates the 2-D water saturation distribution at the economic limit of waterflood
for W551 and X550.
497
Total Water Saturation Distribution time 20540.
Water Saturation Distribution time 15640.
12
8
10
6
freq
freq
8
6
4
4
2
2
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Sw
0.2
0.4
0.6
0.8
1.0
SwT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.42: Distribution of water saturation at waterflood economic limit.
Water Saturation time 15640.
Total Water Saturation time 20540.
Sw
SwT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.43: 2-D water saturation distribution at waterflood economic limit.
498
A.3
Continuous CO2 Injection Results
Figure A.44 illustrates the continuous CO2 injection pressures for W551 and X550.
Injection Pressure psia
5000
5000
4000
4000
PBHP ,Pores
PBHP ,Pores
Injection Pressure psia
3000
2000
3000
2000
1000
1000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
15 000
tday
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.44: Continuous CO2 Injection Pressure.
Figure A.45 illustrates the continuous CO2 injection rates for W551 and X550.
Injection Rate RBPD
200
150
150
qw or qg bbl
qw or qg bbl
Injection Rate RBPD
200
100
50
100
50
0
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
tday
5000
10 000
15 000
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.45: Continuous CO2 Injection Rates.
Figure A.46 illustrates the continuous CO2 production pressures for W551 and X550.
Figure A.47 illustrates the continuous CO2 production rates for W551 and X550.
Figure A.48 illustrates the continuous CO2 production ratios for W551 and X550.
Figure A.49 illustrates the continuous CO2 oil production rate minus the waterflood production
rate for W551 and X550.
Figure A.50 illustrates the continuous CO2 nonlinear iteration convergence for W551 and X550.
499
Production Pressure psia
Production Pressure psia
4500
3000
4000
2500
PBHP ,Pores
PBHP ,Pores
3500
3000
2500
2000
2000
1500
1500
1000
1000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
15 000
tday
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.46: Continuous CO2 Production Pressures.
Production Rate RBPD
200
150
qTOT ,qo ,qg ,qw
150
qTOT ,qo ,qg ,qw
Production Rate RBPD
200
100
50
100
50
0
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
15 000
tday
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.47: Continuous CO2 Production Rates.
Production Ratio
1.0
0.8
0.8
qo qT ,qg qT ,qwqT
qo qT ,qg qT ,qwqT
Production Ratio
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.48: Continuous CO2 Production Ratios.
500
35 000
NewOld Production Rate RBPD
NewOld Production Rate RBPD
200
80
60
qo RB
qo RB
150
100
50
40
20
0
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.49: GF − WF Oil Rate .
Average it 4.10857
Average it 36.8546
70
8
60
50
nonlinear it
nonlinear it
6
4
40
30
20
2
10
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
0
5000
10 000
15 000
20 000
25 000
30 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.50: Continuous CO2 nonlinear iteration convergence.
501
35 000
Figure A.51 illustrates the continuous CO2 CFL criteria (Courant et al., 1967) on time step size
for W551 and X550.
Maximum ts from CFL
20
20
15
max ts size day
max ts size day
15
10
5
5
0
10
0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.51: Continuous CO2 time step criteria.
Figure A.52 illustrates the continuous CO2 pressure for cells along the diagonal between wells
for W551 and X550.
Pressure Across All Cells
5000
5000
4000
4000
Ppsi
Ppsi
Pressure Across All Cells
3000
3000
2000
2000
1000
1000
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
tday
5000
10 000
15 000
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.52: Continuous CO2 pressure for cells along diagonal between wells.
Figure A.53 illustrates the continuous CO2 total mass of CO2 for cells along diagonal between
wells for W551 and X550.
Figure A.54 illustrates the continuous CO2 total mass of hydrocarbons (no CO2 ) for cells along
diagonal between wells for W551 and X550.
Figure A.55 illustrates the continuous CO2 saturation for equivalent one cell model for W551
and X550.
