mathematical modeling, analysis and simulation of …

MATHEMATICAL MODELING, ANALYSIS AND SIMULATION

OF THE PRODUCTIVITY INDEX FOR NON-LINEAR FLOW IN POROUS MEDIA,

WITH APPLICATIONS IN RESERVOIR ENGINEERING

by

Simeon Eburi Losoha, B.S.

A THESIS

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

PETROLEUM ENGINEERING

Approved

Lloyd Heinze Chairperson of the Committee

Padmanabhan Seshaiyer

Co-Chairperson of the Committee

Eugenio Aulisa

Akif Ibragimov

Accepted

John Borrelli Dean of the Graduate School

August, 2007

Texas Tech University, Simeon Eburi Losoha, August 2007

ii

ACKNOWLEDGEMENTS

I would like to express my gratitude to Dr. Lloyd Heinze and Dr. Padmanabhan

Seshaiyer for guiding me through my thesis and serving on my thesis committee along

with Dr. Eugenio Aulisa and Dr. Akif Ibragimov. Dr. Heinze has always been a great

motivator and mentor since I transferred to Texas Tech University as an undergraduate.

He encouraged me to get my Master’s degree and emphasized the advantages that it

could bring to my career and future. Dr. Padmanabhan Seshaiyer was my professor in

Numerical Analysis and one of the best professors in Mathematics I have ever had. I

thank him for his time and motivation. I would also like to take this opportunity to thank

Dr. Eugenio Aulisa for his valuable time. None of this would have been possible without

the financial support from the Texas Tech Petroleum Engineering Department which

offered me a graduate assistantship to partially finance my Master’s Degree.

I would like to give special thanks to my parents, Mr. Ceferino Eburi and Mrs.

Manuela Losoha of Eburi for all their love and support. My sisters, Julia, Piedad,

Manuela, Margarita, Esther, Juana Esperanza, and Maria Isabel have all been very

supportive. I appreciate the encouragement and support from all my family, friends, and

family friends for believing in me.


iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS……………………………………………………………….ii

ABSTRACT…………………………………………………………………………….....v

LIST OF TABLES……………………………………………………………………….vii

LIST OF FIGURES……………………………………………………………………..viii

LIST OF ABBREVIATIONS…………………………………………………………...xii

CHAPTER

1. INTRODUCTION…………………………………………………………………….1

2. LITERATURE REVIEW…………………………………………………….……….4

2.1. Fluid Flow in Porous Media……………………………………………………...4

2.2. High Velocity Flow………………………………………………………………6

3. METHODOLOGY ………………………………………………………………….10

3.1. Finite Differences Introduction………………………………………………….10

3.2. Taylor Series…………………………………………………………………….10

3.3. Finite Differences……………………………………………………………….12

3.4. Terminology…………………………………………………………………….13

3.5. Initial and Boundary Conditions………………………………………………...14

3.6. Discretization Methods………………………………………………………….15

3.6.1. Explicit Method………………………………………………………….16

3.6.2. Implicit Method………………………………………………………….17

3.6.3. Crank-Nicholson ………………………………………………………...18

3.6.4. Other Methods…………………………………………………………...18

4. DERIVATION……………………………………………………………………….21

4.1. Darcy’s Equation………………………………………………………………..21

4.1.1. Darcy’s Radial Diffusivity Equation…………………………………….21

4.1.2. Darcy’s Diffusivity Equation in One Dimension………………………..24

4.1.3. Analytical Solution………………………………………………………26

4.1.4. Steady State Solution…………………………………………………….30

4.1.5. Finite Differences Explicit Discretization……………………………….32


iv

4.1.6. Stability Analysis for the Explicit Scheme………..……………………..44

4.1.7. Convergence Analysis for the Explicit Method……..…………………...49

4.1.8. Finite Differences Implicit Discretization……………………………….63

4.1.9. Stability Analysis for the Implicit Scheme……………………..………..72

4.2. Forchheimer’s Equation…………………………………………………………73

4.2.1. Forchheimer Diffusivity Equation……………………………………….74

4.2.2. Analytical Solution………………………………………………………76

4.2.3. Steady State Solution…………………………………………………….81

4.2.4. Finite Differences Explicit Discretization......…………………………...83

4.2.5. Stability Analysis for the Explicit Scheme…………..…………………..90

4.2.6. Convergence Analysis for the Explicit Method………………………….92

4.2.7. Finite Differences Implicit Discretization ……………………………..108

4.2.8. Stability Analysis for the Implicit Scheme……………………………..114

4.3. Darcy-Forchheimer Equation………………………………………………….116

4.3.1. Darcy-Forchheimer Diffusivity Equation………………………………116

4.3.2. Finite Differences Explicit Discretization ……………………………..119

4.3.3. Stability Analysis for the Explicit Scheme……..………………………134

4.3.4. Finite Differences Implicit Discretization……………..……………….138

4.4. Productivity Index……………………………………………………………..157

5. RESULTS…………………………………………………………………………..161

5.1. Finite Elements………………………………………………………………...161

5.1.1. Darcy’s Diffusivity Equation……………………………………….…..161

5.1.2. Darcy-Forchheimer Diffusivity Equation……………………………..164

5.2. Finite Differences……………………………………………………………...168

5.2.1. Constant Pressure Boundaries…………………………………………168

5.2.2. Boundary Dominated Regime…………………………………………..174

5.2.3. Pseudosteady State……………………………………………………...180

6. CONCLUSIONS AND RECOMMENDATIONS…………………………………187

BIBLIOGRAPHY………………………………………………………………………188


v

APPENDIX

A. ADDITIONAL TABLES AND GRAPHS.…...………………………………..190

B. DIMENSIONAL ANALYSIS, CONVERSION FACTORS, AND

GRAPHICAL USER INTERFACE…………...….……………………………198

C. VITA……………………………………………………………..……………..213


vi

ABSTRACT

Fluid flow in porous media has been often modeled using Darcy’s Law11. Because

it has been observed that Darcy’s Law11 is valid only for low velocities, there have been

several attempts to solve this problem by using modifications of Darcy’s equation11.

Forchheimer12 modified Darcy’s equation11 by adding a new term to account for inertia

caused by high velocity flow that was initially thought to occur only in gas reservoirs but

later confirmed that some oil reservoirs may also exhibit non-Darcy flow behavior. This

paper takes a look into high rate oil wells to determine if they may experience

nonlinearity due to non-Darcy flow and to examine the discrepancy between the

differential equation derived from the traditional Darcy’s Law11 and a new approach

using a non-linear Darcy-Forchheimer3 equation.

Mathematical modeling of Darcy11 and Darcy-Forchheimer3 diffusivity equations

will be performed to simulate high velocity flow. Finite differences methods will be used

to approximate the solution of the partial differential equations. In this paper, both

explicit and implicit methods will be used to evaluate the differential equations. Implicit

methods are more commonly used for domain discretization because they are

unconditionally stable. Stability analysis will be conducted to determine the condition for

stability of the explicit method used. The results will then be compared to results from

finite element method software.

The productivity index from the Darcy-Forchheimer3 diffusivity equation will be

calculated with different boundary conditions to evaluate the effect of non-linear flow on

well potential, pressure gradient, and velocity.


vii

LIST OF TABLES

4.1 Convergence Analysis ( )5.0=λ ………………………………………………...56

4.2 Convergence Analysis ( )3.0=λ ………………………………………………..58

4.3 Convergence Analysis ( )51.0=λ ……………………………………………….59

4.4 Convergence Analysis (Quadratic Function), ( )5.0=λ and ( )β ………………101

4.5 Convergence Analysis (Cubic and Fourth Order Functions), ( )5.0=λ and ( )β ..102

4.6 Convergence Analysis ( )3.0=λ and ( )β ……………………………………...104

4.7 Convergence Analysis ( )515.0=λ and ( )β …………………………………...105

A.1 Convergence Analysis ( )5.0=λ and ( )0=β ………….………………………190

A.2 Convergence Analysis ( )5.0=λ and ( )1=β ……….………………………….190

A.3 Convergence Analysis ( )5.0=λ and ( )5=β ………………………………….191

A.4 Convergence Analysis ( )5.0=λ and ( )10=β ………………………………...191

A.5 Convergence Analysis ( )5.0=λ and ( )100=β …………………………….....192

B.1 SI Metric Conversion Factors…………………………………………………..202

B.2 Sample Data………………………………………………………………….....204


viii

LIST OF FIGURES

2.1 Schematic of Darcy’s Experimental Equipment ………………………………….5

3.1 Explicit Method………………………………………………………………….16

3.2 Amplification Factor for Explicit Method……………………………….………17

3.3 Implicit Method………………………………………………………………….18

3.4 Crank-Nicholson Method………………………………………………………...18

3.5 Amplification Factor for Various Methods………………………………………19

4.1 Linear Flow Model………………………………………………………………26

4.2 Grid with N Intervals in x-direction and J Intervals in Time………...…………..34

4.3 Pressure Profile for Darcy (CPB) Explicit ( )5.0=λ …………………………….37

4.4 Pressure Profile for Darcy (BDR) Explicit ( )5.0=λ ……………………………40

4.5 Pressure Profile for Darcy (PSS) Explicit ( )5.0=λ ……………………………..44

4.6 Unstable Pressure Profile for Darcy (CPB) Explicit ( )51.0=λ …………..……..47

4.7 Unstable Pressure Profile for Darcy (BDR) Explicit ( )5025.0=λ ……..……….48

4.8 Unstable Pressure Profile for Darcy (PSS) Explicit ( )5005.0=λ ………..……..49

4.9 Pressure Profile for ( )5.0=λ and ( )14=N ……………………………………..57

4.10 Convergence Rate ( )tΔ for ( )5.0=λ ……………………………………………61

4.11 Convergence Rate ( )xΔ for ( )5.0=λ …………………………………………...63

4.12 Pressure Profile for Darcy (CPB) Implicit ( )125.0=λ ………………………….66

4.13 Pressure Profile for Darcy (BDR) Implicit ( )125.0=λ …………………………69


ix

4.14 Pressure Profile for Darcy (PSS) Implicit ( )125.0=λ …………………………..72

4.15 Pressure Profiles for ( )5.0=λ and ( )10=β ……………………………………103

4.16 Convergence Rate (Cubic Function) ( )xΔ for ( )5.0=λ …………………….....107

4.17 Convergence Rate (Fourth Order Function) ( )xΔ for ( )5.0=λ ….……………108

4.18 Pressure Profile for Darcy-Forchheimer (CPB) Explicit ( )5.0=λ …………….127

4.19 Pressure Profile for Darcy-Forchheimer (BDR) Explicit ( )5.0=λ ……………130

4.20 Pressure Profile for Darcy-Forchheimer (PSS) Explicit ( )5.0=λ ……………..134

4.21 Unstable Pressure Profile for Darcy-Forchheimer (CPB) Explicit ( )1=λ …….136

4.22 Unstable Pressure Profile for Darcy-Forchheimer (BPR) Explicit ( )1=λ …....137

4.23 Unstable Pressure Profile for Darcy-Forchheimer (PSS) Explicit ( )3=λ …….138

4.24 Pressure Profile for Darcy-Forchheimer (CPB) Implicit ( )125=λ ………...….142

4.25 Pressure Profile for Darcy-Forchheimer (BDR) Implicit ( )125=λ ……………145

4.26 Pressure Profile for Darcy-Forchheimer (PSS) Implicit ( )125=λ ...…………..149

5.1 Pressure Profile for Darcy (CPB) Finite Elements……………………………..162

5.2 Pressure Profile for Darcy (BDR) Finite Elements……………………………..163

5.3 Pressure Profile for Darcy (PSS) Finite Elements……………………………...164

5.4 Pressure Profile for Darcy-Forchheimer (CPB) Finite Elements………………165

5.5 Pressure Profile for Darcy-Forchheimer (BDR) Finite Elements………………166

5.6 Pressure Profile for Darcy-Forchheimer (PSS) Finite Elements……………….167

5.7 Effect of β on Pressure (CPB)………………………………………………....168

5.8 Effect of β on Pressure Gradient (CPB)……………………………………….169


x

5.9 Effect of β on Velocity (CPB)………………………………………………...170

5.10 Change of Velocity with Time (CPB)………………………………………….171

5.11 Effect of β on Productivity Index (CPB)………………………………………172

5.12 Effect of β and Permeability on PI (CPB)……………………………………..173

5.13 Effect of β on Pressure (BDR)…………………………………………………174

5.14 Effect of β on Pressure Gradient (BDR) Tf=0.1……………………………….175

5.15 Effect of β on Velocity (BDR) Tf=0.1…………………………………………176

5.16 Change of Velocity with Time (BDR)………………………………………….177

5.17 Effect of β on Productivity Index (BDR)………………………………………178

5.18 Effect of β and Permeability on PI (BDR)……………………………………..179

5.19 Effect of β on Pressure (PSS)……………………………………….………….180

5.20 Effect of β on Pressure Gradient (PSS)………………………………………...181

5.21 Effect of β on Velocity (PSS) Tf=0.1………………………………………….182

5.22 Change of Velocity with Time (PSS)…………………………………………..183

5.23 Effect of β on Productivity Index (PSS)……………………………………….184

5.24 Effect of β and Flow Rate on PI (PSS)………………………………………...185

5.25 Effect of β and Permeability on PI (PSS)……………………………………...186

A.1 Effect of β on Pressure Gradient (BDR) Tf=1…………………………………193

A.2 Effect of β on Pressure Gradient (BDR) Tf=5…………………………………194

A.3 Effect of β on Velocity (BDR) Tf=2…………………………………………...195

A.4 Effect of β on Velocity (BDR)…………………………………………………196


xi

A.5 Effect of β on Velocity (PSS) Tf=2……………………………………………197

B.1 Linear Flow into a Fracture……………………………………………………..205

B.2 Pressure Profile for Darcy (PSS) in Field Units………………………………..206

B.3 Pressure Profile for Darcy-Forchheimer (PSS) in Field Units…………………207

B.4 Productivity Index (PSS) in Field Units………………………………………..208

B.5 Pressure Profile for Darcy (PSS) in SI Units…………………………………...209

B.6 Pressure Profile for Darcy-Forchheimer (PSS) in SI Units………………..…...210

B.7 Productivity Index (PSS) in SI Units…………………………………………...211

B.8 Darcy-Forchheimer Flow Model GUI………………………………………….212


xii

LIST OF ABBREVIATIONS

α = Alpha, function, weighing parameter a = Constant A = Area

..., 21 AA = Constants

rA = Radial area β = Inertia factor b = Constant B = Formation volume factor c = Compressibility

..., 21 CC = Constants ε = Tolerance e = Error E = Maximum error k = Permeability, constant φ = Porosity γ = Gamma, constant

1−γ = Compressibility λ = CFL, constant η = Diffusivity constant μ = Viscosity π = Pi ρ = Density ρ′ = Derivative of density f = Function h = Height I = Imaginary unit L = Length p = Pressure p~ = Pressure approximation

tp = First derivative in time

xp = First derivative in space q = Flow rate r = Radius Re = Reynolds number s = Skin factor t = Time T = Temperature, constant Tf = Final time


xiii

ν = Velocity scalar νr = Velocity vector νr = Magnitude of velocity

V = Volume w = Function W = Width x = Independent variable, distance X = Function y = Independent variable z = Independent variable dr = Radial differential

pΔ = Pressure drop zyx ΔΔΔ ,, = Increment in x, y, z direction

tΔ = Time increment Δ=∇ 2 = Laplace

∇ = Gradient p∇ = Pressure gradient p∇ = Absolute value of pressure gradient

( )pf ∇ = Non-linear diffusion coefficient div = Divergence Dar = Darcy term unit conversion factor Forch = Forchheimer term unit conversion factor


1

CHAPTER I

INTRODUCTION

For a very long time Darcy’s Law11 has been used to explain fluid flow in porous

media, and it is still being used today, particularly in the oil industry. Darcy is considered

a hero when it comes to reservoir engineering even though Darcy’s Law11 is limited to

low velocities and Darcy flow behavior. Through the years, researchers have suggested

modified versions of Darcy’s equation11 to account for some of its limitations.

Forchheimer’s equation12 is commonly accepted and known to address non-Darcy flow

behavior by adding an inertia term with a non-Darcy coefficient or inertia flow parameter

( )β to Darcy’s equation11.

Predicting the behavior of the forces in the reservoir has been of particular

importance in the oil industry where most of the operations are conducted with the

preconception that physical forces present in the reservoir are known. The physical forces

and the fluid flow behavior in the reservoir are represented by equations. The more

accurately the equations describe the real behavior of the reservoir the better the

prediction.

Most of the software used currently for pressure transient analysis and reservoir

simulation use Darcy’s equation11 or various versions, derivations or solutions of Darcy’s

equation11. If it can be shown that there is an equation that better describes the behavior

of fluid flow in porous media at high velocities, the use of that particular equation could

be of great significance in the petroleum industry, thus improving analysis and reservoir

modeling.


2

The motivation for this project came from a well performance analysis project for

high rate offshore wells in Equatorial Guinea, West Africa. In this work, we will analyze

the effect of high velocity flow and analyze equations that are believed to better describe

the characteristics of high velocity flow. The main objective is to demonstrate that some

modifications of Darcy’s equation11 may improve reservoir modeling by providing a

more accurate representation of the physical properties that govern the reservoir

dynamics.

Two differential equations derived from Forchheimer’s equation12, referred as

Forchheimer diffusivity equation7 and Darcy-Forchheimer3 diffusivity equation will be

analyzed. Both equations will be solved using finite differences. The Forchheimer

diffusivity equation7 seems to have been designed for an experimental setting where the

velocity can be held constant. The results of this work will be based only on the Darcy-

Forchheimer3 diffusivity equation because it allows the velocity to change as a function

of the pressure gradient and an easy transition from linear to non-linear flow by just

changing the inertia coefficient β . The results will be compared with finite elements

method.

To check the performance of the mathematical models presented herein, appropriate

computer codes are developed with MATLAB to solve the various partial differential

equations as a part of this work. These codes employ both explicit as well as implicit

finite differences methods. Explicit methods are known to be conditionally stable and

therefore, the stability condition will be derived and incorporated into the computer

program. The results will be compared with COMSOL Multiphysics (Finite Element


3

Method) software.

The productivity index17 (PI) will be calculated at different boundary conditions to

analyze the effect of non-linear flow. The PI is a measure of well potential and can

therefore help us determine the parameters that will have more significant impact on the

ability of a well to produce, especially the effect of a highly non-linear flow in

comparison with linear or Darcy flow. Other parameters like flow rate and permeability

will also be varied to analyze their impact on the productivity index, pressure gradient

and velocity during non-linear flow.


4

CHAPTER II

LITERATURE REVIEW

2.1. FLUID FLOW IN POROUS MEDIA

Fluid flow in porous media is explained through fluid flow equations that are used to

describe the flow behavior in a reservoir. These equations can vary depending on the type

of fluid (i.e. oil, water, gas), the type of flow (i.e. linear, radial, spherical, hemispherical

flow), and the flow regime (i.e. unsteady state, pseudosteady-state, stady-state). By

combining the continuity equation, the transport equation or equation for fluid motion

(Darcy’s Law11) with various equations-of-state, the flow equations can be developed.

Darcy’s Law11 is considered to be the fundamental law of fluid motion in porous

media and therefore deserves special attention. Henri-Philibert-Gaspard Darcy11 was a

French Engineer who’s considered to be the first experimental reservoir engineer. In

1856, Darcy published his work in improving the waterworks in Dijon and on the design

of a filter large enough to process the town’s daily water requirements. The mathematical

expression developed states that the apparent velocity of a homogeneous fluid in a porous

medium is directly proportional to the pressure gradient and inversely proportional to the

fluid viscosity. Although Darcy experimented by flowing water through a sand pack and

therefore, fluid properties (i.e. viscosity, density) other than water’s were not analyzed.

The rock properties (i.e. properties of the sand packs) were all combined in a main

constant K .

Darcy’s Law11 applies when the following conditions2 exist:

- Laminar (viscous) flow


5

- Steady-state flow

- Incompressible fluids

- Homogeneous formation.

The original expression of Darcy’s equation11 and a schematic is included below.

Figure 2.1.: Schematic of Darcy’s Experimental Equipment10.

lhK

lhhKu Δ

=−

= 21 ( )1.2

where

=u flow velocity, cm/sec

=Δh difference in manometric levels, cm (water equivalent)

=l total length of the sand pack, cm, and

=K constant.

The constant K in Equation (2.1) above was later found to be the ratio of

permeability k and the fluid viscosity μ . Darcy’s equation11 is now written in the

following form:


6

vkdL

dP μ=− ( )2.2

2.2. HIGH VELOCITY FLOW

Results from the literature review show that the topic of high velocity flow has been

of great interest throughout the years and even today. Every year a small but significant

improvement has been done in this area to fully understand non-Darcy flow. After using

Darcy’s Law11 for almost all the liquid flow problems, for higher rates Forchheimer12

suggested an inertial term in his 1901 publication which would create an additional

pressure drop due to inertial forces acting caused by convective accelerations of the fluid

particles passing through the pore spaces.

2vvkdL

dP βρμ+=− , or in vector form: vvv

kp rrr βρμ

+=∇− ( )3.2

Forchheimer12 later showed that some data sets could not be described by his

quadratic flow equation, and he proposed a cubic term to describe the special cases.

322 vvvkdL

dP γρβρμ++=− ( )4.2

Brinkman8 added another viscous term to Darcy’s equation11 that takes into account

the viscous iterations among fluid particles. Equation (2.5) shows a time-dependent

Brinkman-Forchheimer3 equation.

vvvk

vtvcp a

rrrrr

βρμμρ ++⎟⎠⎞

⎜⎝⎛ Δ−

∂∂

=∇− ( )5.2

The non-Darcy component has been generally used for gas. Ikoku14 mentioned that

the velocity of gas is at least an order of magnitude greater than for oil, and therefore, the


7

high-velocity component is always incorporated in the gas flow equations. Ikoku14 also

attributed the slip effect of Klinkenberg effect as another factor that affects the flow of

gases in porous media and interpreted the slip as the bouncing of the gas molecules on the

wall at low pressures when the mean free path of the molecules becomes the same order

of magnitude as the pore diameter.

Non-Darcy flow can also exist in oil reservoirs i.e. fractured reservoirs. Also, when

correlating the data for high-rate water flow, Forchheimer12 found the relationship for the

inertia component in Darcy’s equation11. Since then, lots of experiments have been

conducted to better understand different aspects of Forchheimer’s equation12 and high-

velocity flow.

There has been a controversy over whether there is turbulence at high velocity and a

several attempts to understand turbulence in porous media. Katz and Lee15 mentioned

that the extra friction represented by the 2v term is the result of shear resistance in the

directions perpendicular to the direction of flow as the cross section of the passage

changes size. Some investigators called the resistance an inertia effect and attributed the

extra shear resistance to two ongoing mechanisms that cause the additional pressure drop:

a longitudinal shear and a transverse shear. The transverse shear is varies with the flow

velocity as the term 2vβρ suggests. It is also mentioned in the discussion that the

separation of the fluid flowing adjacent to the wall may induce secondary flow but not

necessarily turbulence. Turbulence may occur in large vugs, or in matrices with high

permeabilities of 5000md or above.

Scheidegger21 concluded that the non-linearity observed is not primarily due to the


8

onset of turbulence, but due to the emergence of inertia effects in laminar flow due to the

curvature or tortuosity of the flow channels. In an attempt to relate the critical Reynolds

number for turbulence ( )2000Re ≈ for flow in straight tubes to flow in porous media,

Scheidegger21 also concluded that there is not a universal number at which true

turbulence will occur in porous media because different formations may have different

curvature of the flow channels or tortuosity and different porosity. True turbulence may

occur at very high Reynolds numbers and will cause the second change in flow regime

that has been observed experimentally.

Special attention has been paid to differentiate the various flow regimes and how to

determine the exact transition from one type of flow to another. Katz and Lee15 stated that

there is no transition from Darcy flow to the so-called non-Darcy flow. They prefer the

terminology viscous Darcy flow (or Darcy flow) for low flow rate and quadratic Darcy

flow for high-velocity flow. Scheidegger21 recognizes after reviewing correlations that

there appear to be two critical Reynolds numbers at which the flow regime changes,

although these cannot be universally defined. The first change of flow occurs when the

inertia effects in laminar flow become important, and the second occurs when true

turbulence sets in. Huang et al.13 identified six different flow regimes: Pre-Darcy flow,

Darcy Flow, Transition between Darcy and Forchheimer, Forchheimer flow, transition

between Forchheimer and Turbulent flow, and Turbulent Flow.

Several authors5, 6 have also measured the inertia factor experimentally. The inertia

factor is assumed to be constant at all velocities, and it depends on numerous factors.

Values for the inertia factor are available in the literature from correlations6 and specific


9

to types of formation and applications.

Another interesting approach to the problem of non-linear flow is the mathematical

modeling. Belhaj et al.7 developed numerical simulation of both Darcy and Forchheimer

equations to compare the simulation results and also, compare the simulation results for

Forchheimer diffusivity equation7 with experimental results. In this paper, a new non-

linear equation from Aulisa et al.3 that will be referred as Darcy-Forcheimer3 equation

will be solved using finite differences, and the results will be compared with those of

Darcy11 and Forchheimer12 equations.


10

CHAPTER III

METHODOLOGY

3.1. FINITE DIFFERENCES INTRODUCTION

In this project, finite differences approximations to partial derivatives will be used.

