MODELLING SUB-CORE SCALE
PERMEABILITY IN SANDSTONE FOR USE IN
STUDYING MULTIPHASE FLOW OF CO2 AND
BRINE IN CORE FLOODING EXPERIMENTS
A REPORT
SUBMITTED TO THE DEPARTMENT OF ENERGY
RESOURCES ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
By
Michael H. Krause
June 11, 2009
© Copyright by Michael H. Krause 2009
All Rights Reserved
ii
I certify that I have read this report and that in my opinion it is
fully adequate, in scope and in quality, as partial fulfillment of the
degree of Master of Science in Petroleum Engineering.
__________________________________
Prof. Sally M. Benson
(Principal Advisor)
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iv
Abstract
As CO2 capture and storage moves closer to commercialization, the ability to make
accurate predictions regarding storage capacity in saline aquifers becomes more
important. Improving storage capacity estimates can be done by conducting detailed
regional studies on saline aquifers, something which the Department of Energy regional
Carbon Sequestration Partnerships have been aggressively pursuing.
Improving
estimates also requires experimental and theoretical work, to develop a better
understanding of the impacts of heterogeneity on multi-phase systems with unfavorable
mobility ratios.
To study these systems, core flood experiments are conducted by injecting CO2 into a
brine saturated sandstone core at reservoir conditions, simulating injection conditions for
CO2 storage in a saline aquifer. Using an X-ray CT scanner, sub-core scale porosity is
mapped in the core prior to experiments, and sub-core scale CO2 saturation is mapped
during the experiments. The results of these experiments reveal that small variations in
porosity can lead to large spatial variations in CO2 distribution, with a high degree of
small scale spatial contrast in CO2 saturation.
To understand how such variable CO2 distribution occurs, simulations of the
experiment can be conducted to test the sensitivity of CO2 saturation to different fluid
parameters from multiphase-flow theory.
To perform such sensitivity studies,
permeability must first be calculated at the same sub-core scale as porosity and saturation
are measured. However, permeability cannot be directly measured at the sub-core scale,
therefore it must be calculated using other measured data, which has traditionally been
porosity.
Methods for calculating permeability from porosity are common, (Nelson, 1994), and
straightforward to apply in sub-core scale studies because sub-core scale porosity is
measured as part of the experiment. In this study, a specific subset of these porositypermeability relationships have been systematically tested using numerical simulations of
the core flood experiment. Comparing the results of the predicted saturation in the
v
simulations to the measured values in the experiment consistently indicate that while
these methods are very accurate for estimating core-scale properties, they do not
accurately represent the sub-core scale permeability, and the simulations do not replicate
the experimental measurements.
To improve the estimate of sub-core scale permeability, a new method was developed
to take advantage of additional data measured as part of the experiments. The capillary
pressure curve for the sandstone core is measured experimentally for use as input in
simulations. This capillary pressure data can also be integrated with the core flood
experiment saturation measurements to calculate permeability. Using a modified version
of the Leverett J-Function, sub-core scale permeability was calculated using the capillary
pressure, and sub-core scale saturation and porosity measurements.
The results of
simulations using this permeability method show a much improved quantitative match to
experimental saturation measurements over the porosity-only based permeability models.
This new method for calculating permeability shows the potential to greatly advance
the study of sub-core scale phenomena in CO2-brine systems by providing an accurate
sub-core scale permeability representation. Using this method to calculate permeability,
sensitivity studies of other multi-phase flow parameters can be conducted to determine
their effect on CO2 saturation in the presence of heterogeneity.
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“… you are only two questions away from the frontier of knowledge”
Dr. Steven Losh
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Acknowledgments
I would first and foremost like to thank my advisor, Dr. Sally Benson for the many
insightful conversations, for her deep insight, and endless patience in seeking answers to
many difficult questions. The things I have gained from her are endless and beyond
description, and they will come with me to many places, through many journeys, and in
the end, hopefully to make a difference in the world.
I would also like to thank Jean-Christophe Perrin for conducting many excellent
experiments over the course of the last two years, without which, none of this and much
other work would not be possible. I would like to thank Ethan Chabora for his many
stimulating conversations and for attentively proofreading so many abstracts and posters
for me. I also need to thank Obi Isebor for getting me through so many late nights of
studying, Louis-Marie Jacquelin for keeping me company in the lab and Thanapong
Boontaeng for helping me understand multi-phase flow.
I would also like to thank Jonathan Ennis-King and Lincoln Paterson for making my
summer in Australia possible. I would especially like to thank Jonathan for taking the
time to teach me how to use and understand TOUGH2, without which I would no doubt
still be error checking my input files.
I would like to thank Lynn Orr and Hamdi Tchelepi for various suggestions and
contributions that helped this work along and for the insight into future directions for this
and other work. I am very grateful to Karsten Pruess for developing TOUGH2, without
which, many scientists would be doing much different work. I am also very grateful to
Dmitry Silin for providing the capillary pressure function upon which most of this work
is based and for taking the time to provide feedback for this work.
I am indebted to the generous financial support of the Global Climate and Energy
Project (GCEP) and to its sponsors for funding this work and many others in our research
group and around the world. I am also grateful to Supri-C for their support, both
financial and intellectual over the past two years.
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Lastly and mostly, I am most grateful to my father, a man I have admired my whole
life, who never gave up in the face of adversity, a man who would sell the shirt off his
back to see his son succeed, without whom I would definitely not be who I am nor where
I am today.
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Contents
Abstract ............................................................................................................................... v
Acknowledgments.............................................................................................................. ix
Contents ............................................................................................................................. xi
List of Tables .................................................................................................................... xv
List of Figures ................................................................................................................. xvii
Chapter 1 ............................................................................................................................. 1
1
Introduction ................................................................................................................... 1
1.1
Statement of the Problem .................................................................................... 3
1.2
Outline of the Research Approach ...................................................................... 4
1.3
Organization of the Report ................................................................................. 5
Chapter 2 ............................................................................................................................. 7
2
Literature Review.......................................................................................................... 7
2.1
Kozeny-Carman Models ..................................................................................... 8
2.2
Models Based on Surface Area and Saturation ................................................ 10
2.3
Models Based on Pore Dimension .................................................................... 10
2.4
Fractal Models .................................................................................................. 13
Chapter 3 ........................................................................................................................... 17
3
Experimental and Simulation Methods ....................................................................... 17
3.1
Multi-Phase Flow Experiments ........................................................................ 17
3.1.1
Multi-Phase Flow Experimental Facility .................................................. 17
3.1.2
Measuring Relative Permeability and Saturation ..................................... 19
3.1.3
Experimental Results ................................................................................ 21
3.2
Capillary Pressure Measurements ..................................................................... 26
3.3
Simulation Method ........................................................................................... 29
3.3.1
Description of TOUGH2 MP .................................................................... 29
3.3.2
Additional Simulation Comments............................................................. 32
Chapter 4 ........................................................................................................................... 33
4
Evaluation of Existing Methods for Calculating Permeability ................................... 33
4.1
Experimental Data Preparation ......................................................................... 33
xi
4.2
4.1.1
CT Image processing ................................................................................ 33
4.1.2
Upscaling .................................................................................................. 35
4.1.3
Relative Permeability ................................................................................ 37
4.1.4
Capillary Pressure ..................................................................................... 39
4.1.5
Model Validation ...................................................................................... 42
Saturation Results using Kozeny-Carman Models ........................................... 46
4.2.1
Permeability Maps of the Kozeny-Carman Models.................................. 46
4.2.2
Comparison of Kozeny-Carman Model Results with Experiment ........... 49
4.3
Saturation Results of Fractal Models ................................................................ 52
4.4
Discussion of Porosity Based Model Results ................................................... 56
4.4.1
Examination of Core Scale Results........................................................... 56
4.4.2
Experiment Porosity-Saturation Relationship........................................... 58
4.4.3
Conclusions ............................................................................................... 59
Chapter 5 ........................................................................................................................... 61
5
A Proposed Method for Calculating Sub-Core Scale Permeability ............................ 61
5.1
5.2
Using the Leverett J-Function for Calculating Permeability ............................ 61
5.1.1
Previous Investigations ............................................................................. 61
5.1.2
Extension of Calhoun et al. Permeability Equation .................................. 62
5.1.3
Capillary Pressure Curve Fits ................................................................... 64
5.1.4
Permeability Maps .................................................................................... 67
Saturation Results of Modified Leverett J-Function Models............................ 68
5.2.1
Residual Brine Saturation Simulations ..................................................... 68
5.2.2
Zero Residual Brine Saturation Results .................................................... 71
5.2.3
Comparison of Core Average Results....................................................... 73
5.3
Statistical Comparison of Permeability Methods ............................................. 74
5.4
Conclusions ....................................................................................................... 76
Chapter 6 ........................................................................................................................... 79
6
Conclusions ................................................................................................................. 79
6.1
Summary of Findings........................................................................................ 79
6.2
Recommendations for Future Work ................................................................. 80
xii
6.3
Concluding Remarks......................................................................................... 81
Appendix A: A Method for Estimating Specific Surface Area ........................................ 83
Nomenclature .................................................................................................................... 87
References ......................................................................................................................... 91
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xiv
List of Tables
Table 2.1 Coefficients and data range for Huet et al. (2005) ......................................13
Table 3.1 Experimental conditions ..............................................................................21
Table 3.2 Coefficients of Kumagai and Yokoyama viscosity relationship ................22
Table 3.3 Permeability calculation data .......................................................................23
Table 3.4 Experimental data for calculating relative permeability ..............................23
Table 4.1 Simulation initial conditions ........................................................................42
Table 4.2 Simulation grid data .....................................................................................43
Table 4.3 Simulation 1-4 permeability parameters for Kozeny-Carman models ........46
Table 4.4 Simulation 5-6 parameters for fractal models..............................................53
Table 4.5 Core average results using traditional permeability models ........................58
Table 5.1 J-Function fitting parameters used to calculate permeability ......................64
Table 5.2 Core average results using modified Leverett J-Function method ..............74
Table 5.3 Linear trend line data for slice 29 average saturation comparisons .............74
Table 5.4 Linear trend line data for slice average saturation comparisons ..................76
xv
xvi
List of Figures
Figure 1.1 (a) Porosity map and (b) CO2 saturation map of a Berea sandstone core ....3
Figure 2.1 Comparison of piecewise terms and Eq. 2.22 ............................................16
Figure 3.1 Relative permeability experiment diagram (Perrin et al., 2009) ................19
Figure 3.2 Experimental relative permeability measurement ......................................24
Figure 3.3 Porosity map of experiment Berea core .....................................................24
Figure 3.4 Saturation map of experiment Berea core at 100 percent CO2 injection ....25
Figure 3.5 Slice average porosity and saturation .........................................................26
Figure 3.6 Measured capillary pressure curves for the CO2-brine system ..................29
Figure 4.1 CT data visualization software CT-view ....................................................34
Figure 4.2 Image processing software CT-daqs ..........................................................35
Figure 4.3 (a) Experiment porosity map, (b) Upscaled simulation porosity map........36
Figure 4.4 (a) Experiment saturation map (b) Upscaled saturation map .....................37
Figure 4.5 Relative permeability curve fit ...................................................................38
Figure 4.6 Capillary pressure curve fit for medium sized Berea sample (ICP1) .........40
Figure 4.7 Capillary pressure curve fit for small sized Berea sample (ICP2) .............41
Figure 4.8 Grid used for simulations ...........................................................................43
Figure 4.9 Test case results after 4 PVI (a) case 1, (b) case 2, (c) case 3, (d) case 4 ..45
Figure 4.10 Plot of specific perimeter vs. porosity of homogeneous Berea sample ....47
Figure 4.11 Permeability maps using Kozeny-Carman models for (a) Simulation 1 (b)
Simulation 2 (c) Simulation 3 (d) Simulation 4 .........................................49
Figure 4.12 CO2 saturation in slice 29 (a) Experiment (b) Sim. 1 (c) Sim. 2 (d) Sim. 3
(e) Sim. 4 ....................................................................................................51
xvii
Figure 4.13 Simulation vs. experiment saturation in slice 29 for Kozney-Carman
models (in order of saturation contrast) .....................................................52
Figure 4.14 Permeability maps of fractal models (a) Simulation 5 (b) Simulation 6 ..54
Figure 4.15 CO2 saturation in slice 29 (a) Experiment (b) Sim. 5 (c) Sim. 6 ..............55
Figure 4.16 Simulation vs. experimental saturation in slice 29 for fractal models (in
order of saturation contrast) .......................................................................56
Figure 4.17 Comparison of simulation saturation vs. porosity in Slice 29 (in order of
saturation contrast) .....................................................................................58
Figure 4.18 Comparison of experimentally measured saturation vs. porosity ............59
Figure 5.1 Flow chart for calculating permeability using capillary pressure data .......63
Figure 5.2 Capillary pressure fits for Simulation 9 (ICP3) and Simulation 10 (ICP4)
from medium Berea data ............................................................................65
Figure 5.3 Capillary pressure fit for Simulation 11 (ICP5) from small Berea data .....66
Figure 5.4 Relative permeability curve fit for Slr = 0 ..................................................66
Figure 5.5 Permeability maps using modified Leverett J-Function for (a) Simulation 7
& 9 (b) Simulation 8 (c) Simulation 10 and (d) Simulation 11 .................67
Figure 5.6 Comparison of (a) fractal permeability map (Simulation 5) and (b)
modified Leverett J-Function permeability map (Simulation 7 & 9) ........68
Figure 5.7 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 7 (c) Sim. 8 ...............69
Figure 5.8 Simulation vs. experiment saturation in slice 29 for J-Function method ...70
Figure 5.9 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 9 (c) Sim. 10 (d) Sim.
11................................................................................................................72
Figure 5.10 Simulation vs. experiment saturation in slice 29 for J-Function method .73
Figure 5.11 Comparison of slice average saturation of simulations 9-11....................75
Figure 5.12 Comparison of slice average saturation of simulations 1-6......................76
xviii
Chapter 1
1 Introduction
In recent years, CO2 capture and storage (CCS) has gained a great deal of attention as
a strategy with significant potential to greatly reduce anthropogenic carbon dioxide
emissions. CCS works by capturing carbon dioxide or any other greenhouse gas (CH4,
HFC’s, etc) from a point source, transporting the gas to an underground storage site, and
injecting it underground for permanent storage.
These storage sites typically fall into three categories, saline aquifers, depleted oil and
gas reservoirs and unmineable coal seams. Some of the benefits of using saline aquifers
for greenhouse gas storage are: worldwide distribution (Bradshaw and Dance, 2006),
good correlation between emissions sources and storage locations (NETL, 2008), and
very large storage capacities.
The Intergovernmental Panel on Climate Change (IPCC) estimates worldwide storage
capacity in saline formations to be 1,000-10,000 Gt of CO2 (IPCC, 2005), but they note
that the limits on capacity are highly uncertain due to the limited data available on saline
formations. According to the National Energy Technology Laboratory (NETL) (2008),
the US and Canada alone have an estimated capacity of 3,300 – 12,000 Gt of CO2 storage
capacity, enough to sequester at least 400 years of US CO2 equivalent emissions at 2006
emissions rates.
The large range of these estimates illustrates the degree of uncertainty which exists in
making capacity estimates without detailed regional studies. Reasons for this uncertainty
include: unknown aquifer extent, thickness, porosity and permeability, unknown seal
quality, limited knowledge of geological features such as faults and fractures, large scale
heterogeneities, and unknown storage efficiency.
NETL (2008) defines storage
efficiency, Es, as the fraction of a basin’s or region’s total pore volume that the CO2 is
1
2
CHAPTER 1. INTRODUCTION
expected to actually contact, but is more simply defined as the fraction of the total
available pore volume that actually stores CO2.
Many of these uncertainties can be addressed by doing comprehensive regional
studies to more precisely estimate total aquifer size and storage potential, something
which the regional CCS partnerships in the US have been aggressively pursuing. Storage
efficiency is difficult to estimate however as it contains three correction factors related to
the total reservoir pore volume available for CO2, and four correction factors related to
displacement efficiency, each of which have a large range of uncertainty (NETL, 2008).
