stability criteria for the thermal adaptive implicit method

STABILITY CRITERIA FOR THE THERMAL

ADAPTIVE IMPLICIT METHOD

A REPORT

SUBMITTED TO THE DEPARTMENT OF PETROLEUM

ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

By

Anshul Agarwal

September 2005

I certify that I have read this report and that in my opin-

ion it is fully adequate, in scope and in quality, as partial

fulfillment of the degree of Master of Science in Petroleum

Engineering.

Hamdi Tchelepi(Principal advisor)

ii

Abstract

The stability criteria for thermal reservoir displacements are derived using Von Neu-

mann analysis. It is shown that when reduced to the isothermal form, we recover

the stability conditions presented by Coats [5], [6]. The derived stability criteria are

tested by violating the time step size in a 1-D, oil-water thermal simulator coded

using MATLAB. The stability criteria can be used to decide the time step size for an

IMPEST (implicit in pressure, explicit in saturation and temperature) formulation.

The criteria account for explicit treatment of capillary pressure, viscous forces,

and heat convection and conduction terms. The criteria can be used as a switching

rule in an adaptive implicit thermal model.

iii

Acknowledgments

I would like to express my deep sense of gratitude to Prof. Hamdi Tchelepi,

for his invaluable support and guidance throughout the duration of this research

work. I am also highly indebted to him for his constant encouragement by giving

his critical feedback and developing ideas on my work. I wish to thank our SUPRI-B

consortium (Reservoir Simulation Industrial Affiliate Program at Stanford University)

for providing me with the opportunity and necessary funding to get involved with this

research project.

Special thanks go to the faculty at the Department of Petroleum Engineering who

have provided me with interesting technical feedbacks, and my colleagues and friends

for the invaluable human ingredient in realizing this work.

Anshul Agarwal

September 2005

Stanford University

iv

Contents

Abstract iii

Acknowledgments iv

Table of Contents v

List of Figures vii

1 Introduction 1

1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Outline of the Report . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Previous Work on Stability Analysis 4

3 Mathematical formulation 6

3.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Mass Conservation Equations (Isothermal Case) . . . . . . . . . . . . 9

3.3 Mass Conservation Equations (Thermal Case) . . . . . . . . . . . . . 12

3.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Application of Von Neumann Method for Stability Analysis . . . . . . 18

3.6 Stability Analysis for Two Phase Isothermal Systems . . . . . . . . . 20

3.7 Stability Analysis for Two Phase Thermal Systems . . . . . . . . . . 22

4 Verification 25

4.1 Thermal Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Formulation of FIM and IMPEST . . . . . . . . . . . . . . . . . . . . 27

v

4.3 Two phase Isothermal results . . . . . . . . . . . . . . . . . . . . . . 27

4.4 Two phase Thermal results . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4.1 Low Mobility Ratio . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4.1.1 Implicit P, Explicit S and T . . . . . . . . . . . . . . 32

4.4.1.2 Implicit P and T, Explicit S . . . . . . . . . . . . . . 38

4.4.2 High Mobility Ratio . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 Dependence on Conduction parameters . . . . . . . . . . . . . . . . . 43

4.5.1 Effect of CpR. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5.2 Effect of Υc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Conclusions and Recommendations 48

Bibliography 50

A Expressions of the Stability Criteria 52

A.1 Comprehensive Stability Criteria . . . . . . . . . . . . . . . . . . . . 52

A.2 Isothermal Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . 55

B Physical properties used in simulator 57

B.1 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B.2 Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B.4 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B.5 Rock Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

C Fractional flow curves 60

vi

List of Figures

4.1 Saturation fronts in different schemes . . . . . . . . . . . . . . . . . . 29

4.2 Pressure profiles in different schemes . . . . . . . . . . . . . . . . . . 29

4.3 Saturation profiles in different schemes . . . . . . . . . . . . . . . . . 30

4.4 Stability number trend in the reservoir . . . . . . . . . . . . . . . . . 31

4.5 Progress of saturation fronts for low mobility ratios at different times 32

4.6 Progress of heat fronts for low mobility ratios at different times . . . 33

4.7 Pressure profile in the middle grid block . . . . . . . . . . . . . . . . 35

4.8 Saturation profile in the middle grid block . . . . . . . . . . . . . . . 35

4.9 Temperature profile in the middle grid block . . . . . . . . . . . . . . 36

4.10 Stability number trend in the thermal reservoir . . . . . . . . . . . . 36

4.11 Saturation fronts at 110 days in the thermal reservoir . . . . . . . . . 37

4.12 Temperature fronts at 110 days in the thermal reservoir . . . . . . . . 37

4.13 Progress of saturation fronts for high mobility ratios at different times 39

4.14 Progress of heat fronts for high mobility ratios at different times . . . 40

4.15 Pressure profile in the middle grid block . . . . . . . . . . . . . . . . 41

4.16 Saturation profile in the middle grid block . . . . . . . . . . . . . . . 41

4.17 Temperature profile in the middle grid block . . . . . . . . . . . . . . 42

4.18 CFL Trend in the 41st timestep . . . . . . . . . . . . . . . . . . . . . 43

4.19 CFL Trend in the 42nd timestep . . . . . . . . . . . . . . . . . . . . . 44

4.20 Saturation front in the 41st timestep . . . . . . . . . . . . . . . . . . 45

4.21 Temperature front after 200 days of production, zero conduction . . . 47

C.1 Fractional flow as a function of temperature . . . . . . . . . . . . . . 61

C.2 Fractional flow as a function of saturation . . . . . . . . . . . . . . . 62

vii

Chapter 1

Introduction

Conventional primary and secondary recovery operations often leave two thirds

of the oil in the reservoir. In the U.S., enhanced oil recovery (EOR) methods have

the potential to recover an estimated 200 billion barrels of the remaining discovered

oil resource [1]. Without EOR, much of this oil will be left in the ground. Although

often highly effective, EOR methods are more expensive production methods; con-

sequently, during times of low oil prices their application is limited. However, with

an emerging consensus for sustained oil prices and growing concerns over America’s

energy security, interest is being revitalized in EOR technologies for increasing recov-

ery. EOR processes involve injecting a fluid into the reservoir to increase reservoir

pressure or reduce oil viscosity in order to mobilize the oil. Injectants include steam

(thermal processes); polymers and gels (chemical processes); carbon dioxide, nitrogen,

and natural gas (gas processes).

Thermal recovery methods in general and steam injection in particular are the

most popular EOR processes. The governing equations that describe multiphase

flow in reservoirs are nonlinear and coupled. The solution to nonlinear fluid flow

problems in reservoirs using the fully implicit method (FIM) is unconditionally stable

but computationally expensive per time step, although it allows large time steps to

be taken. The IMPES method, on the other hand, decouples the pressure equation

and solves it implicitly. The pressure solution is then used to obtain the saturations.

The method is computationally inexpensive but only conditionally stable, and the

1

CHAPTER 1. INTRODUCTION 2

maximum stable time step may be extremely small. Yet, it gives the most accurate

computation of saturation fronts and breakthrough times.

The Adaptive Implicit Method (AIM) is a reservoir simulation method that treats

some primary variables as implicit while the others are treated explicitly. In other

words, on a grid block basis, it treats some grid blocks with the FIM, while the others

are treated with the IMPES method. For example, the flow around the wellbore

requires rigorous analysis since there are large changes in pressure, saturation and

temperature in the vicinity of the well. Away from the well, these changes tend to be

small enough such that using FIM is a waste of computational effort. In these regions,

the IMPES method should suffice. AIM therefore tends to find a balance between

FIM and IMPES. AIM is conditionally stable and its time steps can be controlled

using a stability condition. We derive the stability conditions for thermal systems of

a 1-D two-component oil-water black oil model using Von Neumann method . The

derived stability conditions are tested using MATLAB.

1.1 Objectives

The objective of this research is to derive comprehensive stability criteria for thermal

systems. The research focuses on the following areas:

• Derivation of stability criteria for thermal problems, taking into account con-

vection and conduction terms and neglecting heat losses.

• Verification of the derived stability condition using a simple thermal simulator.

1.2 Outline of the Report

Chapter 2 briefly summarizes the previous works on stability analysis. Chapter 3 dis-

cusses the mathematical formulation of the governing equations and the application

of von Neumann method to stability analysis, and the expressions of the compre-

hensive stability criteria are derived. Chapter 4 presents the FIM formulation and

the IMPEST (IMplicit Pressure Explicit Saturations and Temperature) method for

thermal problems. It then explains the various results testing the derived stability

CHAPTER 1. INTRODUCTION 3

criteria for both isothermal and thermal cases. Chapter 5 summarizes the results of

this research. Conclusions are drawn from the tested stability criteria. Finally, pos-

sible areas for additional work in the future are suggested. Appendix A contains the

expressions of the comprehensive stability criteria, and it is shown that isothermal

stability criteria are obtained as a special case of the thermal criteria. Appendix B

contains the various physical properties for the rock and fluids used for simulation.

Appendix C contains the fractional flow curves for thermal flow, which are non linear.

Chapter 2

Previous Work on Stability

Analysis

Adaptive implicit methods were introduced in reservoir simulation through the work

of Thomas and Thurnau in 1983 [11]. In their work, they developed the mathematical

procedure that involves labeling some variables as implicit while others are classified as

explicit and then constructing the matrix problem. The method provides a changing

level of implicitness in space and time and is done on a cell-by-cell basis. They

presented the savings in computational time and storage using a black-oil model

example.