This is a very powerful technique, and it is the basis for reservoir simulation. There are

many problems that cannot be solved analytically, including quasilinear and non-linear

partial differential equations (PDEs) and even some linear PDEs. These are some cases

where numerical methods like the finite differences method are used. Partial differential

equations can be Hyperbolic like the wave20 equation ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=∂∂

2

2

2

2

xu

yu , Parabolic like the

heat20 equation ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=∂∂

2

2

xu

yu or Elliptic like Laplace’s20 equation ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∂∂

+∂∂ 02

2

2

2

yu

xu . A

brief review of the mathematical tools that will be needed to understand the finite

differences method will be presented, with special emphasis on the techniques that will

be used in this work.

3.2. TAYLOR SERIES

The approximations are based on Taylor9 series expansions of a function of one or

more variables. The Taylor9 series expansion for a function is given by

( )( ) ( )xf

nhxfhxfhxfhxf n

n1

12

!1...)(

!2)(

!1)()( −

−

−+′′+′+=+ ( )1.3

The reminder is given by


11

( )

!)(

nhf

nn ξ , ),( hxx +∈ξ ( )2.3

When approximating a function of more than one independent variable we have

derivatives replaced by partial derivatives. In the case of two independent variables we

have

),(),(!1

),(),( yxfyxfhyxfkyhxf yx ++=++

...),(!2

),(!2

2),(!2

22

++++ yxfkyxfhkyxfhyyxyxx ( )3.3

The remainder can be written in the form

),(!

1 kyhxfy

kx

hn

n

θθ ++⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂ , 10 ≤≤θ . ( )4.3

In this case, the subscript was used to denote partial differentiation. It will be of

interest to obtain an approximation about the point ),( ji yx and the subscript will be used

to denote the function values at the point, i.e. ),(, jiji yxff = .

The Taylor9 series expansion for 1+if about the point ix is given by

...!3!2

32

1 +′′′+′′+′+=+ iiiii fhfhfhff ( )5.3

The Taylor9 series expansion of 1,1 ++ jif about the point ),( ji yx is given by

...22

)(,

22

11 +⎟⎟⎠

⎞⎜⎜⎝

⎛+++++=++

ji

yyy

xyyxxxx

ijyyxxijji fh

fhhfh

fhfhff ( )6.3

The expansion for jif ,1+ about ),( ji yx is the same as of in the case of a function of

one variable.


12

3.3. FINITE DIFFERENCES

Although an infinite number of difference representations can be found for partial

derivatives of ),( yxf , let us use the following operators:

Forward difference operator jijijix fff ,,1, −=Δ + ( )7.3

Backward difference operator jijijix fff ,1,, −−=∇ ( )8.3

Centered difference operator jijijix fff ,1,1, −+ −=δ ( )9.3

jijijix fff ,2/1,2/1, −+ −=δ ( )10.3

Averaging operator 2)( ,2/1,2/1, jijijix fff −+ +=μ ( )11.3

The centered difference operator can be written as a product of the forward and

backward operator, i.e.:

jixxjix ff ,,2 Δ∇=δ ( )12.3

If we rewrite the right hand side we have:

jijijijijijijijix ffffffff ,1,1,1,,,1,,1 )()( −+−++ −=−−−=−∇ ( )13.3

which agrees with the centered difference operator. This will become important when

trying to approximate ))()(( ′′ xyxp at the point ix to a second order. If we take the forward

difference inside and the backward difference outside or vice versa we get:

⎟⎠⎞

⎜⎝⎛

Δ−

∇ +

xyyp ii

ix1 ( )14.3

If we expand the expression we get:


13

21111

11

1

)()(

xypyppyp

xxyyp

xyyp

iiiiiii

iii

iii

Δ++−

=Δ

Δ−

−Δ−

−−−+

−−

+

( )15.3

Note that if 1)( ≡xp then we get the well known centered difference:

211

)(2)(

xyyyxy iii

Δ+−

=′′ −+ ( )16.3

3.4. TERMINOLOGY

There are important terminology that one must be familiar with to understand finite

differences methods, and how they work.

Truncation Error ( )..ET – It is defined as the difference between the partial derivative

and its finite differences representation. If xΔ is the spacing in the x direction, the term

( )xO Δ means that the truncation error can be expressed as xET Δ≤ κ.. for 0→Δx ,

where κ is a positive real constant. The order of a method is defined as the lowest power

of the mesh size in the truncation error.

Consistency – A difference equation is said to be consistent or compatible with the

partial differential equation when it approaches the difference equation as the mesh sizes

approaches zero: 0.. →ET as mesh size 0→ .

Stability – A numerical scheme is called stable if the errors from any source are not

allowed to grow as the calculation proceeds. There are two methods for checking for

stability of linear differential equations. One is the Fourier or von Neumann19 method

which assumes periodic boundary conditions and will be applied to some of our PDEs to

check for stability and find the stability condition. The second is the matrix method

which takes into account the error from the boundary.


14

Convergence – A scheme is said to be convergent if the solution to the finite

differences equation approaches the exact solution to the PDE with the same initial and

boundary conditions as the mesh sizes approaches zero. It has been proven under the Lax

Equivalence Theorem9 that under appropriate conditions and for linear PDEs, consistent

scheme is convergent if and only if it is stable.

Accuracy – The accuracy of a numerical method will depend on the accuracy of the

input data, the size of the time and space discretization and the scheme used to solve the

equations.

Modified Equation – An equation derived to help analyze the numerical effects of the

discretization. A modified equation can be obtained by starting with the truncation error

and replacing the time derivatives by spatial differentiation using the equation from the

truncation error.

3.5. INITIAL AND BOUNDARY CONDITIONS

An appropriate number of additional conditions should be given for PDEs to be

solved completely. These conditions are referred as initial or boundary conditions. Since

in this project we will be dealing with time-dependent problems, this initial condition will

often refer to the conditions at an initial or reference time ( )0=t . The number of

conditions will depend on the orders of the time derivatives involved. The order of a PDE

is the highest order of the partial derivative in the equation. A first order PDE will have a

first partial derivative as the highest partial, and a second order PDE will have a second

partial derivative as the highest partial. The initial or boundary conditions are conditions


15

that the solution must satisfy. The boundary conditions of a PDE with two independent

variables are x and y , and an unknown dependent variable u can be written in the

following general form:

a γ=∂∂

+nubu ( )17.3

where nu∂∂ is a normal derivative that can be expressed as a partial derivative with respect

to x or y . Depending on the values of different parts of the above general equation we

get specific conditions:

• When 0=γ , we have an homogeneous condition

1. When 0=b , we have a Dirichlet20 boundary condition.

2. When 0=a , we have a Neumann20 boundary condition.

3. When 0≠a and 0≠b , we have a Robbin20 boundary condition.

3.6. DISCRETIZATION METHODS

The first step of the process of applying finite differences is the domain discretization.

We need to determine the mesh or grid, how it will be partitioned, whether a regular or

irregular grid will be used and also, if the mesh size will be constant or variable and

where will the nodes be located. In this work, we will use a regular mesh and constant

mesh size. Once the appropriate mesh or grid has been selected, a variety of discretization

methods can be used.


16

3.6.1. Explicit Method

The explicit method is also known as the Forward in Time Central in Space19 (FTCS)

scheme. The basis of the scheme is to replace the time derivative on a given PDE by the

forward differencing scheme and the space derivative on a given PDE by the central

differencing scheme:

tpp

tp jiji

Δ

−≈

∂∂ + ,1, ,

xpp

xp jiji

Δ

−≈

∂∂ −+

2,1,1 and

( )2,1,,1

2

2 2

x

pppx

p jijiji

Δ

+−≈

∂∂ −+ ( )18.3

where ),(, jiji txpp ≈ . This method is very easy to implement, but it is known for being

conditionally stable, which means that the time stepsize and mesh size should be chosen

carefully to avoid unstable results. The FTCS method requires three known values at a

certain time, to be able to approximate one value at the next time step.

Figure 3.1: Explicit Method

If the exact amplification factor is obtained as the quotient:

( )( )

2

,, λβ−=

Δ+= e

txpttxpAmpExact ( )19.3

we can plot the exact amplification factors for different values of β , comparing exact

values and explicit method approximations for different values of λ , also known as the

i , j+1

i+1 , ji , ji-1 , j


17

CFL (Courant, Friedrich, and Lewy)19 condition, to get Figure 3.2.

Figure 3.2: Amplification Factor for Explicit Method

3.6.2. Implicit Method

The implicit method is also known as the Backward in Time Central in Space19

(BTCS) scheme, and it is known for being unconditionally stable. Although it has this

great advantage, the drawback is that a tridiagonal system must be solved for each time

step.

tpp

tp jiji

Δ

−≈

∂∂ + ,1, ,

xpp

xp jiji

Δ

−≈

∂∂ +−++

21,11,1 and

( )21,11,1,1

2

2 2

x

pppx

p jijiji

Δ

+−≈

∂∂ +−+++ ( )20.3


18

Figure 3.3: Implicit Method

3.6.3. Crank-Nicholson Method

This method is also considered implicit and therefore unconditionally stable. It

combines both the explicit and implicit methods by taking a centered difference in time at

a point ( )2

1, +ji tx and the average of the centered differences at time jt and 1+jt .

Figure 3.4: Crank-Nicholson9 Method

3.6.4. Other Methods

There are numerous methods that can be used to approximate the solution of PDEs,

but all the methods have their advantages and disadvantages. Here we will mention a few

more methods found in the literature. For a parabolic equation like Darcy’s diffusivity

equation the implicit, explicit and Crank-Nicholson9 methods can be written in one single

i-1 , j+1 i , j+1

i , j

i+1 , j+1

i -1 , j+1 i , j+1 i-1 , j+1

i+1 , ji , ji -1 , j


19

expression as follows:

( ) ( )( )( )2

,1,,11,11,1,1,1, 212

x

ppppppt

pp jijijijijijijiji

Δ

+−−++−=

Δ

− −++−++++ σση ( )21.3

The weighing parameter σ will determine the implicity of the method. If 1=σ , the

method is implicit; if 0=σ , the method is explicit; and if 5.0=σ , we have Crank-

Nicholson.

Other methods include the DuFort Frankel9 method which gives an unconditionally

unstable method after using the centered difference in time and space. With the use of a

modified equation, the end result was an unconditionally stable method. Just to mention a

few, there is the Barakat-Clark4 method and the Lax-Wendroff 9 scheme among others.

The amplification factors for different schemes are plotted in Figure 3.5 below to

show how close the different methods are to the exact amplification factor.


20

Figure 3.5: Amplification Factor for Various Methods


21

CHAPTER IV

DERIVATION

4.1. DARCY’S EQUATION

In this section, Darcy’s equation11 will be analyzed by deriving both the radial

diffusivity equation and the diffusivity equation in one dimension. Also, an analytical

solution and a steady state solution will be derived for given boundary and initial

conditions. The finite differences technique will be applied by selecting an explicit and

implicit method. The conditions for stability will be checked and a convergence analysis

will be performed.

4.1.1. Darcy’s Radial Diffusivity Equation

The radial diffusivity1 equation is well known and used in petroleum engineering to

simulate the flow of fluids into the wellbore. The differential equation will later be

derived in Cartesian form which is more commonly used in numerical reservoir

simulation.

Some of the simplifying assumptions in developing the radial diffusivity equation

include the fact that the reservoir is considered homogeneous and that the producing well

is completed across the entire height of the formation to ensure fully radial flow.

If we consider the flow through a control volume of thickness dr located at a

distance r from the center of the radial cell, we can apply the principle of conservation of

mass: mass flow rate IN – mass flow rate OUT = rate of change of mass in control

volume. This translates to the following expression:


22

[ ] [ ] [ ]tttrdrr dVAvtAvt )()( φρφρρρ −=Δ−Δ Δ++ ( )1.4

If we let hdrrA drr )(2 +=+ π , hrAr )(2π= , hrV 2π= and drrhdV )2( π= gives:

[ ]tttrdrr drrhvthrvtdrrh )()()2()(2)()(2 φρφρπρπρπ −=Δ−Δ+ Δ++ ( )2.4

Dividing the above expression by drrh)2( π and simplifying we get:

[ ] [ ]tttrdrr tvrvdrr

rdr)()(1)())((1 φρφρρρ −

Δ=−+ Δ++ ( )3.4

Or in term of partial derivatives:

[ ] )()(1 φρρt

vrrr ∂

∂=

∂∂ ( )4.4

Darcy’s Law11 is needed to relate the fluid velocity to the pressure gradient:

rpkv∂∂

=μ

( )5.4

Combining Darcy’s equation11 with the continuity equation we get the general

differential equation that can describe the flow of any fluid flowing in the radial

direction:

)()(1 φρμ

ρtr

pkrrr ∂

∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ ( )6.4

The right hand side of the above equation can be expanded by using the chain rule:

ttrpkr

rr ∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ ρφφρ

μρ)(1 ( )7.4

If we consider the reservoir to be homogeneous in all rock properties and the fluid to

be slightly compressible and for the fluid properties to be constant over pressure, time

and distance, we can make the simplifying assumption that 0≈∂∂

tφ which leads to the


23

expression:

trpkr

rr ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ ρφ

μρ)(1 ( )8.4

And if we recall that p

c∂∂

=ρ

ρ1 , and

tp

pt ∂∂

∂∂

=∂∂ ρρ the right hand side becomes:

tpc

rpkr

rr ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ ρφ

μρ)(1 ( )9.4

If we take derivatives on the left hand side and simplify, the resulting expression is:

tpc

rp

rp

rrp

rk

∂∂

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ρφρρρ

μ 2

2

( )10.4

Dividing both sides by ρ , using the expressions rp

pr ∂∂

∂∂

=∂∂ ρρ and

pc

∂∂

=ρ

ρ1 we get:

tpc

rp

rpc

rp

rk

∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ φμ 2

221 ( )11.4

The term 2

⎟⎠⎞

⎜⎝⎛∂∂

rpc is considered very small. If the term is ignored the equation can be

rewritten as:

tp

kc

rp

rrp

∂∂

=∂∂

+∂∂ φμ1

2

2

( )12.4

If the diffusivity constant is defined as c

kφμ

η = then the diffusivity equation takes the

form:

tp

rp

rrp

∂∂

=∂∂

+∂∂

η11

2

2

( )13.4


24

This equation can help us to determine the pressure as a function of time t and radial

distance r .

4.1.2. Darcy’s Diffusivity Equation in One Dimension

Combining the conservation of mass, the equation of state and Darcy’s equation11 for

linear flow (transport equation), the diffusivity equation in one dimension can be derived

for a single-phase fluid flowing in the x -direction. Darcy’s diffusivity equation are based

on important assumptions: horizontal and laminar flow, negligible gravity effect, constant

compressibility fluid, single phase fluid, isothermal conditions, homogeneous formation

and constant permeability in all directions or isotropic.

We start with the conservation of mass: Mass rate in – Mass rate out = Mass rate

remaining. This translates in the following equation:

tzyxzyvzyv ttt

xxxxxx Δ−

ΔΔΔ=ΔΔ−ΔΔ Δ+Δ+Δ+

ρρφρρ )()()( ( )14.4

Dividing both sides by zyx ΔΔΔ to get:

txvv tttxxxxxx

Δ−

=Δ

−− Δ+Δ+Δ+ ρρφρρ )()( ( )15.4

After taking limits on both sides as 0→Δx and 0→Δt we get the continuity

equation in one dimension for linear flow:

txv

∂∂

−=∂

∂ ρφρ)( ( )16.4

The left hand side can be expanded using the chain rule:

txv

xv

∂∂

−=∂∂

+∂∂ ρφρρ ( )17.4


25

We now use Darcy’s equation11 for linear flow:

xpk

Aqv

∂∂

−==μ

( )18.4

After substituting Equation (4.18) into Equation (4.16) the result is:

txxpk

xpk

x ∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ ρφρ

μμρ ( )19.4

For constant compressibility, we get the following expression for density:

)(0

0ppce −= ρρ ( )20.4

If we introduce this expression into the continuity equation, it becomes

( ) ( ) ( ))(0

)(0

(0

000 ppcppcppc et

exx

pkxpk

xe −−−

∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂ ρφρ

μμρ ( )21.4

After taking partial derivatives with respect to x and t respectively this becomes:

( ) ( ) ( )tpce

xpce

xpk

xpke ppcppcppc

∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂ −−− )(

0)(

02

2(

0000 ρφρ

μμρ ( )22.4

After simplifying we get:

tp

kc

xpc

xp

∂∂

=⎟⎠⎞

⎜⎝⎛∂∂

+∂∂ μφ2

2

2

( )23.4

If we assume that the product 2

⎟⎠⎞

⎜⎝⎛∂∂

xpc is small, it can be neglected and therefore, this

results in Darcy’s diffusivity equation in one dimension:

tp

kc

xp

∂∂

=∂∂ μφ

2

2

( )24.4

Most of this work is based on a linear flow model where the flow paths are

parallel, the fluid flows in one direction, and the cross-sectional area is assumed to be


26

constant. Linear flow can occur when fluid flows into a hydraulic fracture. Figure 4.1

below shows a linear flow model.

Figure 4.1: Linear Flow Model

4.1.3. Analytical Solution

The analytical solution for Darcy’s diffusivity equation can be found by using the

separation of variables technique. The solution will be found with the given initial and

boundary conditions.

Darcy’s PDE:

tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )25.4

Initial condition:

)()0,( xxp φ= ( )26.4

Inner boundary condition:

0),0( 0 == ptp , 0>t ( )27.4

Outer boundary condition:

Flow

A

bp ap

h

W

L


27

0),1( 1 == ptp , 0>t ( )28.4

We rearrange and let c

kKφμ

= to get:

2

2

xpK

tp

∂∂

=∂∂ ( )29.4

4.1.3.1. Finding Solution to the PDE

Let:

)()(),( tTxXtxp = ( )30.4

And so we get:

)()(

)()(

2

2

tTxXxp

tTxXtp

′′=∂∂

=∂∂ &

( )31.4

Darcy’s equation11 can be rewritten as:

)()()()( tTxXKtTxX ′′=& ( )32.4

We can now rearrange the terms to get:

2

)()(

)()( λ−=

′′=

xXxX

tKTtT& ( )33.4

Each of the equalities in Equation (4.33) above can be analyzed and solved

separately:

2

)()( λ−=tKT

tT& ( )34.4

If we let dtdTT =& , then we get:


28

KTdtdT 2λ−= ( )35.4

Take derivatives on both sides:

∫∫ −= KdtTdT 2λ ( )36.4

After solving we get:

cKtT +−= 2ln λ ( )37.4

This solution can also be expressed as:

KteAtT2

1)( λ−= ( )38.4

The other expression from Equation (4.33) can also be solved as follows:

2

)()( λ−=

′′xXxX ( )39.4

Rearranging terms:

0)()( 2 =+′′ xXxX λ ( )40.4

This can also be rewritten as:

022

2

=+∂∂ X

xX λ or 02 =+′′ yy λ ( )41.4

And if we suppose the solution rxey = , then:

rx

rx

eryrey

2=′′=′

( )42.4

And get the following:


29

irrrr

eer rxrx

λλ

λλλ

±=−±=

−==+

=+

2

22

22

22

00

( )43.4

This gives:

)sin()cos()( 22 xBxAxXy λλ +== ( )44.4

Combining both Equations (4.38) and (4.44) we then have:

[ ])sin()cos(),( 2212

xBxAeAtxp tK λλλ += − ( )45.4

The constants can be recombined to get:

[ ])sin()cos(),( 33

2

xBxAetxp tK λλλ += − ( )46.4

4.1.3.2. Finding Solution to the PDE and Boundary Conditions

If we apply the first boundary condition 0),0( =tp we get:

[ ] 0)0sin()0cos(),0( 33

2

=⋅+⋅= − λλλ BAetp tK ( )47.4

To get 03 =A

If we apply the second boundary condition 0),1( =tp we get:

[ ] 0)sin()cos(),1( 33

2

=+= − λλλ BAetp tK ( )48.4

where 0)sin(3 =λB and 0)sin( =λ . ( 03 =B is not of interest)

...3,2,1

0)sin(

=±==

nnπλ

λ ( )49.4

The nth solution can be expressed as follows:


30

)sin(),(2

xeBtxp ntK

nnn λλ−= ,...3,2,1=n ( )50.4

Each of the family of solutions satisfies the PDE and the boundary conditions. We

can now write the general solution using the Principle of Superposition as:

∑∞

=

−=1

)sin(),(2

n

tKn xneBtxp n πλ ( )50.4

4.1.3.3. Finding Solution to the PDE, Boundary and Initial Conditions

We must now also satisfy the initial condition )()0,( xxp φ= which yields:

∑∞

=

=1

)sin()(n

n xnBx πφ ( )51.4

The coefficients nB can be determined from the orthogonality property of

{ },...3,2,1),sin( =nxnπ to yield:

∫=1

0)sin()(2 dxxnxBn πφ ( )52.4

This combination of equations gives us a series of solutions.

4.1.4. Steady State Solution

There are different flow regimes16 when considering fluid flow in porous media.

Transient flow occurs when a well is initially placed on production and will create a

pressure response throughout the drainage area of the well. Late transient flow occurs

when the pressure response from the producing well reaches a boundary that can be

physical such as a fault or a no-flow boundary generated by the effect of flow reversal

from surrounding producers. Pseudosteady state flow occurs when the pressures at all


31

points are changing at the same rate as a linear function of time. Steady state flow exists

when the pressure drop throughout the system is constant. At the steady state condition,

the pressure does not change with time. This is mathematically expressed as 0=∂∂

tp .

If we start from Darcy’s PDE:

2

2

xp

kc

tp

∂∂

=∂∂ φμ ( )53.4


0),0( 0 == ptp , 0>t ( )54.4

Outer boundary condition

1),1( 1 == ptp , 0>t ( )55.4

For steady state conditions Equation (4.53) becomes:

02

2

=∂∂

xp ( )56.4

And we let:

zxp=

∂∂ ( )57.4

After rearranging and taking the integral both sides:

xzp ∂=∂ ∫∫ ( )58.4

Solving the integral gives us the following function:

czxxp +=)( ( )59.4

If we apply the boundary conditions where 0)0( =p and 1)1( =p we can find the

values for z and c that satisfy the above conditions.


32

0)0()0( =+= czp , 0=c ( )60.4

10)1()1( =+= zp , 1=z ( )61.4

This results in an equation of a unit slope line:

xxp =)( , or xp = ( )62.4

4.1.5. Finite Differences Explicit Discretization

The FTCS discretization is an explicit method, and it is conditionally stable. The

condition for stability will be derived. By using the FTCS discretization we would like to

find a function for pressure ),( txp that will approximate Darcy’s partial differential

equation and also satisfy two boundary conditions (BC) and one initial condition (IC) in

the interval ),( bax∈ .

4.1.5.1. Constant Pressure Boundaries (CPB)

To develop the finite differences approximation a mathematically simplistic scenario

for the boundary conditions will be used. We assume that the pressure is constant at both

boundaries. This means that a well could be producing at constant bottomhole pressure

(BHP) and that the pressure at the extent of the reservoir is kept constant as in the case of

a reservoir pressure maintenance technique or waterflooding. Mathematically, these

conditions can be expressed in the interval ),( bax∈ as follows:

Darcy’s diffusivity equation:

tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )63.4


33

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )64.4


aptap =),( , 0>t ( )65.4


bptbp =),( , 0>t ( )66.4

We first rewrite Equation (4.63) and let c

kKφμ

= to get:

2

2

xpK

tp

∂∂

=∂∂ ( )67.4

The above expression is similar to that of the heat diffusion equation

( xxt uu 2α= where ρ

αck

=2 is the diffusivity constant and tu is a first partial derivative in

time and xxu is a second partial derivative in space) and therefore can be treated similarly.

We first partition the domain [ ]ba, in N equal subintervals in “space” )(x ,

xixi Δ= , Ni ,...,1,0= where ( )N

abx −=Δ ( )68.4

The time interval [ ]Tf,0 is also partitioned let J intervals in “time” )(t :

tjt j Δ= , Jj ,...,1,0= where J

Tft =Δ ( )69.4

A two dimensional grid is included in Figure 4.2 below. The coordinates are

partitioned in uniform grid sizes and time step. The red circle is a point in the grid with

coordinates ( )ji tx , .


34

Figure 4.2: Grid with N Intervals in x-direction and J Intervals in Time

The grey and black circles represent the known initial and boundary conditions

respectively.

We set:

jiji ttxxttxx xpK

tp

==== ∂∂

=∂∂

,2

2

,

,...2,1,0

1,...3,2,1=

−=j

Ni ( )70.4

If we replace the time derivative above by the forward differencing scheme and the

space derivative above by the central differencing scheme and letting ),(, jiji txpp ≈ , we

get:

tΔ

xΔ axx == 0 bxx N ==

0t

Tf

( )ji tx ,


35

( ) ⎥⎦

⎤⎢⎣

⎡Δ

+−=

Δ− +−+

2,1,,1,1, 2

xppp

Kt

pp jijijijiji ( )71.4

After rearranging terms we get:

( )[ ] jijijijiji pppp

xtKp ,,1,,121, 2 ++−

ΔΔ

= +−+ ( )72.4

If we let ( )2x

tKΔΔ

=λ , known as the CFL (Courant, Friedrichs, and Lewy)19 condition

and rearrange terms, we now get the following expression:

( ) ( )jijijiji pppp ,1,1,1, 21 −++ ++−= λλ ( )73.4

where JJj

Ni,1...2,1,01...3,2,1

−=−=

We now evaluate for different values of i and letting 0=j .

12

21

−=−=

==

NiNi

ii

M

( ) ( )( ) ( )

( ) ( )( ) ( )0,20,0,11,1

0,30,10,21,2

0,10,30,21,2

0,00,20,11,1

2121

2121

−−−

−−−−

++−=++−=

++−=++−=

NNNN

NNNN

pppppppp

pppppppp

λλλλ

λλλλ

M ( )74.4

From the boundary conditions we know that app =0,0 and bN pp =0, .