By assigning distributions to the uncertainty in each of the parameters, a storage
efficiency range of 0.01 to 0.04 is determined for confidence intervals of 15 to 85 percent
respectively (NETL, 2008).
The two displacement efficiency correction factors for the vertical and horizontal
displacement efficiency, which are generally functions of porosity and permeability
variations (NETL, 2008), one for the influence of gravity, and one arising from the
fundamental principles which govern fluid flow behavior when more than one fluid is
present, called multi-phase flow. In this system, CO2 is displacing brine, however, CO2
is a lighter, less viscous fluid, which results in an unfavorable mobility ratio, meaning
that CO2 will not efficiently displace brine. In systems with an unfavorable mobility, the
displacing fluid will flow through the zone of highest permeability, possibly bypassing
large portions of the reservoir, leading to inefficient displacement. While this is true in
all systems, as permeability is defined as the ability for porous media to transport fluid, it
is especially true in systems with unfavorable mobility.
Understanding how these four factors affect and influence CO2 storage efficiency is
critical to our ability to predict storage capacity. If the displacement efficiency can be
more precisely characterized, then it might be possible to optimize injection and storage
strategies to increase the storage efficiency, Es, through specific knowledge of the effect
of features such as porosity and permeability contrast on CO2 storage.
3
CHAPTER 1. INTRODUCTION
1.1 Statement of the Problem
To study the effect of heterogeneity on displacement efficiency, core flooding
experiments can be used in conjunction with X-ray CT scanning to measure the CO2
saturation in a rock core. In these experiments, a core is saturated with brine and CO2 is
injected into it, simulating the injection conditions in a saline storage reservoir, then the
sub-core scale CO2 saturation measurement provided by the CT scan can be used to
provide insight into the role of heterogeneity in determining the resulting CO2
distribution. To illustrate the problem, a porosity map of a relatively homogeneous Berea
core imaged in a CT scanner is shown below in Figure 1.1 (a), and the resulting CO2
saturation map after injecting CO2 into the brine saturated core is shown in Figure 1.1 (b).
The figure shows that while porosity varies only slightly in the core, the saturation
distribution varies all the way from zero to 100 percent CO2. The next step then is to
determine what geological properties of the core in (a) give rise to the saturation
distribution in (b).
Figure 1.1 (a) Porosity map and (b) CO2 saturation map of a Berea sandstone core
To study the role of heterogeneity, simulations of the experiment were conducted
using the ECO2N module of the TOUGH2-MP reservoir simulator, which was designed
for the CO2-brine system. The goals of the simulations were to study different factors
which affect the CO2 distribution and to replicate and explain the large spatial saturation
variations measured in the CO2 injection experiment (Benson et al., 2008). Numerous
attempts to model the experiment could not quantitatively reproduce or explain the spatial
location of CO2 or the large spatial CO2 saturation contrast measured during the
experiment.
4
CHAPTER 1. INTRODUCTION
The goals of this research are to study the behavior of the CO2-brine system by
conducting core flooding experiments and to validate our understanding of the results
using numerical simulation. Based on the initial work in Benson et al. (2008), several
factors were identified for further study: absolute permeability, relative permeability and
capillary pressure. These three factors control how fluid moves through and is distributed
in the core, but absolute permeability in particular is of interest in this study.
Permeability cannot be directly measured at the sub-core scale as porosity and saturation
can be, therefore, it must be calculated indirectly from other properties. Permeability is a
unique fundamental input for these simulations, and an accurate representation will
provide subsequent relative permeability and capillary pressure studies with more
confident quantitative results.
Once we can understand the role of these very fine scale heterogeneities, we can
extrapolate our knowledge up to understand more about reservoir scale heterogeneities,
and improve our understanding of fluid interaction and storage capacity estimates in CO2
storage aquifers.
1.2 Outline of the Research Approach
The goal of this effort is to determine an accurate method for calculating sub-core
scale permeability. There are many methods which have been derived and developed for
calculating permeability in a variety of applications. Methods which are appropriate for
use at the sub-core scale were tested in this study by using them to create sub-core scale
permeability maps similar to Figure 1.1, and using those permeability maps as input for
numerical simulations. The results of each simulation were then quantitatively compared
to experimental measurements to determine which methods for calculating permeability
provided the most accurate results.
The research approach presented here is to start with core flooding experiments which
are conducted at reservoir conditions. To conduct the experiment, we saturate a rock core
with brine, inject CO2 and measure the pressure difference to calculate relative
permeability. A CT scanner is then used to measure the sub-core scale CO2 saturation
during the experiment, and is also used prior to the experiment to calculate the sub-core
scale porosity. Capillary pressure is then measured on a rock sample from the same core.
5
CHAPTER 1. INTRODUCTION
The challenge then is to determine an accurate method for calculating permeability within
the core at the same scale as the porosity and saturation are measured.
To calculate sub-core scale permeability, porosity, saturation and capillary pressure
data are available as previously stated. The study starts by examining methods for
calculating permeability based on porosity data, these are the oldest and most common
methods for predicting permeability (Nelson, 1994). Next, methods based on residual
water saturation and capillary pressure data are examined. The last method to be tested
was recently developed and is based on using fractal geometry to represent pore structure.
After these established methods have been tested, a new relationship using the capillary
pressure, saturation and porosity data to calculate permeability is proposed and the results
are analyzed.
1.3 Organization of the Report
Chapter 2 provides a literature survey of common methods used for calculating
permeability. Permeability is important in fields such as groundwater flow, contaminant
remediation, ceramics, powders, membranes, oil and gas recovery, and wastewater
filtration, among others; therefore, there are dozens of methods for calculating
permeability, and so a small, practical subset of suitable methods will be covered.
Chapter 3 outlines the core flooding and capillary pressure experiments which were
conducted.
The chapter describes the experimental setups, and then explains how
porosity, permeability, relative permeability and capillary pressure are measured, and
presents the actual data. The chapter also includes information about the simulator used
for conducting the core flood simulations.
Chapter 4 describes the basic inputs of the numerical simulations, and presents the
saturation results of a selected subset of permeability relationships used for numerical
simulation of the core flood experiment.
These results are then examined both
qualitatively, and quantitatively.
Chapter 5 derives a new method for calculating permeability based on the
experimentally measured capillary pressure data and integrating this information with
saturation and porosity measurement. The results of numerical simulations using this
6
CHAPTER 1. INTRODUCTION
method are also shown and analyzed qualitatively and quantitatively.
A statistical
analysis of every simulation and some discussion follows, outlining the strengths and
potential drawbacks of the existing permeability models in chapter 4, and the new
permeability model in chapter 5.
Chapter 6 contains a summary of the work presented in this study, and also a
summary of the research findings. Lastly, recommendations for future work to improve
the new permeability model are included.
Chapter 2
2 Literature Review
Permeability was first deduced by Henri Darcy as being proportional to the length
and flow rate and inversely proportional to the pressure drop, given by Darcy’s Law in
Eq. 2.1 (D’Arcy, 1856). Darcy did not recognize the permeability is also inversely
proportional to viscosity.
Permeability is a macroscopic empirical parameter which
describes the ability of a fluid to move through porous media such as soil, granular beds
and porous rocks.
𝑢=𝐾∙
∆𝑃
𝐿
2.1
where u is the flow rate through the medium, K is Darcy’s description of permeability,
and ΔP is the pressure drop across a medium of length L. In Eq. 2.1, K = k/μ, which
gives the current form of Darcy’s law. k is typically written with units of darcies or
millidarcies, where 1 darcy is 0.98692 μm2.
There are many equations for calculating permeability based on different sets of
information, such as grain size and sorting, residual water saturations, porosity,
cementation, etc. There are also many different factors which affect permeability, such
as diagenesis, clay inclusions, cementation, etc, which will not be discussed here, but
which Nelson (1994) provides a good overview. Nelson organizes permeability models
into several categories: Carman-Kozeny models, models based on grain size and
mineralogy, models based on surface area and water saturation, well log models and
models based on pore dimension. Models based on grain size and mineralogy and well
log models will not be included in this study. Grain size and mineralogy models require
either destructive grain size and sorting analysis, or microtomographic measurements of
the grain and pore structure, neither of which is practical for our research. Well log
7
8
CHAPTER 2. LITERATURE REVIEW
models require well log data, and are not appropriate for use in fundamental core scale
studies. Not included in Nelson’s paper are fractal models, which are also examined in
this study. The models which are included in this study all incorporate data which is
already measured as a part of our experiments, and are thus readily applicable for
calculating permeability.
2.1 Kozeny-Carman Models
Kozeny-Carman based models are the most common and oldest models used for
estimating permeability. These models treat porous media as a bundle of capillary tubes
of equal length and constant cross section. Kozeny derived Eq. 2.2 for permeability by
solving the Navier-Stokes equation for all tubes passing through a point (Bear, 1972).
The equation contains the terms k, which is permeability in millidarcies (md), co, which
is Kozeny’s constant, M is the specific surface area per unit volume, and ϕ is the rock
porosity. The constant co has values dependent on the flow channel shape, where 0.5
corresponds to a circle, 0.562 for a square, 0.597 for an equilateral triangle and 0.667 for
a strip. Permeability (k) will be in millidarcies from here on unless noted.
𝜙3
𝑘 = 𝑐𝑜 2
𝑀
2.2
Carman (1937) extended Kozeny’s equation to the widely recognized CarmanKozeny equation by writing the specific surface area in units of surface area per grain
volume (av) rather than bulk volume (M).
𝜙3
𝑘= 𝑆 2
𝑎𝑣 1 − 𝜙
2
2.3
where av is the specific surface area per grain volume in a unit volume, recognizing that
the bottom term of Eq. 2.3 gives M2 from Eq. 2.2. S is called the shape factor, but serves
the same function as Kozeny’s constant for predicting permeability. Where data is
available, S is a calibration parameter used to match predicted permeability values to
experimental measurements.
CHAPTER 2. LITERATURE REVIEW
9
By assuming spherical grain shape, the parameter av in Eq. 2.3 can be derived in
terms of average grain diameter. A slightly modified form of Eq. 2.3 to account for the
length of a tortuous capillary tube is shown in Eq. 2.4 as given by Panda and Lake
(1994), where Dp is the average grain diameter and τ is the tortuosity. Tortuosity is
defined as the ratio of the length of a flow tube to the length of the core, generally taken
to be around 2 (Carman, 1937).
𝐷𝑝2 𝜙 3
𝑘= 𝑆
72𝛵 1 − 𝜙
2.4
2
A simple version of Eq. 2.3 can be used as a first estimate on permeability by
including the surface area term in the constant S, given by Eq. 2.5 (Benson et al., 2009).
In this equation, S is calibrated to the measured core average permeability and porosity.
𝑘= 𝑆
𝜙3
1−𝜙
2.5
2
Mavko and Nur (1997) provide an additional modification of the Carman-Kozeny
equation by suggesting that one must account for a known lower bound on porosity at
which the pores become disconnected and flow is no longer possible, called the
percolation threshold. The form is given in Eq. 2.6 where ϕc is the percolation porosity
constant and can be measured experimentally, but is generally between 2 and 5 percent
(Mavko and Nur, 1997). One can see from the equation that when porosity is equal to ϕc,
permeability is equal to zero. The authors provide evidence that this becomes more
important at low porosities, where the standard Carman-Kozeny fails to accurately
predict permeability.
𝜙 − 𝜙𝑐 3
𝑘= 𝑆 2
𝑎𝑣 1 − 𝜙 + 𝜙𝑐
2
2.6
An empirical model based on the Kozeny-Carman form in Eq. 2.3 is shown in Eq. 2.7
below. This model uses a variable power of porosity in the numerator to provide a better
match between experimentally measured and predicted permeability (Mavko and Nur,
1997).
10
CHAPTER 2. LITERATURE REVIEW
𝑘= 𝑆
𝜙𝑛
𝑎𝑣2 1 − 𝜙
2.7
2
2.2 Models Based on Surface Area and Saturation
A commonly used model in the oil and gas industry was proposed by Timur (1968) to
include information about residual water saturation, Swr.
He proposed a general
functional form based on previous work, suggesting the general empirical model in Eq.
2.8, based loosely on the semi-theoretical bundle of capillary tubes model of Kozeny (Eq.
2.2) by empirically replacing the surface area term in the denominator with residual water
saturation.
𝑘=𝑎
𝜙𝑏
𝑐
𝑆𝑤𝑟
2.8
where the coefficients a, b and c are determined statistically and Swr is the residual water
saturation.
By comparing
residual water saturation, permeability and porosity
measurement on 155 samples from three US oil fields, Timur determined values of 0.136,
4.4 and 2 for a, b and c respectively.
This equation was extended by Coates by
multiplying the numerator by (1-Swr) to ensure that permeability goes to zero as the
residual liquid phase goes to unity (Nelson, 1994).
This model has proved to be popular in industry as both residual water saturation and
porosity can be easily estimated using well logging techniques, therefore, incorporating
more information to theoretically improve the estimate of permeability.
2.3 Models Based on Pore Dimension
Nelson (1994) explains that it is the pore dimensions which control permeability, not
porosity or residual water saturation, thereby making the claim that all previous methods
are indirect measurements of permeability. Direct information about the connectivity and
dimensions of the pore network will yield the most direct relationship with permeability.
One straight forward manner of doing this is by using capillary pressure data, which
CHAPTER 2. LITERATURE REVIEW
11
relates the pore radius to the capillary pressure through the Washburn (1921) equation,
shown below.
𝑃𝑐 =
2𝜎 𝑐𝑜𝑠 𝜃
𝑅
2.9
where σ is the interfacial tension and θ is the contact angle between two fluids, R is the
tube radius and Pc is the capillary pressure.
Purcell (1949) was the first investigator to derive the fundamental relationship
between permeability and capillary pressure using a bundle of capillary tubes by
recognizing that permeability is the sum of the permeance of each individual tube in the
bundle. By using Poiseuille’s law for fluid flow in tubes and Darcy’s law for fluid
though through porous media, he showed the relationship in Eq. 2.10 can be used to
estimate permeability.
2
𝑘 = 𝛼 𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃 𝜙
1
0
1
𝑑𝑆
𝑃𝑐2 𝑤
2.10
where α is a fitting factor called the Purcell Lithology Factor, which also includes unit
conversions. In this manner, Purcell showed that permeability could be calculated by
integrating the capillary pressure curve with respect to the wetting phase saturation, Sw.
In order to simplify this correlation, Calhoun et al. (1949) sought to relate
permeability to the displacement pressure, which is the minimum pressure required for a
non-wetting fluid phase to invade a saturated porous media, and to the value of the
Leverett J-Function, J(Sw), as defined by Leverett (1940, 1942) and shown in Eq. 2.11
below. The J-Function is evaluated at wetting phase saturation of 1.0 to be consistent
with defining the permeability in terms of displacement pressure, which is also defined at
wetting phase saturation of 1.0.
𝐽 𝑆𝑤 =
𝑃𝑐
𝑘
𝜎 𝑐𝑜𝑠 𝜃 𝜙
2.11
12
CHAPTER 2. LITERATURE REVIEW
𝑘=
1
𝐽 𝑆𝑤
𝑝𝑑2
2
𝑆𝑤 =1.0
𝜎 𝑐𝑜𝑠 𝜃 2 𝜙
2.12
where σ and θ are the interfacial tension and contact angle between the wetting and non
wetting fluids, and pd is the displacement pressure. The J-Function will be explained in
more detail in Chapters 5, but it is a dimensionless function which was shown by Leverett
(1940, 1942) to reduce to the same dimensionless curve for rocks of different
permeability and porosity, but of similar geological character. From this property of the
J-Function, the curve for J(Sw) may be calculated using one set of capillary pressure
measurements, and can then be applied to calculate the capillary pressure curve for other
rocks of similar geologic character using only porosity and permeability data. Therefore
once J(Sw)|1.0 is known, it can be used in Eq. 2.12 to predict permeability in similar rocks.
Many other authors have used various forms like this, Nelson (1994) and Huet et al.
2005 provide summaries of additional methods. In their paper, paper, Nakornthap and
Evan (1986) derive a new form of Eq. 2.10 by substituting Corey’s (1966) capillary
pressure curve solution for Pc in the equation, shown in Eq. 2.13, then integrating to get
the solution, which Huet et al. (2005) write into the form of Eq. 2.14.