Forsyth and Sammon in 1986 [7], noted that the criteria presented by Thomas and

Thurnau for selecting implicit cells are not straightforward. They presented methods

utilizing different degrees of implicitness on a cell by cell basis, and they developed an

iterative matrix solution technique for the adaptive implicit Jacobian using incomplete

LU factorization. Forsyth and Sammon showed that an approximate Jacobian can be

constructed which could be used to solve the fully implicit system, instead of using

the fully implicit Jacobian. They showed significant computing time reductions.

Russell in 1989 [10], developed a switching criterion for black-oil models based

on the CFL stability condition that determines whether an unknown is implicit or

explicit. He indicated that the CFL criterion can also be used for timestep control

in IMPES models and is extensible to compositional simulation. In his work he

observed that AIM provided more accurate answers than FIM through the reduction

4

CHAPTER 2. PREVIOUS WORK ON STABILITY ANALYSIS 5

of numerical dispersion. He treated pressure implicitly owing to the near elliptic

nature of the equation, whereas saturations and compositions were treated explicitly,

as the equations are near hyperbolic in nature. He compared the methods by Thomas

and Thurnau and Forsyth and Sammon, and he concluded that both methods yield

the same answers as the FIM.

Young and Russell [13], applied AIM to compositional simulation and observed

that greater efficiency is obtained when less than 5 percent of the grid blocks are

treated implicitly. The switching logic was based on concepts adapted from Russell

[10]. They tested various problems and concluded that AIM did not show dramatic

efficiency improvement compared to IMPES, but that AIM is a good alternative to

FIM for problems where IMPES is inadequate. They also concluded that AIM appears

to benefit compositional models more than black oil models.

Coats in 2001 [5], derived IMPES stability criteria for multidimensional three-

phase flow for black-oil and compositional models. He expressed the stability con-

ditions in terms of a function Fi which had a different expression for different flow

mechanisms.

Recently, Hui Cao in his Ph.D dissertation [3] developed techniques for general

purpose simulators. He developed GPRS, the General Purpose Reservoir Simula-

tor developed at Stanford University, where he proposed and implemented new AIM

techniques that make use of FIM, IMPES and IMPSAT (implicit pressure and sat-

uration, explicit compositions) combinations in the simulator. GPRS is a general

purpose reservoir simulator that can handle both black-oil and compositional models

for isothermal simulation on unstructured grid.

Wan et al in 2005 [12], presented general stability criteria for compositional and

black-oil models taking into account all of the major mechanisms governing fluid

flow, i.e., convection, diffusion, capillary forces, gravity, fluid and rock compressibility,

vapor-liquid equilibrium in compositional models and solubility in black-oil models.

The black-oil models were treated as special cases of compositional models.

Chapter 3

Mathematical formulation

This chapter explains the mass conservation and energy conservation equations for

the purpose of doing stability analysis. We make the following assumptions:

• Source and sink terms are treated implicitly always, hence they are dropped out

of these equations

• Density ρ and viscosity µ are functions of pressure and temperature

• Specific heat is constant

• Relative permeability is a function of saturation only

• There is no heat loss to the reservoir surroundings

3.1 Governing equations

The equations for non-isothermal oil and water flow are given as follows:

6

CHAPTER 3. MATHEMATICAL FORMULATION 7

uw = −k krw

µw

(∇Pw − γw ∇Z) , (3.1)

uo = −k kro

µo

(∇Po − γo ∇Z) , (3.2)

ut = uw + uo, (3.3)

and Pcwo = Po − Pw, (3.4)

where subscripts w and o refer to the water and the oil phase, respectively, k is the

permeability and kr is the relative permeability. The above equations in 1D horizontal

flow, ignoring the effect of gravity can be expressed as follows:

uw = −k krw

µw

(∂Po

∂x− ∂Pcwo

∂x

), (3.5)

and uo = −k kro

µo

(∂Po

∂x

). (3.6)

The mean mobility is defined as follows:

λ =λw λo

λw + λo

. (3.7)

Using the above definition of mean mobility, uw and uo can be expressed in terms of

fractional flow coefficients, fw and fo respectively and ut.

uw = ut fw − k λ∂Pcwo

∂x, (3.8)

and uo = ut fo + k λ∂Pcwo

∂x, (3.9)

where Pcwo is a function of Sw only. In Eqs.(3.8) and (3.9), fw and fo denote the

fractional flow coefficients of water and oil, respectively, which are given by


fw =uw

ut

, (3.10)

and fo =uo

ut

. (3.11)

We can write ∂Pcwo

∂xas follows:

∂Pcwo

∂x= P ′

c

∂Sw

∂x, (3.12)

where P ′c is the derivative of Pcwo with respect to water saturation Sw. The differential

form of the mass conservation equations for the water, w, and the oil, o, in a black

oil model are as follows:

∂

∂t(φ ρw Sw) +

∂

∂x(ρw uw) = 0, (3.13)

and∂

∂t(φ ρo So) +

∂

∂x(ρo uo) = 0, (3.14)

where φ is the porosity of the rock. The differential form of the energy conservation

equation is given as:

φ∂

∂t(ρw Sw Uw + ρo So Uo) + (1− φ)

∂

∂t(ρR CpR

[T − Ti] )− ∂

∂x

(Υc

∂T

∂x

)

+∂

∂x(ρw uw Hw + ρo uo Ho) = 0, (3.15)

where CpRis the rock heat capacity, Υc is the conduction transmissibility, Hn is the

enthalpy, Un is the internal energy and Ti is the initial temperature in the block. The

internal energy and enthalpy are defined as follows:

Un = Cpn (T − Ti) , (3.16)

and Hn = Un +Po

ρn

, (3.17)

where n refers to the phase (w or o), and Cpn is the heat capacity of the fluid which

is assumed to be constant.


The energy conservation equation is analogous to the mass conservation equations.

In mass conservation equations, the transmissibilities are calculated as their upstream

grid block values. In the energy conservation equation, the corresponding flow term

is a weighted average of phase transmissibilities with their respective enthalpy. The

enthalpy term is calculated in the upstream grid block [4].

3.2 Mass Conservation Equations (Isothermal Case)

In order to derive the stability criteria for IMPES systems, the saturation equation

must be derived from Eq. (3.13) and Eq. (3.14). This is because it is saturation which

is treated explicitly. However, in order to do that, a few assumptions are made. The

phase densities ρn, are functions of average pressure within the grid block. This

implies that ρn is independent of the spatial variable x, but its time dependence is

retained. This is required as we are only concerned with a stability analysis in time.

The density, ρn of the phases is a function of Pn and T only, not Sn. Also, the rock

porosity, φ, is constant with respect to time, t. Thus Eq. (3.13) becomes

−ρw∂uw

∂x= φ

∂

∂t(ρw Sw) , (3.18)

and Eq. (3.14) becomes

−ρo∂uo

∂x= φ

∂

∂t(ρo So) . (3.19)

The 1st term on the LHS of Eq. (3.13) can be written as:

A∂Sw

∂t+ B

∂Pw

∂t,

where

A = φ ρw,

and B = φSw∂ρw

∂Pw

.


Similarly, the 1st term on the LHS of Eq. (3.14) can be written as:

C∂Sw

∂t+ D

∂Po

∂t,

where

C = −φ ρo,

and D = φSo∂ρo

∂Po

.

The LHS of Eq. (3.18) can be expanded using Eq. (3.8) and Eq. (3.12) as follows:

−ρw∂uw

∂x= −ρw ut

∂fw

∂Sw

∂Sw

∂x+ ρw k λ P ′

c

∂2Sw

∂x2. (3.20)

Similarly, the LHS of Eq. (3.19) can be expanded using Eq. (3.9) and Eq. (3.12) and

the fact that fo = 1− fw as follows:

−ρo∂uo

∂x= ρo ut

∂fw

∂Sw

∂Sw

∂x− ρo k λ P ′

c

∂2Sw

∂x2. (3.21)

We can now express Eq. (3.13) as

−ρw ut∂fw

∂Sw

∂Sw


c

∂2Sw

∂x2= φ ρw

∂Sw

∂t+ φSw

∂ρw

∂Pw

∂Pw

∂t,

which can be written as

−ρw ut∂fw

∂Sw

φSw∂ρw

∂Pw

∂Sw

∂x+

ρw k λ P ′c

φSw∂ρw

∂Pw

∂2Sw

∂x2=

ρw

Sw∂ρw

∂Pw

∂Sw

∂t+

∂Pw

∂t. (3.22)

Similarly, Eq. (3.14) takes the form

ρo ut∂fw

∂Sw

∂Sw


c

∂2Sw

∂x2= −φ ρo

∂Sw

∂t+ φSo

∂ρo

∂Po

∂Po

∂t,

which becomes

ρo ut∂fw

∂Sw

φSo∂ρo

∂Po

∂Sw


c

φSo∂ρo

∂Po

∂2Sw

∂x2= − ρo

So∂ρo

∂Po

∂Sw

∂t+

∂Po

∂t. (3.23)


Since,∂Po

∂t− ∂Pw

∂t= P ′

c

∂Sw

∂t, (3.24)

we can eliminate ∂Pn

∂tterms by subtracting Eq. (3.22) from Eq. (3.23) and using Eq.