A matrix can be built as follows:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

b

N

a

N

N

pp

ppp

pp

ppp

0,1

0,2

0,1

1,

1,1

1,2

1,1

1,0

10000021000

0021000021000001

MMOOOMMM

λλλ

λλλλλλ

( )75.4

The above matrix can be written in vector form:


36

)0()1( PBPrr

= ( )76.4

With the help of a computer program, in this case MATLAB, the linear algebra

problem can be analyzed by using an iterative process to generate a sequence of solutions

at different times. As a result, a pressure profile can be generated and for some arbitrary

values for our fluid and rock properties, we get the pressure profile in Figure 4.3.

In this approach, it is assumed that the reservoir is originally at some high initial

constant pressure 1=ip and that the pressure at the wellbore is some low pressure

0=wfp . With time, the reservoir will lose pressure due to drawdown until it reaches

steady state. To get a smooth pressure profile, a function that satisfies the boundary

condition was found and used for initial condition which becomes:

( )( )1000arctan1000arctan)0,( xxp = for all [ ]bax ,∈ ( )77.4

Note that for 0== ax we get 0)0,0( =p and for 1== bx we get 1)0,1( =p . It can

be also shown that no matter what initial conditions used, the pressure will always

converge to steady state.


37

Figure 4.3: Pressure Profile for Darcy (CPB) Explicit ( )5.0=λ

4.1.5.2. Boundary Dominated Regime (BDR)

For this case, we will assume constant bottomhole pressure during production and no

flow across the reservoir’s outer boundary at a distance bxx N == . These conditions can

be expressed mathematically as follows:


tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )78.4

Initial condition:


38

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )79.4


aptap =),( , 0>t ( )80.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )81.4

To satisfy the outer boundary condition we must have: ( ) 02

,1,1 =Δ

− −+

xpp jNjN and

therefore jNjN pp ,1,1 −+ = . We note that jNp ,1+ is located at a node outside the interval

[ ]ba, . To calculate the pressure gradient at the boundary we will assume that the pressure

outside the interval is symmetric to the no-flow boundary and therefore jNjN pp ,1,1 −+ = .

This is commonly used when evaluating reservoirs with no-flow boundaries such as

sealing faults and zones of flow reversal. The pressure drop due to the no-flow boundary

is calculated through the superposition principle by placing an image well identical to the

actual well, on the other side of the no-flow boundary at the same distance and symmetric

with respect to the boundary.

We now incorporate the new conditions to our finite differences approach starting

from the general differences expression for Darcy’s diffusivity equation:

( ) ( )jijijiji pppp ,1,1,1, 21 −++ ++−= λλ ( )82.4

where JJj

Ni,1...2,1,0

...3,2,1−=

=



39

NiNi

ii

=−=

==

1

21

M

( ) ( )( ) ( )

( ) ( )( ) ( )0,10,10,1,

0,20,0,11,1

0,10,30,21,2

0,00,20,11,1

2121

2121

−+

−−−

++−=++−=

++−=++−=

NNNN

NNNN

pppppppp

pppppppp

λλλλ

λλλλ

M ( )83.4

Because jNjN pp ,1,1 −+ = , we now get for Ni = :

( ) ( )0,10,1, 221 −+−= NNN ppp λλ ( )84.4

A new matrix can be built:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

0,

0,1

0,2

0,1

1,

1,1

1,2

1,1

1,0

212000021000

0021000021000001

N

N

a

N

N

pp

ppp

pp

ppp

MMOOOMMM

λλλλλ

λλλλλλ

( )85.4


)0()1( PCPrr

= ( )86.4

Similarly, the pressure profile for the boundary dominated period can be generated

(see Figure 4.4) with MATLAB or other suitable computer program. The fluid and rock

properties values used are the same arbitrary values used to generate Figure 4.3. The

initial condition is also the same; the only differences are in the outer boundary condition

and the fact that this simulation run was let run four times longer.


40

Figure 4.4: Pressure Profile for Darcy (BDP) Explicit ( )5.0=λ

4.1.5.3. Pseudosteady State Flow Regime (PSS)

We will now assume constant rate production and no flow across the reservoir’s outer

boundary at a distance bx = . For a one dimensional scenario we let qv −= . These

conditions can be expressed mathematically as follows:


tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )87.4

Initial condition:


41

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )88.4


jttxxxpkq

==⎥⎦⎤

⎢⎣⎡∂∂

=,0

μ, 0>t ( )89.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )90.4

To satisfy the inner boundary condition we must calculate the pressure gradient at the

inner boundary ( axx == 0 ):

( )xpp

xp jj

j Δ

−=⎥⎦

⎤⎢⎣⎡∂∂ −

2,1,1

,0

( )91.4

If we substitute the pressure gradient in the boundary condition equation we get:

( ) ⎥⎦

⎤⎢⎣

⎡Δ

−= −

xppkq jj

2,1,1

μ ( )92.4

We notice that the pressures jp ,1− are located outside the boundary [ ]ba, . We can

rearrange the above equation to get an expression for jp ,1− :

( )xkqpp jj Δ−=−μ2

,1,1 ( )93.4

For the outer boundary condition we again take jNjN pp ,1,1 −+ = .We now incorporate

the conditions to our finite differences approach starting from the general differences

expression for Darcy’s Law11:


42

( ) ( )jijijiji pppp ,1,1,1, 21 −++ ++−= λλ ( )94.4

where JJj

Ni,1...2,1,0

...3,2,1,0−=

=


NiNi

ii

=−=

==

1

10

M

( ) ( )( ) ( )

( ) ( )( ) ( )0,10,10,1,

0,20,0,11,1

0,00,20,11,1

0,10,10,01,0

2121

2121

−+

−−−

−

++−=++−=

++−=++−=

NNNN

NNNN

pppppppp

pppppppp

λλλλ

λλλλ

M ( )95.4

For 0=i we get:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ−++−= x

kqpppp μλλ 221 0,10,10,01,0

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ−+−= x

kqppp μλλ 2221 0,10,01,0 ( )96.4

For Ni = we get:

( ) ( )0,10,1, 221 −+−= NNN ppp λλ ( )97.4


( )

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡ Δ−

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

00000

2

212000021000

00210000210000221

0,

0,1

0,2

0,1

0,0

1,

1,1

1,2

1,1

1,0 xkq

pp

ppp

pp

ppp

N

N

N

N

μλ

λλλλλ

λλλλλλ

λλ

MMOOOMMM


EPDPrrr

+= )0()1( ( )98.4


43

The pressure profile for the pseudosteady state regime can be generated with a similar

computer code. The fluid and rock properties values used are the same arbitrary values

from Figures 4.3 and 4.4. The initial conditions are now different to honor the assumption

that at initial conditions the reservoir is said to have a constant pressure. And so, we write

our initial condition as:

100)0,( =xp for all [ ]bax ,∈ ( )99.4

Note that we arbitrarily chose the initial value of 100 to avoid the simulated pressure

values to drop to a point that it shows negative values. It is also worth mention that with

the previous cases (CPB and BDP) an initial constant pressure throughout the reservoir

could not be fully applied because a pressure difference was needed to initiate the finite

differences approximation. If all the pressure values where the same initially, the method

would always get the same pressure as a result and therefore, the final result would be

exactly the same as the initial values of pressure at any time.

Other differences include the fact that the pressure is no longer constant at the

boundaries and that a much longer time was allowed. Figure 4.5 shows the pressure

profile for the PSS case.


44

Figure 4.5: Pressure Profile for Darcy (PSS) Explicit ( )5.0=λ

4.1.6. Stability Analysis for the Explicit Scheme

The Fourier or Von Neumann19 method will be applied to the explicit scheme. Let us

suppose that the solution of the difference equations is of the form:

tjxIi eetxp ΔΔ= λβ),( ( )100.4

where 1−=I and we examine the behavior of this solution as ∞→t or ∞→j for a

suitable choice ofλ . We notice that if 1>Δtje λ , the solution becomes unbounded and so

we want to study 1≤Δtje λ .


45

We now consider the explicit difference equation of the PDE which is:

( ) ( )jijijiji pppp ,1,1,1, 21 −++ ++−= λλ ( )101.4

After substituting we get:

( ) ( ) ( ) )()21( 111 tjxiItjxiItjxIitjxIi eeeeeeee ΔΔ−ΔΔ+ΔΔΔ+Δ ++−= λβλβλβλβ λλ ( )102.4

Dividing both sides by tjxIi ee ΔΔ λβ gives:

))1(())1(())1(( )21( iixIiixIjjt eee −−Δ−+Δ−+Δ ++−= ββλ λλλ ( )103.4

After simplifying:

xIxIt eee Δ−ΔΔ ++−= ββλ λλλ)21( ( )104.4

The rest of the derivation goes as follows:

( ) ( )( ) ( )( ) ( )

( )⎟⎠⎞

⎜⎝⎛ Δ

−=

Δ−−=−+Δ=

−+Δ+Δ+Δ−Δ=−++=

Δ

Δ

Δ

Δ

ΔΔ−Δ

2sin41

cos12121cos2

21sincossincos21

2 xe

xexe

xIxxIxeeee

t

t

t

t

xIxIt

βλ

βλλβλ

λββββλλλ

λ

λ

λ

λ

ββλ

( )105.4

The term te Δλ is the amplification factor. Obviously, 1≤Δteλ since 02

sin4 2 ≥⎟⎠⎞

⎜⎝⎛ Δxβλ

for real β . So we now look to satisfy:

1−≥Δtje λ ( )106.4

Therefore we have:


46

21

2sin

22

sin4

12

sin41

2

2

2

≤⎟⎠⎞

⎜⎝⎛ Δ

−≥⎟⎠⎞

⎜⎝⎛ Δ

−

−≥⎟⎠⎞

⎜⎝⎛ Δ

−

x

x

x

βλ

βλ

βλ

( )107.4

Since this inequality is true for all values of xΔ , it should be true for ⎟⎠⎞

⎜⎝⎛ Δ

2sin2 xβ

close to 1. Then we must have 21

≤λ for stability.

Figures 4.6, 4.7 and 4.8 below illustrate the pressure profiles when the stability

requirement is not met. Note that in some cases even if the CFL condition is exceeded by

a small value the result may become unstable.


47

Figure 4.6: Unstable Pressure Profile for Darcy (CPB) Explicit ( )51.0=λ


48

Figure 4.7: Unstable Pressure Profile for Darcy (BDR) Explicit ( )5025.0=λ


49

Figure 4.8: Unstable Pressure Profile for Darcy (PSS) Explicit ( )5005.0=λ

4.1.7. Convergence Analysis for the Explicit Method

Consider the finite differences representation of Darcy’s PDE:

( ) ( )jijijiji pppp ,1,1,1, 21 −++ ++−= λλ ( )108.4

The exact solution can be written as follows:

( ) ( ) ttxptxptxptxp jijijiji Δ+++−= −++ τλλ ),(),(),(21),( 111 ( )109.4

where tΔτ is an error term. The error at any time can be found to be:

),(: 11,1, +++ −= jijiji txppe ( )110.4

( )( ) ( ) ( )( ) ttxpptxpptxppe jijijijijijiji Δ−−+−+−−= −−+++ τλλ ),(),(),(21 1,11,1,1,


50

The maximum error can be expressed as follows:

( ) ( ) teeee jijijiijiiΔ−++−= −++ τλλ ,1,1,1, 21maxmax ( )111.4

If we let jiij eE ,max= we now get:

( ) teeeE jiijiiijiij Δ+++−≤ −++ τλλ ,1,,1 maxmaxmax21 ( )112.4

( ) tEEE jjj Δ++−≤+ τλλ 2211 ( )113.4

We now use the result from the derived stability condition 21

≤λ to show that,

02112≥−≤λλ

( )114.4

This allows us to continue the derivation as follows:

( ) ( ) tEEE jjj Δ++−≤+ τλλ 2211 ( )115.4


tEE jj Δ+≤+ τ1 ( )116.4

We can now use that expression to develop the error at the previous time step:

tEE jj Δ+≤ − τ1 ( )117.4

The previous two equations may be recombined to give:

tEE jj Δ+≤ −+ τ211 ( )118.4

If 0E is the maximum error at a known initial time ( )0=t we get:

ttEE j Δ++≤+ τ)1(01 ( )119.4

If the value at a specific time is known, there is no error at that point so 00 =E :


51

ttE j Δ≤ τ ( )120.4

The term τt can be a final time Tf to give:

tTfE j Δ≤ ( )121.4

This shows that the error decreases as the time step decreases. We will now illustrate

convergence of the computer code developed by applying different time steps to the

explicit method for Darcy’s diffusivity equation.

2

2

xpK

tp

∂∂

=∂∂ ( )122.4

Recall that the finite differences approximation for Darcy’s diffusivity equation had

the following form:

( ) ⎥⎦

⎤⎢⎣

⎡Δ

+−=

Δ− +−+

2,1,,1,1, 2

xppp

Kt


We will first analyze the behavior of Darcy’s diffusivity equation by applying Taylor9

expansions to both the first order derivative in time and second order derivative in space.

For convenience, we will refer to the first derivative in time as tp and the second

derivative in space as xxp . We can then rewrite Darcy’s diffusivity equation in the

following form:

xxt Kpp = ( )124.4

The partial differences equation can also be expressed in the in the following form:

( ) ⎥⎦

⎤⎢⎣

⎡

ΔΔ++−Δ−

=Δ

−Δ+2

),(),(2),(),(),(x

txxptxptxxpKt

txpttxp ( )125.4

The Taylor9 expansions for the pressure at the next time step ),( ttxp Δ+ and the


52

current time step ),( txp can be generated as follows:

( ) ( ) ( )L+

=

Δ+

Δ+

Δ+Δ+=Δ+

),(),(2462

),(),(432

txptxp

tptptptptxpttxp tttttttttt ( )126.4

Using the above expansions, we can get an expression for the first derivative in

time:

( ) ( )L+

Δ+

Δ+

Δ+=

Δ−Δ+

2462),(),( 32 tptptpp

ttxpttxp

tttttttttt ( )127.4

The Taylor9 expansions for the pressure at the next node ),( txxp Δ+ , the current

node ),( txp , and the previous node ),( txxp Δ− can be written as follows:

( ) ( ) ( )

( ) ( ) ( )L

L

+Δ

+Δ

+Δ

+Δ+=Δ+

=

±Δ

+Δ

−Δ

+Δ−=Δ−

2462),(),(

),(),(2462

),(),(

432

432

xpxpxpxptxptxxp

txptxp

xpxpxpxptxptxxp

xxxxxxxxxx

xxxxxxxxxx

( )128.4

The second derivative in space can be approximated using the above developed

Taylor9 expansions to obtain the following:

( )( )

L+Δ

+=Δ

Δ−+−Δ+12

),(),(2),( 2

2

xppx

txxptxptxxpxxxxxx ( )129.4

The first derivative in time and the second derivative in space can be rewritten as:

( ) ( )L−

Δ−

Δ−

Δ−

Δ−Δ+

=2462

),(),( 32 tptptpt

txpttxpp tttttttttt ( )130.4

( )( )

L−Δ

−Δ

Δ−+−Δ+=

12),(),(2),( 2

2

xpx

txxptxptxxpp xxxxxx ( )131.4

Therefore, the truncation error becomes:


53

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡+

Δ−−⎥

⎦

⎤⎢⎣

⎡−

Δ−

Δ−

Δ−=− LL

122462

232 xpKtptptpKpp xxxxtttttttttxxt ( )132.4

If we choose only the expressions with the lowest powers of xΔ and tΔ we get:

( )L±

Δ+

Δ−=− xxxxttxxt KpxtpKpp

122

2

( )133.4

The modified equation can be found by eliminating the time derivatives and replacing

them by spatial differentiation:

xxt Kpp =

( )txxtt Kpp =

( )xxttt pKp =

( )xxxxtt KpKp =

xxxxtt pKp 2= ( )134.4

After substituting Equation (4.134), we get the following modified equation:

( )L+⎥

⎦

⎤⎢⎣

⎡ Δ+Δ−=− xxxxxxt pKxtKKpp

1221 2

2 ( )135.4

To measure the error generated by the approximation of the numerical method, we

must have an exact solution of the differential equation in question to compare the

approximation. Both the exact solution and approximations must apply to the same

boundary and initial conditions.

Consider the following PDE:

Fx

pKtp

+∂∂

=∂∂

2

2

( )136.4


54

Initial condition:

0)()0,( 0 == xpxp for all [ ]bax ,∈ ( )137.4


0),0( 0 == ptp , 0>t ( )138.4


1),1( 1 == ptp , 0>t ( )139.4

If the desired solution to the PDE is a quadratic function, say ( )txxtxp −= 1),( ,

which satisfies the initial and boundary conditions, we can find the corresponding

function F . We first find the partial derivatives of the desired solution of the PDE

( )txxtxp −= 1),( :

( )xxtp

−=∂∂ 1 ( )tx

xp 21−=∂∂ t

xp 22

2

−=∂∂ ( )140.4

Then, the function F should be:

( ) ( ) KtxxtKxxF 221 21 +−=−−−= ( )141.4

Now, the PDE becomes:

Ktxxx

pKtp 22

2

2

+−+∂∂

=∂∂ ( )142.4

The solution to this PDE can be approximated by modifying the finite differences

code developed for the original Darcy’s diffusivity equation, and the results can be

compared to the exact solution:

( )txxtxp −= 1),( ( )143.4

The relative error between exact solution p and approximation p~ can be calculated


55

as follows:

%100~

_2

2 ×−

=p

pperrorrelative ( )144.4

We now repeat the procedure with a cubic function ( )txxtxp −= 1),( 2 as our desired

solution to the PDE, and we can find another function F :

( )xxtp

−=∂∂ 12 ( )txx

xp 232 −=∂∂ ( )tx

xp 622

2

−=∂∂ ( )145.4


( ) ( )txKxxF 62122 −−−= ( )146.4

The PDE becomes:

( ) ( )txKxxx

pKtp 6212

2

2

−−−+∂∂

=∂∂ ( )147.4

The exact solution is given by:

( )txxtxp −= 1),( 2 ( )148.4

The relative error can also be calculated with Equation (4.144).

We now chose a fourth power polynomial ( )txxtxp −= 1),( 3 as the desired solution

to the PDE and find the corresponding function F :

( )xxtp

−=∂∂ 13 ( )txx

xp 32 43 −=∂∂ ( )txx

xp 22

2

126 −=∂∂ ( )149.4


( ) ( )txxKxxF 233 1261 −−−= ( )150.4

The PDE becomes:


56

( ) ( )txxKxxx

pKtp 23

2

2

1261 −−−+∂∂

=∂∂ ( )151.4

The relative error can be calculated from Equation (4.144) as before for all three

polynomials. All the results are included in Table 4.1. We notice that for the second and

third degree polynomials the relative error is almost zero; there is some evidence of some

round-off error, but the relative error is more significant for the fourth degree polynomial.

We also notice that the relative error decreases as tΔ and xΔ decrease.

Table 4.1: Convergence Analysis ( )5.0=λ

N xΔ tΔ Relative Error (%)

( )txxtxp −= 1),(

Relative Error (%) ( )txxtxp −= 1),( 2


2 0.5000 0.2500 0 0 82.84168695795140

4 0.2500 0.0625 0 0 15.67863569562948

6 0.1667 0.0278 0.00000000000001 0.00000000000003 6.88127700351245

8 0.1250 0.0156 0 0 3.86465607329600

10 0.1000 0.0100 0.00000000000003 0.00000000000003 2.47288870518810

12 0.0833 0.0069 0.00000000000001 0.00000000000003 1.71734566499858

14 0.0714 0.0051 0.00000000000003 0.00000000000001 1.26181351325892

16 0.0625 0.0039 0 0 0.96614031773398

18 0.0556 0.0031 0.00000000000002 0.00000000000004 0.76341205265011

20 0.0500 0.0025 0.00000000000004 0.00000000000003 0.61839092338659


57

As illustrated in Figure 4.9 below, the exact values of the polynomials and their

approximated values are plotted. The plot confirms that the error for the quadratic and

cubic function is very small while the error from the fourth order approximation can be

noticed in the graph.

Figure 4.9: Pressure Profile for ( )5.0=λ and ( )14=N

In Table 4.2, we repeat the process with a lower CFL condition and we can see that

the relative error values are slightly lower compared to those of Table 4.1.


58



( )txxtxp −= 1),(



2 0.5000 0.1500 0.00000000000001 0.00000000000001 82.11985972129230

4 0.2500 0.0375 0.00000000000001 0.00000000000001 15.65042930710459

6 0.1667 0.0167 0.00000000000001 0.00000000000003 6.87911851801523

8 0.1250 0.0094 0.00000000000001 0.00000000000001 3.86298519579571

10 0.1000 0.0060 0.00000000000003 0.00000000000003 2.47220754238725

12 0.0833 0.0042 0.00000000000002 0.00000000000003 1.71721331438360

14 0.0714 0.0031 0.00000000000002 0.00000000000003 1.26163687342525

16 0.0625 0.0023 0.00000000000002 0.00000000000001 0.96603686598178

18 0.0556 0.0019 0.00000000000002 0.00000000000004 0.76338598504818

20 0.0500 0.0015 0.00000000000004 0.00000000000003 0.61834859249007

We now choose a CFL condition slightly higher than the stability limit as in Table 4.3

to analyze the results. All the relative error values are lower than those of Tables 4.1 and

4.2 except the last value when N=20 where the relative error increases suddenly. This

phenomenon is due to instability.


59



( )txxtxp −= 1),(



2 0.5000 0.2550 0.00000000000001 0.00000000000001 80.98792030989775

4 0.2500 0.0638 0.00000000000001 0.00000000000001 15.64733704248511

6 0.1667 0.0283 0.00000000000001 0.00000000000003 6.87180944810920

8 0.1250 0.0159 0.00000000000001 0.00000000000001 3.86219804301619

10 0.1000 0.0102 0.00000000000006 0.00000000000002 2.47274029613424

12 0.0833 0.0071 0.00000000000004 0.00000000000006 1.71700069092848

14 0.0714 0.0052 0.00000000000012 0.00000000000047 1.26164863032906

16 0.0625 0.0040 0.00000000000114 0.00000000000572 0.96583989302120

18 0.0556 0.0031 0.00000000012735 0.00000000002540 0.76335576877073

20 0.0500 0.0026 0.00000079888275 0.00000009869283 7.76758505349834

To understand the behavior of both the quadratic and cubic functions, and why their

relative error is a lot lower compared to the fourth order function, we must recall the

Taylor9 expansion for the second spatial differentiation:

( )( )

L+Δ

+=Δ

Δ−+−Δ+12

),(),(2),( 2

2

xppx


We notice that the term in the right hand side has a second order derivative and error

terms, the first error term with a fourth order derivative. To further check the


60

approximation to the second derivative, we test the polynomials 2xp = and 3xp = .

( )( ) ( )

( )2222)(2)(

2

22222

2

222

=Δ

Δ+Δ−+−Δ+Δ+=

ΔΔ−+−Δ+

xxxxxxxxxx

xxxxxx ( )153.4

The second derivative can be calculated by xp 2=′ and 2=′′p . We notice that the

approximation gives us the exact value of the derivative. For 3xp = we have:

( )2333 )(2)(

xxxxxx

ΔΔ−+−Δ+ ( )154.4

( ) ( ) ( ) ( )( )

xx

xxxxxxxxxxxxx 6332332

322333223

=Δ

Δ−Δ+Δ−+−Δ+Δ+Δ+=

The second derivative can be calculated as before by 23xp =′ and xp 6=′′ . The

approximation gives us the exact value of the derivative. For a fourth degree polynomial,

4xp = we will get:

( )( )22

2

444

212)(2)( xxx

xxxxxΔ+=

ΔΔ−+−Δ+ ( )155.4

when the exact answer should be just 212xp =′′ . This is the source of the error, and it

decreases as the grid size ( )xΔ decreases.

To calculate the convergence rate, we write the error at two different time steps in the

following form:

( )α11 ~ tce Δ and ( )α22 ~ tce Δ ( )156.4

After taking the ratio of both error terms we get:

α

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

2

1

2

1 ~tt

ee

( )157.4


61

If we take logarithms on both sides, we get the equation of a straight line in the form

mxy = :

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛

2

1

2

1 log~logtt

ee

α ( )158.4

The convergence rate α can be found by plotting the ratio of relative errors against

the ratio of their corresponding time steps in a Log-log plot (see Figure 4.10). The slope

of the line is the convergence rate. We can also plot simply:

( ) ( )11 log~log te Δα ( )159.4

Figure 4.10: Convergence Rate ( )tΔ for ( )5.0=λ


62

The convergence rate related to the grid size can be found similarly by writing the

error at two different grid sizes in the following form:

( )α11 ~ xce Δ and ( )α22 ~ xce Δ ( )160.4

After taking the ratio of both error terms we get:

α

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

2

1

2

1 ~xx

ee ( )161.4

If we take logarithms on both sides, we get the equation of a straight line in the form

mxy = :

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛

2

1

2

1 log~logxx

ee

α ( )162.4

The convergence rate α can be written in the following form:

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

2

1

2

1

log

log

xx

ee

α ( )163.4

By plotting the ratio of relative error terms against the ratio of their corresponding

time steps in a Log-log plot (Figure 4.11), the convergence rate α can be found. The

slope of the straight line is the convergence rate.


63

Figure 4.11: Convergence Rate ( )xΔ for ( )5.0=λ

4.1.8. Finite Differences Implicit Discretization

The BTCS discretization is an implicit method, and it is unconditionally stable. A

stability analysis will be later applied using the Fourier or Von Neumann19 method to

show that it is in fact unconditionally stable. The implicit BTCS method will now be

applied to Darcy’s differential equation, and the results will be later compared with those

of the explicit FTCS method.


