MATHEMATICAL APPROACH IN DIFFERENT... · A Thesis submitted to Gujarat Technological University For...

MATHEMATICAL APPROACH IN DIFFERENT

PHENOMENA ARISING IN MUTLIPHASE FLOW IN

POROUS MEDIA

A Thesis submitted to Gujarat Technological University

For the Award of

Doctor of Philosophy

In

Science - Maths

Researcher

Mansiben Kishorbhai Desai

Enrollment No. : 149997673008

Supervisor

Dr. Shailesh S. Patel

Professor and Head

ASH Department

GIDC Degree Engineering College, Navsari

GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

JULY 2020

MATHEMATICAL APPROACH IN DIFFERENT

PHENOMENA ARISING IN MULTIPHASE FLOW

IN POROUS MEDIA

A Thesis submitted to Gujarat Technological University

For the Award of

Doctor of Philosophy

In

Science - Maths

Researcher

Mansiben Kishorbhai Desai

Enrollment No. : 149997673008

Supervisor


Professor and Head

ASH Department

GIDC Degree Engineering College, Navsari


AHMEDABAD

JULY 2020

ii

©Mansiben Kishorbhai Desai

iii

DECLARATION

I declare that the thesis entitled “Mathematical Approach in Different Phenomena

Arising in Multiphase Flow in Porous Media” submitted by me for the degree of

Doctor of Philosophy is the record of research work carried out by me during the period

from 2014 to 2019 under the supervision of Dr. Shailesh S. Patel, Professor and Head

of the ASH Department, GIDC Degree Engineering College, Abrama, Navsari and this

has not formed the basis for the award of any degree, diploma, associate ship,

fellowship, titles in this or any other University or other institution of higher learning.

I further declare that the material obtained from other sources has been duly

acknowledged in the thesis. I shall be solely responsible for any plagiarism or other

irregularities, if noticed in the thesis.

Signature of the Research Scholar: Date: 04/07/2020

Name of Research Scholar: MANSIBEN KISHORBHAI DESAI

Place: Bilimora

iv

CERTIFICATE

I certify that the work incorporated in the “Mathematical Approach in Different

Phenomena Arising in Multiphase Flow in Porous Media” submitted by Dr.

Shailesh S. Patel was carried out by the candidate under my supervision/guidance. To

the best of my knowledge: (i) the candidate has not submitted the same research work

to any other institution for any degree/diploma, Associate ship, Fellowship or other

similar titles (ii) the thesis submitted is a record of original research work done by the

Research Scholar during the period of study under my supervision, and (iii) the thesis

represents independent research work on the part of the Research Scholar.

Signature of Supervisor: Date: 04/07/2020

Name of Supervisor: Dr. Shailesh S. Patel

Place: Navsari

v

COURSE-WORK COMPLETION CERTIFICATE

This is to certify that Ms. Mansiben Kishorbhai Desai, Enrolment no.

149997673008 is a PhD scholar enrolled for PhD program in the branch Science -

Maths of Gujarat Technological University, Ahmedabad.

(Please tick the relevant option(s))

He/She has been exempted from the course-work (successfully completed

during M.Phil Course)

He/She has been exempted from Research Methodology Course only

(successfully completed during M. Phil Course)

He/She has successfully completed the PhD course work for the partial

requirement for the award of PhD Degree. His/ Her performance in the course work

is as follows-

Supervisor’s Sign:


Grade Obtained in Research

Methodology

Grade Obtained in Self Study

Course (Core Subject)

(PH001) (PH002)

BC AB

vi

ORIGINALITY REPORT CERTIFICATE

It is certified that PhD Thesis titled “Mathematical Approach in Different

Phenomena Arising in Multiphase Flow in Porous Media” by Mansiben

Kishorbhai Desai has been examined by us. We undertake the following:

a. Thesis has significant new work / knowledge as compared already published or are

under consideration to be published elsewhere. No sentence, equation, diagram, table,

paragraph or section has been copied verbatim from previous work unless it is placed

under quotation marks and duly referenced.

b. The work presented is original and own work of the author (i.e. there is no plagiarism).

No ideas, processes, results or words of others have been presented as Author own

work.

c. There is no fabrication of data or results which have been compiled / analysed.

d. There is no falsification by manipulating research materials, equipment or processes, or

changing or omitting data or results such that the research is not accurately represented

in the research record.

e. The thesis has been checked using Urkund (copy of originality report attached) and

found within limits as per GTU Plagiarism Policy and instructions issued from time to

time (i.e. permitted similarity index <=10%).

Signature of the Research Scholar : Date: 04/07/2020

Name of Research Scholar: Mansiben Kishorbhai Desai

Place: Bilimora

Signature of Supervisor : Date: 04/07/2020

Name of Supervisor: Dr. Shilesh S. Patel

Place: Navsari

viii

PHD THESIS NON-EXCLUSIVE LICENSE TO


In consideration of being a PhD Scholar at GTU and in the interests of facilitation of

Research at GTU and elsewhere, I, Mansiben Kishorbhai Desai having enrolment no.

149997673008 hereby grant a non-exclusive, royalty free and perpetual license to GTU

on the following terms:

a) GTU is permitted to archive, reproduce and distribute my thesis, in whole or in part,

and/or my abstract, in whole or in part (referred to collectively as the “Work”)

anywhere in the world, for non-commercial purposes, in all forms of media;

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mentioned in paragraph (a);

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authority of their “Thesis Non-Exclusive License”;

d) The Universal Copyright Notice (©) shall appear on all copies made under the authority

of this license;

e) I undertake to submit my thesis, through my University, to any Library and Archives.

Any abstract submitted with the thesis will be considered to form part of the thesis.

f) I represent that my thesis is my original work, does not infringe any rights of others,

including privacy rights, and that I have the right to make the grant conferred by this

non-exclusive license.

g) If third party copyrighted material was included in my thesis for which, under the terms

of the Copyright Act, written permission from the copyright owners is required, I have

Obtained such permission from the copyright owners to do the acts mentioned

in paragraph (a) above for the full term of copyright protection.

ix

h) I retain copyright ownership and moral rights in my thesis, and may deal with the

copyright in my thesis, in any way consistent with rights granted by me to my

University in this non-exclusive license.

i) I further promise to inform any person to whom I may hereafter assign or license my

copyright in my thesis of the rights granted by me to my University in this non-

exclusive license.

j) I am aware of and agree to accept the conditions and regulations of PhD including all

policy matters related to authorship and plagiarism.

Signature of the Research Scholar:

Name of Research Scholar: Mansiben Kishorbhai Desai

Date: 04/07/2020 Place: Bilimora

Signature of Supervisor:


Date: 04/07/2020 Place: Navsari

Seal:

x

THESIS APPROVAL FORM

The viva-voce of the PhD Thesis submitted by Kum. Mansiben Kishorbhai Desai

(Enrollment No. 149997673008) entitled “Mathematical Approach in Different

Phenomena Arising in Multiphase Flow in Porous Media” was conducted on

04/07/2020 at Gujarat Technological University.

(Please tick any one of the following option)

The performance of the candidate was satisfactory. We recommend that he/she be

awarded the PhD degree.

Any further modifications in research work recommended by the panel after 3

months from the date of first viva-voce upon request of the Supervisor or request of

Independent Research Scholar after which viva-voce can be re-conducted by the

same panel again.

The performance of the candidate was unsatisfactory. We recommend that he/she

should not be awarded the PhD degree.

(briefly specify the modifications suggested by the panel)

(The panel must give justifications for rejecting the research work)

xi


Name and Signature of Supervisor with Seal

1) (External Examiner:1) Name and Signature

2) (External Examiner 2) Name and Signature

3) (External Examiner 3) Name and Signature

xii

ABSTRACT

The proposed thesis entitled, “Mathematical approach in different phenomena arising in

multi-phase flow in porous media” is based on the investigation on theoretical aspects

of the mathematical modelling of different physical phenomena and solutions of these

Mathematical models arising in fluid flow through homogenous porous media. This

study is primarily concerned with the modelling of the flow of immiscible fluids

through homogenous porous media. The motivation behind studying these flows lies in

the oil recovery process known as immiscible displacement, in which water is injected

into the oil formatted region. The fluid flow through porous media is the most

important concepts in many research areas of applied science and engineering such as

hydrogeology, petroleum engineering, water resource engineering, soil mechanics,

environmental engineering, chemical engineering, construction engineering, civil

engineering, geophysics, biophysics etc.

The fluid flow in porous media has gained extensive attention due to its broad range of

applications in science and industry. In last many years, extensive research works have

been carried out to study the fluid flow through porous media. In particular, the

modelling of fluid flow through porous media is a central problem within the field of

various applications in such areas. The scope of the present study lies in increasing

importance of the hydrodynamics of single phase flow and multiphase flow through

porous media. Due to the vast scope of multiphase flow through porous media, the

specific problems are almost unlimited and therefore it is reasonable to select such

types of problems for discussion here. Accordingly, a selection of more interesting

problems of current interest has been made for mathematical treatment in the work. The

investigated problems of the study are concerned with the flow of immiscible and

incompressible fluids.

The physical phenomena like as fingering phenomenon, imbibition phenomenon,

fingero-imbibition phenomenon, infiltration phenomenon arise in fluid flow through the

porous medium which is encountered in many fields of science and engineering. The

mathematical problems of different physical phenomena give us one dimensional

nonlinear partial differential equations. These equations are solved using Homotopy

xiii

Perturbation Method. The solution of the problems has been studied numerically and

graphically with the help of Microsoft Excel 2010 and Origin Lab.

The thesis is comprised of seven chapters. First and second chapters physically and

mathematically describe basic introductory details as well as theoretical aspects of the

research work carried out. Chapter three to six deal with different phenomena of single

and double phase flow in homogenous porous media and chapter seven include ground

water recharge in porous media.

As the succession of various chapters,

Chapter 1 describes an introductory chapter provides general ideas about fluid flow in

porous media as well as the brief introduction of the problems that the researcher has

tried to study. It also includes objectives, scope, significant as well as the contribution

that the researcher will do through this research. It also includes discussion on the

method through which the problem was formulated.

Chapter 2 is chosen in order to build up a stronger Structure in a logical manner to

provide knowledge of fundamentals of porous media, which is an essential part of the

study to have a better understanding of flow in porous media. This chapter deals with

the necessity of the study of flow through porous media with basic definitions, physical

properties of porous media, Darcy’s law and limitation of Darcy’ law. This chapter

also includes an explanation of the difference between miscible and immiscible flows.

The topics considered are meant to be a general background for the researcher

unfamiliar with fluid flow through porous media. Brief description of Homotopy

Perturbation Methods has been also included in this chapter and linear equation solved

using this method.

Chapter 3 discusses the theoretical studies involved in the development of the

differential equation which describe the spontaneous imbibition of water by oil-

saturated rock. The basic assumption underlying in the present investigation is that the

oil and water form two immiscible liquid phases and the later represents preferentially

wetting phase. The saturation of injected water is calculated by Homotopy Perturbation

Method for the nonlinear differential of imbibition phenomenon under the assumption

xiv

that saturation is decomposed into the saturation of different fingers. Numerical values

and graphical representation have been done by Microsoft Excel, Origin Lab and final

solution physically interpreted. It is concluded that the saturation of injected water is

increasing with different distance and time which is physically consistent with the real

phenomena.

