RS Reader 2010

We Bring

PDEs to Life

In Reservoir Simulation

An Undergraduate Reservoir Simulation Course Uses Comsol Multiphysics to Help Students Unlock the Mystery of Flow Equations in Real Reservoirs.

Dr Jishan LiuSchool of Mechanical EngineeringThe University of Western [email protected]

We Bring

PDEs to Life

In Reservoir Simulation

An Undergraduate Reservoir Simulation Course Uses Comsol Multiphysics to Help Students Unlock the Mystery of Flow Equations in Real Reservoirs.

Dr Jishan LiuSchool of Mechanical EngineeringThe University of Western [email protected]

School of Mechanical Engineering

Faculty of Engineering, Computing and Mathematics

UNIT OUTLINE Semester 2 2010

PETR4511/8522

Reservoir Simulation Unit Coordinator: Jishan Liu

Page 2 of 3

TEACHING STAFF

The lecturer(s) for this unit and their contact details are below:

Insert Photo here if available

Your lecturers: Jishan Liu and Jian Guo Wang

Email: [email protected]

[email protected]

Phone: 6488 7205 (Liu) and 6488 8158 (Wang)

Fax:

Building: Civil & Mech Engineering

Room: 2.76 (Liu) and 2.16 (Wang)

Contact Hours: Wednesday

Insert Photo here if available

Your tutor Hongyan Qu

Email: [email protected]

Phone:

Fax:

Building: Civil and Mechanical Engineering

Room: 2.69

Contact Hours: Wednesday

To contact your lecturer or tutor, please stop by their offices during contact hours. No appointments are required during contact hours.

[Insert unit name here] Unit Outline

Page 3 of 3

UNIT DESCRIPTION This unit presents the theoretical basis for numerical simulation of fluid flow in petroleum Reservoirs. The partial differential equations required for single-phase and multiphase fluid flow in porous media are developed, as well as numerical solutions for these equations using Comsol Multiphysics. Input data requirements and applications of simulation models for evaluation of field performance will be discussed.

Learning Outcomes This unit develops students' knowledge of the fundamentals of fluid flow equations that describe petroleum recovery processes in

porous media, ability to derive numerical solutions by use of a ready-to-use computer package, ability to relate numerical solutions to the practical use of reservoir simulation for making reservoir

performance predictions. Class assignments and the final exam are designed to reinforce these objectives. Unit Structure The Unit will consist of 26 Lectures (17:30 ~ 19:00, Wednesday, ELT1 ) 20 Computer Lab Hours (19:00 ~ 21:00, MCL1.23, Math Building) ASSESSMENT Assessment Overview Continuous assignments and an end-of-semester examination will be used to assess students' performance. Assessment Mechanism

Component Weight Issue Date Due Date 3 Projects 60% TBC TBC Final Exam 40% End of Semester Exam Period End of Semester Exam Period

Penalties Late assignments will incur a deduction of 5% per day from the original mark. Assignments more than one week late without approval from the lecturer or a medical certificate will automatically receive a mark of 0. Assessment Items Description: Simulation projects and final exam are designed to strengthen students’ understanding on the fundamental principles and processes. They help students develop essential knowledge and skills for evaluation of reservoir performances by using numerical solutions. Through completing the assignments and the final exam, students develop the following attributes: ability to apply knowledge of basic science and engineering fundamentals ability to communicate effectively, not only with engineers but also with the community at large in-depth technical competence in at least one engineering discipline ability to undertake problem identification, formulation and solution ability to utilise a systems approach to design and operational performance Recommended / Required Text(s) Course Reader: Petroleum Reservoir Simulation 2010 Software Requirements COMSOL MULTIPHYSICS – available in the UWA School of Mechanical Engineering Computer Labs and MCL, not available for student’s personal machines.

Reservoir Simulation - Course Reader 2010

Saturday, February 27, 2010

1

PREFACE

In recent years, reservoir simulation has unquestionably played a major role in

reservoir engineering. This technique transformed the material balance based

methods that engineers once used in evaluating the reservoir performance into

simulation-based real-time reservoir management. Once the exclusive domain of

skilled researchers in numerical science, these advanced simulation techniques are

now available for all levels of engineers through ready-to-use software packages such

as COMSOL Multiphysics. As a result of this software, the focus on simulation has

shifted from solving the problems to evaluating the solutions obtained from analysis.

When I began my preparations for the teaching of Reservoir Simulation in 2005, I

found that all the reservoir simulation books dealt to a large extent either with

simulation theory and solution development or with practical considerations. It has

been very clear to me that these books are not appropriate for the undergraduate

teaching because they failed to take the shift of reservoir simulation focus into

consideration. The emphasis on evaluation of solutions from analysis has become my

primary motivation to write my own course reader for Reservoir Simulation.

Although personal computers have brought major changes to higher education, a

debate continues as to when is the appropriate time to introduce certain subjects that

seriously rely on computational power. For instance, is simulating partial differential

equations (PDEs) using finite-element analysis (FEA) suitable for an undergraduate

class? My recent experiences with Comsol Multiphysics show that it can be done.

Such an approach not only gives the students an introduction to a new tool and new

knowledge but also motivates them to master these concepts when they later study

them in detail. Through my teaching experience in 2006, it’s clear to me that bringing

Comsol Multiphysics into Reservoir Simulation has tremendous advantages. In fact,

we have brought abstract PDEs into life!

I look forward to studying your evaluations at the end of this semester.



2

TEACHING HISTORY 2006 Reservoir Simulation for UWA Undergraduate Students 2007 Reservoir Simulation for UWA Undergraduate Students 2008 Reservoir Simulation for UWA Undergraduate Students 2008 Reservoir Simulation Short Course for China Hehai University 2008 Reservoir Simulation Short Course for CUMT 2009 Reservoir Simulation for UWA Undergraduate Students 2009 Reservoir Simulation for UWA Postgraduate Students 2009 Reservoir Simulation for Curtin Postgraduate Students 2010 Reservoir Simulation for UWA Undergraduate Students 2010 Reservoir Simulation for UWA Postgraduate Students 2010 Reservoir Simulation for Curtin Postgraduate Students



1

PETR4511/PETR8522 Reservoir Simulation

Second Semester 2010

Reservoir Simulation Course Codes

UWA PETR4511

UWA PETR8522

CURTIN PEng612

This course aims to provide students with the theoretical basis for numerical

simulation of fluid flow in petroleum reservoirs and with essential skills required to

undertake sound reservoir simulations.

The topics covered in the course include:

Mass Conservation Law for Fluid Flow in Rocks;

Flow Classifications

Flow Visualizations of Single Phase Flow

Flow Visualizations of Multiphase Flow

Reservoir Simulation Protocol

On completion of the course, the student should be capable of demonstrating an

understanding of:

Fundamental principles of single phase flow

Fundamental principles of multiphase flow

An ability to perform a sound reservoir simulation by using any industry-standard

simulation tool

Lecturer

Professor Jishan Liu [email protected] Tel: 6488 7205

Teaching Assistant

Hongyan Qu [email protected]



CONTENTS

Preface

Chapter 1 Introduction

1.1 What is Reservoir Simulation?

1.2 Blackbox Model and Glass Model

1.3 What Questions can A Computer Model Answer?

1.4 Impacts of Reservoir Simulation

1.5 History of Developments

1.6 Scope of This Unit

1.7 A Professional Warning

Chapter 2 Principles

2.1 Conservation of Mass

2.2 Conservation of Momentum

2.3 Flow Equation

2.4 Constitutive Equations

2.5 Examples

Chapter 3 Simulation Concept

3.1 Flow Equation in Comsol Multiphysics

3.2 Reservoir Simulation Concept

3.3 Simulation Examples

Chapter 4 Flow Classifications

4.1 Your Classification Based on Compressibility

4.2 Your Classification Based on Deformability

4.3 Your Classification Based on Phase Numbers

4.4 Your Classification Based on Time Dependency

Chapter 5 Simulation of Slightly Compressible Flow

5.1 Simulation Example 1: Hydraulic Diffusivity

5.2 Variable Density Flow



5.3 Flow to Wells

Chapter 6 Simulation of Compressible Flow

6.1 Basic Theory

6.2 Simulations

Chapter 7 Dual Porosity Flow

7.1 Two Physics and Two Overlapping Porosity Model

7.2 Single Physics and Two Interweaving Porosity Model

Chapter 8 Two Phase Flow Simulation

8.1 Buckley-Leverett Theory

8.2 More Vigorous Derivation

8.3 Analytical Solutions

8.4 Numerical Solutions

Chapter 9 Multiphase Flow Simulation

9.1 Phase Mass Accumulation

9.2 Phase Mass Flow Rate

9.3 Multiphase Flow Equations

9.4 Boundary Conditions

9.5 Initial Conditions

9.6 Location Dependent Variables

9.7 Pressure Dependent Variables

9.8 Saturation Dependent Variables

9.9 Well Data

9.10 Field Studies

Appendix A Sample Simulation Project



1

Chapter 1

INTRODUCTION

Reservoir simulations are widely used to study reservoir performance and to

determine methods for enhancing the ultimate recovery of hydrocarbons from the

reservoir. They play a very important role in the modern reservoir management

process, and are used to develop a reservoir management plan and to monitor and

evaluate reservoir performance during the life of the reservoir, which begins with

exploration leading to discovery, followed by delineation, development, production,

and finally abandonment.

1.1 What is Reservoir Simulation?

Reservoir simulation mimics the behavior of a real reservoir system (geology +

engineering) through a model (physical, analog, electrical and numerical) based on

realistic assumptions. In this unit, we focus on reservoir numerical simulation.

Reservoir numerical simulation can be close to reality but it is never the reality (should

approach reality with time).

Simulation requires a computer, and compared to most other reservoir calculations,

large amounts of data. Basically, the simulation model requires that the field under

study be described by a grid system, usually referred to as cells or gridblocks. Each

cell must be assigned reservoir properties to describe the reservoir. The computer

simulator will allow us to describe a fully heterogeneous reservoir, to include varied

well performance, and to study different recovery mechanisms. Reservoir simulation

usually includes the following components.

Geometric Model: The physical (reservoir) system must be constructed outside

of the earth system. The communication conditions between the reservoir system and

its surroundings will be substituted by a set of boundary conditions. This substitution is



2

one source of errors.

Mathematical Model: The physical system to be modeled must be expressed in

terms of appropriate mathematical equations. This process almost always involves

assumptions. The assumptions are necessary from a practical standpoint in order to

make the problem tractable.

Numerical Model: The equations constituting a mathematical model of the

reservoir almost always too complex to be solved by analytical methods.

Approximation must be made to put the equation in a form that is amenable to solution

by computers. Such a set of equations forms a numerical model.

Computer Model: A computer program or a set of programs written to solve the

equations of the numerical model constitutes a computer model of the reservoir. The

use of a computer model to solve practical problems is referred to Reservoir

Simulation.

Figure 1: Construction of a Reservoir Simulation Model

Ground Surface

Oil Reservoir

Reservoir Simulation Model

Ground Surface

Oil Reservoir

Reservoir Simulation Model



3

1.2 Blackbox Model and Glass Model

In reservoir engineering, we treat the reservoir on a gross average basis (tank model) and do

not account adequately for variations in reservoir and fluid parameters in space and time. We

term this approach as Blackbox Model. In reservoir simulation, we can conduct a more

detailed study of the reservoir by dividing the reservoir into a number of blocks (sometimes

several thousands) and applying fundamental equations for flow in porous media to each

block. In this approach, we need to quantify all of physical processes during the production.

We call this approach as Glass Model. Their difference is illustrated in Figure 2.

Figure 2: Illustration of Two Different Approaches in Reservoir

Engineering and Reservoir Simulation.

1.3 What Questions can a computer Model Answer?

Reservoir simulation aims at understanding and handling the geological complexity

(reservoir characterization) and engineering complexity (physics). Computer models

can be valuable tools for the petroleum engineer attempting to answer the following

questions:

How should a field be developed and produced in order to maximize the economic

Ground Surface

Reservoir Storage Capacity

Reservoir Liquids Expansion

Petroleum Liquids Production

Pro

du

ctio

n W

ell

Re

se

rv

oir

En

gin

ee

rin

g

Gro

ss A

vera

ge

Ap

pro

ac

h

Vo

lum

e b

eyo

nd

th

e R

ese

rvo

ir C

ap

acit

y =

Pro

du

cti

on

Re

se

rv

oir

Sim

ul

at

ion

Flo

w P

roce

sses

Ba

sed

Ap

pro

ac

h

Ma

ss D

iffe

ren

ce

be

twee

n F

low

-In

an

d F

low

-Ou

t =

A

cc

um

ula

tio

n

Ground Surface

Reservoir Storage Capacity

Reservoir Liquids Expansion

Petroleum Liquids Production

Pro

du

ctio

n W

ell

Re

se

rv

oir

En

gin

ee

rin

g

Gro

ss A

vera

ge

Ap

pro

ac

h

Vo

lum

e b

eyo

nd

th

e R

ese

rvo

ir C

ap

acit

y =

Pro

du

cti

on

Re

se

rv

oir

Sim

ul

at

ion

Flo

w P

roce

sses

Ba

sed

Ap

pro

ac

h

Ma

ss D

iffe

ren

ce

be

twee

n F

low

-In

an

d F

low

-Ou

t =

A

cc

um

ula

tio

n



4

recovery of hydrocarbons?

What is the best enhanced recovery scheme for the reservoir? How and when

should it be implemented?

Why is the reservoir not behaving according to predictions made by previous

reservoir engineering or simulation studies?

What is the ultimate economic recovery for the field?

What type of laboratory data is required? What is the sensitivity of model

predictions to various data?

Is it necessary to do physical model studies for the reservoir? How can the results

be scaled up for field applications?

What are the critical parameters that should be measured in the field application of

a recovery scheme?

What is the best completion scheme for wells in a reservoir?

From what portion of the reservoir is the production coming?

These are some general questions; many more specific questions may be asked

when one is considering a particular simulation study. Defining the objectives of the

study to be conducted and carefully stating the questions to be answered is an

extremely important step in conducting any simulation studies.

