SEMI-ANALYTICAL PRESSURE TRANSIENT MODEL FOR
COMPLEX WELL – RESERVOIR SYSTEMS
by
Flavio Medeiros Junior
A thesis submitted to the Faculty and the Board of Trustees of the Colorado
School of Mines in partial fulfillment of the requirements for the degree of Doctor of
Philosophy (Petroleum Engineering).
Golden, Colorado
Date: ____________________
Signed: __________________________
Flavio Medeiros Junior
Approved: __________________________
Dr. Erdal Ozkan
Thesis Advisor
Golden, Colorado
Date: ____________________
_________________________________
Dr. Ramona Graves
Professor and Interim Head
Department of Petroleum Engineering
ii
ABSTRACT
This research presents a semi-analytical model to obtain the pressure-transient
response for vertical, horizontal, or multilateral wells in heterogeneous reservoirs. This
model is based on the Green’s function solution of the three-dimensional pressure
diffusivity equation for single-phase flow in a bounded and homogeneous reservoir,
following a methodology similar to the boundary element method. Reservoir
heterogeneity is addressed by discretizing the reservoir into homogeneous subsections
and imposing both flux and pressure continuity at the interface between contiguous
subsections. Results from this semi-analytical model are compared and validated against
results obtained from both the finite differences and the finite elements methods. It is
demonstrated, through validation examples, that the semi-analytical model provides
accurate results for heterogeneous reservoirs. This semi-analytical model also requires
less discretization than either the finite difference or finite element methods in order to
obtain the same level of accuracy at early times. The model has also proven helpful in
computing the pressure response for tight reservoirs with localized heterogeneity in the
near-wellbore region. Applications in production data analysis, identification of flow
regimes, and pressure transient analysis for a hydraulic fractured horizontal well in a tight
reservoir contacting a region with natural fractures are presented. The pressure response
for hydraulically fractured horizontal wells contacting a natural fracture network
indicates that the hydraulic fracture(s) control(s) the drainage of the region within natural
iii
fractures, whereas the drainage of the tight reservoir is controlled by the naturally
fractured region and reservoir permeability.
iv
TABLE OF CONTENTS
ABSTRACT....................................................................................................................... iii
TABLE OF CONTENTS.....................................................................................................v
LIST OF FIGURES ......................................................................................................... viii
LIST OF TABLES............................................................................................................. xi
ACKNOWLEDGEMENT ................................................................................................ xii
CHAPTER 1
INTRODUCTION ................................................................................... 1
1.1 Motivation..........................................................................................................1
1.2 Background ........................................................................................................4
1.3 Objectives ..........................................................................................................7
1.4 Method of Study ................................................................................................8
1.5 Contributions of the Study ...............................................................................10
1.6 Organization of Dissertation ............................................................................11
CHAPTER 2
LITERATURE REVIEW ...................................................................... 13
CHAPTER 3
MATHEMATICAL MODEL DEVELOPMENT ................................. 24
3.1 Pressure-Transient Solution for a Reservoir Subsection .................................26
3.2 Coupling of Multiple Reservoir Subsections (Blocks) ....................................32
3.3 Green’s Function for a Reference Time...........................................................40
3.4 Remarks on the Boundary Element Method (BEM)........................................42
CHAPTER 4
MATHEMATICAL MODEL − SOURCE FUNCTIONS .................... 43
4.1 Source Function for a Plane-Source Segment (Outer Boundary)....................47
4.2 Source Function for a Horizontal or Vertical Line-Source Segment...............49
4.3 Source Function for a Slanted Line-Source Segment ......................................51
v
4.4 Source Function for a Deviated Line-Source Segment....................................52
4.5 Source Function for a Generic Line-Source Segment .....................................54
CHAPTER 5
GAS FLOW AND WELLBORE EFFECTS ......................................... 57
5.1 Gas Flow ..........................................................................................................58
5.2 Skin Effect .......................................................................................................59
5.3 Wellbore Storage .............................................................................................61
5.4 Wellbore Friction .............................................................................................63
5.5 Matrix Equations..............................................................................................65
CHAPTER 6
COMPUTATIONAL ASPECTS.......................................................... 69
6.1 Preliminary Mathematical Results...................................................................70
6.2 Fully Penetrating Plane Source........................................................................72
6.3 Partially Penetrating Plane Source...................................................................76
6.4 Horizontal Line Source ....................................................................................78
6.5 Slanted or Deviated Line Sources....................................................................82
6.6 Generic Line Source ........................................................................................91
6.7 Convergence Criteria for Series.......................................................................94
6.8 Coordinates to Compute Wellbore Pressure ....................................................95
CHAPTER 7
RESULTS AND VALIDATION........................................................... 97
7.1 Model Validation Problem 1: Horizontal Well in a Homogeneous
Reservoir ....................................................................................................97
7.2 Model Validation Problem 2: Vertical Well in a Heterogeneous
Reservoir ..................................................................................................102
7.3 Model Validation Problem 3: Horizontal Well near a Sealing Barrier..........110
7.4 Remarks on the Computation of Results .......................................................115
CHAPTER 8
APPLICATIONS ................................................................................. 118
8.1 Diagnostic Plots for Near-Wellbore Heterogeneity.......................................118
8.2 Production Data Analysis − Field Example...................................................125
8.3 Pressure Transient Analysis − Field Example ...............................................133
CHAPTER 9
CONCLUSIONS AND RECOMMENDATIONS .............................. 139
9.1 Conclusions....................................................................................................139
vi
9.2 Recommendations..........................................................................................144
NOMENCLATURE ....................................................................................................... 146
REFERENCES ............................................................................................................. 152
APPENDIX A EQUIVALENT COORDINATE SYSTEM TO COMPUTE
SOURCE FUNCTIONS ...................................................................... 158
APPENDIX B DERIVATION OF SOURCE FUNCTIONS ...................................... 163
B.1 Fully Penetrating Plane Source .....................................................................164
B.2 Partially Penetrating Plane Source ................................................................167
B.3 Horizontal Line Source .................................................................................171
B.4 Slanted Line Source ......................................................................................177
APPENDIX C INPUT DATA AND RESULTS .................................................CD-ROM
APPENDIX D FORTRAN CODE ......................................................................CD-ROM
vii
LIST OF FIGURES
Figure 3.1 – General domain, boundary, points and outward unit vectors for the
Green’s function solution of the diffusivity equation. ...........................................27
Figure 3.2 – Domain, internal and external boundaries for the Green’s function
solution of the diffusivity equation. .......................................................................29
Figure 3.3 – Illustration of discretization procedure..........................................................33
Figure 3.4 – Horizontal well in a two-block reservoir.......................................................34
Figure 3.5 – Discretization sketch demonstrating the coupling of two reservoir
blocks penetrated by a horizontal well...................................................................35
Figure 3.6 – Matrix-vector form of the linear, two-reservoir-block, four-wellbore-segment, and two-interface-segment system.................................................37
Figure 4.1 – Rectangular parallelepiped representing the reservoir block for the
point source solution. .............................................................................................44
Figure 4.2 – Schematic of the dual-porosity medium used in Warren and Root
(1963) model..........................................................................................................46
Figure 4.3 – Configuration to compute the source function for a plane segment at
the domain’s outer boundary. ................................................................................48
Figure 4.4 – Configuration to compute the source function for a horizontal line
segment. .................................................................................................................50
Figure 4.5 – Configuration to compute the source function for a slanted line
segment. .................................................................................................................51
Figure 4.6 – Configuration to compute the source function for a deviated line
segment. .................................................................................................................53
Figure 4.7 – Configuration to compute the source function for a generic line
source segment.......................................................................................................54
Figure 5.1 – Time discretization of boundary conditions. .................................................68
Figure 6.1 – Sketch for a dual-lateral well.........................................................................88
Figure 6.2 – Results for the dual-lateral well.....................................................................90
viii
Figure 7.1 – Sketch for a homogeneous, isotropic, and rectangular parallelepiped
reservoir with a horizontal well; Model Validation Problem 1 .............................98
Figure 7.2 – Reservoir subsections in the x direction; Model Validation Problem
1..............................................................................................................................99
Figure 7.3 – Discretization schemes at the block interfaces for Verification Cases
1 through 6 in Tables 7.2 and 7.3; Model Validation Problem 1...........................99
Figure 7.4 – Verification plot for a horizontal well in a homogeneous reservoir
(Case two; two horizontal well segments per block and four block
interface segments). Model Validation Problem 1 ..............................................102
Figure 7.5 – Vertical well in a composite reservoir.........................................................103
Figure 7.6 – Permeability contrast and logarithmic grid for numerical models;
Model Validation Problem 2................................................................................103
Figure 7.7 – Permeability contrast and grid for the semi-analytical model; Model
Validation Problem 2. ..........................................................................................104
Figure 7.8 – Pressure and pressure derivative results for a composite reservoir
with an internal permeability greater than the external permeability; Model
Validation Problem 2. ..........................................................................................105
Figure 7.9 – Pressure and pressure derivative results for a composite reservoir
with an external permeability greater than the internal permeability; Model
Validation Problem 2. ..........................................................................................107
Figure 7.10 – Horizontal well in a reservoir with a sealing barrier; Model
Validation Problem 3. ..........................................................................................111
Figure 7.11 – Discretization grid for a horizontal well in a reservoir with a sealing
barrier; Model Validation Problem 3...................................................................113
Figure 7.12 – Pressure transient response - horizontal well in a reservoir with a
sealing barrier; Model Validation Problem 3.......................................................114
Figure 7.13 – Flow distribution - horizontal well in a reservoir with a sealing
barrier; Model Validation Problem 3...................................................................114
Figure 7.14 – Options for radial flow gridding................................................................116
Figure 8.1 – Horizontal well with a longitudinal fracture in a reservoir with a
localized naturally fractured region. ....................................................................121
Figure 8.2 – Diagnostic plot for a longitudinally fractured horizontal well
contacting a region with natural fractures............................................................122
ix
Figure 8.3 – Horizontal well with two large-spaced (L) transverse fractures in a
reservoir with a localized naturally fractured region. ..........................................123
Figure 8.4 – Horizontal well with two short-spaced (S) transverse fractures in a
reservoir with a localized naturally fractured region. ..........................................124
Figure 8.5 – Diagnostic plot for a horizontal well with multiple transverse
hydraulic fractures in a region with natural fractures. .........................................125
Figure 8.6 – Horizontal well with a longitudinal hydraulic fracture in a naturally
fractured reservoir................................................................................................129
Figure 8.7 – Flow rate and bottom-hole pressure for a horizontal well with a
longitudinal hydraulic fracture in a naturally fractured reservoir........................129
Figure 8.8 – Productivity index match for production data analysis – large
reservoir. ..............................................................................................................130
Figure 8.9 – Matching configuration for production data analysis – large
reservoir. ..............................................................................................................131
Figure 8.10 – Productivity index match for production data analysis – small
reservoir. ..............................................................................................................132
Figure 8.11 – Pressure buildup data.................................................................................134
Figure 8.12 – Diagnostic plot for pressure build-up. .......................................................135
Figure 8.13 – Match with buildup data............................................................................136
Figure 8.14 – Reservoir configurations from buildup data match. ..................................137
Figure A. 1 – Configuration for sources parallel to yD axis...........................................159
Figure A. 2 – Configuration for line segment parallel to zD axis....................................160
Figure A. 3 – Configuration for a plane segment parallel to ( xD, yD ) plane. ...............161
Figure A. 4 – Configuration for a deviated line segment. ...............................................161
Figure B. 1– Equivalence between dimensional and dimensionless coordinate
systems.....................................................................................................178
x
LIST OF TABLES
Table 6.1 – Parameters for the horizontal branch. .............................................................89
Table 6.2 – Parameters for the deviated branch.................................................................89
Table 6.3 – Reservoir parameters – dual lateral well case.................................................89
Table 7.1 - Well and reservoir properties for Model Validation Problem 1......................98
Table 7.2 – Results for two well segments in each subsection; Model Validation
Problem 1. ............................................................................................................100
Table 7.3 – Results for four well segments in each subsection; Model Validation
Problem 1. ............................................................................................................101
Table 7.4 – Reservoir and well data the composite reservoir case; Model
Validation Problem 2. ..........................................................................................107
Table 7.5 – Grid coordinates, unknowns, and time steps for the composite
reservoir case; Model Validation Problem 2. ......................................................108
Table 7.6 – CPU times for Model Validation Problem 2. ...............................................110
Table 7.7 – Input data for a horizontal well with a sealing barrier; Model
Validation Problem 3. ..........................................................................................112
Table 8.1 - Reservoir properties for the diagnostic plots.................................................119
Table 8.2 – Hydraulic fractures properties for the diagnostic plots.................................120
Table 8.3 – Naturally fractured reservoir properties for the diagnostic plots..................120
Table 8.4 – Reservoir parameters from production data match.......................................132
Table 8.5 – Match parameters for pressure transient analysis application ......................138
xi
ACKNOWLEDGMENT
The accomplishments of this research were achieved with the support from people
who have demonstrated a great attitude towards my work at Colorado School of Mines.
First, I recognize the extensive support of my employer, Petrobras, during the
period of this research. I also express my gratitude to my former managers at Petrobras,
Carlos Siqueira, Fernando Afonso Lima, and Solange da Silva Guedes – all of whom
helped make this project a reality.
I truly acknowledge the help from my advisor, Dr. Erdal Ozkan, in all phases of
this research, which included discussion on mathematical issues, suggestions for model
applications, review of my dissertation and conference papers, and the sharing of his
FORTRAN routines. I also recognize the pro-active support from Dr. Hossein Kazemi in
reviewing our papers and illuminating my ideas about naturally fractured reservoirs.
I am especially thankful to the members of my thesis committee: Dr. Turhan
Yildiz, Dr. D.V. Griffiths, and Dr. Paul Martin, all of whom have provided important
suggestions for improving the performance of the model developed in this work and they
also provided advice on how to select numerical models for comparison with the results
obtained from this semi-analytical model.
I would like to say thanks to fellow graduate students at the Marathon Center of
Excellence for Reservoir Studies, who have helped me in many ways, namely: Benjamin
xii
Ramirez, Osama Raba, Peggy Brown, Mohammed Al-Kobaisi, Basak Kurtoglu, and
Mahmood Ahmadi.
I also acknowledge the work of the kind personnel at the writing center who
reviewed most of the text in this dissertation.
Finally, my greatest gratitude goes to my family: my wife Glaucia, and my sons
Daniel and Lucas, all of whom were able to understand and to support my full time
dedication to my PhD degree at Colorado School of Mines.
xiii
1
CHAPTER 1
INTRODUCTION
This dissertation presents the results of a PhD study conducted at Marathon
Center of Excellence for Reservoir Studies in the Petroleum Engineering Department of
Colorado School of Mines. The PhD research reported in this dissertation has led to a
semi-analytical technique to simulate single-phase fluid flow in locally heterogeneous
porous media and, subsequently, the model’s application in unconventional reservoirs
consisting of tight formations produced by fractured horizontal wells surrounded by
natural fractures. Both the semi-analytical simulation approach, and the interpretation of
recovery mechanisms in fractured horizontal wells in tight, locally fractured formations,
are new and constitute the major contributions of this PhD study. Motivation,
background, and approach for the PhD research, and the organization of this dissertation,
are presented below.
1.1 Motivation
The exploration and production division of the oil and gas business aims to
discover and produce hydrocarbons in an economic fashion. In this context, reservoir
engineering is a branch of petroleum engineering dedicated to study, develop, and apply
techniques to optimize hydrocarbon production and reservoir drainage. Mathematical and
experimental models of the physical phenomena governing fluid flow in reservoirs are
2
powerful tools to aid reservoir engineers in their work to maximize recovery from
hydrocarbon resources. Construction of models capable of handling the complexity of the
well-reservoir systems used in the hydrocarbon recovery has been a serious challenge to
reservoir engineers in recent years.
In the last decade, developing unconventional oil and gas reservoirs and the
associated modeling problems have become one of the major challenges facing reservoir
engineers. Most unconventional reservoirs have very low permeability, as in the case of
tight gas sands and fractured shale formations. To achieve economical production in
these reservoirs a large reservoir rock surface area contact per well is normally required.
Current approaches for achieving higher well productivity by creating greater reservoir to
well surface contact areas are to drill multilateral or horizontal wells, and to then
stimulate these wells with hydraulic fractures. Furthermore, these wells only need a
single main vertical wellbore connecting to the surface, allowing for field development
using less surface wellhead equipment. This feature could reduce costs and
environmental risks related to land foot-prints in onshore fields, as well as significantly
cut investment expenditures in offshore projects. Productivity may be further improved if
the hydraulically fractured horizontal well is connected to an active natural fracture
network. This fracture network may preexist in a naturally fractured reservoir (dualporosity system), or it may be generated or reactivated locally, by hydraulic fracturing
around the well. The combination of diverse wellbore geometries, hydraulic fractures,
and natural fracture networks leads to a complex well-reservoir system and complicates
the task of reservoir fluid-flow modeling.
3
Reservoir fluid-flow modeling studies are based on solutions of the threedimensional diffusivity equation for fluid flow in porous media. The choice of the
mathematical method to obtain these solutions depends on the constraints of the
application. Analytical methods are available to obtain the pressure-transient response
and well performance for single-phase flow in homogeneous porous media and can be
extended to cover some limited forms of heterogeneity, such as layered and naturally
fractured systems. Numerical methods, on the other hand, are more suitable to compute
the pressure response and fluid saturation in heterogeneous porous media.
Many analytical and semi-analytical models have been developed to obtain the
pressure-transient responses (PTR) in reservoirs [Gringarten and Ramey (1973), Kuchuk
et al. (1990), Odeh and Babu (1990), and Ozkan and Raghavan (1991a)]. These models
present solutions for specific wellbore geometries in a homogeneous reservoir, or for
simple combinations of well geometry and reservoir heterogeneity. These limitations in
analytical and semi-analytical models lead to the use of finite-difference numerical
methods. However, finite-difference models for generating pressure-transient responses
or for modeling flow convergence around horizontal, multilateral, and fractured wells in
complex heterogeneous reservoirs require discretization of the solution domain into a
fine-grid system. Together with the requirement of small time steps for temporal
discretization, finite-difference models are expensive to build and run. Moreover,
especially in the early developmental stage of the reservoir, there are not enough data to
sufficiently characterize the heterogeneity in unconventional reservoirs. Therefore,
sometimes there is a need to build simpler models to obtain faster and less expensive
initial estimates of the reservoir performance.
4
The research reported in this dissertation was motivated to explore the extension
of analytical fluid flow solutions into more general forms of heterogeneity. Considering
the cost and data requirements of finite-difference simulation, semi-analytical simulation
may provide an efficient alternative for a large variety of practical reservoir conditions.
The semi-analytical simulation approach also provides for accurate computation of flow
toward complex wells and is well suited for modeling transient flows. Some potential
areas of application for this semi-analytical method may be single-phase flow in layered,
locally heterogeneous, or compartmentalized reservoirs, horizontal and multilateral wells
penetrating different sections of a reservoir, hydraulically fractured horizontal wells
surrounded by local natural fractures, and reservoirs with high- or low-permeability
streaks.
1.2 Background
A review of numerical reservoir simulators indicates that finite-difference
approximation of fluid-flow equations is the most common numerical modeling
technique in petroleum engineering. The use of finite-element method to numerically
approximate the flow equations has increased recently, especially for complex reservoir
geometries. Other numerical methods, including the boundary-element method, are used
to a lesser extent in reservoir engineering problems.
This work is not the first attempt to utilize analytical solutions in modeling fluid
flow in complex, heterogeneous formations. There have been many studies reporting
different approaches with the objective of using analytical solutions to model fluid flow
5
with certain forms of heterogeneity [Kuchuk (1996), Kikani and Horne(1993), and Sato
and Horne (1993)] and complex boundary shapes [Kikani and Horne (1993)]. In all these
approaches, some form of discretization of the domain and/or its boundaries is required.
Therefore, although the bases of these approaches are analytical, numerical computations
require some sort of semi-analytical evaluation of the solution and can be called semianalytical reservoir simulation for pressure response.
A significant portion of the semi-analytical simulation approaches reported in the
literature use the boundary element method derived from the Garlerkin’s weighted
residual statement. We exemplify this method by applying the inverse weighted residual
statement [Cartwright (2001)]
∂W
∂~
p( M )
2
~
~
(
)
∇
−
(
)
+
dB
W
p
M
WdV
p
M
∫V
∫B B ∂n B
∫B ∂nB B dB = 0 ,
(1.1)
to obtain the approximate solution for the steady state diffusivity equation in a bounded
and isotropic medium
∇ 2 p( M ) = 0 .
(1.2)
p ( M ) is the approximate solution for the pressure p(M ) at point M ( x, y, z ) ,
In Eq. 1.1, ~
W is the weighting function, V is the domain volume, B represents the domain boundary,
and n B the orthogonal outward direction at boundary B .
Applying the free space Green’s function solution of Eq. 1.2, H ( M , M ′) , as the
weighting function in Eq. 1.2, we obtain the internal point equation for the Galerkin’s
boundary element method:
p ( M ) = ∫ H ( M , M B′ )
B
∂H ( M , M B′ )
∂p ( M B )
dB − ∫ p ( M B )
dB .
∂n B
∂n B
B
(1.3)
6
Applying a generic function N (M ) as the weighting function in Eq.1.1 we obtain
the fundamental equation for Galerkin’s finite element method:
~
∂p ( M B )
~
dB = 0 .
∫ (∇p (M )) • (∇N (M ))dV − ∫ N (M )
V
B
∂nB
(1.4)
Observe that both the Galerkin’s boundary elements method and the Galerkin’s
finite elements method may be derived from the inverse weighted residual statement. The
difference between the two methods comes from the characteristics of the weighting
functions. As presented in Eq. 1.3 and 1.4, the boundary element method requires
evaluation of the Green’s function only at the boundaries while the finite element method
requires computation of the shape function at the boundary and also at points inside the
domain.
The finite difference method is formulated in a distinct way. It may be derived by
writing a Taylor series expansion for different points inside the domain V [Aziz (1979)].
Truncating the series we obtain an approximate expression for the pressure gradient in
the format of a difference equation. The set of difference equations are solved
algebraically to yield the pressure response at specified points in the domain.
From the above-mentioned numerical methods, only the boundary element
method applies a mathematical function that is related to the analytical solutions for the
partial differential equation.
In petroleum engineering literature, the boundary element method has been used
to preserve the analytical nature of the fluid flow solution and eliminate the numerical
dispersion and grid-orientation problems [Kikani and Horne (1992)]. It has been noted,
however, that finding the free-space Green’s function required by this approach may not
be always possible in heterogeneous reservoirs [Sato and Horne (1993a)]. Some
7
perturbation boundary element techniques have been proposed to alleviate the problems
in finding the exact free-space Green’s function and some applications have been
demonstrated [Sato and Horne (1993a), (1993b)].
1.3 Objectives
The main objective of this research is to provide a general, semi-analytical model
to simulate single-phase fluid flow in locally heterogeneous porous media with complex
well configurations, such as horizontal, multilateral, and fractured wells in layered,
compartmentalized, and naturally fractured porous media. The semi-analytical model has
been developed in the Laplace domain to easily handle variable production rates, unstable
pressure conditions, and to incorporate standard dual porosity models for naturally
fractured reservoirs [Barenblatt et al. (1960), Warren and Root (1963), Kazemi (1969),
and deSwaan-O, (1976)]. Moreover, calculations in the Laplace domain eliminate the
need for time discretization and the sequential solution in time. Our objectives include the
validation of the semi-analytical model and discussion of its advantages and weaknesses
for reservoir engineering applications.
The semi-analytical model is required to have high accuracy during transient flow
periods in addition to stabilized flow periods for use in tight, unconventional formations
where long transient flow periods dominate the performance and economics of the field
and also to help in the interpretation of pressure- and rate-transient data in such
formations. The model is also intended to be a tool to evaluate the sensitivity of
parameters that affect well performances in complex and unconventional formations.
8
Moreover, it is desired that the model can serve as a history-matching tool to provide
information about reservoir properties and the nature of reservoir heterogeneity, which
are both fundamental for optimizing reservoir production and hydrocarbon recovery.
The second important objective of this study is to provide better understanding of
production mechanisms, drainage areas, and overall performances of fractured horizontal
wells in tight sand or shale. The accomplishment of this objective is expected from the
application of the semi-analytical model to theoretical, as well as practical, field cases.
This study should lead to better understanding of production mechanisms, drainage areas,
and overall performances of fractured horizontal wells in tight sand or shale. The results
of this work are expected to improve reservoir management in unconventional reservoirs
by providing tools for better performance prediction and enhancing our ability to analyze
well-performance and pressure/rate-transient data.
1.4 Method of Study
The method of this research is both analytical and numerical with applications on
theoretical and practical field data. The method of sources and sinks, Green’s functions,
Laplace transformations, and superposition principle for linear partial differential
equations are all used to form the mathematical basis for the semi-analytical simulation
approach developed in this study. The semi-analytical simulation approach is based on
the Green’s function formulation of the solution for the diffusion equation. Because the
domain boundaries are discretized and the solution is defined in terms of the boundary
values of the problem, this approach is closely related with the boundary-element
9
method. Similar mathematical formulations can also be derived from Galerkin’s weighted
residual method (Cartwright, 2001), leading to a more standard boundary element
formulation of the solution. The main difference between our approach and the standard
application of the boundary element method is the use of source functions for bounded
domains in our approach instead of the free-space Green’s function in the boundary
element approach.
The model presented in this research divides the reservoir into subsections. Each
subsection is characterized by uniform average properties. An analytical pressuretransient solution is used for each subsection using the Neumann condition at the outer
subsection boundaries. The inner boundary, representing the well-reservoir interface, is
adjusted so that it is appropriate for the particular well type and operating condition. The
analytical solutions for individual subsections are coupled by imposing pressure and flux
continuity at the interfaces between adjoining subsections. By adding the production
constraint at the well-reservoir interface, a linear system of equations is obtained that may
be solved for the wellbore pressure together with the pressures and fluxes at the
boundaries of the subsections. Once the flux and pressure values at the boundaries of the
subsections are determined, pressure distribution within the reservoir can also be obtained
from the analytical solutions.
The computational code was written in FORTRAN 90 and model results were
verified against results of standard analytical or semi-analytical solutions available in the
petroleum engineering literature for some relatively simple asymptotic cases. The results
were also compared against the results of in-house and commercial numerical models.
10
1.5 Contributions of the Study
The primary product of this research is a semi-analytical simulation approach for
complex well-reservoir systems suitable for heterogeneous reservoirs. It is a helpful tool
for obtaining estimations of reservoir parameters using pressure transient analysis. The
model may also be applied in production decline analysis in order to obtain a production
forecast for complex well geometries in heterogeneous reservoirs.
This mathematical formulation of the semi-analytical simulation approach is
related to the boundary element technique. The formulation used in this work, however,
has not been reported in the literature and constitutes the primary new contribution of this
dissertation. Computational procedures required to numerically evaluate the solution for
the applications of interest are essential for the successful use of the proposed semianalytical simulation approach and also constitute significant contributions to the
literature.
To demonstrate the utility of the semi-analytical approach introduced in this
dissertation, examples for horizontal wells in compartmentalized reservoirs, and in
reservoirs with partial communication in the vertical direction, are presented.
Applications for fractured horizontal wells surrounded by a local natural fracture network
in a tight reservoir are also illustrated. These applications are common in many
unconventional reservoirs including tight gas and shale oil reservoirs. Discussion of the
production mechanisms in these systems, and field examples with the analyses of
production and pressure-transient data, constitute the second important and new
contribution of this dissertation.
11
1.6 Organization of Dissertation
This dissertation is divided into nine chapters and four appendices. Chapter 1
introduces the motivation, background, objective, and method of research. The significant
new contributions of the dissertation are also highlighted in Chapter 1.
Chapter 2 presents the literature review with pertinent papers and publications
related to this research.
Chapter 3 addresses the assumptions, mathematical background, and derivation of
the model solution for heterogeneous reservoirs.
Chapter 4 presents the geometry and mathematical equations for line and planar
surface sources that may be needed in the model solution presented in Chapter 3.
Chapter 5 describes how the model can be applied for the flow of a real gas, and
how wellbore storage, wellbore friction, and skin effects are incorporated in the basic
model presented in Chapter 3.
Chapter 6 discusses computational issues and presents alternative expressions to
obtain accurate and time efficient computations of the source functions presented in
Chapter 4.
Chapter 7 shows the model validation: results computed in this work are
compared to reference analytical solutions for homogeneous reservoirs and also
compared to finite difference and finite element numerical methods for heterogeneous
reservoirs.
Chapter 8 presents applications of the model in unconventional tight reservoirs.
We present applications to productivity and build-up analysis of field data for a
12
hydraulically fractured horizontal well contacting a limited naturally fractured region
around the wellbore.
Chapter 9 summarizes the main conclusions and recommendations for future
research related to the subject presented in this dissertation.
Appendix A shows how the equations for source functions parallel to x direction
may be applied to source functions parallel to the y or z directions.
Appendix B presents a more detailed mathematical derivation of equations
presented in Chapter 6.
Appendix C provides the input data and results for all the examples presented in
this work.
Appendix D documents the FORTRAN source code for the semi-analytical model
developed in this research.
13
CHAPTER 2
LITERATURE REVIEW
The semi-analytical model developed in this research is based on previous and
well-known analytical solutions for the diffusivity equation governing fluid flow in
porous media. Neumman (flux specified) or Dirichlet (potential specified) are the
standard boundary conditions applied to obtain these solutions.
The fundamental work of Carslaw and Jaeger (1959) presents numerous analytical
solutions for three-dimensional (3D) heat flow problems. These solutions have been
adapted to solve problems related to oil flow towards a vertical well in reservoir rocks.
However, many solutions derived from heat transfer problems consider constant media
properties, such as heat conductivity and heat accumulation. These constant property
assumptions can be applied only for slightly compressible liquids, such as oil or water.
For these fluids, viscosity and compressibility are assumed to have a weak dependency
with pressure. On the other hand, gas properties are strongly dependent upon pressure.
Thus, the pressure diffusivity equation for gas flow in porous media is a nonlinear
equation.
Al Hussainy and Ramey (1966) and Al Hussainy et al. (1966) presented a variable
transformation to eliminate the non-linearity in the diffusivity equation for gas flow in
porous media. This new variable was defined as the real gas pseudo-pressure or real gas
potential. By using the real gas pseudo-pressure, analytical solutions derived for liquid
flow (oil or water) can be extended to gas flow as well.
14
Gringarten and Ramey (1973) introduced the use of Green’s function, with source
and sink concepts, to solve transient flow problems in homogeneous porous media. They
presented instantaneous plane, line, and point sources for infinite slab, and bounded
porous media with constant pressure, constant flow rate, or mixed boundary conditions.
The combination of individual sources using Newman’s product method generates
solutions for more complex well-reservoir geometries such as partially penetrating
vertical wells, horizontal wells, and hydraulic fractures. Instantaneous solutions are
presented in the time domain, which requires integration over time to obtain a continuous
solution.
Green’s function solutions for the diffusivity equation in bounded porous media
contain an infinite series summation for each of the coordinate axes. These infinite series
must converge to obtain an accurate solution; this may take an excessive amount of time.
Thompson et al. (1991) presented a methodology for improving the computational time
required to obtain the pressure transient response from horizontal wells in a bounded
reservoir. The proposed methodology works with two sets of equivalent infinite series for
computing the pressure response in the time domain. The first set converges faster at
early times, while the second set converges faster at late times. By switching between the
two sets of series, according to time value, the overall computational time can be
substantially improved. Thompson et al. (1991) also presented a procedure to compute
the solution for horizontal wells in dual-porosity reservoirs. Their procedure applies
numerical integration to compute the Laplace transform of the time domain solution.
Dual-porosity reservoir properties are handled in the Laplace domain, and the Laplace
15
pressure response is converted back to the time domain using Stehfest’s algorithm
(1970).
Marshall 1 applied Poisson’s Summation Formula and the Fourier-Bessel integral
in an effort to improve the convergence of the Fourier series that characterizes the steady
state solution of the potential diffusivity equation in bounded electrochemical cells. This
approach can also be adapted to improve the computing time of the Fourier series in a
Green’s function solution for transient flow in porous media.
Ozkan and Raghavan (1991a) applied Green’s function theory to develop
solutions for the diffusivity equation in the Laplace domain. They derived a continuous
point source solution with unit strength in the Laplace domain. Integrating the point
source for different source geometries, they presented solutions for various wellbore
geometries and boundary conditions.
The diffusivity equation solution for naturally fractured reservoirs can be easily
computed in the Laplace domain by applying either Warren and Root (1963) or Kazemi
(1969) dual-porosity idealizations. Diffusivity equation solutions presented in the Laplace
domain can easily handle boundary conditions such as variable flow rate, wellbore
storage, and constant-pressure production. Moreover, large and short time asymptotic
solutions can be promptly derived from full solutions in the Laplace domain. Solutions
are inverted to real time by using a numerical Laplace transform inversion algorithm,
such as Stehfest (1970), as mentioned previously.
Diffusivity equation solutions presented in Ozkan and Raghavan (1991a),
specifically for horizontal and partially penetrating vertical wells in bounded reservoirs,
1
Marshall, S.L. Rapidly-Converging Modified Green’s Function for Laplace’s Equation in Three-
Dimensional Regions with Rectangular Boundaries. The reference for this paper was not found.