502
Total CO2 Across All Cells
Total CO2 Across All Cells
15 000
20 000
lbmol CO2
lbmol CO2
15 000
10 000
10 000
5000
5000
0
0
5000
10 000
15 000
20 000
25 000
30 000
0
35 000
0
5000
10 000
15 000
tday
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.53: Continuous CO2 total mass of CO2 for cells along diagonal between wells.
Total HC no CO2 Across Diagonal Cells
Total HC no CO2 Across Diagonal Cells
14 000
12 000
12 000
lbmol HC no CO2 trap mobile
14 000
lbmol HC
10 000
8000
6000
4000
2000
0
10 000
8000
6000
4000
2000
0
5000
10 000
15 000
20 000
25 000
30 000
0
35 000
0
5000
10 000
15 000
tday
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.54: Continuous CO2 total mass of hydrocarbons (no CO2 ) for cells along diagonal between
wells.
Saturation for Equivalent OneCell Model
1.0
0.8
0.8
0.6
0.6
Total S
Total S
Saturation for Equivalent OneCell Model
1.0
0.4
0.4
0.2
0.2
0.0
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.55: Continuous CO2 saturation for equivalent one cell model. Purple is trapped water,
blue is mobile water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile
gas.
503
Figure A.56 illustrates the continuous CO2 total mole fraction in the reservoir for W551 and
X550.
Mole Fraction in Reservoir
1.0
0.8
0.8
CH4 , nC4 , nC10 , CO2
CH4 , nC4 , nC10 , CO2
Mole Fraction in Reservoir
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
0.0
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.56: Continuous CO2 total mole fraction in the reservoir.
Figure A.57 illustrates the continuous CO2 recovery factor for W551 and X550.
0.6
0.4
PV RCF
0.8
produced oil RCF
0.8
0.6
0.4
RF
PV RCF
Recovery Factor Econ Limit 0.714275
1.0
RF
produced oil RCF
Recovery Factor Econ Limit 0.847951
1.0
0.2
0.0
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0.0
tday
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.57: Continuous CO2 recovery factor.
Figure A.58 illustrates the continuous CO2 compositional recovery factor for W551 and X550.
Figure A.59 illustrates the continuous CO2 storage of CO2 for W551 and X550.
Figure A.60 illustrates the continuous CO2 utilization of CO2 for W551 and X550.
Figure A.61 illustrates the distribution of pressures at the economic limit of continuous CO2
for W551 and X550.
Figure A.62 illustrates the 2-D pressure distribution at the economic limit of continuous CO2
for W551 and X550.
504
Produced Fraction by Component
1.0
0.8
0.8
CH4 , nC4 , nC10
CH4 , nC4 , nC10
Produced Fraction by Component
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
30 000
0.0
35 000
0
5000
10 000
15 000
tday
20 000
25 000
30 000
35 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.58: Continuous CO2 compositional recovery factor.
CO2 Storage 0.876518
CO2 Storage 0.894284
0.8
CO2 Storage lbmollbmol
CO2 Storage lbmollbmol
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.6
0
5000
10 000
15 000
20 000
25 000
30 000
0.0
35 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.59: Continuous CO2 storage of CO2 .
CO2 Utilizaiotn Econ Limit 4.89089
50
40
CO2 Utilizatoin MCFRB
CO2 Utilizatoin MCFRB
40
30
20
10
0
CO2 Utilizaiotn Econ Limit 24.5112
50
30
20
10
0
5000
10 000
15 000
20 000
25 000
30 000
35 000
0
0
tday
5000
10 000
15 000
20 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.60: Continuous CO2 utilization of CO2 .
505
25 000
35 000
Presure Distribution time 25450.
Presure Distribution time 25520.
0.0030
0.005
0.0025
0.004
0.0020
freq
freq
0.003
0.0015
0.002
0.0010
0.001
0.0005
0.0000
0.000
1000
2000
3000
4000
5000
1000
2000
3000
4000
5000
P psia
P psia
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.61: Distribution of pressures at Continuous CO2 economic limit.
Presure time 25450.
Presure time 25520.
P psia
P psia
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.62: 2-D pressure distribution at Continuous CO2 economic limit.