Chapter 4 discusses the solution of Burger’s equation which arises into the

phenomenon of Fingering in double phase flow through homogenous porous media by

using Homotopy Perturbation Method. The basic assumptions underlying in the present

investigation is that there is uniform water injection into an oil saturated porous

medium. The injected water shoots through the oil formation and gives rise to

protuberance. The mathematical formulation yields to the non-linear partial differential

equation in burger’s equation form. The Saturation of injected water is calculated by

Homotopy Perturbation Method for Burger’s Equation of Fingering phenomenon under

the assumption that Saturation is decomposed into saturation of different levels. A

result clearly shows the saturation of water is increased with specific space. The

obtained results as compared with previous works are highly accurate. Also, Homotopy

Perturbation Method provides a continuous solution in contrast to finite difference

method, which only provides discrete approximations. Numerical values and graphical

illustration have been done by Origin Lab and Microsoft Excel 2010.

In Chapter 5, The mathematical model of fingero-imbibition phenomenon of Time –

Fractional type in double phase flow in homogenous porous media in the secondary oil

recovery process is studied. In this problem, we have considered the non-linear partial

differential equation of time – fractional type describing the spontaneous imbibition of

water by an oil-saturated rock (double phase flow in porous media). The fact that oil

and water form two immiscible liquid phases and water represents preferentially

wetting the phase are the basic assumption of this work. The Homotopy Perturbation

Method is used to obtain the saturation of injected water. We obtained the graphical

representation of a solution using Microsoft Excel 2010 and Origin Lab.

Chapter 6 describes the solution of one-dimensional mathematical model for counter-

current water imbibition phenomena arising fluid flow through in porous media with an

inclination and gravitational effect during the secondary oil recovery process. The

xv

problem has been solved by Homotopy Perturbation Method. Counter-current

imbibition phenomenon occurs due to the difference of viscosity of the injected fluid

and native fluid. The solution in the form of an approximate analytical series represents

the saturation of injected fluid in counter-current imbibition phenomenon with

inclination and gravitational effect solved by Homotopy perturbation Method.

Numerical values and graphical representation of the solution has been obtained using

Microsoft Excel 2010 and Origin Lab. The graphical representation shows the

saturation of water increases with distance X at different time T and different angle .

Chapter 7 deals with Uni – Dimensional vertical ground water recharge through

porous media has been solved using Homotopy Perturbation Method. The ground water

is recharged by spreading of the water in downward direction and the moisture content

of soil increases. The theoretical formulation of the problem gives a nonlinear partial

differential equation for the moisture content. This equation is solved by Homotopy

Perturbation Method. Numerical values and graphical representation of the solution has

been obtained using Microsoft Excel 2010 and Origin Lab. It is concluded that the

moisture content of soil increases with the depth Z and increasing time T.

To sum up, the subject matter of this thesis is to provide research to make an original

contribution and to develop an interpretation of known facts thereby extending the

scope of mathematical study in this area. The solutions obtained in this thesis would be

useful in determining the amount of water required for injection in the secondary oil

recovery process and useful for increasing in the fertility of soil in groundwater

recharge process. Many important analysis and modification are still possible but this is

our humble attempt to make some contribution in this field.

xvi

ACKNOWLEDGEMENT

My name appears on the cover of this thesis, but a great many people have contributed

to its production. I owe my gratitude to all those people who have made this thesis

possible and because of whom my research experience has been one that I will cherish

forever.

First and foremost I would like to thank Almighty, who has given me the strength and

patience during this research work. I am sure without his sheer blessing, it is impossible

for me to complete this research work. He has always showered his choicest blessings

on me, to enable me, for the work.

My deepest gratitude is to my supervisor, Dr. Shailesh S. Patel Professor and Head of

ASH Department, GIDC Degree Emceeing College, Abrama, Navsari. I am deeply

indebted to him for his valuable suggestions, constructive criticism, watchful interest

and invaluable guidance throughout the work and above all for inculcating a spirit of

independent research in me. His guidance, encouragement, during the preparation of

work has motivated me to a great extent. The completion of this thesis would not have

been possible without substantial guidance of my research supervisor. I am really

indebted to Shailesh Sir for his intellectual and emotional support that made possible

this work to survive.

I would also like to acknowledge Dr. Jayesh Dodhiya, Associate Professor, SVNIT,

Surat and Dr. Amit Parikh, Principal, Mehsana Urban Institute of Science, who have

always been there to listen the difficulties and give suggestions. I am deeply grateful to

them for the long discussions that helped me sort out the technical and methodological

details of my work.

A special thanks to my family, to whom this thesis is dedicated to, who has been a

constant source of love, concern, support and strength. My beloved husband Mr. Nimit

Raval who always guides and supports me when there is no one to answer my queries.

xvii

Words cannot express how grateful I am to my father Mr. Kishorbhai C. Desai, who is

not with me but he is always watching me and giving me blessing each and every

moment, my mother Daxaben K. Desai who helped me in every difficult situations, My

brothers Mr. Mehulbhai Mehta and Meet Gohil, My sisters Vipra Mehta and Devanshi

K. Desai, My in laws Mr. Jayeshbhai Raval and Mrs. Jayanaben Raval, for all of the

sacrifices that they have made only for my success. Your prayer for me was what

sustained me thus far.

I would also like to thank all of my friends and my staff members who supported me in

writing specially Dr. Nisha Parekh, who incanted me to strive towards my goal. Many

friends have helped me stay sane through these difficult years. Their support and care

helped me to overcome setbacks and stay focused on my study. I greatly value their

friendship and I deeply appreciate their belief in me.

I would also like to thank our Principal Dr. N.D. Sharma, my dear colleagues of GIDC

Degree Engineering College, Navsari for their support and encouragement.

Besides this, several people have knowingly and unknowingly helped me in the

successful Completion of this project. I doubt that I will ever be able to convey my

appreciation fully, but I owe such people my eternal gratitude.

Mansi K. Desai

xviii

CONTENTS

Abstract xii

Acknowledgement xvi

List of Symbols xxi

List of Figures xxiii

List of Tables xxv

1. Introduction

1

1.1 Introduction……………………………………………………... 2

1.2 Outline of the thesis....................................................................... 3

1.3 Brief description on the state of the art of the research topic….... 6

1.4 Definition of the Problem……………………………………….. 8

1.5 Objective of the work……………………………………………. 9

1.6 Original contribution by the thesis………………………………. 9

1.7 Methodology of Research and Results………………………….. 10

1.8 Achievements from the thesis……………………….................. 10

1.9 Conclusion………………………………………………………. 11

2. Fundamentals of Fluid Flow through Porous Media and Homotopy

Perturbation Method

12

2.1 Fluid flow through porous media……………………………….. 13

2.2 Porous Media……………………………………………………. 14

2.3 Types of Fluid Flow and porous matrix properties ……............ 15

2.4 Classification of Fluid…………………………………………... 17

2.5 Density…………………………………………………………... 18

2.6 Porous Media Properties………………………………………… 19

2.7 Darcy’s law…………………………………………………….... 24

2.8 Limitations of Darcy’s law……………………………………… 27

2.9 Brief Descriptions of Homotopy Perturbation Method………… 27

2.10 Solution of Linear Equation by Homotpy Perturbation Method.. 30

xix

3. Solution of Imbibition Phenomenon by Homotopy Perturbation

Method

35

3.1 Introduction……………………………………………………... 36

3.2 Mathematical Formulation of the Problem…………………….... 37

3.3 Solution by Homotopy Perturbation Method………………….... 40

3.4 Numerical Values and Graphical Presentation………………..... 42

3.5 Conclusion………………………………………………………. 44

4. Mathematical Modelling of Burger’s Equation of Instability

Phenomenon by Homotopy Perturbation Method

45

4.1 Introduction…………………………………………………….... 46

4.2 Mathematical Formulation of the Problem……………………… 47

4.3 Solution by Homotopy Perturbation Method……………………. 51

4.4 Numerical Values and Graphical Presentation………………...... 53

4.5 Conclusion..................................................................................... 54

5. Simulation of the Fingero-Imbibition Phenomenon in Double

Phase Flow in Porous Media solved by Homotopy Perturbation

Method

55

5.1 Introduction…………………………………………………….... 56



5.4 Numerical Values and Graphical Presentation………………...... 64

5.5 Conclusion………………………………………………………. 65

6. Counter-current Imbibition Phenomenon with Effect of Inclination

and Gravitational solved by Homotopy Perturbation Method

66

6.1 Introduction……………………………………………………… 67



6.4 Numerical Values and Graphical Presentation………………….. 73

6.5 Conclusion………………………………………………………. 80

xx

7. Ground Water Recharge in Vertical Direction in an Uni

Dimensional thorough Porous Media solved by Homotopy

Perturbation Method

80

7.1 Introduction……………………………………………………… 80



7.4 Numerical Values and Graphical Presentation………………….. 87

7.5 Conclusion………………………………………………………. 90

7.6 Utilities of the Problem…………………………………………. 90

References 91

List of Publications 100

xxi

LIST OF SYMBOLS

Permeability of the porous media

Relative permeability of injected fluid (water)

Relative permeability of native fluid (oil)

Porosity of the porous medium

q Flow rate

Density

Constant viscosity of oil

Constant viscosity of water

Pressure of injected fluid (water)

Pressure of native fluid (oil)

Darcy velocity of injected fluid (water)

Darcy velocity of native fluid (oil)

Saturation of injected fluid (water)

Saturation of native fluid (oil)

Acceleration due to gravity

Density of injected fluid (water)

Density of native fluid (oil)

Constant co-efficient

Viscosity of injected fluid (water)

Viscosity of native fluid (oil)

Capillary pressure

Inclination angle

Linear operator

Embedding parameter

Distance

Dimensional less Variable of distance

Time

T Dimensional less Variable of time

Depth

xxii

LIST OF FIGURES

2.1 Categorization of the fluid flow through porous media………........... 13

2.2 Representation of porous media……………………………………... 14

2.3 Single phase flow……………………………………………………. 16

2.4 Multi-phase flow…………………………………………………….. 16

2.5 Different types of porosities………………………………………..... 19

2.6 Fluid passing through solid………………………………………….. 20

2.7 Representation of permeability……………………………………… 20

2.8(i) Representation of saturated flow……………………………………. 21

2.8(ii) Representation of saturated flow……………………………………. 22

2.9 Representation of unsaturated flow…………………………………. 22

2.10 Contact angles of different phase……………………………………. 23

2.11 Darcy’s law…………………………………………………………... 25

2.12 Graph of for η = 0.05 ,0 ≤ x ≤1, 0 ≤ t ≤1……………………. 33

2.13 Graph of for η = 0.1 ,0 ≤ x ≤1, 0 ≤ t ≤1……………………... 34

3.1 Representation of imbibition phenomenon………………………….. 36

3.2 Schematic representation of imbibition phenomenon……………….. 36

3.3 Graphical representation of versus distance X for different

time T = ……………………………………….

43

4.1 Representation of fingering phenomenon…………………………… 46

4.2 Schematic representation of fingering Phenomenon………………… 46

4.3 Graphical representation of for distance different values

for time …………………………………………………………...