64

4.1.8.1.Constant Pressure Boundaries


tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )164.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )165.4


aptap =),( , 0>t ( )166.4


bptbp =),( , 0>t ( )167.4

We first rewrite Equation (4.164) and let c

kKφμ

= to get:

2

2

xpK

tp

∂∂

=∂∂ ( )168.4

We set:

1,2

2

,+==== ∂

∂=

∂∂

jiji ttxxttxx xpK

tp ( )169.4

If we consider the following Backward in Time Central in Space (BTCS) differencing

scheme and letting ),(, jiji txpp ≈ , we get:

( ) ⎥⎦

⎤⎢⎣

⎡Δ

+−=

Δ− ++++−+

21,11,1,1,1, 2

xppp

Kt


After rearranging terms we get:


65

( )[ ] 1,1,11,1,12, 2 +++++− ++−

ΔΔ

−= jijijijiji ppppxtKp ( )171.4

If we let ( )2x

tKΔΔ

=λ and rearrange, we now get the following expression:

( ) ( )1,11,11,, 21 +−+++ +−+= jijijiji pppp λλ ( )172.4

where JJj

Ni,1...2,1,01...3,2,1

−=−=


12

21

−=−=

==

NiNi

ii

M

( ) ( )( ) ( )

( ) ( )( ) ( )1,21,1,10,1

1,31,11,20,2

1,11,31,20,2

1,01,21,10,1

2121

2121

−−−

−−−−

+−+=+−+=

+−+=+−+=

NNNN

NNNN

pppppppp

pppppppp

λλλλ

λλλλ

M ( )173.4

From the boundary conditions we know that app =0,0 and bN pp =0, .

A matrix can be built as follows:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−

−+−−+−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

1,

1,1

1,2

1,1

1,0

0,

0,1

0,2

0,1

10000021000

0021000021000001

N

N

N

N

a

pp

ppp

pp

ppp

MMOOOMMM

λλλ

λλλλλλ

( )174.4


( ) )1()0( PBPrr

∗= ( )175.4

To solve the above equation, we rewrite the pressures at the next time step in terms of

the pressures at the previous time step:


66

( ) )0(1)1( PBPrr −∗= ( )176.4

A computer code can be written, and the code will be very similar to the explicit

method, with some differences. The method is unconditionally stable and therefore the

time step and mesh size can be made a lot smaller with no risk of instability. The pressure

profile is illustrated in Figure 4.12. We can see identical results with the plots from the

explicit method but with more resolution. We should also notice that in Figure 4.12 the

limit for instability of the explicit method is exceeded CFL=125 but we get stable results.

Figure 4.12: Pressure Profile for Darcy (CPB) Implicit ( )125=λ


67

4.1.8.2. Boundary Dominated Regime

We will now evaluate the boundary dominated period using an implicit method. The

same assumptions and techniques used for the boundary conditions will be applied, but

the main difference will be in the finite differences technique itself.


tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )177.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )178.4


aptap =),( , 0>t ( )179.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )180.4


from the general differences expression for Darcy’s diffusivity equation:

( ) ( )1,11,11,, 21 +−+++ +−+= jijijiji pppp λλ ( )181.4

where JJj

Ni,1...2,1,0

...3,2,1−=

=



68

NiNi

ii

=−=

==

1

21

M

( ) ( )( ) ( )

( ) ( )( ) ( )1,11,11,0,

1,21,1,10,1

1,11,31,20,2

1,01,21,10,1

2121

2121

−+

−−−

+−+=+−+=

+−+=+−+=

NNNN

NNNN

pppppppp

pppppppp

λλλλ

λλλλ

M ( )182.4

Because now we let jNjN pp ,1,1 −+ = , we get for Ni = :

( ) ( )1,11,0, 221 −−+= NNN ppp λλ ( )183.4


⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−+−

−+−−+−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

1,

1,1

1,2

1,1

1,0

0,1

0,2

0,1

212000021000

0021000021000001

N

N

b

N

a

pp

ppp

pp

ppp

MMOOOMMM

λλλλλ

λλλλλλ

( )184.4


( ) )1()0( PCPrr

∗= ( )185.4

To solve Equation (4.185) above, we again rewrite the pressures at the next time step

in terms of the pressures at the previous time step:

( ) )0(1)1( PCPrr −∗= ( )186.4

Similarly, we can develop a code for these initial and boundary conditions. The

pressure profile is illustrated in Figure 4.13. Because the same arbitrary values are used

for both the implicit and explicit methods, we get an identical graph. We note that here

we also have CFL=125 with no signs of instability.


69

Figure 4.13: Pressure Profile for Darcy (BDR) Implicit ( )125=λ

4.1.8.3. Pseudosteady State Flow Regime

For the steady state flow regime, we will now apply the implicit method, but the same

assumptions and techniques used for the boundary conditions in the explicit method will

be applied.


tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )187.4

Initial condition:


70

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )188.4


jttxxxpkq

==⎥⎦⎤

⎢⎣⎡∂∂

=,0

μ, 0>t ( )189.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )190.4

We now incorporate the conditions to our finite differences approach starting from

the general differences expression for Darcy’s differential equation:

( ) ( )1,11,11,, 21 +−+++ +−+= jijijiji pppp λλ ( )191.4

where JJj

Ni,1...2,1,0

...3,2,1,0−=

=


NiNi

ii

=−=

==

1

10

M

( ) ( )( ) ( )

( ) ( )( ) ( )1,11,11,0,

1,21,1,10,1

1,01,21,10,1

1,11,11,00,0

2121

2121

−+

−−−

−

++−=++−=

+−+=+−+=

NNNN

NNNN

pppppppp

pppppppp

λλλλ

λλλλ

M ( )192.4

We recall:

( )xkqpp jj Δ−=−μ2

,1,1 ( )193.4

For 0=i we get:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ−+−+= x

kqpppp μλλ 221 1,11,11,00,0


71

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ−−+= x

kqppp μλλ 2221 1,11,00,0 ( )194.4

For Ni = we get:

( ) ( )1,11,0, 221 −−+= NNN ppp λλ ( )195.4


( )

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡ Δ

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−+−

−+−−+−

−+

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

00000

2

212000021000

00210000210000221

1,

1,1

1,2

1,1

1,0

0,1

0,2

0,1

xkq

pp

ppp

pp

ppp

N

N

b

N

aμλ

λλλλλ

λλλλλλ

λλ

MMOOOMMM ( )196.4


( ) ∗∗ += EPDPrrr

)1()0( ( )197.4



( ) ( )∗−∗ −= EPDPrrr

)0(1)1( ( )198.4

For pseudosteady state, we also get an identical result of the explicit method, with the

implicit method giving us more flexibility and no stability concerns. Figure 4.14 shows

the corresponding pressure profile, and we can see the pressure changing in both

boundaries with time.


72

Figure 4.14: Pressure Profile for Darcy (PSS) Implicit ( )125=λ

4.1.9. Stability Analysis for the Implicit Scheme

The Fourier or Von Neumann19 method will be applied to the implicit scheme. Let us

suppose again that the solution of the difference equations is of the form:



suitable choice ofλ . We notice that if 1>Δtje λ , the solution becomes unbounded and so



73

We now consider the implicit difference equation for the PDE which is:

( ) ( )1,11,11,, 21 +−+++ +−+= jijijiji pppp λλ ( )200.4


( ) ( ) ( ) ( ) ( ) )()21( 11111 tjxiItjxiItjxIitjxIi eeeeeeee Δ+Δ−Δ+Δ+Δ+ΔΔΔ +−+= λβλβλβλβ λλ ( )201.4


( )))1(())1(())1(())1(())1(()21(1 jjtiixIjjtiixIjjt eeeee −+Δ−−Δ−+Δ−+Δ−+Δ +−+= λβλβλ λλ ( )202.4

After simplifying:

( )txItxIt eeeee ΔΔ−ΔΔΔ +−+= λβλβλ λλ)21(1 ( )203.4


( ) ( )( )( ) ( )( )( )( )

( )( )

( )xe

xexe

xIxxIxeeee

t

t

t

t

xIxIt

Δ−+=

Δ−+=Δ−+=

Δ−Δ+Δ+Δ−+=+−+=

Δ

Δ

Δ

Δ

Δ−ΔΔ

βλ

βλβλλ

ββββλλλλ

λ

λ

λ

λ

ββλ

cos1211

cos1211cos2211

sincossincos211211

( )204.4

where te Δλ is the amplification factor. We can see that 1≤Δteλ for any value of λ , and so

the method is unconditionally stable.

4.2. FORCHHEIMER’S EQUATION

In this section, Forchheimer’s equation12 will be analyzed by deriving a Forchheimer

diffusivity equation7 in one dimension that will be solved analytically. The steady state

solution will also found for given boundary and initial conditions. Both explicit and

implicit finite differences techniques will be applied. The conditions for stability will be


74

checked and a convergence analysis will be performed for both finite differences methods

to learn some of the properties of Forchheimer’s equation12.

4.2.1. Forchheimer Diffusivity Equation

Following the derivation of Darcy’s equation11, we can use a similar approach to

develop the Forchheimer12 diffusivity equation.

Just like with Darcy’s diffusivity equation, we can start with the conservation of

mass: Mass rate in – Mass rate out = Mass rate remaining:

tzyxzyvzyv ttt

xxxxxx Δ−

ΔΔΔ=ΔΔ−ΔΔ Δ+Δ+Δ+

ρρφρρ )()()( ( )205.4

Dividing both sides by zyx ΔΔΔ to get:

txvv tttxxxxxx

Δ−

=Δ

−− Δ+Δ+Δ+ ρρφρρ )()( ( )206.4

After taking limits on both sides as 0→Δx and 0→Δt we get the partial difference

known as the continuity equation in one dimension:

txv

∂∂

−=∂

∂ ρφρ)( ( )207.4

The left hand side can be expanded using the chain rule:

txv

xv

∂∂

−=∂∂

+∂∂ ρφρρ ( )208.4

We now use Forchheimer’s equation12 instead:

2vvkx

p βρμ−−=

∂∂ ( )209.4

Taking derivatives of both sides of the above equation with respect to x , we get:


75

xvv

xv

kxp

∂∂

−∂∂

−=∂∂ βρμ 22

2

( )210.4

Substitute ⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−=∂∂

xv

txv ρρφ

ρ1 from continuity equation:

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

=∂∂

xv

tv

xv

tkxp ρρφβρρφ

ρμ 22

2

( )211.4

This equation can be simplified to:

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

∂∂

xv

tv

kxp ρρφβ

ρμ 22

2

( )212.4

Substituting xp

px ∂∂

∂∂

=∂∂ ρρ and

tp

pt ∂∂

∂∂

=∂∂ ρρ from chain rule:

pxpv

tpv

kxp

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

∂∂ ρφβ

ρμ 22

2

( )213.4

Substituting ρρ cp=

∂∂ and rearranging we get:

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )214.4

Note that if 0=β the velocity must be expressed as xpkv∂∂

−=μ

from Darcy’s

equation11 to get the following:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+∂∂

⎟⎠⎞

⎜⎝⎛=

∂∂

xp

xpk

tp

kc

xp

μφμ

2

2

( )215.4

After rearranging and combining terms we get:


76

2

2

2

⎟⎠⎞

⎜⎝⎛∂∂

−∂∂

=∂∂

xpc

tp

kc

xp μφ ( )216.4

If we assume that the pressure gradient is small, then the term 2

⎟⎠⎞

⎜⎝⎛∂∂

xpc is very small

and the whole term 2

⎟⎠⎞

⎜⎝⎛∂∂

xpc can be neglected to get Darcy’s diffusivity equation:

tp

kc

xp

∂∂

=∂∂ μφ

2

2

( )217.4

4.2.2. Analytical Solution

Forchheimer diffusivity equation7:

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )218.4

Initial condition:

)()0,( xxp φ= ( )219.4


0),0( 0 == ptp , 0>t ( )220.4


0),1( 1 == ptp , 0>t ( )221.4

After rearranging terms of Equation (4.218) and simplifying we get:

( ) xpv

xp

vctp

k ∂∂

−∂∂

+=

∂∂

φβρφ μ 2

2

21 ( )222.4


77

If we assume the velocity is constant, we let ( )vcA

k βρφ μ 21+

= and φvB = to give:

xpB

xpA

tp

∂∂

−∂∂

=∂∂

2

2

( )223.4

The above expression is similar to a diffusion-convection equation xxxt vuuu −= 2α

or the linearized Burger’s equation20 xxxt ucuu μ=+ and therefore can be treated

similarly.

We first apply the following transformation to Forchheimer’s equation12:

( ) ( ) ( )txwetxp AxB Bt

,, 22−= ( )224.4

If we take a partial derivative of pressure with respect to time tp and a first and

second partial derivatives of pressure with respect to space ( xp and xxp respectively):

( )( ) ( ) ( )( ) tAxBAxB

t wetxwBA

BepBtBt 22 22 ,

22−− +⎟

⎠⎞

⎜⎝⎛−= ( )225.4

( )( ) ( ) ( )( ) xAxBAxB

x wetxwA

BepBtBt 22 22 ,

2−− += ( )226.4

( )( ) ( ) ( )( ) ⎥⎦⎤

⎢⎣⎡ += −−

xAxBAxB

xx wetxwA

BeA

BpBtBt 22 22 ,

22

( )( ) ( )( ) ⎥⎦⎤

⎢⎣⎡ ++ −−

xxAxB

xAxB wew

ABe

BtBt 22 22

2 ( )227.4

We now substitute the corresponding values in the PDE, xxxt BpApp −= which after

rearranging and simplifying yields the following:

xxt Aww = ( )228.4

This equation has the same form of Darcy’s diffusivity equation, and its analytical


78

solution can be derived similarly with given initial and boundary conditions.

Initial condition:

)()0,( xxw φ= ( )229.4


0),0( 0 == wtw , 0>t ( )230.4


0),1( 1 == wtw , 0>t ( )231.4

4.2.2.1. Finding Solution to the PDE

Let:

)()(),( tTxXtxw = ( )232.4

And so we get:

)()(

)()(

2

2

tTxXxw

tTxXtw

′′=∂∂

=∂∂ &

( )233.4

The PDE can be rewritten as:

)()()()( tTxXAtTxX ′′=& ( )234.4

We can now rearrange the terms to get:

2

)()(

)()( λ−=

′′=

xXxX

tATtT& ( )235.4

Each of the above equalities can be analyzed and solved separately:


79

2

)()( λ−=tAT

tT& ( )236.4

If we let dtdTT =& :

ATdtdT 2λ−= ( )237.4

Take derivatives on both sides:

∫∫ −= AdtTdT 2λ ( )238.4

After solving we get:

cAtT +−= 2ln λ ( )239.4

This solution can also be expressed as:

AteBtT2

1)( λ−= ( )240.4

The other equality from Equation (4.235) can also be solved as follows:

2

)()( λ−=

′′xXxX ( )241.4

Rearranging terms:

0)()( 2 =+′′ xXxX λ ( )242.4

Can also be rewritten as:

022

2

=+∂∂ X

xX λ or 02 =+′′ yy λ ( )243.4


rx

rx

eryrey

2=′′=′

( )244.4


80


irrrr

eer rxrx

λλ

λλλ

±=−±=

−==+

=+

2

22

22

22

00

( )245.4

This gives:

)sin()cos()( 22 xCxBxXy λλ +== ( )246.4

Combining both solutions we then have:

[ ])sin()cos(),( 221

2

xCxBeBtxw tA λλλ += − ( )247.4

The constants can be recombined to get:

[ ])sin()cos(),(2

xCxBetxw tA λλλ += − ( )248.4

4.2.2.2. Finding Solution to the PDE and Boundary Conditions

If we apply the first boundary condition 0),0( =tw we get:

[ ] 0)0sin()0cos(),0(2

=⋅+⋅= − λλλ CBetw tA ( )249.4

To get 0=B

If we apply the second boundary condition 0),1( =tw we get:

[ ] 0)sin()cos(),1(2

=+= − λλλ CBetw tA ( )250.4

where 0)sin( =λC and 0)sin( =λ . ( 0=C is not of interest)

...3,2,1

0)sin(

=±==

nnπλ

λ ( )251.4


81

The nth solution can be expressed as follows:

)sin(),(2

xeCtxw ntA

nnn λλ−= ,...3,2,1=n ( )252.4

Each of the family of solutions satisfies the PDE and the boundary conditions. We

can now write the general solution using the Principle of Superposition as:

∑∞

=

−=1

)sin(),(2

n

tAn xneCtxw n πλ ( )253.4

4.2.2.3. Finding Solution to the PDE, Boundary and Initial Conditions

We must now also satisfy the initial condition )()0,( xxw φ= which yields:

∑∞

=

=1

)sin()(n

n xnCx πφ ( )254.4

The coefficients nC can be determined from the orthogonality property of

{ },...3,2,1),sin( =nxnπ to yield:

∫=1

0)sin()(2 dxxnxCn πφ ( )255.4

This combination of equations gives us a series of solutions.

4.2.3. Steady State Solution

It is also of special interest to determine the behavior of Forchheimer’s differential

equation7 as it reaches steady state ⎟⎠⎞

⎜⎝⎛ =∂∂ 0

tp . This will provide us with a mathematical

expression of the pressure profile when the well has produced long enough and the

pressure drop is constant throughout the reservoir.


82

If we start from Forchheimer PDE7:

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )256.4


0),0( 0 == ptp , 0>t ( )257.4


1),1( 1 == ptp , 0>t ( )258.4

Combining the constants as before, for steady state conditions we get:

02

2

=∂∂

−∂∂

xpB

xpA ( )259.4

The above equation is a homogeneous linear equation with constant coefficients, and

can be rewritten as:

0=′−′′ yByA ( )260.4


rx

rx

eryrey

2=′′=′

( )261.4


( )

ABr

rBArr

BrArBreeAr rxrx

=

==−=+

=−

2

1

2

2

000

0

( )262.4

The general solution will have the form xrxr eCeCxy 2121)( += and so we write the


83

general pressure function as:

xAB

eCCxp 21)( += ( )263.4

If we apply the some boundary conditions where 0)0( =p and 1)1( =p the values for

the constants 1C and 2C can be found:

0)0( 21 =+= CCp , 21 CC −=

1)1( 21 =+= ABeCCp , A

BeCC 21 1−=

ABe

C−

=1

11 and

11

2 −=

ABe

C ( )264.4

This results in the following equation:

AB

AB

AB e

ee

xp x

−+

−=

11

11)( ( )265.4


The finite differences FTCS discretization will be applied to Forchheimer PDE7 in

one dimension for different boundary and initial conditions.

4.2.4.1. Constant Pressure Boundaries

To develop the finite differences approximation, we will first assume constant



⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )266.4

Initial condition:


84

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )267.4 ( )267.4


aptap =),( , 0>t ( )268.4


bptbp =),( , 0>t ( )269.4

From rearranging and grouping constant terms in Equation (4.266) we get the

following:

jijiji ttxxttxxttxx xpB

xpA

tp

====== ∂∂

−∂∂

=∂∂

,,2

2

,

( )270.4

If we replace the time derivative above by the forward differencing scheme and the

space derivative above by the central differencing scheme, we get:

( ) ⎥⎦

⎤⎢⎣

⎡Δ

−−⎥

⎦

⎤⎢⎣

⎡

Δ

+−=

Δ

− −++−+

xpp

Bx

pppA

tpp jijijijijijiji

22 ,1,1

2,1,,1,1, ( )271.4

After rearranging:

( )[ ] [ ] jijijijijijiji ppp

txBppp

xtAp ,,1,1,1,,121, 2

2 +−ΔΔ

−+−ΔΔ

= −++−+ ( )272.4

If we combine like terms we get:

( ) ( ) ( ) jijijiji pxtB

xtAp

xtAp

txB

xtAp ,12,2,121, 2

212 +−+ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ΔΔ

−ΔΔ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

ΔΔ

−+⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

+ΔΔ

= ( )273.4

Let ( )2x

tAΔΔ

=λ and xtB

ΔΔ

=2

γ to give:

( ) ( ) ( ) jijijiji pppp ,1,,11, 21 +−+ −+−++= γλλγλ ( )274.4


85

where MMj

Ni,1...2,1,01...3,2,1

−=−=


12

321

−=−=

===

NiNi

iii

M

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 0,0,10,21,1

0,10,20,31,2

0,40,30,21,3

0,30,20,11,2

0,20,10,01,1

2121

212121

NNNN

NNNN

pppppppp

pppppppppppp

γλλγλγλλγλ

γλλγλγλλγλγλλγλ

−+−++=−+−++=

−+−++=−+−++=−+−++=

−−−

−−−−

M ( )275.4

Recall from the boundary conditions that app =0,0 and bN pp =0, . A matrix can be

built as follows:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−+

−−+−−+

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

b

N

a

N

N

pp

ppp

pp

ppp

0,1

0,2

0,1

1,

1,1

1,2

1,1

1,0

10000021000

0021000021000001

MMOOOMMM

γλλγλ


( )276.4


)0()1( PFPrr

= ( )277.4


We will assume constant bottomhole pressure during production and no flow across

the reservoir’s outer boundary at a distance bxx N == . These conditions can be

expressed mathematically as follows:



86

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )278.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )279.4


aptap =),( , 0>t ( )280.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )281.4

To satisfy the outer boundary condition we must have: ( ) 02

,1,1 =Δ

− −+

xpp jNjN and

therefore jNjN pp ,1,1 −+ = . We note that jNp ,1+ is located at a node outside the interval

[ ]ba, . To calculate the pressure gradient at the boundary we will assume that the pressure



from the general differences expression for Forchheimer diffusivity equation7:


where JJj

Ni,1...2,1,0

...3,2,1−=

=



87

NiNi

iii

=−=

===

1

321

M

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 0,10,0,11,

0,0,10,21,1

0,40,30,21,3

0,30,20,11,2

0,20,10,01,1

2121

212121

+−

−−−

−+−++=−+−++=

−+−++=−+−++=−+−++=

NNNN

NNNN

pppppppp

pppppppppppp



M ( )283.4


( ) ( ) ( ) 0,10,0,11, 21 −− −+−++= NNNN pppp γλλγλ

( ) ( )0,10,1, 221 −+−= NNN ppp λλ ( )284.4

We now build the matrix:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−+

−−+−−+

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

b

N

a

N

N

pp

ppp

pp

ppp

0,1

0,2

0,1

1,

1,1

1,2

1,1

1,0

212000021000

0021000021000001

MMOOOMMM

λλγλλγλ


( )285.4


)0()1( PGPrr

= ( )286.4



boundary at a distance bxx N == . These conditions can be expressed mathematically as

follows:



88

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )287.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )288.4


βρ

βρμμ

2

4,

2

0 jttxxxp

kkq

==⎥⎦⎤

⎢⎣⎡∂∂

+⎟⎠⎞

⎜⎝⎛±−

= , 0>t ( )289.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )290.4



( )xpp

xp jj

j Δ

−=⎥⎦

⎤⎢⎣⎡∂∂ −

2,1,1

,0

( )291.4

If we substitute the pressure gradient in the Forchheimer’s equation12 and letting

qv −= , we get:

( )2,1,1

2qq

kxpp jj βρμ

+=⎥⎦

⎤⎢⎣

⎡Δ

− − ( )292.4

We notice that the pressures jp ,1− are located outside the interval [ ]ba, . We can

rearrange the above equation to get an expression for jp ,1− :

( ) ( ) 2,1,1 22 qxq

kxpp jj βρμ

Δ−Δ−=− ( )293.4


89



expression for Forchheimer diffusivity equation7:


where JJj

Ni,1...2,1,0

...3,2,1,0−=

=


NiNi

iii

=−=

===

1

210

M

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 0,10,0,11,

0,0,10,21,1

0,30,20,11,2

0,20,10,01,1

0,10,00,11,0

2121

212121

+−

−−−

−

−+−++=−+−++=

−+−++=−+−++=−+−++=

NNNN

NNNN

pppppppp

pppppppppppp



M ( )295.4

For 0=i we get:

( ) ( ) ( ) ( ) ( ) 0,10,02

0,11,0 2122 ppqxqk

xpp γλλβρμγλ −+−+⎟⎠⎞

⎜⎝⎛ Δ−Δ−+=

( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ+Δ−+−= 2

0,10,01,0 22221 qxqk

xppp βρμλλ ( )296.4

For Ni = we get:

( ) ( )0,10,1, 221 −+−= NNN ppp λλ ( )297.4

A new matrix can be built: ( )298.4


90

( ) ( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛ Δ+Δ−

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−+

−−+−−+

−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

00000

22

212000021000

00210000210000221 2

0,1

0,2

0,1

1,

1,1

1,2

1,1

1,0 qxqk

x

pp

ppp

pp

ppp

b

N

a

N

N

βρμλ

λλγλλγλ


λλ

MMOOOMMM


FPHPrrr

+= )0()1( ( )299.4


The Fourier or Von Neumann19 method will be applied to the explicit scheme. Let us




suitable choice of λ . We notice that if 1>Δtje λ , the solution becomes unbounded and so


We now consider the explicit difference equation of the PDE which is:



( ) ( ) ( ) ( ) ( ) ( ) tjxiItjxIitjxiItjxIi eeeeeeee ΔΔ+ΔΔΔΔ−Δ+Δ −+−++= λβλβλβλβ γλλγλ 111 21 ( )302.4


( ) ( ) ( ) ( ) ( ) ( )iixIiixIjjt eee −+Δ−−Δ−+Δ −+−++= )1()1()1( 21 ββλ γλλγλ ( )303.4


91

After simplifying:

( ) ( ) ( ) xIxIt eee ΔΔ−Δ ++−++= ββλ γλλγλ 21 ( )304.4


( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( )xIxexIxe

xIxeeexIxxIxe

eeeee

t

t

t

xIxIt

xIxIxIxIt

Δ−−Δ+=−+Δ−Δ=

−+Δ−+Δ=−+−+Δ+Δ+Δ−Δ=

−+−++=

Δ

Δ

Δ

ΔΔ−Δ

ΔΔ−ΔΔ−Δ

βγβλλβγβλ

λβγβλλγββββλ

λγλ

λ

λ

λ

ββλ

ββββλ

sin21cos2121sin2cos2

21sin2cos221sincossincos

21

( )305.4

where te Δλ is the amplification factor, and we must find a condition for 1≤Δteλ . Here

are a few suggestions.