Despite the power of reservoir simulation, it can be a dangerous tool. It will calculate

meaningless results with incredible precision. Many individuals and companies have

been burned when inappropriate use was made of this tool. As a result, some

managers became disenchanted with simulation, and they were no longer interested

in being fooled by this technique. With significant improvements in simulation

techniques, this attitude has diminished gradually. In reality, failures occurred because

the people applying this technology either did not understand or properly

communicate their assumptions and corresponding limitations of the results. It takes

discipline to realize numerical modeling has both strengths and weaknesses and is

only one of many tools available.



5

1.4 Impacts of Reservoir Simulation

Reservoir simulation is a key technology for real-time reservoir management.

Real-time reservoir management systems are rapidly becoming best practice in the

industry. This technology has the potential to prolong the life of the reservoir and boost

recovery rates above 60% (Baird, WPC 2002).

Figure 3: Simulation Cycle – from Data to Decision.

Sound reservoir management practice involves goal setting, planning, implementing,

monitoring, evaluating, and revising unworkable plans. Reservoir simulation plays a

key role in providing reservoir performance analysis, which is needed to develop a

management plan, as well as to monitor, evaluate, and operate the reservoir. A major

breakthrough in reservoir simulation has occurred with the advent of integrated

geoscience (reservoir description) and engineering (reservoir production) software

designed to manage reservoirs more effectively and efficiently, as shown in Figure 3.

GeoscienceReservoirEngineering

ReservoirSimulation

Data CollectionInterpretationIntegration

Well PlanningSpecificationsLimitations

History MatchingPredictions

AnalysisOptimizationControl


ReservoirSimulation






ReservoirSimulation






ReservoirSimulation







6

1.5 History of Developments

Reservoir simulation has a history almost as old as the history of computers. Many of

the basic numerical techniques for solving simple problems of flow through porous

media developed in the 1950's continue to be used. In the 1960's techniques started

appearing for the solution of three-phase, three-component problems that form the

basis of modern black-oil models. The next major shift occurred in the 70's and early

80's, as simulators became robust enough to move out of the research laboratories

and onto the desktops of practicing reservoir engineers. Since the mid 80's, reservoir

geostatistics has created a new opportunity in the science of modeling reservoir

performance. Geology can now be cast in a numerical framework that is easy for the

engineer to understand and use. For the first time, uncertainty can be quantified.

During the 1970s and early 1980s, reservoir simulation developed quite a mystique

because it was a newest tool for the evaluation of a reservoir performance. Much of

this has worn off, and reservoir simulation has become quite commonplace. As shown

in Figure 4, reservoir simulation has now become a relative mature technology. It has

become a key technology for real-time reservoir management.

Figure 4: Evolution of Reservoir simulation Technology

An integrated reservoir model is the key part of the real-time reservoir management.

Time

Dif

fu

sio

n o

f T

he

Sim

ul

at

ion

T

ec

hn

ol

og

y

Development PeriodTurning Point Deployment Period

2007

Integration of Geoscience and Engineering

1970 1980 1990 2000 Time

Dif

fu

sio

n o

f T

he

Sim

ul

at

ion

T

ec

hn

ol

og

y

Development PeriodTurning Point Deployment Period

2007

Integration of Geoscience and Engineering

1970 1980 1990 2000



7

Traditionally, data of different types have been processed separately, leading to

several different models – a geological model, a geophysical model, and a

production/engineering model. The reservoir model is not just an engineering or a

geoscience model – rather it is an integrated model, prepared jointly by geoscientists

and engineers.

Geoscience: Geoscientists probably play the most important role in developing a

reservoir model. The distributions of the reservoir rock types and fluids determine the

model geometry and model type for reservoir characterization.

Engineering: After identifying the geological model, additional

engineering/production data is necessary for completion of the reservoir model. The

engineering data includes reservoir fluid and rock properties, well location and

completion, well-test pressures, and effective permeability. Material balance

calculations can provide the original oil in place, and natural producing mechanisms –

including gas cap size and aquifer size and strength.

Integration: Integration of geoscience and engineering data is required to

produce the reservoir model, which can be used to simulate realistic reservoir

performance.

1.6 Scope of This Unit

This unit presents the theoretical basis for numerical simulation of fluid flow in

petroleum Reservoirs. The partial differential equations required for single-phase and

multiphase fluid flow in porous media are developed, as well as numerical solutions

for these equations using Comsol Multiphysics. Input data requirements and

applications of simulation models for evaluation of field performance will be

discussed.



8

1.7 A Professional Warning

We must guard against bad data and/or wrong physics in, pretty pictures out!

Figure 5: An Example of Pretty Picture Outputs.



2-1

Chapter 2

Fundamental Principles for Flow in Rocks



2-2

MASS CONSERVATION LAW – A Universal Rule

Volume

Coordinate

Pressure Fluid

Area Sectional Cross

Velocity Flow Fluid

Time

PorosityRock

Density Fluid

Saturation Fluid

V

x

p

A

u

t

S

i

i

i

i

i

x

Mass Accumulation Rate

Mass Flow‐In Rate

Mass Flow‐Out Rate

iixSAt

iixiiin AuQM x

x

MMM in

inout

x

Figure 2.1 Illustration of Mass Conservation Law

Write down your derivation of mass conservation law here ……

Applying the Darcy’s law to your mass conservation law:



2-3

i

i

i

ii x

pku

Write down your final form of mass conservation law here …..



2-4

2.1 A Simple Application

Governing equations for flow in porous media are based on a set of mass, momentum

and energy conservation equations, and constitutive equations both for fluid and for

porous media. For simplicity, we assume isothermal conditions, so that we do not

need an energy conservation equation. However, in cases of changing reservoir

temperature, such as in the case of cold water injection into a warmer reservoir, this

may be of importance.

Conservation of mass is the guiding principle of reservoir simulation. As shown in

Figure 1, the accumulation of mass in a control volume is equal to the difference

between the mass entering and leaving. This principle applies to each component

(rock, water, gas, and oil). Mathematically, the conservation of mass for each

component in a unit control volume can be written as

t

SMM ii

outin

(2.1)

Where inM and outM are the mass flow rate entering into and leaving from the

control volume, respectively; is the porosity, i is the density of the ith component,

iS is the saturation of the ith component, and t is time.

The mass flow rate entering into the control volume, inM , and the mass flow rate

leaving from the control volume, outM , are defined as

iixiiin AuQM (2.2)

xx

MMM in

inout

(2.3)

Where iQ is the volumetric flow rate, ixu is the Darcy velocity and A is the cross area.



2-5

Figure 2.2. Mass relations within a reservoir with one unit of

volume

Substituting Equations (2) and (3) into (1) gives

t

S

x

u iiiix

(2.4)

2.2 Conservation of Momentum

Conservation of momentum is governed by the Darcy’s Law,

x

pk

A

Qu i

i

iiix

(2.5)

Where ik is permeability, i is viscosity, and ip is pressure.

2.3 Flow Equation

Substituting (5) into (4) gives

iii

i

ii Stx

pk

x

(2.6)

A Reservoir with

One Unit of Volume t

Rate Change Mass

Mass Fluid Petroleum

Flow-In Mass Rate Flow-Out Mass Rate

inM outM

A

x

A Reservoir with


Rate Change Mass



inM outM

A

x

A Reservoir with


Rate Change Mass



inM outM

A Reservoir with


Rate Change Mass



inM outM

A

x



2-6

Assuming:

1S

Constant

Constant

Constantk

Constant

Under these assumptions, the flow equation (6) becomes

02

2

x

p (2.7)



2-7

Apply equation (7) to the following two examples:

Example 1: A 1D flow system with constant permeability and viscosity is shown in

Figure 3. Constant pressures are applied at both ends and no flow conditions are

specified for both the upper and the lower boundaries. Apply equation (7) to determine

the pressure distribution along the flow path.

Figure 2.3 1D flow example

Write down your solution here …

Pap 7

1 102 Pap 7

2 10

mL 10

Initially saturated with water

x

Pap 7

1 102 Pap 7

2 10

mL 10


x



2-8

Example 2: A 1D flow system with variable permeability and viscosity is shown in

Figure 4. Constant pressures are applied at both ends and no flow conditions are

specified for both the upper and the lower boundaries. Apply equation (7) to determine

the pressure distribution along the flow path.

Figure 2.4 1D flow system with variable permeabilities where p is the pressure at

x=5m.

Write down your solution here …

Pap 7

1 102 Pap 7

2 10

mL 5 xmL 5

,k ,2kp

Pap 7

1 102 Pap 7

2 10

mL 5 xmL 5

,k ,2k

Pap 7

1 102 Pap 7

2 10

mL 5 xmL 5

,k ,2kp



2-9

Example 3: If a flow region is divided into a number of blocks (sometimes several

thousands), the fundamental equation for flow can be applied to each block as

demonstrated in Figure 6. Computer programs that perform the necessary

computations to do such model studies are called computer models. The use of a

computer model to solve flow problems is called flow simulation.

Figure 2.5. Illustration of computer simulation processes

From the examples above, we see three basic components for the definition of a flow

problem, i.e.,

Flow Domain - Geometrical Model;

Pressure Condition at Boundaries - Boundary Conditions;

Governing Equation – Physical Model.

Can you find a solution???

1p

2p

x

Solutions at interfaces

Nodes Elements

Solution Interpolations

1p

2p

x


Nodes Elements




1

Chapter 3

Simulation Concept



2

Example 2.3: If a flow region is divided into a number of blocks (sometimes several

thousands), the fundamental equation for flow can be applied to each block as

demonstrated in Figure 6. Computer programs that perform the necessary

computations to do such model studies are called computer models. The use of a

computer model to solve flow problems is called flow simulation.

Figure 2.5. Illustration of computer simulation processes

From the examples above, we see three basic components for the definition of a flow

problem, i.e.,

Flow Domain - Geometrical Model;

Pressure Condition at Boundaries - Boundary Conditions;

Governing Equation – Physical Model.

Can you find a solution???

1p

2p

x


Nodes Elements


1p

2p

x


Nodes Elements




3

3.1 Physical Model

We know the general flow equation for a single component as defined:

ty

pk

yx

pk

x

(3.1)

The accumulation term can be written as

t

pcc

t

p

pt

p

pttt lp

11

(3.2)

We define the storativity as

ip ccS (3.3)

Substituting (3.3) into (3.2) gives

t

pS

y

pk

yx

pk

x

(3.4)

We re-write equation (3.4) as

0

y

pk

yx

pk

xt

pS

(3.5)

By using y

jx

i

for gradient and yx

for

divergence, equation (3.5) can be written as

0

pk

t

pS

(3.6)

If we add a fluid source, sQ , equation (3.6) becomes



4

sQpk

t

pS

(3.7)

Scaling Coefficients: In Comsol Multiphysics, the Darcy’s Law application

mode provides optional scaling coefficients to facilitate advanced analyses and

iterative or parametric simulations. The types of analyses that the scaling coefficients

enable include dual domain systems involving relatively fast flow in fractures,

multiphase problems, and density dependencies to name a few. With the optional

coefficients the flow equation takes the following form:

sQKS Qpk

t

pS

(3.8)

For a slightly incompressible fluid, moves outside the divergence operator, and the

flow equation takes the common form shown by default in the user interface:

sQKS Qpk

t

pS

(3.9)

Boundary Conditions: A unique solution to the flow equation requires boundary

conditions for all models as well as initial conditions if the problem is transient or time

dependent. The Darcy’s law application mode of the Earth Science Module provides a

number of boundary conditions. We also can specify unique conditions by entering

expressions in the boundary settings dialogs and/or altering the boundary mechanics

in the equation systems dialogs.

In many cases, the distribution of pressure is known. This is a Dirichlet condition given

by

0pp (3.10)

Where 0p is a known pressure, given as a number, a distribution, or an expression

involving time t, for example.

Fluid does not move across impervious boundaries. This is represented by the zero

flux condition



5

0 pk

n (3.11)

Where n is the vector normal to the boundary. While this Neumann condition specifies

zero flow across the boundary it allows for movement along it. In this way the equation

for the zero flux condition also describes symmetry about an axis or a flow divide, for

example.

Often the fluid flux can be determined from pumping rate or known from

measurements. With the inward boundary condition, positive values correspond to

flow into the model domain.

0Npk

n (3.12)

Initial Conditions: The initial condition specifies the initial state of the primary

variables of the system. For the simple case, a constant initial pressure may be

specified as 00 ptp .



6

3.2 Example

Simulation Example 1: A 1D flow system with constant permeability and

viscosity is shown in Figure 7. Constant pressures are applied at both ends and no

flow conditions are specified for both the upper and the lower boundaries. Use

Comsol Multiphysics to determine the pressure distribution along the flow path.

Figure 3.1. Simulation Model of Example 1.

Model Definition: A simulation study usually consists of Geometric Model

(simulation domain), Physical Model (simulation domain), and Boundary Conditions

(boundary specifications).

Physical Model: The general flow equation in Comsol Multiphysics is defined as

sQpk

t

pS

(3.13)

For this example, we assume 0S and 0sQ . These conditions specify a steady

state flow problem without the source or sink term. The explicit form of equation (29) is

expressed as

0

y

pk

yx

pk

x

(3.14)

In equation (30), we need to solve for the variable p assuming that we know the

Pap 7

1 102 Pap 7

2 10

mL 10


x

Pap 7

1 102 Pap 7

2 10

mL 10


x



7

permeability, k, the viscosity, , and the density, . For this example, we assume:

1000

1

1

k

Geometric Model: The simulation domain and boundary conditions are defined in

Figure 8. No flow boundary conditions are specified at upper and lower boundaries

and constant pressures on the lateral boundaries.