506
Figure A.63 illustrates the distribution of oil saturations at the economic limit of continuous
CO2 for W551 and X550.
Oil Saturation Distribution time 25450.
60
Total Oil Saturation Distribution time 25520.
20
50
15
freq
freq
40
30
10
20
5
10
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
So
0.2
0.4
0.6
0.8
1.0
SoT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.63: Distribution of oil saturation at Continuous CO2 economic limit.
Figure A.64 illustrates the 2-D oil saturation distribution at the economic limit of continuous
CO2 for W551 and X550.
Oil Saturation time 25450.
Total Oil Saturation time 25520.
So
SoT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.64: 2-D oil saturation distribution at Continuous CO2 economic limit.
Figure A.65 illustrates the distribution of gas saturations at the economic limit of continuous
CO2 for W551 and X550.
Figure A.66 illustrates the 2-D gas saturation distribution at the economic limit of continuous
CO2 for W551 and X550.
507
Gas Saturation Distribution time 25450.
40
Total Gas Saturation Distribution time 25520.
14
12
30
10
freq
freq
8
20
6
4
10
2
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
SgT
Sg
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.65: Distribution of gas saturation at Continuous CO2 economic limit.
Gas Saturation time 25450.
Total Gas Saturation time 25520.
Sg
SgT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.66: 2-D gas saturation distribution at Continuous CO2 economic limit.
508
Figure A.67 illustrates the distribution of water saturations at the economic limit of continuous
CO2 for W551 and X550.
Water Saturation Distribution time 25450.
Total Water Saturation Distribution time 25520.
30
10
25
8
freq
freq
20
15
6
4
10
2
5
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Sw
0.2
0.4
0.6
0.8
1.0
SwT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.67: Distribution of water saturation at Continuous CO2 economic limit.
Figure A.68 illustrates the 2-D water saturation distribution at the economic limit of continuous
CO2 for W551 and X550.
Water Saturation time 25450.
Total Water Saturation time 25520.
Sw
SwT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.68: 2-D water saturation distribution at Continuous CO2 economic limit.
A.4
WAG Results
Figure A.69 illustrates the WAG injection pressures for W551 and X550.
Figure A.70 illustrates the WAG injection rates for W551 and X550.
Figure A.71 illustrates the WAG production pressures for W551 and X550.
509
Injection Pressure psia
Injection Pressure psia
10 000
5000
8000
PBHP ,Pores
PBHP ,Pores
4000
3000
6000
4000
2000
2000
1000
0
5000
10 000
15 000
20 000
0
25 000
5000
10 000
15 000
20 000
25 000
30 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.69: WAG Injection Pressure.
Injection Rate RBPD
200
150
150
qw or qg bbl
qw or qg bbl
Injection Rate RBPD
200
100
50
100
50
0
0
0
5000
10 000
15 000
20 000
25 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.70: WAG Injection Rates.
Production Pressure psia
Production Pressure psia
3500
10 000
8000
2500
PBHP ,Pores
PBHP ,Pores
3000
2000
6000
4000
1500
2000
1000
0
5000
10 000
15 000
20 000
25 000
0
5000
10 000
15 000
20 000
25 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.71: WAG Production Pressures.
510
30 000
Figure A.72 illustrates the WAG production rates for W551 and X550.
Production Rate RBPD
200
150
qTOT ,qo ,qg ,qw
150
qTOT ,qo ,qg ,qw
Production Rate RBPD
200
100
50
100
50
0
0
0
5000
10 000
15 000
20 000
25 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.72: WAG Production Rates.
Figure A.73 illustrates the WAG production ratios for W551 and X550.
Production Ratio
1.0
0.8
0.8
qo qT ,qg qT ,qwqT
qo qT ,qg qT ,qwqT
Production Ratio
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.73: WAG Production Ratios.
Figure A.74 illustrates the WAG oil production rate minus the waterflood production rate for
W551 and X550.
Figure A.75 illustrates the WAG nonlinear iteration convergence for W551 and X550.