𝑆𝑤 − 𝑆𝑤𝑟
𝑃𝑐 = 𝑃𝑑
1 − 𝑆𝑤𝑟
𝑘 = 10.66𝛼 𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃
2
1
−
𝜆
1 − 𝑆𝑤𝑟
2.13
4
𝜙2
1
𝜆
2 𝜆+2
𝑝𝑑
2.14
where λ is called the pore geometry factor by Brooks and Corey, and 10.66 is for unit
conversion, which can change depending on what units for interfacial tension and
displacement pressure are preferred. Huet et al. (2005), then rewrite Eq. 2.14 into a
general power law relationship, shown in Eq. 2.15, grouping all the scalar constants into
a1.
𝑘 = 𝑎1
1
𝑝𝑑
𝑎2
𝜆
𝜆+2
𝑎3
1 − 𝑆𝑤𝑟
𝑎4
𝜙𝑎5
2.15
CHAPTER 2. LITERATURE REVIEW
13
This final form was fitted to 89 data sets of varying properties from low to relatively
high porosity, permeability and residual wetting phase saturation.
Using regression
analysis, the empirical solutions for the coefficients a1, a2, a3, a4 and a5 are given in Table
2.1. The authors also sought to find conformance with other models, specifically, a
general form of Timur’s equation, shown in Eq. 2.8. This showed that general solutions
such as Eq. 2.8 work just as well as Eq. 2.14, however, they must be calibrated to every
data set, while Eq. 2.14 is considered by Huet et al. (2005) to be a general solution
applicable to rocks with characteristics falling within the range of those given in the
table.
Table 2.1 Coefficients and data range for Huet et al. (2005)
Coefficients of Eq. 2.15
a1
1017003.24
a2
1.7846
a3
1.6575
a4
a5
0.5475
1.6498
Data Range for Calc. Coefficients
Parameter
Min.
Max
k (md)
ϕ (%)
0.0041
8340
0.3
34
Swr (%)
Pd (psia)
0.7
33
2.32
2176
2.4 Fractal Models
The last group of models to be considered is the so-called fractal models. Fractal
shapes can be used to describe porous media by using characteristic radii to model
features of different scale. The size and geometry of these features can be calculated
using specific surface area measurements, such as using the Brunnauer-Emmett-Teller
(BET) method of measuring nitrogen adsorption onto a grain surface (Pape et al., 2000).
This method is explained in more detail as it is relatively new compared to previously
discussed models and may not be familiar to most readers.
Several authors have derived different methods of incorporating fractal shapes into
permeability models, Xu and Yu (2007) derive a fractal model for calculating KozenyCarman constant in Eq. 2.3, while Civan (2001) derives a power law correlation which
uses fractals to describe the pore volume to solids ratio. The most practical model for
this work however is the model by Pape et al. (2000), who derives a power law
14
CHAPTER 2. LITERATURE REVIEW
relationship to porosity for different sandstones using fractal geometry to describe the
pore structure.
Pape et al. (1999) start with a modified version of the Kozeny-Carman equations,
shown in Eq. 2.16, where T is tortuosity and reff is a characteristic effective pore throat
radius. They then use fractal geometry to derive formulas for tortuosity, porosity, and
effective pore throat radii in terms of a characteristic grain radius.
These fractal
equations are then combined and reduced to give tortuosity and effective pore throat
radius as in Eq. 2.17 and Eq. 2.18 respectively (Pape et al., 1999).
2
𝑟𝑒𝑓𝑓
𝑘=
𝜙
8𝑇
𝑇=
2.16
0.67
𝜙
2.17
2
2
𝑟𝑒𝑓𝑓
= 𝑟𝑔𝑟𝑎𝑖𝑛
2𝜙
8
2.18
By combining Eq. 2.17 and 2.18 into Eq. 2.16, and selecting a characteristic grain
radius, the general formula in Eq. 2.19 can be derived, where β is determined from the
characteristic grain radius. Pape et al. (1999) show that Eq. 2.16 is only accurate for
rocks with porosity greater than 10 percent. For low porosity samples, the mean effective
radius is calculated from permeability measurements using Eq. 2.16 and Eq. 2.17. Then,
Eq. 2.16 and 2.17 are used to derive Eq. 2.20, where γ is determined from the measured
mean effective radius. This equation is given by Pape et al. (1999) to be valid for rocks
with porosity of 1 to 10 percent. For rocks with very low porosity, less than 1 percent,
the formulation is rederived using an absolute minimum effective radius, which they
show reduces to Eq. 2.21, and is valid for rocks with porosity less than 1 percent.
𝑘 = 𝛽 10𝜙
10
2.19
CHAPTER 2. LITERATURE REVIEW
15
𝑘 = 𝛾𝜙 2
2.20
𝑘 = 𝛿𝜙
2.21
Rather than using three different equations for calculating permeability, Pape et al.
(1999) make the case that a simple linear combination of all three may be used instead
since the contribution of each equation outside its given range is negligible.
By
averaging over many sandstone samples, an average grain diameter of 200,000 nm, an
effective pore throat radius of 200 nm, an absolute minimum pore throat radius of 50 nm
are used to determine the constants in Eqs. 2.19 – 2.21 respectively, to derive Eq. 2.22 for
an average sandstone. To show that assuming a simple linear combination of all three
equations to get Eq. 2.22 is valid, Figure 2.1 shows the permeability calculated from Eqs.
2.19-2.21 in their respective porosity ranges, and also shows the permeability calculated
from Eq. 2.22 across all porosity ranges from 0 to 25 percent. The figure shows Eq. 2.22
deviates most from the piecewise construction at very low porosity, but matches very
well in the range of 1 to 25 percent porosity, which is within the range of interest for this
study. Eq. 2.23 is derived in the same way as Eq. 2.22, but is for Berea sandstone.
𝑘 = 31𝜙 + 7463𝜙 2 + 191 10𝜙
10
2.22
𝑘 = 6.2𝜙 + 1493𝜙 2 + 58 10𝜙
10
2.23
16
CHAPTER 2. LITERATURE REVIEW
Figure 2.1 Comparison of piecewise terms and Eq. 2.22
Chapter 3
3 Experimental and Simulation Methods
3.1 Multi-Phase Flow Experiments
To study sub-core scale multi-phase flow phenomena, a facility has been developed
to co-inject brine and CO2 into a rock core at reservoir conditions. During the coinjection experiment, X-ray computed tomography (CT) scanning is used to measure the
saturation of the fluids in the core. This saturation data can then be combined with
concurrent measurements of, pressure, temperature and flow rate to study the sub-core
scale multi-phase flow behavior.
3.1.1
Multi-Phase Flow Experimental Facility
First, a 2-inch diameter rock core is placed in an oven for at least 12 hours at 600˚F to
stabilize any clays which might be present in the core. Then the core is set inside a
Teflon® sleeve, placing the two ends of the core against inlet and outlet plates. The
Teflon sleeve is used to seal the ends of the core against these plates, and the core with
the end plates is placed inside an aluminum core holder. Then the end plates are bolted to
the core holder, sealing the core inside, and water is allowed to surround the Teflon
sleeve, and is pressurized to simulate reservoir confining pressure. Next, the core holder
is placed inside a CT scanner and connected to a pair of dual syringe injection pumps and
a fluid separator, then the core is evacuated with a vacuum pump. At this time, the core
is aligned in the CT scanner and the scanner resolution is set. Then an initial scan of the
core is taken before any fluids have been injected, called the dry scan. A schematic of
the system is shown in Figure 3.1.
After the dry scan has been conducted, CO2 is pumped into the entire system using
one of the dual syringe pumps and brought up to reservoir temperature and pressure. A
17
18
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
pump is used to maintain backpressure, which is the reservoir pore pressure. To maintain
reservoir temperature, the CO2 passes through a heat exchanger before entering the core,
and two electric heaters maintain reservoir temperature in the core. After the system
reaches reservoir conditions, a second CT scan of the core is taken, called the CO2
saturated scan. Next, the CO2 is evacuated from the whole system, the other dual syringe
pump is used to flood the whole system with brine. The system is again brought to
reservoir conditions in the same manner as previously described, and a third CT scan is
taken, called the brine saturated scan.
After these scans are complete, the brine saturated core is disconnected from the
system and CO2 and brine are flowed simultaneously through a closed loop connecting
the two dual syringe pumps and the separator. The two fluids are gravity separated in the
separator, from which, the CO2 dual syringe pump refills by drawing CO2 from the top of
the separator, and the brine dual syringe pump refills by drawing brine from the bottom
of the separator. This closed loop circulation is conducted until the CO2 and brine are in
equilibrium with each other, that is, until the CO2 is saturated with brine, and the brine is
saturated with CO2. Once the two fluid components are saturated with each other, they
are referred to as phases, where the brine is an aqueous, or liquid phase composed of
liquid brine and aqueous CO2, and the CO2 is a supercritical, or gas phase, composed of
gaseous CO2 and dissolved brine. Brine and CO2 hereafter refer only to phases, not
components.
If the phases are not in equilibrium with each other, dryout could occur near the inlet
of the core, which is caused by brine evaporating into the CO2 phase. The undersaturated aqueous phase could also dissolve CO2 from the core, giving erroneous
saturation results. Once the phases are in equilibrium, the lines are flushed of CO2 and
reattached to the core holder, bringing the system back up to reservoir pressure.
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
19
Figure 3.1 Relative permeability experiment diagram (Perrin et al., 2009)
3.1.2
Measuring Relative Permeability and Saturation
The experiment starts by injecting 100 percent brine (brine fractional flow of 1) at a
set flow rate to calculate the absolute permeability of the whole core. This is easily
calculated by measuring the pressure drop across the core and using Darcy’s law in Eq.
3.1.
𝑘=
𝑄∙𝜇
𝐴 ∙ 𝛥𝑃
3.1
where Q is the flow rate, μ is the viscosity of brine, L is the length of the core, A is the
cross sectional area of the core, and Δp is the pressure drop across the core. After the
absolute permeability of the core has been measured, the fractional flow of CO 2 being
injected is increased in stepwise increments, waiting at each step until steady state is
reached to take a CT scan of the core, until 100 percent CO2 is being injected. Steady
state is defined as the time after which saturation is no longer changing in the core. At
20
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
each step, the pressure drop across the core is measured at steady state, and Darcy’s law
for multi-phase flow in Eq. 3.2 is used to calculate the relative permeability.
𝑘𝑟,𝑗 =
𝑘𝐿
𝑄∙𝜇
𝐴 ∙ 𝛥𝑃
𝑗
3.2
where j denotes phase j for CO2 and brine, and kr, j is the relative permeability of the
given phase.
The brine saturated and dry scans of the core taken before the experiment started are
used to generate the porosity map of the core using Eq. 3.3. The CO2 saturated scan and
the scan taken at each fractional flow after steady state was reach are used to generate a
saturation map corresponding to that specific fractional flow of CO2 using Eq. 3.4 (Akin
and Kovscek, 2003). In the equations, CTi refers to the absolute CT number of voxel i in
the core. Water by definition has a CT number of 0 and air has a CT number of -1000.
𝑑𝑟𝑦
𝐶𝑇𝑖𝑏𝑟𝑖𝑛𝑒 − 𝐶𝑇𝑖
𝜙𝑖 =
𝐶𝑇𝑏𝑟𝑖𝑛𝑒 − 𝐶𝑇𝑎𝑖𝑟
3.3
where CTiBrine is the CT number measured in voxel i when the core is saturated with
brine, CTidry is the CT number measured in voxel i before injecting any fluids, and CTBrine
and CTair are the previously defined CT values of water and air respectively (brine taken
same as water).
𝑆𝐶𝑂2 ,𝑖 =
𝐶𝑇𝑖𝑒𝑥𝑝 − 𝐶𝑇𝑖𝑏𝑟𝑖𝑛𝑒
𝐶𝑂2
𝐶𝑇𝑖
− 𝐶𝑇𝑖𝑏𝑟𝑖𝑛𝑒
3.4
where CTiexp is the CT number measured in voxel i during the experiment and CTiCO2 is
the CT number measured in voxel i in the CO2 saturated core. In this manner CO2
saturation maps of the core are calculated for each fractional flow rate in the experiment.
We can also use this information to integrate over the whole core to calculate the average
saturation.
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
3.1.3
21
Experimental Results
3.1.3.1 Experimental Conditions
For this investigation, the most homogeneous Berea sandstone core available was
selected. The experimental conditions, core properties and CT scanning information for
this experiment are shown below in Table 3.1. The experimental conditions have been
selected to replicate a saline aquifer storage reservoir where the CO2 would be in
supercritical state. The salt mass fraction is below the US minimum total dissolved solids
of 10,000 ppm which defines a saline aquifer because previous experiments were
conducted to replicate aquifer conditions at the Otway Basin Pilot Project in Australia,
where there is no defined salinity limit for saline aquifers. In that project, CO2 is being
injected into an aquifer with salinity of 6500 ppm.
Table 3.1 Experimental conditions
Experimental Conditions
P (MPa)
T (˚C)
12.41
xNaCl (ppm)
Qt (ml/min)
6500
50
3
Core Description
CT Scanning Information
Diameter (cm)
5.08
Voxel Length (mm)
1
Length (cm)
20.2
Voxel Width (mm)
0.254
Permeability (md)
Porosity (%)
85
Slice Gap (mm)
0.5
18.5
Number of Sices
132
The scanning resolution is fixed by selecting a field of view at the time of the scan,
the field of view has fixed resolution of 512 by 512 pixels. The voxel length is the
thickness of a single scan slice, and was selected at the beginning of the experiment from
a choice of 1, 3 or 5 mm scan length. A voxel length of 1 mm was selected, with gap of
0.5 mm between slices. A gap is not desirable, but due to cooling constraints, it was
necessary in order to limit the number of slices so the entire core could be measured at
one time.
3.1.3.2 Absolute Permeability
To calculate permeability, a value of viscosity must be selected, as seen from Eq. 3.1,
for consistency, the viscosity relationship used by the simulator module ECO2N is used
to calculate brine viscosity. It should be noted however, that introducing small amounts
of NaCl to the system has an almost negligible effect on viscosity. Philips et al. (1981)
give the relationship in Eq. 3.5 for viscosity, which is calibrated to experimental data sets
22
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
and considered accurate within ± 2 percent at temperatures of 10˚C to 350˚C and
pressures of 0.1 MPa to 50 MPa.
𝜇𝐵𝑟𝑖𝑛𝑒
= 1 + .0816𝑚 + .0122𝑚2 + .000128𝑚3
𝜇𝐻2 𝑂
+ .000629𝑇 1 − 𝑒
3.5
−.7𝑚
where m is the molal concentration of NaCl in g-moles NaCl per kg H2O, T is the system
temperature in ˚C and μ is viscosity in centipoises (cp). Because CO2 and brine are
allowed to circulate in the experiment so that they are in equilibrium with each other, the
brine is also saturated with CO2, which has a small effect on viscosity. Kumagai and
Yokoyama (1999) present a correlation shown in Eq. 3.6 for calculating the viscosity of
brine saturated with CO2 for pressure in the range of 0.1 to 30 MPa and temperature in
the range of 0 to 5 ˚C. This correlation has not been verified experimentally at the
temperatures of our study, therefore it will not be used for viscosity calculations.
Furthermore, the effect of CO2 on viscosity is negligible, amounting to at most a few
percent decrease.
1
2
𝜇𝐵𝑟𝑖𝑛𝑒 = 𝑎 + 𝑏𝑇 𝑀𝑁𝑎𝐶𝑙 + 𝑐 + 𝑑𝑇 𝑀𝑁𝑎𝐶𝑙
+ 𝑒 + 𝑓𝑇 𝑀𝐶𝑂2
2
+ 𝑔 + 𝑕𝑇 𝑀𝐶𝑂
+ 𝑖 𝑃 − 0.1 + 𝜇𝐻2 𝑂
2
3.6
(𝑇,𝑃=0.1)
Table 3.2 Coefficients of Kumagai and Yokoyama viscosity relationship
a
b
c
3.8597100
-0.0132561
-5.3753900
d
e
f
0.0190621
8.7955200
-0.0317229
g
h
i
-7.2276900
0.0264498
-0.0016996
where μ is viscosity in mPa·s, T is temperature in K, P is pressure in MPa, Mi is the
molarities of CO2 and NaCl in moles per kilogram H2O. The permeability was calculated
by averaging the calculation over four different flow rates to verify the accuracy of the
measurements, the data is shown Table 3.3 below, with an average permeability of 84.7
millidarcies determined for the core.