( )( ) ( )222)sin(21)cos(21 xIxe t Δ−+−Δ+=Δ βγβλ

( ) ( )( ))(sin41cos(41)cos(41 2222 xxx Δ+−Δ+−Δ+= βγβλβλ ( )306.4

And since we want 12≤Δteλ , we get the following:

( ) ( )( ) 1)(sin41cos(41)cos(41 2222 ≤Δ+−Δ+−Δ+ xxx βγβλβλ ( )307.4

After simplifying we get the inequality:

( ) ( ) 0)(sin1)cos(1)cos( 2222 ≤Δ+−Δ+−Δ xxx βγλβλβ ( )308.4

Because it has the form of a quadratic equation, 02 =++ cbxax it can be solved

similarly and get the following after simplifying.

( ))cos(12)(sin411 22

2,1 xx

Δ−Δ−±

≤β

βγλ ( )309.4

We would like the second term in the numerator to satisfy 0)(sin41 22 ≥Δ− xβγ ,


92

and so we must have 0)(sin41 22 ≥Δ− xβγ .

This implies:

1)(sin4 22 ≤Δxβγ ( )310.4

)(sin 2 xΔβ can be close to 1 then we need:

14 2 ≤γ

412 ≤γ

21

≤γ ( )311.4

This gives us a condition for gamma γ .

4.2.6. Convergence Analysis for the Explicit Method

Consider the difference representation of Forchheimer PDE7:


The exact solution can be written as follows:

( ) ( ) ( ) ttxptxptxptxp jijijiji Δ+−+−++= +−+ τγλλγλ ),(),(21),(),( 111 ( )313.4

where tΔτ is an error term. The error at any time can be found to be:

),(: 11,1, +++ −= jijiji txppe ( )314.4

( )( ) ( )( ) ( )( ) ttxpptxpptxppe jijijijijijiji Δ−−−+−−+−+= ++−−+ τγλλγλ ),(),(21),( 1,1,1,11,

The maximum error can be expressed as follows:

( ) ( ) ( ) teeee jijijiijiiΔ−−+−++= +−+ τγλλγλ ,1,,11, 21maxmax ( )315.4


93

If we let jiij eE ,max= we now get:

teeeE jiijiijiij Δ+−+−++≤ +−+ τγλλγλ ,1,,11 maxmax21max ( )316.4

tEEEE jjjj Δ+−+−++≤+ τγλλγλ 211 ( )317.4

If we allow all the terms with absolute values to be positive (i.e. 0≥ ), we get:

γλγλγλγλ

λλ

−≥⇒≥+≥⇒≥−

≤⇒≥−

00

21021

( )318.4

The bottom inequality may be rewritten as γλ ≤− to get the condition:

λγλ ≤≤− or λγ ≤ ( )319.4

This shows that for convergence we must have21

≤λ and λγ ≤ .

Now, if λ21− , γλ − and γλ + are all positive we can then write the following

expression:

( ) ( ) ( ) tEEEE jjjj Δ+−+−++≤+ τγλλγλ 211 ( )320.4


tEE jj Δ+≤+ τ1 ( )321.4

We can now use that expression to develop the error at the previous time step:

tEE jj Δ+≤ − τ1 ( )322.4

The previous two equations may be recombined to give:

tEE jj Δ+≤ −+ τ211 ( )323.4

If 0E is the maximum error at a known initial time ( )0=t we get:


94

ttEE j Δ++≤+ τ)1(01 ( )324.4

If the value at a specific time is known, there is no error at that point so 00 =E :

ttE j Δ≤ τ ( )325.4

The term τt can be a final time Tf to also give:

tTfEt Δ≤ ( )326.4

Again, just like we were able to see with the convergence analysis for Darcy’s

diffusivity equation, we can also see that for Forchheimer diffusivity equation7, the error

decreases as the time step decreases. We will now illustrate convergence of the computer

code developed by applying different time steps and grid sizes to the explicit method of


xpB

xpA

tp

∂∂

−∂∂

=∂∂

2

2

( )327.4

Recall that the finite differences approximation for Forchheimer diffusivity equation7

had the following form:

( ) ⎥⎦

⎤⎢⎣

⎡Δ

−−⎥

⎦

⎤⎢⎣

⎡

Δ

+−=

Δ

− −++−+

xpp

Bx

pppA

tpp jijijijijijiji

22 ,1,1

2,1,,1,1, ( )328.4

We will first analyze the behavior of Forchheimer diffusivity equation7 by applying

Taylor9 expansions to the first order derivative in time and the first and second order

derivatives in space. For convenience, we will refer to the first derivative in time as tp

and the first and second derivative in space as xp and xxp , respectively. We can then

rewrite Forchheimer diffusivity equation7 in the following form:

xxxt BpApp −= ( )329.4


95

The partial differences equation can also be expressed in the in the following form:

=Δ

−Δ+t

txpttxp ),(),(

( ) xtxxptxxpB

xtxxptxptxxpA

ΔΔ−−Δ+

−⎥⎦

⎤⎢⎣

⎡

ΔΔ++−Δ−

=2

),(),(),(),(2),(2

( )330.4

The Taylor9 expansions for the pressure at the next time step ),( ttxp Δ+ and the

current time step ),( txp can be generated as before:

( ) ( ) ( )L+

=

Δ+

Δ+

Δ+Δ+=Δ+

),(),(2462

),(),(432

txptxp

tptptptptxpttxp tttttttttt ( )331.4

Using the above expansions, we can get an expression for the first derivative in

time:

( ) ( )L+

Δ+

Δ+

Δ+=

Δ−Δ+

2462),(),( 32 tptptpp

ttxpttxp

tttttttttt ( )332.4

The Taylor9 expansions for the pressure at the next node ),( txxp Δ+ , the current

node ),( txp , and the previous node ),( txxp Δ− can be written as before:

( ) ( ) ( )

( ) ( ) ( )L

L

+Δ

+Δ

+Δ

+Δ+=Δ+

=

±Δ

+Δ

−Δ

+Δ−=Δ−

2462),(),(

),(),(2462

),(),(

432

432

xpxpxpxptxptxxp

txptxp

xpxpxpxptxptxxp

xxxxxxxxxx

xxxxxxxxxx

( )333.4

The first and second derivatives in space can be approximated using the above

developed Taylor9 expansions to obtain the following:

( )L+

Δ+=

ΔΔ−−Δ+

62),(),( 2xpp

xtxxptxxp

xxxx ( )334.4


96

( )( )

L+Δ

+=Δ

Δ−+−Δ+12

),(),(2),( 2

2

xppx


The first derivative in time and the first and second derivatives in space can be

rewritten as:

( ) ( )L−

Δ−

Δ−

Δ−

Δ−Δ+

=2462

),(),( 32 tptptpt

txpttxpp tttttttttt ( )336.4

( )L−

Δ−

ΔΔ−−Δ+

=62

),(),( 2xpx

txxptxxpp xxxx ( )337.4

( )( )

L−Δ

−Δ

Δ−+−Δ+=

12),(),(2),( 2

2

xpx

txxptxptxxpp xxxxxx ( )338.4

Therefore, the truncation error becomes:

( ) ( )⎥⎦

⎤⎢⎣

⎡−

Δ−+⎥

⎦

⎤⎢⎣

⎡−

Δ−−⎥⎦

⎤⎢⎣⎡ −

Δ−=+− LLL

6122

22 xpBxpAtpBpApp xxxxxxxttxxxt ( )339.4

If we choose only the expressions with the lowest powers of xΔ and tΔ we get:

( ) ( )L±

Δ+

Δ−

Δ−=+− xxxxxxxttxxxt ApxBpxtpBpKpp

1262

22

( )340.4

The modified equation can be found by eliminating the time derivatives and replacing

them by spatial differentiation:

xxxt BpApp −=

( )txxxtt BpApp −=

( ) ( )txtxxtt BpApp −=

( ) ( )xtxxttt pBpAp −=

( ) ( )xxxxxxxxxtt BpApBBpApAp −−−=


97

xxxxxxxxxxxxtt pBABpABppAp 22 +−−=

xxxxxxxxxtt pBABppAp 22 2 +−= ( )341.4

After substituting, we get the following modified equation:

( ) ( ) ( )xxxxxxxxxxxxxxxxxxxt ApxBpxtpBABppABpApp

12622

2222 Δ

+Δ

−Δ

+−−=+− ( )342.4

If we group like terms we get:

( ) ( )xxxxxxxxxxxxt ptAxpxBtABptBBpApp ⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−

Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=+−

21262

2222

( )343.4

and let:

( )2xtA

ΔΔ

=λ and xtB

ΔΔ

=γ ( )344.4

we note that γ is different from that of the explicit method because the number two in

the denominator of the first order derivative in space ( )xΔ2 was simplified during the

Taylor9 expansions. The modified equation becomes:

( ) ( )xxxxxxxxxxxxt pxBpxBptBBpApp ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

Δ+⎟

⎠⎞

⎜⎝⎛ −

Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ−=+−

γλ

γλλ

222

6122

1332

( )345.4

We now find an exact solution by modifying the differential equation with an

additional function F for the same initial and boundary conditions as before:

FxpB

xpA

tp

+∂∂

−∂∂

=∂∂

2

2

( )346.4

Initial condition:

0)()0,( 0 == xpxp for all [ ]bax ,∈ ( )347.4



98

0),0( 0 == ptp , 0>t ( )348.4


1),1( 1 == ptp , 0>t ( )349.4

Three desired solutions to the PDE will be considered: a quadratic function

( )txxtxp −= 1),( , a cubic function ( )txxtxp −= 1),( 2 , and a fourth order function

( )txxtxp −= 1),( 3 . The correspondent values for F to make this polynomial exact can

be found as follows. For the quadratic function we get:

( )xxtp

−=∂∂ 1 ( )tx

xp 21−=∂∂ t

xp 22

2

−=∂∂ ( )350.4


( ) ( ) AttxBxxF 22114 +−+−= ( )351.4

Now, the PDE becomes:

( ) AttxBxxxpB

xpA

tp 2)21(12

2

+−+−+∂∂

−∂∂

=∂∂ ( )352.4

The solution to this PDE can be approximated by modifying the finite differences

code developed for the original Forchheimer diffusivity equation7, and the results can be

compared to the exact solution:

( )txxtxp −= 1),( ( )353.4

The relative error between exact solution p and approximation p~ can be calculated

from:

%100~

_2

2 ×−

=p

pperrorrelative ( )354.4


99

We now find another function F for the cubic function ( )txxtxp −= 1),( 2 to be the

exact solution:

( )xxtp

−=∂∂ 12 ( )txx

xp 232 −=∂∂ ( )tx

xp 622

2

−=∂∂ ( )355.4


( ) ( ) ( )txAtxxBxxF 62321 225 −−−+−= ( )356.4

The PDE becomes:

( ) ( ) ( )txAtxxBxxxpB

xpA

tp 62321 22

2

2

−−−+−+∂∂

−∂∂

=∂∂ ( )357.4

The exact solution is given by:

( )txxtxp −= 1),( 2 ( )358.4

We now find another function F for the cubic function ( )txxtxp −= 1),( 3 to be the

exact solution:

( )xxtp

−=∂∂ 13 ( )txx

xp 32 43 −=∂∂ ( )txx

xp 22

2

126 −=∂∂ ( )359.4


( ) ( ) ( )txxAtxxBxxF 23236 126431 −−−+−= ( )360.4

The PDE becomes:

( ) ( ) ( )txxAxxBxxx

pKtp 2323

2

2

126431 −−−+−+∂∂

=∂∂ ( )361.4

The relative error can be calculated as before but for Forchheimer diffusivity

equation7, a special attention must be paid to the problem of stability. Here stability

depends on two terms λ and γ ; it has been noticed through error analysis (see Appendix


100

A) that if we proceed similarly to the case of Darcy’s diffusivity equation by choosing the

grid size from partitioning the domain in N intervals with L4,3,2=N , we get unstable

results as values for the inertia factor β increase and as the powers of the polynomials

increase. We therefore applied the convergence conditions derived previously and let

21

≤λ and λγ ≤ to get an expression for the grid size:

⎟⎠⎞

⎜⎝⎛ +

≤Δv

kvc

xβρμ 2

2 ( )362.4

This condition has been applied to the computer code developed for Forchheimer

diffusivity equation7 for stability. Table 4.4 shows the relative error of the quadratic

function with CFL=0.5 for arbitrary values of the fluid and rock properties by keeping

them constant and changing only the inertia factor β . We notice that the relative error

values are nearly zero.


101

Table 4.4: Convergence Analysis (Quadratic Function), ( )5.0=λ and ( )β

β xΔ tΔ γ Relative Error (%)

( )txxtxp −= 1),(

1 0.3333 0.3333 0.5000 0.00000000000002

2 0.2000 0.200 0.5000 0.00000000000001

3 0.1429 0.1429 0.5000 0.00000000000001

4 0.1111 0.111 0.5000 0.00000000000002

5 0.0909 0.0909 0.5000 0.00000000000001

10 0.0476 0.0476 0.5000 0.00000000000001

20 0.0244 0.0244 0.5000 0.00000000000001

50 0.0099 0.0099 0.5000 0.00000000000001

100 0.0050 0.0050 0.5000 0.00000000000001

200 0.0025 0.0025 0.5000 0.00000000000001

When the relative error values for the cubic and fourth order polynomials are

calculated for the same CFL condition as Table 4.4, we notice more significant relative

error values as Table 4.5 suggests. The cubic polynomial shows lower relative error

values than the values from the fourth order polynomial. We also notice that the CFL and

gamma conditions are equal γλ = and that the values for the time step and grid size are

also equal tx Δ=Δ for the same values for β .


102

Table 4.5: Convergence Analysis (Cubic and Fourth Order Functions), ( )5.0=λ and ( )β


( )txxtxp −= 1),( 2


1 0.3333 0.3333 0.5000 34.74969301789464 37.71083802383927

2 0.2000 0.2000 0.5000 14.27979562386444 16.49836277465184

3 0.1429 0.1429 0.5000 7.72195565412219 9.12636046451933

4 0.1111 0.1111 0.5000 4.82369437590741 5.76321779345298

5 0.0909 0.0909 0.5000 3.29521404215027 3.96130540133563

10 0.0476 0.0476 0.5000 0.94377205838511 1.14707353027556

20 0.0244 0.0244 0.5000 0.25327517009019 0.30925421939874

50 0.0099 0.0099 0.5000 0.04232632894041 0.05180595302110

100 0.0050 0.0050 0.5000 0.01073799616597 0.01315264432408

200 0.0025 0.0025 0.5000 0.00270435398190 0.00331366278072

Figure 4.15 again shows the pressure profiles for the exact and approximate values of

the quadratic, cubic and fourth order polynomials. We can see an appreciable error on the

cubic and fourth order functions while the approximation of the quadratic function is

almost identical to the exact solution.


103

Figure 4.15: Pressure Profiles for ( )5.0=λ and ( )10=β

We now test for CFL=0.3 to analyze the behavior of the computer code with respect

to the corresponding exact solutions and the results are shown in Table 4.6. All the

relative error values are lower than their corresponding values from Table 4.5. We also

notice that the CFL and gamma conditions have remained equal, but now the time step

and grid size are no longer equal for the same values of the inertia factor β , with the

time step being smaller.


104

Table 4.6: Convergence Analysis ( )3.0=λ and ( )β


( )txxtxp −= 1),( 2


1 0.3333 0.2000 0.3000 34.54238008418665 37.48969523188482

2 0.2000 0.1200 0.3000 14.00968649147083 16.31305115598478

3 0.1429 0.0857 0.3000 7.66050315678343 9.08402853832296

4 0.1111 0.0667 0.3000 4.81294682873217 5.75308014696563

5 0.0909 0.0545 0.3000 3.26801381927890 3.94589009131221

10 0.0476 0.0286 0.3000 0.94283556366445 1.14626634918437

20 0.0244 0.0146 0.3000 0.25273028599849 0.30899346807501

50 0.0099 0.0059 0.3000 0.04228961505895 0.05178915247693

100 0.0050 0.0030 0.3000 0.01073685546106 0.01315173586584

200 0.0025 0.0015 0.3000 0.00270376524815 0.00331339981606

If we allow the CFL condition to be slightly greater than 0.5 like in Table 4.7, we get

lower relative error values than both Table 4.5 and Table 4.6 for all values of β except

for 200=β where the relative error seems to be increasing. Again, we attribute this

behavior to instability.

Even though the lower values for the inertia factor are stable and more accurate than

CFL=0.5 or CFL=0.3, at some point the instability will become significant and the

relative error values can get extremely high.


105

Table 4.7: Convergence Analysis ( )515.0=λ and ( )β


( )txxtxp −= 1),( 2


1 0.3333 0.3433 0.5150 32.63111650330222 36.17403864005965

2 0.2000 0.2060 0.5150 13.87982195892397 16.26647708578263

3 0.1429 0.1471 0.5150 7.59977306500535 9.06365535220007

4 0.1111 0.1144 0.5150 4.77760101445805 5.74142531187053

5 0.0909 0.0936 0.5150 3.27625719468587 3.95288439614659

10 0.0476 0.0490 0.5150 0.93766807811311 1.14467076869192

20 0.0244 0.0251 0.5150 0.25263169398202 0.30902310240013

50 0.0099 0.0102 0.5150 0.04231867924093 0.05180362052093

100 0.0050 0.0051 0.5150 0.01073543698805 0.01315181896916

200 0.0025 0.0026 0.5150 0.13845357927513 0.21392438445845

Comparing Table 4.5 to Table 4.1 from Darcy’s diffusivity equation, we can now

start seeing significant values for the relative error from the cubic polynomial, even

though it is slightly lower than the relative error from the fourth order polynomial. The

source of the error can be shown by analyzing the first partial derivative in space:

( )L+

Δ+=

ΔΔ−−Δ+

62),(),( 2xpp

xtxxptxxp

xxxx ( )363.4

The right hand side has a first order derivative in space and error terms, the first error

term with a third order derivative. To further check the approximation to the first


106

derivative, we test a cubic polynomial 3xp = .

xxxxx

ΔΔ−−Δ+

2)()( 33

( )364.4

( ) ( ) ( ) ( ) ( )2232233223

32

3333 xxx

xxxxxxxxxxxxΔ+=

ΔΔ+Δ−Δ+−Δ+Δ+Δ+

=

The first derivative is 23xp =′ . The approximation gives us the exact value of the

derivative and an error term. This is the source of the error for the cubic polynomial, and

it decreases as the grid size ( )xΔ decreases.

To find the convergence rate for the time step we can use the following expressions

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛

2

1

2

1 log~logtt

ee

α ( )365.4

( ) ( )11 log~log te Δα ( )366.4

For the convergence rate for the mesh size the corresponding expressions are:

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛

2

1

2

1 log~logxx

ee

α ( )367.4

( ) ( )11 log~log xe Δα ( )368.4

For the cubic function ( )txxtxp −= 1),( 2 , we can plot relative error values and their

corresponding mesh sizes in a Log-log plot, we get Figure 4.16.


107

Figure 4.16: Convergence Rate (Cubic Function) ( )xΔ for ( )5.0=λ

For the fourth order polynomial ( )txxtxp −= 1),( 3 , the corresponding plot, Figure

4.17 will be very similar because of the error values are close to those of the cubic

polynomial.


108

Figure 4.17: Convergence Rate (Fourth Order Function) ( )xΔ for ( )5.0=λ


The finite differences BTCS discretization will be applied to Forchheimer PDE7 in

one dimension. The implicit method is known to be unconditionally stable and therefore

the Fourier or Von Neumann method will be applied to check for stability of the implicit

method.


109





⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )369.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )370.4


aptap =),( , 0>t ( )371.4


bptbp =),( , 0>t ( )372.4

From rearranging and grouping constant terms from Equation (4.369) we get the

following:

jijiji ttxxttxxttxx xpB

xpA

tp

====== ∂∂

−∂∂

=∂∂

,,2

2

,

( )373.4

If we replace the time derivative above by the backward differencing scheme and the

space derivative above by the central differencing scheme, we get:

( ) ⎥⎦

⎤⎢⎣

⎡Δ−

−⎥⎦

⎤⎢⎣

⎡Δ

+−=

Δ− +−++++++−+

xpp

Bx

pppA

tpp jijijijijijiji

22 1,11,1

21,11,1,1,1, ( )374.4

After rearranging:

( )[ ] [ ] 1,1,11,11,11,1,12, 2

2 ++−++++++− +−ΔΔ

++−ΔΔ

−= jijijijijijiji ppptxBppp

xtAp ( )375.4


110

If we let ( )2x

tAΔΔ

=λ and xtB

ΔΔ

=2

γ this results in:

( ) ( ) ( ) 1,11,1,1, 21 ++++− −+++−= jijijiji pppp λγλγλ ( )376.4

where MMj

Ni,1...2,1,01...3,2,1

−=−=


12

321

−=−=

===

NiNi

iii

M

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 1,1,11,20,1

1,11,21,30,2

1,41,31,20,3

1,31,21,10,2

1,21,11,00,1

2121

212121

NNNN

NNNN

pppppppp

pppppppppppp

λγλγλλγλγλ

λγλγλλγλγλλγλγλ

−++++−=−++++−=

−++++−=−++++−=−++++−=

−−−

−−−−

M ( )377.4


built as follows:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−−

−+−−−+−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

1,

1,1

1,2

1,1

1,0

0,1

0,2

0,1

10000021000

0021000021000001

N

N

b

N

a

pp

ppp

pp

ppp

MMOOOMMM

λγλγλ


( )378.4


( ) )1()0( PFPrr

∗= ( )379.4

To solve the above Equation (4.379) above, we again rewrite the pressures at the next

time step in terms of the pressures at the previous time step:

( ) )0(1)1( PFPrr −∗= ( )380.4


111


The implicit method will also be applied to the boundary dominated period.


⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )381.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )382.4


aptap =),( , 0>t ( )383.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )384.4


from the general differences expression for Forchheimer diffusivity equation7:

( ) ( ) ( ) 1,11,1,1, 21 ++++− −+++−= jijijiji pppp λγλγλ ( )385.4

where JJj

Ni,1...2,1,0

...3,2,1−=

=


NiNi

iii

=−=

===

1

321

M

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 1,11,1,10,

1,1,11,20,1

1,41,31,20,3

1,31,21,10,2

1,21,11,00,1

2121

212121

+−

−−−

−++++−=−++++−=

−++++−=−++++−=−++++−=

NNNN

NNNN

pppppppp

pppppppppppp



M ( )386.4


112


( ) ( ) ( ) 1,11,1,10, 21 −− −++++−= NNNN pppp λγλγλ

( ) ( )1,11,0, 221 −−+= NNN ppp λλ ( )387.4


⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−+−−

−+−−−+−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

1,

1,1

1,2

1,1

1,0

0,1

0,2

0,1

212000021000

0021000021000001

N

N

b

N

a

pp

ppp

pp

ppp

MMOOOMMM

λλλγλγλ


( )388.4

The matrix can be written in vector form:

( ) )1()0( PGPrr

∗= ( )389.4



( ) )0(1)1( PGPrr −∗= ( )390.4


The implicit BTCS scheme will be applied to the pseudosteady state flow regime with

the following initial and boundary conditions.


⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ +=

∂∂

xpv

tpv

kc

xp φβρμ 22

2

( )391.4

Initial condition:


113

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )392.4


βρ

βρμμ

2

4,

2

0 jttxxxp

kkq

==⎥⎦⎤

⎢⎣⎡∂∂

+⎟⎠⎞

⎜⎝⎛±−

= , 0>t ( )393.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )394.4

We now incorporate the conditions to our finite differences approach starting from

the general differences expression for Forchheimer diffusivity equation7:

( ) ( ) ( ) 1,11,1,1, 21 ++++− −++++−= jijijiji pppp λγλγλ ( )395.4

where JJj

Ni,1...2,1,0

...3,2,1,0−=

=


NiNi

iii

=−=

===

1

210

M

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) 1,11,1,10,

1,1,11,20,1

1,31,21,10,2

1,21,11,00,1

1,11,01,10,0

2121

212121

+−

−−−

−

−++++−=−++++−=

−++++−=−++++−=−++++−=

NNNN

NNNN

pppppppp

pppppppppppp



M ( )396.4

Recall:

( ) ( ) 2,1,1 22 qxq

kxpp jj βρμ

Δ−Δ−=− ( )397.4

For 0=i we get:


114

( ) ( ) ( ) ( ) ( ) 1,11,02

1,10,0 2122 ppqxqk

xpp λγλβρμγλ −+++⎟⎠⎞

⎜⎝⎛ Δ−Δ−+−=

( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ+Δ−−+= 2

1,11,00,0 22221 qxqk

xppp βρμλλ ( )398.4

For Ni = we get:

( ) ( )1,11,0, 221 −−+= NNN ppp λλ ( )399.4


( ) ( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛ Δ+Δ

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−+−−

−+−−−+−−

−+

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

00000

22

212000021000

00210000210000221 2

1,

1,1

1,2

1,1

1,0

0,1

0,2

0,1

qxqk

x

pp

ppp

pp

ppp

N

N

b

N

a βρμλ

λλλγλγλ


λλ

MMOOOMMM


( ) ∗∗ += FPHPrrr

)1()0( ( )401.4



( ) ( )∗−∗ −= FPHPrrr

)0(1)1( ( )402.4

4.2.8. Stability Analysis for the Implicit Scheme

The Fourier or Von Neumann19 method will be applied to the implicit scheme. Let us




115


suitable choice of λ . We notice that if 1>Δtje λ , the solution becomes unbounded and so


We now consider the implicit difference equation of the PDE which is:

( ) ( ) ( ) 1,11,1,1, 21 ++++− −++++−= jijijiji pppp λγλγλ ( )404.4


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) tjxiItjxIitjxiItjxIi eeeeeeee Δ+Δ+Δ+ΔΔ+Δ−ΔΔ −++++−= λβλβλβλβ λγλγλ 11111 21 ( )405.4


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )jjtiixIjjtjjtiixI eeeee −+Δ−+Δ−+Δ−+Δ−−Δ −++++−= )1()1()1()1()1( 211 λβλλβ λγλγλ ( )406.4

After simplifying:

( ) ( ) ( ) txIttxI eeeee ΔΔΔΔΔ− −++++−= λβλλβ λγλγλ 211 ( )407.4


( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )( ) ( ) ( )( )

( )

( ) ( )xIxe

xIxexIxe

eexIxxIxeeeeee

t

t

t

xIxIt

xIxIxIxIt

Δ+Δ−+=

++Δ+Δ−=++Δ+Δ−=

++−+Δ+Δ+Δ−Δ−=++−++−=

Δ

Δ

Δ

Δ−ΔΔ

Δ−ΔΔΔ−Δ

βγβλ

λβγβλλβγβλ

λγββββλλγλ

λ

λ

λ

ββλ

ββββλ

sin2cos1211

21sin2cos2121sin2cos21

21sincossincos1211

( )408.4

where te Δλ is the amplification factor. We can also see that 1≤Δteλ for any value of λ ,

and so the method is unconditionally stable.