Figure A.76 illustrates the WAG CFL criteria (Courant et al., 1967) on time step size for W551
and X550.
Figure A.77 illustrates the WAG pressure for cells along the diagonal between wells for W551
and X550.
Figure A.78 illustrates the WAG total mass of CO2 for cells along diagonal between wells for
W551 and X550.
511
NewOld Production Rate RBPD
200
150
qo RB
150
qo RB
NewOld Production Rate RBPD
200
100
100
50
50
0
0
0
5000
10 000
15 000
20 000
25 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.74: WAG − WF Oil Rate .
Average it 34.8999
70
6
60
5
50
nonlinear it
nonlinear it
Average it 3.39745
7
4
3
40
30
2
20
1
10
0
0
5000
10 000
15 000
20 000
0
25 000
0
5000
10 000
15 000
20 000
25 000
30 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.75: WAG nonlinear iteration convergence.
Maximum ts from CFL
20
20
15
max ts size day
max ts size day
15
10
5
5
0
10
0
0
5000
10 000
15 000
20 000
25 000
0
5000
10 000
15 000
20 000
25 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.76: WAG time step criteria.
512
30 000
Pressure Across All Cells
Pressure Across All Cells
10 000
5000
8000
Ppsi
Ppsi
4000
3000
6000
4000
2000
2000
1000
0
5000
10 000
15 000
20 000
0
25 000
5000
10 000
15 000
20 000
25 000
30 000
tday
tday
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.77: WAG pressure for cells along diagonal between wells.
Total CO2 Across All Cells
Total CO2 Across All Cells
20 000
14 000
12 000
15 000
lbmol CO2
lbmol CO2
10 000
10 000
8000
6000
4000
5000
2000
0
0
5000
10 000
15 000
20 000
25 000
0
0
tday
5000
10 000
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.78: WAG total mass of CO2 for cells along diagonal between wells.
513
Figure A.79 illustrates the WAG total mass of hydrocarbons (no CO2 ) for cells along diagonal
between wells for W551 and X550.
Total HC no CO2 Across Diagonal Cells
Total HC no CO2 Across Diagonal Cells
14 000
12 000
12 000
lbmol HC no CO2 trap mobile
14 000
lbmol HC
10 000
8000
6000
4000
2000
0
10 000
8000
6000
4000
2000
0
5000
10 000
15 000
20 000
0
25 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.79: WAG total mass of hydrocarbons (no CO2 ) for cells along diagonal between wells.
Figure A.80 illustrates the WAG saturation for equivalent one cell model for W551 and X550.
Saturation for Equivalent OneCell Model
1.0
0.8
0.8
0.6
0.6
Total S
Total S
Saturation for Equivalent OneCell Model
1.0
0.4
0.4
0.2
0.2
0.0
0
5000
10 000
15 000
20 000
25 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.80: WAG saturation for equivalent one cell model. Purple is trapped water, blue is mobile
water, cyan is trapped oil, green is mobile oil, yellow is trapped gas, red is mobile gas.
Figure A.81 illustrates the WAG total mole fraction in the reservoir for W551 and X550.
Figure A.82 illustrates the WAG recovery factor for W551 and X550.
Figure A.83 illustrates the WAG compositional recovery factor for W551 and X550.
Figure A.84 illustrates the WAG storage of CO2 for W551 and X550.
Figure A.85 illustrates the WAG utilization of CO2 for W551 and X550.
Figure A.86 illustrates the distribution of pressures at the economic limit of WAG for W551
and X550.
514
Mole Fraction in Reservoir
1.0
0.8
0.8
CH4 , nC4 , nC10 , CO2
CH4 , nC4 , nC10 , CO2
Mole Fraction in Reservoir
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
0.0
25 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.81: WAG total mole fraction in the reservoir.
0.8
0.6
0.6
0.4
RF
0.4
PV RCF
0.8
produced oil RCF
PV RCF
Recovery Factor Econ Limit 0.709605
1.0
RF
produced oil RCF
Recovery Factor Econ Limit 0.853358
1.0
0.2
0.0
0.2
0
5000
10 000
15 000
20 000
0.0
25 000
0
5000
10 000
15 000
tday
20 000
25 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.82: WAG recovery factor.