16
may not be suitable for straightforward computation. However, Ozkan and Raghavan
(1991b) subsequently presented a detailed procedure to accurately compute the solution
for a horizontal well in a bounded reservoir. Their procedure recasts the complete
solution as a summation of terms for bounded and infinite domain effects. Alternative,
quickly converging, equivalent series may be applied to reduce computational time,
similar to the method described previously.
Goode and Thambynayagam (1987) presented a methodology to analyze
drawdown and buildup pressure tests in horizontal wells drilled in an undersaturated oil
zone. They obtained a solution for the pressure diffusivity equation by using Laplace and
Fourier transformations, and presented time limits for each flow regime in the reservoir.
They then developed mathematical equations and interpretation procedures according to
reservoir flow regime and test type (drawdown or buildup).
Cinco-Ley et al. (1975) derived an analytical solution to obtain the pressure
response for a fully penetrating, slanted well in an infinite slab reservoir with
homogeneous properties. This type of well has a larger contact area than a fully
penetrating vertical well with the same wellbore radius. The larger contact area leads to
higher productivity for the slanted well than for the vertical well. Ozkan and Raghavan
(1998) extended the analysis presented by Cinco-Ley et al. (1975) for wells with high
inclination angles and partial penetration.
Yildiz and Ozkan (1994) presented an investigation of the pressure transient
response for a horizontal well with selective completion in a homogeneous reservoir.
They have shown that conventional solutions for an open-hole horizontal well will not
17
provide satisfactory results when applied to a well with partial or selective completion,
especially when non-uniform skin distribution is present along the well length.
Larsen and Hegre (1991, 1994) investigated the pressure transient behavior of a
horizontal well, with finite-conductivity vertical fractures, in a homogeneous, infinite
slab reservoir. They proposed an analytical solution for the system and discussed flow
regimes for longitudinal hydraulic fractures that are parallel to the well axis, as well as
transverse hydraulic fractures that are orthogonal to the well axis. Horne and Temeng
(1995) and Raghavan et al. (1994) investigated the pressure transient response and the
productivity of a horizontal well with multiple fractures in a bounded homogeneous
reservoir. Both works addressed interference effects among multiple transverse fractures,
and also discussed how to optimize hydraulic fracture spacing. Al-Kobaisi et al. (2004)
developed a hybrid numerical-analytical model for a horizontal well intercepted by
vertical fractures in an infinite slab reservoir. By adjusting the size and the properties of
the grid blocks representing the hydraulic fractures, Al-Kobaisi et al. (2004) presented the
pressure response and analysis for cases not covered in the previous analytical models.
Ozkan et al. (1994) presented an analytical solution in the Laplace domain for
dual-lateral horizontal wells in a homogeneous infinite slab reservoir. They used this
solution to investigate the sensitivity of wellbore pressure to various well and reservoir
parameters. Phasing and distance between laterals, as well as anisotropic horizontal
permeability, were the dominant parameters in the sensitivity analysis. Ozkan et al.
(1994) showed that anisotropic horizontal permeability is the fundamental parameter
controlling the pressure response for dual-lateral wells. Unfavorable anisotropy can
drastically reduce the effective length of a horizontal well and, therefore, the productivity
18
of a lateral branch. For reservoirs with isotropic horizontal permeability, increasing the
separation between the lateral wells will increase productivity; for the same separation
between laterals, productivity will also increase as the phasing angle changes from 0 to
±180°. Yildiz (2003) extended the analysis presented in Ozkan et al. (1994) for the case
of a multilateral-well with an arbitrary number of lateral branches in the horizontal plane.
Basquet et al. (1999a) presented a semi-analytical model to compute the pressure
response, and hence the productivity, of a well with complex geometry in a stratified
bounded reservoir of rectangular shape. The wellbore is assumed to have infinite
conductivity; variable skin is also allowed along the well path. Results are computed at
discrete points along the well path by a point source solution in the Laplace domain.
Layered boundaries are considered as no flow, constant pressure, or of mixed properties,
allowing interlayer cross-flow between adjacent layers. Cross-flow is accounted for by
the use of the quadripole method derived from heat transfer solutions [Carslaw and
Jaeger (1959)]. Basquet et al. (1999b) applied the same methodology as in Basquet et al.
(1999a) to compute the pressure transient response and productivity for multifractured
wells in bounded reservoirs with multiple layers. In this treatment, fractures are
considered as plane sources with infinite conductivity, and the wellbore connects to the
reservoir only through the fracture flow path.
Ouyang and Aziz (2001) developed a model to couple flow in the reservoir with
flow in multilateral wells. The reservoir flow model considers any type of well located in
a homogeneous, anisotropic, and bounded reservoir of parallelepiped shape. Wells are
represented by line sources in the time domain – this representation was obtained by
integrating instantaneous point sources over time and space, as documented in Gringarten
19
and Ramey (1973). The wellbore flow model coupled with the reservoir model developed
by Ouyang and Aziz (2001) considers the effects of wall friction, gravity, and fluid
acceleration.
Yildiz (2005) also presented results to determine the productivity of multilateral
horizontal wells in an anisotropic homogeneous reservoir. The results were computed
with three-dimensional analytical solutions and compared with experimental results
based on an electric analog apparatus. Good agreement between the analytical solution
and the experimental data was demonstrated.
At this point, it is important to notice that all models and solutions described
above are valid for single-phase flow, either gas or oil, in a reservoir (or layer, in case of
a multi-layer reservoir) with uniform initial pressure distribution. The models also
assume that the coordinate system is aligned with the directions of principal
permeabilities, and that bounded reservoirs have a parallelepiped shape with faces
parallel to the coordinate axes. These papers developed models and derived diffusivity
equation solutions for the determination of the pressure transient response in
homogeneous reservoirs. Next, we review a few papers addressing the use of Green’s
function to obtain the solution of the pressure diffusivity equation in heterogeneous
reservoir.
Kuchuk and Wilkinson (1991) presented a Green’s function solution to obtain
pressure transient response for well producing from commingled reservoirs. This solution
considers non-uniform initial pressure condition, mixed boundary conditions (pressure or
flux), and no interlayer cross flow between individual reservoirs.
20
Kuchuk et al. (1996) presented an approximate analytical method to obtain
pressure transient response in heterogeneous reservoirs. Their formulation assumes that a
homogeneous reservoir has a region with permeability and porosity anomaly. Then, using
a nonlinear approximation for pressure and pressure gradient inside the anomaly and
applying the Green’s function solution of the diffusivity equation, they derived an
analytical expression for pressure response in the region outside the anomaly. They
pointed out that the proposed approximation provides good results when the ratio
between permeabilities is large than the unit value. However, to obtain a better
performance of the nonlinear approximation certain geometric conditions need to be
satisfied.
Kuchuk and Habashy (1997) derived the Green’s function for a point source in a
three-dimensional laterally composite reservoir. The proposed solution is able to compute
pressure response for cases where reservoir permeability changes along y direction in
the x − y plane.
Kikani and Horne (1992) used a two-dimensional free space Green’s function and
the boundary element method to obtain the pressure transient response for an arbitrarily
shaped reservoir. Kikani and Horne (1993) extended this approach to sectionally
homogeneous reservoirs.
Sato and Horne (1993a) applied perturbation methods to overcome the difficulty
in obtaining a free space Green’s function to be used with the boundary element method
in order to obtain the steady-state pressure response in heterogeneous reservoirs. Sato and
Horne (1993b) extended this approach to the transient case.
21
The works of Kikani and Horne (1992, 1993) and Sato and Horne (1993a, 1993b)
are based in a two-dimensional Green’s function that limits their application for twodimensional flow.
Deng and Horne (1993) suggested methods to obtain Green’s functions in
heterogeneous reservoirs and also discussed the validity of the Principle of reciprocity in
such reservoirs. The general solution for an anisotropic and heterogeneous reservoir is
presented with the assumptions that the three-dimensional problem can be decomposed in
three one-dimensional sub-problems.
Pecher and Stanislav (1996) presented an application of the boundary element
method to solve the two-dimensional diffusivity equation for single-phase flow in
heterogeneous reservoirs with complex geometries. They solved the boundary integral
equation using the collocation method, and discussed results obtained with linear,
quadratic, and cubic isoparametric boundary elements.
Jongkittinarukom and Tiab (1998) applied the boundary element method with the
three-dimensional free space Green’s function to obtain the solution for a horizontal well
in a multilayer reservoir. The boundary element solution was applied for each individual
layer by using domain decomposition. Then, the unknowns for each individual layer were
assembled into a global matrix, which was solved to obtain the pressure at the horizontal
well.
Wolfsteiner et al. (1999) presented a model for productivity of non-conventional
wells in heterogeneous reservoirs. They applied a semi-analytical approach, as in Ouyang
and Aziz (2001), to model flow towards the wellbore. Reservoir heterogeneities are
represented by a near wellbore effective skin term. The effective skin is computed by
22
local integration of the permeability field in the near-wellbore region. Thus, reservoir
heterogeneity is converted into an effective skin concept; the semi-analytical model for a
homogeneous reservoir is then applied to compute the pressure response.
Archer and Horner (2000) presented a hybrid boundary element method (Green
element method) to model the pressure transient response for single-phase flow. Using
this methodology, they modified the diffusivity equation to obtain an equivalent equation
capable of handling permeability heterogeneity. They compared their results to a finite
difference numerical simulator, and concluded that both methods yielded accurate
pressure results, however only the boundary element method accurately computed the
pressure derivative.
Cartwright (2001) presented a derivation of the Green’s function solution for the
steady state potential diffusivity equation using a boundary element approach based on
Galerkin’s weighted residual statement. This approach uses the three-dimensional free
space Green’s function (or the fundamental) solution as the weighted function in
Galerkin’s statement. The derivation steps in Cartwright’s (2001) work can be used to
quickly obtain the Green’s function solutions for the transient case in the time or the
Laplace domains.
Sutradhar et al. (2001) derived the Green’s function (point source solution) for the
three-dimensional transient heat diffusivity equation in a media where the heat
conductivity varies exponentially in one direction. This solution was presented in the
Laplace domain, requiring a numerical inversion algorithm to compute values in the time
domain. Berger et al. (2005) derived a steady state Green’s function for the same problem
addressed in Sutradhar et al. (2001). The Green’s function solutions presented in
23
Sutradhar et al. (2001) and Berger et al. (2005) can be applied in the boundary element
method to solve reservoir problems where permeability varies exponentially in one
direction.
Yildiz (2002) presented a model and discussed parameters affecting long-term
performance of multilateral wells in commingled reservoirs. The model considers
isolated layers with differing initial pressures, sizes, and petrophysical properties. Results
and discussions are presented for a case where there is a single lateral well per layer.
Effects of differences in layer permeability, layer thickness, lateral well length, and in
non-uniform formation damage (skin effect), as well as cross flow between layers in the
wellbore are documented.
24
CHAPTER 3
MATHEMATICAL MODEL DEVELOPMENT
Fluid flow in porous media is governed by the diffusivity equation, which is
derived from the mass balance equation and Darcy’s law. It relates pressure, time, and
space coordinates, as well as rock and fluid properties. For a homogeneous and
anisotropic porous media, the diffusivity equation for flow of a slightly compressible
fluid is given by
(
)
div K grad ( p) = φµct
∂p
.
∂t
(3.1)
In Eq. 3.1, φ is porosity, µ is fluid viscosity, ct is total reservoir compressibility,
and K is the permeability tensor
⎡ k xx
⎢
K = ⎢k yx
⎢ k zx
⎣
k xy
k yy
k zy
k xz ⎤
⎥
k yz ⎥ .
k zz ⎥⎦
(3.2)
For three-dimensional flow of a slightly compressible fluid in porous media, the threedimensional permeability tensor is replaced by a diagonal tensor where the diagonal
elements represent the principal permeabilities of the anisotropic porous medium
⎡k x
K = ⎢⎢ 0
⎢⎣ 0
0
ky
0
0⎤
0 ⎥⎥ .
k z ⎥⎦
(3.3)
This representation is justified by the depositional process by which most reservoir rocks
are created. The deposition of sediments creates reservoir rocks in nearly horizontal beds.
25
These sediments are then buried to actual reservoir depth. Because of this process, it is
assumed that the vertical overburden from the weight of the rock above the reservoir
layer defines the minimum value in the main diagonal of the permeability tensor. The
other two permeability components are assumed to lie in the horizontal depositional
plane, with directions related to the minimum and maximum horizontal in-situ stress.
Using these assumptions, the diffusivity equation given by Eq. 3.1 can be recast in the
familiar three-dimensional, Cartesian form as follows
kx
∂2 p
∂2 p
∂2 p
∂p
,
+
k
+
k
= φµct
y
z
2
2
2
∂t
∂x
∂y
∂z
(3.4)
where the coordinate system axes are aligned with the main permeability directions k x ,
k y , and k z . In Eq. 3.4, φ is the porosity of the medium and ct = c + c f is the total
compressibility of the system where c and c f correspond, respectively, to the
compressibilities of the fluid and the pore space. The compressibilities are assumed to be
small and constant and the pore space is completely occupied by a single-phase fluid of
constant viscosity, µ .
The solution of Eq. 3.4 requires spatial boundary conditions and a temporal
condition; it provides the pressure distribution as a function of time in porous media,
which is the fundamental information for pressure transient analysis and well
performance prediction. The degree of difficulty of finding the solution to Eq. 3.4
depends on the geometry of the domain and the sources. To find solutions for complex
well and reservoir geometries, the method of sources and sinks, Green’s functions,
integral transforms, and superposition principle have been used in petroleum engineering.
The same mathematical methods are used in this research as well.
26
In this chapter, we will first present the Green’s function formulation of the
pressure-transient solution for a locally homogeneous reservoir region with arbitrary flux
condition at the boundaries. Then, we describe the methodology to couple solutions for
individual regions to obtain the pressure-transient solution for the combined system. In
the following, we refer to the solution for a reservoir subsection as the block solution
whereas the solution for the coupled subsections will be referred as the model solution.
3.1 Pressure-Transient Solution for a Reservoir Subsection
The mathematical model used in this study is based on the following Green’s
function solution of the three-dimensional diffusivity equation for fluid flow in isotropic
and homogenous porous media [Carslaw and Jaeger (1959), Gringarten and Ramey
(1973)],
∆p ( M , t ) = pi − p ( M , t )
,
⎡
∂p ( M ′,τ )
∂G ( M , M ′, t − τ ) ⎤
′
= −η ∫ ∫ ⎢G ( M , M ′, t − τ )
− p ( M ′,τ )
d
M
d
τ
⎥
∂n B
∂n B
⎦ M ′∈B
0 B⎣
t
(3.5)
where B is the boundary of the domain, D , that consists of a porous medium with
uniform properties as sketched in Figure 3.1. In Eq. 3.5, nb is the outward normal
direction of the boundary surface, B . The points M and M ′ denote the observation and
source locations, respectively, G (M , M ′, t , τ ) is the Green’s function, p (M , t ) is the
pressure, pi is the initial pressure which is assumed to be uniform throughout the
solution domain, and η is the diffusivity constant given by
27
η = 2.637 × 10 −4
k
.
φµc
t
(3.6)
The numeric constant in Eq. 3.6 is required to express the parameters on the right hand
side of Eq. 3.6 in field units.
nb
Domain - D
M
nb
M'
nb
Boundary - B
nb
Figure 3.1 – General domain, boundary, points and outward unit vectors for the Green’s
function solution of the diffusivity equation.
For an anisotropic porous medium, the domain of Eq. 3.5 can be transformed into
an equivalent isotropic system by using the following coordinate transformation
~
ξ =ξ
k
for ξ = x, y, or z ,
kξ
(3.7)
where the uniform permeability, k , corresponds to the equivalent isotropic-system
permeability defined by
k = 3 kxkykz .
(3.8)
In petroleum engineering, most of the Green’s function solutions of the diffusivity
equation are presented in terms of dimensionless variables [Raghavan (1993)]. Therefore,
we choose to express the solutions presented here in terms of dimensionless space and
time variables, allowing for direct comparison with previous works. Dimensionless
28
variables also simplify the equations for the source functions presented in Chapter 4. We
define dimensionless space and time variables by Eqs. 3.9 and 3.10, respectively
ξD =
~
ξ
l
for ξ = x, y , or z
(3.9)
and
t D = 2.637 × 10 −4
k
t.
φµct l 2
(3.10)
In Eqs. 3.9 and 3.10, l is an arbitrary reference length in the system. Then, Eq. 3.5 can
be presented in the dimensionless space-time domain as follows
∆p ( M D , t D ) =
tD
⎡
.
∂G ( M D , M D′ , t D − τ ) ⎤
∂p( M D′ , τ )
dM D′ dτ
− ∫ ∫ ⎢G ( M D , M D′ , t D − τ )
− p( M D′ , τ )
⎥
∂n BD
∂n BD
⎥⎦ M D′ ∈BD
0 BD ⎢
⎣
(3.11)
If we consider the reservoir block in Figure 3.2, the domain boundary, BD , in Eq.
3.11 can be divided into the inner and outer boundaries denoted by BwD and Be D ,
respectively. In this representation, BwD corresponds to the well surface and Be D is the
outer surface of the block. For the reservoir block in Figure 3.2, Eq. 3.11 can be written
as follows
29
∆p ( M D , t D ) =
∂p ( M D′ , τ ) ⎤
⎡
⎢G ( M D , M D′ , t D − τ ) ∂n
⎥
BwD
⎢
⎥
−∫ ∫
⎢
⎥
′
G
M
M
t
∂
−
(
,
,
τ
)
D
D D
0 BwD
⎢ − p ( M D′ , τ )
⎥
∂n BwD
⎣
⎦
tD
∂p( M D′ , τ ) ⎤
⎡
⎢G ( M D , M D′ , t D − τ ) ∂n
⎥
BeD
⎢
⎥
−∫ ∫
⎢
⎥
′
G
(
M
,
M
,
t
τ
)
∂
−
D
D D
0 BeD
⎢ − p( M D′ , τ )
⎥
∂n BeD
⎣
⎦
dM D′ dτ
(3.12)
M D' ∈ B
wD
tD
dM D′ dτ .
M D′ ∈BeD
Inner Boundary - Bw
(well-bore sandface)
M – point inside
domain
Mw – point at inner boundary
Me – point at outer boundary
D -Porous medium
domain
Outer Boundary - Be
(substructure external faces)
Figure 3.2 – Domain, internal and external boundaries for the Green’s function solution
of the diffusivity equation.
Let us now require that the derivative of the Green’s function vanish on the
boundaries. Then, the Green’s function satisfies the homogeneous Neumann boundary
condition at the inner and outer boundaries; that is,
∂G ( M D , M D′ , t D − τ )
∂n BwD
=
M D′ ∈BwD
∂G ( M D , M D′ , t D − τ )
∂n BeD
= 0.
M D′ ∈BeD
(3.13)
30
Using Darcy’s Law to represent the normal derivatives of pressure on the boundary in
terms of flux, we have
dq ( M D′ , t D )
dq ( M ′, t D ) ν w
∂p ( M D′ , t D )
=−
=−
l = −lν w q~w (M ′, t D )
∂nBwD
dBwD
dBw
(3.14)
∂p( M D′ , t D )
dq ( M D′ , t D )
dq ( M ′, t D ) ν e
=−
=−
l = −lν e q~e (M ′, t D ) .
∂nBeD
dBeD
dBe
(3.15)
and
In Eqs. 3.14 and 3.15, ν ξ is the dimension of the inner ( ξ = w ) and the outer ( ξ = e )
boundary surfaces that takes on the values of 0, 1, and 2 for a point-, line-, or surfaceboundary, respectively. Using Eqs. 3.13 through 3.15, we can write Eq. 3.12 as follows:
∆p ( M D , t D ) =
tD
∫ ∫ q~
w
( M D' , τ )G ( M D , M D' , t D − τ ) dM ′dτ
0 Bw
(3.16)
tD
+ ∫ ∫ q~e ( M , τ )G ( M D , M , t D − τ ) dM ′dτ .
'
D
'
D
0 Be
We can divide the inner and outer boundary surfaces Bw and Be into n and m
segments, respectively, as follows:
n
Bw = ∑ Bwi ,
(3.17)
i =1
and
m
Be = ∑ Bej .
j =1
Then, assuming uniform flux in each segment, Eq. 3.16 can be written as
(3.18)
31
⎤
⎡n
∆p( M D , t D ) = ∫ ⎢∑ q~wi (τ ) ∫ G ( M D , M D′ , t D − τ )dM ′⎥ dτ
⎢ i =1
0 ⎣
Bwi
⎦⎥
tD
⎡m
⎤
+ ∫ ⎢∑ q~ej (τ ) ∫ G ( M D , M D′ , t D − τ )dM ′⎥ dτ .
⎥⎦
0 ⎢
Bej
⎣ j =1
tD
(3.19)
Also, defining a source function by
Sξ ( M D , t D ) = ∫ G ( M D , M D′ , t D ) dM ′ for ξ = wi or ej ,
(3.20)
Bξ
we can rewrite Eq. 3.19 as follows:
n tD
m
t
∆p ( M D , t D ) = ∑ ∫ q~wi (τ ) S wi ( M D , t D − τ ) dτ + ∑ ∫ q~ej (τ ) S ej ( M D , t D − τ ) dτ . (3.21)
i =1 0
j =1 0
Equation 3.21 is the three-dimensional Green’s function solution for a reservoir
homogeneous subsection (block solution) including sources and with prescribed flux
condition on the subsection boundaries. This solution is in the real-time domain. In this
work, however, we opted to develop the solution in the Laplace transform domain. There
are two advantages of having the solution in the Laplace domain: First, we can easily
implement variable rate production conditions and incorporate simple forms of
heterogeneity, such as dual-porosity representation of naturally-fractured reservoirs.
Second, the Laplace transformation converts the convolution integrals in Eq. 3.21 into
algebraic expressions and eliminates the need to compute the solution in a sequence of
time steps. However, it has the disadvantage that the results need to be inverted back into
the time domain by using a numerical inversion algorithm, such as the one proposed by
Stehfest (1970).
Applying the Laplace transformation to Eq. 3.21 yields the following expression
32
n
m
i =1
j =1
∆p( M D , s ) = ∑ q~ wi ( s ) S wi ( M D , s ) + ∑ q~ ej ( s ) S ej ( M D , s ) .
(3.22)
The bar sign over the functions in Eq. 3.22 indicates their Laplace transforms with
respect to the time variable, t D , and s is the Laplace transform parameter. The source
function in the Laplace domain, Sξ , is given by
Sξ ( M D , s ) = ∫ G ( M D , M D′ , s ) dM ′
for ξ = wi or ej .
(3.23)
Bξ
In Eq. 3.23, G ( M D , M D′ , s ) is the Green’s function in the Laplace domain.
Equation 3.22 is the fundamental step of our solution method. It requires that
either the source function, Sξ , for the inner ( ξ = wi ) and outer ( ξ = ej ) boundaries, or
the Green’s function, G , for the domain (see Eq. 3.23) be known in the Laplace domain,
together with the fluxes on all discretized segments of the inner and outer boundaries
( q~wi and q~ej , respectively). The inner boundary surface, Bw , corresponds to the sandface
of the wellbore, and the sum of the flow components crossing the inner boundary is
therefore equal to the sandface production rate of the well, q(t ) . It should be noted that
the Laplace-domain source functions for most common source (well) and boundarysurface geometries have been reported in the petroleum engineering literature [Ozkan and
Raghavan (1991a)].
3.2 Coupling of Multiple Reservoir Subsections (Blocks)
The solution given by Eq. 3.22 can be applied to obtain the pressure transient
response in a heterogeneous porous medium. To do this, we assume that the
33
heterogeneous porous medium is continuous and can be decomposed into discrete blocks
with uniform properties. Figure 3.3 illustrates the discretization process for a single-layer,
heterogeneous reservoir where the permeability of each subsection is uniform (but may
be anisotropic) and the permeability of the reservoir may change between subsections in
the x and y directions.
z
k3
k(x,y)
y
Discretization
k1
k4
k2
x
Figure 3.3 – Illustration of discretization procedure.
To demonstrate coupling of multiple reservoir blocks in the application of Eq.
3.22 to obtain the solution for the heterogeneous reservoir, we will consider two
rectangular reservoir subsections that are in series in the x direction as sketched in
Figure 3.4. We will assume that the reservoir blocks are penetrated by a horizontal well.
In this example, all block boundaries are assumed to be impermeable, except at the
interface between the blocks. Physically, this case corresponds to a horizontal well
penetrating a closed, rectangular reservoir, with two homogeneous substructures of
different properties. For simplicity, we will use the discretization scheme shown in Figure
3.5 that includes two well segments in each block and two interface segments between
blocks. However, the procedure can be applied to any number of segments, either for the
well, or for the block interface. (Discretization of the wellbore is required to implement
the appropriate wellbore hydraulics and compute flux distribution into the wellbore. In
this procedure, each well segment has uniform flux and as the number of segments
34
increased, a better approximation to the correct solution is obtained. For uniform-flux
wells, discretization is not required and the solution is simplified.)
z
Reservoir 1
y
Reservoir 2
Horizontal well
x
Figure 3.4 – Horizontal well in a two-block reservoir.
From Eq. 3.22, the pressure drop at the center of a well or interface segment i , in
block k , is given by:
2
2
j =1
l =1
k
∆p( M Di , s ) = ∑ q~wjk ( s ) Siwj
( M Di , s ) + ∑ q~ el ( s ) Sielk ( M Di , s ) ,
k
(3.24)
where M Di indicates the mid-point of segment i . Writing Eq. 3.21 in the center of all
well and interface segments shown in Figure 3.5 yields the following set of eight linear
k
k
equations with 16 unknowns, ∆p wi , ∆p ei , q~wik , and q~eik for i, k = 1,2 :
k
k
~k k
~k k
~k k
q~wk1S wiw
1 + qw 2 S wiw 2 + qe1 S wie1 + qe 2 S wie 2 − ∆p wi = 0 ; for i , k = 1,2 ,
(3.25)
k
k
~k k
~k k
~k k
q~wk1S eiw
1 + qw 2 S eiw 2 + qe1 S eie1 + qe 2 S eie 2 − ∆p ei = 0 ; for i , k = 1,2 .
(3.26)
and
35
Block 1
ze1
q~w11
q~w12
q~e12
ze2
q~e11
Block 2
q~e22
q~w21
q~w22
q~e21
ye1
ye2
xe1
q~e12
q~w11
q~w1 2
L1h1
L1h 2
q~e11
2yf
xe2
2zf2
2zf1
q~e22
q~w21
q~e21
L2h1
q~w22
L2h 2
2yf
Figure 3.5 – Discretization sketch demonstrating the coupling of two reservoir blocks
penetrated by a horizontal well.
In order to match the number of equations with the number of unknowns, we use
continuity of pressure and flux at the interface between Blocks 1 and 2. The continuity of
pressure and flux requires, respectively,
1
2
∆p ei = ∆p ei ; for i = 1 ,2 ,
(3.27)
1
2
q~ ei = − q~ ei ; for i = 1,2 .
(3.28)
and
Then from Eq. 3.26, we can write
1
2
2
~1 1
~1 1
~1 1
q~w11 S eiw
1 + qw 2 S eiw 2 + qe1 ( S eie1 + S eie1 ) + qe 2 ( S eie 2 + S eie 2 )
.
− q~ 2 S 2 − q~ 2 S 2 = 0 ; for i = 1,2
w1 e 2 w1
w2
(3.29)
e2 w2
This decreases the number of unknowns by six and the number of equations by two,
leading to a linear system of six equations and 10 unknowns. Three additional equations
36
can be obtained by imposing a pressure condition for the fluid flow inside the wellbore.
The wellbore may be either an infinite-conductivity (no friction loss) or a finiteconductivity (with friction loss) wellbore. If there is no pressure drop inside the wellbore,
then the infinite conductivity wellbore assumption requires
k
k
1
2
∆p w1 − ∆p w 2 = 0 ; for k = 1,2
(3.30)
and
∆p w 2 − ∆p w1 = 0 .
(3.31)
The infinite-conductivity wellbore assumption, as expressed by Eq. 3.30 and 3.31, can be
easily added to the existing system of linear equations. The finite conductivity wellbore
assumption, however, requires a more elaborate approach to be modeled as a set of
equations in the Laplace domain. This approach will be presented in Chapter 5, where we
incorporate wellbore storage, skin effect, and wellbore friction effects into the model.
Finally, the system of linear equations is completed by using the mass balance
equation, which requires the sum of the fluxes entering the wellbore be equal to the
production rate at the sandface, q(t ) ,
2
2
∑∑ q~
k =1 j =1
k
wj
Lhj = q .
(3.32)
The linear system defined by Eqs. 3.25 and 3.29-3.32 now has 10 equations and 10
unknowns. It can be represented in the matrix-vector form
[A] × {x} = {b},
as shown in Figure 3.6.
(3.33)
37
A1
A2
x1
X
b1
=
x2
b2
Figure 3.6 – Matrix-vector form of the linear, two-reservoir-block, four-well-boresegment, and two-interface-segment system.
The components of the coefficient matrix, [A] = [A1 , A2 ] , are given by:
⎡ S 1w1w1
⎢ 1
⎢ S w 2 w1
⎢ S 1e1w1
⎢ 1
⎢ S e 2 w1
⎢
[A1 ] = ⎢ 0
⎢ 0
⎢
⎢ 0
⎢ 0
⎢
⎢ 0
⎢⎣ L1h1
1
S w1w 2
1
S w2 w2
1
S e1w 2
1
S e2 w2
0
0
0
0
0
L1h 2
1
S w1e1
1
S w 2 e1
1
2
S e1e1 + S e1e1
1
2
S e 2 e1 + S e 2 e1
2
− S w1e1
2
− S w 2 e1
0
0
0
0
1
S w1e 2
1
S w2e2
1
2
S e1e 2 + S e1e 2
1
2
S e2e2 + S e2e2
2
− S w1e 2
2
− S w2e2
0
0
0
0
0
0
2
− S e1w1
2
− S e 2 w1
2
S w1w1
2
S w 2 w1
0
0
0
L2h1
0 ⎤
⎥
0 ⎥
2
− S e1w 2 ⎥
⎥
2
− S e2 w2 ⎥
2
S w1w 2 ⎥
⎥
2
S w2 w2 ⎥
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
L2h 2 ⎥⎦
(3.34)
and
0
0⎤
⎡− 1 0
⎢ 0 −1 0
0⎥
⎥
⎢
0
0
0⎥
⎢0
⎥
⎢
0
0
0⎥
⎢0
⎢0
0 −1 0 ⎥
[A2 ] = ⎢
⎥.
−
0
0
0
1
⎥
⎢
⎢ 1 −1 0
0⎥
⎥
⎢
1 −1 0 ⎥
⎢0
⎢0
0
1 − 1⎥
⎥
⎢
0
0
0 ⎦⎥
⎣⎢ 0
(3.35)
The solution vector, {x} = {x1 , x 2 }, has the following components
{ x1 } = {q~w11
q~w1 2
q~e11
q~e12
q~w21
q~w22 },
(3.36)
38
and
{ x2 } = {∆pw1 1
∆p w1 2
∆p w2 1
∆p w2 2 }.
(3.37)
The components of the right-hand side vector, {b} = {b1 , b2 }, are
{ b1 } = {0
0 0 0 0 0},
(3.38)
{ b2 } = {0
0 0 q }.
(3.39)
and
Any matrix inverter can be used solve this linear system. However, the computed
solution vector
{x} will be in the Laplace domain, and therefore needs to be numerically
inverted into the real-time domain. As customary in petroleum engineering literature, we
have used the Stehfest Algorithm [Stehfest (1970)] for the numerical inversion of Laplace
transforms in this research.
The solution vector, {x}, given in Eqs. 3.36 through 3.37, considers that the total
flow rate, q(t ) , is specified, as is the case for a well producing at a known flow rate,
normally constant in well testing operations. However, it is also common to produce a
well with a specific pressure in lieu of a constant production rate.
The solution for a specific pressure condition can be easily obtained from our
model by noticing that the wellbore pressure is known in this instance while the total
flow rate is unknown. Therefore, the solution for a specific pressure condition is obtained
by recasting the linear system, given by Eqs. 3.34 through 3.39. Assuming an infiniteconductivity wellbore, the components of the linear system, [A] × {x} = {b}, become:
39
⎡ S 1w1w1
⎢ 1
⎢ S w 2 w1
⎢ S 1e1w1
⎢
[A] = ⎢ S 1e 2 w1
⎢
⎢ 0
⎢ 0
⎢ 1
⎣ Lh1
1
S w1w 2
1
S w2 w2
1
S e1w 2
1
S e 2 w2
0
0
L1h 2
{x} = {q~w11
q~w1 2
{b} = {∆pw
∆p w
q~e11
1
S w1e1
1
S w 2 e1
1
2
S e1e1 + S e1e1
1
2
S e 2e1 + S e 2 e1
2
− S w1e1
2
− S w 2e1
0
q~e12
q~w21
1
S w1e 2
1
S w2e 2
1
2
S e1e 2 + S e1e 2
1
2
S e 2e 2 + S e 2e 2
2
− S w1e 2
2
− S w2e 2
0
0
0
2
− S e1w1
2
− S e 2 w1
2
S w1w1
2
S w 2 w1
L2h1
0
0
2
− S e1w 2
2
− S e2 w2
2
S w1w 2
2
S w2 w2
L2h 2
0⎤
⎥
0⎥
0⎥
⎥
0⎥,
⎥
0⎥
0⎥
⎥
− 1⎦
(3.40)
q~w22
q },
(3.41)
∆p w
0}.