53

4.4 Graphical representation of for distance different values

for time …………………………………………………………...

54

5.1 Representation of Fingero -Imbibition phenomenon………………... 56

5.2 The Fingero – Imbibition phenomena in fractured reservoir………... 57

5.3 Graphical representation of versus distance X for

xxiii

different Time T……………………………………………………... 64

5.4 Graphical representation of versus Time T for Distance

X……………………………………………………...........................

65

6.1 Schematic diagram of the problem…………………………………... 68

6.2 Saturation versus X and Time T for ………………………. 74




6.6 Saturation versus X and Time T for ……………………… 77




7.1 Representation of groundwater recharge phenomenon……………… 82

7.2 Moisture content θ vs. time T and depth Z at γ = 0.05…………….... 88

7.3 Moisture content θ vs. time T and depth Z at γ = 0.05…………….... 88

7.4 Moisture content θ vs. time T and depth ξ ………………………….. 89

xxiv

LIST OF TABLES

2.1 Table of for different values of x and t for …………….. 32

2.2 Table of for different values of x and t for ……………….. 33

3.1 Saturation of injected water for different distance X and time

T…………………………………………………………………………...

43

4.1 Saturation of injected water for different values of and ……... 53

5.1 Saturation of injected water for different values of X and Time

T…………………………………………………………………………...

64

6.1 Numerical values of saturation of injected water at ………... 74




7.1 Numerical values table of moisture content Z at different Time T and =

0.05………………………………………………………………………...

87

CHAPTER 1

INTRODUCTION

2

CHAPTER 1

Introduction

1.1 Introduction:

The present research work covers the study of fluid flow in porous media. The theory of

fluid flow in porous media plays a very significant role in many branches of Engineering

including material science, petroleum industries, soil mechanics, hydrology, filtration,

water resource engineering, etc. Petroleum engineering observes the movement of oil and

natural gas in the reservoirs, drilling. Hydrology studies the water in the earth and sand

structures, water bearing formations and drinking water purification in filter beds. The

main concern of filtration studies is to determine how fluid moves through the porous

structure leaving behind unwanted material. The mathematical formulations of the physical

phenomena lead to the partial differential equations whose solutions are obtained using the

Homotopy perturbation method with appropriate conditions.

Here, the researcher has tried to find out an exact and approximate analytical solution of

current interest problems in two phase flow system through homogeneous or heterogeneous

porous media. Theoretical research in fluid flow in porous media has received increased

attention during the past five decades. We have observed these models with the basics of

porous media. We have studied five different models in detail and solved them

approximately. The models of the various phenomena have been solved by Homotopy

Perturbation method which gave an exact solution and an approximate analytical solution

of the problems.

To forecast the problems like the saturation of the injected water have been solved by

Homotopy Perturbation method with the help of appropriate initials and boundary

conditions. The common study of the subject is fluid flow in porous media and it will play

a significant role in future also.

3

Outline of the Thesis

1.2 Outline of the Thesis:

The thesis has been divided into seven chapters as described below:

Chapter ONE “Introduction”:

The introductory chapter provides general ideas about fluid flow in porous media as well

as the brief introduction of the problems that the researcher has tried to study. It also

includes objectives, scope, significant as well as the contribution that the researcher will do

through this research. It also includes discussion on the method through which the problem

was formulated.

Chapter TWO “Fundamentals of Fluid Flow in Porous Media and Homotopy

Perturbation Method”:

Second chapter discusses the fundamental laws and the basic terminology of the theory of

porous media and it has also included a brief description of the Homotopy Perturbation

Method. It is chosen to build up a stronger Structure in a logical manner to provide

knowledge of fundamentals of porous media, which is an essential part of the study to have

a better understanding of flow in porous media. This chapter deals with the necessity of the

study of flow through porous media with basic definitions, physical properties of porous

media, Darcy’s law and limitation of Darcy’ Law. This chapter also includes an

explanation of the difference between miscible and immiscible flows. The topics

considered are meant to be a general background for the researcher unfamiliar with fluid

flow through porous media. Brief description of Homotopy Perturbation Methods has been

also included in this chapter and linear equation solved using this method.

Chapter THREE “Solution of Imbibition Phenomenon by the Homotopy

Perturbation Method”:

This chapter covers the mathematical model of Imbibition Phenomenon of immiscible

fluid flow in a homogeneous porous medium. The imbibition phenomenon equation has

been solved with suitable initial and boundary conditions using Homotopy Perturbation

4

Introduction

Method. The theoretical studies involved in the development of the differential equation

which describe the spontaneous imbibition of water by oil-saturated rock. The basic

assumption underlying in the present investigation is that the oil and water form two

immiscible liquid phases and the latter represents preferentially wetting phase. The

saturation of injected water is calculated by Homotopy Perturbation Method for the

nonlinear differential of imbibition phenomenon under the assumption that saturation is

decomposed into the saturation of different fingers. Numerical Values and graphical

representation has been done by Microsoft Excel 2010 and Origin Lab and final solution

physically interpreted. It is concluded that the saturation of injected water is increasing

with different distance and time which is physically consistent with the real phenomena.

Chapter FOUR “Mathematical Modelling of Burger’s Equation of Instability

Phenomenon solved by Homotopy Perturbation Method”:

It discusses the solution of Burger’s equation which arises into the phenomenon of

Fingering in double phase flow through homogenous porous media by using Homotopy

Perturbation Method. The basic assumptions underlying in the present investigation is that

there is uniform water injection into an oil saturated porous medium. The injected water

shoots through the oil formation and gives rise to protuberance. The mathematical

formulation yields to non-linear partial differential equation in burger’s equation form. The

saturation of injected water is calculated by Homotopy Perturbation Method for Burger’s

equation of fingering phenomenon under the assumption that Saturation is decomposed

into saturation of different levels. The obtained results as compared with previous works

are highly accurate. Also, Homotopy Perturbation Method provides the continuous solution

in contrast to the finite difference method, which only provides discrete approximations.

Numerical values and graphical illustration has been done by Microsoft Excel 2010 and

Origin Lab.

Chapter FIVE “Simulation of the Fingero-Imbibition Phenomenon in Double

Phase Flow in Porous Media solved by Homotopy Perturbation Method”:

5

Outline of the Thesis

This chapter discusses the fingero-imbibition phenomenon of Time – Fractional type in

double phase flow in homogenous porous media in the secondary oil recovery process is

studied. In this problem, we have considered the non-linear partial differential equation of

time – fractional type describing the spontaneous imbibition of water by an oil-saturated

rock (double phase flow in porous media). The fact that oil and water form two immiscible

liquid phases and water represents preferentially wetting the phase are the basic

assumption of this work. The Homotopy Perturbation Method is used to obtain the

saturation of injected water. We got an approximate analytical solution and draw a graph

using Origin Lab, from that we conclude that the saturation of water decrease with

different space X and different time T.

Chapter SIX “Counter-current Imbibition Phenomenon with effect of inclination and

gravitational solved by Homotopy Perturbation Method”:

This chapter six describes the solution of one-dimensional mathematical model for

counter-current water imbibition phenomena arising fluid flow through in porous media

with the inclination and gravitational effect during the secondary oil recovery process. This

problem has been solved by Homotopy Perturbation Method. Counter-current imbibition

phenomenon occurs due to the difference of viscosity of the injected fluid and native fluid.

The solution in the form of an approximate analytical series represents the saturation of

injected fluid in counter-current imbibition phenomenon with the inclination and

gravitational effect solved by Homotopy perturbation Method. This equation is solved by

Homotopy Perturbation Method. Numerical values and graphical representation of the

solution has been obtained using Microsoft Excel 2010 and OriginLab. The graphical

representation shows the saturation of water increases with different distances at

different time T and different angle .

Chapter SEVEN “Ground Water Recharge in the vertical direction in a Uni

dimensional Thorough Porous Media solved by Homotopy Perturbation Method”:

Chapter seven deals with Uni – Dimensional vertical ground water recharge through

6

Introduction

porous media have been solved using Homotopy Perturbation Method. The ground water is

recharged by spreading of the water in downward direction and the moisture content of soil

increases. The average diffusivity coefficient over the whole range of moisture content is

regarded as constant and a parabolic variation of permeability with moisture content is

assumed. The theoretical formulation of the problem gives a nonlinear partial differential

equation for the moisture content. This equation is solved by Homotopy Perturbation

Method. Numerical values and graphical representation of the solution has been obtained

using Microsoft Excel 2010 and OriginLab. It is concluded that the moisture content of soil

increases with the depth Z and increasing time T.

At last, the researcher has enlisted the references which have been used to base this study

on.

1.3 Brief Description of the State of the Art of the Research Topic:

The basic rout of the fluid flow in porous media is the historical development of the ground

water. It is well known that an interconnect pores of Homogenous or Heterogeneous

porous media constitute capillary with irregular walls and fluid flowing in the

interconnected capillaries is called fluid flow in porous media. Scheidegger [2] is the

inventor of the fluid flow in porous media. The series of fluid flow in porous media had

been developed between 1856-1955, which helped to establish the principals of

groundwater evaluation. Porous media is very important for many engineering branches

like petroleum industry, Chemical engineering, and biomechanics. The base of

mathematical investigation of fluid flow in porous media may be initiated in 1856 when

the French hydraulic engineer, Darcy (1803-1858) [42], attempted to develop the water

supply project for the city of Dijon(France), carried out experiment, pipes filled with sand

and to give relation. The fluid flow in porous media is divided into two parts, single phase

flow and multiphase flow. In recent years, researches of single and double flow are

increasing continuously due to the growth of technological applications. Among these the

nineteenth century saw the development of the basis for the quantitative description of

ground water motion.

7

Brief Description of the State of the Art of the Research Topic

Due to many different applications, the study about hydrodynamics in porous media

became necessary for engineering. The hydrodynamic is classified into two parts that is

single phase flow and multiphase flow.

When a phenomenon occurring due to the difference in wetting abilities of the phases is

said to be an imbibition phenomenon. Imbibition - phenomenon is applicable in an oil

recovery process, food industry, biological sciences, surface chemistry, composite

materials, textiles and construction. The phenomenon of imbibition has been formally

discussed by many others; particularly noticeable contribution has been made by Graham

and Richardson. Verma has suggested with suitable conditions that the phenomenon of

fingering and imbibition occur simultaneously in displacement processes provided that

displacing fluid is preferentially wetting and less viscous.

The problem of the fingering phenomenon frequently occurs in petroleum technology.

When fluids which are having greater viscosity flowing in a porous medium is displaced

by another fluid of lesser viscosity then, instead of regular displacement of the whole front,

protuberances take place which shoots through the porous medium at a relatively very high

speed, and fingers are developed during this process. This phenomenon is known as

fingering (instability) phenomenon. It is important in the secondary oil recovery process of

petroleum technology. Scheidegger [2] has obtained condition for fingering by assuming

the Muskat Aronofsky model of oil water displacement.

Fingero-imbibition phenomenon happens in the secondary oil recovery process. When

fingering and imbibition phenomena occur simultaneously is said to be fingero-imbibition

phenomenon and it has been discussed by Verma [21]. An experimental study of co-

current and counter-current flow in natural porous media is done by Bourblaux and

Kalaydjian [28].