Produced Fraction by Component
1.0
0.8
0.8
CH4 , nC4 , nC10
CH4 , nC4 , nC10
Produced Fraction by Component
1.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0
5000
10 000
15 000
20 000
25 000
0.0
0
tday
5000
10 000
15 000
20 000
25 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.83: WAG compositional recovery factor.
515
30 000
CO2 Storage 0.843604
CO2 Storage 0.89084
0.8
CO2 Storage lbmollbmol
CO2 Storage lbmollbmol
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.6
0
5000
10 000
15 000
20 000
0.0
25 000
0
5000
10 000
tday
15 000
20 000
25 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.84: WAG storage of CO2 .
CO2 Utilizaiotn Econ Limit 3.74611
50
40
CO2 Utilizatoin MCFRB
CO2 Utilizatoin MCFRB
40
30
20
10
0
CO2 Utilizaiotn Econ Limit 30.6331
50
30
20
10
0
5000
10 000
15 000
20 000
0
25 000
0
5000
10 000
tday
15 000
20 000
25 000
30 000
tday
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.85: WAG utilization of CO2 .
Presure Distribution time 25300.
Presure Distribution time 26190.
0.0020
0.004
0.0015
freq
freq
0.003
0.002
0.0010
0.0005
0.001
0.0000
0.000
1000
2000
3000
4000
5000
1000
2000
3000
4000
P psia
P psia
(b) Most trapping base, X550.
(a) Least trapping base, W551.
Figure A.86: Distribution of pressures at WAG economic limit.
516
5000
Figure A.87 illustrates the 2-D pressure distribution at the economic limit of WAG for W551
and X550.
Presure time 25300.
Presure time 26190.
P psia
P psia
5000
5000
4000
4000
3000
3000
2000
2000
1000
1000
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.87: 2-D pressure distribution at WAG economic limit.
Figure A.88 illustrates the distribution of oil saturations at the economic limit of WAG for
W551 and X550.
Total Oil Saturation Distribution time 26190.
Oil Saturation Distribution time 25300.
60
5
50
4
freq
freq
40
30
3
2
20
1
10
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
So
0.2
0.4
0.6
0.8
1.0
SoT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.88: Distribution of oil saturation at WAG economic limit.
Figure A.89 illustrates the 2-D oil saturation distribution at the economic limit of WAG for
W551 and X550.
Figure A.90 illustrates the distribution of gas saturations at the economic limit of WAG for
W551 and X550.
517
Oil Saturation time 25300.
Total Oil Saturation time 26190.
So
SoT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.89: 2-D oil saturation distribution at WAG economic limit.
Total Gas Saturation Distribution time 26190.
40
30
30
freq
freq
Gas Saturation Distribution time 25300.
40
20
20
10
10
0
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
SgT
Sg
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.90: Distribution of gas saturation at WAG economic limit.
518
1.0
Figure A.91 illustrates the 2-D gas saturation distribution at the economic limit of WAG for
W551 and X550.
Gas Saturation time 25300.
Total Gas Saturation time 26190.
Sg
SgT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.91: 2-D gas saturation distribution at WAG economic limit.
Figure A.92 illustrates the distribution of water saturations at the economic limit of WAG for
W551 and X550.
Water Saturation Distribution time 25300.
Total Water Saturation Distribution time 26190.
5
20
4
15
freq
freq
3
10
2
5
1
0
0.0
0.2
0.4
0.6
0.8
1.0
0
0.0
Sw
0.2
0.4
0.6
0.8
1.0
SwT
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.92: Distribution of water saturation at WAG economic limit.
Figure A.93 illustrates the 2-D water saturation distribution at the economic limit of WAG for
W551 and X550.
519
Water Saturation time 25300.
Total Water Saturation time 26190.
Sw
SwT
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
(a) Least trapping base, W551.
(b) Most trapping base, X550.
Figure A.93: 2-D water saturation distribution at WAG economic limit.
520