(3.42)
and
0 0 ∆p w
In Eq. 3.42, ∆pw is the Laplace transform of the specified wellbore pressure. Solutions for
specified wellbore pressure condition are useful for production decline and rate transient
analysis (RTA) applications.
The solutions for the linear system of equations discussed above provide both the
fluxes and pressures for the wellbore segments, as well as the fluxes for the segments at
block-interface. When the fluxes and the source functions are known, pressures at any
point in the reservoir can also be computed from Eq. 3.22.
The coefficient matrices, shown in Eq. 3.34 and 3.40, require that the source
functions be known in the Laplace domain. The derivation of the source functions for
well and interface segments will be presented in Chapter 4.
40
3.3 Green’s Function for a Reference Time
The issue of representing the Green’s function solution with respect to a common
reference time for all reservoir blocks arises because of the use of dimensionless time in
the solution. To allow direct implementation of the Green’s and source function solutions
presented in the petroleum engineering literature, which are mostly in terms of
dimensionless time, in this work we chose to present our solution in terms of
dimensionless time.
To explain the issue of common reference time, we consider the diffusion
equation in terms of dimensionless time. Using the pressure difference
∆p( M , t ) = pi − p( M , t ) and applying the dimensionless variables as defined in Eqs. 3.9
and 3.10 simplifies Eq. 3.4 as follows
∂ 2 ∆p ∂ 2 ∆p ∂ 2 ∆p ∂∆p
.
+
+ 2 =
∂xD2
∂y D2
∂z D
∂t D
(3.43)
In Laplace domain, Eq. 3.43 yields:
∂ 2 ∆p ∂ 2 ∆p ∂ 2 ∆p
+
+
= s∆p ,
∂xD2
∂y D2
∂z D2
(3.44)
where we have used p( M , t = 0) = pi and thus ∆p( M , t = 0) = 0 .
The Laplace transform parameter, s , in Eq. 3.44 is related to dimensionless time
t D , which depends on the properties of the porous medium (Eq. 3.10). Therefore, when
combining reservoir blocks with different properties, the dimensionless pressure
definition for each block and thus the Laplace transform parameter, s , are related to the
properties of this particular block. In other words, for each reservoir block, there is a
distinct relation between the Laplace transform parameter, s , and the real time, t .
41
However, the continuity expressed in Eqs. 3.27 and 3.28 will hold only if a unique
dimensionless reference time is used. We address this issue by using a dimensionless
reference time t Dref given by
t Dref = 2.637 × 10 −4η ref / l 2 ,
(3.45)
Using the dimensionless reference time, Eq. 3.44 becomes
η
∂ 2 ∆p ∂ 2 ∆p ∂ 2 ∆p
+
+ 2 = s ref ∆p .
2
2
∂xD
∂y D
∂z D
η
(3.46)
Since the left hand side in Eq. 3.44 and 3.46 are the same, it indicates we can compute the
Green’s function referring to t Dref using the same mathematical expression for the
Green’s function derived for t D , provided that we replace s by sη ref / η . An alternative
way to obtain the Green’s function for t Dref , from that for t D , is to apply the similarity
property of the Laplace transformation [Ozkan (2003)2],
⎡ ⎛ t ⎞⎤
F ⎢ f ⎜ ⎟⎥ = cf ( cs ) .
⎣ ⎝ c ⎠⎦
(3.47)
Note that using the similarity property of the Laplace transformation, the Green’s
function in terms of real time can be obtained from that in terms of dimensionless time.
Then, the solution procedure formulated above can be recast in terms of real time and the
issue of common dimensionless reference time may be avoided.
2
Ozkan, E. (2003) Applied Mathematics for Fluid Flow in Porous Media Class Notes, Department of
Petroleum Engineering, Colorado School of Mines, Golden, Colorado.
42
3.4 Remarks on the Boundary Element Method (BEM)
The Green’s function solution in Eq. 3.12 can be derived by applying Galerkin’s
residual statement to Eq. 3.4 and its boundary conditions [Cartwright (1991)]. This
approach is also known as the boundary element method, or BEM, with prescribed
boundary conditions. In this case, the Green’s function is the solution for an
instantaneous point source in an infinite domain, which is also known as the fundamental,
solution or free-space Green’s function.
The fundamental solution, as presented in Cartwright (1991), is relatively easy to
compute, while the Green’s function used in the formulation presented above requires
more effort to be accurately computed [Ozkan and Raghavan (1991a)]. However, the
BEM requires the computation of not only the fundamental solution but also its
derivative at the domain boundaries, while the methodology in this research requires only
the computation of the Green’s function itself. Moreover, for a single reservoir
subsection, all subsection boundaries must be discretized under the BEM. On the other
hand, for the approach developed in this research, only the discretization of the inner
boundary is required.
43
CHAPTER 4
MATHEMATICAL MODEL − SOURCE FUNCTIONS
In Chapter 3, the solution for pressure distribution in a locally heterogeneous
porous medium has been formulated in the form of a linear system of equations. Solving
the linear system of equations defined by Eqs. 3.34 through 3.39 or Eqs. 3.40 through
3.42 requires the knowledge of source functions in the Laplace domain. These source
functions can be obtained by integration of the appropriate Green’s function over the
source geometry as defined by Eq. 3.23.
In the solution presented in Chapter 3, the wellbore surface constitutes the inner
boundary of the solution domain. For most of our applications, we replace cylindrical
wellbores with line sources. Upon discretization of the inner boundary, wells are
represented by line-sources with uniform flux distribution. Line-source well segments
can be either parallel to one of the coordinate axes (vertical or horizontal wells), make an
angle with the vertical coordinate axis (slanted wells), make an angle with one of the
horizontal coordinate axes (deviated wells), or make an angle with both the vertical and
one of the horizontal coordinate axes (generic wells).
The outer boundaries of the solution domain for each reservoir block are either
the physical boundaries of the reservoir or planar interfaces between contiguous
subsections of the reservoir. When the boundaries corresponding to the interfaces are
discretized, they form plane segments, which are modeled as plane sources with uniform
flux distribution.
44
To derive the source functions required by the model solution, we use the Green’s
function for a rectangular parallelepiped (Figure 4.1) satisfying the adjoint diffusion
equation with vanishing flux at the boundaries of the domain as required by Eq. 3.13.
This Green’s function corresponds to the continuous point source solution in the Laplace
domain for a rectangular parallelepiped [Ozkan and Raghavan (1991a)] as presented in
Eq. 4.1.
hD = zeD
zD
0
M(xD,yD,zD)
yeD
M'(x'D,y'D,z'D)
yD
xD
xeD
Figure 4.1 – Rectangular parallelepiped representing the reservoir block for the point
source solution.
y D1 ) + ch ( u ~
yD2 )
141.2πµ ⎧ ch( u ~
⎨
klxeD hD ⎩
u × sh( u yeD )
∞
k πx D
kπx D′ ch(ε k ~
y D1 ) + ch (ε k ~
yD2 )
+ 2∑ cos(
) cos(
)
xeD
xeD
ε k × sh(ε k y eD )
k =1
∞
nπz D
nπz D′ ⎡ ch (ε n ~
y D1 ) + ch(ε n ~
yD2 )
) cos(
)⎢
+ 2∑ cos(
ε n × sh(ε n yeD )
zeD
zeD ⎣
n =1
∞
y D1 ) + ch(ε k ,n ~
y D 2 ) ⎤ ⎫⎪
k πx D
kπx ′D ch(ε k ,n ~
+ 2∑ cos(
) cos(
)
⎥⎬
xeD
xeD
ε k ,n × sh(ε k ,n yeD )
k =1
⎦ ⎪⎭
G ( M D , M D′ , s ) =
(4.1)
In Eq. 4.1,
hD = zeD =
h k
,
l kz
~
y D1 = yeD − y D − y′D ,
(4.2)
(4.3)
45
~
y D 2 = yeD − y D + y′D ,
(4.4)
ε n = u + (nπ hD )2 ,
(4.5)
ε k = u + (kπ xeD )2 ,
(4.6)
ε k ,n = u + (kπ xeD )2 + (nπ hD )2 .
(4.7)
and
In Eqs. 4.1 and 4.5 through 4.7, u has been introduced to incorporate the dual-porosity
idealization of naturally fractured reservoirs [Ozkan and Raghavan (1991a, 1991b)]. The
definition of u is given by
for homogeneous reservoirs
⎧s
.
u=⎨
⎩sf (s ) for naturally fractured reservoirs
(4.8)
Appropriate expressions for f (s ) in Eq. 4.8 can be found in the literature [Warren
and Root (1963), Kazemi (1969), Raghavan (1993)]. In this work, parameters used to
model the dual-porosity regions are the ones defined by Warren and Root (1963), except
for the shape factor σ . Consider a dual-porosity medium as sketched in Figure 4.2.
Using the properties of a matrix block and the surrounding fractures, the storativity ratio,
ω , and interporosity flow parameter (or transmissivity ratio), λ , are defined,
respectively, by
ω = (φ f ctf ) /(φ f ctf + φm ctm ) ,
(4.9)
and
λ =σ
km 2
l .
k eff
(4.10)
46
In Eq. 4.10 σ is the geometric shape factor, defined by Kazemi et al. (1976), which
depends on the dimensions of the matrix block, given by
σ = 4(
1
1
1
+ 2 + 2 ),
2
Lx L y Lz
(4.11)
where Lx , L y , and Lz represent the dimensions of the matrix blocks.
Note that in Eq. 4.10, k m is the matrix permeability and k eff represents the
effective permeability of the natural fractures. This effective permeability is the actual
permeability of the natural fractures scaled by the fractional volume of the fractures with
respect to the bulk volume.
Dual Porosity Medium
Matrix
Block
Fractures
wf
Lz
z
wf
y
x
Lx
Ly
wf
Figure 4.2 – Schematic of the dual-porosity medium used in Warren and Root (1963)
model.
Integrating the Green’s function in Eq. 4.1 over the source geometry as required
by Eq. 3.23, we derive the source functions for the entire boundary geometries presented
in this work.
47
The normal derivative of the Green’s function in Eq. 4.1 vanishes at the domain
boundaries. Therefore, the use of this Green’s function in the general Green’s function
formulation given by Eq. 3.12 simplifies the boundary terms and yields Eq. 3.16.
Moreover, it does not require any discretization in the external boundaries to compute the
results for bounded homogeneous reservoirs (or reservoir subsections). However, it is
only applicable to reservoirs (or reservoir subsections) with parallelepiped shape; that is,
the use of the Green’s function in Eq. 4.1 requires that the reservoir is divided into
rectangular blocks (subsections) to account for local heterogeneities. In addition, the
numerical evaluation of the Green’s function in Eq. 4.1 may require that alternative
computational forms of Eq. 4.1 be developed [Ozkan and Raghavan (1991a)]. The
alternative computational forms of Eq. 4.1 are discussed in Chapter 6.
4.1 Source Function for a Plane-Source Segment (Outer Boundary)
Following the procedure outlined in the previous section, the source function for a
plane-source segment having the configuration in Figure 4.3 is obtained. The plane
segment is located at the outer boundary of the rectangular reservoir block in the x-z
plane; the segment is centered at point ( xejD , yejD , zejD ) and it has sides with length of 2 x fj
and 2 z fj in x D and z D directions, respectively. Applying Eq. 3.18 to the plane segment
sketched in Figure 4.3, yields
xej + x fj zej + z fj
S ej ( M D , s ) =
∫ ∫ G (x
xej − x fj zej − z fj
D
, y D , z D , x ′D , y ejD , z ′D , s ) dz ′dx ′ ,
(4.12)
48
Note that by changing the orientation of the coordinate system, Eq. 4.9 can be applied for
any plane segment on each one of the six faces of the rectangular parallelepiped
representing the domain. This procedure is presented in detail in Appendix A.
zD
2 x fj
yD
2 z fj
M ej ( xejD , yejD , zejD )
xD
Figure 4.3 – Configuration to compute the source function for a plane segment at the
domain’s outer boundary.
Using the Green’s function given by Eq. 4.1 in Eq. 4.12, we obtain the source
term of the plane segment sketched in Figure 4.3,
S ej ( M D , s ) = 4Cx fj z fj
ch ( u ~
y D1 j ) + ch ( u ~
yD2 j )
u × sh( u y eD )
ch (ε k ~
y D1 j ) + ch (ε k ~
yD2 j )
k πx
kπx ′
cos(
) cos(
) dx ′
+ 4Cz fj ∫ ∑
ε k × sh(ε k y eD )
xe
xe
xej − x fj k =1
xej + x fj ∞
+ 4Cx fj
+ 4C
zej + z fj ∞
∫ ∑
zej − z fj n =1
zej + z fj ∞
∫ ∑ cos(
zej − z fj n =1
×
xej + x fj ∞
∫ ∑
xej − x fj k =1
where
ch(ε n ~
y D1 j ) + ch (ε n ~
yD2 j )
ε n × sh(ε n y eD )
cos(
nπz
nπz ′
) cos(
) dz ′
ze
ze
nπz
nπz ′
) cos(
)dz ′
ze
ze
ch (ε k ,n ~
y D1 j ) + ch (ε k ,n ~
yD2 j )
ε k ,n × sh(ε k ,n y eD )
cos(
k πx
k πx ′
) cos(
) dx ′
xe
xe
(4.13)
49
C=
141.2πµ
.
klxeD hD
(4.14)
The source function in Eq. 4.13 is for a partially penetrating planar surface. If the
height of the plane source, 2 z fj , becomes equal to the height of the block, ze , the
integrals on z ′ vanishes and Eq. 4.13 yields the source function for a fully penetrating
planar surface.
4.2 Source Function for a Horizontal or Vertical Line-Source Segment
We present this derivation considering a horizontal line source segment in a
homogeneous reservoir block, as sketched in Figure 4.4. In this case, the inner boundary,
Bw , is a line equal to the length of the horizontal segment, Lh . Then, from Eq. 3.23, we
have
S HWi ( M D , s ) =
x wi + Lhi 2
∫ G (x
D
, y D , z D , x D′ , y wiD , z wiD , s ) dx ′ ,
(4.15)
x wi − Lhi 2
where, xwi , ywi , and z wi are the coordinates of the mid-point and Lhi is length of the ith
horizontal line segment. Substituting the Green’s function, given by Eq. 4.1 into Eq. 4.15
yields
50
ch( u ~
y D1i ) + ch( u ~
y D 2i )
u × sh( u y eD )
x +L / 2
∞
ch(ε k ~
y D1i ) + ch(ε k ~
y D 2i ) wi hi
k πx
kπx ′
cos(
) cos(
) dx ′
+ 2C ∑
∫
ε k × sh(ε k y eD )
xe
xe
k =1
xwi − Lhi / 2
S HWi ( M D , s ) = CLhi
∞
+ 2CLhi ∑ cos(
n =1
∞
nπz wiD ⎡ ch (ε n ~
y D1i ) + ch(ε n ~
y D 2i ) ⎤
nπz D
) cos(
)⎢
⎥
ε n × sh(ε n y eD )
z eD
z eD ⎣
⎦
nπz wiD
nπz D
) cos(
)
z eD
z eD
~
y ) + ch (ε ~
y ) xwi + Lhi / 2
,
(4.16)
+ 4C ∑ cos(
n =1
∞
×∑
k =1
ch(ε k ,n
D1i
k ,n
ε k ,n × sh(ε k ,n y eD )
D 2i
∫
cos(
xwi − Lhi / 2
k πx
k πx ′
) cos(
) dx ′
xe
xe
zD
yD
M wi ( xwiD , y wiD , zwiD )
Lhi / 2
Lhi / 2
xD
Figure 4.4 – Configuration to compute the source function for a horizontal line segment.
Equation 4.16 was derived for the case where a horizontal line segment is parallel
to the x D coordinate axis. However it can be readily applied for a segment parallel to any
of the coordinate axes by an appropriate replacement of the variables. This procedure is
demonstrated in Appendix A.
51
4.3 Source Function for a Slanted Line-Source Segment
Here we assume that a slanted line-source segment lies in a plane that is
orthogonal to the horizontal plane ( x - y ) and parallel to the vertical plane ( x - z ); the
segment also makes an angle ϕ with the negative direction of the vertical coordinate axis
( z ). The segment is centered at coordinates ( xwi , ywi , zwi ) with length, Lsi . This
configuration for a slanted line-source segment is illustrated in Figure 4.5.
In this case, the inner boundary, Bw , is a line which has the same length of the
slanted segment, Lsi . The integration given by Eq. 3.23 should be performed along the
length of the source in the slanted direction, L . Thus, Eq. 3.23 for a slanted line segment
takes the form given in Eq. 4.17.
Lsi 2
S SWi ( M D , s ) =
∫ G (x
D
, y D , z D , x D′ , y wiD , z ′D , s ) dL′ .
(4.17)
− Lsi 2
ϕ
zD
M wi ( xwiD , y wiD , zwiD )
Lsi / 2
Lsi / 2
yD
L
xD
Figure 4.5 – Configuration to compute the source function for a slanted line segment.
Equation 4.14 indicates that the source coordinates, x ′D and y ′D , change as the
variable of integration moves along the path of integration. Applying parametric
52
integration [Finney and Thomas (1990)] and trigonometric equivalences, we can evaluate
the integral in Eq. 4.17 to obtain
ch( u ~
y D1i ) + ch( u ~
y D 2i )
u × sh( u yeD )
L /2
∞
kπx
ch(ε k ~
y D1i ) + ch(ε k ~
y D 2i ) si
kπ ( xwi + Lsi′ cos α )
cos(
) cos(
) dL ′
+ 2C ∑
∫
sh
(
y
)
x
x
ε
ε
×
k =1
k
k eD
e
e
− Lsi / 2
S SWi ( M D , s ) = CLsi
L /2
ch(ε n ~
y D1i ) + ch(ε n ~
y D 2i ) si
nπ ( z wi + Lsi′ sin α )
nπz
cos(
) cos(
) dL ′ ,
+ 2C ∑
∫
ze
ze
ε n × sh(ε n yeD )
n =1
− Lsi / 2
∞
(4.18)
Lsi / 2
nπ ( z wi + Lsi′ sin α )
nπz
) cos(
)
ze
ze
n =1 − Lsi / 2
~
~
∞ ch(ε
kπ ( xwi + Lsi′ cos α )
kπx
k ,n y D1i ) + ch(ε k ,n y D 2 i )
×∑
cos(
) cos(
) dL ′
xe
xe
ε k ,n × sh(ε k ,n yeD )
k =1
∞
+ 4C ∑
∫
cos(
where
α = ϕ − π / 2 , for 0 < ϕ < π .
(4.19)
4.4 Source Function for a Deviated Line-Source Segment
A deviated line-source segment is assumed to lie in a plane that is orthogonal to
the vertical plane ( x - z ) and parallel to the horizontal plane ( x - y ); it also makes an
angle θ , (0 < θ < π ) with the positive direction of the horizontal coordinate axis ( x ). The
segment is centered at coordinates ( xwi , ywi , zwi ) with length, Ldi . The configuration for a
deviated line source segment is shown in Figure 4.6.
53
Ldi / 2
zD
L
Ldi / 2
θ M wi ( xwiD, ywiD, zwiD)
yD
xD
Figure 4.6 – Configuration to compute the source function for a deviated line segment.
For this configuration, the source coordinate z ′D = z wiD remains constant and the
source coordinates x ′D and y ′D change as we move along the deviated segment length,
Ldi . Comparing Figures 4.5 and 4.6, we can see that the source configuration in Figure
4.6 is the same as that in Figure 4.5 if the system of coordinates is rotated (the
appropriate rotation of the coordinate axes is explained in Appendix A). Therefore, the
source function for a deviated well can be written from Eq. 4.18 with the appropriate
rotation of the coordinate axes as follows
ch ( u ~
z D1i ) + ch ( u ~
z D 2i )
u × sh( u z eD )
L /2
∞
ch (ε k ~
z D1i ) + ch (ε k ~
z D 2i ) di
kπ ( x wi + Ldi′ cos θ )
kπ x
+ 2C ∑
cos(
) cos(
) dL ′
∫
xe
xe
ε k × sh(ε k z eD )
k =1
− Ldi / 2
S dWi ( M D , s ) = CLsi
∞
+ 2C ∑
n =1
ch (ε~n ~
z D1i ) + ch (ε~n ~
z D 2i )
~
~
ε × sh(ε z )
n
n eD
Ldi / 2
∫
− Ldi / 2
cos(
nπ ( y wi + Ldi′ sin θ )
nπ y
) cos(
) dL ′ ,
ye
ye
(4.20)
Ldi / 2
nπ ( y wi + Ldi′ sin θ )
nπ y
) cos(
)
ye
ye
n =1 − Ldi / 2
∞
ch (ε~k ,n ~
z D1i ) + ch (ε~k ,n ~
z D 2i )
kπ ( x wi + Ldi′ cos θ )
k πx
×∑
cos(
) cos(
) dL ′
~
~
xe
xe
ε k ,n × sh(ε k ,n z eD )
k =1
∞
+ 4C ∑
∫
cos(
where
~
z D1 = zeD − z D − z D′ ,
(4.21)
54
~
z D 2 = zeD − z D + z ′D ,
(4.22)
ε~n = u + ( nπ y eD ) 2 ,
(4.23)
ε~k ,n = u + ( kπ xeD ) 2 + ( nπ y eD ) 2 .
(4.24)
and
4.5 Source Function for a Generic Line-Source Segment
A generic line-source segment is assumed to lie in a plane which makes an angle,
θ , with the positive direction of the x coordinate axis. The line segment also makes an
angle, ϕ , with the negative direction of the z coordinate axis. The segment is centered at
( xwi , ywi , zwi ) with length, Lwi . The configuration for a generic line-source segment is
shown in Figure 4.7.
zD
yD
ϕ
Lwi / 2
M wi ( xwiD , y wiD , zwiD )
Lwi / 2
L
θ
xD
Figure 4.7 – Configuration to compute the source function for a generic line source
segment.
55
For this configuration, all three coordinates of the source change as we move
along the length of the source, significantly increasing the complexity of the analytical
evaluation of the integral in Eq. 3.23. Therefore, in this work, the source function for a
generic line-source segment is computed by numeric integration. The expression to
compute the generic line-source function is given by Eq. 4.25 where the integration
parameter is L′ .
S wi ( M D , s ) = C
ch ( u ~
y D1i ) + ch ( u ~
y D 2i )
dL′
∫
u
sh
(
u
y
)
×
− Lwi / 2
eD
Lwi / 2
+ 2C
ch(ε k ~
y D1i ) + ch(ε k ~
y D 2i )
kπx
kπx ′
cos(
) cos(
) dL′
ε k × sh(ε k y eD )
xe
xe
− Lwi / 2 k =1
+ 2C
ch(ε n ~
y D1i ) + ch(ε n ~
y D 2i )
nπz
nπz ′
cos(
) cos(
) dL′ ,
∑
∫
ε n × sh(ε n y eD )
ze
ze
− Lwi / 2 n =1
Lwi / 2
∞
∫ ∑
Lwi / 2
Lwi / 2
∞
(4.25)
nπz
nπz ′
) cos(
)
ze
ze
− Lwi / 2 n =1
∞
ch(ε k ,n ~
y D1i ) + ch (ε k ,n ~
y D 2i )
k πx
kπx ′
cos(
) cos(
) dL′
×∑
ε k ,n × sh(ε k ,n y eD )
xe
xe
k =1
+ 4C
∞
∫ ∑ cos(
where
x′ = xwi + L′ sin ϕ cosθ ,
(4.26)
y ′ = y wi + L′ sin ϕ sin θ ,
(4.27)
z′ = zwi + L′ cosϕ ,
(4.28)
with
− Lwi / 2 ≤ L′ ≤ − Lwi / 2 ,
(4.29)
0 < ξ < π , for ξ = ϕ ,θ
(4.30)
and
56
Efficient and accurate computation of the source function presented in this chapter
requires special techniques to evaluate the ratios of the hyperbolic functions and infinite
summations in Eqs. 4.13, 4.16, 4.18, 4.20 and 4.25. These issues are addressed in Chapter
6.
57
CHAPTER 5
GAS FLOW AND WELLBORE EFFECTS
The basic semi-analytical simulation approach presented in Chapter 3 used the
slightly compressible fluid assumption. Significant applications of the semi-analytical
simulation approach, however, are expected in unconventional gas reservoirs. Therefore,
one of the additional features to be discussed in this chapter is the extension of the model
approach to dry-gas reservoirs.
This chapter also explains how to incorporate wellbore storage and skin effects in
the basic semi-analytical simulation model presented in Chapter 3. Wellbore storage
effect is important if the semi-analytical simulator is to be used for modeling and analysis
of pressure-transient responses. Similarly, skin effect significantly influences the
production and pressure-transient performances of wells and has to be incorporated in
simulators to obtain realistic estimates of reservoir performances.
Another useful feature to include in the semi-analytical simulator is the ability to
account for frictional pressure losses in horizontal wellbores. When dealing with long
horizontal wells, frictional pressure drop in the wellbore may become significant at high
flow rates and infinite-conductivity wellbore assumption may not be appropriate. To
avoid generating deceiving results, the semi-analytical simulator should take into account
the effect of wellbore pressure drop on the performance of the well.
58
5.1 Gas Flow
Real gases have high compressibility and low viscosity when compared to liquids.
These properties have strong dependency on pressure, thereby causing the diffusivity
equation to become non-linear. Al Hussainy and Ramey (1966) and Al Hussainy et al.
(1966) proposed the concept of real gas pseudo-pressure (or real gas potential), in order
to obtain a linear equation for gas flow in porous media. The real gas pseudo-pressure is
given by
m( p ) = 2 ∫
p
p ref
p′
dp ′ .
µg zg
(5.1)
Using the dimensionless variables defined by Eqs. 3.9 and Eq. 3.10, the diffusivity
equation for gas flow in porous media can be written in terms of pseudo-pressure as
follows
∂ 2 ∆m( p) ∂ 2 ∆m( p ) ∂ 2 ∆m( p) ∂∆m( p)
,
+
+
=
∂xD2
∂y D2
∂z D2
∂t D
(5.2)
where
∆m( p ) = m( pi ) − m( p ) ,
(5.3)
2.637 × 10−4 k
t.
φ ( µct )i l 2
(5.4)
and
tD =
Eq. 5.2, with t D defined as in Eq. 5.4, works properly for times where the product of
viscosity and total formation compressibility, ( µct ) i , remains approximately constant;
this happens when the average reservoir pressure ( p av ) does not drop significantly from
the initial reservoir pressure ( pi ) . However, when the average reservoir pressure is
59
significantly lower than the initial pressure, a correction in the dimensionless time is
required to account for changes in the viscosity and compressibility of the real gas
[Raghavan (1993)]. Hence, Eq. 5.2 should be solved by using real gas pseudo-time
[Agarwal (1979)]
t
t ps = ∫
0
dt ′
.
µct
(5.5)
The dimensionless form of real gas pseudo-time is given by
t psD = 2.637 × 10 − 4
k
t ps .
φl 2
(5.6)
Based on Eq. 5.2, we can apply the solutions presented in Chapter 3 for liquid
flow, provided that we compute the real gas pseudo-pressure instead of the gas pressure
and use the appropriate dimensionless time definition. The source functions to obtain the
gas pseudo-pressure are the same as presented in Chapter 4 for the liquid case, except the
constant C must be replaced by
C=
p sc Tres
π
×
,
−5
1.988 × 10 Tsc klhD xeD
(5.7)
where Tres is the reservoir temperature and Tsc is the temperature at standard conditions,
both in degrees Rankine, °R.
5.2 Skin Effect
Skin effect is a concept used to model changes in permeability in a thin region, or
skin zone, in the near wellbore vicinity. Whenever a skin zone is present, it significantly
affects the pressure measured in the wellbore. Skin is represented as the additional
60
pressure drop in a well when compared to an identical well with no skin. A positive skin
effect indicates a restriction of the fluid flow into the wellbore and the wellbore is said to
be damaged. A negative skin represents an increase in flow capacity from the reservoir to
the wellbore and the wellbore is said to be stimulated. The skin effect is obtained through
pressure transient response analysis and is used to determine the need for stimulation to
improve well flow capacity. The skin effect was first defined by van Everdingen and
Hurst (1949) to be the dimensionless value of the pressure drop for a cylindrical skin
zone in a fully penetrating vertical well; that is,
S SK =
kr h
∆pSK ,
141.2qµ
(5.8)
and
p w = p sf − ∆p SK .
(5.9)
In Eqs. 5.8 and 5.9, q is the well flow rate in reservoir conditions, µ is the fluid
viscosity, h is the reservoir thickness, k r is the radial permeability in the plane orthogonal
to the well axis, S SK is the skin effect, and p sf is the pressure at the interface (sand face)
between the skin zone and the reservoir. Note that Eq. 5.8 represents the overall effect of
the skin zone under uniform flux distribution. For long or complex wells with nonuniform flux distribution, we use Eq. 5.8 for each individual segment where it is
reasonable to assume uniform flux distribution. We then define a skin factor by
ssd ( wi ) =
kr
∆psd ( wi ) ,
141.2 q~( wi ) µ
(5.10)
61
where ssd ( wi ) is the skin factor, ∆psd ( wi ) is the pressure drop in the damaged zone, and
q~( wi ) is the flux for the i th well segment expressed in reservoir conditions. Then, applying
the Laplace transformation and using ∆p instead of pressure, we obtain
∆p( wi ) = ∆p sf ( wi ) + C sk ( wi ) q~( wi ) .
(5.11)
Where
C sk ( wi ) =
141.2 µ
s sd ( wi )
k r ( wi )
(5.12)
for liquid and
Csk ( wi ) =
Tres psc
ssd ( wi )
1.988 × 10 −5 Tsc k r ( wi )
(5.13)
for gas. Note that the skin effect (Eq.5.8) and the skin factor (Eq. 5.10) are the same only
if the radial permeability and the flux are uniform along the well length; this condition is
not likely to occur in long horizontal wells. Al-Otaibi and Ozkan (2005) presented a
detailed analysis of the skin factor and skin effect in long horizontal wells.
The skin concept is incorporated in the model by using equation Eq. 5.11 with
constant skin, defined by Eq. 5.12 and Eq. 5.13 for liquid and gas, respectively. Since the
model computes the solution in an equivalent isotropic system, the radial permeability k r
is replaced by the average permeability k in Eqs. 5.8 and 5.10.
5.3 Wellbore Storage
Wellbore storage represents the contribution of the fluid contained in the wellbore
to the total flow rate of the well. It may either be due to fluid expansion/compression or
62
due to changes in the liquid level inside the wellbore [Raghavan (1993), Horne (2005)].
In both cases, the volume coming from wellbore storage is related to changes in wellbore
pressure by the wellbore storage coefficient
Vwb = C wb × ∆pw .
(5.14)
The derivative of Eq. 5.14, with respect to time, gives the flow rate due to wellbore
storage, which, in the Laplace domain, yields:
qwb = sC wb × ∆p w .
(5.15)
The mass balance in the well-reservoir system requires that the total flow rate leaving the
wellbore (q ) must equal the reservoir flow rate (q sf ) plus the flow rate due to wellbore
storage (qwb ) ; thus
q = qsf + qwb .
(5.16)
Considering a particular well segment (wi) with uniform flux, we substitute Eq. 5.15 into
Eq. 5.16 and obtain
q(wi) = (Lq~ )(wi) + sC wb(wi) ∆p(wi) ,
(5.17)
where L is the length of the well segment (wi) . Equation 5.17 may be used to incorporate
the wellbore storage effect into the semi-analytical simulator. In Eq. 5.17, C wb ( wi ) is the
storage coefficient based on the volume of each individual well segment, wi and is
defined by [Horne (2005), Raghavan (1993)]
C wb (wi) = 6.3288 × 10 − 3
ηref
l2
Cliq
(5.18)
for liquids and
C wb (wi) = 3.1622 × 10 − 6
µi Tsc ηref
C gas
p scTres l 2
(5.19)
63
for gases. In Eqs. 5.18 and 5.19,
⎛ ∆V ⎞
⎟⎟ , for ξ = (liq, gas )
Cξ = ⎜⎜
⎝ ∆p ⎠ ξ
(5.20)
Equation 5.17 assumes that time variations in pressure at the center of a well segment are
the same as the time variations in the average pressure of the segment.
It should be noted that under the assumptions of constant wellbore pressure and
infinite conductivity, the wellbore storage effect vanishes in each segment. Hence,
theoretically, the wellbore storage effect does not affect the rate response at constant
pressure. In practice, operational aspects may require a sudden change in wellbore
pressure before the establishment of a constant pressure condition; this could introduce a
wellbore storage effect into the constant pressure response.
5.4 Wellbore Friction
The infinite conductivity wellbore assumption used to obtain the matrix solution
in Chapter 3 implies that the pressure drop along the wellbore is negligible compared to
the pressure drop required to move fluid in the reservoir. This assumption is acceptable to
compute solutions in most well-reservoir systems. However, long wells producing at high
flow rates from high permeability reservoirs may have friction losses which impact the
system’s performance.
We incorporate the finite conductivity wellbore condition into the semi-analytical
simulator by requiring that the pressure in the upstream segment of the wellbore be
greater than the pressure in the downstream segment and that the difference between the
64
two is the amount equal to the frictional pressure drop. The frictional pressure drop is a
nonlinear function of flow rate in the wellbore and flow rate in the wellbore depends on
the reservoir pressure drop. This coupled effect leads to a non-linear problem that cannot
be handled in the Laplace domain. An alternative, approximate approach is to compute
the frictional pressure drop in the real-time domain, using fluxes computed for the
previous time step. If this approach is used, the calculation may be started assuming
uniform flux for the first time step. The frictional pressure drop can then be added to the
pressure drop for each individual wellbore segment in the Laplace domain.