Infiltration process, in which the water on ground surface fills into soils and passing into

solid rocks in pore spaces and gaps. Many researchers have studied the groundwater

infiltration phenomena with different aspects [57, 70, 78, 83, 97].

The groundwater recharge problem is related to hydrology, environment, engineering, soil

mechanics, water resource, etc. The flow of water in unsaturated soil has been considered

8

Introduction

with some specific assumptions. The problem of groundwater flow has been discussed by

many researchers with different aspects, like as Klute [58] reduced the diffusion equation

to an ordinary differential equation and applied a forward integration and iteration method,

Verma [11] obtained the solution of a one dimensional groundwater recharge for constant

diffusivity and linear conductivity by Laplace transform, Joshi [73] obtained the solution

of Uni dimensional vertical groundwater recharge by group theoretical approach. Patel

and Mehta [54] analytically discussed the phenomenon of imbibition in two immiscible

fluids flow in porous media and found an exact solution of the partial differential equation

arising in this phenomenon. Desai [75] had discussed an exact solution of the linear and

non-linear diffusion equations are obtained by the Homotopy perturbation method. Gupta

[102] had applied the Homotopy perturbation transform method for solving heat and like

equations.

The main purpose of the present work is to give a standard presentation of the

mathematical modeling for physical phenomena to arise in single phase flow and in double

phase flow in homogenous porous media and also the form of a nonlinear partial

differential equation.

1.4 Definition of the Problem:

Many physical phenomena have been solved by different authors with different

mathematical methods. Phenomena like an imbibition phenomenon, fingering

phenomenon, fingero–imbibition Phenomena, the co-current imbibition phenomenon, the

counter-current imbibition phenomenon, which occur in the fluid flow through the porous

medium. The researches of different phenomena have focused on four principal aspects of

fracture flow: Development of conceptual models, Development of Exact, approximate

analytical and numerical solutions, Description of fracture hydraulic characteristics in

static and Deforming media. The mathematical models of flow in porous medium have

been studied with some specific assumptions in the present work. The aim of the present

study was to investigate the behavior of the saturation of injected water in different

physical phenomena which are arising in the fluid flow through homogeneous as well as

9

Problem Definition

the heterogeneous porous medium. An infiltration phenomenon in the unsaturated porous

medium has been studied by Homotopy perturbation method. The objective of the work is

to study the behavior of the moisture content of the soil in the groundwater flow when

excess water on the ground surface is spreading in the vertical direction through the

unsaturated porous medium.

1.5 Objectives of the Work:

The main goal is to study the versatility of phenomena arising in multiphase flow through

porous media and apply the particular method to find the injected water saturation which

helps to forecast the amount of oil recovered and also to obtain the solution of one

dimensional nonlinear partial differential equation by the Homotopy perturbation method.

To study the saturation of the injected water which helps us to predict the amount of

water required to inject for recovering oil.

To study the moisture content this helps to forecast the amount of water spread in the

unsaturated soil.

This type of mathematical model is useful for predicting oil recovery from petroleum

reservoir and for predicting moisture content increase in unsaturated soil. The scope of the

current work is to study the problems of fluid flow in porous media.

1.6 Original Contribution to the Thesis:

The main contribution of the thesis is to obtain the solution of one dimensional

mathematical modeling of different phenomena arising in fluid flow in porous media by

the Homotopy perturbation method. Exact and approximate analytical solutions have been

solved to find saturation of the injected water with respect to time and space with suitable

initial and boundary conditions. The goal is to obtain the saturation distribution of the

injected water which may help to predict the amount of oil recovered and moisture content of

soil increases.

10

Introduction

The solution of various phenomena like co-current imbibition phenomenon, the counter-

current imbibition phenomenon with inclination and gravitational effect, Instability

(fingering) phenomenon of time fractional type which arises in fluid flow through porous

media and groundwater recharge are discussed using the Homotopy perturbation method

with suitable initial and boundary conditions. The study will surely provide new ideas as

well as solutions to solve various problems arise in petroleum engineering, groundwater

porous media, etc.

1.7 Methodology of Research and Results:

We observed various physical phenomena of fluid flow through porous media. To

understand these phenomena, we have referred many articles related to fluid flow through

porous media and we have done a comparative analysis to find out research gap and

problem statement. The literature survey helped to find out some models of phenomena

which the researcher tried to solve by using the mathematical method. Microsoft Excel

2010, Mat lab and Origin Lab have been used as a tool to draw graphs and make tables for

nonlinear boundary value problems. We have achieved the exact and approximate

analytical series solutions of nonlinear partial differential equations arising during oil

recovery process with appropriate conditions using Homotopy Perturbation Method.

We have studied various problems of fluid flow through porous medium after keen

observation of the research gap and the mathematical model for different physical

phenomena arising in two phase flow in porous medium during an oil recovery process.

The infiltration phenomenon through the unsaturated porous medium has been discussed.

The problem of groundwater recharge in the vertical direction is discussed. The water of

ground is recharged by spreading water in vertical direction which increases the moisture

content of the soil.

1.8 Achievements from the Thesis:

The mathematical models of different phenomena which arise in fluid flow through porous

media have been studied in detail and their solutions are obtained under suitable initial and

boundary conditions using the Homotopy perturbation method.

11

Achievements from the thesis

The following mathematical models are studied and solved:

Imbibition phenomenon in homogenous porous media.

Instability phenomenon in homogeneous porous media.

Fingero-Imbibition phenomenon of time fractional type in homogenous porous media.

Counter-current imbibition phenomenon with the effect of inclination and gravitational

in porous media.

Uni-dimensional vertical groundwater recharges through porous media.

1.9 Conclusion:

The various mathematical models of different phenomena like imbibition phenomenon, the

counter-current imbibition phenomenon, the instability (fingering) phenomenon, the

finger-imbibition phenomenon have been solved successfully using Homotopy

Perturbation Method. The phenomenon of uni dimensional vertical groundwater recharge

through unsaturated porous medium has been studied too. The analytical solution of the

type approximate solution and exact solution have been solved using appropriate initial and

boundary conditions. Homotopy Perturbation Method has been used to solve all the above

problems. The solutions are explained graphically as well as numerically using Microsoft

excel 2010, Mat lab and Origin Lab and hence get the exact idea of saturation of water in

oil recovery process and observed that saturation of water increases then oil come out of

with the water at each different space and time. The solutions will be useful to determine

the amount of water required for injection and for the prediction of the oil recovered.

Through the study, the researcher came to the conclusion that the saturation of water level

increases with time and space as we inject the water and saturation of water decreased with

time and space in some examples as per our assumptions. At the same time, the moisture of

soil also gets increased when the saturation of water gets increased.

CHAPTER 2

FUNDAMENTALS OF FLUID FLOW IN

POROUS MEDIA AND HOMOTOPY

PERTURBATION METHOD

13

CHAPTER 2

Fundamentals of Fluid Flow in Porous Media and

Homotopy Perturbation Method

2.1 Fluid Flow through Porous Media:

The interconnected pores in homogenous and heterogeneous porous media comprise

capillaries with uneven walls and fluid flowing through the interconnected capillaries is

called fluid flow through porous media. The common phenomenon, Fluid flow through

porous media is used for many fields in science and engineering. In an oil recovery

process, the common problem is the amount of unrecovered oil left in oil reservoirs is

investigated by many new and traditional recovery processes.

FIGURE 2.1: Categorization of the Fluid Flow through Porous Media.

14

Fundamentals of Fluid Flow in Porous Media and Homotopy Perturbation Method

The various applications of flow in porous media allowed investigating the porous media

and gave details about the properties of porous media. The figure (2.1) is the chart of

categorization of the flow in porous media in various branches of science, engineering and

other branches of the real world.

2.2 Porous Media:

Solid materials consist of the interconnected pores in it, generally known as a porous

medium and that pore space can be filled with more than one fluid like oil, water and gas.

Examples of the porous medium are soil, sand, cemented sandstone, bread, foam rubber,

lungs or kidneys. Porous medium refers to a solid body which is having void space (pore

space) in it. Pores are the complex network of void spaces of various sizes and shape

circulated more or less often throughout the substance and that substance is called Porous.

A porous medium is a substance having voids (pores). The holes (voids or pores) usually

pored with a fluid like liquid or gas. The porous medium is considered as a solid body with

pores.

The nonsolid space in a solid body is called the pore space which can see in below figures.

FIGURE 2.2: Representations of Porous Media

15

Porous media

Porous media is divided into two parts, one is consolidated and other is an unconsolidated.

Examples of an unconsolidated medium are glass beads, catalyst pellets, beach sand,

column packing, soil, gravel and packing, charcoal.

Examples of consolidated media are naturally occurring in rocks like the sandstones and

limestones. Cement, concrete, paper, bricks, clothes, etc. are made by humans. Human

lungs and wood are also examples of consolidates media.

2.3 Types of Fluid Flows and Porous Matrix Properties:

2.3.1 Steady and Unsteady Flow:

Fluid flow has two types of flow the same as steady or unsteady, depend on fluid

properties. The flow which is not varying with time at a point is a steady flow and the flow

which is varying with time at a time is unsteady.

2.3.2 Uniform and Non-uniform flow:

Flow is known as uniform flow, when the velocity of flow remains unchanged either in

direction or in magnitude at any point in a flowing fluid, for a given time.

Flow is known as non-uniform, when the velocity of the flow changes with different points

in a flowing fluid, for a given time.

2.3.3 Compressible and Incompressible flow:

A fluid is classified into two parts that are compressible or incompressible, depending upon

the level of variation in density during fluid flow. The density defers with each points then

that type of flow is known as compressible flow. Gases are highly compressible fluid.

The density remains unchanged with each point then that type of fluid is called an

incompressible flow. When an observing rockets spacecraft involve high-speed gas flows

is an incompressible flow.

16


2.3.4 Single Phase Flow:

It is a concern with the flow of a single fluid through the porous medium. Darcy’s law is

expressed Single Phase flow accurately.

FIGURE 2.3: Single Phase Flow

2.3.5 Multi Phase Flow:

Multiphase system is when the void space is occupied by more than two fluids that are

immiscible with each other, i.e. they maintain a distinct boundary between them (e.g. water

and oil).

FIGURE 2.4: Multi Phase Flow

17

Classification of Fluids

2.3.6 Miscible Flow:

Single–phase system the void space of the porous medium is filled by a single fluid (e.g.

water) or by several fluids completely miscible with each other (e. g. freshwater and salt

water). There may only be one gaseous phase since gases are always completely miscible.

Miscible fluids are considered dealing with pollutant transport, enhanced oil recovery in

petroleum engineering or chemical grouting.

2.3.7 Immiscible Flow:

Immiscible fluids are insolvable with each other. Immiscible liquids cannot be

homogeneously mixed because they are very different in overall net molecular polarity.

For Example, benzene and water, kerosene and water, etc.

Immiscible fluid is appropriate for reservoir simulation or unsaturated soils.

2.4 Classification of Fluids:

2.4.1 Ideal Fluid:

An ideal fluid has no viscosity and it is incompressible. Ideal fluid is an imaginary fluid as

all the fluid has little or more viscosity. Imaginary fluid is the type of ideal fluid because

this type of fluids has little or more viscosity.