(3.24) to obtain the following equation in Sw:

(ρo ut

∂fw

∂Sw

φSo∂ρo

∂Po

+ρw ut

∂fw

∂Sw

φSw∂ρw

∂Pw

)∂Sw

∂x−

(ρo k λ P ′

c

φSo∂ρo

∂Po

+ρw k λ P ′

c

φSw∂ρw

∂Pw

)∂2Sw

∂x2

= −(

ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

− P ′c

)∂Sw

∂t, (3.25)

which we write as,

∂Sw

∂t= −C′ ∂Sw

∂x+ D′ ∂

2Sw

∂x2, (3.26)

where

C′ =ut

∂fw

∂Sw

φ

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

− P ′c

, (3.27)

D′ =k λ P ′

c

φ

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

− P ′c

. (3.28)

If the density is not a function of pressure, the term containing the partial deriv-

ative of density with respect to pressure is absent, and the form of these equations

changes slightly. This can be observed by taking the limit of ∂ρo

∂Poand ∂ρw

∂Pwapproaching

zero. The second term in the above expressions become unity on taking this limit.

The coefficient C′ reduces tout

∂fw∂Sw

φ, and D′ reduces to k λ P ′c

φ. The same result is also

obtained if either one of Eq. (3.22) or Eq. (3.23) is used to perform the stability

analysis.


3.3 Mass Conservation Equations (Thermal Case)

When temperature is no longer constant, the equations shown in section 3.2 get

modified due to the temperature dependent terms. We will express the equations

shown in section 3.2 with temperature, T , treated explicitly. Later, it will be simple

to observe the changes if T is implicit.

Eq. (3.13) becomes

−ρw∂uw

∂x− uw

∂ρw

∂T

∂T

∂x=

∂

∂t(φ ρw Sw) , (3.29)

and Eq. (3.14) becomes

−ρo∂uo

∂x− uo

∂ρo

∂T

∂T

∂x=

∂

∂t(φ ρo So) . (3.30)

The 1st term on the LHS of Eq. (3.13) can be written as:

A∂Sw

∂t+ B

∂Pw

∂t+ C

∂T

∂t,

where

A = φ ρw,

B = φSw∂ρw

∂Pw

,

and C = φSw∂ρw

∂T.

Similarly the 1st term on the LHS of Eq. (3.14) can be written as:

D∂Sw

∂t+ E

∂Po

∂t+ F

∂T

∂t,


where

D = −φ ρo,

E = φSo∂ρo

∂Po

,

and F = φSo∂ρo

∂T.

The first term on the LHS of Eq. (3.29) can be expanded using Eq. (3.8) and Eq.

(3.12) as follows:

−ρw∂uw

∂x= −ρw ut

∂fw

∂Sw

∂Sw


c

∂2Sw

∂x2− ρw ut

∂fw

∂T

∂T

∂x. (3.31)

Similarly, the first term on the LHS of Eq. (3.30) can be expanded using Eq. (3.9)

and Eq. (3.12) and the fact that fo = 1− fw as follows:

−ρo∂uo

∂x= ρo ut

∂fw

∂Sw

∂Sw


c

∂2Sw

∂x2+ ρo ut

∂fw

∂T

∂T

∂x. (3.32)

We can now express Eq. (3.13) as

−ρw ut∂fw

∂Sw

∂Sw


c

∂2Sw

∂x2−

(uw

∂ρw

∂T+ ρw ut

∂fw

∂T

)∂T

∂x

= φ ρw∂Sw

∂t+ φSw

∂ρw

∂Pw

∂Pw

∂t+ φSw

∂ρw

∂T

∂T

∂t,

which can be written as,

−ρw ut∂fw

∂Sw

φ Sw∂ρw

∂Pw

∂Sw

∂x+

ρw k λ P ′c

φSw∂ρw

∂Pw

∂2Sw

∂x2− uw

∂ρw

∂T+ ρw ut

∂fw

∂T

φSw∂ρw

∂Pw

∂T

∂x

=ρw

Sw∂ρw

∂Pw

∂Sw

∂t+

∂Pw

∂t+

∂ρw

∂T∂ρw

∂Pw

∂T

∂t, (3.33)


and Eq. (3.14) can be expressed as

ρo ut∂fw

∂Sw

∂Sw


c

∂2Sw

∂x2−

(uo

∂ρo

∂T− ρo ut

∂fw

∂T

)∂T

∂x

= −φ ρo∂Sw

∂t+ φSo

∂ρo

∂Po

∂Po

∂t+ φSo

∂ρo

∂T

∂T

∂t,


ρo ut∂fw

∂Sw

φSo∂ρo

∂Po

∂Sw


c

φSo∂ρo

∂Po

∂2Sw

∂x2− uo

∂ρo

∂T− ρo ut

∂fw

∂T

φSo∂ρo

∂Po

∂T

∂x

= − ρo

So∂ρo

∂Po

∂Sw

∂t+

∂Po

∂t+

∂ρo

∂T∂ρo

∂Po

∂T

∂t. (3.34)

From Eq. (3.24), we know∂Po

∂t− ∂Pw

∂t= P ′

c

∂Sw

∂t.

We can eliminate the ∂Pn

∂tterms by subtracting Eq. (3.33) from Eq. (3.34) and using

Eq. (3.24) to obtain the following equation in Sw and T :

(ρo ut

∂fw

∂Sw

φSo∂ρo

∂Po

+ρw ut

∂fw

∂Sw

φSw∂ρw

∂Pw

)∂Sw

∂x−

(ρo k λ P ′

c

φSo∂ρo

∂Po

+ρw k λ P ′

c

φSw∂ρw

∂Pw

)∂2Sw

∂x2

+

((ρo ut

∂fw

∂T

φSo∂ρo

∂T

+ρw ut

∂fw

∂T

φSw∂ρw

∂T

)−

(uo

∂ρw

∂T

φSo∂ρo

∂T

− uw∂ρw

∂T

φSw∂ρw

∂T

) )∂T

∂x

= −(

ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

− P ′c

)∂Sw

∂t+

(∂ρo

∂T∂ρo

∂Po

−∂ρw

∂T∂ρw

∂Pw

)∂T

∂t,


A′ ∂Sw

∂t+ B′ ∂T

∂t= −C′ ∂Sw

∂x+ D′ ∂

2Sw

∂x2− E′ ∂T

∂x, (3.35)


where

A′ =ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

− P ′c, (3.36)

B′ =∂ρw

∂T∂ρw

∂Pw

−∂ρo

∂T∂ρo

∂Po

, (3.37)

C′ =ut

∂fw

∂Sw

φ

(ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

), (3.38)

D′ =k λ P ′

c

φ

(ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

), (3.39)

and E′ =ut

∂fw

∂T

φ

(ρo

So∂ρo

∂T

+ρw

Sw∂ρw

∂T

)− 1

φ

(uo

∂ρo

∂T

So∂ρo

∂T

− uw∂ρw

∂T

Sw∂ρw

∂T

). (3.40)

3.4 Conservation of Energy

A similar expansion of the derivative terms in Eq. (3.15) provides the second equation

in terms of Sw and T . Before we start expanding the energy balance given by Eq.

(3.15), it is useful to keep in mind the following relations defined by Eq. (3.16) and

Eq. (3.17):

Un = Cpn (T − Ti) ,

and Hn = Un +Po

ρn

,

The first two terms on the LHS of Eq. (3.15) can be written as:

L∂Sw

∂t+ M

∂Po

∂t+ N

∂T

∂t,

where


L = φ

(−ρo Uo + ρw Uw − Uw Sw P ′

c

∂ρw

∂Pw

),

M = φ

(Uo So

∂ρo

∂Po

+ Uw Sw∂ρw

∂Pw

+1− φ

φCpR

(T − Ti)∂ρR

∂Po

),

and N = φ

(Uo So

∂ρo

∂T+ ρo So

∂Uo

∂T+ Uw Sw

∂ρw

∂T+ ρw Sw

∂Uw

∂T

)

+ (1− φ) CpR

((T − Ti)

∂ρR

∂T+ ρR

).

Assuming the conduction transmissibility coefficient, Υc, to be constant, the last two

terms on the LHS of Eq. (3.15) can be expanded as follows:

G∂Sw

∂x+ I

∂2S

∂x2+ J

∂T

∂x+ K

∂2T

∂x2,

where

G = ut∂fw

∂Sw

(ρo Ho − ρw Hw) ,

I = −k λ P ′c (ρo Ho − ρw Hw) ,

J = ut∂fw

∂T(ρo Ho − ρw Hw)−Ho uo

∂ρo

∂T−Hw uw

∂ρw

∂T− ρo uo

∂Ho

∂T− ρw uw

∂Hw

∂T,

and K = Υc.

Thus, the energy equation takes the form:

G

M

∂Sw

∂x+

I

M

∂2Sw

∂x2+

J

M

∂T

∂x+

K

M

∂2T

∂x2=

L

M

∂Sw

∂t+

∂Po

∂t+

N

M

∂T

∂t. (3.41)

We can see that the energy equation is very similar in form to the mass conserva-

tion equations. The conduction transmissibility term appearing in the coefficient K

of the energy equation is analogous to the capillary pressure term in the saturation

equation, Eq. (3.26). We can eliminate ∂Po

∂tterms by subtracting Eq. (3.34) from Eq.