The frictional pressure drop for each segment in the wellbore may be computed
using [Ozkan et al. (1995) and Ozkan et al. (1999)]
∆p fri = Ei f i qci2 Lwi
(5.21)
where
⎧~
⎪q L +
qci = ⎨ wi wi
⎪q~ L
⎩ wi wi
n
∑ q~
j = i +1
Ei = 9.117 × 10−13
wj
⎫
; for i = 1,2, n − 1⎪
⎬ ,
⎪
; for i = n
⎭
Lwj
ρ
π r
2 5
wi
,
(5.22)
(5.23)
and f i is the Fanning friction factor, which is a function of the Reynolds number:
(N Re )i
= 6.175 × 10− 2
ρ qci
.
µ rwi
(5.24)
Equations 5.21 and 5.22 consider n segments numbered in the upstream direction of the
well. After the frictional pressure drop is computed using the fluxes from the previous
time step, the finite-conductivity wellbore condition is implemented in the Laplace
domain, as follows:
65
∆pi −1 − ∆pi =
∆p fr( i −1)
s
; for i = 2, n .
(5.25)
The procedure to compute the wellbore friction loss, given in Eqs. 5.21 through
5.25, requires that the fluxes and pressure drops be computed sequentially in time. In this
case, the solution, even in the Laplace domain, needs to be computed in a sequence of
time steps.
5.5 Matrix Equations
In order to introduce the effects presented in the prior sections of this chapter, the
matrix equations presented in Chapter 3 must be modified. Here, we consider the
example of the two-block system considered in Chapter 3 (Figure 3.5) and present the
linear system of equations by incorporating the effects of skin, wellbore storage, and
friction.
First, we consider the case for a specified wellbore flow rate. Without the effects
of the wellbore storage, skin, and wellbore friction, the linear system of equations have
been presented in Chapter 3 by Eqs. 3.34 through 3.39. The linear system of equations
have been also presented in matrix-vector form by Eq. 3.33 and illustrated in Figure 3.6.
After the introduction of the wellbore storage, skin, and wellbore friction effects, only the
[A1 ] component of the coefficient matrix and the right-hand side vector, {b2 }, change as
follows
66
⎡ ⎛ S 1w1w1 ⎞
⎟
⎢⎜
1
⎜
⎢ ⎝ + Cskw1 ⎟⎠
⎢
⎢ 1
⎢ S w 2 w1
⎢
⎢ S 1e1w1
⎢
1
⎢ S e 2 w1
⎢
[A1 ] = ⎢⎢ 0
⎢
⎢
0
⎢
⎢
0
⎢
⎢
0
⎢
⎢
0
⎢⎛ L1h1
⎞
⎟
⎢⎜
1
⎜
⎢⎣⎝ + sCstw1 ⎟⎠
1
1
S w1w 2
S w1e1
⎛ S 1w 2 w 2 ⎞
⎟
⎜
⎜ + C1 ⎟
skw 2 ⎠
⎝
S w 2e1
1
S w1e 2
1
1
S w2e 2
1
S e1e1 + S e1e1
S e2 w2
1
S e 2e1 + S e 2 e1
0
− S w1e1
0
S e1w 2
0
1
2
S e1e 2 + S e1e 2
1
2
S e 2e 2 + S e 2e 2
1
2
1
2
2
2
− S e1w1
2
− S e 2 w1
⎛ S 2w1w1 ⎞
⎟
⎜
⎜+ C2 ⎟
skw1 ⎠
⎝
2
− S w1e 2
− S w 2e1
2
− S w2e 2
S w 2 w1
0
0
0
0
0
0
0
0
0
0
0
0
0
⎛L
⎞
⎜
⎟
1
⎜ + sC ⎟
stw 2 ⎠
⎝
1
h2
0
2
2
0
⎛L
⎞
⎜
⎟
2
⎜ + sC ⎟
stw1 ⎠
⎝
2
h1
⎤
⎥
⎥
⎥
⎥
0
⎥
⎥
2
− S e1w 2 ⎥
⎥
2
− S e2 w2 ⎥
⎥
2
S w1w 2 ⎥
⎥
⎥
⎛ S 2w 2 w 2 ⎞ ⎥
⎟
⎜
⎜+ C2 ⎟ ⎥
skw 2 ⎠ ⎥
⎝
0
⎥
⎥
0
⎥
⎥
0
2
⎛ Lh 2
⎞⎥
⎜
⎟⎥
⎜ + sC 2 ⎟⎥
stw 2 ⎠ ⎦
⎝
0
(5.26)
and
⎧⎪ ∆p1fr1
{b2 } = ⎨
⎪⎩ s
∆p1fr, 22
∆p 2fr3
s
s
⎫⎪
q⎬ .
⎪⎭
(5.27)
The other components of the linear system remain unchanged.
Next, we consider the case for a specified wellbore pressure. After the effects of
skin, wellbore storage, and friction are incorporated, the linear system in Eqs. 3.40
through 3.42 becomes
67
⎡ ⎛ S 1w1w1 ⎞
⎟
⎢⎜
1
⎜
⎢ ⎝ + Cskw1 ⎟⎠
⎢
⎢ 1
⎢ S w 2 w1
⎢
⎢
1
⎢ S e1w1
⎢
⎢
[A] = ⎢⎢ S 1e 2 w1
⎢
⎢
⎢
0
⎢
⎢
⎢
0
⎢
⎢
⎢⎛ 1
⎞
⎢⎜ Lh1
⎟
1
⎢⎜ + sCstw1 ⎟
⎠
⎣⎝
1
1
1
S w1w 2
S w1e1
⎛ S 1w1w 2 ⎞
⎜
⎟
⎜ + C1 ⎟
skw 2 ⎠
⎝
S w 2 e1
S w2e 2
⎛ S 1e1e1 ⎞
⎜
⎟
⎜ 2 ⎟
⎝ + S e1e1 ⎠
⎛ S 1e 2 e1 ⎞
⎜
⎟
⎜ 2 ⎟
⎝ + S e 2e1 ⎠
⎛ S 1e1e 2 ⎞
⎜
⎟
⎜ 2 ⎟
⎝ + S e1e 2 ⎠
⎛ S 1e 2e 2 ⎞
⎟
⎜
⎜ 2 ⎟
⎝ + S e 2e 2 ⎠
S
S w1e 2
1
1
e1w 2
1
S e 2 w2
0
0
−S
2
w1e1
−S
2
w 2 e1
⎛ L1h 2
⎞
⎜
⎟
1
⎜ + sC ⎟
stw 2 ⎠
⎝
1
0
−S
2
w1e 2
−S
2
w2e 2
0
0
0
0
0
2
− S e1w 2
− S e 2 w1
2
− S e2 w2
⎛ S 2w1w1 ⎞
⎜
⎟
⎜+ C2 ⎟
skw1 ⎠
⎝
S w1w 2
− S e1w1
S
2
w 2 w1
⎛ L2h1
⎞
⎜
⎟
2
⎜ + sC ⎟
stw1 ⎠
⎝
2
2
2
⎛ S 2w1w 2 ⎞
⎜
⎟
⎜+ C2 ⎟
skw 2 ⎠
⎝
2
⎛ Lh 2
⎞
⎜
⎟
2
⎜ + sC ⎟
stw 2 ⎠
⎝
⎤
0⎥
⎥
⎥
⎥
0⎥
⎥
⎥
0⎥
⎥
⎥
⎥
0 ⎥,
⎥
⎥
0⎥
⎥
⎥
⎥
0⎥
⎥
⎥
− 1⎥
⎥
⎦
(5.28)
{x} = {q~w11
q~w1 2
q~e11
q~e12
q~w21
q~w22
q },
(5.29)
and
⎧∆pw
⎫
1
⎪
⎪
∆p fr1
⎪∆pw −
⎪
s
⎪
⎪
⎪⎪0
⎪⎪
{b} = ⎨0
⎬.
⎪
⎪
1
1, 2
⎪∆pw − ∆p fr1 + ∆p fr2 / s
⎪
⎪∆p − ∆p1 + ∆p1, 2 + ∆p 2 / s ⎪
fr1
fr2
fr3
⎪ w
⎪
⎪⎩0
⎪⎭
(
(
)
(5.30)
)
It is important to emphasize that Eqs. 5.26 through 5.30 work only when the
Laplace variable, s , is related to the time step, ∆t , instead of to the total time, t .
Consequently, if the specified boundary conditions, q or ∆pw , are not constant over
time, they need to be discretized over time as shown in Figure 5.1.
68
q, ∆pw
q j , ∆ p wj
∆t j
t
Figure 5.1 – Time discretization of boundary conditions.
Therefore, to compute the solution for the time interval, ∆t j , the boundary
conditions in Eqs. 5.27 and 5.30 are given by:
q = qj / s
(5.31)
∆pw = ∆pwj / s ,
(5.32)
and
respectively.
69
CHAPTER 6
COMPUTATIONAL ASPECTS
As presented in Chapter 4, the model developed in this work uses a point-source
solution in a parallelepiped reservoir to obtain source functions for different source
geometries. Point-source solutions for parallelepiped reservoirs are usually derived by
applying the method of images to the solutions for infinite domain [Gringarten and
Ramey (1974), Ozkan (1988)] or by applying Fourier transformation to solve the
diffusion equation with specified boundary conditions. Both options lead to solutions that
include infinite summations of eigenfunctions, which generally have slow convergence
characteristics. This issue has been addressed in Marshall3 for steady state diffusion
problems in electrochemical cells, Thompson et al. (1991) for transient pressure problems
in the time domain, and Ozkan and Raghavan (1991a, 1991b), and Raghavan and Ozkan
(1994) for transient pressure problems in the Laplace transform domain.
Since the solutions presented in this work are in the Laplace transform domain,
we follow the approach suggested by Ozkan and Raghavan (1991b) to improve the
computation of the source functions given in Chapter 4. The main idea presented in
Ozkan and Raghavan (1991b) is to separate in the infinite-acting and boundarydominated flow contributions in the solution; then recast the slow converging
3
Marshall, S.L. Rapidly-Converging Modified Green’s Function for Laplace’s Equation in Three-
Dimensional Regions with Rectangular Boundaries. The reference for this paper was not found.
70
summations and computationally difficult integrals into computationally more efficient
forms.
In this chapter, we first present some useful formulas to be used in our
derivations. We then start our discussion with the simpler case of a fully penetrating
planar source. Our discussions continue with the presentation of the computational forms
for a partially penetrating planar source and for line sources with different orientations in
the coordinate geometry (horizontal, vertical, slanted, deviated, etc.). The chapter
concludes with a note about the convergence criteria applied to compute the source
functions. Appendix B presents detailed derivations of the equations presented in this
chapter.
6.1 Preliminary Mathematical Results
The computationally efficient forms of the solutions presented in the following
sections are derived by using a few important mathematical results [Ozkan (1988), Ozkan
and Raghavan (1991a, 1991b, 1998) ]. These results are presented here as background
and will be used to simplify the mathematical notation in the following sections.
The first result converts the ratio of hyperbolic functions into an infinite
summation of exponential terms:
ch ( ξ ~
y D1 ) + ch ( ξ ~
yD2 )
sh( ξ y e )
{
= e−
ξ ( y D + ywD )
+e
yD1 )
− ξ ( yD + ~
+e
yD 2 )
− ξ ( yD + ~
+e
− ξ y D − ywD
}
∞
⎡
× ⎢1 + ∑ e −2 m
⎣ m=1
ξ yeD ⎤
⎥
⎦
.
(6.1)
71
In the following we refer to the left hand side of Eq. 6.1 as Fcsht [ξ ] . This
expression can be decomposed into two terms
Fcsht [ξ ] = Fcshp [ξ ] + e
− ξ yD − ywD
,
(6.2)
where
{
Fcshp [ξ ] = e −
{
+ e
− ξ ( yD + ywD )
ξ ( yD + ywD )
+e
+ e−
− ξ ( yD + ~yD 1 )
ξ ( yD + ~yD1 )
+e
+ e−
− ξ ( yD + ~
yD 2 )
ξ ( yD + ~yD 2 )
}× ⎡⎢1 + ∑ e
∞
⎣
+e
− ξ yD − ywD
}
−2 m ξ yeD
m =1
⎡∞
× ⎢∑ e −2 m
⎣ m=1
⎤
⎥
⎦
ξ yeD
⎤
⎥
⎦
.
(6.3)
Using the results in Eq. 6.1, the term in the left hand side, hence the source functions, can
be accurately computed for t AD ≥ 10 −2 [Ozkan and Raghavan (1991a)], where t AD is given
by
t AD =
tD
.
xeD yeD
(6.4)
The term, Fcsht [ξ ] , appears in our solutions multiplying cosine functions in
infinite series. Equation 6.3 indicates that at early times, the argument of e
− ξ y D − y wD
tends to zero and Fcsht [ξ ] approaches the unit value quickly. This slows down the
convergence of the cosine series and affects the accuracy of the computations at early
times. This issue was addressed by Ozkan (1988) using the following result:
∞
∑ cos(nπz ) cos(nπz
n =1
[
w
)
e
− u + ( nπ / hD ) 2 + a 2 y D − y wD
u + ( nπ / h D ) 2 + a 2
⎧ +∞
2 2
2
2
hD ⎪ ∑ K 0 ( z − z w − 2n ) hD + ( y D − y wD ) u + a
=
⎨n = −∞
2π ⎪
2 2
2
2
⎩+ K 0 ( z + z w − 2n ) hD + ( y D − y wD ) u + a
[
]
]⎫⎪ − e
⎬
⎪
⎭
− u + a 2 y D − y wD
2 u + a2
.
(6.5)
72
6.2 Fully Penetrating Plane Source
This case corresponds to the computation of the source function given in Eq. 4.13,
when 2 z f = ze . For this condition, Eq. 4.13 simplifies to:
ch( u ~
y D1 ) + ch( u ~
yD 2 )
~
u × sh( u yeD )
.
xe + x f ∞
ch(ε k ~
y D1 ) + ch(ε k ~
yD 2 )
kπx
kπx′
cos(
) cos(
) dx ′
+ 4Cz f ∫ ∑
x
x
ε × sh(ε ~y )
S FP = 4Cx f z f
xe − x f k =1
k
k
eD
e
(6.6)
e
In Eq. 6.6, the subscript FP stands for fully penetrating planar source. In the following
derivations, the coordinates of the central point in a source segment will be denoted by
the subscript w . This notation will be applied to all equations in this chapter.
Using Eq. 6.3, we recast Eq. 6.6 as
S FP = 4Cx f z f
Fcsht [u ]
+ 4Cz f
u
xw + x f
∞
1
∫ ∑ε
xw − x f
k =1
k
cos(
kπx
kπx′ Fcsht [ε k ]
) cos(
)
dx ′ ,
xe
xe
εk
(6.7)
To identify the terms that contribute to the solution at early and late times, we write Eq.
6.7 in the following form:
S FP = S FPb1 + S FPb 2 + S FPb3 + S FP inf ,
(6.8)
where
S FPb1 = 4Cx f z f
Fcsht [u ]
.
u
(6.9)
The terms S FPb 2 , S FPb 3 , and S FP inf are related to the infinite summation in Eq.6.7 and
are computed differently at early and late times. At early times,
S FPb 2 =
8Cz f x e
π
∞
1
∑ kε
k =1
sin(
k
kπ x f
xe
) cos(
kπ x w
kπ x
) cos(
)Fcshp [ε k ] ,
xe
xe
(6.10)
73
xeD
S FPb 3 = 2Cz f x f
+ 2Cz f x f
− 2Cz f x f
xeD
π
e
π
∫K [
+1
0
]
( x D + xwD − x fD β ) 2 + ( y D − y wD ) 2 u dβ
−1
∑∫K [
∞ +1
k =1 −1
0
]
( x D m xwD m 2kxeD − x fD β ) 2 + ( y D − y wD ) 2 u dβ ,
(6.11)
− u y D − y wD
u
and
S FP inf = 2Cz f x f
xeD
π
∫K [
+1
0
]
( xD − xwD − x fD β ) 2 + ( yD − ywD ) 2 u dβ ,
(6.12)
−1
where
x fD = x f k / k x / l .
(6.13)
Note that, following Ozkan and Raghavan (1991a), in Eq.6.11, we used the m sign to
present the equation in a compact form. For a generic function, f , the m sign represents
the following sum of the terms:
f (a m b m c) = f (a − b − c) + f (a − b + c) + f (a + b − c) + f (a + b + c) .
(6.14)
At late times, we have
S FPb 2 =
8Cz f x e
π
∞
1
∑ kε
k =1
sin(
k
kπ x f
xe
) cos(
k πx w
k πx
) cos(
)Fcsht [ε k ] ,
xe
xe
(6.15)
and S FPb3 = S FPb inf = 0 . The terms, S FPb1 , S FPb 2 , and S FPb3 represent the contribution
from the boundaries, while S FP inf is the solution for a fully penetrating plane source in an
infinite slab reservoir.
We note that the ratio
Fcsht [u ]
in S FPb1 (Eq. 6.9) becomes difficult to compute at
u
late times ( u → 0 ) when yeD becomes small because of slowly converging series. This
situation arises in thin rectangular porous media used to model hydraulic fractures in the
74
reservoir. We have found that the infinite series in the term Fcsht of Eq. 6.9 converges
faster when
u yeD > 5 × 10 −4 .
(6.16)
If the condition given by Eq. 6.16 is not satisfied, we apply the identity [Gradshtein and
Ryzhik (1965)]
∞
cos kx
π ch[a(π − x)] 1
=
− 2 , (0 ≤ x ≤ 2π ) ,
2
2
+a
2a sh(aπ )
2a
∑k
k =1
(6.17)
and recast Eq. 6.9 as [Ozkan (1988)]:
S FPb1
⎧
⎪
⎪ 2
= 4Cx f z f ⎨
⎪ yeD
⎪
⎩
We apply Eq. 6.18 for
∞
∑
cos[kπ (
y − ywD
y D + ywD
)] + cos[kπ D
]
yeD
yeD
⎛ kπ ⎞
⎟⎟
u + ⎜⎜
⎝ yeD ⎠
k =1
2
⎫
⎪
2 ⎪
+
⎬.
uyeD ⎪
⎪
⎭
(6.18)
u y eD ≤ 5 × 10 −4 . Eq. 6.18 can be further simplified if
2
⎛ kπ ⎞
⎟⎟ .
u << ⎜⎜
⎝ yeD ⎠
(6.19)
In these cases, we have the following relation for the argument of the summation in Eq.
6.18:
cos[kπ (
y − ywD
y D + ywD
)] + cos[kπ D
]
yeD
yeD
⎛ kπ ⎞
⎟⎟
u + ⎜⎜
⎝ yeD ⎠
2
⎧
y − ywD
y D + ywD
)] + cos[kπ D
⎪ cos[kπ (
y ⎪
yeD
yeD
≅ eD2 ⎨
2
π ⎪
k
⎪⎩
⎫
]⎪
⎪
⎬
⎪
⎪⎭
.
(6.20)
75
Using Eq. 6.20 in Eq. 6.18 and knowing the summation formula [Gradshtein and Ryzhik
(1965)],
cos kx π 2 πx x 2
=
− + , (0 ≤ x ≤ 2π ) ,
∑
6
2 4
k2
k =1
∞
(6.21)
we obtain the following expression to compute S FPb1 [Ozkan (1988)]:
⎧⎪ 2
⎡ 1 ⎛ y + ywD + y D − ywD
+ 2 yeD ⎢ − ⎜⎜ D
S FPb1 = 4Cx f z f ⎨
2 yeD
⎢⎣ 3 ⎝
⎪⎩ uyeD
2 ⎞⎤ ⎫
⎞ ⎛ y D2 + ywD
⎟⎥ ⎪⎬ .
⎟+⎜
⎟ ⎜ 2 y2
⎟
yeD
⎠ ⎝
⎠⎥⎦ ⎪⎭
(6.22)
u y eD ≤ 5 × 10 −4 and u / (kπ / y eD ) < 0.01 .
We use Eq. 6.22 when
The integrals in Eqs. 6.11 and 6.12 are computed by numeric integration using
Simpson or Gauss-Legendre quadrature rules [e.g. Griffiths (1991)]. However, the
numerical evaluation of the integrals in Eq. 6.11 poses difficulties when x D = xwD and
y D = y wD . For these cases, we evaluate the integrals by using the following formulas
[Ozkan and Raghavan (1991a)]:
+a
∫−a K 0 ⎡⎢⎣b
(x D − cξ )2 ⎤⎥dξ =
1 ⎡
⎢
bc ⎣⎢
b ( x D + ac )
(x D − cξ )2 ⎤⎥dξ =
1 ⎡
⎢
bc ⎢⎣
b ( ac + x D )
⎦
∫ K 0 (β )dβ −
b ( x D − ac )
⎤
(
)
K
β
d
β
⎥, ( x D ≥ ac ) ,
0
∫0
⎦⎥
0
(6.23)
and
a
∫−a K 0 ⎡⎣⎢b
⎦
∫ K 0 (β )dβ +
b ( ac − x D )
0
⎤
∫ K (β )dβ ⎥⎥, ( x
0
0
⎦
D
≤ ac ) .
(6.24)
Use of Eqs. 6.9 through 6.24 with their appropriate time conditions makes the
computation of Eq. 6.6 more efficient and accurate.
76
6.3 Partially Penetrating Plane Source
This case corresponds to the computation of the source function in Eq. 4.13 when
2 z f < ze . Comparing Eq. 4.13 with the solution for the fully penetrating fracture given in
Eq. 6.6, we re-write Eq. 4.13 as follows:
zw + z j
S PP = S FP + 4Cx f
∞
+ 4C ∑
n =1
∞
∫ ∑ cos(
z w − z f n =1
nπz
nπz′ Fcsht [ε n ]
dz′
) cos(
)
ze
ze
εn
⎧⎪ z w + z f
⎫⎪ ∞
nπz
nπz′
′
) cos(
)dz ⎬ × ∑
⎨ ∫ cos(
ze
ze
⎪⎩ z w − z f
⎪⎭ k =1
⎧⎪ x w + x f
kπx
kπx′ Fcsht [ε k , n ] ⎫⎪
dx′⎬
) cos(
)
⎨ ∫ cos(
x
x
ε
e
e
k ,n
⎪⎩ x w − x f
⎪⎭
.
(6.25)
Equation 6.25 presents the solution for a partially penetrating plane source, S PP ,
as the summation of the solution for a fully penetrating plane source, S FP , and an
additional term, S PSF , that physically represents the partial penetration pseudoskin.
Hence,
S PP = S FP + S PSF
(6.26)
where
zw + z j
S PSF = 4Cx f
∞
+ 4C ∑
n =1
∞
∫ ∑ cos(
z w − z f n =1
nπz
nπz′ Fcsht [ε n ]
dz′
) cos(
)
ze
ze
εn
⎧⎪ z w + z f
⎫⎪ ∞
nπz
nπz′
) cos(
) dz′⎬ × ∑
⎨ ∫ cos(
ze
ze
⎪⎩ z w − z f
⎪⎭ k =1
⎧⎪ x w + x f
kπx
kπx′ Fcsht [ε k , n ] ⎫⎪
dx′⎬
) cos(
)
⎨ ∫ cos(
xe
xe
ε k ,n
⎪⎩ x w − x f
⎪⎭
.
(6.27)
Following the same procedure as in Section 6.2, we split Eq. 6.27 into terms
representing the contributions of the early- and late-time flow periods as follows
S PSF = S PSFb 4 + S PSFb5 + S PSFb6 + S PSF inf ,
(6.28)
77
At early time, we have
S PSFb 4 =
8Cx f ze
∞
π
S PSFb5 = 16C
1
∑ n sin(
nπz f
ze
n =1
∞
ze xe
π
1
∑ n sin(
2
) cos(
nπ z f
ze
n =1
nπz D
nπz wD Fcshp [ε n ]
) cos(
)
,
εn
zeD
zeD
) cos(
nπ z
nπ z w
) cos(
)
ze
ze
k πx f
1
kπ x
kπxw Fcshp [ε k , n ]
) cos(
) cos(
)
× ∑ sin(
ε k ,n
xe
xe
xe
k =1 k
∞
x eD
S PSFb 6 = 2Cx f z e
π
2
∞
1
∑ n sin(
nπz f
ze
n =1
[
) cos(
,
(6.29)
(6.30)
nπz w
nπz
) cos(
)
ze
ze
]
⎤
⎡ +1
2
2
⎥,
⎢ ∫ K 0 ( x D + x wD − x fD β ) + ( y D − y wD ) ε n dβ
−1
⎥
⎢
×
⎥
⎢ ∞ +1
⎢ + ∑ ∫ K 0 ( x D m x wD m 2kx eD − x fD β ) 2 + ( y D − y wD ) 2 ε n dβ ⎥
⎥⎦
⎣⎢ k =1 −1
[
(6.31)
]
and
xeD
S PSF inf = 2Cx f ze
+1
π
[
2
∞
1
∑ n sin(
nπz f
ze
n =1
) cos(
nπz
kπz
) cos( w )
ze
ze
]
.
(6.32)
× ∫ K 0 ( xD − xwD − x fD β ) + ( yD − ywD ) ε n dβ
2
2
−1
At late times
S PSFb 4 =
8Cx f h
π
S PSFb 5 = 16C
∞
1
∑ n sin(
n =1
ze xe
π
2
∞
1
nπz f
ze
∑ n sin(
n =1
) cos(
nπ z f
ze
nπzD
nπzwD Fcsht [ε n ]
,
) cos(
)
εn
zeD
zeD
) cos(
nπ z
nπ z w
) cos(
)
ze
ze
k πx f
kπ x
kπxw Fcsht [ε k , n ]
1
× ∑ sin(
) cos(
) cos(
)
xe
xe
xe
ε k ,n
k =1 k
∞
,
(6.33)
(6.34)
and
S FPb 6 = S PF inf = 0 .
(6.35)
78
The computation of the integrals in Eq. 6.31 and Eq. 6.32 follows the same lines
as in Section 6.2.
6.4 Horizontal Line Source
In this section, we present the computational aspects for a line source along the
horizontal axis, xD . The approach used here to obtain computationally more efficient
forms of the line source solution is the same as that used for a partially penetrating planar
source in Section 6.3.
For early times ( t AD < 10−2 ), the horizontal line-source function, S HW , given by
Eq. 4.16 can also be written as the sum of the fully penetrating planar source solution,
S FPFHW , and the horizontal well pseudoskin, S PSHW [Ozkan and Raghavan (1991a)]:
S HW = S FPFHW + S PSHW .
(6.36)
At early times, S FPFHW is the summation of the components
S HW = S HWb1 + S HWb 2 + S HWb3 + S HW inf ,
(6.37)
where
S HWb1 = CLH
S HWb 2 =
Fcsht [u ]
,
u
4Cxe
π
∞
∑
k =1
1
kπLH
k πx
kπxw Fcshp [ε k ]
sin(
) cos(
) cos(
)
,
2 xe
εk
k
xe
xe
(6.38)
(6.39)
79
+1
[
]
xeD
K 0 ( xD + xwD − xHD β ) 2 + ( yD − ywD ) 2 u dβ
2π −∫1
S HWb3 = CLH
[
+1
∞
]
xeD
K 0 ( xD m xwD m 2kxeD − xHD β ) 2 + ( yD − ywD ) 2 u dβ ,
∫
k =1 2π −1
+ CLH ∑
− CLH
e
(6.40)
− u y D − y wD
u
and
+1
S HW inf
[
]
x
= CLH eD ∫ K 0 ( xD − xwD − xHD β ) 2 + ( yD − ywD ) 2 u dβ .
2π −1
(6.41)
The term x HD in Eqs. 6.40 and 6.41 is defined by
xHD = ( LH / 2) k / k x / l .
(6.42)
We can write the pseudoskin term, S PSHW , in Eq. 6.36 as follows:
S PSFHW = S HWb 4 + S HWb5 + S HWb 6 + S PSHW inf ,
(6.43)
Proceeding in the same way we used for S FPFHW above, we obtain the following earlytime computational forms for S HWb 4 , S HWb5 , S HWb6 , and S PSHW inf :
∞
S HWb 4 = 2CLH ∑ cos(
n =1
S HWb5 = 8C
xe
π
∞
nπz D
nπzwD Fcshp [ε n ]
) cos(
)
,
zeD
zeD
εn
∑ cos(
n =1
nπzD
nπzwD
) cos(
)
zeD
zeD
1
kπLH
k πx
kπxw Fcshp [ε k , n ]
) cos(
) cos(
)
× ∑ sin(
2 xe
ε k ,n
xe
xe
k =1 k
∞
,
(6.44)
(6.45)
80
S HWb6 = CLH
x eD
π
∞
∑ cos(
n =1
[
nπz wD
nπz D
) cos(
)
z eD
z eD
]
⎧+1
⎫
2
2
⎪ ∫ K 0 ( x D + x wD − x HD β ) + ( y D − y wD ) ε n dβ
⎪,
⎪ −1
⎪
×⎨
⎬
+
1
∞
2
2
⎪+
K 0 ( x D m x wD m 2kx eD − x HD β ) + ( y D − y wD ) ε n dβ ⎪
∫
⎪ ∑
⎪
⎩ k =1 −1
⎭
[
(6.46)
]
and
S PSHW inf = CLH
x eD
π
∞
∑ cos(
n =1
nπz wD
nπ z D
) cos(
)
z eD
z eD
+1
× ∫ K 0 ⎡ [( x D − x wD − x HD β )] + ( y D − y wD ) 2 ε n ⎤ dβ
⎢⎣
⎥⎦
−1
.
(6.47)
2
The series in Eq. 6.47 converges slowly for small arguments of the modified Bessel’s
function, K 0 , when xD − xwD ≤ xHD [Raghavan and Ozkan (1994)]. For this case, the
series in Eq. 6.47 can be broken into a summation of two functions in the following form:
S PSHW inf = CLH
xeD
π
( F1 − F2 ) ,
(6.48)
nπz D
nπz wD − ε n ( yD − ywD )
) cos(
)e
zeD
zeD
(6.49)
where
F1 =
π
xHD
∞
1
∑ε
n =1
n
cos(
and
F2 =
1 ∞ 1
nπz D
nπz wD
cos(
) cos(
)
∑
xHD n =1 ε n
zeD
zeD
∞
⎧
⎫
2
K 0 ⎡ β 2 + [ε n ( yD − ywD )] ⎤ dβ ⎪ .
⎪
∫
⎢
⎥⎦
⎪ε n ( x HD + x D − x wD ) ⎣
⎪
×⎨
⎬
∞
2⎤
⎪+
2
⎡
K 0 β + [ε n ( yD − ywD )] dβ ⎪⎪
∫
⎪
⎢⎣
⎥⎦
⎩ ε n ( x HD − x D + x wD )
⎭
(6.50)
81
The function F1 can be computed more efficiently if alternate expressions are used for
large and small times [Raghavan and Ozkan (1994)]. Using the formula in Eq. 6.5, we
obtain the following alternate expression for small times:
F1 =
[
]
− u y − ywD
+∞
D
hD
e
× ∑ K 0 ( z D m z wD − 2nzeD ) 2 + ( y D − ywD ) 2 −
2 xHD n=−∞
2 u
.
(6.51)
For large times, we add and subtract the asymptotic form of the function F1 as u → 0
and write Eq. 6.45 as follows [Raghavan and Ozkan (1994)]:
F1 =
π
− ε ( y −y )
− nπ ( yD − ywD ) / hD
⎤
nπz D
nπz wD ⎡ e n D wD
e
−
cos(
)
cos(
)
⎢
⎥
∑
(nπ / hD ) ⎦⎥
εn
zeD
zeD ⎣⎢
n =1
∞
xHD
⎧ ⎡
⎛ π
⎞ −π ( y − y ) / h ⎤ ⎫
−π ( y − y ) / h
( z D + z wD ) ⎟⎟ + e D wD D ⎥ ⎪ .
⎪ln ⎢1 − 2e D wD D cos⎜⎜
⎝ zeD
⎠
h ⎪ ⎣
⎦ ⎪
− D⎨
⎬
4π ⎪
⎡
⎛ π
⎞ −π ( yD − ywD ) / hD ⎤ ⎪
−π ( yD − ywD ) / hD
cos⎜⎜
( z D − z wD ) ⎟⎟ + e
⎥⎪
⎪+ ln ⎢1 − 2e
z
⎝ eD
⎠
⎣
⎦⎭
⎩
(6.52)
Application of Eq. 6.50 and Eq. 6.51 or 6.52 improves significantly the computation of
the source term for a horizontal or vertical line segment at early time.
At late time, we can use the following expressions for more efficient numerical
evaluations:
S HWb 2 =
4Cxe
π
∞
∑
k =1
∞
1
kπLH
k πx
kπxw Fcsht [ε k ]
,
sin(
) cos(
) cos(
)
εk
k
2 xe
xe
xe
S HWb 4 = 2CLH ∑ cos(
n =1
S HWb5 = 8C
xe
∞
nπ z D
nπzwD Fcsht [ε n ]
) cos(
)
,
zeD
zeD
εn
∑ cos(
π n =1
kπLH
nπzD
nπzwD
) cos(
)
zeD
zeD
1
kπx
kπxw Fcsht [ε n ]
× ∑ sin(
) cos(
) cos(
)
εn
2 xe
xe
xe
k =1 k
∞
and
,,
(6.53)
(6.54)
(6.55)
82
S HWb3 = S HW inf = S HWb6 = S PSHW inf = 0 .