2.4.2 Real Fluid:

A real fluid has viscosity. All fluids in real practice are real fluids. Real fluids are actually

present in nature and possess properties like viscosity, compressibility, etc. These types of

fluid are also known as viscous fluids.

2.4.3 Newtonian Fluids:

In this fluid, the shear stress is directly proportional to the rate of the shear strain is known

as a Newtonian fluid. Means, the fluids which obey the newton’s law of viscosity are

known as Newtonian fluids. Examples: Water, oil, Kerosene.

18


2.4.4 Non - Newtonian Fluids:

In this fluid, the shear stress is not proportional to the rate of shear strain is a non –

newtonian fluid. Means, the fluid which does not follow the newton’s law of viscosity are

known as non- newtonian fluids. Examples: Milk, Starch, Ink, Honey, pints, toothpaste.

2.4.5 Homogenous Fluid:

Homogenous Fluid is a fluid which has constant density. The homogeneous fluid is a

single fluid phase which may be a gas or a liquid or a mixture of fluids which are miscible

to each other throughout the process. Examples: Water and Salt.

2.4.6. Heterogeneous fluid:

The heterogeneous fluid is a fluid which has density varies to each point. Two immiscible

like Liquid and Gas, if they are present as a dispersed mixture. There are two immiscible

fluids but separated with constant densities by an interface. Examples: Oil and water, Gas

and oil.

2.5 Density:

Density is a characteristic property of Fluids. The density of a substance is the relationship

between the mass of the fluid and how much space it takes up that is volume. Generally, it

changes with temperature ( ) and pressure ( ) according to relation,

( ) ( )

In the physics system (M, L, T), the dimensions of density are M and in

the technical system (F, L, T).

The fluid density is very useful for measuring of Reynolds number and other parameters

which are important to study of fluid flow through porous media.

19

Porous Media Properties

2.6 Porous Media Properties:

2.6.1. Porosity:

Porosity is composed of the tiny spaces in the rock that hold the fluid. Porosity is a

dimensionless quantity. Mathematically, porosity is the ratio of the void space to the bulk

volume.

.

FIGURE 2.5: Different Types of Porosities

The average porosity or simply porosity of a sample of a porous medium is the ratio of

interconnected pore volume to the total volume . The volume includes solid as

well as pore volume. It is denoted by . (Change)

2.6.2 Permeability:

Permeability is a function that permits the oil and gas to flow via the rock. One of the most

important of the oil field is the study of the movement of fluid through porous media in the

secondary oil recovery process, and that is the characteristics of its permeability.

20


Permeability is mostly known as absolute permeability and it depends on the geometry of

the medium only. The relative permeability depends on the fluid flow through porous

media in the porous matrix.

FIGURE 2.6: Fluid Passing through Solid

When it is hard to flow through a rock then takes more pressure to squeeze, that the rock

has low permeability. If oil and water like fluid pass through the rock easily then it has

High Permeability. Permeability shows how easily water is passed through solid or rock.

FIGURE 2.7: Representation of Permeability [27]

21

Saturation

2.6.3 Saturation:

When more than one fluid filled in porous media such as liquids or gases then the

saturation at that point with respect to a certain fluid is defined as the fraction of the pore

volume of the porous medium filled by that certain fluid representative elementary volume

(REV) around the considered point.

Fluid saturation =

The sum of the saturation is unity. Water and oil saturation is written as

Here and are the saturations of the native fluid and the injected fluid respectively.

FIGURE 2.8(i): Representation of Saturated Flow

22


FIGURE 2.8(ii): Representation of Saturated Flow

FIGURE 2.9: Representation of Unsaturated Flow

23

Wettability

2.6.4 Wettability:

It is very important to understand the wettability of an injected fluid when it is reserved in

reservoir rock during an oil recovery process in homogenous and heterogeneous porous

media. Wettability is an important part of the production of gas and oil as it is a major

factor in the fluid flow processes in the reservoir rock.

To understand of wettability, take a liquid drop on a solid plane surface which can take a

different shape. The Shapes like a pearl or a flat depends on the wettability of the solid

surface.

A fluid angle is known as the wetting phase and is known as the non-

wetting phase. The figure shows the wettability of water and oil.

FIGURE 2.10: Contact angles of Different Phases

2.6.5 Capillary Pressure:

Two immiscible fluids which are in contact that interstices of a porous medium, a

discontinuity in pressure arises across the interface to separate them. The pressure in

between the wetting and non-wetting fluid is called Capillary Pressure.

By the definition of capillary pressure,

Where pressure of the wetting phase and is pressure of the non-wetting phase.

24


2.6.6 Viscosity:

The viscosity of a fluid is a measure of its resistance to deformation at a given rate. It is

direct contact to the informal concept of "thickness" in liquids. Viscosity is the property

which resistance to the movement of one layer of the liquid (fluid) to another adjacent

layer of the liquid (fluid). Property of fluid E.g. Syrup viscosity is higher than water.

Viscosity of fluid is very important for any transportation phenomenon in multiphase flow.

Different fluids have different viscosity and because of the difference in viscosity,

multiphase flow is divided in two parts: miscible or immiscible. Hence it is important

property of fluid flow through porous media. There are two types of viscosity: One is

Dynamic viscosity which is measured in Pascal seconds, and Second is kinematic viscosity,

measured in metres per second squared.

The word viscous comes from the Latin root viscum, meaning sticky.

2.7 Darcy’s Law:

The French civil engineer Henry Darcy formulated the basic law of the flow of fluid

through porous media which is known as Darcy’s Law on the basis of his experiments on

vertical water filtration through sand bends. The flow of ground water or any other fluid

moving through rock (oil, water) is governed by an empirical law, one derived from

experimental observation, not from theory. This expression for ground water flow is

known as Darcy’s Law. Darcy became very knowledgeable about the flow of fluids in

pipes. He also worked with the flow of fluids through pipes filled with sand. The sand

acted as a purification system. An example of a pipe which can be used to demonstrate

Darcy’s Law experiment:

https://en.wikipedia.org/wiki/Fluid

https://en.wikipedia.org/wiki/Drag_(physics)

https://en.wikipedia.org/wiki/Syrup

https://en.wikipedia.org/wiki/Water

https://simple.wikipedia.org/wiki/Fluid_dynamics

https://simple.wikipedia.org/wiki/Pascal_(unit)

https://simple.wikipedia.org/wiki/Kinematics

https://simple.wikipedia.org/wiki/Metres

https://simple.wikipedia.org/wiki/Second

https://simple.wikipedia.org/wiki/Square_(mathematics)

https://simple.wikipedia.org/wiki/Latin

25

Darcy’s Law

FIGURE 2.11: Darcy’s Law

Cylinder filled with sand. Circular area = = A.Water flows through the cylinder at a

constant rate Q, maintained by keeping ha constant (constant head). Head loss is produced

across the sand tube (L).

Darcy noted that if he doubled Q, that the head loss also doubled. This implies that there is

a direct relationship between Q and the gradient.

Q is proportional to the head loss divided by the flow length is called the hydraulic

gradient. Q as well is directly proportional to the cross section area. If a tube of larger

diameter is used, then a larger Q must be maintained to keep the same head.

We can write,

26


Darcy called this proportionality constant K, the permeability of the medium. K depends

on the size, shape, packing and orientation of the material in the sand tube. When these

portions are combined, we are left with Darcy’s Law to describe fluid flow through porous

media.

( )

Where,

Q = volumetric flow rate [ /s]

k = constant of proportionality [ /Pascal]

A = cross-sectional area of sample [ ]

L = length of sample [m]

= Measure the pressure at A

= Measure the pressure at B

The negative sign is used by convention to note that water is flowing from highest to

lowest hydraulic head.

Darcy’s Law states that fluid will flow through a porous media at a rate which is

proportional to the product of the cross sectional area through which flow can occur, or

the hydraulic gradient and the hydraulic conductivity. Hydraulic Conductivity is a term

which has replaced what Darcy called the permeability.

Permeability has units but in petroleum engineering it is conventional to use “Darcy”

units, defined by

27

Darcy’s Law

Q as well is directly proportional to the cross section area. If a tube of larger diameter is

used, then a larger Q must be maintained to keep the same head.

2.8 Limitation of Darcy’s Law:

Darcy’s Law established in certain circumstances: Laminar flow in saturated media, Flow

conditions of steady state, to consider the fluid homogenous, incompressible, kinematic

energy is being neglected. Still, because of its average character which is based on the

representative continuum and the little effect of other factors, the Darcy’s Law is used for

many situations which do not correspond to these primary assumptions:

Saturated flow and unsaturated flow;

Transient flow and steady State flow;

Flow in aquifers;

Homogenous system flow & Heterogeneous system flow;

Anisotropic media & Isotropic flow.

Darcy’s law is valid only for laminar flow condition in the soil mass. Coefficient of

Permeability. The coefficient of permeability is defined as the average velocity of flow that

will occur through the total cross-sectional area of the soil mass under a unit hydraulic

gradient.

2.9 Brief description of Homotopy Perturbation Method:

The idea of Homotopy Perturbation Method was first introduced by Liao in 1992.

Homotopy Perturbation Method is the straight forward method and it is a very convenient

method to solve linear and nonlinear problems. Homotopy Perturbation Method is the

mixture of Homotopy method and Perturbation method. Homotopy Perturbation Method

has been applied on the ordinary differential equation, nonlinear partial differential

equation and other fields also. We can directly apply this method to solve the nonlinear

partial differential equation except linearizing the problem and solve them easily.

28


Here we will first introduce a new perturbation technique which coupled with the

homotopy technique. In topology, the word "homo-topic" is defined as two continuous

functions from one topological space to another topological space. Mostly, a Homotopy

between two continuous function f and g from a topological space X to a topological space

Y is defined as a continuous function

And

( ) ( ) and ( ) ( )

The homotopy perturbation method is not depended on a small parameter. In topology, a

homotopy constructs which an embedding parameter considers as a small

parameter.

2.9.1. Concept of Homotopy

Homotopy describes as a continuous deformation in mathematics. e.g., The shape and size

of a coffee cup can continuously deform into the shape of a doughnut (paczki), a circle can

continuously deform into an ellipse but the shape of a teacup cannot be continuously

deformed into the shape of a circle. So the Homotopy interprets a relation among different

things in mathematics.

2.9.2 Concept of Perturbation Theory

Perturbation theory is a collection of different methods and it is used to find the

approximate solution to a problem for which the exact solution is solved in closed form.

Here, a parameter is introduced by using some strategy into the problem, so that the

confusing by adding parameter and the solution is to be determined to the original problem

as a power series in,

( ) ( ) ( )

29


2.9.3 Homotopy Perturbation Method

Consider the below function to explain the method,

( ) ( ) (2.1)

With boundary condition

(

) (2.2)

Here,

= Differential operator,

= Boundary operator,

( ) = Analytic function,

A will be divided into two parts which are linear and nonlinear, say ( ) and ( ).