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
23
Table 3.3 Permeability calculation data
Flow Rate
(ml/min)
ΔP (psi)
Flow Rate
(m3/s)
ΔP (Pa)
Permebility
(m2)
Permeability
(md)
1.00
1.52
1.67E-08
10480
8.761E-14
88.77
2.00
3.15
3.33E-08
21718
8.455E-14
85.67
3.00
4.83
5.00E-08
33302
8.271E-14
83.81
4.00
6.70
6.67E-08
46195
7.950E-14
80.56
Average
84.70
3.1.3.3 Relative Permeability
After absolute permeability has been measured, relative permeability was measured
as previously described. The fractional flows at which data were recorded are shown in
Table 3.4 below. The experiment was conducted using a total flow rate of 3 ml/min into
the core, starting with 100 percent brine, and gradually increasing the CO2 fractional flow
in the steps shown in the table. At each step, the relative permeability of each phase was
calculated using Eq. 3.2 using the measured pressure drop across the core after steady
state was achieved. The viscosity for brine was calculated using Eq. 3.5 and the viscosity
of CO2 comes from the National Institute for Standards and Technology (NIST)
webbook.
Relative permeability is typically shown as a function of saturation, which in this case
is the average saturation in the entire core. CO2 saturation is calculated for each voxel in
the core at each fractional flow using Eq. 3.4, and then averaged over the whole core.
The resulting core average saturation at each fractional flow is also included in Table 3.4.
The relative permeability data plotted as a function of their corresponding saturation are
shown in Figure 3.2.
Table 3.4 Experimental data for calculating relative permeability
CO2 Flow Rate Brine Flow Rate
(ml/min)
(ml/min)
0.00
3.00
0.15
2.85
0.45
2.55
0.75
2.25
1.20
1.80
1.50
1.50
1.95
1.05
2.35
0.00
fCO2
fBrine
SCO2
SBrine
ΔPss (Pa)
krCO2
krBrine
0.00
0.05
0.15
0.25
0.40
0.50
0.65
1.00
1.00
0.95
0.85
0.75
0.60
0.50
0.35
0.00
0.00
0.04
0.07
0.09
0.12
0.14
0.16
0.51
1.00
0.96
0.93
0.91
0.88
0.86
0.84
0.49
32833
36103
39230
38377
42499
49748
41220
7107
0.0000
0.0038
0.0105
0.0179
0.0258
0.0275
0.0432
0.3021
1.0000
0.8640
0.7114
0.6417
0.4636
0.3300
0.2788
0.0000
24
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
Chart Title
1.00
Brine Relative Permeability
0.90
CO2 Relative Permeability
Relative Permeability
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Brine Saturation
Figure 3.2 Experimental relative permeability measurement
3.1.3.4 Porosity and Saturation Maps
Using Eq. 3.3 and Eq. 3.4, porosity and saturation maps of the core are created, as
shown in Figure 3.3 and Figure 3.4. For expediency, only the saturation map of CO2
measured at 100 percent CO2 injection is shown. The porosity map in Figure 3.3 shows
that there is no apparent structured heterogeneity. There does appear to be a slight
porosity gradient along the core however.
Figure 3.3 Porosity map of experiment Berea core
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
25
The injection direction and gravity vector shown in Figure 3.4 are consistent with all
of the following core images in this report. The figure shows a slight saturation gradient
along the core, with higher average saturation near the inlet and lower average saturation
near the outlet. The figure also shows that while the spatial contrast in porosity is
relatively low, on the order of 10 percent porosity, the spatial saturation contrast is
extremely high, from near zero up to 100 percent CO2.
Injection
𝒈
Figure 3.4 Saturation map of experiment Berea core at 100 percent CO2 injection
The slice average values of porosity and saturation for the core are shown in Figure
3.5. The figure confirms from the saturation map that a small saturation gradient does
exist along the core. This saturation gradient however, is not due to the influence of
gravity as gravity override can be easily distinguished in saturation maps (see Perrin et
al., 2009).
26
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
Slice Average Values
0.25
0.80
0.70
0.60
0.19
0.50
0.16
0.40
0.13
Average Saturation
Average Porosity
0.22
0.30
Porosity
Saturation
0.10
0.20
0
20
40
60
80
100
120
140
Slice Number
Figure 3.5 Slice average porosity and saturation
3.2 Capillary Pressure Measurements
Capillary forces exist when two or more fluids are present in a system due to the
interfacial tension that exists between them. The interface is curved, creating a pressure
difference between them, this pressure difference is termed the capillary pressure.
Capillary pressure can be measured dynamically or statically, however, Brown (1951)
showed that these two methods yield identical results.
The dynamic method uses a centrifuge to simulate large gravitational forces on fluid
in saturated rock samples (Hassler and Brunner, 1945). During the experiment, the
centrifuge does not stop and readings of the amount of fluid displaced from a rock sample
are taken electronically by measuring fluid levels in a collection chamber attached to the
outside of the rock sample.
Static methods consist of the restored state method and mercury intrusion (Hassler
and Brunner, 1945, Purcell, 1949). In the restored state method, a saturated sample is
placed on top of a membrane to which that liquid is wetting and permeable. The
saturated sample is then surrounded by another liquid to which the membrane is not
wetting.
The pressure in the liquid surrounding the sample is then incremented
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
27
successively, while the liquid in the sample is forced out through the membrane at the
bottom. Each time the pressure in the non wetting liquid is incremented, the system is
allowed to come into equilibrium and the saturation in the sample is measured by mass
balance (Hassler and Brunner, 1945). One of the limitations of this method is that true
capillary equilibrium can take a long time to achieve, and a full capillary pressure curve
can take weeks to attain. Another limitation of this method is the maximum pressure that
can be imposed before the non wetting fluid can enter the membrane is typically much
lower than pressures attainable by mercury intrusion.
Mercury intrusion, or mercury injection capillary pressure (MICP), is the most direct
way to measure capillary pressure.
A clean and dry rock sample of any shape or
geometry is placed in a sample holder, the sample is evacuated to very low absolute
pressure and mercury is allowed to surround the sample.
The data is obtained by
successively increasing the mercury pressure and measuring the amount of mercury
intruded into the sample at each pressure interval (Purcell, 1949). At each step a short
interval of time is required for pressure to reach equilibrium, typically less than a minute.
The entire test is usually finished in less than two hours and the maximum pressure
attainable with this method is 60,000 psi or higher.
Mercury intrusion is used to measure capillary pressure on rock samples in this study.
We use a Micromeritics Autopore IV which can measure capillary pressure up to 30,000
psi. When the test is conducted, the measured capillary pressure is for the mercury-air
system, where mercury is the non wetting phase and air is the wetting phase. The
following conversion must be applied to convert the pressure readings to their CO2-brine
system equivalents.
𝑃𝑐,𝐶𝑂2 −𝑏𝑟𝑖𝑛𝑒
𝜎𝐶𝑂2 −𝑏𝑟𝑖𝑛𝑒 𝑐𝑜𝑠 𝜃𝐶𝑂2 −𝑏𝑟𝑖𝑛𝑒
=
𝑃𝑐,𝐻𝑔−𝑎𝑖𝑟
𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃𝐻𝑔−𝑎𝑖𝑟
3.7
where σi and θi are the interfacial tension and contact angle of fluid system i, and Pc,Hg-air
is the set of measured capillary pressure data points. The interfacial tension and contact
angle of mercury-air are relatively well established, although with a range of uncertainty,
but were taken to be 485 dynes/cm and 130˚ respectively. The contact angle between
brine and CO2 was not well established at the initiation of this study and was taken to be
28
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
180˚. The interfacial tension between CO2 and brine is also not well established as a
function of pressure, temperature and salinity, however, Chalbaud et al. (2008) report
values, from which an interpolation of 28.5 dynes/cm was made.
Two capillary pressure curves were generated using two samples of different size, the
small sample weight was 1.870g and the medium sample weight was 3.317g, the sizes
were intentionally different to determine the resolution of the test.
The amount of
mercury intruded into the sample is determined by automatically measuring the amount
of mercury in a penstock attached to the sample holder, which is filled at the time that the
sample is surrounded by mercury at the beginning of the test. Acceptable precision
requires that at least 20 percent of the penstock volume is used up by the end of the test,
otherwise the resolution between data points can have excessive experimental error. The
minimum recommended intrusion volume is 20 percent, (Micromeritics, 2008). The
medium sample used 25 percent of a medium sized penetrometer, and the small sample
used 44 percent of a smaller sized penetrometer stem volumes, so both tests are
considered acceptable.
The capillary pressure results have already been converted to the CO2-brine system
using Eq. 3.7 and are shown for both samples in Figure 3.6. The two tests have the same
characteristic shape and are very close to one another, however, there is some difference,
which must be accounted for by doing a sensitivity analysis on the capillary pressure
curve fit in the following series of numerical simulations.
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
29
Figure 3.6 Measured capillary pressure curves for the CO2-brine system
3.3 Simulation Method
The compositional simulator TOUGH2 MP was used to conduct the simulations of
the core flooding experiment. The simulator was originally developed at Lawrence
Berkeley National Laboratory (Pruess et al., 1999).
The simulator has undergone
significant extension and modification by including new fluid flow modules for different
systems, such as geothermal, hydrology, condensable and non-condensable gas flows,
and a variety of additional fluid systems (Pruess et al. 1999). The code has also been
extended to a parallel version for use on clusters or servers, called TOUGH2 MP (Zhang
et al., 2008).
3.3.1
Description of TOUGH2 MP
TOUGH2 MP is the massively parallel version of TOUGH2 V2.0 and works by
subdividing up the main domain into a series of smaller domains and solving a local flow
problem on each subdomain (Zhang et al, 2008). In each subdomain, the accumulation
and source/sink terms (i.e. injection blocks) are solved locally, then the flow terms are
solved, such that the boundary cells of a given subdomain communicate with the
30
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
boundary cells of the adjacent subdomains, ensuring conservation of mass for each
subdomain.
3.3.1.1 Mass and Energy Balance Equations
In each grid block TOUGH2 is solving a mass flow balance such that all mass flows
and accumulation are conservative. This is solved simultaneously for each grid block,
shown in residual form below (Zhang et al., 2008).
𝑅𝑖𝑘 𝑥 𝑛+1 = 𝑀𝑖𝑘 𝑥 𝑛 +1 − 𝑀𝑖𝑘 𝑥 𝑛
−
𝛥𝑡
𝑉𝑖
𝐴𝑖𝑗 𝐹𝑖𝑗𝑘 𝑥 𝑛+1 + 𝑉𝑖 𝑞𝑖𝑘,𝑛+1 = 0
3.8
𝑗
where the vectors xn, n+1 correspond to the primary variables at the current time step n and
the next time step n+1, i corresponds to block i and k corresponds to component k. M
corresponds to accumulation of component k in block i, F corresponds to mass or energy
flows from blocks j into block i across the interface area A between the two blocks. V
corresponds to the block i volume and q is the injection or production rate in block i, Δt is
the current time step.
After the residual vector is calculated for each component and each block, Newton’s
method is used as shown below in Eq. 3.9 to drive the residual to zero, indicating
convergence (Zhang et al. 2008).
−
𝑚
𝜕𝑅𝑖𝑘,𝑛+1
𝜕𝑥𝑚
𝑥𝑚 ,𝑝+1 − 𝑥𝑚 ,𝑝 = 𝑅𝑖𝑖,𝑛+1 𝑥𝑚 ,𝑝
3.9
𝑝
where m indicates the mth primary variable, p is the current iteration, and the solution is
for iteration p+1 by solving the system for xm, p+1.
3.3.1.2 Thermodynamic Variables
The primary variables in TOUGH2 are dependent upon the flow module being used.
The ECO2N module has been developed for the brine-CO2-NaCl system and uses the
following four primary variables for isothermal systems: pressure, NaCl mass fraction,
CO2 gas (or supercritical fluid) saturation and temperature in ˚C.
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
31
The Redlich-Kwong cubic equation of state (EOS) is used to determine component
partitioning into the two phase gas-aqueous mixture and is correlated to experimental
data as reported by Pruess (2005). Thermodynamic limits of the ECO2N module include
system temperatures of 12 ˚C ≤ T ≤ 110 ˚C, and it cannot represent two or three phase
mixtures of CO2 since the simulator does not make any distinction between liquid, gas or
supercritical CO2 (Pruess, 2005).
Using the cubic EOS, the mole fraction of each component is calculated in each
phase, from which the molality of CO2, n, is calculated and used to calculate the mass
fraction of each component, k, in each phase.
The procedure is illustrated in the
equations below using the mole fraction of CO2 in the aqueous phase, x2 and mole
fraction of H2O in the gas phase, y1, provided from the cubic EOS (Pruess, 2005).
𝑥2 2𝑚 + 1000 𝑀
𝐻2 𝑂
𝑛=
1 − 𝑥2
3.10
𝑋2 =
𝑛𝑀𝐶𝑂2
1000 + 𝑚𝑀𝑁𝑎𝐶𝑙 + 𝑛𝑀𝐶𝑂2
3.11
𝑌1 =
𝑦1 ∙ 𝑀𝐻2 𝑂
𝑦1 ∙ 𝑀𝐻2 𝑂 + 1 − 𝑦1 𝑀𝐶𝑂2
3.12
where m is the molality of NaCl in the brine, which is required as an input, Mk is the
molecular weight of component k, and X2 and Y1 are the mass fractions of CO2 in the
aqueous phase and H2O in the gas phase respectively.
3.3.1.3 Thermophysical Data
The thermophysical data required for simulation are density, viscosity and specific
enthalpy. These properties for CO2 are calculated using experimental data over a range
of pressure and temperature and provided with the ECO2N module as part of the regular
input files. The CO2 density and viscosity calculations assume that the gas phase is pure
CO2.
32
CHAPTER 3. EXPERIMENTAL AND SIMULATION METHODS
Brine density is correlated to experimental data for a range of pressure, temperature
and salinity, and is calculated using the additive densities of brine and dissolved CO2 as
given in Eq. 3.13 (Pruess, 2005).
1
1 − 𝑋2
𝑋2
=
+
𝜌𝑎𝑞
𝜌𝑏
𝜌𝐶𝑂2
3.13
where ρCO2 is calculated as given by Garcia (2001). This formulation neglects the
pressure dependence of partial density of dissolved CO2 since it amounts to only a few
percent of the total aqueous density (Pruess, 2005).
The brine viscosity uses Eq. 3.5 developed by Philips et al. (1981) and is considered
valid for salinities up to 5 molal (230,000 ppm) and assumes that brine viscosity is
independent of dissolved CO2 (Pruess, 2005).
3.3.2 Additional Simulation Comments
The version of TOUGH2 MP used by our research group has additional custom
modifications. A keyword has been inserted into the mesh file which tells the simulator
to keep the outlet slice of the core out of capillary contact with the rest of the core by
setting the capillary gradient between the last two slices to zero (Benson et al., 2009).
This has been shown by trial and error to best represent the experimental conditions
measured in the lab.
In addition to this, a modification to the code has been made to include an additional
capillary pressure function developed by Silin et al. (2009), which is not available in the
commercial release TOUGH2 MP.
Chapter 4
4
Evaluation of Existing Methods for Calculating
Permeability
Using the experimental data that has been obtained, simulations have been conducted
using two types of permeability relationships discussed in Chapter 2. First, the most
common relationship, that of Kozeny-Carman and its various forms are used to calculate
permeability for a series of simulations. Second, the fractal models are used to calculate
permeability in a series of simulations. Finally, some analysis and discussion of the
results of those models is presented.