Figure 8: Simulation model

We implement the simulation model into Comsol Multiphysics. Simulation results are

shown in the following graphs. Important steps include:

Model navigator: Darcy Flow, Stationary Linear

Draw mode: Geometry

Physics mode: Pressure inlet and outlet, insulation, permeability

Mesh mode: Mesh

Solve problem Solve problem

Post-processing Pressure, velocity

aP11 p 02 p

1m

Flow Domain:• Length = 1m• Height = 0.1m

Fluid Properties:• Viscosity=0.001Pa.s• density =1000kg/m^3

Boundary Conditions:• Inlet Pressure, P1• Outlet Pressure, P2

aP11 p 02 p

1m

Flow Domain:• Length = 1m• Height = 0.1m

Fluid Properties:• Viscosity=0.001Pa.s• density =1000kg/m^3

Boundary Conditions:• Inlet Pressure, P1• Outlet Pressure, P2



8

NAVIGATOR:

Choose new from the file menu to open the model navigator

On the new page, select dimension 2D double-click on earth science module,

double-click fluid flow and Darcy’s law and then select pressure analysis.



9

Draw Mode

Draw a rectangle R1 from (0,0) to (1,0.1)

Double-click on R1 to modify the properties of R1 if it is necessary

Press ok to close the dialog box



10

Physics Mode

Choose boundary settings from the physics menu, select boundaries 1, 2, 3, 4 in

the domain selection and select the appropriate condition (1: p=1; 4: p=0, 2 and 3: no

flow). Then apply and press ok to close the dialog box.



11

Subdomain Mode:

Choose subdomain settings from the subdomain menu to open a dialog box. Then

enter the dynamic viscosity 0.001 Pa.s and the permeability k = 1. Then apply and

press ok to close the dialog box.

Mesh the subdomain by click the mesh button

Solve the problem by click the = sign.



12

Post-processing Mode

Choose plot parameters from the postprocessing menu to open a dialog box. For

example, select surface plot and pressure (p) and height as pressure (p) on the

surface menu and apply.



13

Simulation Example 2: We will use a similar case as example 1 with two

simulation domains. The permeabilities of two domains are 1.0 and 0.1, respectively.

All the other conditions remain the same.

Numerical solutions are shown in the following.

0 10

2

No Flow Boundary

No Flow Boundary

71 10p

62 10pDomain 1 Domain 2

PR

ES

SU

RE

VELOCITY

Numerical solutions are shown in this graph.

The pressure distribution in each region is still linear but the pressure gradients are different.

The velocity distribution in each region is identical.

Questions:

Why does the spatial variation in permeability affect the pressure distribution only?

PR

ES

SU

RE

VELOCITY

Numerical solutions are shown in this graph.

The pressure distribution in each region is still linear but the pressure gradients are different.

The velocity distribution in each region is identical.

Questions:

Why does the spatial variation in permeability affect the pressure distribution only?

Reservoir Simulation PETR4511/PETR8522 Course Reader 2009


1

Chapter 4

FLOW CLASSIFICATION

t

Sii

??

1

?

1

1

3

1

2

1

t

St

S

S

t

St

S

S

t

t

t

S

ii

ii

ii

ii

ii

ii

i

i

i

i

Change Phase with- 7 Case

Change Phase No - 6 Case

Change Phase with- 5 Case

Change Phase No - 4 Case

3 Case

2 Case

1 Case

pS

pp

i

ii

t

Sii

0

t

Sii

MASS CONSERVATION LAW:

DensityPermeability ViscosityPressure

Storage Component Flow Component

0

y

pk

yx

pk

xS

ti

i

iii

i

iiii

Saturation




0

y

pk

yx

pk

xS

ti

i

iii

i

iiii

Saturation



2

The general flow equation in porous media is defined as

iii

i

iii

i

ii Sty

pk

yx

pk

x

(4.1)

Where

Time

sCoordinate

Fluid the of Saturation

Medium the ofPorosity

Fluid the of Pressure Fluid the ofDensity

Fluid the of Viscosity

Fluid the ofty Permeabili

t

yx

S

p

k

i

i

i

i

i

,

Fluid flows in porous media can be classified based on:

Compressibility of the Fluid

Saturation of the Fluid

Deformability of the Porous Medium

Number of Phases

Time Dependency of Pressure and Saturation



3

4.1 Your Classification Based on Compressibility

We study fluid flow through idealized fluids that represent important classes of fluids.

They are characteristically described in terms of simple equations of state (EOS) and

the expressions of isothermal compressibility. Equations of state are mathematical

expressions that relate fluid density to pressure. We consider four idealized fluids: (1)

an incompressible fluid; (2) a constant compressibility fluid; (3) a slightly compressible

fluid; and (4) an ideal gas.

The isothermal compressibility is defined as

pc

1

(4.2)

Pressure Fluid

Density Fluid

ilityCompressib Isothermal

p

c

Incompressible Fluid: The density for an incompressible fluid is constant. Thus,

the applicable EOS is 0 , where 0 is a constant. The fluid compressibility for an

incompressible fluid is zero, c=0. In practice, no reservoir fluid is incompressible;

however, water and oil approach incompressibility over small pressure ranges.

Constant-Compressibility Fluid: If a fluid has a constant compressibility, c,

Equation (2) can be integrated to give the following EOS:

00 exp ppc (4.3)

Where 0 is the reference density at reference pressure, 0p . In practice, most fluids

that exhibit a constant compressibility also have a small compressibility. These liquids

are termed as slightly compressible liquids.

Slightly Compressible Liquid: Liquids often exhibit exceedingly small

compressibilities. If the term 0ppc in Equation (3) is less than 0.1, then it simplies

to

00 1 ppc (4.4)

As a rough guide, a liquid can generally be treated as slightly compressible if c is



4

constant and less than 15105 psi .

Ideal Gas: The EOS for an ideal gas is

RT

p (4.5)

Noting that the derivative RTp

1

and substituting this into Eq. (2) yields p

c1

.

This equation shows that the compressibility of an ideal gas is a strong hyperbolic

function of pressure.

Real Fluid: A real fluid is not an idealized fluid. Real fluids rarely have simple

EOS’s; instead, they either have complicated EOS’s or require elaborate correlations.

As an example, consider a real gas. The density of a real gas is correlated with a

z-factor, defined by

zRT

p (4.6)

Where z is a strong function of temperature and pressure that is determined from

correlation. The isothermal compressibility of a real gas is defined as

p

z

zpc

11

(4.7)



5

4.2 Your Classification Based on Deformability

In the general flow Equation (1), the storage term is related to porosity, , and the

flow term to permeability, k. Both parameters are dependent on the deformability of

the porous media.

The isothermal compressibility in terms of porosity is defined as

pcm

1

(4.8)

Pressure Fluid

Porosity Medium

ilityCompressib Isothermal Medium

p

cm

Incompressible Medium: The porosity for an incompressible medium is constant.

Thus, the applicable EOS is 0 , where 0 is a constant. The medium

compressibility for an incompressible medium is zero, 0mc .

Constant-Compressibility Medium: If a porous medium has a constant

compressibility, mc , Equation (8) can be integrated to give the following EOS:

00 exp ppcm (4.9)

Where 0 is the reference density at reference pressure, 0p .

Slightly Compressible Porous Medium: If the term 0ppcm in Equation

(8) is less than 0.1, then it simplies to

00 1 ppc (4.10)

As a rough guide, a porous medium can generally be treated as slightly compressible

if mc is constant and less than 15105 psi .

Deformable Porous Medium: In a host of engineering phenomena, porosity is

a function of effective stress, chemical process and temperature. In these situations,

the flow equation needs to be solved in conjunction with other physics.



6

4.3 Your Classification Based on Phase Numbers

One-Phase System: The term single phase applies to any system with only one

phase present in the reservoir. In some cases it may also apply where two phases are

present in the reservoir, if one of the phases is immobile, and no mass exchange

takes place between the fluids. This is normally the case where immobile water is

present with oil or with gas in the reservoir. By regarding the immobile water as a fixed

part of the pores, it can be accounted for reducing porosity and modifying rock

compressibility correspondingly.

Normally, in one phase reservoir simulation we would deal with one of the following

fluid systems:

One phase gas

One phase water

One phase oil

Before proceeding to the flow equations, we will briefly recall the fluid models for

these three systems.

One Phase Gas: The gas must be single phase in the reservoir, which means that

crossing the dew point line is not permitted in order to avoid condensate fallout in the

pores. The relation between the gas density at the reservoir condition and that at the

surface condition is defined as

g

SCg

RCg B,

,

(4.11)

One Phase Water: The relation between the water density at the reservoir

condition and that at the surface condition is defined as

w

SCw

RCw B,

,

(4.12)

One Phase Oil: In order for the oil to be single phase in the reservoir, it must be

undersaturated, which means that the reservoir pressure is higher than the bubble

point pressure. The relation between the oil density at the reservoir condition and that



7

at the surface condition is defined as

o

SCo

RCo B,

,

(4.13)

In equations (11) to (13), oil water,gas,iBi is the formation volume factor. For

single phase flow equation, the fluid density at the reservoir condition can be replaced

by the formation volume factor.

Multiphase System: Reservoir models can be classified based on the number of

phases and the number of components:

Water

Oil

Gas

PhasesN

...........

nHydrocarboLight

nHydrocarboteIntermedia

nHydrocarboHeavy

ComponentsM

If N=3 and M=2 (Heavy Hydrocarbon and Light Hydrocarbon), it is called Black-Oil

Model (BOM); if N=3 and M>2, it is called Compositional Reservoir Model (CRM).

Apparently, BOM is the simplest compositional model.

For a black-oil model, the phases and the components are defined, respectively, as

Water

Oil

Gas

PhasesN

nHydrocarboLight

nHydrocarboHeavyComponentsM

The black-oil model is the simplest case of compositional models: it concerns the flow,

through the porous medium, of one heavy hydrocarbon component (the “oil”), one



8

light hydrocarbon component (“the gas”) and of water. Depending on the pressure and

temperature conditions, the light component can eventually be completed dissolved in

the heavy one (then one has a single liquid hydrocarbon phase), and conversely it

could happen that the heavy component vaporizes completely (one would then have a

single hydrocarbon gaseous phase). For intermediate conditions, one has two

hydrocarbon phases (liquid and gas), each of which contains the two components in

variable proportions.

In contrast to compositional models with three or more components, where the mass

conservations of the components are usually taken as main unknowns, the black-oil

models are usually solved in terms of phase saturations and pressures.



9

4.4 Your Classification Based on Time Dependency

There are basically three types of flow regimes that must be recognized in order to

describe the fluid flow behavior and reservoir pressure distribution as a function of

time. There are three flow regimes:

Unsteady-state flow

Pseudo steady-state flow

Steady State flow

Steady-State Flow: The flow regime is identified as a steady-state flow if the

pressure at every location in the reservoir remains constant, i.e., does not change with

time. Mathematically, this condition is expressed as:

0

it

p (4.14)

The above equation states that the rate of change of pressure p with respect to time t

at any location i is zero. In reservoirs, the steady-state flow condition can only occur

when the reservoir is completely recharged and supported by strong aquifer or

pressure maintenance operations.

Unsteady-State Flow: The unsteady-state flow (frequently called transient flow)

is defined as the fluid flowing condition at which the rate of change of pressure with

respect to time at any position in the reservoir is not zero or constant. This definition

suggests that the pressure derivative with respect to time is essentially a function of

both position i and time t, thus

tift

p

i

,

(4.15)

Pseudo Steady State Flow: When the pressure at different locations in the

reservoir is declining linearly as a function of time, i.e., at a constant declining rate, the

flowing condition is characterized as the pseudo-steady-state flow. Mathematically,

this definition states that the rate of change of pressure with respect to time at every

position is constant, or



10

Constant

it

p (4.16)

It should be pointed out that the pseudo steady-state flow is commonly referred to as

semi steady-state flow and quasisteady-state flow. Figure 1 shows a schematic

comparison of the pressure declines as a function of time of the three flow regimes.



1

Chapter 5

SIMULATIONS OF SLIGHTLTY COMPRESSIBLE FLOW




0

y

pk

yx

pk

xt

Slightly Compressible Flow:

Flow leCompressibSlightly


Flow ibleIncompress

00

00

0

exp

1

ppc

ppc

000

pk

t

pS

011 0000

pk

ppct

pppcS

0expexp 0000

pk

ppct

pppcS




0

y

pk

yx

pk

xt

Slightly Compressible Flow:



Flow ibleIncompress

00

00

0

exp

1

ppc

ppc

000

pk

t

pS

011 0000

pk

ppct

pppcS

0expexp 0000

pk

ppct

pppcS



2

We write the single-phase flow equation as

0

pk

t

pS KS

(1)

Flow leCompressib


Flow ibleIncompress

00

00

0

exp

1

ppc

ppc

KS

KS

KS

For the incompressible flow, equation (1) becomes

000

pk

t

pS

(2)

For the slightly compressible flow, equation (1) becomes

011 0000

pk

ppct

pppcS

(3)

For the compressible flow, equation (1) becomes

0expexp 0000

pk

ppct

pppcS

(4)



3

5.1 Simulation Example 1: Hydraulic Diffusivity

CASE 1: When both permeability and viscosity are constant, we can write the flow

equation (2) as:

t

p

y

p

x

p

1

2

2

2

2

(E1)

Where is defined as the hydraulic diffusivity:

tc

k (E2)

In this example, we use a 2D flow system to investigate the impact of the hydraulic

diffusivity on the flow regimes.

Model Definition: The simulation model is shown in figure E1.

It should be 2D

Other inputs are as follows:

5.0102.01010 6612 qck t

41 101p 2

2 101p

No Flow Boundary

No Flow Boundary

Figure E1: 1D Oil Reservoir Model

1ppini 4

1 101p 22 101p

No Flow Boundary

No Flow Boundary


1ppini



4

The following graph shows the evolution of oil pressure. Explain

why?

Initial Pressure

Steady State Solution

Transient Solutions

Initial Pressure


Transient Solutions



5

CASE 2: The simulation model is shown in figure E2.


5.0102.01010 6612 qck t

The evolution of pressure is given below. Explain why?