(3.41) to obtain the following second equation in Sw and T , the first being Eq. (3.35)

obtained in section 3.3:


(G

M− ρo ut

∂fw

∂Sw

φSo∂ρo

∂Po

)∂Sw

∂x+

(I

M+

ρo k λ P ′c

φSo∂ρo

∂Po

)∂2Sw

∂x2

+

(J

M+

uo∂ρo

∂T− ρo ut

∂fw

∂T

φSo∂ρo

∂Po

)∂T

∂x+

K

M

∂2T

∂x2

=

(L

M+

ρo

So∂ρo

∂Po

)∂Sw

∂t+

(N

M−

∂ρo

∂T∂ρo

∂Po

)∂T

∂t,


F′∂Sw

∂t+ G′ ∂T

∂t= −H′ ∂Sw

∂x+ I′

∂2Sw

∂x2− J′

∂T

∂x+ K′ ∂

2T

∂x2, (3.42)

where

F′ =ρo

So∂ρo

∂Po

+−ρo Uo + ρw Uw − Uw Sw P ′

c∂ρw

∂Pw

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

, (3.43)

G′ =Uo So

∂ρo

∂T+ ρo So

∂Uo

∂T+ Uw Sw

∂ρw

∂T+ ρw Sw

∂Uw

∂T+ 1−φ

φCpR

((T − Ti)

∂ρR

∂T+ ρR

)

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

−∂ρo

∂T∂ρo

∂Po

, (3.44)

H′ = ut∂fw

∂Sw

(ρo

φSo∂ρo

∂Po

− ρo Ho − ρw Hw

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

), (3.45)

I′ = k λ P ′c

(ρo

φSo∂ρo

∂Po


Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

), (3.46)


J′ = −ut∂fw

∂T(ρo Ho − ρw Hw)− uo

(Ho

∂ρo

∂T+ ρo

∂Ho

∂T

)− uw

(Hw

∂ρw

∂T+ ρw

∂Hw

∂T

)

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

+ρo ut

∂fw

∂T− uo

∂ρo

∂T

φSo∂ρo

∂Po

, (3.47)

and K′ =Υc

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

. (3.48)

3.5 Application of Von Neumann Method for Sta-

bility Analysis

The Von Neumann analysis is used to study the stability characteristics of our finite

difference scheme. The procedure is to perform a spatial Fourier transform along all

spatial dimensions, thereby reducing the finite difference scheme to a time recursion

in terms of a spatial Fourier transform of the system. The system is stable if this

time recursion is at least marginally stable.

If we apply von Neumann analysis to the following finite difference scheme,

yn+1,m = yn,m+1 + yn,m−1 − yn−1,m, (3.49)

where m refers to the spatial coordinate, and n refers to the temporal coordinate.

There is only one spatial dimension, so we only need a single 1D Discrete Time

Fourier Transform (DTFT) along m. For a length N complex sequence x(n), n =

0, 1, 2, ..., N − 1, the discrete Fourier transform (DFT) is defined by


X (ωk) ,N−1∑n =0

x (n) e−j ωk tn

=N−1∑n =0

x (n) e−j 2π k n/N , k = 0, 1, 2, ...N − 1, (3.50)

tn , nT = nthsampling instant(sec), (3.51)

ωk , k Ω = kthfrequency sample(rad/sec), (3.52)

T , 1/fs = time sampling interval(sec), (3.53)

and Ω , 2 π fs/N = frequency sampling interval(rad/sec). (3.54)

The Discrete Time Fourier Transform (DTFT) can be viewed as the limiting form

of the Discrete Fourier Transform (DFT) when its length N is allowed to approach

infinity:

X (ω) ,∞∑

n =−∞x (n) e−j ω n, (3.55)

where ω ∈ [−π , π ) denotes the continuous radian frequency variable, and x (n) is

the signal amplitude at sample number n. The Shift Theorem is stated as follows:

For any x ∈ CN and any integer ∆,

DFTk [ SHIFT∆ (x)] = exp−j ωk ∆ X (k) . (3.56)

Using the shift theorem for the DTFT, we obtain

Yn+1 (k) =(ejkX + e−jkX

)Yn (k)− Yn−1 (k)

= 2 cos (kX) Yn (k)− Yn−1 (k)

, 2 ck Yn (k)− Yn−1 (k) , (3.57)

where k = 2 π / λ denotes radian spatial frequency (wave number).

When the time recursion is first order, we can compute the amplification factor


as the complex gain G (k) in the relation

Yn+1 (k) = G (k) Yn (k) . (3.58)

The finite difference scheme is then stable if |G (k)| 6 1 for all spatial frequencies k.

3.6 Stability Analysis for Two Phase Isothermal

Systems

We use the Von Neumann method to derive the required stability criteria for non-

oscillatory stability. The term non-oscillatory stability refers to the case in which all

eigenvalues are positive and less than unity. For example, the following 1D water

saturation equation

∂Sw

∂t= −C

∂Sw

∂x+ D

∂2Sw

∂x2, (3.59)

can be discretized explicitly as follows:

Sn+1wj

− Snwj

∆ t= −C

[Sn

wj− Sn

wj−1

∆ x

]+ D

[Sn

wj+1− 2 Sn

wj+ Sn

wj−1

∆ x2

]. (3.60)

The discrete value in the difference equation is replaced by its generalized Fourier

component:

Snwj

= ξn ei β j. (3.61)

This results in the following equation:

ξn+1

ξn= 1−

(C ∆ t

∆ x+

2 D ∆ t

∆ x2

)(1− cos β) − i

C ∆ t

∆ xsin β. (3.62)

For non-oscillatory stability, the sufficient condition is:

∣∣∣∣ξn+1

ξn

∣∣∣∣ < 1, (3.63)

where ξn+1

ξn is the amplification factor.


From Eq. (3.62), this amplification factor is derived as:

|λ|2 =

∣∣∣∣ξn+1

ξn

∣∣∣∣2

=

[1−

(C ∆ t

∆ x+

2 D ∆ t

∆ x2

)(1− cos β)

]2

+

[C ∆ t

∆ xsin β

]2

. (3.64)

The maximum value of this factor is obtained by using the criteria:

d

d β

(|λ|2) = 0, (3.65)

which is satisfied for

β = nπ. (3.66)

Combining Eq. (3.63) and Eq. (3.64) we have:

∣∣∣∣ 1− 2

(C ∆ t

∆ x+

2 D ∆ t

∆ x2

)∣∣∣∣ < 1. (3.67)

We thus arrive at the following stability condition:

0 <C ∆ t

∆ x+

2 D ∆ t

∆ x2< 1. (3.68)

Eq. (3.59) has the same form as Eq. (3.26). As a result, coefficients C and D can be

replaced with C′ from Eq. (3.27) and D′ from Eq. (3.28) respectively and the general

stability condition for isothermal two-phase oil-water flow in the presence of capillary

effects can be obtained as follows:

0 <

[qt

∂fw

∂Sw∆ t

V φ+

2 ΥG λ P ′c ∆ t

V φ

]

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

− P ′c

< 1, (3.69)

where

qt = Aut,

V = A ∆ x,

and ΥG =k A

∆ x,

where A is the cross-sectional area and ΥG is the geometric transmissibility of the


system.

If the capillary pressure term is dropped out, this expression becomesqt

∂fw∂Sw

∆ t

V φ,

which is the well known CFL number criteria as given by Coats [5], [6] in its simplest

form. From this expression we can calculate the maximum allowed time step size.

3.7 Stability Analysis for Two Phase Thermal Sys-

tems

The stability analysis for a two phase thermal system is similar to that for a three

phase isothermal system.

Eq. (3.35) and Eq. (3.42) can be summarized as follows:

A′ ∂Sw

∂t+ B′ ∂T

∂t= −C′ ∂Sw

∂x+ D′ ∂

2Sw

∂x2− E′ ∂T

∂x,

F′∂Sw

∂t+ G′ ∂T

∂t= −H′ ∂Sw

∂x+ I′

∂2Sw

∂x2− J′

∂T

∂x+ K′ ∂

2T

∂x2.