4.1 Experimental Data Preparation
4.1.1
CT Image processing
To prepare the experimental data, several graphically interactive programs have been
developed to help process and evaluate large volumes of data in an efficient manner. The
CT scanner takes the image of the core in slices, which are reconstructed using these
programs to create 3-dimensional images. To reconstruct the composite image, first, the
center of the core in the CT field of view is determined by visually examining the CT
image of any slice in the core. The radius of the core in pixels is also selected visually
after the center has been determined. To facilitate this, the program CT-view, shown in
Figure 4.1, displays the absolute CT values of a single slice in any CT dataset and
updates the view automatically as the image center and radius are adjusted.
33
34
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Figure 4.1 CT data visualization software CT-view
After the core radius and center are determined, the program CT-Daqs is used to
process the CT files to create different types of output files. The program, with a
screenshot of the input shown in Figure 4.2, is used to load the dry core CT image, brine
saturated CT image, CO2 saturated CT image and experimental CT images discussed in
chapter 3, and uses the image center and radius information from the previous step. The
program CT-daqs assumes that all the CT images have the same center and radius, and
that every slice in the composite image of the core also shares the same center and radius.
Then using Eq. 3.3 and Eq. 3.4, CT-daqs creates a map of the porosity and saturation data
by selecting the Tecplot® or Upscaled Tecplot checkbox.
To calculate permeability, one of the options is selected from the dropdown menu
shown in Figure 4.2, the permeability methods discussed in this study have already been
installed in the program. To assign porosity and permeability to a mesh file for the
simulations, the “Save Slice Porosity and Permeability Files” checkbox is selected; these
output files are used by the program varmesh, developed at LBNL, to assign the geologic
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
35
properties to a generic mesh file. The varmesh program also creates an initial conditions
file, an injection conditions file, and assigns boundary conditions to the mesh file.
Figure 4.2 Image processing software CT-daqs
4.1.2
Upscaling
To keep the simulations tractable, the original CT grid must be averaged, or upscaled
into a courser grid. In addition to this, it is also important to keep the mesh refined
enough to retain the same order of contrast measured at the experimental scale to study
the effect of heterogeneity. For this study, a transverse (in a single slice) upscaling factor
of 5 was selected, meaning that the properties in 25 cells in a slice are averaged into one.
A longitudinal upscaling factor of two was selected, meaning the properties of two slices
were averaged into one. Transverse upscaling is done arithmetically so that for a factor
of five, 25 cells are arithmetically averaged together. Longitudinal upscaling also uses
arithmetic averaging for porosity and saturation, but harmonically for permeability.
36
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Saturation and porosity are scalar values which are directly measured, and arithmetic
averaging is appropriate for upscaling these properties. Permeability however, represents
the ability for porous media to transport fluid, and averaging in the direction
perpendicular to flow is done harmonically. This can be shown by solving Eq. 3.1 for the
pressure drop across each layer perpendicular to flow in a multi-layer system, and then
determining the effective permeability required to transmit a constant amount of fluid, q.
The result of this upscaling procedure on the porosity and 100 percent CO2 injection
saturation maps are shown below in Figure 4.3 and Figure 4.4. Note that the original
experimental maps are also upscaled by a factor of 2 to 1 in the slice plane because of the
very large size of the data files. The figures show that the upscaling procedure has
reduced the spatial contrast in the porosity and saturation maps. The porosity map in
Figure 4.3 (a) has a relatively narrow range of values, and while the spatial distribution of
porosity in (b) has been smoothed out, the level of contrast in the core is comparable to
the original image.
Figure 4.3 (a) Experiment porosity map, (b) Upscaled simulation porosity map
In contrast to the porosity map, the saturation map in Figure 4.4 (a) shows values in
the range of zero to 100 percent CO2 saturation, and the upscaled image in (b) shows a
significant amount of smoothing compared to the original image. It is therefore possible
that when averaging adjacent cells with very large differences in CO2 saturation, the
upscaled cell may have a very different value than any of the original cells.
The
sensitivity of the simulations to this upscaling procedure has not been evaluated in this
study, but numerical effects of the upscaling will be discussed with the simulation results.
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
37
Figure 4.4 (a) Experiment saturation map (b) Upscaled saturation map
4.1.3
Relative Permeability
TOUGH2 offers a number of built in functions for fitting relative permeability
curves, however, there is no option to use a table of data points, therefore, a built in
function which best matches the data must be used. Our version of TOUGH2 has some
custom features, one of which is an additional relative permeability function which is not
available in the general release. The functions we use are shown below in Eq. 4.1 for
brine relative permeability and Eq. 4.2 for CO2 relative permeability.
𝑘𝑟,𝐵𝑟𝑖𝑛𝑒
𝑆𝐵𝑟𝑖𝑛𝑒 − 𝑆𝑙𝑟
=
1 − 𝑆𝑙𝑟
𝑘𝑟,𝐶𝑂2 =
𝑆𝐶𝑂2 − 𝑆𝑔𝑟
1 − 𝑆𝑙𝑟𝑛
𝑛 𝐵𝑟𝑖𝑛𝑒
𝑛 𝐶𝑂 2
4.1
4.2
where nBrine and nCO2 are exponential fitting parameters, Slr and Sgr are the residual liquid
and gas (CO2) saturation respectively and Slrn is an adjustable parameter which allows the
endpoint gas relative permeability to be less than 1, an option not available in other curve
fits.. The brine and CO2 relative permeability fits using the above equations with the
corresponding parameters are shown below in Figure 4.5.
The data is shown as a
function of normalized brine saturation, given by Eq. 4.3, which is the standard way of
displaying relative permeability and capillary pressure data. The fit does not correspond
exactly with the data because the form of the functions above do not allow a perfect fit,
however, the functions do provide a better fit than other options in TOUGH2. The
38
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
parameters were also adjusted slightly from the R2 value closest to one after several
simple history matching tests on a homogeneous core to get a better match to the average
saturation value.
𝑆𝐵𝑟𝑖𝑛𝑒 ,𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 = 𝑆 ∗ =
𝑆𝐵𝑟𝑖𝑛𝑒 − 𝑆𝑙𝑟
1 − 𝑆𝑙𝑟
4.3
Figure 4.5 Relative permeability curve fit
nBrine
4.20
Relative Permeability Fit Parameters
nCO2
Sgr
Slr
2.00
0.20
0.0
Slrn
0.0
The value of residual liquid phase saturation, Slr, was selected based on previous
work by Kuo et al. (2009), who found that the residual liquid phase saturation measured
in the relative permeability experiment does not necessarily represent the true value. In
Figure 4.5, the residual liquid saturation is the data point at the lowest brine saturation on
the brine relative permeability curve (blue) because at this point, injecting 100 percent
CO2 does not further reduce the brine saturation. However, work by Perrin et al. (2009)
has shown that increasing the flow rate can reduce this residual liquid saturation,
therefore, its true value should be determined by other methods such as history matching,
Slr of 0.20 was selected based on the several history matching simulations previously
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
39
mentioned and should be the same for both the capillary pressure and relative
permeability functions.
4.1.4
Capillary Pressure
4.1.4.1 Capillary Pressure Curve Fits
Leverett (1941) showed that scaling capillary pressure data by the dimensional
group 𝑘 𝜙 /𝜎, on the left side of Eq. 4.4, and plotting the results vs. wetting phase
saturation, resulted in a curve, denoted by J(Sw) and called the J-Function. Furthermore,
he showed that the capillary pressure data for cores of different permeability and porosity
collapsed to a single J(Sw) curve when plotted in this non-dimensional form. From this,
we can assume that if we know the functional form of the J-Function, we can infer the
specific capillary pressure curve for rocks of different properties, given by Eq. 4.5, which
includes an additional cos 𝜃 term not in Leverett’s original definition to account for the
contact angle between the two fluids.
𝑃𝑐
𝜎
𝑘
= 𝐽 𝑆𝑤
𝜙
𝑃𝑐,𝑖 = 𝜎 𝑐𝑜𝑠 𝜃
4.4
𝜙𝑖
𝐽 𝑆𝑤
𝑘𝑖
4.5
A number of investigators have developed functional forms for J(Sw) (Brooks and
Corey, 1966, Van Genuchten, 1980), and some of these are available in TOUGH2. A
new form which provides a better curve fit to typical sandstone capillary pressure curves
was developed by Silin et al. (2009) and is available in the version of TOUGH2 used for
this study, it is shown below in Eq. 4.6. The parameters A, B and λi are empirical fitting
parameters determined by the user to provide the best curve fit. The curve fits of the two
capillary pressure shown in Figure 3.6 using this functional form for J(Sw) are shown in
Figure 4.6 and Figure 4.7.
𝐽 𝑆𝑤 = 𝐴
1
𝜆
𝑆∗ 1
− 1 + 𝐵 1−
𝜆
𝑆∗ 2
1
𝜆2
4.6
40
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Figure 4.6 Capillary pressure curve fit for medium sized Berea sample (ICP1)
A
.004
ICP1 Capillary Pressure Fit Parameters
B
Slr
λ1
0.2
0.20
3.4
λ2
2.8
One can see from the figures that the curve fits still do not match all of the data
points, particularly at the ends. The capillary pressure curve shape is typical of most
rocks and has proven difficult to precisely fit using functional forms for the J-Function
because of its distinctive shape. Due to the curve shape and the number of fitting
parameters in Eq. 4.6, using a simple fitting procedure to vary the parameters and set the
coefficient of variation (R2) to 1 does not work, therefore the parameters are manually
adjusted to get a subjective curve fit. The curve fits in these figures were selected to best
match the middle range of the capillary pressure data, where most of the saturation values
in the simulations are expected to be. The important difference between the two curve
fits is that ICP1 in Figure 4.6 has a much flatter plateau region than ICP2.
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
41
Figure 4.7 Capillary pressure curve fit for small sized Berea sample (ICP2)
A
.01
ICP2 Capillary Pressure Fit Parameters
B
Slr
λ1
0.2
0.20
2.92
λ2
2.0
4.1.4.2 Unique Capillary Pressure Curves
One can see from Eq. 4.4, that the subscript i has been added, this signifies that every
voxel in the simulation mesh has a unique capillary pressure curve, calculated by the
function J(Sw) and scaled to that voxels unique porosity and permeability values. Recall
that it was stated that J(Sw) is dimensionless and was shown by Leverett (1941) to be the
same for cores of different properties. Therefore, we assume that each voxel also has the
same J(Sw) function.
Therefore, once we have determined the J-Function fitting
parameters for ICP1 and ICP2, it is possible to directly calculate each voxels unique
capillary pressure curve.
This implies that permeability has the additional function of scaling the original
capillary pressure data to determine each voxels unique capillary pressure curve. From
this relationship, we can see that each permeability relationship also carries with it, a new
and unique set of capillary pressure curves. It is this concept which we hypothesize is
partly responsible for the large variations in saturation distribution as a result of only
small variations in geologic parameters. This is also the reason that finding an accurate
42
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
method to calculate permeability is the most important step in studying these sub-core
scale saturation phenomena.
4.1.5 Model Validation
4.1.5.1 Base Case Model
To ensure that the model is running appropriately, a series of four simulations have
been conducted to validate the model using Eq. 2.5 as the test case for permeability.
These simulations were also conducted to determine a base case for the actual study. As
stated in section 4.1.3, several quick simulations were conducted on a homogeneous core
to establish a good relative permeability fit based on the core average saturation in the
simulation and correlated to the experimentally measured average for the 100 percent
CO2 injection case, therefore, RP1 will not be examined again here.
Next, the base case was set up using the parameters shown in Table 4.1. In the table,
most of the information is given in Chapter 2, but the average core permeability, which
was given as 85 md in Table 3.3 and is shown as 89 md below, this is due to a minor
discrepancy made in the initial calculation, however, the difference of 4 md should not
affect the outcome. To determine the amount of CO2 that dissolves into the brine at
phase equilibrium, trial and error was used to determine the dissolved CO2 mass fraction
where CO2 began to evolve out of the brine before any injection occurred. The interfacial
tension (σ) is calculated by interpolating the data of Chalbaud et al. (2009). The injection
fractional flow for all of the simulations was selected to be 100 percent CO2. The
injection of CO2 does not account for brine dissolved in the CO2 because the amount is
very insignificant and dry out near the core inlet should not be a problem for such low
injection volumes.
Table 4.1 Simulation initial conditions
Simulation Conditions
T (˚C)
P (MPa)
50
12.41
Thermophysical Data
init
Injection Conditions
0.04873
QCO2-Gas (kg/s)
3.04E-05
3
608.38
QCO2-Aq (kg/s)
0.00E+00
3
Dissolved CO2
(mf)
ρCO2 (kg/m )
xNaCl (ppm)
6500
ρH2O (kg/m )
993.33
QH2O-Gas (kg/s)
0.00E+00
φave
0.185
σCO2-Brine (N/m)
0.0285
QH2O-Aq (kg/s)
0.00E+00
QNaCl (kg/s)
0.00E+00
kave (md)
89
Injection Rate (ml/m)
3
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
43
The base mesh is shown in Figure 4.8 with the summary of the grid shown in Table
4.2.
The cell dimensions are not exactly cubic because that would have required
excessive upscaling in the planar direction. Also notice that the number of cells in each
slice is the number of cells in the circular slice, not the number of cells in a square in
which the circle is inscribed, in this sense, the entire mesh is actually a cylinder. The
number of slices in the CT data set given in Table 3.1 was 132, the upscaling factor in the
longitudinal direction is 2:1, which gives 66 active simulation slices, however there is
also an inlet and outlet slice added to the core, giving 68 total slices. The inlet slice is
where the injection occurs. Fluids are injected into these grid elements and allowed to
intrude the inlet end of the core in whichever flow paths the simulator finds. The outlet is
maintained out of capillary contact with the core by setting the capillary gradient between
the outlet slice and the last slice of the core to zero. The outlet grid elements have very
large volumes so that the initial pressure is maintained during the simulation, which is the
same outlet pressure condition as the experiment.
Injection
𝒈
Figure 4.8 Grid used for simulations
Table 4.2 Simulation grid data
Y-Z Upscaling
X Upscaling
5:1
2:1
Grid Dimensions
3
Cell Size (mm )
36x36x68
Cells/Slice
936
1.27 x 1.27 x 3
Total Cells
63648
4.1.5.2 Base Case Results
The model validation consisted of four simulations, test case one uses capillary
pressure curve ICP2 on a heterogeneous core, test case two uses capillary pressure curve
ICP2 on a homogeneous core, test case three uses capillary pressure curve ICP2 on a
44
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
heterogeneous core without gravity and test case four uses capillary pressure curve ICP1
on a heterogeneous core. The main concern in the method we use to conduct the
experiments is gravity override, since the core is horizontal, the system is unstable, if
injection were to stop, the fluids would redistribute themselves due to buoyancy forces
caused by the density difference between brine and CO2. Often core flood experiments
are done vertically, with the lighter fluid injected in the top, this leads to a stable system
because if injection stops, the fluids, theoretically, will not redistribute themselves,
however, experimentally, this is much harder to conduct due to the constraints of the CT
scanner position.
The results of the four simulations are shown below in Figure 4.9 for injection time of
6000s, or dimensionless time of 4 pore volumes injected (PVI). Injection time tau (τ), is
typically reported in terms of PVI since it has more physical meaning than the actual
injection time in seconds does. The results are shown in the cross section of the plane
passing vertically down the length of the core to highlight any gravity effect.
Test case two in (b) is the homogeneous core, which would show the strongest
gravity effect, however, there is no obvious separation of phases present in the
simulation, thereby confirming that the flow rate selected should be free of gravity
effects. Comparing image (a) and (c) also confirms that gravity does not have an effect
on the simulation results because the results of the no gravity case in (c) are the same as
when gravity is present in (a). Lastly, comparing image (a) with image (d), whose only
difference is the capillary pressure relationship, we can see that using ICP1 in (d) results
in greater saturation contrast than using ICP2 in (a).
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
45
Figure 4.9 Test case results after 4 PVI (a) case 1, (b) case 2, (c) case 3, (d) case 4
The results show that the simulations may not have reached steady state because there
is a saturation gradient across the core in all of the images in the figure.