22 101p

No Flow Boundary

No Flow Boundary


1ppini No Flow

Initial Pressure


Transient Solutions

Initial Pressure


Transient Solutions



6



100102.01010 6612 qck t




7


It should be 2D


100102.01010 6612 qck t


q

No Flow Boundary

No Flow Boundary


1ppini No Flow



8

5.2 Variable Density Flow

Model Definition: We will consider the following three situations:

Flow leCompressib


Flow ibleIncompress

00

00

0

exp

1

ppc

ppc

KS

KS

KS

Input parameters include

Pressure Reference

ilityCompressib

Density Reference

yStorativit

PressureBoundary PressureBoundary

ViscosityFluid

tyPermeabili Rock

30

6

5

52

71

8

10

10

10000

10

1010

1

10

p

c

rho

S

pp

mu

perm

s

2p

No Flow Boundary

No Flow Boundary

Figure E5: 1D Oil Reservoir Model under Variable Densities

1ppini 1p



9

5.3 Simulation Example 3: Flow to Wells

Analyzing fluid flow to and from wells is critical in petroleum engineering. This

example defines transient flow to a well of finite radius in an oil reservoir. The reservoir

is of infinite horizontal extent and is confined above and below by impermeable layers.

As the well fully penetrates the reservoir, withdraws are uniform from its length,

making flow entirely horizontal. Storage in the well is neglected. Fluids are released

instantaneously from storage in the reservoir. Prior to pumping, the flow field is at

steady state. Flow is horizontal, does not vary with depth.

Input parameters include

Pressure Initial

Rate Pumping

Density Fluid

HeadHydraulic for StorageSpecific

Pressure for tCoefficien Storage

Thickness

tyConductiviHydraulic

Radius Well

1250

13

3

15

21

1082.9

05.0

1000

10

,

50

/0001.0

1.0

mkgsp

smW

kgm

mS

mskgg

SS

mb

smK

mr

f

s

s

w

Model Definition: The flow equation is defined as

sQpk

t

pS

The drawdown, rd is defined as

g

ppdr

0

Fluid moves into the well with velocity described by



10

br

Wp

g

K

w 2

The boundary and initial conditions for this model are

0

2

0

0

tpp

ppbr

Wp

g

K

w

Boundary Outside

Well

n



1

Chapter 6

SIMULATION OF COMPRESSIBLE FLOW




0

y

pk

yx

pk

xt

Compressible Flow:

z

p

RT

M

Molecular Weight Gas Pressure

Gas Constant Gas Temperature

Deviation Factor

z

p

RT

M

ty

pk

z

p

RT

M

yx

pk

z

p

RT

M

x




0

y

pk

yx

pk

xt

Compressible Flow:

z

p

RT

M



Deviation Factor

z

p

RT

M



Deviation Factor

z

p

RT

M

ty

pk

z

p

RT

M

yx

pk

z

p

RT

M

x



2

Gas flow in porous media differs from liquid flow because of the large gas

compressibility and pressure-dependent effective permeability. In this chapter, we will

simulate the highly compressible gas flow in a rigid porous medium.

6.1 Basic Theory

In this chapter, we assume the fluid is gas which is very compressible. It is further

assumed that the change in porosity is negligible. Gases are highly compressible and

the above equations for slightly compressible liquids are not applicable. We write the

general flow equation as

ty

pk

yx

pk

x

(1)

In equation (1), the density, , is pressure-dependent and can be defined through the

equation-of-sate. The PVT behavior of a gas is given by the real gas law,

znRTpV (2)

Where z is the gas deviation factor, V is the volume of the gas, T is the temperature

(assumed to be a constant), n is the number of moles of gas, R is the universal gas

law constant.

Since,

M

mn (3)

Where m is the mass of the gas and M is the gas molecular weight, the real gas law

may be written as

z

p

RT

M (4)



3

Substituting equation (4) into (1) gives

z

p

RT

M

ty

pk

z

p

RT

M

yx

pk

z

p

RT

M

x

(5)

Assuming ,,,, TRkM are constants, equation (5) can be simplified as

pt

p

Mk

RT

y

p

z

p

yx

p

z

p

x

(6)

From the definition of compressibility,

z

p

RT

Mcc

p

(7)


t

p

kz

cp

z

p

RT

Mc

t

p

Mk

RT

y

p

z

p

yx

p

z

p

x

(8)

In order to solve equation (8) by using COMSOL Multiphysics, we write equation (8)

as

t

p

z

cp

y

p

z

pk

yx

p

z

pk

x

(9)

For an ideal gas, z=1 and 1 pc , equation (9) becomes

t

p

y

pp

k

yx

pp

k

x

(10)

For steady state gas flow, equation (10) becomes



4

0

y

pp

k

yx

pp

k

x (11)

If k and are constant, equation (11) becomes

0

y

pp

yx

pp

x (12)

For general cases, equation (12) can be written as

z

cppS

z

pkpK

t

ppS

y

ppK

yx

ppK

x

(13)

Equation (13) can be implemented directly into COMSOL MULTIPHYSICS.



5

6.2 Simulations

A gas field with one central producer is illustrated in Figure 1. Assuming the gas

density is directly proportional to the pressure, i.e., Cp (C is a constant), the gas

flow equation can be defined as

sQpg

K

t

pS

(14)

We solve this equation for two examples, one for 1D flow and other for 2D flow. They

are presented in details as follows.

5.2.1 Steady State Flow

Model Definition: The steady state gas flow is defined as

p

y

pk

yx

pk

x

K

KK

0 (E1)

Figure E1: A Gas Reservoir Model

We apply Equation (E1) to the 2D gas reservoir as show in Figure E1, with boundary

conditions as specified in Figure E2.

Large Gas Reservoir

Small Gas Reservoir

Production Tunnel

Large Gas Reservoir

Small Gas Reservoir

Production Tunnel



6

Simulation Results: We will analyze the distributions of gas pressure, volumetric

flow velocity, and the mass flow rate.

The distribution of gas pressure is shown in the following graph.

Unlike the incompressible flow, the pressure distribution is not a

straight line. Why?

71 101p 6

2 101p

No Flow Boundary

No Flow Boundary

Figure E2: 1D Gas Reservoir Model

71 101p 6

2 101p

No Flow Boundary

No Flow Boundary

Figure E2: 1D Gas Reservoir Model



7

The following graph shows the distribution of gas mass flow

rate. What can you conclude from this distribution?



8

The following graph shows the distribution of gas volumetric

flow rate. Unlike the incompressible flow, the distribution is not

linear. Why?



9

6.2.2 Unsteady State Flow


S

p

t

pS

y

pk

yx

pk

x

K

KK

(E2)

We apply Equation (E2) to the same reservoir model as in Figure E2.

Simulation Results: We will analyze the distributions of gas pressure, volumetric

flow velocity, and the mass flow rate.



10

The following graph shows the distribution of gas pressure at

different times. Please explain the evolution.



11

The following graph shows the evolution of mass and volumetric

flow rates. Explain why?



12

The following graph shows the spatial variations of mass and

volumetric flow rates. Explain why?



13

6.2.3 Steady State Flow – 2D


S

p

t

pS

y

pk

yx

pk

x

K

KK

(E3)

We apply Equation (E3) to the same reservoir model as in Figure E3. A gas field with

one central producer is illustrated in Figure E3. Assuming the gas density is directly

proportional to the pressure, i.e., Cp (C is a constant), complete the following

questions:

Assuming PapsmKPaSsmu i

725 10;/105.1;/10;/1.0 , use

Comsol Multiphysics to solve the problem as defined above;

Investigate the sensitivity of the gas pressure to S, K, and u.

Discuss the implications for the validity of material balance equation.

Figure E3: 2D Gas Flow

R = 1m

X

Y(-25,25)

(-25,-25) (25,-25)

(25,25)

No Flow

No Flow

No Flow

No Flow

uR = 1m

X

Y(-25,25)

(-25,-25) (25,-25)

(25,25)

No Flow

No Flow

No Flow

No Flow

u



14

The following graph shows the distribution of gas pressure at

st 810 . Explain why?



15

The following graph shows the distribution of gas volumetric

flow rate at st 810 . Explain why?



16

The following graph shows the evolution of gas pressure.

Explain why?



1

Chapter 7

DUAL POROSITY FLOW




0

y

pk

yx

pk

xt

Dual-Porosity Flow:

SystemMatrix mmmm

m Qpg

K

t

pS

System Fracturemfff

f Qpg

K

t

pS

Exchange Between Systems




0

y

pk

yx

pk

xt

Dual-Porosity Flow:

SystemMatrix mmmm

m Qpg

K

t

pS

System Fracturemfff

f Qpg

K

t

pS

Exchange Between Systems



2

As illustrated in Figure 1, the reservoir is conceptualized as a series of sugar cubes. It

is assumed that the fractures extend infinitely. In addition, it is assumed that the

fractures are evenly spaced. All flow to the well takes place in the fractures. The

fractures are, in turn, supported by a matrix of lower permeability.

Figure 1: Illustration of A Dual Porosity Model

7.1 Two-Physics and Two Overlapping Porosity Model

A fractured carbonate reservoir can be described by two overlapping porosity

(dual-porosity) systems: one for fractures and the other for rock matrixes. It is believed

that flow takes place predominantly in fractures while the storage effects take place

primarily in rock matrixes. Therefore, flow behaviors in the fracture system is different

from the flow behaviors in the matrix system (two physics). Geometrical definitions are



3

shown in Figure 2.

Figure 2: Illustration of A Two Porosity Model.

THEORY AND PROCEDURE: The hydraulic conductivity of the fracture system can

be defined as

s

gbK

s

gbK

xfy

yfx

12

123

3

(1)

Where fyfx KK and are the hydraulic conductivities in the x- and y-directions,

respectively. If fyfx KK , the fractured system is anisotropic and the anisotropy ratio

is defined as

xb

yb

0X

Y

s

sFractureRock

MatrixRock

ff

mm

SK

SK

xb

yb

0X

Y

s

sFractureRock

MatrixRock

ff

mm

SK

SK



4

fx

fyr K

KA (2)

Governing equations for this kind of reservoir can be defined as

mmmm

m Qpg

K

t

pS

(3)

mfff

f Qpg

K

t

pS

(4)

Where fm SS and are storage coefficients for rock matrix and fracture systems,

respectively; fm KK and are hydraulic conductivities for rock matrix and fracture

systems, respectively; fmp p and are liquid pressures for rock matrix and fracture

systems, respectively, and mQ represents the flow exchange between two systems

as defined as

fmm ppQ (5)

Where can be defined as a function of the fracture spacing, s:

g

K

s

k

sm

m

m

22

218

218 (6)

Combining equations (1) through (6) gives

fmmmmmmm

m ppg

K

sy

p

g

K

yx

p

g

K

xt

pS

2

218 (7)

fmmffyffxf

f ppg

K

sy

p

g

K

yx

p

g

K

xt

pS

2

218 (8)



5

Equations (7) and (8) are the equation-of-continuity for the rock matrix system and the

fracture system, respectively. These two systems are fully coupled through the source

term.

It is apparent that the source term is controlled by three major factors:

s , the fracture spacing

mK , the hydraulic conductivity of rock matrix

fm pp , the pressure difference between two overlapping systems

For the purpose of discussion, the source term is re-written as

g

K

sm

2

218 (9)

Influence of s: A smaller s (higher fracture frequency) contributes to larger surface

area over which the matrix/fracture fluid transfer takes place. A larger area

corresponds to longer time for reaching the equilibrium state because of the increase

in the fracture hydraulic conductivity. Similarly, a smaller fracture width leads to lower

fracture conductivity, which retards fluid transfer to and from the fracture as well as

through the fracture. The net result of this low fracture conductivity is a delay in

reaching the steady state.

Influence of mK : A higher rock matrix hydraulic conductivity accelerates the fluid

transfer to and from the matrix as well as through the matrix. The net result of this high

matrix conductivity is an advance in reaching the steady state.

Influence of Pressure Difference: The pressure difference is determined by a

combination of fmfm SSKK ,,, and the initial conditions. When 0 fm pp , the

two overlapping systems reach the steady state and no fluid transfer takes place

between these two systems.



6

These conceptual understandings are substantiated through investigating the

evolution of two pressure systems for the reservoir as illustrated in Figure 2.

Figure 3: A Base Simulation Model for the Fractured Carbonate

Reservoir.

X-Coordinate

Y-C

oord

inat

e

Pap

Pap

i

w

7

4

10

10

96

39

1010

:SystemMatrix Rock

1010

:System Fracture

mm

ff

KS

KS

1310

pi

pi

pi

pi

pw

X-Coordinate

Y-C

oord

inat

e

Pap

Pap

i

w

7

4

10

10

96

39

1010

:SystemMatrix Rock

1010

:System Fracture

mm

ff

KS

KS

1310

pi

pi

pi

pi

pw

X-Coordinate

Y-C

oord

inat

e

Pap

Pap

i

w

7

4

10

10

96

39

1010

:SystemMatrix Rock

1010

:System Fracture

mm

ff

KS

KS

1310

pi

pi

pi

pi

pw



7

SIMULATION STRATEGY: The following cases are conducted as shown in Table 1.

Simulation results are illustrated in Figure (4) through (11).

Table 1: Simulation Cases

fS fKmS mK

rA

910 310 610 910 1310 1

1210

710

510

710

Base Case

Case 5

Case 2

Case 3

Case 4

25.0Case 6

Smaller Fracture Spacing

Larger Fracture Spacing or Smaller Aperture

Larger Fracture Storativity

Larger Matrix Conductivity

Anisotropy

710Case 1 Smaller Matrix Storativity

Table 1: Simulation Cases

fS fKmS mK

rA

910 310 610 910 1310 1

1210

710

510

710

Base Case

Case 5

Case 2

Case 3

Case 4

25.0Case 6

Smaller Fracture Spacing

Larger Fracture Spacing or Smaller Aperture

Larger Fracture Storativity

Larger Matrix Conductivity

Anisotropy

710Case 1 Smaller Matrix Storativity



8

Figure 4 shows the evolution of the pressure in the fracture system and the pressure

in the matrix system. It is obvious that two pressure profiles are dramatically different.