We can solve for ∂Sw

∂tand ∂T

∂tto obtain the following equations:

∂Sw

∂t= −A1

∂Sw

∂x+ A2

∂2Sw

∂x2−B1

∂T

∂x+ B2

∂2T

∂x2, (3.70)

and∂T

∂t= −C1

∂Sw

∂x+ C2

∂2Sw

∂x2−D1

∂T

∂x+ D2

∂2T

∂x2, (3.71)

where the coefficients A1, A2, B1, B2, C1, C2, D1 and D2 are defined as:

X =

(A′ B′

F′ G′

)−1 (−C′ D′ −E′ 0

−H′ I′ −J′ K′

)=

(−A1 A2 −B1 B2

−C1 C2 −D1 D2

)

Using a similar approach to that for two-phase isothermal flow, the following coupled

system of error propagation equations is obtained from Eq. (3.70) and Eq. (3.71):


ξn+1w = ξn

w

[1−

(A1 ∆ t

∆ x+

2 A2 ∆ t

∆ x2

)(1− cos βw) − i

A1 ∆ t

∆ xsin βw

]

+ ξnT

[−

(B1 ∆ t

∆ x+

2 B2 ∆ t

∆ x2

)(1− cos βT ) − i

B1 ∆ t

∆ xsin βT

],(3.72)

and

ξn+1T = ξn

T

[1−

(D1 ∆ t

∆ x+

2 D2 ∆ t

∆ x2

)(1− cos βT ) − i

D1 ∆ t

∆ xsin βT

]

+ ξnw

[−

(C1 ∆ t

∆ x+

2 C2 ∆ t

∆ x2

)(1− cos βw) − i

C1 ∆ t

∆ xsin βw

],(3.73)

where the subscript w refers to water equation, and the subscript T refers to energy

equation. Again, the maximum eigenvalues are obtained for β = nπ. Thus the above

system is reduced to the following:

ξn+1w = ξn

w

[1− 2

(A1 ∆ t

∆ x+

2 A2 ∆ t

∆ x2

)]+ ξn

T

[−2

(B1 ∆ t

∆ x+

2 B2 ∆ t

∆ x2

)], (3.74)

and

ξn+1T = ξn

w

[−2

(C1 ∆ t

∆ x+

2 C2 ∆ t

∆ x2

)]+ ξn

T

[1− 2

(D1 ∆ t

∆ x+

2 D2 ∆ t

∆ x2

)]. (3.75)

Writing the above equations in matrix form gives:

[ξn+1w

ξn+1T

]=

(1− A −B

−C 1−D

) [ξnw

ξnT

]. (3.76)

The non-oscillatory stability condition requires that the spectral radius of the above

error propagation matrix be less than unity. Thus we have:

∣∣∣∣ 2 − A − D ±√

(A−D)2 + 4 B C

∣∣∣∣ < 2. (3.77)

This reduces to:

A + D ±√

(A−D)2 + 4 B C < 4. (3.78)


The various parameters in the above equation are complicated expressions obtained

by Eqs.(3.36)-(3.40) in section 3.3 and Eqs.(3.43)-(3.48) in section 3.4. Appendix A

gives the expressions for the stability criteria of thermal two-phase flows.

Chapter 4

Verification

4.1 Thermal Simulator

The thermal simulator was built using MATLAB [8]. The primary variables are oil

pressure P , water saturation S, and temperature T . The conservation equations in

discrete form are given as follows:

Oil conservation equation:

Υoi− 1

2

(Pi−1 − Pi) + Υoi+1

2

(Pi+1 − Pi) =V

∆ t

[(φSo ρo)

n+1 − (φSo ρo)n]+ ρo

o qo, (4.1)

Water conservation equation:

Υwi− 1

2

(Pi−1 − Pi) + Υwi+1

2

(Pi+1 − Pi)−Υwi− 1

2

Pcwoi− 1

2

(Si−1 − Si)

− Υwi+1

2

Pcwoi+1

2

(Si+1 − Si) =V

∆ t

[(φSw ρw)n+1 − (φSw ρw)n

]+ ρo

w qw, (4.2)

25

CHAPTER 4. VERIFICATION 26

and Energy conservation equation:

Υi− 12(Pi−1 − Pi) + Υi+ 1

2(Pi+1 − Pi) + Υc

i− 12

(Ti−1 − Ti) + Υci+1

2

(Ti+1 − Ti)

− ΥHi+1

2

Pcwoi+1

2

(Si+1 − Si)−ΥHi− 1

2

Pcwoi− 1

2

(Si−1 − Si)

=V

∆ t

( [(φ (Sw ρw Uw + So ρo Uo))

n+1 − (φ (Sw ρw Uw + So ρo Uo))n]

+[((1− φ) (CpR

ρR (T − Ti)))n+1 − ((1− φ) (CpR

ρR (T − Ti)))n] )

+ ρow qw Hw + ρo

o qo Ho, (4.3)

where subscripts o and w refer to the oil and water phases, superscript o refers to

standard conditions, and we have the following definitions:

Υni− 1

2

= ρon

(k A

∆ x

)

i− 12

(krn

Bn µn

)

i− 12

, (4.4)

ρn =ρo

n

Bn

, (4.5)

Υi− 12

= Υwi− 1

2

Hwi− 1

2

+ Υoi− 1

2

Hoi− 1

2

, (4.6)

Hni− 1

2

= Cpn (T − Ti) +

(P Bn

ρon

)

i− 12

, (4.7)

and Un = Cpn (T − Ti), (4.8)

where Υn is the transmissibility of the water or oil phase, ρn is the density of the

phase at the given pressure and temperature conditions. The other symbols have

been defined earlier.

The residual equations for oil, water and energy are obtained from Eqs.(4.1)-

(4.3). The nonlinear terms involve the relative permeability and enthalpy. These are

calculated as their upstream grid block values along the same principles described

in [2]. The assumptions are stated at the beginning of Chapter 3. The physical

properties data used for oil, water and rock are given in Appendix B.

The verification results of the derived stability criteria are shown in the following

sections of this chapter. The approach used is to test the stability condition by violat-

ing the time step size and observing the oscillations produced in pressure, saturation

and temperature profiles.


From the derived stability criteria we can calculate the maximum stable time step

size. In an AIM formulation, we choose some grid blocks IMPEST (IMplicit Pressure

Explicit Saturations and Temperature). We use the maximum allowed time step

for such grid blocks. This will not only ensure a stable solution, but also allow the

largest possible time steps to be taken thereby optimizing on the CPU time required

for simulation.

4.2 Formulation of FIM and IMPEST

The Residual is defined as Net inflow - Accumulation + Source. In symbolic form, it

can be written as follows:

Rn+1 = Υγ(Xn+1 −Xn)− ∆t M

∆ t+ qn+1, (4.9)

where X is a vector of unknowns (P , S, T ) in the grid blocks. The term implicit refers

to the evaluation of inter block flow terms and production rates at the new time level,

n + 1. Therefore, γ = n + 1 for FIM formulation, whereas γ = n for IMPEST.

The capillary pressure terms are contained in the transmissibility Υ. When cap-

illary pressure is present, the basic assumption of the IMPEST method is that the

capillary pressure in the flow terms does not change over a time step. Thus in the

discretized form of Eq. (3.12), the saturation, S is evaluated at the previous time

step.

4.3 Two phase Isothermal results

The two phase oil-water isothermal mass conservation equations including the effect

of capillary pressure in discrete form are given by Eq. (4.1) and Eq. (4.2). The

stability number taken from Eq. (3.69) derived in section 3.6 is:

CFL =

[qt

∂fw

∂Sw∆ t

V φ+

2 ΥG λ P ′c ∆ t

V φ

]

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

ρo

So∂ρo∂Po

+ ρw

Sw∂ρw∂Pw

− P ′c

. (4.10)

The capillarity effect is contained in the term P ′c. Pc being inversely related with


water saturation, Sw, P ′c is negative. Therefore, the stability number computed by

Eq. (4.10) will be smaller when capillary effects are present than when they are

not. This implies that the system becomes more unstable with explicit treatment of

capillarity when it is present, and the maximum allowed time step size is smaller.

To test the derived stability criteria, a test case is set up which is a 1-D horizontal

reservoir 500 ft in length divided equally into 10 grid blocks, with water injection at

one end and oil production at the other end. Water is injected at a constant rate,

and oil is produced from the production well under bottom hole pressure control.

We violate the maximum allowed time step size and observe the behavior of the

saturation front. A slight violation is induced after 50 days of production by injecting

1.5 times the cell pore volume (∆ t = 15 days) when only 10 days is allowed from Eq.

(4.10). The results are shown in Fig. (4.1). We observe the stability number crossing

the limit of unity in a few grid blocks, but we do not observe drastic oscillations in

pressure and saturation histories or the saturation fronts. The profiles tend to deviate

from the FIM solution slightly. In the second experiment, we violated the time step

drastically, injecting 4.5 times the cell pore volume (∆ t = 45 days) after 50 days

of production. In this case, we observe oscillations in the pressure and saturation

profiles. The saturation front looks far from the fully implicit result. This is observed

in Fig. (4.1).

In order to study the time step violation trend, a third experiment corresponding

to injecting 3 times the cell pore volume (∆ t = 30 days) was conducted. This is

labeled as intermediate violation in Figs.(4.1)-(4.4). We observe clearly that as the

magnitude of the violation increases, the intensity of oscillations in the pressure and

saturation profiles also increases.

The pressure and saturation profiles for the fully implicit, IMPES, and violated

IMPES are shown in Figs.(4.2) and (4.3). Also, observe the result of the fully implicit

solution with large time steps. It shows a close match with the regular FIM. The only

errors being those due to time truncation errors when discretizing the equations.

We plotted the trend of CFL numbers in the grid blocks in Fig. (4.4) at time

equal to 50 days. There is a linear trend between CFL number and ∆t implied from

Eq. (4.10). It is interesting to note that the saturation front in Fig. (4.1) corresponds

with the CFL trend. We observe the sharp bump in saturation fronts at the same


Saturation Fronts at 110 days

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

Sw

FIM

FIM - large steps

IMPES

IMPES - large violation

IMPES - small violation

Figure 4.1: Saturation fronts in different schemes

Pressure in the injection block vs time

2100

3100

4100

5100

6100

7100

8100

0 50 100 150 200 250 300 350 400 450

time (days)

Pre

ssu

re (

psi

a)

IMPES

FIM

FIM - large steps

IMPES - large viol

IMPES - small viol

IMPES - intermed viol

Figure 4.2: Pressure profiles in different schemes


Saturation in the production block vs time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350 400 450

time (days)

Sw

IMPES

FIM

FIM - large steps

IMPES - large viol

IMPES - small viol

IMPES - intermed viol

Figure 4.3: Saturation profiles in different schemes

spatial location where the corresponding CFL numbers are largest.