The
homogeneous core should not have a saturation gradient at steady state, and therefore has
not yet reached it. CO2 dissolves on the order of 0.3 percent brine by mass (Pruess,
2005), which amounts to slightly less than 0.7g of brine at 4τ, or about 11 percent of the
brine in the inlet slice, which means dryout may be important at early times near the inlet,
and will certainly have an effect at times longer than 4τ. Additional simulations on the
homogeneous core showed that at 8τ, the change in average saturation in the core is only
3 percent, therefore, the effect is not large.
The heterogeneous cases in Figure 4.9 also appear to have a saturation gradient;
however, this is not necessarily due only to dry out. Recall Figure 3.5, which showed
that the experimental results had a saturation gradient; in addition the figure shows that
the core has lower average porosity near the outlet. It will be shown later that saturation
is closely related to porosity in these simulations, therefore, some saturation gradient
should be expected. To reduce the chance that steady state will not be reached however,
simulation time was increased to 5.3τ for the remaining simulations.
Lastly, the results in Figure 4.9 show that using ICP1 resulted in more saturation
contrast than ICP2 by comparing image (d) with image (a) respectively. It was stated in
Chapter 1 that the saturation contrast in the experiment was very high, therefore, this is
desirable if we want to match the experiment as best as possible, therefore, ICP1 will be
used for the cases presented in this chapter.
46
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
4.2 Saturation Results using Kozeny-Carman Models
4.2.1 Permeability Maps of the Kozeny-Carman Models
Four of the Kozeny-Carman type models were selected for the first set of simulations.
The four permeability models are shown in Eq. 4.7 – 4.10, with accompanying
parameters shown in Table 4.3. The table shows that the value of S, the empirical shape
factor, varies greatly from one relationship to another, this is in some part caused by the
upscaling, since it is harmonic, small permeability values have a disproportionate effect
on the upscaled permeability, resulting in a large range of S depending on the
permeability relationship selected. The value of ϕc for Eq. 4.10 in the table was selected
from a range of typical values given by Mavko and Nur (1997) and was not determined
exactly for this Berea core.
𝜙𝑖3
𝑘𝑖 = 𝑆
1 − 𝜙𝑖
𝑆
𝜙𝑖3
𝑘𝑖 = 2
𝑎𝑣 1 − 𝜙𝑖
𝜙𝑖5
𝑘𝑖 = 𝑆
1 − 𝜙𝑖
𝑘𝑖 = 𝑆
4.7
2
4.8
2
4.9
2
𝜙𝑖 − 𝜙𝑐 3
1 − 𝜙𝑖 + 𝜙𝑐
4.10
2
Table 4.3 Simulation 1-4 permeability parameters for Kozeny-Carman models
Simulation
1
Equation
4.7
2
4.8
3
4.9
4
4.10
Parameter(s)
S
S
av
S
S
ϕc
Value
9253.5
650.4
Eq. 4.11
260846
28431
0.04
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
47
4.2.1.1 Description of Calculating Specific Surface Area
To calculate the parameter av, a custom program was written using Matlab’s® image
processing toolbox. The details of the method are explained in Appendix A for the
interested reader, but a quick explanation is given here.
Thin sections were acquired of the Berea core used in the experiment. Using Matlab,
the image is converted to binary format (black and white) and the pore space is analyzed
to determine the porosity and perimeter in a small sample area of the thin section called a
region of interest (ROI). This process is repeated for many ROI’s, in each one, producing
a data point of perimeter per grain area (specific perimeter) vs. porosity, with the
composite data set of the thin section shown in Figure 4.10.
Figure 4.10 Plot of specific perimeter vs. porosity of homogeneous Berea sample
By fitting a curve to the data in Figure 4.10, we can get an equation for specific
perimeter as a function of porosity. If we assume that the amount of perimeter per unit
grain area is directly proportional to the amount of surface area per unit grain volume,
then av can be written as the curve fit in the figure, where the constant multiplier from the
proportionality assumption can be combined with S in Eq. 4.8. This process is explained
in more detail in Appendix A. By combining the Eq. 4.11 for av with Eq. 4.8, and
reducing, the result in Eq. 4.12 is an equation only in terms of porosity, which was
48
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
previously stated to be the easiest form to calculate permeability because the porosity of
the core is easily measured by the CT scanner.
𝑎𝑣,𝑖 = .033𝜙𝑖.79
𝜙𝑖1.42
𝑘𝑖 = 𝑆
1 − 𝜙𝑖
4.11
2
4.12
4.2.1.2 Permeability Maps
Using the equations for permeability given above and using the experiment porosity
map from Figure 4.3(a), the permeability was calculated for these four simulations and
upscaled in the manner previously described, the results are shown below in Figure 4.11.
The figure shows that the main difference between the permeability maps is the level of
contrast between the high and low permeability values, which is expected since each
relationship is only a function of porosity.
Based on the figure, (b) is the low
heterogeneity case and (c) is the high heterogeneity case, with the maps in (a) and (d)
showing relatively little qualitative difference, although (d) does appear to have more
contrast than (a).
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
49
Figure 4.11 Permeability maps using Kozeny-Carman models for (a) Simulation 1 (b)
Simulation 2 (c) Simulation 3 (d) Simulation 4
4.2.2
Comparison of Kozeny-Carman Model Results with Experiment
To better compare the simulation and experimental results, a single slice, from the
experiment and each simulation is shown, but the qualitative and quantitative match to
the experiment can be shown to be the same in all slices in the core in every simulation in
this report. Slice 29 was selected because it has the same average saturation as the whole
core and is far away from any end effects present in the experiment and simulations.
The results are shown in Figure 4.12 for simulations 1-4. The results show that none
of the Kozeny-Carman models matches the experimentally measured saturation values
very well. Qualitatively, the match is poor, both in absolute value and in the spatial
distribution of CO2. The experiment clearly has the largest amount of spatial contrast in
CO2 saturation, while simulation 3, the model with the highest level of CO2 saturation
contrast, does not approach the level of contrast in the experiment. Moreover, the model
which includes information about the specific perimeter actually does the worst job of
predicting saturation distribution, it appears almost homogeneous using the selected
saturation scale.
50
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
51
Figure 4.12 CO2 saturation in slice 29 (a) Experiment (b) Sim. 1 (c) Sim. 2 (d) Sim. 3 (e) Sim.
4
52
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
To have a quantitative understanding of how well each simulation is able to predict
the experimentally measured saturation at the sub-core scale, we can plot the simulation
saturation results vs. the experimentally measured results in this slice, where a perfect
correlation results in a 45 degree line across the graph. The results of this are shown in
Figure 4.12, plotted in order of CO2 contrast, with the perfect correlation line shown in
purple. The figure shows that there is no observable correlation in the spatial value of
saturation for any of the simulations. The figure also shows that the range of saturations
measured in the experiment is much larger than the range of saturations predicted in the
simulations, even in the highest contrast case. Additional discussion is found in section
4.4 and at the end of chapter 5.
Figure 4.13 Simulation vs. experiment saturation in slice 29 for Kozney-Carman models (in
order of saturation contrast)
4.3 Saturation Results of Fractal Models
The previous simulations showed that the Kozeny-Carman models do not adequately
predict permeability at the sub-core scale, therefore, a different type of model built on the
same principle of predicting permeability from porosity was selected for investigation.
These models are described in detail in Chapter 2, but two models are taken directly from
Pape et al. (2000). The models are shown below, where Eq. 4.13 is used to calculate
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
53
permeability in an average sandstone and Eq. 4.14 is used to calculate permeability
specifically for a Berea sandstone (see Pape et al., 2000), ki is given in nm2 rather than
md for the fractal models. The fitting parameters S, for the equations are shown in Table
4.4.
𝑘𝑖 = 31𝜙𝑖 + 7463𝜙𝑖2 + 191 10𝜙𝑖
10
4.13
𝑘𝑖 = 6.2𝜙𝑖 + 1493𝜙𝑖2 + 58 10𝜙𝑖
10
4.14
Table 4.4 Simulation 5-6 parameters for fractal models
Simulation
5
6
Equation
4.13
4.14
Parameter
S
S
Value
0.000799
0.002923
Calculating permeability in the same manner as described in section 4.2 gives the
resulting permeability maps in Figure 4.14 using Eq. 4.13 and Eq. 4.14 for simulations 5
and 6 respectively. The figure shows that the two relationships are nearly identical for
this core, however, the amount of contrast in permeability is much larger than the
Kozeny-Carman permeability maps in Figure 4.11. This is due to the nonlinearity of the
dependence of permeability on porosity in the fractal models, which is much greater than
the Kozeny-Carman models simply by observation of the powers to which porosity is
raised in the fractal models.
54
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Figure 4.14 Permeability maps of fractal models (a) Simulation 5 (b) Simulation 6
The simulation results for slice 29 using the permeability maps in Figure 4.14 are
shown in Figure 4.15. The two models show nearly identical saturation results, which is
expected because of the very similar permeability maps. From the figure, The level of
spatial saturation contrast in these models is much greater than the Kozeny-Carman
models and is much closer to the contrast measured in the actual experiment.
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
55
Figure 4.15 CO2 saturation in slice 29 (a) Experiment (b) Sim. 5 (c) Sim. 6
Plotting the simulation results for slice 29 vs. the experimental measurements can
again show the level of accuracy of the spatial saturation prediction from the simulation.
Plotting the simulation 5 and 6 results in Figure 4.16 in order of saturation contrast, it is
apparent that the spatial saturation prediction from these simulations still does not match
the experimental measurements; simulation 1 is also shown on the figure for reference.
The figure shows that the range of saturations in simulations 5 and 6 is greater than in
simulation 1-4 (Figure 4.13), which quantitatively confirms the qualitative contrast seen
in Figure 4.15.
56
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Figure 4.16 Simulation vs. experimental saturation in slice 29 for fractal models (in order of
saturation contrast)
4.4 Discussion of Porosity Based Model Results
4.4.1 Examination of Core Scale Results
The results presented in the previous sections consistently indicate that traditional
porosity based permeability methods for simulation of sub-core scale phenomena do not
accurately reproduce experimentally measured saturation.
While this study is not
completely exhaustive, there is an indication that extrapolating permeability from
porosity using a power law relationship is not accurate enough for sub-core scale
permeability prediction, and that another approach may be required.
There are many equations for predicting permeability using information in addition to
porosity, some of the general forms are presented in Chapter 2. If we consider the
general form derived by Huet et al. (2005), shown below in Eq. 4.15, we see that it
includes three additional parameters, displacement pressure at 100 percent wetting phase
saturation, pd, index of pore size distribution, λ, and residual liquid saturation, Swr.
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
𝑘 = 𝑎1
1
𝑝𝑑
𝑎2
𝜆
𝜆+2
𝑎3
1 − 𝑆𝑤𝑟
𝑎4
𝜙𝑎5
57
4.15
If we consider the equation however, it is apparent that once values of pd, λ and Swr
are determined for a rock type, these are simply constants which are raised to some
respective power ai, and multiplied by porosity raised to some power a5, which reduces to
a general power low for permeability as a function of porosity. The same can also be said
about all of the equations presented in Chapter 2, therefore, these permeability
relationships are unlikely to significantly improve the correlation between simulation
predicted saturation and experimentally measured saturation.
These permeability equations are usually calibrated to core scale measurements of
permeability and other parameters to determine accurate correlations, not to sub-core
scale studies. The equations have often been shown to be quite accurate at predicting
core scale and larger permeability (see Nelson, 1994), but it has never been shown that
they are accurate at predicting sub-core scale permeability and it may be inappropriate
use them to extrapolate down to this scale.
The experimentally measured core average values of CO2 saturation and pressure
drop across the core are 50.26% and 7059 Pa respectively.
Comparing these
experimental values with the simulation values in Table 4.5, we can see that the
simulations to an excellent job of predicting the average CO2 saturation, and an
acceptable of predicting the pressure drop. The simulations predict a larger pressure drop
than was measured, this could indicate that the relative permeability relationship is
incorrect, or that the core average permeability calculation is incorrect, each of which
will have a strong influence on the pressure drop, therefore, a more thorough history
match for the base case could further improve the results.
58
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Table 4.5 Core average results using traditional permeability models
Simulation
1
2
3
4
5
6
Average CO2
Saturation
0.5007
0.4957
0.5053
0.5027
0.5036
0.4945
Saturation
Error (%)
0.3780
1.3729
0.5372
0.0199
0.1990
1.6116
Average ΔP
(Pa)
9222
9372
9155
9157
9638
9100
Pressure
Error (%)
30.64
32.77
29.69
29.72
36.53
28.91
4.4.2 Experiment Porosity-Saturation Relationship
To understand why these relationships fail to accurately predict the spatial saturation
distributions, we can examine the relationship between porosity and saturation in the
simulation and the experiment.
Plotting the simulation saturation values vs. their
corresponding porosity values in Figure 4.17 for slice 29 using selected permeability
relationships, we can see that there is a clear relationship suggested between saturation
and porosity.
The results in the figure also show that there is a very important
relationship between the degree of permeability contrast and corresponding CO2 contrast
when using porosity based permeability models.
Figure 4.17 Comparison of simulation saturation vs. porosity in Slice 29 (in order of
saturation contrast)
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
59
Making the same plot of the experimentally measured saturation for three different
slices near the inlet, middle and outlet end of the core, we can see in Figure 4.18 that
there is no discernable relationship between CO2 saturation and porosity.
This
relationship has been shown to be consistent for several different cores tested under very
similar conditions, with only a very heterogeneous core with structured heterogeneity
showing only a very weak relationship between saturation and porosity (Perrin et al.,
2009).
From the figure, it is apparent that there is not necessarily a direct relationship
between saturation and porosity, but using porosity-based permeability predictions causes
such a relationship to exist in numerical simulations. Therefore, while porosity-based
permeability estimation may be very useful for core scale predictions, these relationships
do not appear to be appropriate for calculating sub-core scale permeability.
Figure 4.18 Comparison of experimentally measured saturation vs. porosity
4.4.3
Conclusions
Based on this analysis, we conclude that if porosity based permeability models force
saturation to be a function of porosity, and experimental results show no distinguishable
relationship between saturation and porosity, another approach is required. While there
are many different permeability relationships in the literature that were not discussed in
60
CHAPTER 4. EXISTING PERMEABILITY METHOD RESULTS
Chapter 2, many of them require complex grain analysis and many reduce down to a
function of porosity.
Additionally, there are other models which take advantage of capillary pressure data
to predict permeability because, as Nelson (1994) explains, it is the pore throats, not the
pores themselves which control permeability. Since capillary pressure data is a direct
measurement of pore throat structure, it is possible to use the data to improve
permeability predictions. This approach is the subject of the next chapter.
Chapter 5
5
A Proposed Method for Calculating Sub-Core Scale
Permeability
As Nelson (1994) explains, it is the pore throats, not the pores themselves which
control how fluid moves through porous media. Chapter 2 refers to several investigators
(Purcell, 1949, Calhoun et al., 1949, Huet et al., 2005) who have used capillary pressure
data to derive information about the pore throats to create permeability relationships. In
this chapter, a new method is proposed for calculating permeability using capillary
pressure data, which builds on the work of previous investigators.
5.1 Using the Leverett J-Function for Calculating Permeability
5.1.1
Previous Investigations
5.1.1.1 Purcell’s Permeability Equation
From chapter 2, Purcell (1949) proposed that permeability could be directly
calculated using a capillary bundle model and integrating over the inverse of the capillary
pressure curve squared. For reference, Purcell’s equation is shown again below in Eq.
5.1. Using the capillary pressure curves in chapter 3, it is possible to do this integration
numerically for the whole core. However, since capillary pressure is only measured on a
representative sample of the whole core, no information is available for the unique
capillary pressure curve for each voxel. Therefore, the integration is not unique to a
voxel and a different approach is required.
2
𝑘 = 𝛼 𝜎𝐻𝑔−𝑎𝑖𝑟 𝑐𝑜𝑠 𝜃 𝜙
1
0
61
1
𝑑𝑆
𝑃𝑐2 𝑤
5.1
62
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
5.1.1.2 Calhoun et al. Permeability Equation
Calhoun et al. extended Purcell’s work by determining a theoretical form for the
Lithology Factor constant, shown as α in Eq. 5.1, introduced by Purcell. In the course of
this, Calhoun showed that permeability can be calculated as a function of the capillary
pressure by solving Leverett’s J-Function (Eq. 5.2) at 100 percent wetting phase
saturation, as shown below in Eq. 5.4.