The pressure in the fracture system reaches the steady state value almost instantly

while the pressure in the matrix system takes a much longer time to reach the steady

state. When both pressures reach their steady state, there is no mass transfer

between two systems.

Figure 4: Evolutions of Matrix and Fracture Pressures with Time

at the Point of X=10 and Y=0 for the Base Case.

Matrix Pressure

Fracture Pressure

Matrix Pressure

Fracture Pressure



9

Figure 5 shows the evolution of the Source Term with Time at the Point of X=10 and

Y=0 for the Base Case. This is consistent with the pressure evolutions. When the

pressure difference is zero, the source term is equal to zero. When the pressure

difference reach the maximum value, the source term also reach the peak.

Figure 5: Evolution of the Source Term with Time at the Point of

X=10 and Y=0 for the Base Case.



10

As shown in Figure 6, a larger value of the matrix storativity contributes to a larger

amount of liquid to be transferred between two systems and takes a longer time to

reach the steady state.

Figure 6: Comparison of Source Term Evolutions with Time at the

Point of X=10 and Y=0 between the Base Case of 610mS and the

New Case of 710mS .

710mS

610mS

710mS

610mS



11

As shown in Figure 7, a lager value of the fracture storativity corresponds to the

reduction in the storativity difference between two systems. This reduction leads to a

short time to reach the steady state.


Point of X=10 and Y=0 between the Base Case of 910fS and the

New Case of 710fS .

710fS

910fS

710fS

910fS



12

As shown in Figure 8, a smaller value of the fracture hydraulic conductivity

corresponds to lager fracture spacing or smaller fracture aperture, and to the

reduction in the hydraulic conductivity difference. This reduction leads to a shorter

time to reach the steady state. It impacts on the source term in the early stage.


Point of X=10 and Y=0 between the Base Case of 310fK and the

New Case of 510fK .

510fK

310fK

510fK

310fK



13

As shown in Figure 9, a higher rock matrix hydraulic conductivity corresponds to a

reduction in the hydraulic conductivity difference between two systems. This leads to

a slightly shorter time to reach the steady state.


Point of X=10 and Y=0 between the Base Case of 910mK and the

New Case of 710mK .

710mK

910mK

710mK

910mK



14

As shown in Figure 10, a larger value of the mass transfer coefficient contributes to a

larger matrix/fracture fluid transfer and a quicker time to reach the steady state.


Point of X=10 and Y=0 between the Base Case of 1310 and the

New Case of 1210 .

1210

1310

1210

1310



15

As shown in Figure 11, the anisotropy of hydraulic conductivity causes the anisotropy

of the source term.


Point of X=10 and Y=0 between the Base Case of 1rA and the

New Case of 25.0rA .

pi

i

i

pw

pi

i

i

pw

pi

i

i

pw

LINE I

LINE II

25.0rA

1rA

pi

i

i

pw

pi

i

i

pw

pi

i

i

pw

LINE I

LINE II

25.0rA

1rA



16

7.2 Single-Physics Two Interweaving System Model

We write the flow equation and boundary conditions in the following form

sQKS QgDpk

t

pS

(10)

0pp (11)

0 Dgpk

fn (12)

In these equations, both S and k could be space-dependent. Their spatial variations

may have tremendous impacts on the solution. In this chapter, we will introduce the

digital imaged based approach to incorporate the material property map directly into

the finite element analysis.

7.2.1 DIB Approach

Sectional images of the core are obtained via X-ray CT scanning. For this kind of

RGB image, at each pixel there are three integer values to represent the red, green

and blue, so the color image data consist of three discrete functions, fk(i,j), where k =1,

2 or 3, in the i and j Cartesian coordinate system:

MNfNfNf

Mfff

Mfff

jif k

,2,1,

,22,21,2

,12,11,1

,

(k = 1, 2, and 3) (13)

where i varies from 1 to N, and j from 1 to M. M and N are the number of pixels in the

horizontal and vertical directions, respectively.

In this work we use MATLAB to read the image file in JPEG format to obtain the

discrete function, fk(i,j). As an alternative to the RGB color space, the HSI color



17

space may be substituted, as it is close to how humans perceive colors. HSI is an

acronym for hue, saturation, and intensity. The hue component (H) represents

repression related to the dominant wavelength of the color stimulus. Therefore, the

hue is the domain color perceived by human beings. The saturation component (S)

signals how much the color is polluted with white color. The intensity component (I)

stands for brightness or lightness and is irrelevant to colors. In general, hue,

saturation, and intensity are obtained by different transformation formulae through

converting numerical values of R, G, and B in the RGB color space to the HSI color

space.

Distinct microstructures (such as fractures and minerals) in the rock sample are

acquired according to the values of H, S, or I of individual pixels, and the different

material properties (such as permeability) are specified for each pixel. If the material

properties of different minerals or structures are known in advance, by this means, the

relation between values of H (S, or I) of the digital image and its materials properties

can be uniquely established.



18

7.2.2 Simulation Example

As shown in Figure 12, we simulate a fractured reservoir.

Figure 12: Simulation Model for the Fractured Reservoir.

Production Well

Production Well

H=100m

H0 = 1000m

No Flow

No FlowN

o F

low

No

Flo

w

Production Well

Production Well

H=100m

H0 = 1000m

No Flow

No FlowN

o F

low

No

Flo

w



19

As shown in Figure 13, we assign much higher values of hydraulic conductivity to the

fractures and lower values to the rock matrix system.

Figure 13: Hydraulic Conductivity Map



20

As shown in Figure 14, we assign much higher values of storativity to the matrixes

and lower values to the rock fracture system.

Figure 14: Storativity Map.



21

As shown in Figure 15, the hydraulic heads in the rock matrixes are much higher than

those in the rock fractures.

Figure 15: Distribution of Hydraulic Head



22

As shown in Figure 16, flow takes place primarily in the fractures.

Figure 16: Distribution of Flow Velocity and Hydraulic Head.

Reservoir Simulation PETR4511/8522 Course Reader 2009


1

CHAPTER 8

TWO PHASE FLOW SIMULATION

Fluid displacement processes require contact between the displacing fluid and the

displaced fluid. The movement of the interface between displacing and displaced

fluids and the breakthrough time associated with the production of the injected fluid at

producing wells are indicators of sweeping efficiency.



2

8.1 Mass Conservation Law

We start with two phase (oil and water) flow by applying the mass conservation law to

each phase:

0

y

pk

yx

pk

xt

S ww

w

www

w

www

(1)

0

y

pk

yx

pk

xt

S oo

o

ooo

o

ooo

(2)

Equations (1) and (2) are identical with the single-phase flow. What makes them

different from the single-phase flow equation is that these two equations are coupled

through a number of things!

Can you identify the cross-couplings?

Two pressures are dependent on each other due to the capillary effect;

The capillary pressure is affected by saturations;

Saturations are affected by the interfacial tension and by the wettability;

……

Once we work out all of these cross-couplings, we can implement them into Comsol

Multiphysics and solve the equations.

Before we formulate the coupling relations, let us re-arrange the equations as

0

y

pk

yx

pk

xtS

t

S ww

w

www

w

www

ww

(3)

0

y

pk

yx

pk

xtS

t

S oo

o

ooo

o

ooo

oo

(4)



3

If we assume both oil and water are incompressible, equations (3) and (4) can be

simplified as

0

y

pk

yx

pk

xt

p

p

S w

w

ww

w

wc

c

w

(5)

0

y

pk

yx

pk

xt

p

p

S o

o

oo

o

oc

c

o

(6)

In order to solve these equations, we need to define the following terms as a function

of pressures:

cww pSS

coo pSS

coowo pkppk ,

cwoww pkppk ,

All these terms are related to the capillary pressure. Therefore, we start our

discussions from the concept of capillary pressure.



4

8.2 Capillary Pressure Concept:

rp

rp

prr

gh

ghpp

ghpp

ppp

hgrr

awawcaw

c

c

aa

CwAa

BwAac

cos2

cos2

0cos2

0cos2

2

,,

,,

2

AB

C

Figure 1. Pressure relations in a capillary tube for air-water system.

For example, the capillary pressure of a water-oil system is defined as

rppp owow

wocow

cos

Phase Wetting

Phase wetting-Non

PressureCapillary

cos

w

nw

cr

p nwwnwwcnww



5

8.3 Dependence of Capillary Pressure on Rock and Fluid Properties

Figure 2. Dependence of capillary pressure on wetting characteristics and pore size

(tube radius).



6

8.4 Capillary Pressure and Saturation History

Figure 3. Relationship between capillary pressure and saturation history

rp nwwnww

cnww

cos

Pore Geometry

WettabilityInterfacial Tension

Figure 4. Relation between capillary pressure, interfacial tension, wettability and pore

geometry.



7

Figure 5. Schematic representation of water-wet and oil-wet pore spaces.

8.5 Capillary Pressure & Permeability: The J Function

The capillary pressure data measured in the lab are normally based on individual core

plug samples, representing an extremely small part of the entire reservoir. Moreover,

because of the rock heterogeneity, no single capillary pressure curve can be used for

the entire reservoir. Therefore, it becomes necessary to combine all the capillary

pressure data to classify a particular reservoir. The approach that is commonly used

in the petroleum industry is actually based on a dimensionless function, called the J

function (sometimes referred to as Leverett’s J function) and defined as

2/1

kp

SJ cw (7)



8

Figure 6. J function vs. water saturation for North Sea reservoir rock samples

8.6 Fluid Contacts

g

ph

ghr

p

c

c

cos



9

Figure 7. Fluid distributions and contacts based on the capillary pressure or height vs.

water saturation data

The oil-water contact (OWC) is defined as the uppermost depth in the reservoir where

a 100% water saturation exists, or in other words, the OWC and 100% water

saturation point on the height or depth saturation curve is represented by the point

%100; wdc Spp .

In terms of height above the FWL:

dp

OWC144

(8)

In terms of depth above the FWL:

dp

FWLOWC144

(9)



10

8.7 Relative Permeability

Figure 8. A typical gas-oil relative permeability curve



11

Figure 9. A typical oil-water relative permeability curve

1poreV

wiS

orS

wmS

omS1 omwmorwr SSSS

Irreducib

le water saturatio

n

Resid

ual oil w

ater saturation

Mo

vable w

ater saturation

Mo

vable o

il saturation

1

1

1

1

**

*

*

ow

orwr

orwo

orwr

wrww

SS

SS

SSS

SS

SSS

1

wo

oromo

wrwmw

SS

SSS

SSS

Figure 10. Saturations and Their Relations

8.8 Capillary Models

Corey Model - An analytical expression for the wetting and nonwetting phase relative permeabilities may be

obtained if capillary pressure curves can be represented by a simple mathematical function. Corey (1954) found



12

that oil-gas capillary pressure curves could be expressed approximately using the following linear relation:

*1w

c

CSp

where C is a constant and *wS is the normalized wetting phase saturation. Corey (1954) obtained the

following equations to calculate the wetting (water) and nonwetting (oil) phase relative permeabilities for

drainage cases:

2*2*

4*

11 wwrnw

wrw

SSk

Sk

Brooks-Corey Model - Because of the limitation of Corey's model, Brooks and Corey (1966) modified the

representation of capillary pressure function to a more general form as follows:

/1* wec Spp

Where ep is the entry capillary pressure and is the pore size distribution index.

Brooks and Corey (1966) derived equations to calculate the wetting and nonwetting phase relative

permeabilities

as follows:

2*2*

32*

11 wwrnw

wrw

SSk

Sk

When is equal to 2, the Brooks-Corey model reduces to the Corey model.

We use a capillary pressure model

* 1mn

w cS ah

, (10)

where , ,a n m are parameters, /( )c c wh p g is the capillary pressure head and

*wS is defined by



13

*

1w wr

wor wr

S SS

S S

. (11)

The relative permeability curves are given by the following formula

2

*1/ 2 *1/, 1 1

mmr w w wK S S

, (12)

1/ 2 2* *1/, 1 1

mmr o w wK S S . (13)



14

8.9 Buckley-Leverett Theory

One of the simplest and most widely used methods of estimating the advance of a

fluid displacement front in an immiscible displacement process is the

Buckley-Leverett method. The BL Theory (1942) estimates the rate at which an

injected water bank moves through a porous medium. The approach uses fractional

flow theory and is based on the following assumptions:

Flow is linear and horizontal

Water is injected into an oil reservoir

Oil and water are both incompressible

Oil and water are immiscible

Gravity and capillary pressure effects are negligible

A

x

Rock

Figure 11. Flow Geometry

Darcy's equation is a macroscopic equation based on average quantities and derived

for one phase flow. The difficulties first arise when describing two-phase flow in

porous media. The microscopic interactions between the two liquids (capillary

pressure) cause local fluctuations in the pressure gradient inside the sample.

Moreover, when two fluids are simultaneously present the ability to flow of one fluid

depends on the local configuration of the other fluid. The different microscopic effects

have, in a variety of ways, been incorporated in Darcy's equation. One such famous

method describing two-phase flow in porous media, is the Buckley-Leverett

displacement which we briefly discuss below.

Consider a reservoir saturated with oil (o) which is going to be displaced by water (w).

The absolute permeability of the reservoir is k and the viscosities of oil and water



15

are denoted by o and w respectively. In the beginning only the displaced phase (o)

flows out of the medium until the displacing phase (w) breaks through, and both

phases are produced. Assume that a steady state is reached where both oil and

water flow through the medium in a fixed configuration. Thus, water and oil flow in an

effective porous medium that does not have the full pore space available. Let

wu denote the production of water and ou denote the flow of oil, then Darcy's equation

for the two phases becomes:

x

pkku w

w

rww

(14)

x

pkku o

o

roo

(15)

when the gravity effects are ignored. rwk and rok indicate the relative permeabilities of

the two fluids and wp and op are the pressures in water and oil respectively. The

pressure in water differs from that in oil, but the exact difference is only known when

there is no flow. Then the meniscuses between the fluids are adjusted due to the

interfacial tension between the water and oil phases. The pressure difference defines

the capillary pressure cp which becomes woc ppp .