In general, from all the figures in this section, Figs.(4.1)-(4.4), we observe that the

larger the violation in the time step size, there is more oscillation in the profiles, and

the saturation fronts are further away from the fully implicit solution.

4.4 Two phase Thermal results

The two phase thermal results are divided into two categories. The low mobility

ratio case with M ∼ 10, or oil is less viscous, and the high mobility ratio case with

M ∼ 100, or oil being highly viscous. Since, the viscosities of both oil and water

depend on pressure and temperature, which are not constant, the mobilities are also

variable.

In this section, we analyze two schemes. The first one is labeled as IMPEST,

which means implicit in pressure, and explicit in both saturation and temperature.

The stability criteria are given in Appendix A. We should note that here the maximum


Trend of CFL number with distance

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

CF

L #

CFL - small voilation

CFL - large violation

CFL - intermediate violation

Figure 4.4: Stability number trend in the reservoir

allowed stability number using the relation of Eq. (3.78) is four instead of unity in

the isothermal case given by Eq. (3.69).

The other scheme of solving the nonlinear system of equations is termed as

IMPTES, which is implicit in both pressure and temperature, but explicit in sat-

uration. Since, now only one variable is explicit, Eq. (3.69) is applicable and we

consider a violation if CFL is more than unity. Comparing IMPEST and IMPTES,

we can better understand the effect of making temperature explicit.

A third scheme could be IMPSET, implying implicit in pressure and saturation

but explicit in temperature. It was found that it was very tough to violate the time

step size in this case. This is because the flow terms which are largely saturation

dependent, are now implicit. The effect of temperature in flow terms gets included

through enthalpy, which is a linear function of temperature in this case because specific

heats are assumed to be constant. It would be interesting to study this case when

enthalpy is a nonlinear function of temperature.


Progress of saturation fronts (low mobility ratio)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

Sw

time increasing

Figure 4.5: Progress of saturation fronts for low mobility ratios at different times

4.4.1 Low Mobility Ratio

4.4.1.1 Implicit P, Explicit S and T

We set up the test case in a reservoir of length 800 ft divided equally into 10 grid

blocks. Hot water at a fixed temperature and constant rate is injected at one end and

oil is produced at the other end. This test is similar to that of the isothermal case.

The reservoir has a rock heat capacity, and it retains heat. This heat is transmitted

through the conduction process. Thus conduction mainly depends on two parameters,

the rock heat capacity CpRand conduction transmissibility Υc. However, heat transfer

from one end of the reservoir to the other takes place more effectively by means

of convection, the heat carried with the moving fluid front, which is a dominant

mechanism in thermal displacements.

We considered a case when heat transfer is by convection only (i.e. the rock


Progress of heat fronts (low mobility ratio)

290

310

330

350

370

390

410

430

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

T (

K)

time increasing

Figure 4.6: Progress of heat fronts for low mobility ratios at different times


does not retain any heat, CpR= 0, and the rock conduction transmissibility is zero).

We observed that heat and saturation fronts begin to proceed together, but the heat

front lags behind as time progresses. This can be clearly observed from Figs.(4.5) and

(4.6). The two fluids considered here are oil and water. Oil has a lower specific heat

as compared to water. Thus we observe that heat fronts lag behind even in the pure

heat convection process. Therefore, when conduction is also present, heat fronts will

definitely lag behind the saturation fronts. This idea will be helpful in understanding

the results of the complete system later on.

As we did in section 4.3, we violate the time step size indicated by Eq. (3.78) by

small and large amounts, and we study the effect on the obtained solutions. In the

first case we violate the time step after 50 days of production, by injecting an amount

equivalent to the cell pore volume (∆ t = 20 days). In the second case, we inject

twice the cell pore volume (∆ t = 40 days) after 70 days of production. We study

the production history in the middle of the reservoir for a total of 240 days from the

start. In both cases, the pressure and saturation values begin to oscillate at the time

when we violated the criteria. Larger deviations from the FIM result are seen with

the larger time step.

When the time step is violated by a larger amount, the system fails to converge

after 200 days of production, and we do not get any data output after that. This

diverging behavior for the small violation case comes after a longer time. The os-

cillations are also stronger for the larger violation case as observed from Figs.(4.7)

and (4.8). The temperature profile does not show any oscillations, but the solution

is completely wrong, and we end up with lower temperatures than the fully implicit

solution. This is shown in Fig. (4.9).

When an equivalent amount of cell pore volumes is injected (∆ t = 20 days), the

magnitude of the stability numbers is considerably smaller than when twice the cell

pore volumes are injected (∆ t = 40 days). Fig. (4.10) shows the oscillations in the

saturation fronts of Fig. (4.11). We observe a peak in the saturation front corre-

sponding to the location where the stability numbers are maximum in the violated

IMPEST case. The saturation fronts in Fig. (4.11) are plotted at 110 days.

The temperature fronts shown in Fig. (4.12) look almost the same for all the

five cases shown, however they differ slightly from each other. There is no sharp


Pressure in middle block

2200

2300

2400

2500

2600

2700

2800

2900

3000

0 50 100 150 200 250

time (days)

P (

Psi

a)

FIM

IMPEST

IMPEST - small viol

FIM - large steps

IMPEST - large viol

Figure 4.7: Pressure profile in the middle grid block

Saturation in middle block

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250

time (days)

Sw

FIM

IMPEST

IMPEST - small viol

FIM - large steps

IMPEST - large viol

Figure 4.8: Saturation profile in the middle grid block


Temperature in middle block

300

305

310

315

320

325

330

335

340

345

0 50 100 150 200 250

time (days)

T (

K)

FIM

IMPEST

IMPEST - small viol

FIM - large steps

IMPEST - large viol

Figure 4.9: Temperature profile in the middle grid block

CFL Trend (low mobility ratio)

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600 700 800

Distance (ft)

CF

L #

40 day step, 110 days




Figure 4.10: Stability number trend in the thermal reservoir


Saturation Fronts (low mobility ratio)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600 700 800

Distance (ft)

Sw

IMPEST - large viol

IMPEST - small viol

IMPEST

FIM

FIM - large steps

Figure 4.11: Saturation fronts at 110 days in the thermal reservoir

Temperature Fronts (low mobility ratio)

296

316

336

356

376

396

416

436

456

0 100 200 300 400 500 600 700 800

Distance (ft)

T (

K)

IMPEST - large viol

IMPEST - small viol

IMPEST

FIM

FIM - large steps

Figure 4.12: Temperature fronts at 110 days in the thermal reservoir


contrast in this case as it was with saturation. The primary reason for this is that

a lot of heat is retained in the rock, and conduction is not strong. Therefore the

temperature gradients in the fluid are not strong enough to show sharp differences

in the temperature fronts. As we showed earlier in this chapter, the temperature

fronts lag behind saturation fronts. Therefore, the temperature fronts in all solution

methods look alike in Fig. (4.12). The physical properties used for the experiments

are listed in Appendix B.

4.4.1.2 Implicit P and T, Explicit S

The only difference in the Jacobian of the above IMPEST system and this system

is that the terms ∂Re

∂Ti−1and ∂Re

∂Ti+1are non zero and equal to Υc. Here Re refers to

the energy residual. But as we have observed, physically reasonable values of rock

transmissibility do not have a large impact on the stability behavior. Therefore, all

the profiles and histories look the same as in section 4.4.1.1. The stability number

in this case is only due to water saturation, Sw, being explicit. Therefore, stability

numbers greater than unity are considered to be a violation of the criteria in this

case, as opposed to CFL > 4 in section 4.4.1.1.

The stability numbers for IMPEST and IMPTES cases are roughly the same for

all practical purposes, though they are slightly less in the case when temperature

is implicit. This implies that larger timesteps can be taken without disturbing the

stability of the solution. However in this case the nonlinearities dependent on temper-

ature were not very strong as discussed earlier, therefore we expect larger differences

in stability number for the two cases in more realistic problems.

4.4.2 High Mobility Ratio

The saturation and heat fronts shown in Figs.(4.13) and (4.14) for M ∼ 100 can be

compared with Figs.(4.5) and (4.6), which are for a low mobility ratio, M ∼ 10. The

frontal saturation in the case of high mobility ratios is much less than that in low

mobility ratio case. Correspondingly, the front temperature in Fig. (4.14) is less than

the front temperature in Fig. (4.6).