(6.56)
The computation of the integrals in Eqs. 6.40, 6.41, 6.46, 6.47 and 6.50 follows
the same lines as in Section 6.3.
6.5 Slanted or Deviated Line Sources
The expression for a slanted or deviated line source presented in Eq. 4.18 or 4.20
respectively, differs from the horizontal line source presented in Eq. 4.13 in that the
integration is performed over the direction L that makes an angle ϕ with the negative
direction of the vertical coordinate axis z for a slanted line source or an angle θ with the
positive direction of the horizontal coordinate axis x for a deviated line source. This
introduces additional complexity to the computation of the source function for slanted or
deviated line-sources.
Because the computational procedures follow the same lines for slanted and
deviated line sources, here we only present the details for slanted line sources (Eq. 4.18).
First we address the fact that the slant angle, ϕ̂ , the auxiliary angle, αˆ = ϕˆ − π / 2 , and
the length of the slanted well, LSD , are dependent on permeability anisotropy. They can
be computed as given in Ozkan et al. (1998), Ozkan and Raghavan (1998), and Yildiz
(2003):
LSD =
LS
l
k
k
cos 2 α + sin 2 α ,
kx
kz
⎛ kx
⎞
tan α ⎟⎟ ,
⎝ kz
⎠
αˆ = tan −1 ⎜⎜
(6.57)
(6.58)
83
and
⎛ kz
⎞
tan ϕ ⎟⎟ .
⎝ kx
⎠
ϕˆ = tan −1 ⎜⎜
(6.59)
As in Section 6.4, we first recast the slanted line-source solution (Eq. 4.18) in the
following form:
S SW = S FPFSW + S PSSW .
(6.60)
Since the well is slanted in the x − z plane, the S FPFSW term will have components for x
and z directions; thus,
S FPFSW = SSWb1 + S SWb 2 x + S SWb3 x + S SW inf x + S SWb 2 z .
(6.61)
The early-time computational forms of the terms in the right-hand side of Eq. 6.61 are
given by
S SWb1 = CLS
S SWb 2 x =
S SWb 3 x
Fcsht [u ]
,
u
(6.62)
4CLS xeD ∞
kπLSxD
kπxD
kπxwD Fcshp [ε k ]
,
sin(
) cos(
) cos(
)
∑
πLSD cos αˆ k =1
εk
xeD
xeD
xeD
2CLS
=
LSD cos αˆ
+1
xeD
∫ 2π
−1
K0
[
[ (x
D
(6.63)
]
+ x wD − LSxD β ) 2 + ( y D − y wD ) 2 u dβ
]
⎧ x eD ∞ +1
⎫
2
2
−
+
−
K
x
x
kx
L
y
y
u
d
(
m
m
2
β
)
(
)
β
⎪
⎪,
∑
0
D
wD
eD
SxD
D
wD
∫
2CLS ⎪ 2π k =1 −1
⎪
+
⎨ − u y −y
⎬
D
wD
LSD cos αˆ ⎪ e
⎪
⎪−
⎪
u
⎩
⎭
(6.64)
S SW inf x =
and
CLS xeD +1 ⎡
2
K o [( x D − x wD − LSxD β )] + ( y D − y wD ) 2 u ⎤ dβ , (6.65)
∫
⎢
⎣
⎦⎥
ˆ
πLSD cos α −1
84
S SWb 2 z =
4Cze
π sin αˆ
∞
1
∑ n sin(
n =1
nπLSzD
nπzD
nπzwD Fcshp [ε n ]
) cos(
) cos(
)
.
εn
zeD
zeD
zeD
(6.66)
The parameters LSxD and LSzD are defined by
LSxD = (LSD / 2 ) cos α̂ ,
(6.67)
LSzD = (LSD / 2 ) sin α̂ .
(6.68)
and
The pseudoskin term, S PSSW , in Eq. 6.60 is given by
∞
S PSSW = 4 FH ∑ cos(
n =1
nπz D ∞
kπx D Fcsht [ε k ,n ]
)∑ cos(
)
z eD k =1
x eD
ε k ,n
LS / 2
⎡
nπ ( z w + LS′ sin α ) D
kπ ( x w + LS′ cos α ) D ⎤
) cos(
)⎥ dLS′
× ∫ ⎢cos(
z eD
x eD
⎦
− LS / 2 ⎣
.
(6.69)
Making the integration variable dimensionless, Eq. 6.64 becomes
S PSSW =
4CLS ∞
nπz D ∞
kπxD Fcsht [ε k , n ]
)∑ cos(
)
× ∑ cos(
LSD n =1
zeD k =1
xeD
ε k ,n
'
⎡
nπ ( z wD + LD′ sin αˆ )
kπ ( xwD + LD′ cos αˆ ) ⎤
) cos(
)⎥ dLD′
× ∫ ⎢cos(
z
x
eD
eD
− ( LS / 2 ) D ⎣
⎦
( LS / 2 ) D
.
(6.70)
The integral in Eq. 6.70 has a closed analytical solution [Gradshtein and Ryzhik (1965)],
which is presented in Appendix B. However, the use of the analytical solution of the
integral makes the convergence of the series in Eq. 6.70 extremely difficult. To alleviate
this problem, we proceed as in Section 6.4 and write
S PSSW = S PSSWb 4 + S PSSWb5 + S PSSW inf
where
(6.71)
85
4CLS ∞
nπz D ∞
kπxD Fcshp [ε k , n ]
)∑ cos(
)
× ∑ cos(
LSD n =1
zeD k =1
xeD
ε k ,n
S PSSWb 4 =
( LS / 2 ) D
'
⎡
nπ ( z wD + LD′ sin αˆ ) D
kπ ( xwD + LD′ cos αˆ ) D ⎤
) cos(
)⎥ dLD′
× ∫ ⎢cos(
z
x
eD
eD
− ( LS / 2 ) D ⎣
⎦
×
∫
− ( LS / 2 ) D
(6.72)
nπz D
4CLS ∞
)
× ∑ cos(
LSD n =1
zeD
S PSSWb 6 =
( LS / 2 ) D
,
nπ ( z wD + LD′ sin αˆ )
⎡
)
⎢cos(
zeD
⎢
xeD ⎢ ⎛
K ( xD + ( xwD + LD′ cos αˆ )) 2 + ( y D − ywD ) 2 ε n
2π ⎢ ⎜ 0
⎢× ⎜ ∞
2
2
⎢ ⎜⎜ + ∑ K 0 ( xD m ( xwD + LD′ cos αˆ ) m 2kxeD ) + ( y D − ywD ) ε n
⎣ ⎝ k =1
[
]
[
⎤
⎥
,
⎥
⎞⎥ dLD′
⎟⎥
⎟⎥
⎟⎟⎥
⎠⎦
]
(6.73)
and
S PSSW inf =
×
( LS / 2 ) D
∫
− ( LS / 2 ) D
4CLS ∞
nπz D
× ∑ cos(
)
LSD n =1
zeD
nπ ( z wD + LD′ sin αˆ )
⎡
cos(
)
⎢
xeD
zeD
2π ⎢
⎢× K ( x − ( x + L′ cos αˆ )) 2 + ( y − y ) 2 ε
0
D
wD
D
D
wD
n
⎣
( [
⎤
.
⎥
⎥ dLD′
⎥
⎦
(6.74)
])
The integrals in Eqs. 6.72 through 6.74 require special attention to be computed with high
accuracy.
We first discuss the integral in Eq. 6.72. Upon changing the integration variable,
expanding the cosine terms, and applying the properties of even functions, we obtain
S PSSWb 4 =
4CLS
LSD sin αˆ
∞
1
∑ nπ cos(
n =1
nπzD
nπzwD
) cos(
)
zeD
zeD
k πx D
kπxwD Fcshp [ε k , n ]
) cos(
)
× ∑ cos(
× Ib4
ε k ,n
xeD
xeD
k =1
∞
In Eq. 6.75,
.
(6.75)
86
⎡ sin[γ (1 + τ )] sin[γ (1 − τ )] ⎤
+
I b4 = ⎢
(1 − τ ) ⎥⎦
⎣ (1 + τ )
nπzwD
kπxwD ⎡ sin[γ (1 − τ )] sin[γ (1 + τ )] ⎤
+ tan(
−
) tan(
)
zeD
xeD ⎢⎣ (1 − τ )
(1 + τ ) ⎥⎦
,
(6.76)
where
γ = nπLSzD / zeD ,
(6.77)
τ = kzeD / nxeD tan αˆ .
(6.78)
and
Equation 6.76 is valid for τ ≠ 1 . When τ = 1 , we use the following expressions:
I b4 =
nπzwD
kπxwD ⎤
sin(2γ ) ⎡
) tan(
)⎥ for τ = 1
⎢1 − tan(
zeD
xeD ⎦
2 ⎣
(6.79)
I b4 =
nπzwD
kπxwD ⎤
sin(2γ ) ⎡
) tan(
)⎥ for τ = −1 .
⎢1 + tan(
zeD
xeD ⎦
2 ⎣
(6.80)
and
The integrals in Eqs. 6.73 and 6.74 are evaluated numerically. We obtained the
best results by fitting Chebyshev polynomials [Abramowitz and Stegun(1972)] to the
integrands of the integrals and then using Chebyshev quadratures to integrate the
polynomial functions. Also, the convergence of the cosine series in Eqs.6.73 and 6.74
may be improved by changing the integration variable [Ozkan and Raghavan (1998)].
Hence, we recast Eqs. 6.73 and 6.74 as
87
S PSHWb 6 =
2CLS xeD zeD ∞ 1
nπz D
cos(
)
∑
2
π LSD sin αˆ n =1 n
zeD
[
]
⎧
⎡ K ( x + x + ω ξ )2 + ( y − y )2 ε
D
wD
z
D
wD
n
⎪
⎢ 0
nπz wD
× ∫ ⎨cos(
+ξ)× ⎢ ∞
2
2
zeD
−γ ⎪
⎢+ ∑ K 0 ( xD m ( xwD + ω zξ ) m 2kxeD ) + ( yD − ywD ) ε n
⎣ k =0
⎩
γ
[
⎤⎫ ,
⎥⎪
⎥ ⎬ dξ
⎥⎪
⎦⎭
]
(6.81)
and
S PSHW inf =
2CLS xeD zeD ∞ 1
nπz D
cos(
)
∑
2
π LSD sin αˆ n =1 n
zeD
[
γ
]
nπz wD
× ∫ cos(
+ ξ ) K 0 ( xD − xwD − ω zξ ) 2 + ( yD − ywD ) 2 ε n dξ
zeD
−γ
.
(6.82)
To compute pressure at slanted or deviated wellbores, Eq. 6.82 may be approximated by
S PSHW inf =
4CLS xeD zeD ∞ 1
nπz D
cos(
)
∑
2
π LSD sin αˆ n =1 n
zeD
[
γ
]
nπz wD
× ∫ cos(
) cos(ξ ) K 0 ( xD − xwD − ω zξ ) 2 + ( y D − ywD ) 2 ε n dξ
zeD
0
.
(6.83)
In Eqs. 6.81 through 6.83, ω z = zeD / nπ tan αˆ . Note that Eqs. 6.76 through 6.83 require
application of limits to compute the solution for horizontal ( αˆ = 0 ) or vertical ( αˆ = π / 2 )
wells.
Following the same procedure as in the previous section, we obtain the
computational forms of the right-hand side terms of Eq. 6.61 for late times:
and
S SWb 2 x =
4CxeD
π cos αˆ
S SWb 2 z =
4Cze
π sin αˆ
∞
1
∑ k sin(
k =1
∞
1
∑ n sin(
n =1
kπLSxD
k πx D
kπxwD Fcsht [ε k ]
) cos(
) cos(
)
xeD
xeD
xeD
εk ,
(6.84)
nπLSzD
nπ z D
nπzwD Fcsht [ε n ]
) cos(
) cos(
)
zeD
zeD
zeD
ε n ,.
(6.85)
88
S SWb 3 x = S SW inf x = 0 .
(6.86)
The late time terms in Eq.6.71 are computed as
S PSSWb 4 =
nπz D
nπz wD
4CLS ∞ 1
cos(
) cos(
)
∑
LSD sin αˆ n =1 nπ
zeD
zeD
kπxD
kπxwD Fcsht [ε k , n ]
× ∑ cos(
× Ib4
) cos(
)
ε k ,n
xeD
xeD
k =1
∞
,
(6.87)
and
S PSSWb6 = S PSSW inf = 0 .
(6.88)
Because the integrals in Eqs. 6.81 through 6.83 are computed numerically, we
have run a case to verify the accuracy of Eqs. 6.81 through 6.83. We selected a dual
lateral well configuration in an anisotropic reservoir as presented in Aguilar (2005). This
configuration is shown in Figure 6.1. The reservoir properties are shown in Table 6.1.
The parameters of the horizontal and deviated wells are presented in Tables 6.2. and 6.3,
respectively.
y
ye
Deviated
branch
LD=1350 ft
60°
LH=780 ft
Horizontal branch
xe
x
Figure 6.1 – Sketch for a dual-lateral well.
89
Table 6.1 – Parameters for the horizontal branch.
PARAMETERS
Horizontal branch
Length, LH, ft
Wellbore radius, rw , ft
Wellbore center in x, xw , ft
Wellbore center in y, yw , ft
Wellbore center in z, zw , ft
780
0.25
1120
1120
135
Table 6.2 – Parameters for the deviated branch.
Deviated branch
Length, LH, ft
Wellbore radius, rw , ft
Wellbore center in x, xw , ft
Wellbore center in y, yw , ft
Wellbore center in z, zw , ft
Azimuth angle, α, degree
1350
0.25
1017.5
1754.5
135
60
Table 6.3 – Reservoir parameters – dual lateral well case.
Reservoir
Formation thickness, h, ft
Reservoir size in x-direction, xe, ft
Reservoir size in y-direction, ye, ft
Production rate, q, rbbl/day
Viscosity, µ, cp
Porosity, φ
Total system compressibility, ct, psi-1
Reservoir permeability, kx, md
Reservoir permeability, ky, md
Reservoir permeability, kz, md
270
2080
3600
356
1.0
0.2
1.4 x 10-5
10
10
5
Results computed with Eqs.6.75 through 6.83 are presented in Figure 6.2, together
with the results of the finite-difference simulator of Aguilar (2005). In Figure 6.2, solid
markers indicate pressures and empty markers indicate pressure derivatives.
90
Circular markers show the results when the deviated branch is discretized in two
segments whereas square markers show results for a discretization with four segments for
the same, deviated branch. In both cases, the horizontal well branch is discretized in four
segments and the two branches are located in the same, single reservoir block. Figure 6.2
indicates good agreement between the results computed with the semi-analytical model
and the finite-difference model denoted by the triangular markers. However, the
derivative curves computed with the semi-analytical model display some oscillations.
Comparing the results for two and four segments, it appears that increasing the
discretization in the deviated branch would improve the stability of pressure derivative.
However, further increasing the number of segments in the deviated branch may actually
increase the oscillation of the pressure derivative values due to the numerical errors in the
computation of the integrals in Eqs. 6.75 through 6.83.
1.E+03
Aguilar (2005)
2 Deviated segments
∆p, d(∆p)/dlnt, psi
1.E+02
2 Deviated segments grid
4 Deviated segments
1.E+01
1.E+00
1.E-01
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
Time, hours
Figure 6.2 – Results for the dual-lateral well.
1.E+04
91
We have found that the derivative curve becomes more stable when the reservoir
is discretized into blocks and the horizontal and deviated branches are positioned in
different blocks. This result is presented by diamond markers in Figure 6.2. However, the
reservoir discretization introduces a distortion of the derivative responses in the transition
between the early-time radial flow and intermediate linear flow periods. This distortion
disappears as the well spacing increases.
6.6 Generic Line Source
The last case in this chapter is the generic configuration for a line source. The
source function for this configuration is given in Eq. 4.25. As sketched in Figure 4.7, all
three source-coordinates change as we move along the integration direction, L . In this
configuration, as in the case for a slanted line segment, the use of dimensionless space
variables changes the geometry of the line segment. We compute the dimensionless
length, LwD , the deformed inclination angle, ϕ̂ , and the deformed azimuth angle, θˆ , by:
LWD =
LW
l
k
k
k
sin 2 ϕ cos2 θ + sin 2 ϕ sin 2 θ + cos2 ϕ ,
kx
ky
kz
⎛ kz
k
⎜
sin 2 ϕ cos2 θ + z sin 2 ϕ sin 2 θ
ky
⎜ kx
ϕˆ = tan −1 ⎜
cos ϕ
⎜
⎜
⎝
⎞
⎟
⎟
⎟,
⎟
⎟
⎠
(6.89)
(6.90)
and
⎛ kx
⎞
tan θ ⎟ .
⎟
⎝ ky
⎠
θˆ = tan −1 ⎜⎜
(6.91)
92
For this case, the integrals in Eq. 4.25 are computed numerically following the
same procedure applied to the slanted line source solution in Section 6.5.
To improve the computation of these integrals at early times, we recast their
arguments in alternative forms that converge faster at early and late times. Then, we write
Eq. 4.25 as:
SW = C
LH / 2
∫G
W
dL' .
(6.92)
− LH / 2
Making the variable of integration dimensionless, Eq. 6.84 becomes
CL
SW = W
LWD
( LW / 2 ) D
∫G
W
− ( LW / 2 ) D
dξ .
(6.93)
Then we write
G W = G1 + G 2 + G 3 + G 4
(6.94)
where,
G1 =
Fcsht [u ]
.
u
(6.95)
The remaining terms, G 2 , G 3 , and G 4 are written as follows:
G2 = Gxb 2 + Gxb 3 + Gx inf ,
(6.96)
G3 = Gzb 2 ,
(6.97)
G4 = Gxzb 2 + Gxzb 3 + Gxz inf .
(6.98)
and
For early times, the terms in Eqs. 6.95 through 6.97 are computed with the
following expressions:
93
k πx D
kπx′D Fcshp [ε k ]
) cos(
)
,
xeD
xeD
εk
∞
Gxb 2 = 2∑ cos(
k =1
[
(6.99)
]
⎧ K ( x + x′ ) 2 + ( y − y ′ ) 2 u +
⎫ −
D
D
D
D
xeD ⎪ 0
⎪ e
Gxb 3 =
⎨∞
⎬−
π ⎪∑ K0 ( xD m xD′ m 2kxeD )2 + ( y D − yD′ )2 u ⎪
⎩ k =1
⎭
[
]
u y D − y ′D
,
u
(6.100)
Gx inf =
xeD
π
[
]
K 0 ( xD − x′D ) 2 + ( yD − y D′ )2 u ,
∞
Gzb 2 = 2∑ cos(
n =1
nπzD
nπz′D Fcshp [ε n ]
) cos(
)
,
zeD
zeD
εn
∞
Gxzb 2 = 4∑ cos(
n =1
xeD
Gxzb 3 = 2
π
[
(6.101)
∞
∞
nπzD
nπz′D
k πx D
kπx′D Fcshp [ε k , n ]
,
) cos(
) × ∑ cos(
) cos(
)
zeD
zeD
xeD
xeD
ε k ,n
k =1
∑ cos(
n =1
(6.102)
nπzD
nπzD′
) cos(
)
zeD
zeD
]
[
(6.103)
]
∞
⎡
⎤
× ⎢ K 0 ( xD + x′D ) 2 + ( y D − y′D ) 2 ε n + ∑ K 0 ( xD m x′D m 2kxeD ) 2 + ( y D − y′D ) 2 ε n ⎥
k =1
⎣
⎦
,
(6.104)
and
Gxz inf = 2
xeD
π
∞
∑ cos(
n =1
[
]
nπ z D
nπz′D
) cos(
) K 0 ( xD − x′D ) 2 + ( y D − y ′D ) 2 ε n ,
zeD
zeD
(6.105)
At late times, the counterparts of Eqs. 6.99 through 6.105 are given by
∞
Gxb 2 = 2∑ cos(
k =1
∞
Gzb 2 = 2∑ cos(
n =1
∞
k πx D
kπx′D Fcsht [ε k ]
) cos(
)
,
xeD
xeD
εk
(6.106)
nπ z D
nπz′D Fcsht [ε n ]
,
) cos(
)
εn
zeD
zeD
(6.107)
Gxzb 2 = 4∑ cos(
n =1
∞
nπzD
nπzD′
kπxD
kπx′D Fcsht [ε k , n ]
,
) cos(
) × ∑ cos(
) cos(
)
zeD
zeD
xeD
xeD
ε k ,n
k =1
(6.108)
94
and
Gxb 3 = Gx inf = Gxzb 3 = Gxz inf = 0 .
(6.109)
Note that in Eqs. 6.99 through 6.108 the source coordinates ( x′D , y ′D , z′D ) are
functions of the integration variable ξ and are given by
x′D = xwD + ξ sin ϕˆ cos θˆ .
(6.110)
y ′D = y wD + ξ sin ϕˆ sin θˆ .
(6.111)
z′D = zwD + ξ cos ϕˆ .
(6.112)
and
The numerical integration in Eq. 6.69 takes more time than its analytical
counterpart presented in Section 6.5. It indicates that if either the inclination angle, ϕ , or
the azimuth angle, θ , is small, computations can be improved by replacing the generic
well response with the response for the slanted or deviated well.
6.7 Convergence Criteria for Series
All source functions presented in this chapter contain infinite exponential or
trigonometric series. In practice, these series are truncated after computing a finite
number of terms. The number of terms used in the evaluation of the series governs the
accuracy of the computations and affects the computing time. In this work, the series are
truncated when an absolute error less than 1×10−7 is achieved. For the exponential series,
the absolute error corresponds to the last computed term in the series. For the
trigonometric series, the absolute error corresponds to the partial summation of the last
95
eight computed terms in the series. The above mentioned convergence criteria have been
successfully applied to compute accurate results for horizontal wells [Ozkan (1988)].
6.8 Coordinates to Compute Wellbore Pressure
Wellbore pressures are computed at points located at a distance equal to the radius
of the well measured from the center of the well segment in a direction orthogonal to the
well axis. In the case of an isotropic reservoir, pressures are approximately the same
around the wellbore circumference. However, permeability anisotropy may cause
significant difference in pressures computed along the circumference of the wellbore. In
these cases, coordinates to compute the wellbore pressure for a slanted well are defined
as follows
xD = xwD − rw k
y D = y wD − rw
k
z D = z wD ± rw k
sin α .
kx
.
ky
kz
cos α .
(6.113)
(6.114)
(6.115)
For a horizontal segment, wellbore pressures are computed at
z D = zwD + rw k
kz
.
(6.116)
Theses coordinates were applied to match the results from the numerical model
presented in Aguilar (2005). The location of the point to compute the correct wellbore
pressure in anisotropic porous media has been addressed in many publication in the
Petroleum Engineering literature [Cinco-Ley et al. (1975), Besson (1990), and Ozkan and
96
Raghavan (1998)]. This issue is outside the objectives of this study and was not
investigated.
97
CHAPTER 7
RESULTS AND VALIDATION
In this chapter, we present the results that have been generated using the semianalytical simulator developed in this work. These results have been validated by
comparison to established analytical or numerical models and methods that are available
in the Petroleum Engineering literature. Because, in practice, pressure or flow rate
responses are measured at wellbores, we present the results for wellbore performances.
However, by using Eq. 3.22, the semi-analytical simulator can be used to compute the
pressure at any point in the reservoir. We start our discussion with the results for a
horizontal well in a homogeneous reservoir. The discussion then continues on to show
results in more complex reservoir-well systems.
7.1 Model Validation Problem 1: Horizontal Well in a Homogeneous Reservoir
The first model validation problem is relatively simple, consisting of a horizontal
well located at the center of a closed homogenous and isotropic reservoir, with a
rectangular parallelepiped shape, as sketched in Figure 7.1. We show that the results
computed with this model match those obtained from the existing analytical solution for
the same problem [Ozkan and Raghavan (1991b)]. The properties of the well and the
reservoir are given in Table 7.1.
98
z
HW
y
40 ft
200 ft
k=100 mD
400 ft
x
400 ft
Figure 7.1 – Sketch for a homogeneous, isotropic, and rectangular parallelepiped
reservoir with a horizontal well; Model Validation Problem 1
Table 7.1 - Well and reservoir properties for Model Validation Problem 1.
Horizontal well length, Lh, ft
Wellbore radius, rw , ft
Reference length, ℓ, ft
Formation thickness, h, ft
Reservoir size in x-direction, xe, ft
Reservoir size in y-direction, ye, ft
Surface production rate, q, stb/d
Formation volume factor, B, bbl/stb
Viscosity, µ, cp
Porosity, φ
Total system compressibility, ct, psi-1
Reservoir permeability, k, md
200
0.01
100
40
400
400
200
1
0.6
0.17
8 x 10-6
100
We first generate the pressure-transient responses for the system shown in Figure
7.1 by using the analytical solution presented by Ozkan and Raghavan (1991b). The
infinite-conductivity wellbore assumption is incorporated into the analytical solution by
dividing the horizontal wellbore into segments and then imposing the equality of
pressures in all of these segments [Rosa and Carvalho (1989)]. We consider these results
to be reference responses, and will use them as the basis for comparison.
To verify the segmented-reservoir solution used in the semi-analytical simulator
proposed in this work, we divide the reservoir into three blocks in the x direction as
99
shown in Figure 7.2. The first block is 200 feet in length and the other blocks are 100 feet
each. The horizontal well penetrates the first two blocks.
z
Block 1
q11
y
Block 2
q21
100 ft
Block 3
40 ft
q12 q22 HW
100 ft
k=100 mD
400 ft
x
400 ft
Figure 7.2 – Reservoir subsections in the x direction; Model Validation Problem 1.
Tables 7.2 and 7.3 present the results of six cases to compare different
discretizations of the horizontal well and the interfaces between the blocks. Figure 7.3
shows the discretization schemes used on the block interfaces.
2 Interface Segments
(m = 2)
4 Interface Segments
(m = 4)
z
9 Interface Segments
(m = 9)
z
y
z
y
y
Figure 7.3 – Discretization schemes at the block interfaces for Verification Cases 1
through 6 in Tables 7.2 and 7.3; Model Validation Problem 1.
100
In Table 7.2, the results are for two horizontal-well segments (n = 2) in each
subsection and for two, four, and nine interface segments (m) corresponding to Cases 1,
2, and 3, respectively.
Table 7.2 – Results for two well segments in each subsection; Model Validation Problem
1.
Reference Case
tD
Case 1 (n=2,m=2)
Case 2 (n=2,m=4)
Case 3 (n=2,m=9)
∆p
(psi)
1 x 10-3
1 x 10-2
1 x 10-1
1 x 100
1 x 101
d∆p/dlntD
d∆p/dlntD
d∆p/dlntD
d∆p/dlntD
∆p
∆p
∆p
(psi)
(psi)
(psi)
(psi)
(psi)
(psi)
(psi)
5.2195
0.4233
5.2195
0.4236
5.2195
0.4236
5.2195
0.4236
6.1800
0.4188
6.1961
0.4360
6.1961
0.4360
6.1593
0.3774
7.6356
0.9718
7.7601
1.0828
7.7601
1.0828
7.5197
0.8943
10.9630 2.0711 11.5510 2.2349 11.5500
2.2348
10.6790
2.0468
25.7850 16.1970 26.7670 16.6440 26.7670 16.6440 25.8090 16.6411
In Table 7.3, we use four horizontal-well segments per subsection and consider
the effect of two, four, and nine segments at the block interfaces (for Cases 4, 5, and 6,
respectively). For comparison, the reference responses (the analytical solution) are also
reported in both tables.
101
Table 7.3 – Results for four well segments in each subsection; Model Validation Problem
1.
Reference Case
tD
1 x 10-3
1 x 10-2
1 x 10-1
1 x 100
1 x 101
Case 4 (n=4,m=2)
Case 5 (n=4,m=4)
Case 6 (n=4,m=9)
d∆p/dlntD
d∆p/dlntD
d∆p/dlntD
d∆p/dlntD
∆p
∆p
∆p
∆p
(psi)
(psi)
(psi)
(psi)
(psi)
(psi)
(psi)
(psi)
5.2195
0.4233
5.2195
0.4235
5.2195
0.4235
5.2196
0.4238
6.1800
0.4188
6.1888
0.4295
6.1889
0.4295
6.1587
0.3999
7.6356
0.9718
7.7381
1.0748
7.7381
1.0748
7.4321
0.8754
10.9630 2.0711 11.5050 2.2289 11.5050
2.2288
10.5420
2.0359
25.7850 16.1970 26.7140 16.6440 26.7096 16.6436 25.6860 16.6410
Tables 7.2 and 7.3 indicate that even with a small number of horizontal well and
interface segments, it is still possible to closely match the reference responses; the
maximum difference between the pressure responses for the reference and verification
cases is less than 6.7%. As expected from most numerical methods, the errors are smaller
in pressure responses than in the accompanying derivative responses.
As an example, Figure 7.4 shows the comparison of the results for Case 2 (two
horizontal well segments and four block interface segments) with the reference solution.
As noted above, the agreement is acceptable for all practical purposes. Since the
differences among the six cases considered in Tables 7.2 and 7.3 are too small to be
distinguished on a log-log plot, they are not presented graphically.
An important point to be noted regarding the results in Tables 7.2 and 7.3 is the
effect of the number of segments used in the discretization. In general, Tables 7.2 and 7.3
indicate that increasing the number of segments used in the discretization would improve
the accuracy of the results obtained from the semi-analytical simulation model. However,
as in any numerical approximation, numerical dispersion errors may adversely affect the
accuracy when too many segments are used.
102
1.E+03
Reference
Case 2 (n=2,m=4)
∆p, d∆p/dlnt, psi
1.E+02
1.E+01
1.E+00
1.E-01
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Dimensionless time, tD
Figure 7.4 – Verification plot for a horizontal well in a homogeneous reservoir (Case
two; two horizontal well segments per block and four block interface segments). Model
Validation Problem 1
7.2 Model Validation Problem 2: Vertical Well in a Heterogeneous Reservoir
In this example for the validation of the semi-analytical simulation model, we
consider a fully penetrating vertical well in a composite reservoir as shown in Figure 7.5.
We present the results for two cases: In the first case, the internal permeability ( kint ) is
larger than the external permeability ( k ext ) and in the second case, kint is smaller than
k ext . Since no analytical solution exists for these cases, we compare the results with a
finite element (FE) model [e.g. Smith and Griffiths (2004)] and with a commercial finite
difference (FD) model (Eclipse 100®).
103
15 ft
30 ft
φ= 0.17
ct=8 x10-6 1/psi
µ=0.8 cp
rw =0.25 ft
qB =2 rbbl/day
h=40 ft
kext
Vertical
Well
kint
10 ft
15 ft
30 ft
10 ft
20 ft
Figure 7.5 – Vertical well in a composite reservoir.
Vertical
well
Figure 7.6 – Permeability contrast and logarithmic grid for numerical models; Model
Validation Problem 2
104
The numerical results were computed with a logarithmic grid as shown in Figure
7.6. The grid system was generated to ensure accurate computation of early-time radial
flow responses. Figure 7.6 is color coded to display the permeability contrast between the
internal (red) and external (blue) zones. Figure 7.7 shows the grid blocks used in the
semi-analytical model presented in this research.
Vertical
well
Figure 7.7 – Permeability contrast and grid for the semi-analytical model; Model
Validation Problem 2.
Figure 7.8 presents the results for the first case where the internal permeability is
larger than the external permeability by a factor of ten; kint = 2md and k ext = 0.2md .
Solid markers show the results for pressure, while open markers show the results for
pressure derivative. The curves in Figure 7.8 indicate a good agreement between the
results of the numerical and semi-analytical models for times greater than 0.1 hours. For
times smaller than 0.01 hours, the pressure results from the numerical models show a
105
small discrepancy with the semi-analytical model. This difference is magnified in the
pressure derivative. This is indicative of the difficulty in obtaining accurate results during
early time radial flow with numerical models. Flow regime characteristics displayed by
the semi-analytical results, on the other hand, are consistent with our expectations and
this verifies the results of the semi-analytical model at early times. In general, the results
shown in Figure 7.8 demonstrate that the late-time responses, affected by reservoir
heterogeneity, are correctly handled by the semi-analytical model. Moreover, the semianalytical model provides more accurate results at early times than either the FD or FE
numerical models.
1.E+03
Finite element
∆p, d(∆p)/dlnt, psi
Finite diference
1.E+02
Semi-analytical
1.E+01
1.E+00
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
Time, hours
Figure 7.8 – Pressure and pressure derivative results for a composite reservoir with an
internal permeability greater than the external permeability; Model Validation Problem 2.
Next, we consider the same composite reservoir system as in Figure 7.5 but in this
case, the permeability of the external zone is larger than the permeability of the internal
106
zone by a factor of ten, kint = 0.2md and k ext = 2md . Results for this case are presented
in Figure 7.9 and were computed with the same grids as those in Figures 7.6 and 7.7.
As expected, the results in Figure 7.9 show that the early-time radial flow for this
case of less permeable inner zone lasts longer than the previous case of more permeable
inner zone shown in Figure 7.8. For this period, the early time pressure response from the
FE model shows the same behavior as that from the FD response, and diverges from the
semi-analytical model at times less than 0.01 hour. The pressure derivative from the FE
model indicates the presence of early-time radial flow, while the FD model fails to
capture the characteristics of early-time radial flow. All three models show good
agreement in pressure for times greater than 0.01 hours; however, the pressure derivative
from the FD model deviates from the other models in the interval between 0.1 and 1 hour.