The saturations of water and oil in the system are defined as wS and oS where the

saturation of a liquid is given by the total amount of the liquid in the system divided by

the pore volume. The effective permeabilities depend on the corresponding

saturations and if water displaces all oil so that 1wS we expect 1rwk and 0rok .

The total pore volume is conserved giving:

1 ow SS

As long as the fluids flow in a steady state the above equations are valid. In

Buckley-Leverett displacements, one attempts to use the equations outside the

stationary regime and solve them for one dimensional flow where the saturations and

the flow rates are functions of the position along the medium. To keep track of the

changes in saturation, the mass balance equations for water and oil are required:



16

0

x

pkk

xt

S

x

u

t

S w

w

rwwww

(16)

0

x

pkk

xt

S

x

u

t

S o

o

roooo

(17)

There are four unknowns: owow ppSS ,,, . Thus, we need two more relations to solve

the set of equations. These two equations are

wowc

ow

ppSp

SS

1

Buckley-Leverett Equation:

We assume that the capillary pressure is negligible, ppp ow . Water flow

equation can be written as

0

x

pkk

xt

S

w

rww

(18)

0

x

pkk

xt

S

o

roo

(19)

Let us assume w

rww

kk

and

o

ronw

kk

. Darcy velocities for water and oil can be

defined as

x

pu ww

(20)

x

pu nwo

(21)

We define fractional flow of water as



17

orw

wronww

w

ow

ww

k

kuu

uf

1

1 (22)

tw

orw

wro

t

orw

wro

oww uf

k

ku

k

kuu

u

11 (23)

Substituting into Eq. (16)

0

dx

dS

dS

dfu

t

S

x

uf

t

S w

w

wt

wtww

(24)

0dx

dS

dS

dfu

t

S w

w

wtw

(25)

Assume wt f

uf

, Eq. (25) becomes

0dx

dS

dS

df

t

S w

w

w

(26)

Typical relative permeabilities are shown in Figure . When only oil is produced and no

water flows out of the system 0rwk and rok is close to 1. After the critical saturation

of water wrS , that is when water first breaks through, rwk starts to increase towards 1.

As water displaces oil the relative permeability of oil tends to zero and when all the oil

has been displaced 1rwk and 0rok . Often some oil is left in the system and at the

saturation ocS , water is not able to displace more oil causing the residual saturation

orS of oil in the system to become wror SS 1 .



18

Figure 12: The relative permeabiliy of water and oil as a function of water saturation.

Assume that initial and boundary values of saturation at ( , ) (0,0)x t be constants, we

have

W

w

W S

dfQLx t

A dS , (24)

where

1

1w

w ro

o rw

fk

k

.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sw

f w

fw

Figure 13. Relation between fractional flow to water and saturation



19

If give relations between relative permeabilities and saturation, we can know

saturation changes as time increases.

From the equation in hyperbolic form, the initial and boundary values

( ,0( , 0)w wS x t S and ,( 0, )w w cS x t S constants), we know there is a shock when

constant saturations propagate in porous media. The saturation in the region ahead

the shock doesn’t change and is the same as the initial value.

8.11 Numerical Simulations

We may use an empirical formula to give a function relation between the capillary

pressure and the water saturation. Then we can solve the two phase flow equations

with the initial and boundary conditions numerically by the finite element method

based on COMSOL multiphysics.

Model Description:

Water injection Oil + Water

x

Figure 14. Numerical Simulation Model

Governing Equations

The problem describing the two phase flow is shown in Figure 1(a). The fluid flow is

governed by the equations, i.e.,

0

y

pk

yx

pk

xt

p

p

S w

w

ww

w

wc

c

w

(27)

0

y

pk

yx

pk

xt

p

p

S o

o

oo

o

oc

c

o

(28)

Substituting wo SS 1 into Eq. (28) gives

0

y

pk

yx

pk

xt

p

p

S o

o

oo

o

oc

c

w

(29)



20

In order to solve these equations, we must have a capillary model linking the wetting

phase saturation, wS , to the capillary pressure, cp , and a relative permeability model

linking the relative permeability to the saturation.

Capillary Model: We use a capillary pressure model

* 1mn

w cS ah

, (30)

where , ,a n m are parameters, /( )c c wh p g is the capillary pressure head and

*wS is defined by

*

1w wr

wor wr

S SS

S S

. (31)

The Relative Permeability Model: The relative permeability curves are given by the

following formula

2

*1/ 2 *1/, 1 1

mmr w w wK S S

, (32)

1/ 2 2* *1/, 1 1

mmr o w wK S S . (33)

In addition, we need boundary conditions and initial conditions.

Boundary Conditions: The pressure of the water phase is given and the Darcy flux

of the oil phase equals to zero at the inlet boundary, i.e.,

,w w ip p , at the inlet boundary, (34)

, 0r oo o

o c

kk gn p h

g

, at the inlet boundary ,(35)

where n

is the normal vector. Flow rates for the water and oil phases are specified at

the outlet boundary



21

,r ww w w

w c

kk gn p h Q

g

, at the outlet boundary, (36)

,r oo o o

o c

kk gn p h Q

g

, at the outlet boundary,(37)

where w o TQ Q Q is the total flow rate across the outlet boundary. Generally, wQ

and oQ are two time-dependent functions. In numerical computations, assume

w w TQ f Q and o o TQ f Q .

Initial Conditions: The initial pressures of the water and oil phases are given

0 ,0w t wp p ,

0 ,0o t op p ,

Some simulation results are illustrated in the following graphs.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Sw

t=1

Analytical Numerical

Figure 15. Saturation Profiles: Comparison between analytical solution and numerical

solution at t=1.



22

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Sw

t=2



solution at t=2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Sw

t=3



solution at t=3.



23

APPENDIX - A Detailed Example In order to solve the two-phase flow equations, we need three things:

Flow Equations satisfying the mass conservation law for each phase

A Capillary Model linking wetting phase saturation to capillary pressure

A Relative Permeability Model linking relative permeability to capillary pressure

though wetting phase saturation

Flow Equations:

0

y

pk

yx

pk

xt

p

p

S w

w

ww

w

wc

c

w

(A1)

0

y

pk

yx

pk

xt

p

p

S o

o

oo

o

oc

c

o

(A2)

Capillary Model

Scec

w

c

ewwec pp

p

S

p

pSSpp

32

*2

*2/1* 2 (A3)

Relative Permeability Model

nwK

c

e

c

ewwrnw

wK

c

ewrw

p

p

p

pSSk

p

pSk

4222*2*

84*

1111

(A4)

Substituting Equations (A3) and (A4) into (A1) and (A2) gives

0

y

pk

yx

pk

xt

p wwK

w

wwK

w

cS

(A5)

0

y

pk

yx

pk

xt

p onwK

nw

onwK

nw

cS

(A6)

According to the definition of capillary pressure, we have the following equation

wownwc ppppp (A7) Substituting (A7) into (A5) and (A6) gives



24

0

y

pk

yx

pk

xt

p

t

p wwK

w

wwK

w

wS

oS

(A8)

0

y

pk

yx

pk

xt

p

t

p onwK

nw

onwK

nw

wS

oS

(A9)

Now we can implement (A8) and (A9) into Comsol Multiphysics.



1

CHAPTER 9

MULTIPHASE FLOW




0

y

pk

yx

pk

xS

ti

i

iii

i

iiii

Saturation

Multiphase Flow:

ww

ww

w

rw

gvovg

g

o

og

g

rgvo

oo

ro

osfgso

o

g

go

o

rosg

gg

rg

qB

S

tp

B

kk

qRqRB

S

B

S

tp

B

kkRp

B

kk

qRqRB

S

B

S

tp

B

kkRp

B

kk

1 wgo SSScwoow ppp cgoog ppp




0

y

pk

yx

pk

xS

ti

i

iii

i

iiii

Saturation

Multiphase Flow:

ww

ww

w

rw

gvovg

g

o

og

g

rgvo

oo

ro

osfgso

o

g

go

o

rosg

gg

rg

qB

S

tp

B

kk

qRqRB

S

B

S

tp

B

kkRp

B

kk

qRqRB

S

B

S

tp

B

kkRp

B

kk

1 wgo SSScwoow ppp cgoog ppp



2

We know the general flow equation for a single phase as defined:

iii

i

iii

i

ii Sty

pk

yx

pk

x

(1)

For multiphase flow problems, we assume that there are at most three distinct phases:

oil, water and gas. Usually water is the wetting phase, oil has an intermediate

wettability and gas is the nonwetting phase. Water and oil are assumed to be

immiscible and they do not exchange mass or change phase. Gas is assumed to be

soluble in oil but usually not in water. Furthermore, it is assumed that fluids are at

constant temperature and in thermodynamic equilibrium throughout the reservoir.

Under these conditions, the petroleum liquid PVT properties are defined in Figure 1.

Figure 1: Graphical Definitions of Petroleum Fluid PVT Properties.

In the following sections, we will apply the mass conservation law to each phase:

water, oil, and gas. As shown in Figure 1, we assume the original gas volume in the

reservoir, 1V , and the original oil volume in the reservoir, 2V , are equal to 1 unit. We

now apply the mass conservation law to each phase.

Systems!

gOverlappin Threebulkw

go

VV

VV

6

2oB

6

5

soR

3

1

gB

3

4vR

3

GasOil

Expandedgas

5

6

4

GasOil

Expandedoil

OIL

Mo

veab

le P

isto

n

BPpp

1

2GasOil

BPpp

Reservoir temperature

LEGEND

Stock-tank oilSurface gas

Oil+solution gasl

Gas+volatilized oil

The liquid phase volume at reservoir conditions divided by the liquid volume of the same sample at standard conditions;

The vapor phase volume at reservoir conditions divided by the gas volume of the same sample at standard conditions;

The ratio of the volume of surface gas to stock-tank oil in a reservoir liquid phase at reservoir conditions;

The ratio of the volume of stock-tank oil to surface gas contained in a reservoir vapor phase at reservoir conditions.

oB

soR

gB

vR

6

2oB

6

2oB

6

5

soR6

5

soR

3

1

gB

3

1

gB

3

4vR

3

4vR

3

GasOil

Expandedgas

3

GasOil

Expandedgas

5

6

4

GasOil

Expandedoil

5

6

4

GasOil

ExpandedoilExpandedoil

OIL

Mo

veab

le P

isto

n

BPpp

OIL

Mo

veab

le P

isto

n

BPpp

1

2GasOil

BPpp

1

2GasOil

BPpp


LEGEND


Oil+solution gasl

Gas+volatilized oil





oB

soR

gB

vR

FLOW PROCESSSurface Condition

121 VVAssume:

How to apply the mass balance law to each phase?

Systems!

gOverlappin Threebulkw

go

VV

VV

6

2oB

6

5

soR

3

1

gB

3

4vR

3

GasOil

Expandedgas

5

6

4

GasOil

Expandedoil

OIL

Mo

veab

le P

isto

n

BPpp

1

2GasOil

BPpp


LEGEND


Oil+solution gasl

Gas+volatilized oil





oB

soR

gB

vR

6

2oB

6

2oB

6

5

soR6

5

soR

3

1

gB

3

1

gB

3

4vR

3

4vR

3

GasOil

Expandedgas

3

GasOil

Expandedgas

5

6

4

GasOil

Expandedoil

5

6

4

GasOil

ExpandedoilExpandedoil

OIL

Mo

veab

le P

isto

n

BPpp

OIL

Mo

veab

le P

isto

n

BPpp

1

2GasOil

BPpp

1

2GasOil

BPpp


LEGEND


Oil+solution gasl

Gas+volatilized oil





oB

soR

gB

vR

FLOW PROCESSSurface Condition

121 VVAssume:

How to apply the mass balance law to each phase?



3

9.1 Phase Mass Accumulation

In the reservoir, the gas mass is defined as

RCgRCgRCg VM ,1,, (2)

Where RCgM , is the gas mass in the reservoir, and RCg , is the gas density under

the reservoir condition. On the surface, the gas mass in the reservoir has become two

parts, the gas mass in the volume 3, 3V , and the oil mass in the volume 5, 5V , i.e.,

v

g

SCo

g

SCg

SCoSCgSCg RBB

VVM ,,

5,3,,

(3)

where SCg , is the gas density on the surface condition. According the mass

conservation law, the following relation holds

ogv

g

SCo

g

SCg

RCg MMRBB ,4,3

,,

,

(4)

Similarly, the relation between oil densities is defined as

gos

o

SCg

o

SCo

RCo MMRBB ,5,6

,,

,

(5)

Where RCo , and SCo , are the gas densities under the reservoir condition and under

the surface condition, respectively. From equations (4) and (5), we can define the total

gas mass and the total oil mass under the surface condition. Assuming the reservoir

gas volume and the reservoir oil volume are equal to 1, we can calculate the total gas

volume and the total oil volume in the reservoir as

gg SV (6)



4

oo SV (7)

If we move all of the reservoir fluids (gas and oil) to the ground surface, we calculate

the total gas and the total oil as

o

o

sSCg

g

g

SCg

g SB

RS

BM

,, (8)

g

g

vSCo

o

o

SCo

o SB

RS

BM

,, (9)

For water, there is no phase transformation. The total mass is calculated as

w

w

SCw

w SB

M , (10)

9.2 Phase Mass Flow Rate

The flowing gas consists of two components: the free-phase gas and the gas released

from the oil-phase. Therefore, the gas mass flow rate, gm , is defined as

o

o

sSCg

g

SCg

gg B

R

Bm uuu gg

,, (11)

Similarly, we can calculate the oil mass flow rate, om , as

g

g

vSCo

o

o

SCo

ooo B

R

Bm uuu ,,

(12)

For the water phase, the water mass flow rate is defined as

w

w

SCw

www Bm uu ,

(13)



5

9.3 Multiphase Flow Equations

We re-write equation (1) as

iii St

iu (14)

Substituting equations (8) through (13) into (14) gives

w

w

w

w

v

g

g

o

ov

g

g

o

o

s

o

o

g

g

s

o

o

g

B

S

tB

RB

S

B

S

tR

BB

RB

S

B

S

tR

BB

u

uu

uug

(15)

Adding production terms to (15) gives

w

w

w

w

w

gvov

g

g

o

ov

g

g

o

o

osfgs

o

o

g

g

s

o

o

g

qB

S

tB

qRqRB

S

B

S

tR

BB

qRqRB

S

B

S

tR

BB

u

uu

uug

(16)

Darcy law is defined as



6

i

g

g

rg

g

i

o

o

roo

i

w

w

rw

x

pkk

x

pkk

x

pkk

u

u

uw

(17)


ww

ww

w

rw

gvovg

g

o

og

g

rgvo

oo

ro

osfgso

o

g

go

o

rosg

gg

rg

qB

S

tp

B

kk

qRqRB

S

B

S

tp

B

kkRp

B

kk

qRqRB

S

B

S

tp

B

kkRp

B

kk

(18)

There are SIX unknowns in equation (10): wgowgo pppSSS ,,,,, . Additional three

equations are needed to define the system completely. One is the saturation

condition:

1 wgo SSS (19)

The other two come from the relations between phase pressures.