We studied the effect of pure convection in this experiment. All conduction terms


Progress of saturation fronts (high mobility ratio)

0

0.02

0.04

0.06

0.08

0.1

0.12

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

Sw

time increasing

Figure 4.13: Progress of saturation fronts for high mobility ratios at different times


Progress of heat fronts (high mobility ratio)

290

300

310

320

330

340

350

360

370

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

T (

K)

time increasing

Figure 4.14: Progress of heat fronts for high mobility ratios at different times


Pressure in the middle grid block, high mobility ratio

3000

3500

4000

4500

5000

5500

6000

6500

0 50 100 150 200 250

time (days)

P (

psi

a)

FIM

FIM - large steps

IMPEST

Figure 4.15: Pressure profile in the middle grid block

Saturation in the middle grid block, high mobility ratio

0

0.05

0.1

0.15

0.2

0.25

0.3

0 50 100 150 200 250

time (days)

Sw

FIM

FIM - large steps

IMPEST

Figure 4.16: Saturation profile in the middle grid block


Temperature in the middle block, high mobility ratio

295

300

305

310

315

320

325

330

335

0 50 100 150 200 250

time (days)

T (

K)

FIM

FIM - large steps

IMPEST

Figure 4.17: Temperature profile in the middle grid block

were removed and the test case set up as before. Oscillations in pressure and sat-

uration histories are observed as before when the criteria are violated, as seen in

Figs.(4.15) and (4.16). The temperature profile is seen to be increasingly deviating

from the actual FIM solution. Also observe in Fig. (4.17) that the fully implicit

solution with similar large timesteps is close to the original fully implicit solution.

The small differences are due to time truncation errors.

In order to somehow quantify the individual effects of explicit treatment of satura-

tion and temperature, we decoupled the saturation and temperature equations while

deriving the stability criteria. This means that we no longer have to solve Eq. (3.35)

and Eq. (3.42) simultaneously. We ignore the coefficients B′ and E′ from satura-

tion equation, Eq. (3.35) and obtain the stability numbers corresponding to explicit

saturation only. Similarly, we ignore the coefficients F′, H′ and I′ from the energy

equation, [Eq. (3.42)] and obtain the stability numbers corresponding to explicit tem-

perature only. The overall stability number keeping both saturation and temperature

explicit is also calculated.

It was observed that both the individual stability numbers were less than the


CFL # with distance at 41st time step

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100 120 140 160 180 200

Distance (ft)

CF

L #

CFL (S,T)

CFL (S)

CFL (T)

Figure 4.18: CFL Trend in the 41st timestep

overall stability number. In this case, the CFL number was violated in the 41st

timestep. The CFL trend for that timestep and the next one are shown in Fig. (4.18)

and Fig. (4.19) respectively. From these figures, it might be misleading that CFL(S)

is zero in a part of the reservoir, implying as large timestep as possible. But this

is not the case. The zero stability number is there only because the front has not

reached that place. This can be seen in the saturation front for the 41st timestep in

Fig. (4.20).

Thus we can conclude that temperature plays an important role as far as convec-

tion is concerned. In the next section, we show some results of the effect of conduction

parameters on the system behavior.

4.5 Dependence on Conduction parameters

In order to study the effect of rock heat capacity CpRand rock conduction transmissi-

bility Υc, simulation runs were made using FIM. These show that the contribution of


CFL # with distance at 42nd time step

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140 160 180 200

Distance (ft)

CF

L #

CFL (S,T)

CFL (S)

CFL (T)

Figure 4.19: CFL Trend in the 42nd timestep

the temperature effects is essentially negligible as far as heat conduction is concerned.

The correct treatment of the temperature variable is however important in the flow

terms. This is because the flow term contains enthalpy as defined by Eq. (4.7), which

is a strong function of temperature.

4.5.1 Effect of CpR

The rock heat capacity is the amount of heat needed to raise the temperature of

a unit volume of the rock by a degree. The higher the value of this constant, the

more heat is retained by the rock, which it will conduct depending on the conduction

transmissibility coefficient. The accumulation of heat in the rock as compared to that

in the fluids is given by:

Grock

Gfluids

=(1− φ) ρR CpR

φ (ρw Sw Cpw + ρo So Cpo). (4.11)

On substituting the values from Appendix A, we obtain that the rock accumulates

about 5.6 times the heat retained by fluids.

From the temperature fronts in Fig. (4.21) it is obvious that the fluid is left


Saturation Front at 41st timestep

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 20 40 60 80 100 120 140 160 180 200

Distance (ft)

Sw

Figure 4.20: Saturation front in the 41st timestep

with very little heat when CpRis close to the realistic range of values. There is a

difference of about 25 degrees in temperature in the middle of the reservoir. This

trend is physically intuitive as well. Therefore, the fluid transfers the remaining heat

by convection.

4.5.2 Effect of Υc

In this experiment we fix the CpR, and vary the value of rock conduction transmis-

sibility coefficient, Υc. The injection of hot water is stopped after a few days, and

the reservoir is allowed to equilibrate. The temperature gradients get smaller with

time, but their magnitude depends on Υc. Even with a very high value of Υc, it will

take an infinitely long tome for the system to achieve steady state. This shows that

conduction is a much slower process, and the transport of heat by this means will be

far less significant than that due to actual flow of fluids.

In order to quantify that conduction effects are much smaller than convection

effects, it would be good to observe the ratio of conduction transmissibility to con-

vection transmissibility. This is quantified in the following way:


Υc

Υ=

Υc

Υw Cpw (T − Ti) +(

P Bw

ρow

)+ Υo Cpo (T − Ti) +

(P Bo

ρoo

) (4.12)

On substituting the values in the above expression from Appendix A, we see the value

of the fraction is 0.000156, which is << 1, showing that conduction has much less

effect than convection.

In this chapter we showed the results of the numerical experiments performed to

test various stability criteria and study the effect of some parameters. Some cases were

hypothetical, but nevertheless they help to understand the physics of non-isothermal

flows.


Temperature Profile at 200 days

290

300

310

320

330

340

350

360

370

380

0 50 100 150 200 250 300 350 400 450 500

Distance (ft)

T (

K)

Cpr = 0

Cpr = 0.1

Cpr = 0.5

Cpr = 0.7

Cpr = 1

Figure 4.21: Temperature front after 200 days of production, zero conduction

Chapter 5

Conclusions and Recommendations

We derived the stability criteria for a 1-D thermal oil-water system in the absence of

temperature terms, the criteria reduce to those of the isothermal case. The derived

stability criteria were tested by violating the time step and observing the solutions

for pressure, saturation and temperature in the reservoir. We found that on violating

the stability condition with a greater time step than what is allowed, we observed

oscillations in the pressure and saturation profiles for the isothermal cases, and the

wrong solution for temperature. The magnitude of oscillations increases with higher

violation of the stability number.

The main results of this work are:

1. The temperature effects are contained in the enthalpy term in the energy equa-

tion, which behaves like saturation in the mass conservation equation.

2. The combined stability number containing the effects of both explicit saturation

and temperature is higher than the individual stability numbers. Therefore,

smaller time steps would be allowed if both saturation and temperature are to

be treated explicitly.

3. The non linearity is introduced because of enthalpy which is a strong function

of temperature. The flow transmissibilities are weighted functions of respective

phase enthalpies.

The stability criteria provide a good tool for deciding the switching criteria for an

adaptive implicit method. Adaptive implicit methods not only preserve the robustness

48

CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS 49

of the fully implicit solution, but also give a stable solution in less computational

time. With the increasing need for thermal simulation of steam injection processes

in enhanced oil recovery, an adaptive implicit technique is an extremely valuable

method. The stability criteria is an inevitable ingredient in the implementation.

GPRS, General Purpose Research Simulator, developed in Stanford University is

a comprehensive isothermal simulator, based on an adaptive implicit solution tech-

nique. It is important to extend GPRS to model thermal processes. The stability

criteria derived in this research are the first building blocks towards this goal. More

complicated processes such as dealing with phase change, three phase flow, and com-

positional models are ongoing.

Bibliography

[1] http://www.netl.doe.gov/scngo/Petroleum/.

[2] K. Aziz and A. Settari. Petroleum Reservoir Simulation. Elsevier Applied Sci-

ence, 1986.

[3] H. Cao. Development of Techniques for General Purpose Reservoir Simulators.

Ph.D. Dissertation, 2002.

[4] K.H. Coats. A highly implicit steamflood model. SPE-AIME, 6105, 1978.

[5] K.H. Coats. Impes stability: The stable step. presented at SPE Reservoir Sim-

ulation Symposium, Houston, Feb 2001, SPE Journal, 69225, 2001.

[6] K.H. Coats. Impes stability: Selection of stable timesteps. SPE Journal, 84924,

2003.

[7] P.A. Forsyth Jr. and P.H. Sammon. Practical considerations for adaptive implicit

methods in reservoir simulation. Journal of Computational Physics, 62:265–281,

1986.

[8] William J. Palm. Introduction to MATLAB 7 for engineers. McGraw-Hill, 2005.

[9] M. Prats. Thermal Recovery. Monograph V-7, SPE-AIME, 1982.

[10] T.F. Russell. Stability analysis and switching criteria for adaptive implicit meth-

ods based on the cfl condition. SPE Journal, 18416, 1989.

[11] G.W. Thomas and D.H. Thurnau. Reservoir simulation using an adaptive im-

plicit method. SPE-AIME, 23:759–768, 1983.

50

BIBLIOGRAPHY 51

[12] J. Wan, P. Sarma, A.K. Usadi, and B.L. Beckner. General stability criteria

for compositional and black-oil models. presented at SPE Reservoir Simulation

Symposium, Houston, Jan-Feb 2005, SPE Journal, 93096, 2005.

[13] L.C. Young and T.F. Russel. Implementation of an adaptive implicit method.

presented at SPE Reservoir Simulation Symposium, New Orleans, Feb-Mar 1993,

SPE Journal, 25245, 1993.