116

4.3. DARCY-FORCHHEIMER EQUATION

In this section, the new Darcy-Forchheimer3 equation will be analyzed by deriving a

general Darcy-Forchheimer diffusivity equation3. Both the explicit and implicit finite

differences techniques will be applied. A stability analysis will be conducted, and the

pressure profiles for different initial and boundary conditions will be plotted from the

MATLAB code generated.

4.3.1. Darcy-Forchheimer Diffusivity Equation

A more general expression for Forchheimer diffusivity equation7 can be derived by

writing velocity as a vector and a function of pressure gradient.

We start with the following form of the Forchheimer’s equation3:

vvvk

p rrr βμ+=∇− ( )409.4

where 2

1kFρφβ = and 22

yx vvv +=r

An expression for velocity can be expressed as follows:

( ) ppfv ∇∇−= βr and ( ) ppfv ∇∇= β

r ( )410.4

where f is a non-linear diffusion coefficient.

Substituting vr and vr on Forchheimer’s equation3 we get:

( )( ) ( )( ) ( )( )ppfppfppfk

p ∇∇−∇∇+∇∇−=∇− βββ βμ ( )411.4

Multiplying both sides by )1(− and rearranging:


117

( )[ ] ( )( ) 012 =−∇+∇∇ pfk

ppf ββμβ ( )412.4

Solving the quadratic equation we get:

( )p

pf kk

∇

∇+±−=

β

βμμ

ββ 24

2

, 21 ( )413.4

Using the positive root and multiplying numerator and denominator by

pkk

∇+⎟⎠⎞

⎜⎝⎛+ βμμ 4

2

gives:

( )( ) p

pfkk ∇++

=∇βμμ

β

4

22

( )414.4

The continuity equation applied to a porous medium of porosityφ can be written in

the equation below. Porosity22 is defined as the fraction of the bulk volume of the

reservoir that is not occupied by the solid framework of the reservoir b

p

b

grb

VV

VVV

=−

=φ .

0)( =+∂∂ vdiv

trρρφ ( )415.4

After rewriting and expanding the right hand side we get:

vvdivt

rr ρρρφ ′−−=∂∂ )( ( )416.4

where p∂∂

=′ρρ and using chain rule in the x-direction for

xp

px ∂∂

∂∂

=∂∂ ρρ and

tp

pt ∂∂

∂∂

=∂∂ ρρ

Equation (4.416) above in one dimension translates to:

vxp

pxv

tp

p ∂∂

∂∂

−∂∂

−=∂∂

∂∂ ρρρφ or more generally,


118

pvvdivtp

∇•′−−=∂∂′ rr ρρρφ )( ( )417.4

For an incompressible fluid, we have 0=∂∂

pρ . For a slightly compressible fluid, we

have ρρ cp=

∂∂ . If the expression for velocity is incorporated in the term pv ∇•′rρ , and for

a slightly compressible fluid, we get ( )( )2ppfc ∇∇βρ . Since the product ( )2pc ∇ is

assumed to be small, the term pv ∇•′rρ can be neglected to get:

)(vdivtp rρρφ −=∂∂′ ( )418.4

For a slightly compressible fluid, we have ργρ 1−=′ and ( )oppe −−

=1

0γρρ , where

c=−1γ is the fluid compressibility. The equation above can be written as:

)(1 vdivtp rρρφγ −=∂∂− ( )419.4


)(vdivtp r

φγ

−=∂∂

( )( )ppfdivtp

∇∇=∂∂

βφγ

( )( )ppftp

∇∇∇=∂∂

βφγ ( )420.4

Note that when 0=β we get Darcy’s equation11:

vk

p rμ=∇− ( )421.4

and also Darcy’s diffusivity equation:


119

( )( ) μμμ

βkpf

kk

=+

=∇2

2 ( )422.4

⎟⎟⎠

⎞⎜⎜⎝

⎛∇∇=

∂∂ pk

tp

μφγ ( )423.4

which can be rewritten as:

pktp 2∇=∂∂

μφγ ( )424.4

By rearranging terms, letting 1−= γc and considering only the x-direction we get:

tp

kc

xp

∂∂

=∂∂ φμ

2

2

( )425.4


The finite differences FTCS discretization will be applied to Darcy-Forchheimer3

PDE for different initial and boundary conditions.



boundary conditions. Mathematically, these conditions can be expressed as follows in the

interval ),( bax∈ :

Darcy-Forchheimer3 diffusivity equation:

( )( )ppftp

∇∇∇=∂∂

βφγ ( )426.4

Initial condition:


120

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )427.4


aptap =),( , 0>t ( )428.4


bptbp =),( , 0>t ( )429.4

We will now rename ( ) ( )pfp ∇=∇ βα and will refer to it as the alpha function. If we

consider the Darcy-Forchheimer3 diffusivity equation in 1-D we get:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

xp

xp

xtp α

φγ ( )430.4

This appears to be a more complex problem, and to approximate the solution, a three

step process will be followed. First, the alpha function ( )?α will be assumed to be a

function of distance x only. Then, the alpha function will be assumed to be a function of

pressure ( )xp only. The final approach will be solving the actual problem by letting the

alpha function be a function of the absolute value of pressure gradient xp∂∂ .

4.3.2.1.1. Alpha as a Function as a Function of Distance

Finite differences will be applied to Darcy-Forchheimer3 PDE assuming alpha is a

function of x only (i.e. )(xα ) as a first approach to the non-linear problem. The equation

reduces to:


121

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

xpx

xtp )(α

φγ ( )431.4

With the FTCS discretization we get:

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Δ

⎥⎦⎤

⎢⎣⎡

∂∂

−⎥⎦⎤

⎢⎣⎡

∂∂

=Δ

− −++

xxpx

xpx

tpp jijijiji ,,,1, 2

12

1

)()( αα

φγ ( )432.4

After expanding:

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−

=Δ

−−

−+

++

xxpp

xx

ppx

tpp

jijiji

jijiji

jiji

,1,,

,,1,

,1,

)()(2

12

1 αα

φγ ( )433.4

The right hand side of Equation (4.433) can be expanded to get: ( )434.4

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ

⎥⎦

⎤⎢⎣

⎡

Δ

−−⎥

⎦

⎤⎢⎣

⎡

Δ

−

=Δ

−−−−+++

+

xx

pxpxx

pxpx

tpp

jijijijijijijiji

jiji

,1,,,,,,1,

,1,

)()()()(2

12

12

12

1 αααα

φγ

After simplifying right hand side:

( )( ) ⎥

⎦

⎤⎢⎣

⎡

Δ

++−=

Δ

− −−−++++2

,1,,,,,1,,1, )()()()(2

12

12

12

1

x

pxpxxpxt

pp jijijijijijijijiji αααα

φγ ( )435.4

We now rearrange terms: ( )436.4

( )( )[ ] jijijijijijijijiji ppxpxxpx

xtp ,,1,,,,,1,21, )()()()(

21

21

21

21 +++−

ΔΔ

= −−−++++ ααααφγ


122

Let ( )2x

tΔΔ

=φγλ and expand:

( ) jijijijijijijijiji ppxpxxpxp ,,1,,,,,1,1, )()()()(2

12

12

12

1 +++−= −−−++++ λαααλλα ( )437.4

Combining like terms:

( )[ ] jijijijijijijiji pxpxxpxp ,1,,,,,1,1, )()()(1)(2

12

12

12

1 −−−++++ ++−+= λαααλλα ( )438.4

where MMj

Ni,1,...2,1,01,...3,2,1

−=−=

We now evaluate for different values of i and letting 0=j . ( )439.4

( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 0,20,10,10,10,0,0,1,1

0,30,20,20,20,10,10,11,2

0,20,20,30,20,30,40,31,3

0,10,10,20,10,20,30,21,2

0,00,0,10,0,10,20,11,1

)()()(1)()()()(1)(

)()()(1)()()()(1)(

)()()(1)(

12

321

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

−−−−−−−

−−−−−−−−

++−+=++−+=

++−+=++−+=++−+=

−=−=

===

NNNNNNNN

NNNNNNNN

pxpxxpxppxpxxpxp

pxpxxpxppxpxxpxp

pxpxxpxp

NiNi

iii

λαααλλαλαααλλα

λαααλλαλαααλλαλαααλλα

MM


built as follows: ( )440.4

( )[ ]

( )[ ]⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

b

N

a

NNNN

N

N

pp

pp

xxxx

xxxx

pp

pp

0,1

0,1

0,0,10,0,1

0,10,0,10,

1,

1,1

1,1

1,0

10000)()()(1)(00

00)()()(1)(00001

21

21

21

21

21

21

21

21

MMOOOMM

λαααλλα

λαααλλα


)0()1( PIPrr

= ( )441.4


123

4.3.2.1.2. Alpha Function as a Function of Pressure

Finite differences will be applied to Darcy-Forchheimer3 PDE assuming alpha is a

function of pressure ( ) pxp = only (i.e. )( pα ) as a second approach to the non-linear

problem. The equation reduces to:

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

xpp

xtp )(α

φγ ( )442.4


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Δ

⎥⎦⎤

⎢⎣⎡

∂∂

−⎥⎦⎤

⎢⎣⎡

∂∂

=Δ

− −++

xxpp

xpp


12

1

)()( αα

φγ ( )443.4

After expanding:

( ) ( )

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−

=Δ

−−

−+

++

xxpp

px

ppp

tpp

jijiji

jijiji

jiji

,1,,

,,1,

,1,2

12

1 αα

φγ ( )444.4

The right hand side can be expanded to get: ( )445.4

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ

⎥⎦

⎤⎢⎣

⎡

Δ

−−⎥

⎦

⎤⎢⎣

⎡

Δ

−

=Δ

−−−−+++

+

xx

ppppx

pppp

tpp

jijijijijijijiji

jiji

,1,,,,,,1,

,1,

)()()()(2

12

12

12

1 αααα

φγ

After simplifying right hand side:

( )( ) ⎥

⎦

⎤⎢⎣

⎡

Δ

++−=

Δ

− −−−++++2

,1,,,,,1,,1, )()()()(2

12

12

12

1

x

pppppppt


φγ ( )446.4


124

We now rearrange terms: ( )447.4

( )( )[ ] jijijijijijijijiji pppppppp

xtp ,,1,,,,,1,21, )()()()(

21

21

21

21 +++−

ΔΔ

= −−−++++ ααααφγ

Let ( )2x

tΔΔ

=φγλ and expand:

( ) jijijijijijijijiji ppppppppp ,,1,,,,,1,1, )()()()(2

12

12

12

1 +++−= −−−++++ λαααλλα ( )448.4


( )[ ] jijijijijijijiji pppppppp ,1,,,,,1,1, )()()(1)(2

12

12

12

1 −−−++++ ++−+= λαααλλα ( )449.4

where MMj

Ni,1,...2,1,01,...3,2,1

−=−=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 0,20,10,10,10,0,0,1,1

0,30,20,20,20,10,10,11,2

0,20,20,30,20,30,40,31,3

0,10,10,20,10,20,30,21,2

0,00,0,10,0,10,20,11,1

)()()(1)()()()(1)(

)()()(1)()()()(1)(

)()()(1)(

12

321

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

−−−−−−−

−−−−−−−−

++−+=++−+=

++−+=++−+=++−+=

−=−=

===

NNNNNNNN

NNNNNNNN

pppppppppppppppp

pppppppppppppppp

pppppppp

NiNi

iii



MM



( )[ ]

( )[ ]⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

b

N

a

NNNN

N

N

pp

pp

pppp

pppp

pp

pp

0,1

0,1

0,0,10,0,1

0,10,0,10,

1,

1,1

1,1

1,0

10000)()()(1)(00

00)()()(1)(00001

21

21

21

21

21

21

21

21

MMOOOMM

λαααλλα

λαααλλα

In vector form:

)0()1( PJPrr

= ( )452.4


125

4.3.2.1.3. Alpha Function as a Function of the Absolute Value of Pressure Gradient

Finite differences will be applied to Darcy-Forchheimer3 PDE assuming the alpha

function is a function of the absolute value of the pressure gradient (i.e. ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

xpα ) as the

final approach and actual approximation to the solution of the non-linear problem.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

xp

xp

xtp α

φγ ( )453.4


⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ

⎥⎦

⎤⎢⎣

⎡∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−⎥⎦

⎤⎢⎣

⎡∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=Δ

− −++

x

xp

xp

xp

xp


12

1

αα

φγ ( )454.4

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

−

=Δ

−−−++

+

x

xpp

xpp

xpp

xpp

tpp

jijijijijijijiji

jiji

,1,,1,,,1,,1

,1,

αα

φγ

For simplicity we let ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=xpαα . The right hand side can be expanded to get:

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ

⎥⎦

⎤⎢⎣

⎡

Δ

−−⎥

⎦

⎤⎢⎣

⎡

Δ

−

=Δ

−−−−+++

+

xx

ppx

pp

tpp

jijijijijijijiji

jiji

,1,,,,,,1,

,1,

21

21

21

21 αααα

φγ ( )455.4

After simplifying the right hand side:


126

( )( ) ⎥

⎦

⎤⎢⎣

⎡

Δ

++−=

Δ

− −−−++++2

,1,,,,,1,,1, 21

21

21

21

x

pppt


φγ ( )456.4

We now rearrange terms:

( )( )[ ] jijijijijijijijiji pppp

xtp ,,1,,,,,1,21, 2

12

12

12

1 +++−ΔΔ

= −−−++++ ααααφγ ( )457.4

Let ( )2x

tΔΔ

=φγλ and expand:

( ) jijijijijijijijiji ppppp ,,1,,,,,1,1, 21

21

21

21 +++−= −−−++++ λαααλλα ( )458.4


( )[ ] jijijijijijijiji pppp ,1,,,,,1,1, 21

21

21

21 1 −−−++++ ++−+= λαααλλα ( )459.4

where MMj

Ni,1,...2,1,01,...3,2,1

−=−=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 0,20,10,10,10,0,0,1,1

0,30,20,20,20,10,10,11,2

0,20,20,30,20,30,40,31,3

0,10,10,20,10,20,30,21,2

0,00,0,10,0,10,20,11,1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

111

12

321

−−−−−−−

−−−−−−−−

++−+=++−+=

++−+=++−+=++−+=

−=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppp

pppp

NiNi

iii



MM




127

( )[ ]

( )[ ]⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

b

N

a

NNNN

N

N

pp

pp

pp

pp

0,1

0,1

0,0,10,0,1

0,10,0,10,

1,

1,1

1,1

1,0

10000100

00100001

21

21

21

21

21

21

21

21

MMOOOMM

λαααλλα

λαααλλα


)0()1( PLPrr

= ( )462.4

A computer code can be written and the pressure profile for arbitrary values for

fluid and rock are shown in Figure 4.18.

Figure 4.18: Pressure Profile for Darcy-Forchheimer3 (CPB) Explicit ( )5.0=λ


128


We will assume constant bottomhole pressure during production and no flow across

the reservoir’s outer boundary at a distance bxx N == . These conditions can be

expressed mathematically as follows:


( )( )ppftp

∇∇∇=∂∂

βφγ ( )463.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )464.4


aptap =),( , 0>t ( )465.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )466.4

These boundary and initial conditions will only be applied to the third approach

where the alpha function is a function of the absolute value of the pressure gradient

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

xpα .

To satisfy the outer boundary condition we must have ( ) 02

,1,1 =Δ

− −+

xpp jNjN and

therefore jNjN pp ,1,1 −+ = . Note that jNp ,1+ is located at a node outside the interval [ ]ba,

and consequently, the alpha value jN ,21+α will also be located outside the interval [ ]ba, .


129

To calculate the pressure gradient at the boundary we will assume that the pressure


This same concept will be used to calculate the alpha value at 2

1+Nx :

jNjNNjNjN

jNjN x

ppx

ppxp

,,1,,1

,, 2

1

21

21 −

−+

++ =⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ−

=⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

= ααααα ( )467.4


from the general differences expression for Darcy-Forchheimer3 diffusivity equation:


21

21

21 1 −−−++++ ++−+= λαααλλα ( )468.4

where JJj

Ni,1...2,1,0

...3,2,1−=

=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 0,10,0,0,0,0,10,1,

0,20,10,10,10,0,0,1,1

0,20,20,30,20,30,40,31,3

0,10,10,20,10,20,30,21,2

0,00,0,10,0,10,20,11,1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

111

1

321

−−−+++

−−−−−−−

++−+=++−+=

++−+=++−+=++−+=

=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppp

pppp

NiNi

iii



MM( )469.4


( )[ ] 0,10,0,0,0,0,10,1, 21

21

21

21 1 −−−+−+ ++−+= NNNNNNNN pppp λαααλλα ( )470.4

It was shown above that jNjN ,, 21

21 −+ −= αα and after substituting we get:

( )[ ] 0,10,0,0,0,0,10,1, 21

21

21

21 1 −−−−−− ++−+= NNNNNNNN pppp λαααλλα

( )[ ] ( ) 0,10,0,0,1, 21

21 221 −−− +−= NNNNN ppp λααλ ( )471.4


130

We now build the matrix: ( )472.4

( )[ ]

( )[ ]( )[ ] ⎥

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−

+−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−−−−−

b

N

a

NN

NNNN

N

N

pp

pp

pp

pp

0,1

0,1

0,0,

0,0,10,0,1

0,10,0,10,

1,

1,1

1,1

1,0

21

21

21

21

21

21

21

21

21

21

212000100

00100001

MMOOOMM

αλλαλαααλλα

λαααλλα

This matrix can be abbreviated as follows:

)0()1( POPrr

= ( )473.4

The pressure profile for the Darcy-Forchheimer diffusivity equation is presented in

Figure 4.19 below. It shows a stable pressure profile for CFL=0.5.

Figure 4.19: Pressure Profile for Darcy-Forchheimer3 (BDR) Explicit ( )5.0=λ


131



boundary at a distance bxx N == . These conditions can be expressed mathematically as

follows:


( )( )ppftp

∇∇∇=∂∂

βφγ ( )474.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )475.4


jj ttxxttxx xp

xpfq

====⎥⎦⎤

⎢⎣⎡∂∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

=,, 00

β , 0>t ( )476.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )477.4



( )xpp

xp jj

j Δ−

=⎥⎦⎤

⎢⎣⎡∂∂ −

2,1,1

,0

( )478.4


( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ−

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

+⎟⎠⎞

⎜⎝⎛+

= −

−xpp

xpp

kk

q jj

jj2

24

2 ,1,1

,1,12

βμμ ( )479.4


132

We notice that the pressures jp ,1− are located outside the interval [ ]ba, . We can

rearrange the above equation and after solving for jp ,1− we get:

( ) ( ) 2,1,1 22 qxq

kxpp jj βρμ

Δ−Δ−=− ( )480.4

Note that jp ,1− is located at a node outside the interval [ ]ba, and consequently, the

alpha value j,21−α will also be located outside the interval [ ]ba, . To get the pressure

gradient at the inner boundary, the value derived above for jp ,1− will be used and the

alpha value at 2

1−x will be estimated as follows: ( )481.4

( ) ( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎠⎞

⎜⎝⎛ Δ−Δ−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

= −

−− x

qxqk

xpp

xpp

xp jj

jj

jj

2,1,0

,1,0

,,

22

21

21

βρμ

αααα

These alpha values can be calculated separately at each time step by a computer

program and their values substituted into the main equation.



expression for Darcy-Forchheimer3 diffusivity equation:


21

21

21 1 −−−++++ ++−+= λαααλλα ( )482.4

where JJj

Ni,1...2,1,0

...3,2,1,0−=

=



133

( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 0,10,0,0,0,0,10,1,

0,20,10,10,10,0,0,1,1

0,10,10,20,10,20,30,21,2

0,00,0,10,0,10,20,11,1

0,10,0,00,0,0,10,1,0

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

11

1

1

210

−−−+++

−−−−−−−

−−−

++−+=++−+=

++−+=++−+=++−+=

=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppppppp

NiNi

iii



MM( )483.4

For 0=i we get:

( )[ ] ( ) ( ) ⎟⎠⎞

⎜⎝⎛ Δ−Δ−++−+= −−

20,10,0,00,0,0,10,1,0 221

21

21

21

21 qxq

kxpppp βρμλαααλλα

( )[ ] ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ+Δ−+++−= −−

20,10,0,0,00,0,1,0 221

21

21

21

21 qxq

kxppp βρμααλααλ ( )484.4

For Ni = with jNjN pp ,1,1 −+ = and jNjN ,, 21

21 −+ −= αα we get:

( )[ ] 0,10,0,0,0,0,10,1, 21

21

21

21 1 −−−+−+ ++−+= NNNNNNNN pppp λαααλλα

( )[ ] ( ) 0,10,0,0,0,1, 21

21

21 21 −−−+ ++−= NNNNNN ppp αλααλ ( )485.4


( )[ ] ( )( )[ ]

( )[ ] ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

+−++−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

b

a

NNN p

pp

p

pp

MOOOMM0,1

0,0,

0,10,0,10,

0,0,0,0,

1,

1,1

1,0

21

21

21

21

21

21

21

21

21

21

21200

01001

αλλα

λαααλλαααλααλ

( ) ( )

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛ Δ+Δ−

+

000

22 2qxqk

x βρμλ



134

GPRPrrr

+= )0()1( ( )487.4

As before, a computer program can be generated, and the pressure profile can be

plotted as in Figure 4.20.

Figure 4.20: Pressure Profile for Darcy-Forchheimer3 (PSS) Explicit ( )5.0=λ


It has been proven previously that for Darcy’s diffusivity equation, the condition

5.0≤λ must be met for stability. If we recall Darcy’s diffusivity equation,

2

2

xpK

tp

∂∂

=∂∂ ( )488.4


135

where c

kKμφ

= , we can write the maximum time step for Darcy’s diffusivity equation

as:

( )KxtD

2

max 21 Δ

=Δ ( )489.4

Recall Darcy-Forchheimer3 diffusivity Equation:

( )( )ppftp

∇∇∇=∂∂

βφγ ( )490.4

Because for the explicit method the function non-linear diffusion coefficient ( )pf ∇β

is computed at the previous time step, its value is always known at the time step being

considered; we can refer to it as a function )(xf . The maximum time step for Darcy-

Forchheimer3 diffusivity equation can be expressed as follows.

( )

[ ]bax

F xfxt

,min

2

max )(21

∈

Δ=Δ

φγ ( )491.4

We also recall that the maximum value of the function is μkxf =max)( , and we can

write the following inequality:

( )

[ ]

( )max

2

,min

2

max 21

)(21

D

bax

F tKx

xfxt Δ=

Δ≥

Δ=Δ

∈φγ ( )492.4

The relationship between the time steps for Darcy’s diffusivity equation and Darcy-

Forchheimer3 diffusivity equation is given by:

maxmax DF tt Δ≥Δ ( )493.4

This tells us that the Darcy-Forchheimer3 diffusivity equation allows for a bigger time


136

step than Darcy’s before becoming unstable. We can also show that for Darcy-

Forchheimer3 equation the CFL condition can be greater than Darcy’s.

( )KxtF

2

max 21 Δ

≥Δ

( ) 21

2max ≥

ΔΔ

xtK F

21

≥Fλ ( )494.4

In Figures 4.21, 4.22 and 4.23 we have pressure profiles for low β values; we notice

that instability does not start until values as high as twice the CFL stability condition.

Figure 4.21: Unstable Pressure Profile for Darcy-Forchheimer3 (CPB) Explicit ( )1=λ


137

Figure 4.22: Unstable Pressure Profile for Darcy-Forchheimer3 (BDP) Explicit ( )1=λ


138

Figure 4.23: Unstable Pressure Profile for Darcy-Forchheimer3 (PSS) Explicit ( )3=λ


The implicit BTCS method will be applied to Darcy-Forchheimer3 diffusivity

equation. This version of the diffusivity equation has a non-linear term that must be

handled carefully. Various approaches will be considered to approximate the solution for

this non-linear problem and the discrepancies between the methods will be analyzed.