After the dip in the pressure derivative, which characterizes the effect of the external
zone, all of the models display good agreement on pressure and the pressure derivative.
As in the first case discussed above, this second case also demonstrates that the semianalytical model provides good results during both late times, when the system response
is dominated by the external zone (heterogeneity), and early times, when the system
response is dominated by the inner zone.
The input data used to compute the results in Figures 7.8 and 7.9 are presented in
Tables 7.4 and 7.5. Table 7.4 shows the well and reservoir parameters, while Table 7.5
shows the simulation grid, the number of unknowns, and the time steps for each model.
107
1.E+03
Finite difference
Finite element
∆p, d(∆p)/dlnt, psi
Semi-analytical
1.E+02
1.E+01
1.E+00
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
Time, hours
Figure 7.9 – Pressure and pressure derivative results for a composite reservoir with an
external permeability greater than the internal permeability; Model Validation Problem 2.
Table 7.4 – Reservoir and well data the composite reservoir case; Model Validation
Problem 2.
Well length, L h , ft
Wellbore radius, r w , ft
40
0.25
Formation thickness, h , ft
Reservoir size in x -direction, x e , ft
40
60
Reservoir size in y -direction, y e , ft
60
Production rate, q , rbbl/day
Viscosity, µ , cp
Porosity, φ
Total system compressibility, c t , psi-1
Reservoir permeability, k , md
Reservoir depth, ft
Reservoir pressure, psi
2
0.8
0.17
8.0 x 10-6
0.2 ; 2
3000
4500
108
Table 7.5 – Grid coordinates, unknowns, and time steps for the composite reservoir case;
Model Validation Problem 2.
Finite element
Elements ∆x (ft)
1
14.00
2
6.00
3
3.00
4
1.75
5
1.25
6
0.95
7
0.80
8
0.67
9
0.58
10
0.52
11
0.46
12
0.50
∆y (ft)
14.00
6.00
3.00
1.75
1.25
0.95
0.80
0.67
0.58
0.52
0.46
0.50
Number of unknows
576
Finite difference
Cells
1
2
3
4
5
6
7
8
9
10
11
12
∆x (ft)
13.33
6.00
3.00
1.75
1.25
0.95
0.80
0.67
0.58
0.52
0.46
1.40
∆y (ft)
13.33
6.00
3.00
1.75
1.25
0.95
0.80
0.67
0.58
0.52
0.46
1.40
Number of unknows
529
Time step
Time step
0.001 hr for 0 < t < 1 hr 0.001 hr for 0 < t < 0.01 hr
0.01 hr for t > 1 hr
0.01 hr for 0 < t < 0.1 hr
0.1 hr for 0 < t < 1 hr
1 hr for t > 1 hr
Semi-analytical
Blocks ∆x (ft)
1 to 6
10.00
7 to 12 10.00
13 to 14 10.00
15
20.00
16 to 17 10.00
18 to 23 10.00
24 to 29 10.00
∆y (ft)
14.00
11.00
10.00
10.00
10.00
11.00
14.00
Number of plane sources
per block interface
1
Number of unknows
49
Time step
not required
Note that Table 7.5 shows only one quarter of the grid for the FD and the FE
models; the remaining portions of the grid are symmetric in relation to the central line
(line 12) and central column (column 12). Also notice that the central cell in the FD grid
is larger than the central element in the FE grid. This was imposed by the FD software,
which requires a minimum size for the block containing a wellbore having a radius of
0.25 feet.
109
Although this example aims to demonstrate that the semi-analytical model works
properly in heterogeneous systems, we would like to further examine some of the data in
Table 7.5. First, we note the number of unknowns needed to solve the problem for each
model. The fine, logarithmic grid used in the FD and FE models introduce more
unknowns than the Cartesian grid used in the semi-analytical model. More unknowns
increases the size of the matrix requiring more computational effort in the matrix solver4.
This provides the semi-analytical solution with an advantage in large reservoirs. The
numerical models can be run with coarse grid, thereby easing computational effort,
however, the result would be poor, especially at early times.
Second, we note that both numerical methods present solutions in the time
domain, requiring that the results be computed sequentially in time. To avoid oscillatory
behavior in early times, the numerical models require small time steps. On the other
hand, the semi-analytical model presents the solution in the Laplace domain and is able to
compute responses at any point in time without the need for the results in previous times.
The above results highlight the advantages of the semi-analytical model over the
numerical models under certain conditions. These advantages stem from the fact that the
semi-analytical model is based on a Green’s function solution for the Diffusivity equation
and also computed in the Laplace space. Both of these mathematical tools require
linearity of the problem. In addition, the analytical nature of the method is more
appropriate for single-phase flow applications. These limitations, however, may be
relaxed; the semi-analytical simulation approach, for example, can be used in connection
with streamline models to consider movement of multiple phases in displacement
4
A numerical model with a banded and symmetric coefficient matrix could reduce the computational
effort in the matrix solver.
110
processes. Also, the computation of the responses of finite-conductivity horizontal wells
discussed in Chapter 5 is an example of the extension of the semi-analytical simulation
approach to non-linear problems. In this application, the semi-analytical model can
reduce the number of unknowns by requiring discretization only at block boundaries; it,
however, requires the computation of the solution in a sequential manner that increases
the computational effort.
The CPU times required to compute results for the Model Validation Problem 2
are presented in Table 7.6. The figures in Table 7.6 refers to the specific data set
presented in Table 7.4 and are strongly dependent on the level of discretization required
to computer accurate results.
Table 7.6 – CPU times for Model Validation Problem 2.
Semi-analytical model
Finite elements
Finite differences
0.7 minutes
2.5 minutes
4 minutes
7.3 Model Validation Problem 3: Horizontal Well near a Sealing Barrier.
In this example, we explore the capability of the semi-analytical model to
compute the fluxes along the horizontal well length. Here, we consider a horizontal well
in a homogeneous reservoir with an internal sealing shale barrier. Figure 7.10 shows the
sketch of the reservoir and horizontal well. The sealing barrier extends horizontally
through half of the the reservoir in the direction of the well axis and is positioned at 2/3
of the reservoir height. The horizontal well is parallel to the sealing barrier with a total
111
length of 2000 feet. The first half of the well is located in a region of the reservoir not
affected by the sealing barrier, whereas the second half is in the region below the shale
barrier.
1000 ft
1000 ft
1000 ft
20 ft
4000 ft
2000 ft
40 ft
4000 ft
Horizontal well segments
q1
q2
q3
q4
500 ft
500 ft
500 ft
500 ft
Figure 7.10 – Horizontal well in a reservoir with a sealing barrier; Model Validation
Problem 3.
In this case, the horizontal well length is discretized into four segments of 500
feet each. Moreover, the symmetry in the direction perpendicular to wellbore axis (y
direction) reduces the discretization needed to accurately compute the solution. We
divide the reservoir into eight blocks, using plane sources of different sizes; non-uniform
plane sources are required to optimize computational time. Figure 7.11 presents the
discretization grid applied in this example.
Table 7.7 shows the well, reservoir, and grid parameters for the Model Validation
Problem 3.
112
Table 7.7 – Input data for a horizontal well with a sealing barrier; Model Validation
Problem 3.
Well length, L h , ft
Wellbore radius, r w , ft
Formation thickness, h , ft
Reservoir size in x -direction,
Reservoir size in y -direction,
Production rate, q , rbbl/day
Viscosity, µ , cp
Porosity, φ
Total system compressibility,
Reservoir permeability, k , md
Reservoir depth, ft
Reservoir pressure, psi
Block #
1,3,5,7
2,4,6,8
2000
0.01
60
4000
4000
200
0.6
0.17
-6
8.0 x 10
100
3000
4500
Block dimensions
xe (ft)
ye (ft)
1000
2000
1000
2000
ze (ft)
40
20
Planar sources at block interfaces
Number of
Interface
Side (ft) Height (ft)
sources
480, 480,
Blocks 1 to 2
4
2000
20, 20
520, 480,
Blocks 1 to 3
6
40
Blocks 3 to 5
480, 480,
Blocks 5 to 7
20, 20
520, 480,
Blocks 2 to 4
20
6
Blocks 4 to 6
480, 480,
Blocks 6 to 8
20, 20
520, 480,
Blocks 3 to 4
1000
6
480, 480,
20, 20
Blocks 5 to 6
0
no source
Blocks 7 to 8
113
20 ft
Blk 2
Blk 4
Blk 6
Blk 8
40 ft Blk 1
Blk 3
Blk 5
No flow Blk 7
Interfaces
1000 ft
1000 ft
1000 ft
2000 ft
1000 ft
Figure 7.11 – Discretization grid for a horizontal well in a reservoir with a sealing barrier;
Model Validation Problem 3.
The pressure transient response for this example is shown in Figure 7.12, together
with the homogeneous reservoir response, which is the reference for our analysis. As
expected, for early-time radial flow and for boundary-dominated flow, the pressure and
pressure derivative responses are not significantly influenced by the sealing barrier.
During intermediate times, however, the pressure derivative shows a slightly different
signature when compared to the reference homogeneous case.
A more significant effect of the sealing barrier can be seen in the flow profile
along the horizontal well length as presented in Figure 7.13. During early-time radial
flow, each segment has the same flow rate, which indicates that the sealing barrier is not
affecting the flow convergence. As time increases, however, the flow rates in the well
segments located under the sealing barrier begin to decrease. Eventually, during the
boundary-dominated flow regime, production from these segments becomes 65 % of the
production from segments not affected by the sealing barrier.
114
1.E+02
Reference - Homogeneous reservoir
Sealant barrier
∆p,d∆p/dlntD, psi
1.E+01
1.E+00
Pressure
1.E-01
Derivative
1.E-02
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Dimensionless time, tD
Figure 7.12 – Pressure transient response - horizontal well in a reservoir with a sealing
barrier; Model Validation Problem 3.
75
70
q1
q2
q3
q4
Flow rate, rbbl/d
65
60
55
50
45
40
35
30
25
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Dimensionless time, tD
Figure 7.13 – Flow distribution - horizontal well in a reservoir with a sealing barrier;
Model Validation Problem 3.
115
The computation of flow per well segment, combined with the analysis of
production logs, may be a valuable tool in diagnosing reservoir heterogeneity in the near
wellbore region. Moreover, plots similar to Figure 7.13 may be used in probabilistic
economic analyses in order to optimize horizontal well length.
7.4 Remarks on the Computation of Results
In closing this chapter, we make some remarks on the computation of the results
by using the semi-analytical simulator. First, we emphasize that the results from this
model are based on analytical and numerical components. The Green’s function solution
inside homogeneous subsections represents the analytical component. The continuity of
flux and pressure between the discretized contiguous surfaces at the block interfaces
represents the numerical component.
We note that the calculation of the analytical source functions takes most of the
computational time and we have noticed that some improvements are possible in the
computational speed of the source functions. For example, using a fully penetrating plane
source instead of partially penetrating surface source significantly improves the
computational speed in multi-block grids. This is due to the lack of a pseudoskin term in
the fully penetrating plane source solution. Also, positioning a well segment in the center
of a reservoir block will reduce the number of terms in the source equation, accelerating
the convergence of the series and increasing the speed of computations.
In multi-block problems, continuity conditions are imposed on uniform-flux plane
sources at the block boundaries. This type of source generates a linear (one-dimensional)
116
flow convergence in the vicinity of the source. This may require fine discretization in
order to accurately compute results for two or three dimensional flow regimes, a
difficulty that can be minimized by selecting a grid according to the anticipated flow
regime. As an example, Figure 7.14 shows two possible grids for a fully penetrating
vertical well positioned in a corner of a reservoir, having slightly different permeabilities.
On the left side of Figure 7.14, the reservoir is divided into two blocks along the
plane AA′ . In this case, all sources are parallel to the y axis, requiring fine source
discretization in order to accurately represent radial flow towards the vertical wellbore.
On the right-hand side of Figure 7.14, the plane sources are distributed in the x and y
directions along the BB′ and CC ′ planes. The combination of the resulting linear fluxes in
orthogonal directions leads to accurate results and these results can be obtained with a
coarser source discretization.
y
y
A'
B'
C
C'
Well
A
Well
x
B
x
Figure 7.14 – Options for radial flow gridding.
117
We acknowledge that grid configuration is fundamental to computing accurate
results in problems with complex geometry. Grid optimization would be an interesting
subject to be explored in future research.
118
CHAPTER 8
APPLICATIONS
The model presented in this research was developed with the aim that it be
applied to problems that cannot be handled by fully analytical solutions and, at the same
time, problems that require fine gridding or very small time steps in order to be computed
using numerical models. Therefore, we selected three types of applications where the
semi-analytical approach presented here has been successfully applied.
In the first application, semi-analytical simulation is applied to generate
diagnostic plots for pressure responses and to identify the effects of near wellbore
heterogeneity. In the second application, we use the semi-analytical simulation to match
production history and forecast production for a long, hydraulically fractured horizontal
well in a naturally fractured reservoir. In the final example, we estimate reservoir
parameters by matching a build-up test data with the results obtained from the semianalytical simulator. In all examples considered in this chapter, a naturally fractured
medium is represented with the dual-porosity idealization of Warren and Root (1963).
8.1 Diagnostic Plots for Near-Wellbore Heterogeneity
Here, we investigate a case where a hydraulic fracturing operation was performed
on a horizontal well in a homogeneous reservoir. The hydraulic fracturing process is
assumed to create a network of natural fractures in a limited region around the horizontal
119
well. This assumption is in agreement with field data from micro-seismic surveys. Then,
we discuss the pressure response when we have one longitudinal fracture or multiple
transverse hydraulic fractures. For this example, the homogeneous reservoir, hydraulic
fracture, and the naturally fractured reservoir properties are presented in Table 8.1, 8.2
and 8.3, respectively. Although we use specific data sets in our simulations, the flow
regimes presented, and the conclusions drawn here, are not limited to the particular cases
covered. They can be used as a qualitative reference for similar well-reservoir
configurations.
Table 8.1 - Reservoir properties for the diagnostic plots.
Horizontal well length, Lh, ft
Wellbore radius, rw , ft
Formation thickness, h, ft
Reservoir size in x-direction, xe, ft
Reservoir size in y-direction, ye, ft
Production rate, q, Mscf/day
Viscosity, µ, cp
Porosity, φ
Total system compressibility, ct, psi-1
Reservoir (matrix) permeability, k, md
800
0.3
100
2800
4400
200
0.2
0.10
1.6 x 10-5
0.01
120
Table 8.2 – Hydraulic fractures properties for the diagnostic plots.
Longitudinal Fracture
Half length, xf, ft
Width, wf, ft
Permeability, kf , md
Porosity, φf
Total compressibility, ctf, psi-1
Transverse Fractures
Half length, xf, ft
Width, wf, ft
Permeability, kf , md
Porosity, φf
Total compressibility, ctf, psi-1
400
0.1
1000
0.10
1.6 x 10-5
200
0.1
1000
0.10
1.6 x 10-5
Table 8.3 – Naturally fractured reservoir properties for the diagnostic plots.
Matrix Data
Permeability, km, md
0.01
0.10
Porosity, φm
Total compressibility, ctm
1.6 x 10-5
Block dimensions, Lx=Ly=Lz, ft
10
Fracture Data
Permeability, kf , md
1
0.001
Porosity, φf
-1
Total compressibility, ctf, psi
1.13 x 10-4
Dual-Porosity Parameters
0.12
Shape Factor, σ, ft-1
Storativity, w
0.066
1.08 x 10-4
Transmissivity Ratio, λ
Initially, we consider the case where the hydraulic fracture is positioned along the
horizontal well axis, as sketched in Figure 8.1. The diagnostic plot for this case is
presented in Figure 8.2. Solid markers indicate pressure and open markers correspond to
pressure derivative. The curves for a horizontal well in a reservoir with a localized
naturally fractured region (HW + NF region) and for the same well with a longitudinal
hydraulic fracture (HW + LF + NF region) are shown on the same graph. The pressure
121
derivative responses in Figure 8.2 indicate that there are no significant differences in flow
convergence (flow regimes) between the two cases. At very early times, however, the
longitudinally fractured horizontal well has linear flow towards the hydraulic fracture
(FLF), while the unfractured horizontal well displays early-time radial flow (ERF). After
0.1 hour, the derivative responses indicate that the same flow regimes develop for both
cases (intermediate-time linear flow, ILF, pseudoradial flow, PRF, and boundarydominated flow, BDF). Pressure responses also become virtually the same after 100
hours. This result indicates that the longitudinal hydraulic fracture helps the drainage of
the naturally fractured zone at early times. However, when pressure-transients reach
beyond the naturally fractured zone, flow convergence and well performance are
governed by the properties of the naturally fractured region and the external
homogeneous reservoir.
y
200 ft
4400 ft
Homogeneous reservoir
Hydraulic
fracture
Natural fractures
Natural fractures
800 ft
2800 ft
x
Figure 8.1 – Horizontal well with a longitudinal fracture in a reservoir with a localized
naturally fractured region.
It may be observed in Figure 8.2 that, at intermediate times between 10 and 1000
hours, the characteristics of both the pressure and the derivative responses resemble
boundary-dominated flow and the derivative responses approach a unit-slope behavior.
122
This indicates that flow convergence has reached the boundaries of the naturally fractured
region. This characteristic behavior can be used to compute an approximate value for the
volume of the naturally fractured region [Medeiros et al. (2007a)].
1.E+05
HW + NF region
1.E+04
HW + LF + NF region
∆p, d∆p/dlnt, psi
BDF
PRF
1.E+03
ILF
1.E+02
1.E+01
1.E+00
FLF
1.E-01
1.E-02
1.E+00
1.E+02
1.E+04
1.E+06
Time, hours
Figure 8.2 – Diagnostic plot for a longitudinally fractured horizontal well contacting a
region with natural fractures.
The approximate volume of the region containing natural fractures can be
obtained by applying the boundary dominated flow analysis when the derivative
responses display an approximately unit-slope behavior at intermediate times in the
diagnostic log-log plot (approximately between 10 and 100 hours for the plot in Figure
8.2). This analysis requires making a Cartesian plot of the pressure versus time data
showing the same time interval when the derivative responses display an approximate
unit slope in the log-log plot. Then, the Cartesian plot should display a straight line with
slope [Medeiros et al. (2007c)]:
123
m=
d∆p wf
dt
=
0.234q
.
Ahφct
(8.1)
In Eq. 8.1, the product Ahφ is the approximated porous volume for the naturally
fractured region.
We now consider the case where the horizontal well has multiple transverse
fractures as sketched in Figures 8.3 and 8.4. The diagnostic plot in Fig 8.5 shows the
curves for a horizontal well with two transverse fractures with large spacing (HW +
2TF(L) + NF region), two transverse fractures with small spacing (HW + 2TF(S) + NF
region), and four transverse fractures (HW + 4TF + NF region). The configuration for the
four transverse fractures case may be visualized by merging Figures 8.3 and 8.4. In all
three cases, the flow regimes appear, chronologically, as early-time radial flow (ERF)
followed by the characteristic dip of naturally fractured formations, fracture linear flow
(FLF), pseudoradial flow (PRF), and boundary-dominated flow (BDF).
y
4400 ft
200 ft
Regions with
natural fractures
200 ft
400 ft
HW
200 ft
200 ft
Transverse hydraulic fractures
2800 ft
x
Figure 8.3 – Horizontal well with two large-spaced (L) transverse fractures in a reservoir
with a localized naturally fractured region.
124
4400 ft
y
Region with
natural
fractures
400 ft
200 ft
200 ft
200 ft
HW
Transverse hydraulic fractures
2800 ft
x
Figure 8.4 – Horizontal well with two short-spaced (S) transverse fractures in a reservoir
with a localized naturally fractured region.
In Figure 8.5, there is no noticeable difference in the pressure responses of the
two-transverse-fracture cases with small (S) and large (L) spacing. However, increasing
the number of fractures to four reduces the pressure drop prior to the start of pseudoradial
flow. When pseudoradial flow develops, the difference between the cases with two and
four fractures becomes constant and negligible. This indicates that the drainage of the
tight, homogeneous, exterior reservoir is controlled by the horizontal well and the
naturally fractured regions.
Comparing the results for a longitudinal fracture in Figure 8.2 with those for
transverse fractures in Figure 8.5, it can be seen that pressure derivatives indicate the
same flow regimes for both types of hydraulic fractures.
The pressure transient responses shown in Figures 8.2 and 8.5 indicate that longterm reservoir drainage is controlled by the natural fractures rather than by the hydraulic
fractures. Hydraulic fracturing operations can still be applied, however, as a means to
activate or contact a natural fracture network in the near-wellbore region. A detailed
discussion on diagnostic plots for a hydraulically fractured horizontal well with a near
wellbore, dual-porosity region is presented in Medeiros et al. (2007c).
125
1.E+06
1.E+05
HW + 2TF(S) + NF region
HW + 2TF(L) + NF region
∆p, d(∆p)/dlnt, psi
1.E+04
HW + 4TF + NF region
1.E+03
1.E+02
1.E+01
1.E+00
1.E-01
1.E-02
1.E+00
1.E+02
1.E+04
1.E+06
Time, hours
Figure 8.5 – Diagnostic plot for a horizontal well with multiple transverse hydraulic
fractures in a region with natural fractures.
8.2 Production Data Analysis − Field Example
Production data analysis is a low cost method to estimate reservoir parameters and
forecast well production. It is applied when pressure transient data from well test
operations are not available. The main idea of this analysis consists of matching
production data with a theoretical model; when a satisfactory match is achieved, the
reservoir parameters are obtained from the model and production forecast is possible
based on the theoretical model. The production decline analysis in the example
considered here is based on the transient productivity index concept [Araya and Ozkan
(2002)]. The transient productivity index analysis uses the material balance time,
proposed by Palacio and Blasingame (1993) which was also used by Agarwal et al.
(1999) and Marhaendrajna and Blasingame (2001). The use of material balance time
126
allows for direct comparison between the field transient productivity index, computed for
variable production conditions (pressure, flow rate), and the model transient productivity
index, computed with a constant flow rate.
Following Araya and Ozkan (2002), we define the transient productivity index for
liquid flow as follows:
J (t e ) =
q( t )
,
pav (t ) − p wf (t )
(8.2)
where pav (t ) is the average reservoir pressure, p wf (t ) is the flowing wellbore pressure,
and t e is the material balance time given by [(Raghavan (1993) and Palacio and
Blasingame (1993)]
t
1
Q (t )
,
te =
q(τ )dτ =
∫
q(t ) 0
q( t )
(8.3)
For the constant-rate production mode, the material balance time, t e , is the same
as the actual time, t . For other modes of production, the use of the material balance time
makes the transient productivity index a weak function of the mode of production.
Therefore, transient productivity indices for any mode of production can be correlated by
that of the constant-rate production. This concept has been introduced by Agarwal et al.
(1999) to compute the production decline in hydraulically fractured vertical wells and
used by Araya and Ozkan (2002) to compute the transient productivity index for
horizontal wells.
Equation 8.2 can be also written in the following form
J (t e ) =
where
q( t )
,
∆p wf (t ) − ∆p av (t )
(8.4)
127
∆p wf (t ) = pi − p wf (t ) ,
(8.5)
and
∆p (t ) = pi − pav (t ) =
0.234 BoQ (t )
,
Ahφct
(8.6)
The second equality in Eq. 8.6 follows from the material balance for a fluid of constant
compressibility in a volumetric reservoir. Eqs. 8.4 through 8.6 are useful in the
computation of model (theoretical) and field transient productivity indices.
The transient productivity index concept can be extended to gas reservoirs using
pseudopressure and material balance pseudotime. The definition of the transient
productivity index for gas reservoirs is given by [Araya and Ozkan (2002)]
J (ta ) =
q(t )
,
m( pav ) − m ( p wf )
(8.7)
where the pseudopressure, m( p ) , is defined in Eq. 5.1, and the material balance
pseudotime, t a , is defined by [Raghavan (1993), Palacio and Blasingame (1993), and
Agarwal et al. (1999)]
ta =
(µct )i t
qsc (τ )
dτ .
∫
qsc (t ) 0 µ ( pav )ct ( p av )
(8.8)
Similar to liquid production, for gas production, the use of the transient productivity
index defined by Eq. 8.7, with the material balance pseudotime given in Eq. 8.8,
correlates the results with those of the constant-rate production.
Eq. 8.7 can be rearranged as follows:
J (t a ) =
In Eq. 8.9,
q(t )
.
∆m( p wf ) − ∆m( p av )
(8.9)
128
∆m( p wf ) = m( pi ) − m[p wf (t )]
(8.10)
and
∆m( pav ) = m( pi ) − m[ pav (t )] =
2.356Q (t )T
.
Ahφ ( µc g ) av
(8.11)
For the mass balance expressed by the second equality of Eq. 8.11 to hold [Raghavan
(1993)],
( µc g ) av =
t
1
µ (τ )c g (τ )dτ .
t ∫0
(8.12)
Equations 8.10 through 8.12 should be applied to compute the gas transient productivity
indices for the model and for the field data. The detailed procedure to compute the
transient productivity index from field data and with a theoretical model is explained in
Medeiros et al. (2007b).
In our example, the subject of the production data analysis is a hydraulically
fractured horizontal well in a reservoir of low permeability. The reservoir is expected to
be entirely naturally fractured as in Figure 8.6. Figure 8.7 shows the field data for the
well considered here, including the bottom-hole pressures (computed from wellhead
pressure measurements) and the surface flow rates as a function of time. A fast decline in
the bottom-hole pressures was required to maintain surface flow rates in the range of 300
to 400 stb/d.
129
y
4000 ft
Hydraulic Fracture
9000 ft
x
10200 ft
Figure 8.6 – Horizontal well with a longitudinal hydraulic fracture in a naturally fractured
reservoir.
1.E+04
8000
7000
1.E+02
5000
BHP, psi
qo, STB/d
6000
4000
Rate
3000
BHP
1.E+00
1.E+01
2000
1.E+02
1.E+03
1.E+04
Time, hours
Figure 8.7 – Flow rate and bottom-hole pressure for a horizontal well with a longitudinal
hydraulic fracture in a naturally fractured reservoir.
The transient productivity index computed from the field data is presented by the
open circular markers in Figure 8.8. As presented in Medeiros et al. (2007b), the flat
region shown in the transient productivity index computed from field data is an indication
of an active naturally fractured region in a tight reservoir. Also in Figure 8.8, open
130
triangle and solid diamond markers present the results that were generated with the semianalytical model to match the field data. In both cases, we have adjusted the input
parameters in the semi-analytical model to obtain a match in the early part of the field
data. The results denoted by the triangular markers are for the case where the entire
reservoir has dual-porosity properties, as sketched in Figure 8.6. However, as shown in
Figure 8.8, this reservoir configuration does not match the late time part of the field data
curve. The results presented with diamond markers are for the case where the reservoir
has natural fractures only in a 360-foot region on each side of the horizontal well as
shown in Fig 8.9. In this case, there is a fair agreement at late times between the field
data and the data generated by the semi-analytical simulator.
J, (STB/d)/psi
1.E+00
1.E-01
Field data
Naturally fractured reservoir
Naturally fractured region
1.E-02
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
Mass balance time (te), hours
Figure 8.8 – Productivity index match for production data analysis – large reservoir.
131
y
4000 ft
Dual porosity region
720ft
9000 ft
10200 ft
x
Figure 8.9 – Matching configuration for production data analysis – large reservoir.
The reservoir parameters used to get the match between the generated results
(diamond markers) and field data (open circular markers) in Figure 8.8 are tabulated in
Table 8.4. The permeability of the natural fracture, presented in Table 8.4, is the effective
fracture permeability, which represents the value of the actual natural fracture
permeability scaled to the total flow area of the dual-porosity medium. Hence, the actual
permeability of the thin naturally fractured flow channels is given by [Kazemi (1969)]
k f = keff V f .
(8.13)
where V f is the fractional volume of the fractures with respect to the bulk volume.
Also in Table 8.4, the matrix permeability is three orders of magnitude smaller
than the effective permeability of the natural fractures. This may indicate that the outside
region, shown in Figure 8.8, does not significantly contribute to the production of the
well because it has very low matrix permeability. To verify this hypothesis, we tried to
match the field data by using a smaller dual-porosity reservoir with the dimensions of the
limited dual-porosity region in Fig 8.9. The best match for this case is shown in Figure
132
8.10, where the transient productivity indices computed from the field data (circles) and
generated by the theoretical model (diamonds) are presented.
Table 8.4 – Reservoir parameters from production data match
Fracture permeability, keff, md
Matrix permeability, km, md
Storativity, w
Transmissivity Ratio, λ
2.2 x 10-2
2.2 x 10-5
0.015
1.08 x 10-4
J, (STB/d)/psi
1.E+00
1.E-01
Field data
Naturally fractured reservoir
1.E-02
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
Mass balance time (te), hours
Figure 8.10 – Productivity index match for production data analysis – small reservoir.
The model productivity index in Figure 8.10 displays a flat behavior at
intermediate and late times, which characterizes boundary dominated flow [Araya and
Ozkan (2002), Medeiros et al. (2007a)]. On the other hand, the field data do not show any
trend to be flat. In fact, it displays a downward bend indicating a larger drainage volume
133
than that used to compute the field transient productivity index. We therefore conclude
that the outside region in Figure 8.9 is contributing to the well flow, and that the model
with a large reservoir volume and localized dual-porosity region is a better representation
for the actual reservoir.
This application demonstrates that the semi-analytical model proposed in this
work can be used in combination with the transient productivity index and material
balance concepts to perform the analysis of production data in heterogeneous reservoirs.
8.3 Pressure Transient Analysis − Field Example
This last example application focuses on the use of the semi-analytical simulation
to obtain reservoir parameters and estimate well performance from pressure-transient
analysis. In this example, a buildup test was performed in the horizontal well sketched in
Figure 8.6. The well was open to flow for approximately 120 hours, and then shut-in for a
72-hour pressure build-up period, as shown in Figure 8.11. The oil flow rate prior to shutin is assumed to be constant and equal to 400 reservoir barrels per day (rbbl/d).
Moreover, fluid analysis indicates that the bubble point pressure is below the minimum
pressure measured during the test, meaning that the fluid in the reservoir was moving as
single-phase.
After screening the pressure data set, the starting point for the buildup analysis
was selected as t = 120.48 hours and p = 2951 psi. Figure 8.12 shows the log-log
diagnostic plot, where red, solid circular markers represent pressure and gray, open
134
circular markers indicate the pressure derivative. Note that the pressure derivative curve
was computed numerically with no smoothing factor.
Bottom hole pressure , psi
4500
4000
3500
3000
2500
0
50
100
150
200
Time, hours
Figure 8.11 – Pressure buildup data.
Figure 8.12 also includes the interpretation of the flow regimes based on pressure
derivative behavior. When ∆t < 0.1 hour, the pressure response is dominated by the
wellbore storage effect and the flow regime in the reservoir cannot be inferred from
Figure 8.12. Following the wellbore storage period, the pressure derivative shows a
transitional flow regime, which may represent the combined effects of linear flow in both
the reservoir and the hydraulic fracture(s). This transitional flow lasts until approximately
∆t = 1 hour. In the sequence, the ½-slope in the pressure and the pressure derivative
indicates linear flow in the reservoir. After ∆t = 32 hours, a new transitional flow period
takes place, where the pressure derivative shows an increasing trend. We interpret this
behavior by assuming that the reservoir is a composite medium, similar to the one
135
sketched in Figure 8.9, and that the pressure response after ∆t = 32 hours indicates the
effect of the changes in reservoir properties.
1.E+04
Pressure
Pressure derivative
∆p , psi
1.E+03
1.E+02
Trans.
flow #2
1.E+01
Linear flow
Transitional flow
Wellbore storage
1.E+00
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
∆t, hours
Figure 8.12 – Diagnostic plot for pressure build-up.
Using a configuration similar to Figure 8.9, the input parameters for the semianalytical model were adjusted to match the build-up data in Figure 8.12. The match with
the results from the semi-analytical model is shown in Figure 8.13. Figure 8.13 also
shows the results generated with commercial software for pressure transient analysis.
From this figure, it may be observed that both models show good agreement with field
data for most of the time. However, the commercial software shows a decrease in the
slope of the pressure derivative curve, while field data show an increase. For the same
period, the semi-analytical model shows a derivative with a constant slope, which is
closer to the behavior of the field data. Extrapolating both of the model results to the next
log cycle yields two distinct trends. The commercial model shows the effect of a dualporosity system, while the semi-analytical model reflects the effect of the change in
136
reservoir properties. Note that in Figure 8.13, the field pressure derivative is shown with
a smoothing factor [Horne (2005)].
1.E+04
Field data
Commercial software - Saphir®
∆p , psi
1.E+03
Semi-analytical model
1.E+02
1.E+01
1.E+00
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
∆t, hours
Figure 8.13 – Match with buildup data.
The reservoir configurations obtained from the match with the semi-analytical
model and with the commercial software are presented in Figure 8.14 and match
parameters are tabulated in Table 8.5. The two possibilities to interpret the pressure build
up data in this example reflect the non-uniqueness of the inverse solution procedure
involved in well test interpretation. In this case, performing a longer well test and
gathering additional information from the geologic model or from production history may
provide insights into the selection of the best model to represent the actual reservoir.
137
Lh
xe
Commercial
software match
ydp
ye
Lh
xe
Semi-analytical
model match
Figure 8.14 – Reservoir configurations from buildup data match.
As a final note in this chapter, we computed the results for the pressure transient
analysis reported above by using a grid with 344 unknowns—the largest linear system
used during this research. An LU-decomposition solver was used to compute results with
satisfactory performance.