Appendix A

Expressions of the Stability

Criteria

In expressing the results, we use the same coefficients as obtained in the final derived

form of the conservation equations given by Eq. (3.26), Eq. (3.35) and Eq. (3.42).

It will then be shown how the comprehensive stability criteria reduces to the more

widely known isothermal criteria when temperature terms are dropped.

A.1 Comprehensive Stability Criteria

The comprehensive stability criteria for two explicit variables, here Sw and T , as

derived by Eq. (3.78) in section 3.7 are obtained from the following relation:

A + D ±√

(A−D)2 + 4 B C < 4, (A.1)

where

A = 2

(A1 ∆ t

∆ x+

2A2 ∆ t

∆ x2

), (A.2)

B = 2

(B1 ∆ t

∆ x+

2B2 ∆ t

∆ x2

), (A.3)

C = 2

(C1 ∆ t

∆ x+

2C2 ∆ t

∆ x2

), (A.4)

52

APPENDIX A. EXPRESSIONS OF THE STABILITY CRITERIA 53

D = 2

(D1 ∆ t

∆ x+

2D2 ∆ t

∆ x2

), (A.5)

and

(−A1 A2 −B1 B2

−C1 C2 −D1 D2

)=

(A′ B′

F′ G′

)−1 (−C′ D′ −E′ 0

−H′ I′ −J′ K′

). (A.6)

On expanding the inverse of the matrix, we get the following form:

(−A1 A2 −B1 B2

−C1 C2 −D1 D2

)=

1

Det

(G′ −B′

−F′ A′

) (−C′ D′ −E′ 0

−H′ I′ −J′ K′

),(A.7)

where Det = A′G′ −B′F′ and this leads to the following expressions:

A1 =G′C′ −B′H′

Det,

A2 =G′D′ −B′I′

Det,

B1 =G′E′ −B′J′

Det,

B2 = −B′K′

Det,

C1 = −F′C′ −A′H′

Det,

C2 = −F′C′ −A′H′

Det,

D1 = −F′C′ −A′H′

Det,

and D2 =A′K′

Det.

The coefficients already derived in sections 3.3 and 3.4 are summarized below as:

A′ =ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

− P ′c,


B′ =∂ρw

∂T∂ρw

∂Pw

−∂ρo

∂T∂ρo

∂Po

,

C′ =ut

∂fw

∂Sw

φ

(ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

),

D′ =k λ P ′

c

φ

(ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

),

E′ =ut

∂fw

∂T

φ

(ρo

So∂ρo

∂T

+ρw

Sw∂ρw

∂T

)− 1

φ

(uo

∂ρo

∂T

So∂ρo

∂T

− uw∂ρw

∂T

Sw∂ρw

∂T

),

F′ =ρo

So∂ρo

∂Po

+−ρo Uo + ρw Uw − Uw Sw P ′

c∂ρw

∂Pw

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

,

G′ =Uo So

∂ρo

∂T+ ρo So

∂Uo

∂T+ Uw Sw

∂ρw

∂T+ ρw Sw

∂Uw

∂T+ 1−φ

φCpR

((T − Ti)

∂ρR

∂T+ ρR

)

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

−∂ρo

∂T∂ρo

∂Po

,

H′ = ut∂fw

∂Sw

(ρo

φSo∂ρo

∂Po


Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

),

I′ = k λ P ′c

(ρo

φSo∂ρo

∂Po


Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

),

J′ = −ut∂fw

∂T(ρo Ho − ρw Hw)− uo

(Ho

∂ρo

∂T+ ρo

∂Ho

∂T

)− uw

(Hw

∂ρw

∂T+ ρw

∂Hw

∂T

)

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

+ρo ut

∂fw

∂T− uo

∂ρo

∂T

φSo∂ρo

∂Po

,

K′ =Υc

Uo So∂ρo

∂Po+ Uw Sw

∂ρw

∂Pw+ 1−φ

φCpR

(T − Ti)∂ρR

∂Po

.

The stability numbers can be calculated from the above expressions and used in the

simulator when both Sw and T are explicit.


A.2 Isothermal Stability Criteria

From the final result of section A.1 we can obtain the isothermal stability criteria

and verify that it matches with the general two-phase isothermal stability criteria as

derived by Coats. [5], [6]

Substituting zero for temperature derivative terms, the various expressions become

as follows:

A′ =ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

− P ′c,

B′ = 0,

C′ =ut

∂fw

∂Sw

φ

(ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

),

D′ =k λ P ′

c

φ

(ρo

So∂ρo

∂Po

+ρw

Sw∂ρw

∂Pw

),

E′ = 0,

F′ =ρo

So∂ρo

∂Po

,

G′ = 0,

H′ = ut∂fw

∂Sw

(ρo

φSo∂ρo

∂Po

),

I′ = k λ P ′c

(ρo

φSo∂ρo

∂Po

),

J′ = 0,

and K′ = 0.

Thus Eq. (3.35) and Eq. (3.42) can be summarized as follows:

A′ ∂Sw

∂t= −C′ ∂Sw

∂x+ D′ ∂

2Sw

∂x2, (A.8)

F′∂Sw

∂t= −H′ ∂Sw

∂x+ I′

∂2Sw

∂x2. (A.9)


Each of the two equations above is of the same form as Eq. (3.26) and can be used

to determine the stability criteria. We should note that we need only one equation

for stability analysis because we just have one explicit variable, and that is Sw here.

Thus from Eq. (A.8) we obtain the stability criteria given by Eq. (3.69) obtained in

section 3.6. Therefore, we conclude that the isothermal stability criteria is a special

form of the more comprehensive thermal stability criteria.

Appendix B

Physical properties used in

simulator

B.1 Relative Permeability

The relative permeability functions can be expressed as:

kro = koro

(1− Sw)2, (B.1)

and krw = korw

S2w, (B.2)

where the end point relative permeabilities are koro

= 1 and korw

= 0.6.

B.2 Capillary Pressure

The capillary pressure curve is taken from [9]. Capillary pressure is inversely related

to water saturation. The function is given as:

Pc =1

0.1 + Sw

. (B.3)

57

APPENDIX B. PHYSICAL PROPERTIES USED IN SIMULATOR 58

B.3 Viscosity

The water viscosity is assumed to be dependent on temperature only. The following

function mapped from the data in [9] was used:

µw = 517.68 e−0.0217 T . (B.4)

The oil viscosity is a function of both pressure and temperature. The combined

dependence is evaluated by :

µo = µo(P ) µo(T ), (B.5)

where the functions µo(P ) and µo(T ) are given as:

µo(P ) = 6 ∗ 10−7 P 2 − 0.0046 P + 11.179, (B.6)

µo(T ) = −4.8611 ln(T ) + 29.775. (B.7)

For the higher mobility ratio case, we multiply the viscosity function given by Eq.

(B.5) with a factor of 10.

B.4 Density

The density of both oil and water is a function of pressure and temperature. The

density is evaluated as follows:

ρn =ρo

n

Bn(P ) Bn(T ). (B.8)

For water ρow = 62 lb/ft3, and for oil ρo

o = 45 lb/ft3. The formation volume factors

are given as follows:

APPENDIX B. PHYSICAL PROPERTIES USED IN SIMULATOR 59

Bo(P ) = 1− 5 ∗ 10−6 (P − 14.7), (B.9)

Bo(T ) = 1 + 3.8 ∗ 10−4 (T − 293), (B.10)

Bw(P ) = 1− 2.66 ∗ 10−6 (P − 14.7), (B.11)

Bw(T ) = 1 + 3 ∗ 10−4 (T − 293) + 3 ∗ 10−6 (T − 293)2. (B.12)

B.5 Rock Properties

The density of the rock is a function of pressure, given by the following equation:

ρR =ρo

R

BR(P ), (B.13)

where ρoR = 156 lb/ft3, and BR(P ) = 1 + 4 ∗ 10−6 (P − 3600) + 8 ∗ 10−12 (P −

3600)2. The specific heat of the rock CpR= 0.7 BTU/lb−F , and the rock conduction

transmissibility Υc = 1000 BTU/day − ft− F .

Appendix C

Fractional flow curves

The fractional flow of water fw, is defined as the fraction of water flowing in the total

fluid. Thus,

fw =uw

ut

=λw

λt

(1 +

k λo

ut

∂Pcwo

∂x

). (C.1)

In order to observe the behavior of fw, we neglect the capillary pressure term, and

plot fw = λw

λtwith Sw and T . Therefore Eq. (C.1) reduces to

fw =λw

λt

=krw

µw

kro

µo

. (C.2)

We used the data from one of the thermal simulated cases, which included capillary

effects, and plotted the function given by Eq. (C.2). Since ∂Pcwo

∂xis a positive quantity,

and we had neglected it in Eq. (C.2), we observe that the fractional flow curves do

not approach fw = 1 at larger saturations.

60

APPENDIX C. FRACTIONAL FLOW CURVES 61

Fractional Flow of water

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

300 320 340 360 380 400 420 440

T (K)

fw

Figure C.1: Fractional flow as a function of temperature

APPENDIX C. FRACTIONAL FLOW CURVES 62

Fractional Flow of water

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Sw

fw

Figure C.2: Fractional flow as a function of saturation

stability criteria for the thermal adaptive implicit method

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