139

4.3.4.1. Implicit Discretization – (A)

In this first approach, the alpha function ( )p∇α will be evaluated at some thj time

( )[ ]ji

p,

∇α which can be pre-computed and therefore the alpha function is known at each

time.

4.3.4.1.1. Constant Pressure Boundaries





( )( )ppftp

∇∇∇=∂∂

βφγ ( )495.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )496.4


aptap =),( , 0>t ( )497.4


bptbp =),( , 0>t ( )498.4

If we rewrite the PDE in one dimension we get:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

xp

xp

xtp α

φγ ( )499.4

Again we let ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=xpαα . The right hand side can be expanded to get:


140

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Δ

⎥⎦

⎤⎢⎣

⎡Δ

−−⎥

⎦

⎤⎢⎣

⎡Δ

−

=Δ−

+−−+−+++++

+

xx

ppx

pp

tpp

jijijijijijijiji

jiji

1,1,1,,1,,1,1,

,1,

21

21

21

21 αααα

φγ ( )500.4


( )( ) ⎥

⎦

⎤⎢⎣

⎡

Δ

++−=

Δ− +−−+−+++++

21,1,1,,,1,1,,1, 2

12

12

12

1

xppp

tpp jijijijijijijijiji αααα

φγ ( )501.4


( )( )[ ] 1,1,1,1,,,1,1,2, 2

12

12

12

1 ++−−+−++++ +++−ΔΔ

−= jijijijijijijijiji ppppxtp αααα

φγ ( )502.4

Let ( )2x

tΔΔ

=φγλ and expand:

( ) 1,1,1,1,,,1,1,, 21

21

21

21 ++−−+−++++ +−++−= jijijijijijijijiji ppppp λαααλλα ( )503.4


( )[ ] 1,1,1,,,1,1,, 21

21

21

21 1 +−−+−++++ −+++−= jijijijijijijiji pppp λαααλλα ( )504.4

where MMj

Ni,1,...2,1,01,...3,2,1

−=−=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 1,20,11,10,10,1,0,0,1

1,30,21,20,20,11,10,10,2

1,20,21,30,20,31,40,30,3

1,10,11,20,10,21,30,20,2

1,00,1,10,0,11,20,10,1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

111

12

321

−−−−−−−

−−−−−−−−

−+++−=−+++−=

−+++−=−+++−=−+++−=

−=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppp

pppp

NiNi

iii



MM( )505.4



141

built as follows:

( )[ ]

( )[ ]⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−++−

−++−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

1,

1,1

1,1

1,0

0,0,10,0,1

0,10,0,10,

0,1

0,1

10000100

00100001

21

21

21

21

21

21

21

21

N

NNNNN

b

N

a

pp

pp

pp

pp

MMOOOMM

λαααλλα

λαααλλα


( ) )1()0()0( PPLPrr

= ( )506.4

To solve the above equation, we again rewrite the pressures at the next time step in

terms of the pressures at the previous time step:

( )[ ] )0(1)0()1( PPLPrr −

= ( )507.4

We can see in Figure 4.24 below that we get the same profile as the explicit method

applied before. The main difference is the CFL condition and a better resolution.


142

Figure 4.24: Pressure Profile for Darcy-Forchheimer3 (CPB) Implicit ( )125=λ

4.3.4.1.2. Boundary Dominated Regime

The implicit – (A) method will now be applied to the boundary dominated period for


( )( )ppftp

∇∇∇=∂∂

βφγ ( )508.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )509.4



143

aptap =),( , 0>t ( )510.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )511.4



( )[ ] 1,1,1,,,1,1,, 21

21

21


where JJj

Ni,1...2,1,0

...3,2,1−=

=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 1,10,1,0,0,1,10,0,

1,20,11,10,10,1,0,0,1

1,20,21,30,20,31,40,30,3

1,10,11,20,10,21,30,20,2

1,00,1,10,0,11,20,10,1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

111

1

321

−−−+++

−−−−−−−

−+++−=−+++−=

−+++−=−+++−=−+++−=

=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppp

pppp

NiNi

iii



MM( )513.4

Recall that jNjN pp ,1,1 −+ = , we now get for Ni = :

( )[ ] 1,10,1,0,0,1,10,0, 21

21

21

21 1 −−−+−+ −+++−= NNNNNNNN pppp λαααλλα ( )514.4

It was also shown earlier that jNjN ,, 21


( )[ ] 1,10,1,0,0,1,10,0, 21

21

21

21 1 −−−−−− −+++−= NNNNNNNN pppp λαααλλα

( )[ ] ( ) 1,10,1,0,0, 21

21 221 −−− −+= NNNNN ppp λααλ ( )515.4

We now build the matrix: ( )516.4


144

( )[ ]

( )[ ]( )[ ] ⎥

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−++−

−++−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−−−−−

1,

1,1

1,1

1,0

0,0,

0,0,10,0,1

0,10,0,10,

0,1

0,1

21

21

21

21

21

21

21

21

21

21

212000100

00100001

N

N

NN

NNNN

b

N

a

pp

pp

pp

pp

MMOOOMM


λαααλλα

This matrix can be abbreviated as follows:

( ) )1()0()0( PPOPrr

= ( )517.4

To solve the above equation, we again rewrite the pressures at the next time step in

terms of the pressures at the previous time step:

( )[ ] )0(1)0()1( PPOPrr −

= ( )518.4

The pressure profile for the boundary dominated regime is illustrated in Figure 4.25.

A higher CFL value translates into more refined grid sizes and smaller time steps will not

result in unstable results.


145

Figure 4.25: Pressure Profile for Darcy-Forchheimer3 (BDR) Implicit ( )125=λ

4.3.4.1.3. Pseudosteady State Flow Regime

We can also apply the implicit – (A) method to the pseudoseady state regime.


( )( )ppftp

∇∇∇=∂∂

βφγ ( )518.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )519.4



146

jj ttxxttxx xp

xpfq

====⎥⎦⎤

⎢⎣⎡∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=,, 00

β , 0>t ( )520.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )521.4



( )xpp

xp jj

j Δ−

=⎥⎦⎤

⎢⎣⎡∂∂ −

2,1,1

,0

( )522.4


( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ−

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

+⎟⎠⎞

⎜⎝⎛+

= −

−xpp

xpp

kk

q jj

jj2

24

2 ,1,1

,1,12

βμμ ( )523.4

We can rearrange the above equation and after solving for jp ,1− we get:

( ) ( ) 2,1,1 22 qxq

kxpp jj βρμ

Δ−Δ−=− ( )524.4

To get the pressure gradient at the inner boundary, the value derived above for jp ,1−

will be used and the alpha value at 21−=x can be estimated as follows: ( )525.4

( ) ( )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎠⎞

⎜⎝⎛ Δ−Δ−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

= −

−− x

qxqk

xpp

xpp

xp jj

jj

jj

2,1,0

,1,0

,,

22

21

21

βρμ

αααα

These alpha values can be calculated separately at each time step by a computer

program and their values incorporated into the main equation.


147

( )[ ] 1,1,1,,,1,1,, 21

21

21


where JJj

Ni,1...2,1,0

...3,2,1,0−=

=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 1,10,1,0,0,1,10,0,

1,20,11,10,10,1,0,0,1

1,10,11,20,10,21,30,20,2

1,00,1,10,0,11,20,10,1

1,10,1,00,0,1,10,0,0

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

11

1

1

210

−−−+++

−−−−−−−

−−−

−+++−=−+++−=

−+++−=−+++−=−+++−=

=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppppppp

NiNi

iii



MM( )527.4

For 0=i we get:

( )[ ] ( ) ( ) ⎟⎠⎞

⎜⎝⎛ Δ−Δ−−+++−= −−

21,10,1,00,0,1,10,0,0 221

21

21

21

21 qxq


( )[ ] ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ+Δ−+−++= −−

21,10,0,1,00,0,0,0 221

21

21

21

21 qxq


For Ni = with jNjN pp ,1,1 −+ = and jNjN ,, 21

21 −+ −= αα we get:

( )[ ] 1,10,1,0,0,1,10,0, 21

21

21

21 1 −−−+−+ −+++−= NNNNNNNN pppp λαααλλα

( )[ ] ( ) 1,10,1,0,0,0, 21

21

21 21 −−−+ −++= NNNNNN ppp αλααλ ( )529.4


( )[ ] ( )( )[ ]

( )[ ] ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+−

−+++−++

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

1,

1,1

1,0

0,0,

0,10,0,10,

0,0,0,0,

0,1

21

21

21

21

21

21

21

21

21

21

21200

01001

NNNb

a

p

pp

p

pp

MOOOMM

αλλα



148

( ) ( )

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛ Δ+Δ

+

000

22 2qxqk

x βρμλ

( )530.4

The matrix can be written in vector form:

( ) ∗+= GPPRPrrr

)1()0()0( ( )531.4

To solve the above equation, we rewrite the pressures at the next time step in terms of

the pressures at the previous time step:

( )[ ] ( )∗−−= GPPRPrrr

)0(1)0()1( ( )532.4

The pressure profile for pseudosteady state conditions is included in Figure 4.26. We

note again that the pressure profile is very similar to the profile developed with the

explicit method. This shows that the codes are giving reliable and consistent results for

two different methods.


149

Figure 4.26: Pressure Profile for Darcy-Forchheimer3 (PSS) Implicit ( )125=λ

4.3.4.2. Implicit Discretization – (B)

For this second approach, the alpha function ( )p∇α will be evaluated at time ( )1+j

( )[ ]1, +

∇ji

pα . Now the alpha function is unknown because the pressures at time ( )1+j are

unknown, and to calculate those pressures, the values of alpha are needed, making this

problem fully non-linear. We will proceed and apply the implicit – (B) method to the

various boundary conditions and an algorithm to solve this problem is included later in

the section.


150

4.3.4.2.1. Constant Pressure Boundaries





( )( )ppftp

∇∇∇=∂∂

βφγ ( )533.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )534.4


aptap =),( , 0>t ( )535.4


bptbp =),( , 0>t ( )536.4

If we rewrite Equation (4.533) in one dimension we get:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

xp

xp

xtp α

φγ ( )537.4

Again we let ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=xpαα . The right hand side can be expanded to get: ( )538.4

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Δ

⎥⎦

⎤⎢⎣

⎡Δ

−−⎥

⎦

⎤⎢⎣

⎡Δ

−

=Δ−

+−+−++−+++++++

+

xx

ppx

pp

tpp

jijijijijijijiji

jiji

1,11,1,1,1,1,1,11,

,1,

21

21

21

21 αααα

φγ



151

( )( ) ⎥

⎦

⎤⎢⎣

⎡

Δ

++−=

Δ− +−+−++−+++++++

21,11,1,1,1,1,11,,1, 2

12

12

12

1

xppp

tpp jijijijijijijijiji αααα

φγ ( )539.4


( )( )[ ] 1,1,11,1,1,1,1,11,2, 2

12

12

12

1 ++−+−++−++++++ +++−ΔΔ

−= jijijijijijijijiji ppppxtp αααα

φγ ( )540.4

Let ( )2x

tΔΔ

=φγλ and expand:

( ) 1,1,11,1,1,1,1,11,, 21

21

21

21 ++−+−++−++++++ +−++−= jijijijijijijijiji ppppap λαααλλ ( )541.4


( )[ ] 1,11,1,1,1,1,11,, 21

21

21

21 1 +−+−++−++++++ −+++−= jijijijijijijiji pppap λαααλλ ( )542.4

where MMj

Ni,1,...2,1,01,...3,2,1

−=−=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 1,21,11,11,11,1,1,0,1

1,31,21,21,21,11,11,10,2

1,21,21,31,21,31,41,30,3

1,11,11,21,11,21,31,20,2

1,01,1,11,1,11,21,10,1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

111

12

321

−−−−−−−

−−−−−−−−

−+++−=−+++−=

−+++−=−+++−=−+++−=

−=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppp

pppp

NiNi

iii



MM( )543.4



( )[ ]

( )[ ]⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−++−

−++−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

1,

1,1

1,1

1,0

1,1,11,1,1

1,11,1,11,

0,1

0,1

10000100

00100001

21

21

21

21

21

21

21

21

N

NNNNN

b

N

a

pp

pp

pp

pp

MMOOOMM

λαααλλα

λαααλλα


152


( ) )1()1()0( PPLPrr

∗= ( )545.4

Note that we can no longer write the pressure at the next time step in terms of the

pressures at the previous time step.

4.3.4.2.2. Boundary Dominated Regime

The implicit – (B) method will now be applied to the boundary dominated period


( )( )ppftp

∇∇∇=∂∂

βφγ ( )546.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )547.4


aptap =),( , 0>t ( )548.4


0,

=⎥⎦⎤

⎢⎣⎡∂∂

== jN ttxxxp , 0>t ( )549.4



( )[ ] 1,11,1,1,1,1,11,, 21

21

21


where JJj

Ni,1...2,1,0

...3,2,1−=

=


153


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 1,11,1,1,1,1,11,0,

1,21,11,11,11,1,1,0,1

1,21,21,31,21,31,41,30,3

1,11,11,21,11,21,31,20,2

1,01,1,11,1,11,21,10,1

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

111

1

321

−−−+++

−−−−−−−

−+++−=−+++−=

−+++−=−+++−=−+++−=

=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppp

pppp

NiNi

iii



MM( )551.4

Recall that jNjN pp ,1,1 −+ = , we now get for Ni = :

( )[ ] 1,11,1,1,1,1,11,0, 21

21

21

21 1 −−−+−+ −+++−= NNNNNNNN pppp λαααλλα ( )552.4

It was also shown earlier that jNjN ,, 21


( )[ ] 1,11,1,1,1,1,11,0, 21

21

21

21 1 −−−−−− −+++−= NNNNNNNN pppp λαααλλα

( )[ ] ( ) 1,11,1,1,0, 21

21 221 −−− −+= NNNNN ppp λααλ ( )553.4


( )[ ]

( )[ ]( )[ ] ⎥

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−++−

−++−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−

−−−−−

1,

1,1

1,1

1,0

1,1,

1,1,11,1,1

1,11,1,11,

0,1

0,1

21

21

21

21

21

21

21

21

21

21

212000100

00100001

N

N

NN

NNNN

b

N

a

pp

pp

pp

pp

MMOOOMM


λαααλλα

This can be written as follows:

( ) )1()1()0( PPOPrr

∗= ( )554.4

4.3.4.2.3. Pseudosteady State Flow Regime

We can also apply the implicit – (B) method to the pseudoseady state regime.



154

( )( )ppftp

∇∇∇=∂∂

βφγ ( )555.4

Initial condition:

)()0,( 0 xpxp = for all [ ]bax ,∈ ( )556.4


1010 ,, ++ ====⎥⎦⎤

⎢⎣⎡∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=jj ttxxttxx x

pxpfq β , 0>t ( )557.4


01,

=⎥⎦⎤

⎢⎣⎡∂∂

+== jN ttxxxp , 0>t ( )558.4



( )xpp

xp jj

j Δ−

=⎥⎦⎤

⎢⎣⎡∂∂ +−+

+ 21,11,1

1,0

( )559.4


( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ−

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

+⎟⎠⎞

⎜⎝⎛+

= +−+

+−+xpp

xpp

kk

q jj

jj2

24

2 1,11,1

1,11,12

βμμ ( )560.4

We can rearrange the above equation and after solving for 1,1 +− jp we get:

( ) ( ) 21,11,1 22 qxq

kxpp jj βρμ

Δ−Δ−= ++− ( )561.4

To get the pressure gradient at the inner boundary, the value derived above for 1,1 +− jp

will be used and the alpha value at 21−=x can be estimated as follows: ( )562.4


155

( ) ( )

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Δ

⎟⎠⎞

⎜⎝⎛ Δ−Δ−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=++

+−+

+−+− x

qxqk

xpp

xpp

xp jj

jj

jj

21,11,0

1,11,0

1,1,

22

21

21

βρμ

αααα

Note that these alpha values cannot be calculated directly because the pressures at

time ( )1+j are unknown.

( )[ ] 1,11,1,1,1,1,11,, 21

21

21


where JJj

Ni,1...2,1,0

...3,2,1,0−=

=


( )[ ]( )[ ]( )[ ]

( )[ ]( )[ ] 1,11,1,1,1,1,11,0,

1,21,11,11,11,1,1,0,1

1,11,11,21,11,21,31,20,2

1,01,1,11,1,11,21,10,1

1,11,1,01,1,1,11,0,0

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

21

11

11

1

1

210

−−−+++

−−−−−−−

−−−

−+++−=−+++−=

−+++−=−+++−=−+++−=

=−=

===

NNNNNNNN

NNNNNNNN

pppppppp

pppppppppppp

NiNi

iii



MM( )564.4

For 0=i we get:

( )[ ] ( ) ( ) ⎟⎠⎞

⎜⎝⎛ Δ−Δ−−+++−= −−

21,11,1,01,1,1,11,0,0 221

21

21

21

21 qxq


( )[ ] ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ Δ+Δ−+−++= −−

21,11,1,1,01,1,0,0 221

21

21

21

21 qxq


For Ni = with 1,11,1 +−++ = jNjN pp and 1,1, 21

21 +−++ −= jNjN αα we get:

( )[ ] 1,11,1,1,1,1,11,0, 21

21

21

21 1 −−−+−+ −+++−= NNNNNNNN pppp λαααλλα

( )[ ] ( ) 1,11,1,1,1,0, 21

21

21 21 −−−+ −++= NNNNNN ppp αλααλ ( )566.4


156


( )[ ] ( )( )[ ]

( )[ ] ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+−

−+++−++

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

1,

1,1

1,0

1,1,

1,11,1,11,

1,1,1,1,

0,1

21

21

21

21

21

21

21

21

21

21

21200

01001

NNNb

a

p

pp

p

pp

MOOOMM

αλλα


( ) ( )

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛ Δ+Δ

+

000

22 2qxqk

x βρμλ

( )567.4

This can be written as:

( ) ∗∗ += GPPRPrrr

)1()1()0( ( )568.4

Note that we cannot write the pressure at the next time step in terms of the pressures

at the previous time step.

The solution to the non-linear problem generated as a result of the implicit – (B)

method can be solved for a general case via:

( ) 11 ++= nnn PPAP ( )569.4

Let us define:

( ) ( ) nnnn PPPAPF −= +++ 111 : ( )570.4

So at each time step we need to solve:

( ) 01 =+nPF ( )571.4

This can be solved using Newton’s Method. Here is an algorithm for tolerance ε

(specified) and the Jacobian matrix J (Matrix of partial derivatives that must be


157

computed for each time.

• For each time step MaxTimen :1= do the following

1. Make an initial guess 10+nP

2. Solve the following for each iteration MaxIterk :1=

( ) ( )nk

nk

nk

nk PFPJPP 1

11

111

1−

+−

−+−

+ −= ( )570.4

If ε<⎟⎟⎠

⎞⎜⎜⎝

⎛ −

−

−+

nk

nk

nk

PPPnorm

1

11

then stop and accept 1+nkP

Else go to step (2) for next iteration k .

• Go to next time step n .

4.4. PRODUCTIVITY INDEX

The productivity index17 (PI) is a great indicator of well performance. One well

cannot be compared to another by the reservoir pressure alone or by how much volume

they produce. These indicators can change from well to well and from one reservoir to

another. Also, a well can produce at very high rates for a relatively short period of time or

have a high reservoir pressure that decreases rapidly. Instead, the combination of these

indicators will give us a better idea of a well’s potential. The Productivity index can

change due to increasing damage, reservoir heterogeneities, permeability reduction, and

relative perm effect, among others. The PI is defined17 in field units as:

⎥⎦

⎤⎢⎣

⎡+−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

−=

srC

AB

khpp

qJ

wA

wf

4306.10ln

21

00708.0

2μ ( )571.4


158

As we can see by the right hand side of Equation 4.571, the PI is sometimes related to

the reservoir properties and reservoir geometry. This suggests that an increase

permeability or reservoir thickness should result in an increase in PI, and an increase of

viscosity or formation volume factor should cause the PI to decrease. An increase in

wellbore radius and shape factor AC , which is related to our reservoir geometry with a

centered well in a cylindrical reservoir having the highest value, will also result in an

increase in PI. An increase in area will only decrease the productivity index. But among

all the terms in the equation, we would like to concentrate on the skin factor s .

The skin factor is an indicator of damage or stimulation in the wellbore or the near

wellbore area. It is of great significance in reservoir engineering and specifically in

pressure transient analysis. It can be translated to an additional pressure drop or pressure

improvement which ultimately, affects the ability to produce. Equation (4.571) suggests

that the higher the skin factor or the damage the lower the PI. A positive skin value

indicates damage and a negative skin value indicates improvement. We will now analyze

the skin factor further to understand how and in what ways it affects well potential. Also,

we are interested in learning if it can be related to high velocity or non-Darcy flow.

There are many components that make the total skin factor, which is defined by

Koederitz16 as:

ftespa sssssss +++++= ( )572.4

where the total skin factor s is the result of the skin factor due to the actual formation as ,

the skin factor due to the partial penetration, partial completion, or perforations ps , the

skin factor due to slant or horizontal hole ss , the skin factor due downhole equipment


159

es also known as mechanical skin, the skin factor due to turbulence ts , and the skin factor

due to fractures fs .

Among all the different sources of skin, it seems like the skin factors due to

turbulence and fractures can be related to the issue of high velocity flow or non-Darcy

behavior. It is known that high velocity flow can occur in fractured reservoirs, but

fractures are also considered highly beneficial because they allow more flow to the

wellbore by creating an additional and highly conductive path for the fluid to flow.

Therefore, the skin factor due to fractures is considered highly negative, which means

that fractures improve the reservoir.

If we take a look at the skin factor due to turbulence we learn that a high flow rates in

gas wells turbulence flow may exist, particularly near the wellbore area where for radial

flow will create a region of high pressure drawdown and high velocity. But based on the

literature review discussion in Chapter 2, we also learned that high velocity does not

necessarily translate into turbulence. For true turbulence to occur even in gas wells we

must have very high velocities, which makes it less likely for oil wells. Katz and Lee15

stated that to obtain a quadratic velocity pressure drop it is not necessary to have

turbulence. Scheidegger also stated that the non-linearity observed is not primarily due to

turbulence, but due to emergence of inertia effects.

The additional pressure drop due to inertia effects described by Forchheimer12 that is

well-known to occur has not been clearly incorporated in the definition of skin in

Equation (4.572), especially in oil wells. A new skin factor due to inertia or non-linear

effects should be considered. For gas wells such effects are interpreted as a rate-


160

dependent14 skin factor Dq :

TFkhD

1422= and

⎥⎥⎦

⎤

⎢⎢⎣

⎡×= −

wpw

g

rhT

F 21210161.3

μγβ

( )573.4

where β is defined as a first velocity coefficient and F as the high-velocity or non-

Darcy flow coefficient, psia2/cp-Mcfd2.

In this work, we will attempt to understand the effects of non-linear flow on

productivity index and other significant consequences of the inertia effects. To model PI

on the computer code developed, the following equation was used:

( ) ( )taptxp

qpp

qpp

qPI b

aaavgwf ,, −=

−=

−=

∫ ( )574.4


161

CHAPTER V

RESULTS

The results we have found from this analysis are very satisfactory. First, the results

from the code that was developed with MATLAB to solve PDEs with the finite

differences method will be compared with the results from COMSOL Multyphysics

(Finite Elements Method) software to validate our code. Then, using the computer code

developed, the productivity index will be estimated at each time and various plots will be

generated to understand the effect that certain parameters have on the productivity index.

5.1. FINITE ELEMENTS

All the plots from both the explicit and implicit methods have been included

throughout Chapter 4; we will now present the plots from the finite element method. The

arbitrary values for fluid and rock properties have been kept the same for similar

boundary conditions to allow for a more logical comparison.

5.1.1. Darcy’s Diffusivity Equation

The first three plots from the finite elements software can be found in Figures 5.1, 5.2

and 5.3. They all show similar results to the pressure profiles developed with the implicit

and explicit methods. They have higher resolution and so they are better compared

against the pressure profile from the implicit method which had a finer grid size than the

explicit method.


162

Figure 5.1: Pressure Profile for Darcy (CPB) Finite Elements


163

Figure 5.2: Pressure Profile for Darcy (BDR) Finite Elements


164

Figure 5.3: Pressure Profile for Darcy (PSS) Finite Elements

5.1.2. Darcy-Forchheimer Diffusivity Equation

Again we can see in Figures 5.4, 5.5 and 5.6 that the results from the finite elements

software are identical to the pressure profiles developed with the computer code. All the

constants, fluid and rock properties, and inertia coefficients are the same as the values

used with the computer code for corresponding boundary conditions.


165

Figure 5.4: Pressure Profile for Darcy-Forchheimer3 (CPB) Finite Elements


166

Figure 5.5: Pressure Profile for Darcy-Forchheimer3 (BDR) Finite Elements


167

Figure 5.6: Pressure Profile for Darcy-Forchheimer3 (PSS) Finite Elements

We can see that the graphs are identical to those generated from the finite differences

method. This proves that the code developed generates comparable results. The code will

now be used to analyze productivity index and other plots of interest.


168

5.2. FINITE DIFFERENCES

Now that the computer code has been proved to be reliable, several plots of interest

will be created based on Darcy-Forchheimer diffusivity equation to analyze and interpret

the effect of the inertia factor on different parameters like pressure, velocity, and PI.

5.2.1. Constant Pressure Boundaries

The pressure profiles in Figure 5.7 show that Darcy flow will reach steady state faster

than any other non-linear flow. The higher the inertia factor the more pressure or energy

it will take to get the same amount of production.

Figure 5.7: Effect of β on Pressure (CPB)


169

The effect of the inertia factor on pressure gradient can be seen in Figure 5.8 which

shows the linear relationship between velocity and pressure gradient while there is a clear

increase in non-linearity as the inertia factor increases. This effect has been also observed

experimentally6.