Equation (2.1) can write,

( ) ( ) ( ) (2.3)

He’s constructed a homotopy ( ) which satisfies:

( ) ( ) ( ) ( ) ( ) ( ) (2.4)

Where is a homotopy parameter, and is an initial guess of the

equation (2.1) which satisfies the boundary conditions of the equation (2.4) then we get,

( ) ( ) ( ) & (2.5)

( ) ( ) ( ) (2.6)

30


In topological space, ( ) ( ) is known as deformation. ( ) ( ) is known as

Homotopic. The parameter monotonically increasing from zero to one (unit) to the

problem ( ) and deforms continuously to the original problem ( )

Now, is the embedding parameter which can consider as an expanding

parameter. The perturbation technique will be applied due to the fact that , is

considered as a tiny parameter.

The solution of (2.1) and (2.2) are written as,

(2.7)

Let here in equation (2.7) and we got an approximate solution of the equation,

(2.8)

Equation (2.8) is the proper solution of the equation (2.1) solved by Homotopy

Perturbation Method [85].

2.10 Solution of Linear Equation By Homotopy Perturbation Method:

The linear differential equation is,

(2.9)

Assuming initial condition is,

( ) ( ) (2.10)

And boundary conditions are

( ) ( ) (2.11)

31

Solution of Linear Equation by Homotopy Perturbation Method

To solve this problem we use Homotopy Perturbation Method.

Homotopy ( ) for equation (2.9) is define as

( ) ( ) *

+ + *

+ (2.12)

[ ]

[

]

Comparing the powers of P,

………………………………

Using equation (2.12) for other order of, we can solve the following results:

( ) ( ) ( )

( )

It is obvious that ( ) converges to the exact solution as increasing order of:

32


( ) ( ) ( ) (2.13)

Table and Graph represent the Homotopy perturbation method solution ( ) for

and respectively for and

TABLE 2.1: Table of ( ) for different values of x and t for

X/T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0 0 0 0 0 0 0 0 0 0

0.05 0.30886552 0.253588778 0.208205 0.170943 0.14035 0.115232 0.094609 0.077677 0.063776 0.063776 0.042991

0.1 0.587527526 0.482379475 0.396049 0.32517 0.266975 0.219195 0.145545 0.147758 0.121315 0.121315 0.081778

0.15 0.808736061 0.663998978 0.545165 0.447598 0.367493 0.301724 0.247725 0.203391 0.16699 0.16699 0.112567

0.2 0.950859461 0.780686976 0.64097 0.526257 0.432075 0.354748 0.291259 0.239134 0.196337 0.196337 0.13235

0.25 0.999999683 0.82103272 0.674095 0.553454 0.454404 0.373081 0.306312 0.251492 0.206483 0.206483 0.139189

0.3 0.951351376 0.781090855 0.641301 0.52653 0.432298 0.354931 0.29141 0.239257 0.196438 0.196438 0.132418

0.35 0.809671788 0.664767241 0.545796 0.448116 0.367918 0.302073 0.248012 0.203626 0.167184 0.167184 0.112698

0.4 0.588815562 0.483436995 0.396918 0.325883 0.26756 0.219676 0.180361 0.148082 0.121581 0.121581 0.081957

0.45 0.31037991 0.254832142 0.209226 0.171781 0.141038 0.115797 0.095073 0.078058 0.064088 0.064088 0.043202

0.5 0.001592653 0.001307621 0.001074 0.000881 0.000724 0.000594 0.000488 0.000401 0.000329 0.000329 0.000222

0.55 -0.30735035 -0.25234477 -0.20718 -0.1701 -0.13966 -0.11467 -0.09415 -0.0773 -0.06346 -0.06346 -0.04278

0.6 -0.586238 -0.48132073 -0.39518 -0.32446 -0.26639 -0.21871 -0.17957 -0.14743 -0.12105 -0.12105 -0.0816

0.65 -0.80779828 -0.66322903 -0.54453 -0.44708 -0.36707 -0.30137 -0.24744 -0.20315 -0.1668 -0.1668 -0.11244

0.7 -0.95036513 -0.78028112 -0.64064 -0.52598 -0.43185 -0.35456 -0.29111 -0.23901 -0.19623 -0.19623 -0.13228

0.75 -0.99999715 -0.82103064 -0.67409 -0.55345 -0.4544 -0.37308 -0.30631 -0.25149 -0.20648 -0.20648 -0.13919

0.8 -0.95184088 -0.78149275 -0.64163 -0.5268 -0.43252 -0.35511 -0.29156 -0.23938 -0.19654 -0.19654 -0.13249

0.85 -0.81060546 -0.66553382 -0.54643 -0.44863 -0.36834 -0.30242 -0.2483 -0.20386 -0.16738 -0.16738 -0.11283

0.9 -0.5901021 -0.48449329 -0.39778 -0.32659 -0.26814 -0.22016 -0.18076 -0.14841 -0.12185 -0.12185 -0.08214

0.95 -0.31189351 -0.25607486 -0.21025 -0.17262 -0.14173 -0.11636 -0.09554 -0.07844 -0.0644 -0.0644 -0.04341

1 -0.0031853 -0.00261524 -0.00215 -0.00176 -0.00145 -0.00119 -0.00098 -0.0008 -0.00066 -0.00066 -0.00044

33

Solution by Homotopy perturbation Method

FIGURE 2.12: ( )

TABLE 2.2: Table of ( ) for different values of x and t for

X/T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0 0 0 0 0 0 0 0 0 0

0.05 0.30886552 0.20820475 0.140349813 0.094609129 0.063775555 0.115231825 0.028979885 0.0195352 0.013168584 0.008876878 0.005983861

0.1 0.587527526 0.396049458 0.26697502 0.179966567 0.121314591 0.219195296 0.055125869 0.037160081 0.025049431 0.0168857 0.011382568

0.15 0.808736061 0.545165059 0.367493125 0.247725334 0.166990447 0.301723975 0.075881173 0.051151131 0.03448073 0.023243293 0.015668191

0.2 0.950859461 0.640969754 0.432074605 0.291259398 0.196336548 0.354747501 0.089216167 0.060140186 0.040540208 0.027327958 0.018421644

0.25 0.999999683 0.67409494 0.454404133 0.306311624 0.206483181 0.373080779 0.093826841 0.063248219 0.042635318 0.028740261 0.019373671

0.3 0.951351376 0.641301352 0.432298134 0.291410077 0.196438121 0.354931025 0.089262322 0.060171299 0.040561181 0.027342096 0.018431174

0.35 0.809671788 0.545795829 0.367918323 0.248011959 0.16718366 0.302073077 0.07596897 0.051210314 0.034520625 0.023270186 0.01568632

0.4 0.588815562 0.396917717 0.26756031 0.180361108 0.121580549 0.219675838 0.055246721 0.037241547 0.025104346 0.016922718 0.011407522

0.45 0.31037991 0.209225593 0.141037958 0.095073004 0.064088251 0.115796815 0.029121976 0.019630983 0.01323315 0.008920402 0.0060132

0.5 0.001592653 0.0010736 0.000723708 0.000487848 0.000328856 0.000594188 0.000149434 0.000100732 6.79E-05 4.58E-05 3.09E-05

0.55 -0.30735035 -0.20718338 -0.13966131 -0.09414514 -0.0634627 -0.11466654 0.028837721 0.019439368 -0.01313984 0.008833332 0.005954506

0.6 -0.586238 -0.395180194 -0.26638905 -0.17957157 -0.12104833 -0.2187142 -0.05504877 -0.03707821 0.024994451 0.016848638 0.011357586

0.65 -0.80779828 -0.544532907 -0.36706699 -0.24743808 -0.16679681 -0.30137411 0.075793185 0.051091818 0.034440747 0.023216341 0.015650023

0.7 -0.95036513 -0.640636531 0.431849981 -0.29110779 -0.19623448 -0.35456308 0.089169786 0.060108921 0.040519132 0.027313751 0.018412067

0.75 -0.99999715 -0.67409323 -0.45440298 0.306310847 -0.20648266 -0.37307983 0.093826603 -0.06348058 -0.04263209 0.028740188 0.019373622

0.8 -0.95184088 -0.641631324 0.432520566 0.291560018 -0.19653919 -0.35511365 0.089308251 0.060202259 0.040582051 0.027356164 0.018440658

0.85 -0.81060546 -0.546425214 0.368342589 0.248297954 -0.16737645 -0.30242141 0.076056574 0.051269368 0.034560432 -0.02329702 0.015704408

0.9 -0.5901021 -0.397784969 -0.26814492 0.180755191 -0.1218462 -0.22015582 0.055367434 0.037322919 -0.02515999 0.016959694 0.011432447

0.95 -0.31189351 -0.210245905 0.141725746 0.095536638 -0.06440078 -0.11636151 0.029263992 0.019726715 0.013297683 0.008963904 0.006042524

1 -0.0031853 -0.002147197 0.001447415 0.000975695 -0.00065771 -0.00118838 0.000298867 0.000201465 0.000135806 -9.15E-05 -6.17E-05

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0v

x

T = 0

T = 0.1

T = 0.2

T = 0.3

T = 0.4

T = 0.5

T = 0.6

T = 0.7

T = 0.8

T = 0.9

T = 1

34


FIGURE 2.13: ( )

2.10.1 Conclusion

Homotopy Perturbation Method used to solve the linear equation. Results solved by

Homotopy Perturbation Method are presented in table and graph. The values of table and

graph show that it is increased with space and time initial stage after that it is decreased

and again it is increased after some time. So we can say that the graph fluctuates with time

and space. Homotopy Perturbation Method is a very effective method in solving the linear

equation and solved also for nonlinear partial differential equations which have been

solved in other chapters.

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

V

X

T = 0

T = 0.1

T = 0.2

T = 0.3

T = 0.4

T = 0.5

T = 0.6

T = 0.7

T = 0.8

T = 0.9

T = 1

CHAPTER 3

SOLUTION OF IMBIBITION PHENOMENON

BY THE HOMOTOPY PERTURBATION

METHOD

36

CHAPTER 3

Solution of Imbibition Phenomenon by the


3.1 Introduction:

The imbibition phenomenon is a well-known phenomenon, It happens when porous

medium of length (L) filled with some native liquid (oil) contacts with other injected liquid

(water) initially wet the medium, then there is observed that a spontaneous flow of the

injected fluid into the medium and a counter of the native liquid from the medium, there

occurs a phenomenon is called imbibition phenomenon. The phenomenon imbibition has

been observed by many authors from there different viewpoints. Scheidegger [2], Verma

[15], Mehta [70]. This phenomenon occurs when there are differences between wetting

abilities of water and oil.

FIGURE 3.1: Representation of Imbibition Phenomenon

FIGURE 3.2: Schematic Representation of Imbibition Phenomenon

37

Mathematical Formulation of the Problem

The imbibition phenomenon has been observed by many authors with different approaches.

Moe, Baldwin [7] have observed the effects of an injection rate of an initial saturation of

water and gravity on water injection in little water-wet in fractured porous media. Mehta

[70] have observed the counter-current imbibition in a curved homogeneous porous

medium with the help of integral method. Patel and Maher [43] have obtained an

approximate solution of counter-current imbibition Phenomenon in a Heterogeneous

Porous Media. Patel [94] has solved an exact solution of imbibition phenomena in two

phase flow through homogenous porous media.