𝐽 𝑆𝑤 =
𝐽 𝑆𝑤
𝑘=
𝑃𝑐
𝑘
𝜎 𝑐𝑜𝑠 𝜃 𝜙
𝑆𝑤 =1.0
1
𝜙 𝐽 𝑆𝑤
𝑝𝐷2
5.2
𝑃𝐷
𝑘
𝜎 𝑐𝑜𝑠 𝜃 𝜙
=
1.0
2
5.3
2
𝜎 𝑐𝑜𝑠 𝜃
5.4
5.1.2 Extension of Calhoun et al. Permeability Equation
Equation 5.4 has the same problem as Purcell’s equation in Eq. 5.1 in that J(Sw)1.0 is
the same for each voxel by the definition of the J-Function, and unless displacement
pressure, pD, is known for each voxel, the core average pD must be used to solve Eq. 5.4
for permeability, once again, resulting in permeability as a linear function of porosity.
In the course of the derivation, Calhoun et al., did not specify a theoretical reason for
selecting pD as the value at which to solve J(Sw), it was just convenient for their
derivation. However, Leverett (1941) showed that Eq. 5.2 is true at all saturations,
therefore, Eq. 5.3 may be solved for any value of Sw and its corresponding capillary
pressure. This introduces saturation as a second sub-core scale measured parameter for
use in calculating sub-core scale permeability. Since we do not require that saturation
have any specific value, Eq. 5.4 may be rewritten in the general form shown in Eq. 5.5.
𝑘𝑖 =
1
𝜙 𝐽 𝑆𝑤
𝑝𝑐2 𝑖
2
𝜎 𝑐𝑜𝑠 𝜃
2
5.5
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
63
where the J-Function is given by Eq. 4.6. Substituting Eq. 4.6 into Eq. 5.5 gives Eq. 5.6
for permeability of element i as a function of porosity and saturation of element i,
measured in the experiment.
𝑘𝑖 = 𝑆 ∙ 𝜙𝑖 𝐴
1
𝜆
𝑆∗,𝑖1
𝜆
− 1 + 𝐵 1 − 𝑆∗,𝑖2
1
2
𝜆2
𝜎 𝑐𝑜𝑠 𝜃
𝑃𝑐
2
5.6
The empirical factor S has again been added to the function to ensure that the core
average permeability value is in agreement with the experimentally measured value. The
value of capillary pressure used in Eq. 5.6 is calculated using the given capillary pressure
curve fit, evaluated at the core average saturation as measured in the experiment. This
average value is used because at steady state, capillary pressure must be the same
everywhere in the core except near the ends where there is a minor end effect
(Richardson et al., 1952). If capillary pressure is not the same, pressure gradients would
be induced in the core, and the fluids will redistribute themselves unless a capillary
barrier exists to prevent this. To visualize how all of the experimental data is used to
calculate a permeability map using this method, a flowchart is provided in Figure 5.1.
Figure 5.1 Flow chart for calculating permeability using capillary pressure data
64
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
5.1.3 Capillary Pressure Curve Fits
In Section 4.1.4, it was shown that the parameters A, B, λ1 and λ2 are not unique and
therefore, the continuous function used to describe capillary pressure is subjective. This
creates a uniqueness problem with Eq. 5.6 because there can be many realizations of
permeability from the same dataset depending on how the user chooses to fit the data,
therefore, three different curve fits were selected to test the validity of this method. As an
alternative method, it would be possible to create an interpolation procedure which uses
the experimental data to calculate permeability directly from the measured data. In order
to maintain consistency between the capillary pressure curves used in the simulation and
the permeability calculation, the function in Eq. 5.6, rather than the measured data was
used to calculate permeability.
The selected fitting parameters for these simulations are shown in Table 5.1.
Experimentally measured brine saturation was almost as low as zero in some portions of
the core, therefore, using residual values greater than zero to calculate normalized
saturation (Eq. 4.3) in Eq. 5.6 resulted in nonphysical permeability values, therefore, a
residual brine saturation of zero was used to calculate normalized brine saturation, S*, in
Eq. 5.6. The different values for residual liquid used to calculate permeability and used
in the actual simulation to calculate capillary pressure are designated by S lrk and Slrs
respectively in the table.
Table 5.1 J-Function fitting parameters used to calculate permeability
Simulation
Pc Curve #
A
B
λ1
λ2
Slr
7
8
9
10
11
ICP1
ICP2
ICP3
ICP4
ICP5
0.004
0.01
0.004
0.016
0.025
0.2
0.2
0.2
0.17
0.25
3.4
2.9
3.4
2
2.2
2.8
2
2.8
2.8
1.4
0
0
0
0
0
k
Slr
s
0.2
0.2
0
0
0
S
0.502
0.349
0.502
0.589
0.535
The fitting parameters in simulation 7 and 8 were selected to be the same as ICP1 and
ICP2 in the previous simulations in Chapter 4. The curve fit for simulation 9 and 10
correspond to the capillary pressure data in Figure 4.6 (ICP1), where simulation 9 uses
the same parameters as simulation 7 but with different residual brine saturation in the
simulation. Simulation 10 uses fitting parameters designed to better match the data at
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
65
low and high values of brine saturation. Simulation 11 uses a curve fit to match the
capillary pressure data in Figure 4.7 (ICP2) at low and high values of brine saturation.
The curve fits for simulations 9 and 10 are shown in Figure 5.2 and the curve fit for
simulation 11 is shown in Figure 5.3.
For simulations 9-11, a new relative permeability fit was required because the
normalized brine saturation has been changed. The new relative permeability function is
shown in Figure 5.4 and qualitatively looks the same as Figure 4.5 where Slr = 0.20 cases.
Figure 5.2 Capillary pressure fits for Simulation 9 (ICP3) and Simulation 10 (ICP4) from
medium Berea data
66
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
Figure 5.3 Capillary pressure fit for Simulation 11 (ICP5) from small Berea data
Figure 5.4 Relative permeability curve fit for Slr = 0
nBrine
5.50
Relative Permeability Fit Parameters
nCO2
Sgr
Slr
1.90
0.0
0.0
Slrn
0.0
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
5.1.4
67
Permeability Maps
Permeability maps were generated for Simulations 7-11 by using Eq. 5.6 and the
steady state saturation map for the 100 percent CO2 injection case, shown in Figure 4.4.
The J-Function has very large values as brine saturation goes to zero, which Eq. 5.5 and
5.6 show leads to very high permeability values. To keep permeability bounded within a
reasonable upper limit, a maximum of permeability of 2000 md was allowed for the
simulation grids.
The resulting permeability maps using the parameters in Table 5.1 are shown below
in Figure 5.5, note that the scale is different than in Chapter 4. It is clear from the figure
that using different Eq. 5.6 to calculate permeability dramatically changes the resulting
permeability profile.
It is also clear that these models have a very high level of
permeability contrast compared to those presented in Chapter 4. For comparison, the
model with the highest contrast in Chapter 4, which was the fractal model used for
simulation 5, is shown compared on the same scale to the permeability map for
simulations 7 and 9 in Figure 5.6(a).
Figure 5.5 Permeability maps using modified Leverett J-Function for (a) Simulation 7 & 9
(b) Simulation 8 (c) Simulation 10 and (d) Simulation 11
68
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
Figure 5.6 Comparison of (a) fractal permeability map (Simulation 5) and (b) modified
Leverett J-Function permeability map (Simulation 7 & 9)
5.2 Saturation Results of Modified Leverett J-Function Models
5.2.1 Residual Brine Saturation Simulations
Selecting slice 29 to make a qualitative comparison again, the results of simulations 7
and 8 are shown with the experimental results in Figure 5.7. The figure shows a very
good match for both cases in terms of saturation contrast, with simulation 8 appearing to
have higher average saturation than simulation 7, but about the same factor of contrast
between the high and low CO2 saturation values in both simulations.
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
69
Figure 5.7 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 7 (c) Sim. 8
To see the accuracy of the saturation prediction, we can plot the simulation saturation
values in the slice vs. the experimentally measured values, as in chapter 4.
The
comparison is shown in Figure 5.8, and shows that the correlation between the
simulations and experiment is much improved over the porosity based methods in chapter
4.
The figure shows that on average the high and low saturations appear to be
underpredicted as most of the values in this region fall below the perfect correlation line
given by the purple line. However, the middle range of saturation values are being
70
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
relatively well predicted, falling along the general trend of the perfect correlation line for
both simulation results.
Figure 5.8 Simulation vs. experiment saturation in slice 29 for J-Function method
One of the reasons that the simulations underpredict saturation at high saturation
values is because of the capillary pressure fitting parameters used for ICP1 and ICP2.
These relationships have poor fits at low brine saturations (see Figure 4.6 and Figure 4.7,
respectively) overpredicting capillary pressure by almost an order of magnitude at very
low brine saturations, this is artificially forcing CO2 saturation to be lower. The other
reason is that an artificially imposed residual liquid saturation of 20 percent was used in
these simulations. In order for any cells to have more than 80 percent CO2 saturation, the
residual liquid saturation must be reduced.
These two reasons explain the capillary pressure fitting parameters selected for
simulations 9-11, shown in Table 5.1. In order to determine the effect of residual brine
saturation in simulations 7 and 8, the same J-Function fitting parameters with zero
residual liquid saturation were used for capillary pressure in simulation 9 as in simulation
7. In order to test the importance of an accurate curve fit at low brine saturations, the JFunction fitting parameters in simulation 10 and 11 were selected to better match the data
at very low brine saturations.
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
5.2.2
71
Zero Residual Brine Saturation Results
The resulting CO2 saturation in slice 29 for simulations 9-11 is plotted along with the
experimental results in Figure 5.9. The saturation contrast is higher in these results, so
the scale has been changed from previous comparisons to better highlight the contrast.
The figure shows that simulation 9 actually results in higher contrast than the
experimental measurements using the scale shown in the figure. Simulation 10 has
slightly lower contrast than simulation 9, but still appears to have more than the
experimental data. Simulation 11 shows the best qualitative match to the experiment, but
all three simulations show very good overall qualitative matches to the experimental data.
To determine how well the results compare quantitatively, the simulation results have
been plotted vs. their corresponding experimental results for slice 29 in Figure 5.10. The
figure shows that the maximum CO2 saturation in these simulations is higher than in the
previous cases, which should be expected because the residual liquid saturation has been
set to zero. The data trends of simulations 9-11 in the figure also follow the diagonal
“perfect correlation” line more consistently than simulations 7 and 8 in Figure 5.8.
Despite the improved correlation, it is apparent that there is still a maximum threshold on
CO2 saturation in the simulation near 75 percent. This maximum saturation threshold is
likely due to the very low relative permeability of brine at low saturations, shown in
Figure 5.4. The figure shows that at and below 25 percent brine saturation, the brine
relative permeability is essentially negligible, therefore, even though the residual value is
zero, the brine is essentially immobile at such low brine saturations. The upper bound on
permeability of 2000 md might also have an impact on the maximum saturation. A
history match testing the sensitivity of relative permeability and maximum permeability
limit on these models has the potential to improve the correlation in Figure 5.9 and Figure
5.10.
72
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
Figure
a
b
Simulation Key
k Params Simulation
Figure
Experiment
c
ICP3
9
d
k Params Simulation
ICP4
10
ICP5
11
Figure 5.9 CO2 Saturation in slice 29 (a) Experiment (b) Sim. 9 (c) Sim. 10 (d) Sim. 11
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
73
Figure 5.10 Simulation vs. experiment saturation in slice 29 for J-Function method
5.2.3
Comparison of Core Average Results
These results have shown qualitatively that this modified Leverett method is more
accurate at predicting sub-core scale CO2 saturation, however, the previous chapter
showed in Table 4.5 that the traditional permeability methods were very accurate in
predicting the core average CO2 saturation and relatively accurate at predicting the
average pressure drop across the core.
The average saturation and pressure drop
simulations 7-11 are shown below in Table 5.2. The results indicate that this new method
does a much poorer job of predicting core average values of saturation and pressure drop
than the porosity based permeability models did. The saturation is still relatively good,
within 9 percent in all cases, however, this is a factor of five greater error than the
porosity based methods. The match in pressure drop is also relatively poor, off by an
average of about 100 percent using these methods. The pressure drop however, is
strongly correlated to relative permeability, and a more accurate history match could
improve the prediction.
74
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
Table 5.2 Core average results using modified Leverett J-Function method
Simulation
7
8
9
10
11
Average CO2
Saturation
0.4819
0.4592
0.5066
0.5102
0.4915
Saturation
Error (%)
4.12
8.64
0.80
1.51
2.21
Average ΔP
(Pa)
12016
14242
13618
13229
15567
Pressure
Error (%)
70.22
101.76
92.92
87.41
120.53
5.3 Statistical Comparison of Permeability Methods
As a representative dataset for the core, the coefficient of determination (R2) values of
slice 29 data are shown in Table 5.3. The R2 values are calculated by forcing a linear
trendline through the origin of the data in Figure 4.13, Figure 4.16, Figure 5.8 and Figure
5.10. This fit was selected because the 45 degree perfect correlation line for these plots
passes through the origin and has a fit slope of 1. The table shows that most of the R2
values are actually negative, indicating that assigning the average saturation value to all
the simulation data points would actually perform better than the curve fit. Simulations 9
and 10 are the only models with significant positive R2 values, indicating that these
models best match the experimental values.
Table 5.3 Linear trend line data for slice 29 average saturation comparisons
Simulation
1
2
3
4
5
6
Fit Slope
0.8742
0.8685
0.8793
0.8764
0.8743
0.8589
Fit R
2
-96.05
-722.4
-27.7
-65.26
-5.457
-5.417
Fig. Ref.
4.13
4.13
4.13
4.13
4.16
4.16
Simulation
7
8
9
10
11
Fit Slope
0.8809
0.8504
0.9441
0.9534
0.916
Fit R
2
-0.628
0.0015
0.1757
0.2162
-0.012
Fig. Ref.
5.7
5.7
5.9
5.9
5.9
These qualitative results from using this modified Leverett J-Function method show a
greatly improved visual match to the experimental results, both in contrast (Figure 5.7
and Figure 5.9) and in absolute value (Figure 5.8 and Figure 5.10). However, the results
in Table 5.3 show that even the best sub-core scale saturation match requires
improvement. Therefore, it may be more statistically significant to compare slice average
data because of the incomplete history match.
There is also a certain amount of
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
75
experimental error in measuring saturation at such small scales, however, the slice
average value is very precise, (Perrin and Krevor, personal communication, 2009).
The curve fits for the three models which best match the experiment are shown Figure
5.11. The figure shows that the R2 of the fits using the average values is very good, over
.95 for simulation 11. In addition to this, the slope of each curve fit is nearly 1, which
would be a perfect correlation. The simulations do appear to overpredict the average
saturation of the slices with low experimentally measured saturation, but on average, the
matches to the experimental results are very good.
Figure 5.11 Comparison of slice average saturation of simulations 9-11
The same plot of the slice average values of the porosity based permeability models is
shown in Figure 5.12. The curve fits are again linear and forced to go through the origin,
the corresponding curve fit data is shown in Table 5.4. The figure shows that the low
saturation values are consistently over predicted and the high saturation values are
consistently underpredicted, resulting in poor slice average saturation prediction.
Recall that simulations 5 and 6 had permeability models with the most contrast,
however, the figure shows that these models are the worst at predicting slice average CO2
saturation. Recall also that simulation 2 had the permeability model with the least
76
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
contrast, however, this model actually does the best at predicting slice average saturation
among these porosity based permeability models.