9.4 Boundary Conditions

A unique solution to the governing statements requires boundary conditions for all

models as well as initial conditions if the problem is transient or time dependent. The

Darcy’s law application mode of the Earth Science Module provides a number of

boundary conditions. We can also specify unique conditions by entering expressions

in the boundary settings dialogs and/or altering the boundary mechanics in the



7

equation systems dialogs.

In many cases, the distribution of pressure is known. This is a Dirichlet condition given

by

0ii pp (20)

Where 0p is a known pressure, given as a number, a distribution, or an expression

involving time t, for example.

Fluid does not move across impervious boundaries. This is represented by the zero

flux condition

0 Dgpk

ii

i

i

n (21)

Where n is the vector normal to the boundary. While this Neumann condition specifies

zero flow across the boundary it allows for movement along it. In this way the equation

for the zero flux condition also describes symmetry about an axis or a flow divide, for

example.

Often the fluid flux can be determined from pumping rate or known from

measurements. With the inward boundary condition, positive values correspond to

flow into the model domain.

0NDgpk

ii

i

i

n (22)

9.5 Initial Conditions

Both pressure and saturation distributions are required at t=0:

0,,, tzyxppi (23)



8

0,,, tzyxSSi (24)

In order to solve equations (18) through (24), the following data are required:

Variables required for assignment to each cell (location dependent): porosity,

absolute permeability, initial pressure and saturation;

Variables required as a function of pressure: dissolved gas-oil ratio, formation

volume factors, viscosities, densities, and compressibilities;

Variables required as a function of saturation: relative permeability, capillary

pressure;

Well data: production (or injection) rate, location in grid system, production

limitations.

9.6 Location Dependent Variables

Porosity: Although a reservoir rock looks a solid to the naked eye, a microscopic

examination reveals the existence of voids in the rock. These pores are the ones

where petroleum reservoir fluids are present. This particular storage capacity is called

porosity. The more porous a reservoir rock material is, the greater the amount of voids

it contains, hence greater the capacity to store petroleum reservoir fluids. From a

reservoir engineering perspective, porosity is probably one of the most important

reservoir rock properties.

Porosity, φ is a volumetric fraction defined as the ratio of the pore volume, poreV in a

reservoir rock to the total volume (bulk volume), bulkV :

bulk

pore

V

V (25)



9

The porosity of a rock is a measure of the storage capacity (pore volume) that is

capable of holding fluids. It may by occupied by a single-phase fluid or mixtures. As

the sediments were deposited and the rocks were being formed during past geological

times, some void spaces that developed became isolated from the other void spaces

by excessive cementation. Thus, many of the void spaces are interconnected while

some of the pore spaces are completely isolated. This leads to two distinct types of

porosity, namely:

Absolute porosity

Effective porosity.

bulk

totalporea V

V (26)

bulk

dporeerconnectee V

Vint (27)

The effective porosity is the value that is used in all reservoir engineering calculations

because it represents the interconnected pore space that contains the recoverable

hydrocarbon fluids.

Absolute Permeability: Unlike porosity, permeability is a flow property (dynamic)

and therefore can be characterized only by flow experiments in a reservoir rock.

Permeability is a property of the porous medium that measures the capacity and

ability of the formation to transmit fluids. The rock permeability, k, is a very important

rock property because it controls the directional movement and the flow rate of the

reservoir fluids in the formation. Absolute permeability is the rock permeability when a

reservoir rock is 100% saturated with a given fluid. It should be noted that the absolute

permeability is a property of the rock alone and not the fluid that flows through it,

provided no chemical reaction takes place between the rock and the flowing fluid.

All the equations used to describe fluid flow in reservoirs are based on Darcy’s law.



10

Darcy (1856), investigated the flow of water through sand filters. He observed the

following relationship between velocity and pressure gradient as

x

pk

A

Qux

(28)

Where

direction- xthe in Coordinate

Pressure

flow to open area sectional-Cross

ViscosityFluid

tyPermeabili

RateFlow

direction- xthe invelocity Darcy

x

p

A

k

Q

ux

Basic assumptions:

It is assumed that the porous medium is saturated with a single fluid.

The flowing fluid is incompressible.

The linear dependence of flow velocity on the pressure gradient implies

laminar.

The flow takes place under the viscous regime (i.e., the rate of flow is

sufficiently low so that it is directly proportional to the pressure differential or the

hydraulic gradient).

The flowing fluid does not react with the porous medium.

The negative sign in the above equations indicates that pressure decreases in the

direction of flow. The sign convention is therefore that distance is measured positive in

the direction of flow.

Saturation: Saturation is defined as that fraction, or percent, of the pore volume

occupied by a particular fluid (oil, gas, or water). This property is expressed



11

mathematically by the following relationship:

VolumePore Total

Fluid the of VolumeTotal Saturation Fluid

Assuming wgo SSS ,, represent the oil saturation, gas saturation and water saturation,

respectively, and wgo VVV ,, for the oil volume, gas volume, and water volume,

respectively, and pV the total pore volume, applying the above mathematical concept

to each to each reservoir fluid gives

p

ww

p

g

g

p

oo

V

VS

V

VS

V

VS

(29)

Thus, all saturation values are based on pore volume and not on the gross reservoir

volume. The saturation of each individual phase ranges between zero to 100 percent.

By definition, the sum of the saturations is 100%, therefore

1 wgo SSS (30)

Equation (2.8) is probably the simplest, yet the most fundamental equation in

reservoir engineering, and is used everywhere in reservoir engineering calculations.

Moreover, many important reservoir rock properties, such as capillary pressure and

relative permeability, are actually related or linked with individual fluid-phase

saturations.

The fluids in most reservoirs are believed to have reached a state of equilibrium and,

therefore, will have become separated according to their density, i.e., oil overlain by

gas and underlain by water. In addition to the bottom (or edge) water, there will be

connate water distributed throughout the oil and gas zones. The water in these zones



12

will have been reduced to some irreducible minimum. The forces retaining the water in

the oil and gas zones are referred to as capillary forces because they are important

only in pore spaces of capillary size. Connate (interstitial) water saturation Swc is

important primarily because it reduces the amount of space available between oil and

gas. It is generally not uniformly distributed throughout the reservoir but varies with

permeability, lithology, and height above the free water table.

Pressure: Pressures are required for each cell in a simulator and may be input on a

per cell basis; however, if the simulation begins at equilibrium conditions, it is much

easier to use a pressure at a known datum and calculate pressures for all cells using

a density gradient adjustment:

144

Dpp datum

(31)

Where

3lb/ft Density,

ft Elevation,in Change

psi Pressure, Datum

psi cell,in Pressure

D

p

p

datum

Additionally, in multiphase flow, a pressure for each phase (oil, gas and water) must

be calculated. The pressure in the water phase is related to the oil pressure by the

capillary pressure:

cwoow ppp (32)

And the pressure in the gas phase is related to the oil pressure by:

cgoog ppp (33)



13

9.7 Pressure Dependent Variables

Dissolved Gas-Oil Ratio: sR is required as a function of pressure and based on

the pressure in each cell, the amount of dissolved gas will be calculated for each cell.

It may have units of either SCF of solution gas per STB oil, or MCF solution gas per

STB oil; in the former case, the values should be between 50 and 1400 SCF/STB with

the majority of fields falling between 200 and 1000 over reasonable pressure ranges.

Obviously, for units of MCF/STB, the variations are 0.05 to 1.4. A typical dissolve

gas-oil ratio curve is shown in Figure 2.

Oil Formation Volume Factor: oB relates to a reservoir volume of oil to a

surface volume. The reservoir volume includes the dissolved gas whereas the surface

volume does not. The oil formation volume factor has units of RVB/STB. A reasonable

range is from 1.05 to 1.40. A typical curve is shown in Figure 3.

Figure 2: Dissolved Gas Ratio as A Function of Pressure

sR

p

initialpbublep

Production Path

sR

p

initialpbublep

Production Path



14

Figure 3: Oil Formation Volume Factor as A Function of Pressure

Gas Formation Volume Factor: gB relates to a reservoir volume of gas to a

surface volume. Several units may be applied to the gas formation volume factor:

RCF/SCF, RVB/SCF, or RVB/MCF. For most reservoir pressures encountered, gB

will be between 0.2 and 1.5 RVB/MCF. The gas formation volume factor is readily

calculated from:

p

TzBg

460035.5 (34)

eTemperaturReservoir

FactorDeviation Gas

T

z

oB

p

initialpbublep

Production Path

oB

p

initialpbublep

Production Path



15

A typical curve is shown in Figure 4.

Figure 4. Gas Formation Factor as A Function of Pressure.

9.8 Saturation Dependent Variables

Relative Permeability: rk is a reduction in flow capability due to the presence

of another fluid and is based on

Pore geometry

Wettability

Fluid distribution

Saturation history

Relative permeability is dimensionless and is determine the effective permeability for

flow as follows:

re kkk (35)

gB

p

initialpbublep

Production Path

gB

p

initialpbublep

Production Path



16

Relative permeability data are entered in models as functions of saturation and may

be obtained from laboratory measurements, field data, correlations, or simulation

results of a similar formation. Whether appropriate or not, it is usually the first data to

be modified in a model study. The simplest concept in relative permeability is that of

two-phase flow. For oil reservoirs, the combinations are water-oil and liquid-gas

(usually thought of as oil-gas); for gas reservoirs, gas-water applies; and for

condensate reservoirs, gas-liquid.

Figure 5: Water-oil relative permeability

Water-Oil Relative Permeability: It is usually plotted as a function of water saturation

as illustrated in Figure 5. At the critical (or connate) water saturation (Swc), the water

relative permeability is zero, 0rwk . And the oil relative permeability with respect to

water (or in the presence of water) is some value less than one. At this point, only oil

can flow and the capability of the oil to flow is reduced by the presence of water. Note

that data to the left of the critical water saturation is useless (unless the critical water

becomes mobile). As water saturation increases, the water relative permeability

increases and the oil permeability (with respect to water) decreases. For the oil

reservoir, a maximum water saturation is reached at the residual oil saturation (Sorw);

0 1

1

wcS orwS

Oil Water

rowk rwkrk

oSwS0 1

1

wcS orwS

Oil Water

rowk rwkrk

oSwS



17

however, since models use an average saturation within each cell, oil saturation

values of less than residual oil (in a cell) should be correctly entered.

Wettability: It is a measurement of the ability of a fluid to coat the rock surface.

Classical definitions of wettability are based on the contact angle of water surrounded

by oil and are defined as

ty wettabilimixedor teintermedia90

wet-oil90

wet-water90

0

0

0

Figure 6: Contact angles

Capillary Pressure: cP is required in simulators to determine the initial fluid

distributions and to calculate the pressures of oil, gas and water. It is the difference in

pressure between two fluids due to a limited contact environment. This data is

required as a function of saturations and may be obtained from laboratory

measurements, correlations or estimated to yield the desired fluid distributions. When

laboratory measurements are used, they must be corrected to reservoir conditions:

L

rcLcr PP

(36)

water

water

oil

Oil-wetWater-wet

water

water

oil

Oil-wetWater-wet

water

water

oil

Oil-wetWater-wet



18

fluids lab of tension linterfacia

fluidsreservoir of tension linterfaciaConditionsReservoir at PressureCapillary

conditions Labat PressureCapillary

L

r

cr

cL

P

P

When fluid distributions are known at various depths, capillary pressures may be

estimated from

144

HPc (37)

number) positive (alb/ft fluids, obetween twdensity in difference

ft fluid,densor above zonen transitioofheight 3

H

With rare exceptions (high capillary ranges), capillary pressures have minimal effects

once the reservoir is produced.

9.9 Well Data

Production or injection rates are required for each well to be modeled. For

fluids, the rate is usually in STB/day and for gas, MCF/day. For producing wells, only

one phase production should be specified and that phase is usually the predominant

phase. For example, an oil well would specify oil production, and the appropriate gas

and water producing rates would be calculated by the model. This data is normally

obtained from well files.

Production limitations may be imposed on wells. Some of these may be

bottom-hole pressures, skin factors, maximum GOR or WOR limits, total field

limitations, coning effects and abandonment conditions.

9.10 Field Studies



19

Reservoir engineers usually follow the following steps to conduct a field study:

Definition of Simulation Objective: Reservoir simulation typically begins

with a question about some physical phenomenon within a particular region of

reservoir. Defining the question well typically means the type of information needed to

answer it is obvious.

Identification of Physics: One then identifies which physics actually are

important to answering the question and how they interact.

Mathematical Formulation of the Physics: A mathematical model fitted to

this conceptual model of the physics is defined for the domain of interest. The

mathematical model contains governing equations with boundary conditions and

possibly initial conditions. The boundary conditions detail how the model domain

interacts with the surrounding environment. The initial conditions make up a snapshot

of the physics at some initial time.

Solution and Interpretation: When mathematical model is solved, one

interprets the results in light of the original question. If the mathematical model is

consistent with the one in a commercial simulation tool, the problem can be solved by

using this tool. If the mathematical model is NOT consistent with the one in a

commercial tool, reservoir engineer needs to modify the tool or to develop a new one

for the problem.