138
Table 8.5 – Match parameters for pressure transient analysis application
PARAMETERS
Commercial
Model
Horizontal well length, Lh, ft
9000
Wellbore radius, rw , ft
0.25
Wellbore center in x, xw , ft
4500
Wellbore center in y, yw , ft
2000
Wellbore center in z, zw , ft
15
Wellbore storage, bbl/psi
0.0127
Formation thickness, h, ft
30
Reservoir size in x-direction, xe, ft
10200
Reservoir size in y-direction, ye, ft
4000
Production rate, q, Mscf/day
400
0.2
Viscosity, µ, cp
0.08
Porosity, φ
-1
Total system compressibility, ct, psi
Reservoir (matrix) permeability, k, md
Hydraulic Fracture
Half length, xf, ft
Width, wf, ft
Permeability, kf , md
Porosity, φf
Total compressibility, ctf, psi-1
Dual porosity
Fractures permeability, kf , md
2.4 x 10-2
Storativity, w
0.08
2.94 x 10-7
Transmissivity ratio, λ
Length in y direction, ydp, ft
SemiAnalytical
Model
9000
0.25
4500
2000
15
0.0216
30
10200
4000
400
0.2
0.08
5.33 x 10-6
1.0 x 10-6
4500
0.1
100
0.02
5.33 x 10-6
2.2 x 10-2
0.08
3.0 x 10-7
900
139
CHAPTER 9
CONCLUSIONS AND RECOMMENDATIONS
This chapter summarizes the characteristics, advantages, and weaknesses of the
semi-analytical model developed in this research. It also discusses potential areas for
further studies - not only to improve the model’s performance - but also to investigate
applications not covered in this work.
9.1 Conclusions
First, we summarize the main conclusions of this PhD research. We provide our
conclusions in two categories. The first category includes the conclusions about the semianalytical simulation approach presented in this dissertation. The second category is
concerned with the validation and application examples used in this work and the
information gained through the analyses of these examples.
The conclusions regarding the semi-analytical simulation approach are listed
below:
1.
The semi-analytical model developed in this research uses an approach
similar to the boundary element method in order to compute the
pressure transient response in heterogeneous porous media. This
approach requires only the discretization of the domain’s boundaries,
which is a competitive advantage over numerical methods that require
140
discretization across the entire domain. This method also computes all
of the fluxes over the boundaries of a subsection’s domain, which
allows for the calculation of pressure at any point inside the entire
domain by using analytical solutions. Moreover, the model can handle
problems with either specified flux or specified pressure conditions.
2.
The semi-analytical simulation approach presented in this work
represents a heterogeneous reservoir in terms of rectangular
substructures with different properties. This representation eliminates
the problem of finding the Greens’ function for a heterogeneous
domain, which has limited the use of previous semi-analytical models
based on the boundary-element method. The Green’s function used in
this model, however, is for a rectangular parallelepiped and its
complexity is responsible for most of the computational time.
3.
Because of its analytical basis, the accuracy of the semi-analytical
simulation approaches that of analytical solutions. Also, because the
formulation is in the Laplace domain, solutions can be obtained at
individual time points without requiring the results at the prior time
points. The solutions formulated in the Laplace domain can easily
incorporate wellbore storage and other variable rate production
applications. It is also possible to incorporate naturally fractured
reservoir solutions through dual-porosity idealizations. Both of these
features, however, limit the application of the semi-analytical
simulation presented in this dissertation to linear, single-phase flow
141
problems where the diffusion equation and the boundary conditions can
be linearized and an analytical solution can be obtained. Extensions of
the method to multi-phase flow problems through streamline
formulations are possible. Similarly, some forms of nonlinearity, such
as that included in the problem of frictional pressure drop in horizontal
wells, can be handled by utilizing a sequential application of the method
in time as discussed in Chapter 5.
4.
A standard LU-decomposition matrix solver was used to provide results
presented in this study with up to 350 unknowns without any matrix
inversion problems. Computation of the source functions in the Laplace
domain, however, is responsible for most of the computational time and
difficulty. To improve the accuracy and speed of computations, the
computational considerations presented in Chapter 6 are essential. For
example, the source function for a slanted (or deviated) well in a closed
reservoir cannot be accurately computed using a closed analytical
integration form. It requires a special numerical integration algorithm to
be accurately computed.
The validation and application examples lead to the following conclusions:
1.
Validation and application examples used in this dissertation
demonstrate that the semi-analytical approach can yield the same results
as the fully analytical solution in a homogeneous medium, while it can
also generate responses consistent with results from numerical models
142
in heterogeneous reservoirs. The semi-analytical simulation model has
been successfully applied to compute the pressure response for a
vertical well in a composite reservoir, for a horizontal well in a
reservoir with a sealing fault, and for a hydraulically fractured
horizontal well contacting a region with natural fractures.
2.
The semi-analytical model developed in this research is especially
suitable for investigating the effect of near-wellbore heterogeneity on
pressure-transient responses and well performance.
3.
Numerical gridding (discretization) may significantly impact the results
computed with the semi-analytical model. Improvements in accuracy
and computational time can be achieved if gridding is adjusted in
accordance with the expected flow convergence. Gridding issues,
however, are less severe for the semi-analytical simulation approach
than they are for the finite-element and finite-difference models, which
require fine-scale logarithmic gridding and small time steps to
accurately compute the early-time responses especially.
4.
The pressure-transient response for a horizontal well that partially
penetrates a reservoir section under a sealing barrier may not show a
significant difference from that obtained for the homogeneous case.
However, the long-term, stabilized fluxes along the horizontal-well
sections that are under the sealing barrier are significantly reduced.
5.
Diagnostic plots for wells contacting a limited region of natural
fractures in a tight gas reservoir show a behavior similar to boundary
143
dominated flow at intermediate times. The slope of the pressure
response during these time periods may be used to obtain an
approximate value for the volume of the naturally fractured region.
6.
Hydraulic fractures in wells contacting a naturally fractured region in a
tight reservoir do not improve the drainage of the tight matrix. The
drainage of the tight matrix is controlled by the contrast between the
properties of the naturally fractured zone and the matrix—not the
properties of the hydraulic fractures. However, hydraulic fractures do
improve the drainage of the naturally fractured region.
7.
Horizontal wells in a region with local, natural fractures, display similar
flow regimes at intermediate and late times, regardless of whether they
have longitudinal or multiple, transverse hydraulic fractures. At early
times, however, a longitudinal fracture may display bilinear flow, while
transverse fractures may display radial-linear or bilinear flow behavior.
8.
The transient productivity index and material balance time concepts can
be used to analyze performances of horizontal wells in heterogeneous
reservoirs. These concepts can be applied in production data analysis to
estimate reservoir parameters, reservoir drainage volume, and to
forecast production.
9.
The productivity index for a horizontal well contacting a naturally
fractured region in a tight reservoir displays a flat behavior during early
times. This can be applied in production data analysis to identify the
presence of a network of active natural fractures.
144
10.
The semi-analytical simulation is a useful tool to analyze pressure
build-up responses in a reservoir with near-wellbore heterogeneities.
9.2 Recommendations
The following discussion describes potential areas to improve the performance,
compare the results, and extend the use of the semi-analytical model presented in this
research:
The semi-analytical simulation approach presented in this work differs from
Galerkin’s boundary element method in that the approach used here is based on the
Green’s function for a rectangular parallelepiped. The standard boundary element method
that uses a free-space Green’s function leads to simpler source solutions but they require
additional discretization for no-flow boundaries. Qualitatively, better performance should
be achieved by combining the two approaches; the conventional boundary element
method can be applied for internal blocks with flowing boundaries, while the solution
proposed in this model is applied in peripheral blocks with no-flow boundaries. The
combined approach should also have the flexibility to compute pressure response for a
reservoir with an arbitrary shape.
The model presented in this work can be readily extended to multiple laterals or
multiple independent wells in the reservoir. Another natural extension of the approach
presented here may be the use of semi-analytical simulation to improve the accuracy of
finite-difference simulators in the near vicinity of wells. In these hybrid models, finitedifference simulation can be used to obtain the boundary conditions for a region
145
surrounding the well and the semi-analytical approach can be used to compute flow
convergence toward the wellbore. This approach eliminates the need for approximate
well-index calculations in finite-difference simulators. In addition, the need for fine
gridding around wells and fractures is eliminated because of the semi-analytical
approach. Overall, this hybrid approach should yield more accurate results especially at
early times and in the near wellbore vicinity.
In an effort to improve the performance of the semi-analytical model presented
here, a study to investigate grid optimization based on flow convergence dictated by
source geometry and reservoir heterogeneity is necessary.
The model presented here is limited to linear problems, thus it is not capable of
computing solutions for multiphase flow. Extension of the model to two-phase flow of oil
and water by using a streamline-simulation approach may be the first step towards
multiphase flow computation.
146
NOMENCLATURE
A
Area [ft2]
B
Surface boundary [ft2]
Bo
Formation volume factor [rbbl/STB]
C
Proportional constant for source functions
C gas
Gas storage coefficient [Mscf/psi]
Cliq
Liquid storage coefficient [rbbl/psi]
c
Fluid compressibility [psi-1]
cf
Formation compressibility [psi-1]
ct
Total compressibility [psi-1]
div
Divergence operator
D
Domain
f
Friction factor
F
Laplace transform operator
G
Green’s function
grad
Gradient operator
h
Height [ft]
H
Fundamental solution (free space Green’s function)
k
Permeability [md]
kξ
Main permeability in ξ = x, y , z [md]
147
K
Permeability tensor [md]
L
Length [ft]
M
Observation point
M′
Source point
nB
Outward normal direction of boundary surface B
N
Shape function
N Re
Reynolds number
m( p )
Gas pseudo-pressure [psia2/cp]
p
Pressure [psia]
pref
Reference pressure [psia]
pi
Initial pressure [psia]
psc
Pressure at standard conditions [psia]
q
Volumetric rate [rbbl/day; Mscf/day]
q~e
Plane segment flux [(rbbl/day)/ft2; (Mscf/day)/ft2]
q~w
Line segment flux [(rbbl/day)/ft; (Mscf/day)/ft]
Q(t )
Cumulative production [(STB; Mscf]
rw
Wellbore radius [ft]
s
Laplace parameter
ssk ( wi )
Skin factor
S
Source function
S SK
Skin effect
148
t
Time [hours]
t ps
Pseudo time
T
Temperature [°R]
T sc
Temperature at standard conditions [°R]
Tres
Reservoir temperature [°R]
V
Volume [rbbl ; Mscf]
V fR
Natural fracture volume to total bulk volume ratio
W
Weighting function
x
Point coordinate in x direction [ft]
xe
Subsection dimension in x direction [ft]
xf
Plane source (fracture) half length [ft]
y
Point coordinate in y direction [ft]
ye
Subsection dimension in y direction [ft]
z
Point coordinate in z direction [ft]
ze
Subsection dimension in z direction [ft]
zf
Plane source (fracture) half height [ft]
zg
Real gas compressibility factor
GREEK
α
Auxiliary angle [rads]
149
∆
Difference operator
ϕ
Inclination (slant) angle [rads]
φ
Porosity
λ
Transmissivity ratio
l
Reference length [ft]
∇
Gradient operator
∇2
Three dimensional Laplace operator
µ
Viscosity [cp]
η
Hydraulic diffusivity [ft2/hr]
π
Pi constant
ρ
Fluid density [lbm/ft3]
σ
Shape factor [ft-2]
θ
Azimuth (deviated) angle [rads]
ω
Storativity ratio
SUBSCRIPTS
av
Average value
b, B
Boundary
c
Cumulative
d
Deviated
D
Dimensionless
150
e
External boundary (plane source)
eff
Effective
f ,F
Fracture
fr
Friction
FP
Full penetrating plane source
g
Gas
HW
Horizontal line source
h, H
Horizontal
i
Number for the ith source at the internal boundary
inf
Infinite
j
Number for the jth source at the external boundary
k
Number for the kth reservoir subsection (block)
liq
Liquid
m
Matrix
PP
Partial penetrating plane source
r
Radial direction
ref
Reference value
sf
Sand face
w
Internal boundary (line source)
x, y , z
3D Cartesian-directions
S, s
Slanted
SW
Slanted line source
151
W
Generic line source
wb
Wellbore storage
152
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Temeng K. O. and Horne, R. N. 1995. Relative Productivities and Pressure Transient
Modeling of Horizontal Wells with Multiple Fractures. Paper SPE 29891 presented at
the SPE Middle East Show, Bahrain, 11-14 March.
Thompson, L.G., Manrique, J.L., and Jelmert, T.A. 1991. Efficient Algorithms for
Computing the Bounded Reservoir Horizontal Well Pressure Response. Paper SPE
21827 presented at the Rocky Mountain Regional Meeting and Low Permeability
Reservoir Symposium, Denver, Colorado, 15-17 April.
van Everdingen, A.F. and Hurst, W. 1949. Application of the Laplace Transformation to
Flow Problems in Reservoirs. Trans. AIME 186: 305-324.
Warren, J. E. and Root, P. J. 1963. The Behavior of Naturally Fractured Reservoirs. SPEJ
3 (3): 245-55; Trans. AIME, 228. SPE-426-PA. “DOI:10.2118/426-PA.”
Wolfsteiner, C., Durlofsky, L.J. and Aziz, K. 2000. Approximate Model for Productivity
of Nonconventional Wells in Heterogeneous Reservoirs. SPEJ 5 (2): 218-226. SPE62812-PA. “DOI:10.2118/62812-PA.”
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Yildiz, T. 2003b. Long-Term Performance of Multilaterals in Commingled Reservoirs. J.
Cdn. Pet. Tech. 42 (10): 47-53. Paper SPE 78985.
Yildiz, T. 2005. Multilateral Horizontal Well Productivity. Paper SPE 94223 presented at
the SPE Europec/EAGE Annual Conference, Madrid, Spain, 13-16 June.
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Conference and Exhibition, New Orleans, Louisiana, 25-28 September.
158
APPENDIX A
EQUIVALENT COORDINATE SYSTEMS TO COMPUTE SOURCE
FUNCTIONS
159
APPENDIX A
EQUIVALENT COORDINATE SYSTEMS TO COMPUTE SOURCE
FUNCTIONS
Sources functions presented in Chapter 4 were derived for specific configurations
presented in that chapter. However, Eqs. 4.10, 4.13, and 4.15, can be used to compute
source functions for other source configurations provided we change the orientation of
the coordinate system accordingly.
First, we consider the case where the line or plane segments are parallel to y D
axis as sketched in Figure A-1.
ẑ D
zD
M wi
zeD
yeD
ŷ D
yD
xeD
M ej
x̂D
xD
Figure A. 1 – Configuration for sources parallel to yD axis.
From Figure A.1, we easily observe that for the coordinate system ( xˆ D , yˆ D , zˆ D )
the sources are parallel to the x̂ D axis; this is the same configuration presented in Chapter
4. Therefore, by converting all the coordinates from the original system ( x D , y D , z D ) to
the new system ( xˆ D , yˆ D , zˆ D ), we can use Eqs. 4.10 and 4.13 to compute source functions
160
for plane and line segments parallel to y D axis, respectively. Thus for the subsection
dimensions
xˆ eD = yeD ; yˆ eD = xeD ; zˆeD = zeD ,
(A.1)
and for points inside or at subsection boundaries,
xˆξD = yξD ; yˆ ξD = xeD − xξD ; zˆξD = zξD , for ξ = iw, ej .
(A.2)
Second, we consider the case where the line segment is parallel to z D which is the
real situation for vertical wells. This configuration is shown in Figure A.2.
zD
x̂D
zeD
yeD
ẑ D
yD
xeD
M wi
ŷ D
xD
Figure A. 2 – Configuration for line segment parallel to zD axis.
For this case, we can use Eq. 4.13 with coordinates defined in the ( xˆ D , yˆ D , zˆ D )
coordinate system as
xˆeD = zeD ; yˆ eD = yeD ; zˆeD = xeD , and
(A.3)
xˆiwD = ziwD ; yˆ iwD = yiwD ; zˆiwD = xeD − xiwD .
(A.4)
Next, we consider a configuration where a plane segment is parallel to
the ( x D , y D ) plane; this represents a plane surface at the top or bottom boundary of the
reservoir subsection. This configuration is sketched in Figure A.3.
161
x̂D
zD
ẑ D
M ej
zeD
yeD
yD
xeD
ŷ D
xD
Figure A. 3 – Configuration for a plane segment parallel to ( xD, yD ) plane.
For this configuration the source term can be computed with Eq. 4.10 by using the
( xˆ D , yˆ D , zˆ D ) coordinate system where:
xˆ eD = yeD ; yˆ eD = zeD ; zˆeD = xeD , and
(A.5)
xˆ ejD = yejD ; yˆ ejD = zeD − zej ; zˆejD = xeD − xejD .
(A.6)
The last case is a deviated line segment as sketched in Figure A.4.
ẑ D
yD
zD
M wi
zeD
yeD
ŷ D
ϕ̂
L
θ
x̂D
xeD
xD
Figure A. 4 – Configuration for a deviated line segment.
For this case, using space coordinates relative to the ( xˆ D , yˆ D , zˆ D ) coordinate
system, we can compute the source term for a deviated line segment shown in Figure A.4
162
by using the solution for a slanted well given in Eq. 4.15. The space coordinates and
angles in the system ( xˆ D , yˆ D , zˆ D ) are related to corresponding values in the system
( x D , y D , z D ) by the following
xˆeD = xeD ; yˆ eD = zeD ; zˆeD = yeD ,
(A.7)
xˆ wiD = xwiD ; yˆ wiD = zeD − zwiD ; zˆwiD = ywiD ,
(A.8)
αˆ = ϕˆ − 90 = θ ,
(A.9)
and
Results presented in this work were computed using the Eqs. A.1 trough A.9 to
convert variables between the two coordinate systems, ( x D , y D , z D ) and ( xˆ D , yˆ D , zˆ D ).
However, it is possible to apply a different orientation to the coordinate system
( xˆ D , yˆ D , zˆ D ) and obtain other set of equations to relate space coordinates between
systems.
163
APPENDIX B
DERIVATION OF SOURCE FUNCTIONS
164
APPENDIX B
DERIVATION OF SOURCE FUNCTIONS
This appendix presents the mathematical derivation of equations presented in
Chapter 6. It follows the same presentation sequence in Chapter 6, starting with the fully
penetrating and partially penetrating plane source configurations, followed by horizontal
(or vertical), slanted (or deviated), and finally, generic line segments.
B.1 Fully Penetrating Plane Source
We start our derivation with the source function for a fully penetrating planar
surface given by Eq. 6.6 in the text and repeated here as Eq. B-1 for convenience:
ch( u ~
y D1 ) + ch( u ~
yD 2 )
~
u × sh( u yeD )
.
xe + x f ∞
ch(ε k ~
y D1 ) + ch(ε k ~
yD 2 )
kπx
kπx′
cos(
) cos(
) dx ′
+ 4Cz f ∫ ∑
ε × sh(ε ~y )
x
x
S FP = 4Cx f z f
xe − x f k =1
k
k
eD
e
(B.1)
e
Using the definitions
{
Fcsht [ξ ] = e −
ξ ( y D + y wD )
+ e−
ξ ( y D + ~y D1 )
+ e−
ξ ( y D + ~y D 2 )
+e
− ξ y D − y wD
}⎡⎢⎣1 + ∑ e
∞
m =1
− 2 m ξ y eD
⎤
⎥ (B.2)
⎦
and
{
Fcshp [ξ ] = e −
{
+ e
ξ ( y D + y wD )
− ξ ( y D + y wD )
+e
+ e−
y D1 )
− ξ ( yD + ~
ξ ( y D + ~y D1 )
+e
+ e−
ξ ( y D + ~y D 2 )
yD2 )
− ξ ( yD + ~
} ⎡⎢1 + ∑ e
∞
⎣
+e
− ξ y D − y wD
− 2 m ξ y eD
m =1
}
⎡ ∞ − 2m
⎢∑ e
⎣ m =1
⎤
⎥
⎦
ξ y eD
⎤
⎥
⎦
(B.3)
165
derived using the Eq. 6.1, the first term of the right-hand side of can be written as
Fcsht [u ]
.
u
S FPb1 = 4Cx f z f
(B.4)
Using the result given by Eq. 6.1 in the second term of the right hand side of Eq.
B.1, we find
S FP1 = S FP 2 + S FPb 2 ,
(B.5)
where
xw + x f
∞
∫ ∑
S FP 2 = 4Cz f
e
− u y D − y wD
cos(
εk
x w − x f k =1
kπx
kπx′
) cos(
)dx′ ,
xe
xe
(B.6)
and
S FPb 2
∞
×∑
k =1
{
u ( y D + y wD )
{
u ( y D + y wD )
⎧ −
⎪e
⎪
= 4Cz f ⎨
⎪ e−
⎪
⎩
1
xw + x f
εk x
∫
w
cos(
−x f
+ e−
+ e−
u ( y D + ~y D 1 )
u ( y D + ~y D1 )
+ e−
+ e−
u ( yD + ~
yD2 )
} ⎡⎢1 + ∑ e
∞
⎣
u ( y D + ~y D 2 )
+e
− 2 m u y eD
m =1
− u y D − y wD
⎤
⎥+
⎦
} ⎡⎢⎣1 + ∑ e
∞
m =1
−2m
⎫
⎪
⎪
⎬
u y eD ⎤ ⎪
⎥⎪
⎦⎭
kπx
kπx′
) cos(
)dx′
xe
xe
.
(B.7)
Evaluating the integral in Eq. B.6 and using the definition of Fcshp [u ] given in Eq. B.3,
we obtain
S FPb 2 =
8Cz f x e
π
∞
1
∑ kε
k =1
sin(
k
kπ x f
xe
) cos(
kπ x w
kπ x
) cos(
)Fcshp [ε k ] .
xe
xe
(B.8)
Using the dimensionless variables in Eq. B.6 yields:
S FP 2 =
( xw + x f ) D ∞
kπx D
kπx′D e
cos(
) cos(
)
∑
∫
xeD
xeD
k / k x ( xw − x f ) D k =1
4Cz f l
− ε k y D − y wD
εk
dx′D .
(B.9)
166
Using the result in Eq. 6.5 in Eq. B.7 and making the change of the integration variable,
~
x D′ = x′D /( x f ) D , where x fD = ( x f ) D = x f k / k x / l , we recast Eq. B-7 as follows:
S FP 2
[
~
]
⎫⎪
⎧⎪ x wD +1 x
xD′ − 2kxeD ) 2 + ( yD − ywD ) 2 u d~
xD′ ⎬
= 4Cz f x f ∑ ⎨ ∫ eD K o ( xD m x fD ~
⎪⎭
k = −∞ ⎪
⎩ ~x wD −1 2π
. (B.10)
∞
− 2Cz f x f
e
− u y D − y wD
u
Defining the variable of integration β = ~
x D′ − ~
x wD [Ozkan (1988)],
S FP 2 = 4Cz f x f
− 2Cz f x f
e
[
]
⎧ 1 xeD
⎫
K 0 ( x D m xwD − 2kxeD − x fD β ) 2 + ( y D − y wD ) 2 u dβ ⎬
⎨∫
∑
k = −∞ ⎩ −1 2π
⎭.
∞
− u yD − ywD
u
(B.11)
Then, we break and recast the summation in Eq. B.11 as
S FP 2 = S FPb3 + S FP inf ,
(B.12)
where
[
S FPb 3
]
⎧ +1
⎫
2
2
⎪ ∫ K 0 ( x D + xwD − x fD β ) + ( y D − y wD ) u dβ
⎪
xeD ⎪ −1
⎪
= 2Cz f x f
⎨ ∞ +1
⎬
π ⎪
2
2
+ ∫ K 0 ( x D m xwD m 2kxeD − x fD β ) + ( y D − y wD ) u dβ ⎪ ,
⎪ ∑
⎪
⎩ k =1 −1
⎭
− 2Cz f x f
[
e
]
− u yD − ywD
u
(B.13)
and
S FP inf = 2Cz f x f
xeD
π
∫K [
+1
0
−1
]
( x D − x wD − x fD β ) 2 + ( y D − y wD ) 2 u dβ .
(B.14)
167
Equations B.4, B.8, B.13, and B.14 are the source components for a fully penetrating
planar surface source presented in Chapter 6.
Note that, at late times, we compute
S FPb 2 =
8Cz f x e
π
∞
1
∑ kε
k =1
sin(
k
k πx f
xe
) cos(
k πx w
k πx
) cos(
)Fcsht [ε k ] .
xe
xe
(B.15)
Computing S FPb 2 with Fcsht [ε k ] as in Eq. B.2 , instead of Fcshp [ε k ] as in Eq. B.3, the term
S FP 2 is added to Eq. B.15, hence S FPb 3 = S FP inf = 0 .
B.2 Partially Penetrating Plane Source
The solution for a partially penetrating plane source is presented in Eq. 6.25 as the
summation of the fully penetrating planar surface source and the partial penetration
pseudoskin term. Since the fully penetrating planar surface source has been derived in the
previous section, we present the derivation for the pseudoskin term here. The pseudoskin
term is given in Eq. 6.27 in the text and repeated below for convenience:
zw + z j
S PSF = 4Cx f
zw + z f
+ 4C
∫
zw − z f
∞
∫ ∑ cos(
z w − z f n =1
nπz
nπz ′ Fcsht [ε n ]
dz ′
) cos(
)
ze
ze
εn
xw + x f
nπz
nπz ′ ⎧⎪
cos(
) cos(
)⎨
∑
ze
z e ⎪ xw ∫− x f
n =1
⎩
∞
kπ x
kπx ′ Fcsht [ε k ,n ] ⎫⎪
cos(
)
cos(
)
dx ′⎬dz ′
∑
ε k ,n
xe
xe
k =1
⎪⎭
∞
. (B.16)
We divide the right-hand side of Eq. B.16 in two parts and write
S PSF = S PSF 2 + S PSF 3 ,
where
(B.17)
168
zw + z j
∞
zw − z f
n =1
∫ ∑ cos(
S PSF 2 = 4Cx f
nπz
nπz′ Fcsht [ε n ]
) cos(
)
dz ′
εn
ze
ze
(B.18)
and
zw + z f
S PSF 3 = 4C
∫
zw − z f
xw + x f
nπ z
nπz ′ ⎧⎪
cos(
) cos(
)⎨
∑
ze
z e ⎪ xw ∫− x f
n =1
⎩
∞
∞
∑ cos(
k =1
k πx
kπx ′ Fcsht [ε k ,n ] ⎫⎪
) cos(
)
dx ′ ⎬ dz ′ .
ε k ,n
xe
xe
⎪⎭
(B.19)
Using the result in Eq. 6.5, we write
S PSF 2 = S PSFb 4 + S PSF 21 ,
(B.20)
where
∞
zw + z f
n =1
zw − z f
S PSFb 4 = 4Cx f ∑
∫
cos(
nπz
nπz′ Fcshp [ε n ]
) cos(
)
dz ′
εn
ze
ze
(B.21)
and
zw + z f
S PSF 21 = 4Cx f
∫
zw − z f
∞
∑ cos(
n =1
− ε y − ywD
nπz
nπz′ e n D
) cos(
)
εn
ze
ze
dz ′ .
(B.22)
Evaluating the integral in Eq. B.21 yields
S PSFb 4 =
8Cx f z e
π
∞
1
∑ n sin(
n =1
nπ z f
ze
) cos(
nπz wD Fcshp [ε n ]
n πz D
.
) cos(
)
z eD
z eD
εn
(B.23)
Equation B.23 is Eq. 6.29 in the text. We have kept S PSF 21 in Eq. B.22 unchanged
because it cancels out with another term coming from the other components of the
solution.
The next step in this derivation also uses Eq. 6.5 to write
S PSF 3 = S PSF 31 + S PSFb 5 ,
with
(B.24)
169
S PSF 31 = 4C
zw + z f ∞
∫ ∑ cos(
zw − z f n =1
nπ z
nπz ′
) cos(
)
ze
ze
(B.25)
y −y
−ε
⎧⎪ xw + x f ∞
⎫⎪
kπ x
kπx ′ e k ,n D wD
× ⎨ ∫ ∑ cos(
) cos(
)
dx ′ ⎬ dz ′
ε k ,n
xe
xe
⎪⎩ xw − x f k =1
⎪⎭
and
zw + z f
S PSFb 5 = 4C
∫
zw − z f
xw + x f
nπz
nπz ′ ⎧⎪
cos(
) cos(
)⎨
∑
ze
z e ⎪ xw ∫− x f
n =1
⎩
∞
∞
∑ cos(
k =1
k πx
kπx ′ Fcshp [ε k ,n ] ⎫⎪
dx ′⎬dz ′ .
) cos(
)
ε k ,n
xe
xe
⎪⎭
(B.26)
Evaluation of the integrals in Eq. B.26 yields the final form of S PSFb5 given in Eq. 6.30 in
the text. Using the dimensionless variable, x ′D , we recast Eq. B.25 as
S PSF 31
4Cl
=
k / kx
zw + z f
∞
∫ ∑ cos(
z w − z f n =1
nπ z
nπ z ′
) cos(
)
ze
ze
−ε
y −y
⎧⎪( xw + x f ) D ∞
⎫⎪
kπx D
kπx D' e k ,n D wD
× ⎨ ∫ ∑ cos(
) cos(
)
dx ′D ⎬dz ′
xeD
x eD
ε k ,n
⎪⎩( xw − x f ) D k =1
⎪⎭
.
(B.27)
Using the result in Eq. 6.5 in Eq. B.27 yields
S PSF 31
4Cl
=
k / kx
zw + z f ∞
∫ ∑ cos(
zw − z f n =1
nπ z
nπ z ′
) cos(
)
ze
ze
[
]
⎧ ∞ ⎧⎪( xw + x f )D x
⎫⎪ ⎫
eD
⎪∑ ⎨ ∫
K o ( x D m x ′D − 2kxeD ) 2 + ( y D − y wD ) 2 ε n dx ′D ⎬ ⎪ .
⎪⎪k =−∞ ⎪⎩( xw − x f )D 2π
⎪⎭ ⎪⎪
×⎨
⎬ dz ′
( xw + x f ) D −ε y − y
n D
wD
⎪
⎪
e
dx ′D
⎪
⎪− ∫
⎪⎭
⎪⎩ ( xw − x f )D 2ε n
(B.28)
Equation B.28 is further divided into terms
S PSF 31 = S PSF 31 A + S PSF 31 B ,
(B.29)
170
where
S PSF 31 A
( x w + x f ) D −ε y − y
⎫⎪
nπz
nπz ′ ⎧⎪
e n D wD
′
=−
cos(
)
cos(
)
d
x
⎨
⎬dz ′ ,
∑
D
∫
2ε n
ze
z e ⎪( xw −∫x f ) D
k / k x z w − z f n =1
⎪⎭
⎩
S PSF 31B
4Cl
=
k / kx
zw + z f
4Cl
∞
(B.30)
and
zw + z f ∞
∫ ∑ cos(
zw − z f n =1
nπ z
nπz ′
) cos(
)
ze
ze
[
]
⎧⎪ ∞ ⎧⎪( xw + x f )D x
⎫⎪ ⎫⎪
eD
×⎨∑ ⎨ ∫
K o ( x D m x ′D − 2kxeD ) 2 + ( y D − y wD ) 2 ε n dx ′D ⎬ ⎬dz ′
⎪⎭ ⎪⎭
⎪⎩k =−∞ ⎪⎩( xw − x f )D 2π
.
(B.31)
Evaluation of the integral in x ′D in Eq. B.30 gives the final form of S PSF 31 A as follows:
zw + z f
S PSF 31 A = −4Cx f
∫
∞
∑ cos(
z w − z f n =1
nπz
nπz ′ e
) cos(
)
ze
ze
− ε n y D − y wD
εn
dz ′ .
(B.32)
Note that in the summation to obtain S PSF , Eq. B.32 cancels out with Eq. B.22.
Following the same steps as in Eqs. B.9 and B.11, we recast Eq. B.32 as follows:
S PSF 31B = 4Cx f
~
zw + z f ∞
∫ ∑ cos(
zw − z f n =1
[
nπz
nπ z ′
) cos(
)
ze
ze
]
⎧⎪ xwD +1 x
⎫⎪
× ∑ ⎨ ∫ eD K o ( x D m x fD ~
x D′ − 2kxeD ) 2 + ( y D − y wD ) 2 ε n d~
x D′ ⎬dz ′
⎪⎭
k = −∞ ⎪
⎩ ~xwD −1 2π
∞
.
(B.33)
Evaluating the integral in z ′ , changing the remaining integration variable
to β = ~
x D′ − ~
x wD , and breaking the second infinite summation into two parts, we write Eq.
B.33 as
S PSF 31 B = S PSFb 6 + S PSF inf ,
where
(B.34)
171
S PSFb 6 = 2Cx f ze
xeD
π
2
[
∞
1
∑ n sin(
nπz f
n =1
ze
) cos(
nπz
nπz w
) cos(
)
ze
ze
]
⎡ +1
⎤
2
2
⎢ ∫ K 0 ( x D + x wD − x fD β ) + ( y D − y wD ) ε n dβ
⎥,
⎥
× ⎢ −1 +1
⎢ ∞
⎥
⎢ + ∑ ∫ K 0 ( x D m x wD m 2kxeD − x fD β ) 2 + ( y D − y wD ) 2 ε n dβ ⎥
⎥⎦
⎣⎢ k =1 −1
[
(B.35)
]
and
S PSF inf = 2Cx f ze
+1
[
xeD
π
2
∞
1
∑ n sin(
n =1
nπz f
ze
) cos(
n πz
k πz w
) cos(
)
ze
ze
]
.