Figure 5.12 Comparison of slice average saturation of simulations 1-6
Table 5.4 Linear trend line data for slice average saturation comparisons
Simulation
1
2
3
4
5
6
Fit Slope
0.9973
0.9878
1.0061
1.0011
1.0025
0.9841
Fit R
2
-0.531
0.3326
-2.393
-1.145
-3.61
-6.78
Simulation
7
8
9
10
11
Fit Slope
0.9605
0.9154
1.0082
1.015
0.9788
Fit R
2
0.9161
0.9051
0.791
0.8164
0.9554
5.4 Conclusions
The CO2 saturation prediction has been much improved by the use of this modified
Leverett method of predicting permeability. The qualitative comparisons showed much
improved matches for this method over the porosity-based permeability models discussed
in chapter 4. The quantitative analysis also showed that the results are much better than
the porosity based methods, although the core average saturation and pressure results are
less accurate.
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
77
It has been discussed that a more systematic and complete sensitivity study to history
match average core saturation and pressure drop will likely improve the results from
using this proposed permeability method. A more thorough sensitivity study may also
improve the simulation match of the porosity-based permeability methods, although not
to the extent to which this new permeability method improves the match.
In addition to a sensitivity study, a finer simulation grid could also be used to improve
the results if the simulator could be found to run faster. Simulations where such large
contrast in properties exist at such a small spatial distance take days to run, if a simulator
could be optimized for these sub-core scale simulations, any effect due to upscaling could
be reduced.
There is also some experimental error in measuring saturation at the sub-mm scale,
however, there are techniques to greatly reduce this error, such as longer scan time,
higher scanning power, and multiple scans (Perrin and Krevor, personal communication,
2009) which will be utilized in future studies to improve the accuracy of experimental
results.
78
CHAPTER 5. MODIFIED LEVERETT J-FUNCTION METHOD
Chapter 6
6 Conclusions
6.1 Summary of Findings
Chapter 3 showed that the saturation of CO2 measured at the sub-core scale in a core
flooding experiment can vary dramatically over very small spatial scales. Subsequent
analysis of the porosity map of the core revealed no obvious geological explanation for
this, such as large spatial contrast in porosity, or structured heterogeneity. This gave rise
to the problem of determining what controls the distribution of CO2 at the sub-core scale.
In the absence of gravity and compositional changes, capillary pressure and relative
permeability control the movement of fluid in a multiphase system once the geological
parameters have been determined. Subsequent simulations (Benson et al., 2008) showed
that these two parameters could not accurately replicate the spatial distribution of CO2,
leading to the conclusion that the permeability predictions at the sub-core scale needed
further investigation.
Simulations in chapter 4 showed that using Kozeny-Carman and fractal models for
calculating permeability did not give accurate saturation results at the sub-core scale.
The methods did show good agreement with measured core average saturation and
pressure drop however. It was then discussed how most of the permeability models in
general use reduce down to a function of porosity once other parameters, such as average
grain diameter or residual liquid saturation have been determined.
A new approach was then derived from previous work by Calhoun et al. (1949) to
calculate permeability using Leverett’s J-Function scaling relationship for calculating
capillary pressure. This approach works by combining capillary pressure, saturation and
porosity data into one formula for predicting sub-core scale permeability. Simulations
79
80
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
using this method for calculating permeability showed greatly improved results over
traditional methods in terms of spatial distribution, contrast, and absolute value of CO2
saturation, both at the sub-core scale, and at the slice average scale. These methods do
not do as well as the traditional methods at predicting core average saturation and
pressure drop however.
The findings presented here do not invalidate the traditional permeability models, but
show that they should be used with care at such small scales. Most of the data used to
calibrate and validate these traditional models was collected at the core scale, and
comparing the core average simulation saturation and pressure drop to the experimental
results showed a very good match, while the new modified Leverett method showed a
poorer match to the experiment average.
6.2 Recommendations for Future Work
The findings of this report indicate that a substantial improvement has been made in
predicting sub-core scale permeability in this relatively homogeneous Berea core. With
this new approach proposed, the method should be thoroughly tested under a variety of
conditions to determine the best implementation. After the model has been thoroughly
tested and validated, the permeability map can be used as input to begin testing the effect
of relative permeability and capillary pressure on saturation distribution.
The first recommendation is to test the uniqueness of the permeability map. In this
study, the 100 percent CO2 injection saturation map was used as input to evaluate the
Leverett-J Function and capillary pressure. Saturation maps from different fractional
flow rates are also available from the experiment and should be used as input to
determine how unique the permeability map is. This should only be done on a dataset
which has very high confidence in spatially mapped saturation values. It was stated
previously that there is some error in measuring saturation at such small spatial scales, it
is possible to reduce this error using increased CT voltage, amperage and scan time
(Perrin and Krevor, 2009), and therefore, a highly precise series of experiments should be
conducted to test the validity of this method with respect to uniqueness.
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
81
The second recommendation is that the ability to predict saturation at different
fractional flow rates should be determined. This can be done by simply conducting
simulations using the permeability map generated from the 100 percent CO2 injection
saturation map. Then, simulations at all of the measured fractional flows should be
conducted using permeability maps created from experimental saturation maps measured
at that respective fractional flow, this will further validate any results from the previous
recommendation.
The third recommendation is to investigate the effect of structured heterogeneity on
this method. If one considers a core with a high level of structured heterogeneity, such
that the CO2 is forced to circumvent a portion of the core by sub-core scale geological
features, the CO2 could be artificially forced to bypass a region of the core with high
permeability, however, using this method, due to the low saturation values, it would
appear that the region has low permeability. There is no obvious method to correct for
this effect at this time, however, it might be possible to use some type of mixed
prediction-correction method that could be used with inverse modeling.
Once the first two recommendations have been completed, I believe this method will
be very useful for sub-core scale studies in relatively homogeneous cores. However, I do
believe that some work remains to be done to extend this methods to cores with high
levels of heterogeneity, particularly any structured heterogeneity which might force CO2
to bypass certain portions of the core.
6.3 Concluding Remarks
With this, I would again like to thank everyone for their valuable input in this work,
especially my advisor, Sally Benson, and to the post doc who performed all of the
relative permeability experiments, Jean-Christophe Perrin.
I hope that future
investigators find this method useful and that additional investigations following these
recommendations can further validate this theory and these results.
82
CHAPTER 6. CONCLUSIONS AND FUTURE WORK
Appendix A: A Method for Estimating Specific Surface
Area
Specific surface area, av, is the amount of surface area per unit of grain volume, and is
traditionally measured by doing destructive grain size analysis of a rock sample (Panda
and Lake, 1994), using a scanning electron microscope (Berryman and Blair, 1986) or
using the method of nitrogen surface adsorption (Pape et al., 2000). Here a new method
is proposed using thin sections and image analysis techniques to estimate the specific
surface area.
First a very thin, epoxy impregnated sample of the rock, called a thin section, is
digitally scanned at a desired resolution. The rock grains in a thin section are transparent
and easily distinguishable from the blue epoxy. The color image is converted to gray
scale, and using the measured core average porosity as a threshold, the grayscale image is
converted to binary. This process is illustrated for the thin section shown below in Figure
A.1.
Figure A.1 Thin section conversion to binary image
Berryman and Blair (1986) use a statistical approach using thin section analysis for
calculating specific surface area, but a simpler approach is taken here. The Kozeny-
83
84
APPENDIX A: SPECIFIC SURFACE AREA
Carman permeability equation is derived assuming a bundle of capillary tubes transports
fluid through the porous media, for which specific surface area is linearly proportional to
specific perimeter, or the amount of perimeter per unit grain area.
This linearly
proportionality can be assumed to be true for general porous media (Ross, personal
communication, 2008), therefore, measuring the pore perimeter and grain area can
provide an estimate for specific surface area.
To analyze the thin section, a small sample area, called a region of interest (ROI), is
analyzed in Matlab using the image processing toolbox.
The size of this ROI is
determined by the user, but is generally taken to be on the same scale as the CT
measurements, or the upscaled voxel area. A sample ROI is shown below in Figure A.2
(a) where the porosity is shown in white and the grain area is shown in black, the image
has dimensions of 1.05 mm on each side. The trace of the pore perimeter is shown in
Figure A.2 (b), which is used to calculate the total perimeter contained in the image area.
Figure A.2 (c) is used to calculate the total pore area in the ROI, which is shown outlined
in red, the feature outlined in green is an embedded grain and its area is subtracted from
the total pore area.
Figure A.2 (a) Binary ROI (b) pore perimeter trace of ROI (c) pore identification (red) and
embedded feature (green) identification of ROI (each image is 1.05 mm across)
The white perimeter along the boundary of the ROI in Figure A.2 (a) or (b) is not
included in the total perimeter calculation because does not represent true perimeter, but
is instead the internal portion of a larger pore which has been truncated by the sample
area. The perimeter in each ROI is then converted to specific perimeter by dividing by
APPENDIX A: SPECIFIC SURFACE AREA
85
the fraction of the unit area of the ROI which is grain, giving perimeter per unit grain
area. The procedure for calculating these values for the sample shown in Figure A.2 is
given below.
33
27
𝑃𝑇 =
𝑃𝑖 +
𝑖=1
𝜙=
27
𝑖=1 𝐴𝑖
𝑃𝑖
A.1
𝑖=33
−
𝐴𝑇
33
𝑖=33 𝐴𝑖
𝐴𝐺 = 𝐴𝑇 ∙ 1 − 𝜙
A.2
A.3
𝑃𝑣
𝐴𝑔
A.4
𝑎𝑣 = 𝛼𝑃𝑣
A.5
𝑃𝑣 =
where PT is the total perimeter in the ROI, Pi and Ai are the perimeter and area of each
outlined feature in Figure A.2 (b), where the subscripts, i are given for each feature in
Figure A.2 (c), ϕ is the porosity of the ROI, AG is the total grain area in the ROI, Pv is the
specific perimeter in the ROI, av is the specific surface area, and α is the linear
proportionality factor between av and Pv. Plotting Pv vs. ϕ for each ROI gives the plot in
Figure A.3, which can be used to fit a power law correlation between specific perimeter
and porosity.
86
APPENDIX A: SPECIFIC SURFACE AREA
Figure A.3 Specific perimeter for thin section in Figure A.1
Eq. 4.8 used in the study is given below in Eq. A.6.
𝑆
𝜙3
𝑘= 2
𝑎𝑣 1 − 𝜙
𝑆
𝑘=
0.033𝛼𝜙 0.79
2
A.6
2
𝜙3
1− 𝜙
2
A.7
Since α is a linear proportionality constant, it can be combined with the constant S,
which reduces Eq. A.7 to Eq. A.8, which is given as Eq. 4.12 in the chapter 4
permeability study.
𝜙1.42
𝑘= 𝑆
1− 𝜙
2
A.8
Nomenclature
Abbreviations
CCS -
Carbon capture and storage
CT
Computed Tomography
-
EOS -
Equation of State
ICPj -
Capillary pressure curve fit j
IPCC -
International Panel on Climate Change
LBNL-
Lawrence Berkeley National Laboratory
NETL -
National Energy Technology Laboratory
NIST -
National Institute for Standards and Technology
PVI
Pore Volumes Injected
-
ROI -
Region of Interest
Symbols
a
-
Fitting parameter in Timur (1968) permeability model
(md)
ai
-
Huet et al. (2005) fitting parameter, where i is from 1 to 5
av
-
Specific surface area per unit grain volume
A
-
Area
A
-
Fitting parameter in Silin et al. (2009) J-Function parameterization
(-)
b
-
Porosity fitting in Timur (1968)
(-)
B
-
Fitting parameter in Silin et al. (2009) J-Function parameterization
(-)
c
-
Residual water saturation fitting exponent in Timur (1968)
(-)
(md or none)
(m2/m3)
(m2)
87
88
NOMENCLATURE
co
-
Kozeny’s constant
Dp
-
Average grain diameter
Es
-
Storage efficiency
fj
-
Fractional flow of phase j
F
-
Mass or energy flux in TOUGH2 solver
ḡ
-
Gravitational vector
(m/s2)
J
-
Leverett J-Function
(-)
k
-
Permeability
K
-
Darcy’s original definition of permeability
m
-
Molal concentration of NaCl in brine in Philips et al. (1981)
M
-
Specific surface area per unit bulk volume
Mi
-
Molarities of CO2 and NaCl in Kumagai and Yokoyama (1999)
Mk
-
Molecular weight of component k in ECO2N
n
-
Molality of CO2 in brine
nj
-
Fitting exponent of phase j for relative permeability curve
P
-
Fluid pressure
(Pa)
Pc
-
Capillary pressure
(Pa)
Pd
-
Displacement capillary pressure (entry pressure)
(Pa)
Q
-
Injection rate
R
-
Residual vector of mass and energy balance in TOUGH2
S
-
Shape factor used in Permeability models
Sj
-
Saturation of phase j
(-)
(m2)
(Fraction of void volume)
(Fraction of total volumetric flow rate)
𝑘𝑔
𝑚2𝑠
or
𝐽
𝑚 2𝑠
(md, nm2)
𝑚𝑑
𝑃𝑎 ∙𝑠
𝑔−𝑚𝑜𝑙𝑒𝑠 𝑁𝑎𝐶𝑙
𝑘𝑔 −𝐻2 𝑂
(m2/m3)
𝑚𝑜𝑙𝑒𝑠
𝑘𝑔 −𝐻2 𝑂
(g/mole)
𝑔−𝑚𝑜𝑙𝑒𝑠 𝐶𝑂2
𝑘𝑔 −𝐻2 𝑂
(-)
(kg/s)
(kg or J)
(varies)
(Percent or Fraction)
NOMENCLATURE
89
Slrn
-
Fitting parameter for relative permeability curves
(-)
T
-
Temperature
u
-
Darcy flow velocity
V
-
Volume
x
-
Vector of primary variables in TOUGH2 solver
x1
-
Mole fraction of CO2 in the aqueous phase
(mol/mol)
y2
-
Mole fraction of H2O in the gas phase
(mol/mol)
X1
-
Mass fraction of CO2 in the aqueous phase
(kg/kg)
Y2
-
Mass fraction of H2O in the gas phase
(kg/kg)
(˚C)
(m/s)
(m3)
(varies)
Greek Symbols
Δ
-
Change in Value
(-)
λ
-
Brooks and Corey pore geometry factor
(-)
λ1
-
Fitting parameter in Silin et al. (2009) J-Function parameterization
(-)
λ2
-
Fitting parameter in Silin et al. (2009) J-Function parameterization
(-)
μ
-
Viscosity
ϕ
-
Porosity
ϕc
-
Percolation porosity constant
ρj
-
Density of phase or component j
σ
-
Interfacial Tension
Τ
-
Tortuosity in capillary tube model
τ
-
Dimensionless injection time
θ
-
Contact angle between wetting and non wetting fluids
Subscripts
(Pa·s)
𝑉𝑜𝑖𝑑 𝑉𝑜𝑙𝑢𝑚𝑒
𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
𝑉𝑜𝑖𝑑 𝑉𝑜𝑙𝑢𝑚𝑒
𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
(kg/m3)
𝐷𝑦𝑛𝑒𝑠
𝑐𝑚
(m/m)
(Pore Volumes Injected)
(Degrees)
90
NOMENCLATURE
brine -
Pertaining to aqueous brine wetting phase
CO2 -
Pertaining to CO2 non-wetting gas phase
CO2-brine Pertaining to the CO2 brine system in capillary pressure conversions
eff
-
Effective value
gr
-
Pertaining to residual gas phase saturation
grain -
Rock or mineral grain
hg-air -
Pertaining to the Mercury-air system in capillary pressure measurements
H2O -
Pertaining to water
i
-
Index referring to grid element or CT voxel i
j
-
Index referring to grid element j, or pertaining to phase j
lr
-
Pertaining to residual liquid or aqueous phase saturation
m
-
Indicating the mth primary variable in the TOUGH2 Newton solver
ss
-
Indicates a measurement taken at steady state
NaCl -
Pertaining to sodium chloride
p
-
Iteration counter in TOUGH2 Newton iteration solver
w
-
Pertaining to wetting phase
wr
-
Pertaining to residual wetting phase saturation
Superscripts
*
-
Pertaining to normalized saturation
brine -
Pertaining to the aqueous brine wetting phase
k
-
Indicates a component (Brine, CO2, NaCl)
n
-
Indicates time step n solution in TOUGH2 simulation
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