The current commercial packages such as ECLIPSE can meet most of reservoir

simulation requirements. In other words, the physics for majority of reservoir

engineering problems is contained in these existing models. Reservoir engineers are

not required to modify the physics or implement the physics into a new computer

program. Under this framework, most simulation studies follow a similar format and

basic procedure. The processes have been summarized in the following figure.



20

Figure 7: Processes of a Typical Reservoir Simulation.

Geological Model: Geoscientists probably play the most important role in

developing a reservoir model. The distributions of the reservoir rock types and fluids

determine the model geometry and model type for reservoir characterization. The

development and use of reservoir model should be guided by both engineering and

geological judgments. Geoscientists and engineers need feedbacks from each other

throughout their work. For example, core analyses provide data to verify reservoir rock

types, whereas well test analysis can confirm flow barriers and fractures recognized

by the geoscientists. By discussing all the data as a team, each specialist can

contribute the data he/she has available and can help other team members

understand the significance of that data.

Three-dimensional seismic data can be used to assist in

Defining the geometric framework;

Qualitative and quantitative definition of rock and fluid properties;

Flow surveillance.

G e o lo g ic a l R e v ie w

R e s e r v o ir P e r f o rm a n c e R e v ie w

P r o d u c t io n R e v ie w

D a ta G a th e r in g

S c r e e n in g

R e s e r v o ir M o d e l

S in g le w e l l o r p a t te r n

C r o s s - s e c t io n

F ie ld M o d e l

In i t ia l iz a t io n

H is to r y M a tc h in g

P re d ic t io n s

R e p o r t & P r e s e n ta t io n s

G e o lo g ic a l R e v ie w

R e s e r v o ir P e r f o rm a n c e R e v ie w

P r o d u c t io n R e v ie w

D a ta G a th e r in g

S c r e e n in g

R e s e r v o ir M o d e l

S in g le w e l l o r p a t te r n

C r o s s - s e c t io n

F ie ld M o d e l

In i t ia l iz a t io n

H is to r y M a tc h in g

P re d ic t io n s

R e p o r t & P r e s e n ta t io n s



21

A 3-D seismic survey impacts the original development plan. With the drilling of

development wells, the added information is used to refine the original interpretation.

As time passes and the data builds, elements of the 3-D data that were initially

ambiguous begin to make sense. The usefulness of a 3-D seismic survey lasts for the

life of a reservoir.

Geostatistical modeling of reservoir heterogeneity is playing an important role

ingenerating more accurate reservoir models. It provides a set of spatial data analysis

tools as a probabilistic language to be shared by geologists, geophysicists, and

reservoir engineers, as well as a vehicle for integrating various sources of uncertain

information. Geostatistics is useful in modeling the spatial variability of reservoir

properties and the correlation between related properties such as porosity and

seismic velocity. A geostatistical model can then be used to interpolate a property

whose average is critically important and to stochastically simulate for a property

whose extremes are critically important.

After identifying the geological model, additional engineering/production data are

necessary for completion of the reservoir model. The engineering data include

reservoir fluid and rock properties, well location and completion, well test pressures,

and pulse-test responses to determine well continuity and effective permeability.

Material balance calculations can provide the original oil in place, and natural

producing mechanism – including gas cap size and aquifer size and strength. The use

of injection/production profiles provides vertical fluid distributions. Integration of

geosciences and engineering is required to produce the reservoir model, which can

be used to simulate realistic reservoir performance.

Reservoir Performance: Reviewing the production performance of the

reservoir is an important part of constructing a simulation for the following two

reasons:

It will help determine the correct input data required.

It will give direct clues as to the depletion process, i.e., mechanisms occurring

in the reservoir.



22

Figure 8: Combination of reservoir simulation and material balance equation.

In a number of examples, this process could avoid problems, and in some cases,

production performance data was useful in setting the scope of a simulation project.

We can use material balance equation to identify producing mechanisms. For different

producing mechanism, we can use different plots to evaluate the production

performance. Using solution gas drive as an example, the following plots are usually

used to describe reservoir performance:

A plot of reservoir pressure. Above the bubblepoint, the pressure drops rapidly

with cumulative production, followed by a decrease in pressure decline below

the bubblepoint.

A plot of production. As time goes on, the production rate declines. In fact, for

solution gas drives, production rate normally shows as a relatively straight line

on a semilog plot of production.

A plot of GORs. This is constant or slightly decreasing at first and then rises

with time.

A plot of water cuts. This normally doesn’t increase for solution gas drives,

ignoring potential coning. However, for waterfloods or water drives, water

breakthroughs will occur.

In large part, production performance analysis consists of the reverse of this process.

We plot graphs of the reservoir pressure versus time, or cumulative production, and

production. From this, we can interpret the reservoir mechanism. On the surface, this

etc

SizeAquifer

, iZOOIP

OGIP

Data PVTPHASE-BEHAVIORPACKAGE

COMPOSITIONALRESERVOIRSIMULATOR

vso

go

RR

BB

,

,,MBE

average

ppp

P

WNG ,,,

etc

SizeAquifer

, iZOOIP

OGIP

COMPARE

etc

SizeAquifer

, iZOOIP

OGIP

Data PVTPHASE-BEHAVIORPACKAGE

COMPOSITIONALRESERVOIRSIMULATOR

vso

go

RR

BB

,

,,MBE

average

ppp

P

WNG ,,,

etc

SizeAquifer

, iZOOIP

OGIP

COMPARE



23

is a simple process and this is true in some cases. However, in the majority of cases, it

is not nearly so simple, and experience helps. It is common to plot both the reservoir

as a whole, groups of wells that may be isolated (such as a fault block), and individual

wells. The analysis of production performance data should lead to increased chances

of success in interpreting the drive mechanisms in the reservoir and in the

quality-applicability-correctness of the simulation study subsequently carried out.

Data Gathering: Throughout the life of a reservoir, from exploration to

abandonment, enormous amounts of data are collected. An efficient data

management program, consisting of acquisition, analysis, validating, storing, and

retrieving, can play a key role in reservoir simulation. An effective data acquisition

and analysis program requires careful planning and well-coordinated team efforts

throughout the life of the reservoir. Justification, priority, timeliness, quality, and

cost-effectiveness should be the guiding factors. Field data are subject to many errors,

e.g., sampling, systematic, random, etc. Therefore, the collected data need to be

carefully reviewed and checked for accuracy as well as for consistency.

The reservoir performance should be closely monitored while collecting routine

production and injection data, including reservoir pressures. If past production and

pressure data are available, classical material balance technique and reservoir

simulation can be very useful to validate the volumetric original hydrocarbons-in-place

and aquifer size and strength.

Reservoir Models: Selection of a reservoir model type is usually straightforward

depending study objectives. For examples:

1D models for material balance with water influx

or 3 Phase 2D radial for water and gas coning, thin oil columns

D aerial for thin reservoirs, eg sweep efficiency determination in pattern

waterfloods

2D cross section models for gas override and water underrun

Sector models for isolated fault blocks



24

3D models for full field simulation, large reservoir thickness of area, vertical

heterogeneities, multiple wells

Other factors to consider include

Study objectives

Detail of reservoir description available (Reservoir heterogeneities)

Mechanisms to be modelled (Gravity segregation, Fluid override/underrun

(Coning))

Availability of dynamic data for history matching

Cost and time limitations for study

Initialization: The primary objective of initialization is to establish the initial

pressure field, saturation distributions, and gravity. The majority of simulations are

initialized based on static gravity majority capillary pressure equilibrium. This works

well in the majority of cases. However, there is no requirement the fluids in the earth

are static and in truth, this is rarely completely correct.

Calibration – History Matching: The purpose of calibration is to establish

that the model can reproduce pressures and flows. During calibration a set of values

for reservoir rock and fluid properties is found that approximates field-measured

pressures and flows. Calibration is done by trial-and-error adjustment of parameters

or by an automated parameter estimation code, as shown in Figure 9.



25

Figure 9: A Systematic Approach to History Matching

The calibrated model is influenced by uncertainty owing to the inability to define the

exact spatial and (temporal) distribution of parameter values in the problem domain.

There is also uncertainty over definition of boundary conditions and stresses. A

sensitivity study should be conducted in order to establish the effect of uncertainty on

the calibrated model.

The general strategy is to start with the overall picture and work down to progressively

more detailed matching. No hard-and-fast rules exist to set up a guideline. Probably

the best thing to do is read about field cases in the literature.

Predictions: At the end of the tuning phase the model is usually terminated with a

restart. This input data file contains all of the information necessary to continue a

simulation at a later time. A number of new production scenarios or alternatives are

run from the same time step and compared. With different runs, various injector

patterns, changes in rates, producer-injector locations can be evaluated.

Match Average Reservoir Pressure

Match Average GOR and Water Cut

Match Average Individual Wells

Tune Wells

Rock Compressibility

Pore Volume

Boundary Conditions

Relative Permeability

Curves

Pore Volume Adjustments

Permeability Adjustments

Permeability*Height

Skin

Bottomhole Pressure

START

FINAL HISTORY MATCH

Match Average Reservoir Pressure

Match Average GOR and Water Cut

Match Average Individual Wells

Tune Wells

Rock Compressibility

Pore Volume

Boundary Conditions

Relative Permeability

Curves

Pore Volume Adjustments

Permeability Adjustments

Permeability*Height

Skin

Bottomhole Pressure

START

FINAL HISTORY MATCH

Issued on Wed. 23/8/07; Due Date: 1700 Friday 3/9/07; Submission: webCT

PETR4511 - Simulation Project I – Jishan Liu

1

PROJECT I: Steady State Flow in a Heterogeneous Reservoir (20%)

Project Objectives: ♦ Learn How to Find Numerical Solutions for Steady State Flow ♦ Investigate How the Spatial Variation in Permeability Affects the Pressure Distribution ♦ Investigate How the Spatial Variation in Permeability Affects the Darcy Velocity Distribution Simulation Requirements: In the following three cases assuming incompressible and steady-state flow, you are required to investigate the influences of the spatial variation in permeability both on the pressure distribution and the Darcy velocity distribution. ♦ Scenario 1

214213212

0 101010;1 mmmk −−−==α

♦ Scenario 2

1051;10 214

0 == − αmk

♦ Scenario 3

214213212

0 101010;5 mmmk −−−==α

Assessment Criteria: 1. Problem Identification: Work out the real physics starting from the general flow equation. 2. Implementation in Comsol Multiphysics: Determine input parameters essential for the solutions

through comparing the real physics with the Comsol physics. 3. Validation: Validate your simulation model against a known case such as a homogeneous case. 4. Investigations: Conduct investigations using the information specified for the simulation project. 5. Discussions: Discuss the simulation results through relating the results to the real physics.

♦ Definition of the Problem (2D)

No Flow No Flow

No Flow No Flow

Pap

Pap

k

k

k

k

100

10000

Cdomain

Bdomain

Adomain

2

1

0

1

0

0

==

⎪⎩

⎪⎨

⎧=

−αα

domain A

domain B

0.5m 0.5m

0.2m

0.2m

0.5m 0.5m

0.2m

0.2m

1p 2p

domain B

domain C

♦ Definition of the Problem (2D)

No Flow No Flow

No Flow No Flow

Pap

Pap

k

k

k

k

100

10000

Cdomain

Bdomain

Adomain

2

1

0

1

0

0

==

⎪⎩

⎪⎨

⎧=

−αα

domain A

domain B

0.5m 0.5m

0.2m

0.2m

0.5m 0.5m

0.2m

0.2m

1p 2p

domain B

domain C



2

Reference solution: Marking Compositions: Problem Identification + Discussions = 10 Marks Implementation, Validation and Investigation = 10 Marks Theoretical Analysis through Real physics: The general flow equation for this project is defined as

0=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

y

pk

yy

pk

x μμ (1)

Where

( )

( )

⎪⎪⎩

⎪⎪⎨

⎧

=

=

CDomain 1

BDomain

ADomain 1

0

α

αα

α

f

fkk

(2)

Substituting equation (2) into (1) gives

( ) ( ) 000 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

y

pf

k

yy

pf

k

xα

μα

μ (3)

Because 0k and μ are constants, equation (3) can be simplified as

( ) ( ) 0=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

y

pf

yy

pf

xαα (4)

Equation (4) indicates that the solution for pressure is determined by the function of ( )αf only under specific boundary conditions.

Darcy velocity is defined as

( ) ( )

( ) ( )y

pfK

y

pf

k

y

pku

x

pfK

x

pf

k

x

pku

y

x

∂∂

−=∂∂

−=∂∂

−=

∂∂

−=∂∂

−=∂∂

−=

ααμμ

ααμμ

00

00

(5)

Rearranging equation (5) as



3

( )

( )y

pf

K

u

x

pf

K

u

y

x

∂∂

−=

∂∂

−=

α

α

0

0 (6)

Equation (6) indicates the dimensionless velocities are also determined by the function of ( )αf only under specific boundary conditions.

Implementation into Comsol Multiphysics: The simulation model was implemented into Comsol Multiphysics using the given input numbers as shown in the Definition of the Problem. Validation against a Known Solution: For this simulation project, we know the solution when 1=α , as shown in Figure 1. Figure 1. Pressure solution when 1=α and 2141312

0 10,10,10 mk −−−=

Simulation Scenario I: Figure 1 shows that the pressure solutions are independent of 0k . These results are consistent with the theoretical analysis.

Simulation Scenario II: 1051;10 214

0 == − αmk . In this cases, both

pressure solution and velocity solution changes with the magnitude of α based on the theoretical analysis. This analytical conclusion is confirmed by Figures 2 and 3.



4

Figure 2: Pressure solutions under different alpha values. Figure 3. Velocity solutions under different alpha values Discussions: From Figure 1 to 3, we conclude that simulation results are consistent with the analytical analyses. Simulation Scenario III: Because alpha does not change, the pressure solution and the velocity solution do not change with k0.

5=α 10=α1=α 5=α 10=α1=α

0K

u5=α 10=α1=α

0K

u5=α 10=α1=α

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