Figure 5.8: Effect of β on Pressure Gradient (CPB)


170

The effect of the inertia factor on velocity is illustrated in Figure 5.9. For all the

values of β the velocity increases as we transition from high pressure (right) to low

pressure (left) which simulates the effect of flow to the wellbore and the pressure drop

creates higher velocities near the wellbore. We also notice that the velocities are lower as

the inertia factor increases with the highest velocities being experienced under Darcy

flow.

Figure 5.9: Effect of β on Velocity (CPB)


171

For the same β value, in this case for Darcy 0=β , Figure 5.10 shows that the

velocity stabilizes to a constant value with time. The velocity becomes constant when we

reach steady state.

Figure 5.10: Change of Velocity with Time (CPB)


172

The effect of the inertia factor on the productivity index is shown in Figure 5.11

where we can see a clear decrease in PI as β increases. The PI start at high values early

in time but drops rapidly until it reaches a constant value.

Figure 5.11: Effect of β on Productivity Index (CPB)


173

To analyze the effect of permeability and the inertia factor on the productivity index,

we plotted in Figure 5.12 different arbitrary and increasing values for permeability for

two different inertia factors: one is for Darcy and the other is a very low β value. The

results suggests that for Darcy, there is a clear increase of PI as permeability increases,

but the PI increase is not as significant (though increasing) for β =1. We can then

conclude that for high inertia factor values the PI will not change significally with an

increase of permeability.

Figure 5.12: Effect of β and Permeability on PI (CPB)


174

5.2.2. Boundary Dominated Regime

We will analyze various plots of interest for the boundary dominated regime. Figure

5.13 confirms what we concluded previously for the constant pressure boundaries case –

Darcy flow will reach steady state faster. If two wells that flow naturally and had the

same amount of oil in place and identical properties were left in absolute open flow

(AOF) and the only difference being that one had Darcy flow and the other non-Darcy

flow, the well on Darcy flow will produce more oil at the same time and as a

consequence, will deplete its reservoir faster than the well with non-Darcy flow. The well

on non-Darcy flow will experience and additional pressure drop due to high velocity.

Figure 5.13.: Effect of β on Pressure (BDR)


175

The same profile as in the constant pressure boundaries case has been generated in

Figure 5.14. With increasing β , the pressure gradient increases at a given velocity.

Figure 5.14 was generated at an early time ( )1.0=Tf . Similar plots were generated at

later times Figure A.1 ( )1=Tf and Figure A.2 ( )5=Tf . The results suggest that later in

time the pressure curves tend to become more “linear,” especially close to steady state

conditions.

Figure 5.14: Effect of β on Pressure Gradient (BDR) Tf=0.1


176

As the inertia effect decreases the velocity increases. Velocity also increases as

the fluid flows from high to low pressure. Figure 5.15 was also captured at an early time

( )1.0=Tf , and we notice that the curves follow a sequence based on the inertia value. A

similar graph was plotted at a later time Figure A.3 ( )2=Tf , but we can see a disrupted

sequence. To understand this effect, we plotted in Figure A.4 the change of velocity with

time for all the selected β values.

Figure 5.15: Effect of β on Velocity (BDR) Tf=0.1


177

Figure 5.16 shows the change of velocity with time for Darcy. We notice a very

abrupt jump in velocity near the left boundary early in time, but later it follows a more

predictable pattern.

Figure 5.16: Change of Velocity with Time (BDR) Tf=2


178

For the boundary dominated regime, we can also see lower PI values with increasing

inertia factor, but this time the curves reach a unique asymptotic value for PI as they

reach steady state. The higher the inertia factor the longer it takes to reach the asymptotic

PI value.

Figure 5.17: Effect of β on Productivity Index (BDR)


179

We would also like to analyze the effect of permeability. We notice a similar effect in

Figure 5.18 that we saw in Figure 5.17. For a given permeability value, the corresponding

high inertia curve tends to converge to the same PI that Darcy converges to.

Figure 5.18: Effect of β and Permeability on PI (BDR)


180

5.2.3. Pseudo Steady State

We will now analyze different plots with the pseudosteady state conditions. Figure

5.19 shows higher pressure gradients for high values of β . With time, all the curves

seem to overlap to reach the same values.

Figure 5.19: Effect of β on Pressure (PSS)


181

The same effect as previous velocity versus pressure gradient plots is shown in Figure

5.20. Darcy shows a linear relationship while the other curves show non-linear

relationships.

Figure 5.20: Effect of β on Pressure Gradient (PSS)


182

Again, Darcy flow experiences the highest velocity. The velocity decreases with

increasing β but increases as we get closer the low pressure boundary. Figure 5.21 was

plotted at a time ( )1.0=Tf . Figure A.5 shows the same plot at a later time ( )2=Tf .

Figure 5.21: Effect of β on Velocity (PSS) Tf=0.1


183

Figure 5.22 shows that for Darcy flow, the velocity increases with time until it

becomes constant. Different curves would follow the same trend as Darcy flow but, it

will take longer to reach the same conditions.

Figure 5.22: Change of Velocity with Time (PSS)


184

The PI curves in Figure 5.27 show similar trends of Figure 5.11. The PI decreases

with increasing β but each curve stabilizes at its own value. Unlike the productivity

index from the boundary dominated case, Figure 5.27 suggests that the PI for Darcy will

always be higher than the rest and will never have the same values except very early in

time when all the curves overlap.

Figure 5.23: Effect of β on Productivity Index (PSS)


185

In this case we will analyze the effect of increasing flow rate and inertia effects on

productivity index. In Figure 5.24 we can see that for Darcy flow, an increase in flow rate

will have no significant impact on PI. But for an arbitrary value for the inertia factor we

notice that the PI is lower for high flow rates.

Figure 5.24: Effect of β and Flow Rate on PI (PSS)


186

We can also confirm by Figure 5.25 that permeability increase has a significant

impact on PI, but as the inertia factor increases, an increase in permeability results in

roughly the same low (and slightly decreasing) PI value.

Figure 5.25: Effect of β and Permeability on PI (PSS)


187

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

The results show that the computer code developed to solve PDEs is reliable, and the

results are consistent with the finite elements software package. The results also show

that an increase of the inertia factor will decrease the productivity index of a well. The

higher the inertial factor, the more non-linear the flow becomes and therefore it affects

the PI negatively.

As we would expect, with an increase in permeability we can see an improvement in

PI for Darcy flow, 0=β . But for highly non-linear cases, the permeability change does

not have as much effect on the PI. When changing the flow rates, we were able to see that

for Darcy flow, an increase in flow rate did not affect the PI, but for non-linear flow, high

flow rates resulted in lower PI. These results agree with the pressure transient analysis

theory that relates the high velocity flow effect to an additional pressure drop due to high

flow rate or rate dependent skin. The effects felt in the wellbore are similar to having

mechanical damage, plugged perforations and other types of damage mechanisms that

prevent the fluid to flow at its natural (undamaged) conditions.

For future work, Darcy-Forchheimer equation could be implemented with radial flow,

and also in a 3D model using finite elements method to study more realistic and complex

scenarios. It is of interest to model the near wellbore area for high velocity, but also other

areas in the reservoir where high velocities can be experienced. A graphical user interface

has been developed (see Figure B.8.) for the Darcy-Forchheimer flow model.


188

BIBLIOGRAPHY

1. Ahmed, T. and McKinney, P.D. 2005. Advanced Reservoir Engineering, 1/17,

1/18, 1/56. Amsterdam: Golf Professional Publishing Co. 2. Ahmed, T. 2001. Reservoir Engineering Handbook, 330-332. Boston: Golf

Professional Publishing Co.

3. Aulisa, E., Ibragimov, A., Valko, P. and Walton, J. Mathematical Frame-work for Productivity Index of the Well for Fast Forchheimer (Non-Darcy) Flow in Porous Media. (Unpublished).

4. Barakat, H.Z. and Clark, J.A. 1966. On the Solution of the Diffusion Equation by

Numerical Methods. Journal of Heat Transfer. 421-427.

5. Barree, R.D. 2004. Beyond Beta Factors: A Complete Model for Darcy, Forchheimer, and Trans-Forchheimer Flow in Porous Media. Paper SPE 89325 presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 26-29 September.

6. Belhaj, H.A., Agha, K.R., Nouri, A.M., Butt, S.D., and Islam, M.R. 2003.

Numerical and Experimental Modeling of Non-Darcy Flow in Porous Media. Paper SPE 81037 presented at the SPE Latin American and Caribbean Petroleum Energy Conference, Port-of-Spain, Trinidad, West Indies, 27-30 April.

7. Belhaj, H.A., Agha, K.R., Nouri, A.M., Butt, S.D., Vaziri, H.H., and Islam, M.R. 2003. Numerical Modeling of Forchheimer’s Equation to Describe Darcy and Non-Darcy Flow in Porous Media. Paper SPE 80440 presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Jakarta, Indonesia, 15-17 April.

8. Brinkman, H.C. 1947. A Calculation of Viscous Force Exerted by a Flowing

Fluid on a Dense Swarm of Particles. Appl. Sci. Res. A, 1, 27-34.

9. Burden, R.L. and Faires, D.J. eigth edition. 2004. Numerical Analysis. California: Thomson Brooks/Cole.

10. Dake, L.P. 1978. Fundamentals of Reservoir Engineering, 103, 131. Amsterdam:

Elsevier Scientific Publishing Co.

11. Darcy, H.P.G. 1856. Les Fontaines Publiques de la Ville de Dijon, Exposition et Application des principes a Suivre et des Formules a Employer dans les Questions de Distribution d’Eau. Victor Dalmont, Paris.


189

12. Forchheimer, P. 1901. Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45, 1782-1788.

13. Huang, H. and Ayoub, J. 2006. Applicability of the Forchheimer Equation for

Non-Darcy Flow in Porous Media. Paper SPE 102715 presented at the 2006 SPE Annual Conference and Exhibition, San Antonio, Texas, 24-27 September.

14. Ikoku, C.U. 1984. Natural Gas Reservoir Engineering, 152-155. New York: John

Wiley and Sons.

15. Katz, D.L. and Lee, R.L. 1990. Natural Gas Engineering Production and Storage, 59-66. New York: McGraw-Hill Publishing Co.

16. Koederitz, L.F. Revised edition. 2002. Notes on Well Test Analysis. University of

Missouri, Rolla, Missouri. (Unpublished).

17. Lee, J. 1982. Well Testing, Textbook Series Vol. 1, SPE, Dallas, Texas.

18. MacCluer, C.R. first edition. 1999. Industrial Mathematics: Modeling in Industry, Science and Government, New Jersey: Prentice Hall.

19. Morton, K.W. and Mayers, D.F. second edition. 2005. Numerical Solution of

Partial Differential Equations: An Introduction. Cambridge, England: University Press.

20. Nagle, R.K., Saff, E.B. and Snider, A.D. fourth edition. 2004. Fundamentals of

Differential Equations and Boundary Value Problems, 579, 625, 639, 803. Boston: Addison Wesley.

21. Scheidegger, A.E. third edition. 1974. The Physics of Flow through Porous

Media, 152-167. Toronto, Canada: University of Toronto Press.

22. Tiab, D. and Donaldson, E.C. second edition. 2004. Petrophysics. Theory and Practice of Measuring Reservoir Rock and Fluid Properties, 88. Amsterdam: Golf Professional Publishing Co.

23. Fowler, B.F. 2002. Fundamental Principles of Reservoir Engineering. Textbook

Series Vol. 8, SPE, Richardson, Texas.


190

APPENDIX A

ADDITIONAL TABLES AND GRAPHS

Table A.1: Convergence Analysis ( )5.0=λ and ( )0=β

N xΔ tΔ γ Relative Error (%)

( )txxtxp −= 1),( 2


2 0.5000 0.2500 0.2500 41.42084347897570 0

3 0.3333 0.1111 0.1667 17.37425320257839 7.27774743539849

5 0.2000 0.0400 0.1000 6.13950108473032 2.82612174997150

10 0.1000 0.0100 0.0500 1.52633684117290 0.71978095349406

20 0.0500 0.0025 0.2500 0.02500000000000 0.18065186712518



( )txxtxp −= 1),( 2


2 0.5000 0.7500 0.7500 67.08203932499369 89.44271909999159

3 0.3333 0.3333 0.5000 34.74969301789464 37.71083802383927

5 0.2000 0.1200 0.3000 11.74269168197564 11.50547102748093

10 0.1000 0.0300 0.1500 2.89767510633201 2.66387738805520

20 0.0500 0.0075 0.075 0.72158410261192 0.65115737380176


191



( )txxtxp −= 1),( 2


2 0.5000 2.7500 2.7500 Undefined Undefined

3 0.3333 1.2222 1.8333 0.00000000000001 0.00000000000002

5 0.2000 0.4400 1.1000 17.49467811538338 27.63994112767362

10 0.1000 0.1100 0.5500 3.99572010954212 4.85651995267795

20 0.0500 0.0275 0.2750 0.97418450306405 1.13552537161852



( )txxtxp −= 1),( 2




5 0.2000 0.8400 2.1000 13.80951493365000 28.14256932675863

10 0.1000 0.2100 1.0500 4.47402808290369 5.45396398319529

20 0.0500 0.0525 0.5250 1.04125455843501 1.26935377663790


192



( )txxtxp −= 1),( 2Relative Error (%)

( )txxtxp −= 1),( 3





20 0.0500 0.5250 5.0250 1.16874858541487 2.88976365846586


193

Figure A.1: Effect of β on Pressure Gradient (BDR) Tf=1


194

Figure A.2: Effect of β on Pressure Gradient (BDR) Tf=5


195

Figure A.3: Effect of β on Velocity (BDR) Tf=2


196

Figure A.4: Effect of β on Velocity (BDR)


197

Figure A.5: Effect of β on Velocity (PSS) Tf=2


198

APPENDIX B

DIMENSIONAL ANALYSIS, CONVERSION FACTORS, AND GRAPHICAL USER INTERFACE

B.1. DIMENSIONAL ANALYSIS

The dimensions of the inertia factor ( )β will be determined through dimensional

analysis. We will start with Forchheimer’s equation and express the variables in terms of

their fundamental dimensions, space (or length) [L], time [t], and mass [M].

2vvkx

p βρμ+=

∂∂

− ( )1.B

First, we recall the dimensions of the physical variables in Forchheimer’s equation

except for the inertia factor:

Pressure = 21 −−== tMLAFP ( )2.B

Length = 1Lx = ( )3.B

Viscosity = 11 −−= tMLμ ( )4.B

Permeability = 2Lk = ( )5.B

Velocity = 11 −=ΔΔ= tLtxv ( )6.B

Density = 3−== MLVmρ ( )7.B

We now rewrite Forchheimer’s equation in terms of its dimensions only to get the

following equation:

2]][][[][][][

][][ vv

kxP ρβμ

+= ( )8.B

If we replace the terms in Equation B.8 by their corresponding fundamental units we


199

get the following:

2113112

1121

)(][ −−−−−−−

+= tLMLtLL

tMLL

tML β ( )9.B

After simplifying Equation B.9 we get:

212222 ][ −−−−−− += tMLtMLtML β ( )10.B

We can see that the Forchheimer term must have the same dimensions of both the

Darcy term and the pressure gradient.

2122 ][ −−−− = tMLtML β ( )11.B

The dimensions of the inertia factor can now be found by solving Equation B.11.

1][ −= Lβ ( )12.B

If we recall the expression for the inertia factor, we can also confirm the dimensions:

21k

Fφβ = ( )13.B

21][][][

kφβ = ;

21][][][

2

0

LL

=β ; 11][ −== LL

β ( )14.B

Note that for the units of Equation B.13 to be consistent dimensionally, the

permeability must be expressed in terms of its square root, 21k .

The same analysis will be applied to both the Darcy-Forchheimer equation and the

non-linear function ( )pf ∇ . We first recall the vector form of the Darcy-Forchheimer

equation:

( ) ppfv ∇∇−=r ( )15.B

To write the dimensional equation, we must first find the dimensions of the non-linear


200

function:

( )p

kk

pf

∇+⎟⎠⎞

⎜⎝⎛+

=∇

βρμμ 4

22

( )16.B

We can now rewrite Equation B.16 in one dimension in terms of its dimensions only:

][][]][[

][][

][][

12

xP

kk

xpf

ρβμμ+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ ( )17.B

After substituting Equation B.17 by the corresponding fundamental units we get the

same dimensional equation we would get with a more general non-linear function

( )pf ∇ :

( )

LtMLMLL

LtML

LtML

pf21

312

2

11

2

11

1−−

−−−−−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

=∇ ( )18.B

Equation B.18 can be simplified to give the following equations:

( )( ) 26221313

1−−−−−− ++

=∇tLMtMLtML

pf ( )19.B

( ) 1313

1−−−− +

=∇tMLtML

pf ( )20.B

The dimensions of the non-linear function are as follows:

( ) tLMtML

pf 3113

1 −−− ==∇ ( )21.B

We can now check the dimensions of the Darcy-Forchheimer equation in one

dimension for consistency.


201

][][

][][][

xP

xPfv ⎟⎟⎠

⎞⎜⎜⎝

⎛= ( )22.B

Equation B.22 can be rewritten in terms of the fundamental dimensions.

LtMLtLMv

2131][

−−−= ( )23.B

After simplifying we get the correct dimensions for velocity:

1][ −= Ltv ( )24.B

B.2. CONVERSION FACTORS

The Darcy-Forchheimer equation can be used with any consistent unit system like the

S.I. Unit System. To use oilfield units, we must find the appropriate conversion factors.

We will find the conversion factors for Forchheimer’s equation, the Darcy-Forchheimer

equation, and Darcy-Forchheimer diffusivity equation. The SI Metric System will be

used as the consistent unit system, and Table B.1. shows some conversion factors.


202

Table B.1: SI Metric Conversion Factors23

Oilfield Units Conversion Factors SI Metric Units

bbl 01589873.1 −× E 3m

cp 03−× E * sPa ⋅

ft 03048.3 −× E * m

md 04869233.9 −× E 2mμ

.in 0454.2 −× E * cm

lbm 01535924.4 −× E kg

psi 00894757.6 +× E kPa

*Conversion Factor is Exact.

The unit conversion factors ( A , B , and C ) will be determined from Forchheimer’s

equation as follows:

2vCvk

BxpA βρμ

+=∂∂

− ( )25.B

To find the conversion factors we must convert the desired oilfield units to consistent

units. ( )26.B

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡ −

⎥⎦

⎤⎢⎣

⎡ ⋅

=

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡ −

minsec60min60100.

54.2.12

16869233.9

10

100.54.2.12

1013257.14

2

3

hr

cmm

incm

ftin

hrftv

mdmEmdk

cpsPacp

cmm

incm

ftinftx

atmPa

psiatmpsiP μ


203

2

3

minsec60min60100.

54.2.12

100.54.2.12

4535924.0

100.54.2.121

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

hr

cmm

incm

ftin

hrftv

cmm

incm

ftinft

lbmkglbm

cmm

incm

ftinft

ρβ

The resulting equation after combining all the conversion factors in Equation B.26 is

the following modified Forchheimer’s equation:

2310000003767.01068.8578849759479.22620 vvkx

p βρμ+=

∂∂

− ( )27.B

If we rearrange the constants from Equation B.27, we get:

( ) ( ) 21166542.1495197.3792 vEvkx

p βρμ−+=

∂∂

− ( )28.B

The new conversion factors will be renamed as 495197.3792=Dar for the Darcy

term and 116654.1 −= EForch for the Forchheimer term. The resulting form of the

modified Forchheimer’s equation is as follows:

2vForchvk

Darxp βρμ

+=∂∂

− ( )29.B

If we follow a similar derivation as in Chapter 4.3.1 to get an expression for the

Darcy-Forchheimer equation, we get the following modified non-linear function:

( )pForch

kDar

kDar

pf

∇+⎟⎠⎞

⎜⎝⎛+

=∇

βρμμ 4

22

( )30.B

When the non-linear function from Equation B.30 is used with the Darcy-

Forchheimer equation ( ) ppfv ∇∇−= βr and the Darcy-Forchheimer diffusivity equation

( )( )ppfdivtp

∇∇=∂∂

βφγ , the resulting equations become consistent in oilfield units.


204

Sample data1 in field units will be used to generate plots from a modified MATLAB

code, and the results will be compared with the corresponding plots in SI metric units.

Table B.2: Sample Data1

Symbol Field Units SI Metric Units

q 4900 STB/D 0.0090 sm3

wr 0.354 ft 0.1079 m

tc 22.6E-06 1/psi 3.2779E-9 1/Pa

oμ 0.20 cp 0.0002 sPa ⋅ 2mμ

oB 1.55 bbl/STB 1.55 res 3m /std 3m

wfp 2761 psig 19.036 kPa

φ 0.09 0.09

er 2640 ft 804.672 m

pt 310 hrs 1116000 s

ip 3325 psig 22.925 kPa

β * 2.43406E+15 1/ft 7.98577E+15 1/m

* Calculated from available correlations6 in the literature

A few assumptions will be made to allow us to plot the given data using the modified

computer code. We will assume linear flow into a fracture; the reservoir width, W, will


205

be assumed to be the same as the reservoir height, h, the drainage radius, er , will be

assumed to be the length, L, from the extent of the reservoir to the fracture, and the initial

pressure ip will be assumed to be 3325 psig . Figure B.1 below illustrates the settings.

Figure B.1: Linear Flow into a Fracture

The following plots show various pressure profiles and the productivity index for

Darcy flow and Forchheimer (or non-Darcy) flow during the pseudosteady state flow

regime using both field units and SI units. All models have been run for a time period

(Tf) of 48 hours. We can see that the plots are identical with their corresponding units.

This shows a successful determination and implementation of the conversion factors in

the computer code. We also notice that for the non-Darcy pressure profile, the pressure

drops fast as a result of a very large calculated inertia factor. With no information about

the type of reservoir, this inertia factor is only an assumption. The Darcy pressure profile

shows a more gradual pressure decline.


206

Figure B.2: Pressure Profile for Darcy (PSS) in Field Units


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Figure B.3: Pressure Profile for Darcy-Forchheimer (PSS) in Field Units


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Figure B.4: Productivity Index (PSS) in Field Units


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Figure B.5: Pressure Profile for Darcy (PSS) in SI Units


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Figure B.6: Pressure Profile for Darcy-Forchheimer (PSS) in SI Units


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Figure B.7: Productivity Index (PSS) in SI Units


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B.3. GRAPHICAL USER INTERFACE

A graphical user interface (GUI) has been developed (see Figure B.8) to allow the user to

generate various plots from the solution of the Darcy-Forchheimer diffusivity equation.

The user can input the fluid and rock properties, initial and boundary conditions,

reservoir dimensions for linear flow, and time. The GUI allows the selection between

oilfield units and SI units; this feature allows the user to input and plot in the desired unit

system. Available plots are the pressure profile, productivity index versus time, and

pressure gradient versus velocity. The user is allowed to reset specific graphs while

leaving others intact, reset all the graphs at once or just reset the data. If graphs are not

reset, the new and old graphs will overlap to allow an easy comparison.

Figure B.8: Darcy-Forchheimer Flow Model GUI


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APPENDIX C

VITA

Simeon Eburi Losoha was born in Malabo, Equatorial Guinea. He graduated from

the Spanish High School of Malabo in 2001, and later graduated with honors with an

Associate of Science Degree from Ranger College in 2003. He then transferred to Texas

Tech University where he graduated Magna Cum Laude in 2006 with a Bachelor of

Science Degree in Petroleum Engineering. He also passed the Fundamentals of

Engineering (FE) Exam.

Simeon has been involved in SPE by attending regular meetings, events,

conferences, working as the SPE Petroleum Engineering and Math tutor and participating

in the SPE Student Paper Contest. He placed first place in the 2006 Texas Tech Student

Paper contest, undergraduate division and went on to represent Texas Tech in the 2006

Regional SPE Paper Contest in Texas A&M. In 2007, he placed second in the Texas

Tech Student Paper Contest, graduate division, and went on to represent Texas Tech in

the 2007 Regional SPE Paper Contest in LSU.

Simeon acquired significant experience after working with ExxonMobil

Production Company, Houston for two summer internships in two projects – Pressure

Transient Analysis and Well Performance analysis for Zafiro Field, Equatorial Guinea.

He also worked on the Texas Tech campus as a Math, Physics, Chemistry and

Engineering tutor. He later worked as a graduate assistant for Dr. Ziaja in an

undergraduate production engineering class and as a teaching assistant for Dr. Fasesan in

an undergraduate advanced reservoir engineering class and engineering communications.

Simeon worked closely with Dr. Heinze and Dr. Padmanabhan Seshaiyer and

learned MATLAB and COMSOL (Femlab). He has been very active with student

organizations. He was the president of the African Student Organization in 2005-2006

and the president of the Engineering Ambassadors in 2006-2007, both at Texas Tech

University. He enjoys playing basketball, soccer and sports in general. He has really

enjoyed his academic journey through West Texas.

PERMISSION TO COPY

In presenting this thesis in partial fulfillment of the requirements for a master’s

degree at Texas Tech University or Texas Tech University Health Sciences Center, I

agree that the Library and my major department shall make it freely available for research

purposes. Permission to copy this thesis for scholarly purposes may be granted by the

Director of the Library or my major professor. It is understood that any copying or

publication of this thesis for financial gain shall not be allowed without my further

written permission and that any user may be liable for copyright infringement.

Agree (Permission is granted.)

________________________________________________ ________________ Student Signature Date Disagree (Permission is not granted.) _______________________________________________ _________________ Student Signature Date

Simeon Eburi Losoha 06/28/07

mathematical modeling, analysis and simulation of …

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