(B.36)
× ∫ K 0 ( x D − xwD − x fD β ) 2 + ( y D − y wD ) 2 ε n dβ
−1
Equations B.35 and B.36 are the same as Eqs. 6.31 and 6.32 presented in Chapter 6,
respectively.
The derivation of the late-time solution follows the same lines as for the fully
penetrating plane source presented in Section B.1.
B.3 Horizontal Line Source
For a horizontal line source, the source term given in Eq.4.13 is written as
S HW = S FPFHW + S PSHW ,
(B.37)
where
ch( u ~
y D1 ) + ch( u ~
yD2 )
S FPFHW = 4CLH
~
u × sh( u yeD )
x w + LH / 2 ∞
ch(ε k ~
y D1 ) + ch(ε k ~
yD2 )
kπx
kπx′
cos(
)
cos(
)dx′
+ 4C ∫ ∑
ε × sh(ε ~y )
x
x
x w − LH / 2 k =1
and
k
k
eD
e
e
(B.38)
172
nπz D
nπz wD ⎡ ch(ε n ~
yD1i ) + ch(ε n ~
yD 2i ) ⎤
) cos(
)⎢
⎥
ε n × sh(ε n yeD )
zeD
zeD ⎣
⎦
∞
S PSHW = 2CLH ∑ cos(
n =1
nπz D
nπz wD
) cos(
)
zeD
zeD
x +L / 2
~
y ) + ch(ε ~
y ) w h
∞
+ 4C ∑ cos(
n =1
∞
×∑
ch(ε k , n
k =1
D1i
k ,n
D 2i
ε k , n × sh(ε k , n yeD )
∫
.
cos(
x w − Lh / 2
(B.39)
kπx
kπx′
) cos(
) dx ′
xe
xe
Using the result in Eq. 6.1 and Eqs. B.2 through B.3 in Eq. B.38, the first term in the right
hand side of Eq. B.38 becomes S HWb1 given in Eq. 6.38; the second term in the right-hand
side of Eq. B.38 can be written as
S FPFHW = 4C
xw + LH / 2 ∞
kπx
∫ ∑ cos( x
xw − LH / 2 k =1
) cos(
e
kπx ′ Fcsht [ε k ]
)
dx ′ .
xe
εk
(B.40)
Then, we break the right-hand side of Eq. B.40 into two terms
S FPFHW = S HWb 2 + S HW 3 ,
(B.41)
where
S HWb 2 = 4C
xw + LH / 2 ∞
1
xw − LH / 2 k =1
k
∫ ∑ε
cos(
k πx
kπx ′ Fcshp [ε k ]
dx ′
) cos(
)
xe
xe
εk
(B.42)
and
S HW 3 = 4C
xw + LH / 2 ∞
∫ ∑
e
− ε k yD − ywD
εk
xw − LH / 2 k =1
cos(
k πx
k πx ′
) cos(
) dx′ .
xe
xe
(B.43)
Evaluating the integral in Eq. B.42 yields
S HWb 2 =
4Cxe
π
∞
∑
k =1
1
kε k
sin(
kπx w Fcshp [ε k ]
kπLH
k πx
.
) cos(
) cos(
)
εk
2 xe
xe
xe
Equation B.34 is Eq. 6.39 given in the text.
Applying the dimensionless variables in Eq. B.43 yields
(B.44)
173
S HW 3 =
4Cl
k / kx
( x w + LH / 2) D ∞
∑
∫
e
− ε k y D − y wD
εk
( x w − L H / 2 ) D k =1
cos(
kπxD
kπx′D
) cos(
) dx′D .
xeD
xeD
(B.45)
Then, using the result in Eq. 6.5 in Eq. B.45 and changing the integration variable to
~
x D′ = x ′D /( x H ) D , where x HD = ( x H ) D = ( LH / 2) k / k x / l , we recast Eq. B.45 as
[
~
]
⎧⎪ xwD +1 x
⎫⎪
S HW 3 = CLH ∑ ⎨ ∫ eD K o ( x D m x HD ~
x D′ − 2kxeD ) 2 + ( y D − y wD ) 2 u d~
x D′ ⎬
⎪⎭
k = −∞ ⎪
⎩ ~xwD −1 2π
. (B.46)
− u yD − ywD
e
− CLH
u
∞
Breaking the summation in the right-hand side of Eq. B.46 and changing the integration
variable to β = ~
x D′ − ~
x wD , we write
S HW 3 = S HWb3 + S HW inf ,
(B.47)
where
S HWb 3 = CLH
xeD
2π
+1
∫K
0
[ (x
D
]
+ x wD − x HD β ) 2 + ( y D − y wD ) 2 u dβ +
−1
[
]
−
⎧⎪ x
e
CLH ∑ ⎨ ∫ eD K 0 ( x D m x wD m 2kxeD − x HD β ) 2 + ( y D − y wD ) 2 u dβ −
k =1 ⎪
⎩ −1 2π
∞
+1
u y D − y wD
u
⎫⎪
⎬
⎪⎭
,
(B.48)
and
S HW inf
x
= CLH eD
2π
∫K [
+1
0
]
( x D − x wD − x HD β ) 2 + ( y D − y wD ) 2 u dβ .
(B.49)
−1
Equations B.48 and B.49 correspond to Eqs. 6.40 and 6.41 in the text, respectively.
Now, to obtain the pseudoskin term for the horizontal line source, we apply the same
procedure as we used for partially penetrating plane source in Section B.2. First, we
divide the right-hand side of Eq. B.39 into two parts and write
174
S PSHW = S PSHW 2 + S PSHW 3 ,
(B.50)
where
∞
S PSHW 2 = 2CLH ∑ cos(
n =1
nπz wD Fcsht [ε n ]
nπ z D
) cos(
)
z eD
z eD
εn
(B.51)
and
∞
S PSHW 3 = 4C ∑ cos(
n =1
∞
×∑
k =1
nπz wD
nπ z D
) cos(
)
z eD
z eD
Fcsht [ε k ,n ] xw + Lh / 2
ε k ,n
k πx
k πx ′
cos(
) cos(
)dx ′
∫
x
x
e
e
x w − Lh / 2
.
(B.52)
Using the results in Eq. 6.1 and Eqs. B.2 through B.3, we can write
S PSHW 2 = S HWb 4 + S PSHW 21 ,
(B.53)
where
∞
S HWb 4 = 2CLH ∑ cos(
n =1
nπz w Fcshp [ε n ]
nπz
) cos(
)
ze
ze
εn
(B.54)
and
∞
S PSHW 21 = 2CLH ∑ cos(
n =1
nπz w e
nπz
) cos(
)
ze
ze
− ε n y D − y wD
εn
.
(B.55)
Equation B.54 is Eq. 6.44 given in the text. Note that Eq. B.55 is the counterpart of Eq.
B.22 for horizontal line source; it will cancel out with another term coming from the
other components of the solution.
The next step in this derivation also uses Eq. 6.1, and Eqs. B.2 through B.3 to
write
S PSHW 3 = S PSHW 31 + S HWb5 ,
with
(B.56)
175
∞
S PSHW 31 = 4C ∑ cos(
n =1
x +L / 2
w
H
∞
nπz
nπz w
e
) cos(
)× ∫ ∑
ze
ze
x w − L H / 2 k =1
− ε k ,n y D − y wD
ε k ,n
cos(
kπx
kπx′
) cos(
) dx ′
xe
xe
(B.57)
and
∞
S HWb5 = 4C ∑ cos(
n =1
x +L / 2
nπz w w H ∞
nπ z
k πx
kπx ′ Fcshp [ε k ,n ]
) cos(
) ∫ ∑ cos(
) cos(
)
dx ′ . (B.58)
ε k ,n
ze
z e xw − LH / 2 k =1
xe
xe
Evaluation of the integrals in Eq. B.58 yields the final form of S HWb5 given in Eq. 6.45.
Using the dimensionless variable, x ′D , we recast Eq. B.57 as
S PSHW 31 =
4Cl
k / kx
∞
∑ cos(
n =1
nπz w
nπz
) cos(
)
ze
ze
( x w + LH / 2 ) D ∞
−ε
y − y wD
kπx D
kπx D' e k ,n D
cos(
)
cos(
)
×
∫ ∑
xeD
xeD
ε k ,n
( x w − LH / 2 ) D k =1
.
(B.59)
dx′D
Using the result in Eq. 6.5 in Eq. B.59 yields
S PSHW 31 =
4Cl
k / kx
∞
∑ cos(
n =1
nπ z
nπz w
) cos(
)
ze
ze
[
]
⎧ ∞ ⎧⎪( xw + LH / 2 )D x
⎫⎪ ⎫
eD
⎪∑ ⎨ ∫
K o ( x D m x D′ − 2kxeD ) 2 + ( y D − y wD ) 2 ε n dx ′D ⎬ ⎪ .
⎪⎭ ⎪
⎪k =−∞ ⎪⎩( xw − LH / 2 )D 2π
×⎨
⎬
( x +L / 2)
−ε y − y
⎪ w H D e n D wD
⎪
dx ′D
⎪−
⎪
∫
⎩ −( xw + LH / 2 )D 2ε n
⎭
(B.60)
Equation B.60 is further divided into two terms
S PSHW 31 = S HW 31 A + S HW 31 B ,
(B.61)
with
S HW 3 1 A
( x + L / 2)
− ε y − y wD
nπz w w H D e n D
nπz
=−
cos(
)
cos(
)
∑ z
z e ( xw − L∫H / 2 ) D
2ε n
k / k x n =1
e
4Cl
∞
dx′D ,
(B.62)
176
and
S HW 31B =
4Cl
k / kx
∞
∑ cos(
n =1
nπ z
nπz w
) cos(
)
ze
ze
[
]
⎧⎪( xw + LH / 2 ) D x
⎫⎪
eD
×∑⎨ ∫
K o ( x D m x ′D − 2kxeD ) 2 + ( y D − y wD ) 2 ε n dx ′D ⎬
⎪⎭
k = −∞ ⎪
⎩( xw − LH / 2 )D 2π
∞
.
(B.63)
Evaluation of the x′D integral in Eq. B.62 gives the final form for S HW 31 A :
S HW 31 A
− ε y − ywD
nπz w e n D
nπz
= −2CLH ∑ cos(
) cos(
)
εn
ze
ze
n =1
∞
.
(B.64)
Note that the term in Eq. B.64 cancels out with the term in Eq. B.55.
Following the same steps as in Eq. B.45 and B.46, we recast Eq. B.63 as follows:
∞
S HW 31B = 2CLH ∑ cos(
n =1
nπz
nπ z w
) cos(
)
ze
ze
[
~
xwD +1
]
⎫⎪
⎧⎪
x
x D′ − 2kxeD ) 2 + ( y D − y wD ) 2 ε n d~
x D′ ⎬
× ∑ ⎨ ∫ eD K o ( x D m x HD ~
⎪⎭
k = −∞ ⎪
⎩ ~xwD −1 2π
∞
.
(B.65)
In Eq. B.65, changing the integration variable to β = ~
x D′ − ~
x wD , and breaking the second
infinite summation into two parts, we write Eq. B.65 as
S HW 31 B = S HWb6 + S PSHW inf ,
(B.66)
where,
S HWb 6 = CLH
[
xeD
π
∞
∑ cos(
n =1
nπz
nπz w
) cos(
)
ze
ze
]
⎤
⎡
2
2
⎥,
⎢ ∫ K 0 ( x D + xwD − x HD β ) + ( y D − y wD ) ε n dβ
−1
⎥
⎢
×
⎥
⎢ ∞ +1
⎢ + ∑ ∫ K 0 ( xD m xwD m 2kxeD − x HD β ) 2 + ( y D − y wD ) 2 ε n dβ ⎥
⎥⎦
⎣⎢ k =1 −1
+1
[
and
]
(B.67)
177
S PSHW inf = CLH
+1
[
xeD
π
∞
∑ cos(
n =1
nπz
nπzw
) cos(
)
ze
ze
]
.
(B.68)
× ∫ K 0 ( x D − xwD − x HD β ) + ( y D − y wD ) ε n dβ
2
2
−1
Equation B.67 and Eq. B.68 are Eq. 6.46 and Eq. 6.47 given in the text,
respectively.
The derivation of the late-time solution for the horizontal line source follows the
same lines presented in Section B.2
B.4 Slanted Line Source
The source function for a slanted line source is given in Eq. 4.15. Since the
integration in Eq. 4.15 leads to components in x and z directions, x − z permeability
anisotropy strongly affects the computation of the source function for a slanted well. First
we use trigonometric relations to obtain the deformed angles in the dimensionless
coordinate system. Figure B.1 shows the equivalence for dimensions and angles in the
dimensional and equivalent isotropic dimensionless coordinate systems.
178
z
LS
LS sin α
α
ϕ
LS cos α
x
zD
LSD
LSD sin α̂
α̂
ϕ̂
LSD cos α̂
xD
Figure B. 1– Equivalence between dimensional and dimensionless coordinate systems.
Using the definition of the dimensionless space variables (Eq. 3.7), we have
2 LSxD =
LS cos α k / k x
,
(B.69)
LS sin α k / k z
.
l
(B.70)
l
and
2 LSzD =
Using trigonometric relations
LSD =
LS
l
k
k
cos 2 α + sin 2 α ,
kx
kz
(B.71)
and
⎛ kx
⎞
tan α ⎟⎟ .
⎝ kz
⎠
αˆ = tan −1 ⎜⎜
(B.72)
179
Then, we consider the first term in the right-hand side of Eq. 4.15, which does not
depend on the slant angle, ϕ . The term S SWb1 is given by
S SWb1 = CLS
Fcsht
.
u
(B.73)
We apply the same procedure as in the horizontal line source case to divide the
second term in the right-hand side of Eq. 4.15 into two components, given by
S SWx = S SW 1x + S SWb 2 x ,
(B.74)
with
LS / 2
S SW 1x
kπ ( xw + L' cos α ) e
kπx
= 2C ∫ ∑ cos(
) cos(
)
xe
xe
− LS / 2 k =1
∞
− ε k y D − y wD
εk
dL' ,
(B.75)
and
S SWb 2 x ( M , s ) = 2C
LS / 2 ∞
kπx
∫ ∑ cos( x
− LS / 2 k =1
) cos(
e
kπ ( xw + L′ cos α ) Fcshp [ε k ]
dL′ .
)
εk
xe
(B.76)
Then, using dimensionless variables, we recast Eq. B.76 as follows:
S SWb 2 x = 2C
( LS / 2 ) D
∞
∫ ∑ cos(
− ( L S / 2 ) D k =1
kπxD
kπ ( xwD + ξ cos αˆ ) Fcshp [ε k ] LS
) cos(
)
dξ .
xeD
xeD
εk
LSD
Defining x′ = xwD + ξ cos αˆ and thus dξ ' =
S SWb 2 x =
2CLS
LSD cosαˆ
x wD + L SxD ∞
∫ ∑ cos(
x wD − L SxD
k =1
(B.77)
1
dx′ , we rewrite Eq. B.77 as
cos αˆ
kπxD
kπx′ Fcshp [ε k ]
) cos(
)
dx′ .
xeD
xeD
εk
(B.78)
Evaluating the integral in Eq. B.78 we obtain
S SWb 2 x =
kπLSxD
kπxD
kπxwD Fcshp [ε k ]
4CLS xeD ∞
,
sin(
) cos(
) cos(
)
∑
πLSD cos αˆ k =1
xeD
xeD
xeD
kε k
where LSxD = (LSD / 2)cos α̂ .
(B.79)
180
Equation B.79 is Eq. 6.63 in the text. We apply the same procedure as in Eqs. B.43 and
B. 45 to recast Eq. B.75 as follows:
S SW 1x
2CLS
=
LSD cos αˆ
x wD + L SxD ∞
∑ cos(
∫
x wD − L SxD k =1
kπx′ e
kπxD
) cos(
)
xeD
xeD
− ε k y D − y wD
εk
dx′ .
(B.80)
Using the steps applied to derive Eqs. B.46 through B.49, we obtain the terms
S SW 1x = S SWb 3 x + S SHW inf x
,
(B.81)
where
S SWb 3 x
2CLS
=
LSD cos αˆ
+1
xeD
∫ 2π
−1
[
]
K 0 ( x D + xwD − LSxD β ) 2 + ( y D − y wD ) 2 u dβ
[
]
⎧ xeD ∞ +1
⎫
2
2
−
+
−
K
x
x
kx
L
y
y
u
d
(
m
m
2
β
)
(
)
β
⎪
⎪ , (B.82)
∑
0
D
wD
eD
SxD
D
wD
∫
2CLS ⎪ 2π k =1 −1
⎪
+
⎨ − u y −y
⎬
D
wD
LSD cos αˆ ⎪ e
⎪
⎪−
⎪
u
⎩
⎭
and
+1
S SW inf x
CLS xeD
=
K
πLSD cos αˆ −∫1 o
[ [( x
D
]
− x wD − LSxD β )] 2 + ( y D − y wD ) 2 u dβ .
(B.83)
Equation B.82 and Eq. B.83 are Eq. 6.64 and Eq. 6.65 given in the text,
respectively.
The third term in Eq. 4.15 can also be divided into two terms:
S SWz = S SW 1z + S SWb 2 z ,
(B.84)
where
S SW 1z = 2C
Ls / 2
∫
∞
∑ cos(
− Ls / 2 n =1
and
nπ ( z w + L′ sin α ) e n D
nπz
) cos(
)
εn
ze
ze
− ε y − ywD
dL′ ,
(B.85)
181
S SWb 2 z = 2C
LS / 2 ∞
nπz
∫ ∑ cos( z
− LS / 2 n =1
) cos(
e
nπ ( z w + L′ sin α ) Fcshp [ε n ]
)
dL′ .
εn
ze
(B.86)
Using the dimensionless variables, we recast Eq. B.86 as
S SWb 2 z = 2C
( LS / 2 ) D
∞
∫ ∑ cos(
− ( L S / 2 ) D n =1
nπz D
nπ ( zwD + ξ sin αˆ ) Fcshp [ε n ] LS
) cos(
)
dξ .
zeD
zeD
ε n LSD
Defining z′ = zwD + ξ sin αˆ and thus dξ ' =
S SWb 2 z =
2CLS
LSD sin αˆ
z wD + L SzD ∞
∫ ∑ cos(
z wD − L SzD n =1
(B.87)
1
dz′ , we rewrite Eq. B.86 as
sin αˆ
nπz D
nπz′ Fcshp [ε n ]
) cos(
)
dz′ .
zeD
zeD
εn
(B.88)
Evaluating the integral in Eq. B.88 yields
S SWb 2 z =
2CLS zeD
πLSD sin αˆ
∞
∑ sin(
n =1
nπLSzD
nπzD
nπzwD Fcshp [ε n ]
,
) cos(
) cos(
)
zeD
zeD
zeD
nε n
(B.89)
where LSzD = (LSD / 2)sin α̂ . Following the same lines, we can write Eq. B.85 in terms of
dimensionless variables as follows:
S SW 1z
2CLS
=
LSD sin αˆ
z wD + LSzD ∞
nπz D
nπz ′ e
cos(
) cos(
)
∑
∫
zeD
zeD
z wD − LSzD n =1
− ε n y D − y wD
εn
dz ′ .
(B.90)
In the full solution, the term in Eq. B.90 cancels out with another component of the
solution.
The pseudoskin term for slanted well, S PSSW , corresponds to the last term in the
right-hand side of Eq. 4.15 and is given by
∞
S PSSW = 4 FH ∑ cos(
n =1
LH / 2
yD1 ) + ch(ε k ,n ~
yD 2 )
nπz D ∞
kπxD ch(ε k ,n ~
)∑ cos(
)
~
ε k ,n sh(ε k ,n yeD )
zeD k =1
xeD
⎡
nπ ( z w + LS′ sin α )
kπ ( xw + LS′ cos α ) ⎤ '
) cos(
)⎥ dLS′
× ∫ ⎢cos(
ze
xe
⎦
− LH / 2 ⎣
.
Changing the integration variable to dimensionless form, Eq. B.91 becomes
(B.91)
182
S PSSW =
4CLS
LSD
∞
∑ cos(
n =1
y D1 ) + ch(ε k , n ~
yD 2 )
nπz D ∞
kπxD ch(ε k , n ~
)∑ cos(
)
zeD k =1
xeD
ε k , n sh(ε k , n ~yeD )
( LS / 2 ) D
⎡
nπ ( z wD + ξ sin αˆ )
kπ ( xwD + ξ cos αˆ ) ⎤
) cos(
) ⎥ dξ
× ∫ ⎢cos(
z
x
eD
eD
⎣
⎦
− ( LS / 2 ) D
.
(B.92)
Evaluation of the integral in Eq. B.92 [Gradshtein (1965)] yields,
⎛ nπz wD kπx wD ⎞ ⎡⎛ nπ sin αˆ kπ cos αˆ ⎞ LsD ⎤
⎟ sin ⎢⎜
⎟
cos⎜⎜
−
−
⎥
xeD ⎟⎠ ⎣⎜⎝ z eD
xeD ⎟⎠ 2 ⎦
⎝ zeD
I=
⎛ nπ sin αˆ kπ cos αˆ ⎞
⎜⎜
⎟
−
xeD ⎟⎠
⎝ z eD
⎛ nπz wD kπx w ⎞ ⎡⎛ nπ sin αˆ kπ cos αˆ ⎞ LsD ⎤
⎟ sin ⎢⎜
⎟
cos⎜⎜
+
+
⎥
x eD ⎟⎠ ⎣⎜⎝ z eD
xeD ⎟⎠ 2 ⎦
⎝ z eD
+
⎛ nπ sin αˆ kπ cos αˆ ⎞
⎜⎜
⎟
+
xeD ⎟⎠
⎝ z eD
,
(B.93)
for
2
2
⎛ nπ sin αˆ ⎞ ⎛ kπ cos αˆ ⎞
⎜⎜
⎟⎟ ≠ ⎜⎜
⎟⎟ ,
z
x
eD
eD
⎝
⎠ ⎝
⎠
(B.94)
and
I=
⎛ nπz wD kπx wD ⎞
LsD
⎟
−
cos⎜⎜
xeD ⎟⎠
2
⎝ z eD
⎡ nπz wD kπx wD LsD nπ sin αˆ ⎤
⎡ nπz wD kπx wD LsD nπ sin αˆ ⎤
+
+
− sin ⎢
+
−
sin ⎢
⎥
⎥,
z eD
x eD
z eD
z eD
x eD
z eD
⎣
⎦
⎣
⎦
+
⎛ 4nπ sin αˆ ⎞
⎜⎜
⎟⎟
z eD
⎝
⎠
(B.95)
for
2
2
⎛ nπ sin αˆ ⎞ ⎛ kπ cos αˆ ⎞
⎜⎜
⎟⎟ = ⎜⎜
⎟⎟ .
⎝ z eD ⎠ ⎝ xeD ⎠
(B.96)
The computation of S PSSW using the results in Eqs. B.93 through B.96 may require a long
time due to slow convergence of the series in Eq. B.92.
183
We overcome this difficult by writing
S PSSW = S PSSWb 4 + S PSSW 5 ,
(B.97)
where
S PSSWb 4 =
S PSSW 5
∞
4CLS
LSD
∑ cos(
n =1
nπz D ∞
kπx D Fcshp [ε k ,n ]
× I b 4 ,.
)∑ cos(
)
z dD k =1
x eD
ε k ,n
−ε
y − y wD
nπz D ∞
kπx D e k ,n D
cos(
)∑ cos(
)
∑
ε k ,n
z eD k =1
x eD
n =1
∞
4CLS
=
LSD
(B.98)
( LS / 2 ) D
⎡
nπ ( z wD + ξ sin αˆ )
kπ ( x wD + ξ cos αˆ ) ⎤
) cos(
)⎥ dξ
× ∫ ⎢cos(
z
x
eD
eD
⎦
−( LS / 2 ) D ⎣
,
(B.99)
and
I b4 =
( LS / 2 ) D
⎡
nπ ( z wD + ξ sin αˆ )
kπ ( x wD + ξ cos αˆ ) ⎤
cos(
)
cos(
)⎥ dξ .
⎢
∫
z
x
eD
eD
⎦
−( LS / 2 ) D ⎣
(B.100)
We change the variable of integration in Eq. B.100 to z ′ = ξ sin αˆ and obtain
I b4 =
sin αˆ ( LS / 2 ) D
⎡
nπ ( z wD + z ′)
kπ ( x wD + ( z ′ / tan αˆ )) ⎤ dz ′
) cos(
)⎥
.
⎢cos(
z eD
x eD
⎦ sin αˆ
−sin ˆ ( LS / 2 ) D ⎣
∫
α
(B.101)
z = nπz ′ / z eD , Eq. B.101 yields
Changing the variable of integration again to ~
Ib4 =
zeD
nπ sin αˆ
γ
⎡
∫γ ⎢⎣cos(
−
kπxwD ~ ⎤ ~
nπz wD ~
+ z ) cos(
+ τz )⎥ dz ,
xeD
zeD
⎦
(B.102)
where we defined γ = nπLSzD / z eD and τ = kz eD / nx eD tan αˆ for convenience in the
following presentation of the equations. Expanding the cosine summation in Eq. B.102
and noting that the integral of an odd function vanishes in a symmetric interval around
the origin we have
184
I b4
γ
⎧
⎫
nπz wD
kπxwD
[cos(~z ) cos(τ~z )] d~z ⎪
cos(
)
cos(
)
⎪
∫
zeD
xeD −γ
zeD ⎪
⎪
=
⎨
⎬.
γ
ˆ
nπ sin α ⎪
nπz wD
kπxwD
+ sin(
z ) sin(τ~
z )] d~
z⎪
) sin(
) ∫ [sin( ~
⎪
⎪
z
x
eD
eD
−γ
⎩
⎭
(B.103)
Evaluation of the integral in Eq. B.82 yields
⎡ sin[γ (1 + τ )] sin[γ (1 − τ )] ⎤
+
I b4 = ⎢
(1 − τ ) ⎥⎦
⎣ (1 + τ )
nπzwD
kπxwD ⎡ sin[γ (1 − τ )] sin[γ (1 + τ )] ⎤
+ tan(
) tan(
)
−
(1 + τ ) ⎥⎦
zeD
xeD ⎢⎣ (1 − τ )
,
(B.104)
for τ ≠ 1 .When τ = 1 we use the expressions,
I b4 =
sin(2γ ) ⎡
nπzwD
kπxwD ⎤
) tan(
)⎥ for τ = 1
⎢1 − tan(
2 ⎣
zeD
xeD ⎦
(B.105)
I b4 =
sin(2γ ) ⎡
nπzwD
kπxwD ⎤
) tan(
)⎥ for τ = −1 .
⎢1 + tan(
2 ⎣
zeD
xeD ⎦
(B.106)
and
Now we rearrange the terms in Eq. B.97 to obtain
S PSSW 5 =
4CLS
LSD
∞
∑ cos(
n =1
nπz D
)
zeD
y −y
−ε
⎡
nπ ( z wD + ξ sin αˆ ) ∞
kπxD
kπ ( xwD + ξ cos αˆ ) e k ,n D wD ⎤
)∑ cos(
) cos(
)
× ∫ ⎢cos(
⎥ dξ
zeD
xeD
xeD
ε k ,n
k =1
⎢
− ( LS / 2 ) D ⎣
⎦⎥
(B.107)
( LS / 2 ) D
.
Then, using the result in Eq. 6.5 we recast S PSSW 5 as
S PSSW 5 = S PSSW 51 + S PSSW 5 2 ,
where
(B.108)
185
S PSHW 51 =
4CLS
LSD
∞
∑ cos(
n =1
nπz D
)
zeD
⎧
nπ ( z wD + ξ sin αˆ )
⎡
)
⎪ ( LS / 2) D ⎢cos(
zeD
⎪
×⎨ ∫ ⎢ ∞
xeD
2
2
⎪( − LS / 2) D ⎢×
⎢ ∑ 2π K o ( xD m ( xwD + ξ cos αˆ ) − 2kxeD ) + ( y D − ywD ) ε n
⎪⎩
⎣ k = −∞
[
⎤ ⎫
⎥ ⎪.
⎥ dξ ⎪⎬
⎥ ⎪
⎥ ⎪
⎦ ⎭
]
(B.109)
and
S PSHW 5 2 = −
4CLS
LSD
∞
∑ cos(
n =1
−ε y − y
nπz D S D
nπ ( z wD + ξ sin αˆ ) e n D wD
dξ .
) ∫ cos(
zeD − ( LS / 2) D
zeD
2ε n
( L / 2)
Defining z′ = zwD + ξ sin αˆ and thus dξ ' =
S PSHW 5 2
2CLS
=−
LSD sin αˆ
z wD + LSzD ∞
(B.110)
1
dz′ , we recast Eq. B.110 as
sin αˆ
nπz D
nπz ′ e
cos(
) cos(
)
∑
∫
z eD
z eD
z wD − LSzD n =1
− ε n y D − y wD
εn
dz ′ .
(B.111)
Equation B.111 cancels out with Eq. B.85. Now, we write
S PSSW 5 2 = S PSSWb 6 + S PSSW inf ,
(B.112)
where
S PSHWb 6 =
nπz D
2CLS xeD ∞
cos(
)
∑
πLSD n =1
zeD
nπ ( zwD + ξ sin αˆ )
⎧
cos(
)
⎪
zeD
( LS / 2 ) D ⎪
⎪
× ∫ ⎨ ⎡ K o ( xD + ( xwD + ξ cos αˆ ))2 + ( yD − ywD ) 2 ε n
( − LS / 2 ) D ⎪ ⎢
∞
×
⎪ ⎢+
K o ( xD m ( xwD + ξ cos αˆ ) m 2kxeD ) 2 + ( yD − ywD ) 2 ε n
⎪⎩ ⎢⎣ ∑
k =0
[
and
[
]
⎫
⎪
, (B.113)
⎪
⎪
⎤ ⎬ dξ
⎥⎪
⎥⎪
⎥⎪
⎦⎭
]
186
S PSHW inf =
nπz D
2CLS xeD ∞
cos(
)
∑
zeD
πLSD n =1
[
]
⎧⎪ ( LS / 2 ) D
⎫⎪
nπ ( zwD + ξ sin αˆ )
× ⎨ ∫ cos(
) K o ( xD − ( xwD + ξ cos αˆ ))2 + ( y D − ywD ) 2 ε n dξ ⎬
zeD
⎪⎩( − LS / 2 ) D
⎪⎭
.
(B.114)
Defining ~
z = nπξ sin αˆ / zeD and thus dξ =
zeD
d~
z , we recast Eqs. B.113 and B.114
nπ sin αˆ
as
S PSHWb6 =
2CLS xeD zeD ∞ 1
nπz D
cos(
)
∑
2
zeD
π LSD sin αˆ n =1 n
[
]
2
2
⎧
⎡K (x + x + ω ~
D
wD
z z ) + ( y D − y wD ) ε n
⎪
nπz wD ~ ⎢ 0
× ∫ ⎨cos(
+ z)× ⎢ ∞
2
2
~
zeD
−γ ⎪
⎢+ ∑ K 0 ( xD m ( xwD + ω z z ) m 2kxeD ) + ( yD − ywD ) ε n
⎣ k =0
⎩
γ
[
⎤⎫ ,
⎥⎪ ~
⎥ ⎬ dz
⎥⎪
⎦⎭
]
(B.115)
and
S PSHW inf =
γ
nπz D
2CLS xeD zeD ∞ 1
cos(
)
∑
2
zeD
π LSD sin αˆ n =1 n
[
]
nπz wD ~
× ∫ cos(
+ z ) K 0 ( xD − xwD − ω z ~
z ) 2 + ( yD − ywD ) 2 ε n d~
z
z
eD
−γ
with ω z =
,
(B.116)
zeD
. Equations B.115 and B.116 can be used to compute the solutions for
nπ tan αˆ
points where xD − xWD ≠ 0 . However, Eq. B.116 will not provide good results for early
times when xD − xWD → 0 . In this case, we rewrite the S PSSW inf term by expanding the
cosine summation in Eq. B.116 to obtain
187
S PSHW inf =
2CLS xeD zeD ∞ 1
nπz D
cos(
)
∑
2
π LSD sin αˆ n =1 n
zeD
[
]
⎧γ
⎫
nπz wD
~
z ) 2 + ( yD − ywD ) 2 ε n d~
z ⎪,
) K 0 ( xD − xwD − ω z ~
⎪ ∫ cos z cos(
zeD
⎪− γ
⎪
×⎨ γ
⎬
nπz wD
⎪
⎪
2
2
~
~
~
⎪− ∫ sin z sin( z ) K 0 ( xD − xwD − ω z z ) + ( yD − ywD ) ε n dz ⎪
eD
⎩ −γ
⎭
[
(B.117)
]
The K 0 term in Eq. B.117 behaves as an even function when xD − xWD → 0 . Therefore
the second integral in Eq. B.117 may be dropped and the first integral is multiplied by
two and evaluated over the positive half of the integration path; thus
S PSHW inf =
γ
4CLS xeD zeD ∞ 1
nπz D
cos(
)
∑
2
π LSD sin αˆ n =1 n
zeD
[
]
nπzwD
z cos(
z ) 2 + ( yD − ywD ) 2 ε n d~
z
× ∫ cos ~
) K 0 ( xD − xwD − ω z ~
z
eD
0
,
(B.118)
We have applied Eq. B.118 to compute wellbore pressures of slanted and deviated wells.