This chapter includes, the imbibition phenomenon has been discussed to arise in the flow

of immiscible fluids flow through homogenous porous media with capillary effect and

derived an analytical solution of the standard equation of the imbibition phenomenon with

the help of Homotopy Perturbation Method.

3.2 Mathematical Formulation of the Problem:

Take a cylindrical piece which finite length is L of homogenous porous medium to contain

viscous fluid which completely adjoining by water resistance surface except at one end of

the cylinder which is labelled as the imbibition phase and this end is exposed to an adjacent

formation of injected water. In this problem, an injected liquid and native liquid are two

different liquid of different brininess with small viscosity difference has been assumed.

This arrangement gives rise to the phenomenon of linear counter current imbibition on

both sides of common interface which is a spontaneous linear counter flow of injected fluid

into the heterogeneous porous media into oil formatted area and the counter flow of native

fluid into injected fluid formulated area in opposite direction as shown in figure 3.1.

Water injects at a common interface in homogeneous porous medium to connect native oil

will be displaced by injecting water. Hence, injected water and native oil both satisfies

Darcy’s Law given by Bear [26] which is given velocities of water and oil respectively as,

(3.1)

(3.2)

38

Solution of Imbibition Phenomenon by the Homotopy Perturbation Method

Where

K = Permeability of the homogenous porous medium

constant kinematic viscosity of oil

he constant kinematic viscosity of water

Pressure of oil

Pressure of water

Relative permeability of oil

Relative permeability of water

The flow is counter current of imbibition Phenomena,

= (3.3)

So, from (3.1) and (3.2) we write,

+

= 0 (3.4)

The capillary pressure definition gives,

= (3.5)

Combining equation (3.4) and (3.5), we get

+

+

= 0 (3.6)

Substitute the value of

into equation (3.1), we obtain

(

) (

)

(

)

(3.7)

39


Since an injected water and native oil follow in a porous medium thorough interconnected

capillaries during the imbibition phenomenon because of capillary pressure of an injected

water and native oil.

The continuity equation for the injected water is, ( scheideggers [2] )

+

= 0 (3.8)

Here is the porosity and is the water saturation.

Substitute the value of in to the equation (3.8), it became

+

*

+ = 0 (3.9)

Equation (3.9) is a non-linear partial differential equation, which indicates the counter

current imbibition phenomenon of injected water and native oil like two immiscible fluids

flow thorough homogeneous cylindrical medium.

The fictitious relative permeability is the function of displacing fluid saturation. Now, we

will take the standard forms of the Permeability, Saturation and Capillary pressure phase

saturation, (schidegger[2])

,

Where (3.10)

Here the present study involves an injected water and native oil, so from the Scheidegger

(1960) we can take,

=

(3.11)

Where P is the porosity of medium, K is permeability, and are saturations of

injected water and native oil respectively, is capillary pressure, is capillary pressure

coefficient.

40


Using the relation equation, then (3.9) became,

*

+ (3.12)

Selecting new variables,

and

Equation (3.12) becomes

*

+ (3.13)

This is standard governing equation of imbibition phenomenon in homogenous porous

media, which has been solved with appropriate boundary conditions and with the help of

Homotopy Perturbation Method.

We consider the following initial and boundary conditions to solve equation (3.13)

completely,

;

(3.14)

3.3 Solution by Homotopy Perturbation Method:

We obtain the approximate analytical solution of equation (3.13) subject to the boundary

conditions (3.14), we construct the a Homotopy of the equation [ ]

which satisfies,

*

+ [

(

) ] = 0 (3.15)

[

]

41

Solution by Homotopy Perturbation Method

(

(

)

(

)

(

)

)

(

)

(

)

(

)

(

)

( (

) (

) (

) (

) (

) (

)

(

) (

) (

) (

)

(

) (

) )

Comparing the powers of ,

(

) (

)

(

) (

) (

)

(

) (

) (

) (

)

(

) (

)

42


(

) (

) (

) (

) (

) (

)

(

) (

)

………………………………………..

Solving all above equations, we get

…………………………………………

According to Homotopy Perturbation Method, one can conclude that,

(3.16)

Putting all the values in the equation (3.16) we get,

… (3.17)

Equation (3.17) is the solution of the equation (3.13).

3.4 Numerical Values and the Graphical Presentation:

The standard equation of an imbibition phenomenon has solved with the help of Homotopy

Perturbation Method. Table 3.1 is showing different values of saturation of injected water

of X and T. We can see in the Figure 3.1, the graph of versus X for values of

time . For our convenience, we have taken here

43

Results and Graphical Solution

From the graph, we can say that the saturation of injected water increases as the

space increases for different values of time.

TABLE 3.1: Saturation of Injected Water for Different Distance X and Time T.

X/T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0.01 0.019999 0.029996 0.039989 0.049979 0.059964 0.069943 0.079915 0.089879 0.099833

0.1 0.099833 0.109828 0.119812 0.129784 0.139743 0.149688 0.159618 0.169532 0.179429 0.189308 0.199168

0.2 0.198669 0.208659 0.218628 0.228575 0.2385 0.2484 0.258276 0.268126 0.277949 0.287744 0.29751

0.3 0.29552 0.305505 0.31546 0.325383 0.335273 0.34513 0.354952 0.364739 0.37449 0.384203 0.393877

0.4 0.389418 0.399399 0.409339 0.419239 0.429096 0.438911 0.448682 0.458408 0.468088 0.477721 0.487306

0.5 0.479426 0.489401 0.499328 0.509205 0.519031 0.528806 0.538527 0.548194 0.557807 0.567364 0.576864

0.6 0.564642 0.574614 0.584528 0.594384 0.60418 0.613916 0.62359 0.633203 0.642751 0.652236 0.661655

0.7 0.644218 0.654185 0.664088 0.673923 0.683692 0.693392 0.703022 0.712583 0.722072 0.731489 0.740833

0.8 0.717356 0.72732 0.737211 0.747029 0.756772 0.766439 0.776029 0.785542 0.794976 0.804331 0.813606

0.9 0.783327 0.793288 0.803169 0.81297 0.82269 0.832327 0.841881 0.851351 0.860736 0.870035 0.879247

1 0.841471 0.851429 0.861301 0.871088 0.880787 0.890399 0.899921 0.909353 0.918694 0.927944 0.937101

FIGURE 3.3: Graphical Representation of Versus Distance X for Different Time T

44


3.5 Conclusion:

The solution obtained which is useful in determining the amount of water requiring for oil

recovery process. An oil water imbibition problem in a homogenous porous medium has

been discussed under special boundary conditions, by using Homotopy Perturbation

Method.

We consider that the sides of basins are limited by rigid boundaries and bottom by a thick

layer of water flows only in positive X-direction.

It is interpreted form graph that at particular time level, saturation of injected liquid

increase with increase in value of but as time T is increasing then saturation of injected

fluid is slowly increasing due to short time of imbibition phenomenon at common

interface. This occurs without any external force, so after sometimes length of small

fingers reduce to zero which is physically consistent with real phenomena.

The values if obtained in the table are valid up to our theoretical assumptions which

can be compared, if required, based on other available data.

CHAPTER 4

MATHEMATICAL MODELING OF BURGER’S

EQUATION OF INSTABILITY PHENOMENON

SOLVED BY HOMOTOPY PERTURBATION

METHOD

46

CHAPTER 4

Mathematical Modeling of Burger’s Equation of

Instability Phenomenon Solved by Homotopy

Perturbation Method

4.1 Introduction

Instability Phenomenon (Fingering Phenomenon) is the phenomenon which happens

between two immiscible phase flows through homogeneous porous media. Two different

immiscible fluids in a large medium would be explored if it is not in the same direction.

The phenomenon of fingering happens when a native liquid contained in a porous medium

is displaced by injected liquid with less viscosity which occurred frequently in the problem

of petroleum technology. When the inject liquid (water) injects in oil formatted porous

media then in place of regular movement of common interface protuberance occurs

because of the difference in viscosities of injected liquid (water) and native liquid (oil)

which gives arise to shape of fingers. This phenomenon is called instability phenomenon.

FIGURE 4.1: Representation of Fingering Phenomenon

FIGURE 4.2: Schematic Representation of Fingering Phenomenon

47

Introduction

Scheidegger [3] obtained the uncertain behaviour in flows in homogeneous porous media

using method of characteristics. The statistical behaviour of the instability phenomenon in

a process of displacement through heterogeneous porous media with capillary pressure

effect using perturbation solution has been examined by Verma [13]. Patel [93] has

examined this problem by using method of the advection diffusion concept. The

phenomenon of instability or fingering occurs due to viscosities difference of flowing

fluids. Mehta and Joshi [71] have been solved instability phenomenon in homogenous

porous media by invariant method. Mukherjee [31] have solved the fingering phenomenon

in a homogeneous porous medium by means of calculus of variation and similarity theory.

Many authors have neglected the effect of capillary pressure. Verma [13] and Mehta[70]

both have taken capillary pressure in the analysis of fingers. Scheidegger [3] gave perfect

review of the topic. Therefore, it is necessary in an oil recovery process to stabilize the

fingers.

4.2 Mathematical Formulation of Problem:

We consider a porous matrix of length L having its three sides impermeable except one.

We take vertical cross-sectional area of this porous matrix which is rectangle for

mathematical model. The open end will be the common interface .

Let the water be injected at , then due to the injecting force and viscosity

difference, the fingers or instabilities may arise. The length of the fingers is being

measured in the direction of displacement. Scheidegger and Johnson suggested replacing

these irregular fingers by schematic fingers of rectangular size (Figure 4.2).

By Darcy’s Law, the velocities of injected liquid (water) and native oil are,

(4.1)

(4.2)

Here,

K = Permeability of the homogenous porous medium

he kinematic viscosity of oil

The kinematic viscosity of water

48

Mathematical Modeling of Burger’s Equation of Instability Phenomenon solved by


Pressures of oil

Pressures of water

Relative permeability of oil

Relative permeability of water

The continuity equations of two phase densities defined as constant can be written as,

(

) (

) (4.3)

(

) (

) (4.4)

Here porosity of the medium is P.

Now, saturations of water and oil are unity so we can write,

(4.5)

By the definition of the capillary pressure defined as discontinuity of the flowing phase

across their common interface, is a function of the phase saturation.

It is written as

(4.6)

The saturation of the motion is derived by putting the values of and .

Now the equations (4.1) and (4.2) substituting in to the equations (4.3) and (4.4)

respectively, we get

49


(

)

*(

) (

)+ (4.7)

(

)

*(

) (

)+ (4.8)

Eliminating

form the equations (4.6) & (4.7) we got,

(

) =

*(

) ,(

) (

)-+ (4.9)

From the equations (4.8) & (4.9), and also using equation (4.5), we get

*,(

) (

) - (

) (

) (

) + (4.10)

Taking integration of equation (4.10) with respect to , we get

*(

) (

) + (

) (

) (

) (4.11)

Where is a constant of integration, which can be evaluated later on.

Now simplify (4.11), we got

[(

) (

)] +

(

)

(

)(

) (4.12)

Solving the equation (4.9 to 4.12), then we get

[

(

)(

)

[(

) (

)]

(

)(

) ] (4.13)

The standard value of the pressure of oil is,

50



(4.14)

Here is the constant mean pressure.

Now, for solving the above equation, take differentiation of above equation with respect to

then,

=

(4.15)

Substituting the value of

in the equation (4.11) we can obtained,

C *

(

)

+ (4.16)

On substituting the value of C from the equations (4.16) and (4.13), we can obtained,

* (

)+ (4.17)

*(

)+

Where B (

)

Now, from the equation (4.17) we get,

Where

(4.18)

With the help of Hopf – Cole transformation equation,

(4.19)

The (4.18) becomes,

51


Where

(4.20)

The above equation is known as Burger’s equation of instability phenomena in double

phase flow in porous media.

Now , we take appropriate boundary conditions are,

S and S( )

We choose initial condition is,

S


Homotopy [ ] for the equation (4.20) is define as

*

+ [

] = 0

[

]

(

[

]

)

= [

]

52




+

= 0

+

+

+

+

= 0

…………………..

∑

(4.21)

Solving all above the non – linear partial differential equation we get,

…………………….

Solution the equation (4.20) can be written as,

(4.22)

Equation (4.22) is the solution of equation (4.20).

53

Numerical Values and Graphical Presentation

4.4 Numerical Values and Graphical Presentation:

The nonlinear partial differential equation which is known as burger’s equation arising in

an instability phenomenon has been solved. Table values and graphs have been obtained

using Microsoft Excel 2010 and Origin Lab. Table 4.1 is showing the values of saturation

of injected water for and time t. We can see in the figure 4.1, the saturation of water

decreases with time and Figure 4.2 shows that the saturation of water increases

with distance.

TABLE 4.1: Saturation of Injected Water for Different Values of and t

FIGURE 4.3: Graphical Presentation of for Different Values at and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0 0 0 0 0 0 0 0 0 0

0.1 0.1 0.090909 0.083333 0.076923 0.071429 0.066667 0.0625 0.058824 0.055556 0.052632 0.05

0.2 0.2 0.181818 0.166667 0.153846 0.142857 0.133333 0.125 0.117647 0.111111 0.105263 0.1

0.3 0.3 0.272727 0.25 0.230769 0.214286 0.2 0.1875 0.176471 0.166667 0.157895 0.15

0.4 0.4 0.363636 0.333333 0.307692 0.285714 0.266667 0.25 0.235294 0.222222 0.210526 0.2

0.5 0.5 0.454545 0.416667 0.384615 0.357143 0.333333 0.3125 0.294118 0.277778 0.263158 0.25

0.6 0.6 0.545455 0.5 0.461538 0.428571 0.4 0.375 0.352941 0.333333 0.315789 0.3

0.7 0.7 0.636364 0.583333 0.538462 0.5 0.466667 0.4375 0.411765 0.388889 0.368421 0.35

0.8 0.8 0.727273 0.666667 0.615385 0.571429 0.533333 0.5 0.470588 0.444444 0.421053 0.4

0.9 0.9 0.818182 0.75 0.692308 0.642857 0.6 0.5625 0.529412 0.5 0.473684 0.45

1 1 0.909091 0.833333 0.769231 0.714286 0.666667 0.625 0.588235 0.555556 0.526316 0.5

54



FIGURE 4.4: Graphical Presentation of for Different Values at and

4.5 Conclusion:

The results (4.22) show the saturation of injected water in instability phenomenon in

homogenous porous media during secondary oil recovery process.

In instability phenomenon, when water is injected at common interface during

secondary oil recovery process in homogenous porous media, then initially saturation is

very small and then protuberance take place for The saturation of injected fluid

steadily increasing due to capillary pressure of finger occurs in instability phenomenon.

The saturation of injected fluid occurs due to the interconnected capillaries of fingers

which will be increased as distance increased for differed time which is shown in

figure (4.3). But due to unsaturated homogenous porous media when is increasing the

saturation of injected fluid is slightly decreasing then the previous time. From the figure

(4.4), it has been concluded that the saturation of injected fluid in instability phenomenon

increases with respect to distance form common interface but effect of time is very less

due to external force applied to time of injected water in instability phenomena.

55

Conclusion

From the physical interpretation of the solution (4.22) and graph (4.4), we can conclude

that the saturation of injected water increases for increasing value of for different time .

In this model, we have concluded that the saturation of injected fluid which occupied by

the average cross sectional area of the schematic fingers is increases as distance

increases for which is physically fact in case of instability phenomenon.

.

CHAPTER 5

SIMULATION OF THE FINGERO-IMBIBITION

PHENOMENON IN DOUBLE PHASE FLOW IN

POROUS MEDIA SOLVED BY HOMOTOPY

PERTURBATION METHOD

56

CHAPTER 5

Simulation of the Fingero-Imbibition Phenomenon in

Double Phase Flow in Porous Media solved by


5.1 Introduction:

When two important phenomena, an instability (fingering) and imbibition occur

simultaneously in displacement process is called ‘Fingero-Imbibition’ phenomenon. This

Fingero-Imbibition phenomenon has gained importance by various fields such as geo-

hydrology, geophysics, reservoir, engineering and little attention to the fractional type.

FIGURE 5.1: Representation of Fingero -Imbibition Phenomenon.

The researchers Caputo [68] and He [46] have observed the approach which is useful as an

account of the effects of changing flux is to embody the effects of memory which has to do

with a posing problem in terms of fractional calculus. The levy-flight type of transport is

known for the diffusion process which narrated by the fractional system. Motivated by this

idea, we take a fractional type Fingero-Imbibition phenomena equation in the double phase

flow through porous media and obtain an analytical approximate solution using Homotopy

Perturbation Method.

57

Introduction

We are taking here the non-linear partial differential equation which is the spontaneous

imbibition of injected water through an oil-saturated solid. The hypothesis of this model is

that native oil and injected water are two liquid phases and water is considered as

preferentially wetting phase.

Generally, research has been done for last three to four decades, fractional calculus

considered as very importance part due to its various applications in fluid flow, control

theory of dynamical systems, chemical physics, electrical networks, and so on.


We are considering here a finite cylindrical piece of a homogeneous porous matrix which

is fully saturated with a native liquid (oil) surrounded completely by an impermeable

surface except for one end of the cylinder labelled as the imbibition face . This end is

exposed to an adjacent formation of injected liquid (water). If the injected liquid (water) is

a less viscous and preferentially wetting phase, then the phenomenon of fingering will

occur simultaneously with imbibition phenomenon, and then this arrangement describes a

one dimensional fingero-imbibition phenomenon in which the injection is started by

imbibition and resulting displacement produces instabilities.

FIGURE 5.2: The Fingero – Imbibition Phenomena in Fractured Reservoir [82]

58

Simulation of the Fingero-imbibition Phenomenon in Double Phase Flow of Time

Fractional Type in Porous Media solved by Homotopy Perturbation Method

We are assuming the Darcy’s Law for the double phase flow system, the seepage velocities

of wetting phase ( ) and non-wetting phase ( ) as

(5.1)

(5.2)

Where,

= The relative permeability of injected water,

= The relative permeability of native oil,

= The pressure of injected water,

= The pressure of native oil,

= The kinetic viscosities of injected water,

= The kinetic viscosities of native oil,

K = The permeability of homogeneous medium.

The co-ordinate is measured along the axis of the cylindrical medium, the origin being

located at the imbibition face = 0. We can write,

(5.3)

For counter current flow.

Hence (5.1) & (5.2) give

(5.4)

59


Mehta [105], give us the definition of capillary pressure written as,

(5.5)

That is,

(5.6)

Equation (5.4) & (5.6),

(

)

+

= 0 (5.7)

From here, using (5.7), we can write (5.1) as

= (

) (

)

(

)

(5.8)

The equation of continuity is given by,

(5.9)

Where is the wetting saturation and is the porosity medium.

Substituting the value of in to (5.9), we obtain

*

+ = 0 (5.10)

Equation (5.10) is known as the non-linear partial differential equation which indicates that

the finger-imbibition phenomenon of two immiscible fluids flows through the

homogeneous porous cylindrical medium.

60



We assume standard forms for the analytical relationship between the relative

permeability, Phase saturation and capillary pressure. Relative permeability is the function

of displacing fluid saturation.

Where = (5.11)

(5.12)

(5.13)

We are considering the injected water and viscous oil, so according to Scheidegger [2].

We get,

(5.14)

Hence by substituting (5.14), (5.13),(5.12) and (5.11) into (5.10) , we got,

*

+ (5.15)

We are taking the new variables for a dimensionless form.

&

We got,

=

*( )

+ (5.16)

We are choosing initial condition is due to the fact that the saturation of injected water

decreases exponentially when increases (Mehta [70]).

61

Solution by Homotopy perturbation method

Here, the Fingero-imbibition phenomena equation through porous media using HPM to

obtain analytical solution.

(

)

(5.17)

5.3 Solution by Homotopy Perturbation method :

We choose,

And boundary conditions are,

Here and are the water saturation at common interface and saturation

of injected water at the end of the matrix length X = 1 (i.e. L)

Now using Homotopy Perturbation Method,

Homotopy [ ] for equation (5.17) is defined as

*

+ [

(

) ] = 0

*

+

(

(

)

(

)

( )

(

)

(

)

)

62



–

–

(

)

(

)

+ (

)

+ (

)

+ (

)

=

–

–

(

)

+

+

+ ((

) (

)

(

) (

) (

) (

) (

) )

63



(

) (

)

(

) (

) (

)

(

) (

) (

) (

)

(

) (

)

……………………………

Now solving above all equations we are getting,

……………………………. (5.18)

Solution of the equation (5.17) is written as,

+ (5.19)

Putting all the values in the equation (5.19) we get,

( ) + ( )

+

( )

+ ………………..

64




The nonlinear partial differential equation of finger imbibition phenomenon of time

fractional type has been solved using Homotopy Perturbation Method. Table values and

graph have been obtained using Microsoft Excel 2010 and Origin Lab. We can see in the

figure 5.1, the graphical interpretation of the saturation versus distance (space)

for fixed values of time . Numerical values of Figure 5.3 show

the graph with respect to different time and different space.

TABLE 5.1: Saturation of Injected Water for Different Values of and Time

X/T 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 1 0.987184 0.973136 0.957856 0.941344 0.9236 0.904624 0.884416 0.862976 0.840304 0.8164

0.1 0.904837 0.894877 0.883252 0.869961 0.855004 0.838383 0.820096 0.800143 0.778525 0.755242 0.730293

0.2 0.818731 0.810966 0.80106 0.789012 0.774822 0.758491 0.740018 0.719403 0.696647 0.671749 0.64471

0.3 0.740818 0.734716 0.725961 0.714554 0.700493 0.683779 0.664413 0.642393 0.617721 0.590395 0.560417

0.4 0.67032 0.665452 0.657391 0.646137 0.63169 0.61405 0.593218 0.569193 0.541975 0.511564 0.477961

0.5 0.606531 0.602551 0.594815 0.583324 0.568076 0.549072 0.526312 0.499797 0.469525 0.435497 0.397713

0.6 0.548812 0.545444 0.537738 0.525694 0.509312 0.488591 0.463532 0.434134 0.400398 0.362324 0.319912

0.7 0.496585 0.493609 0.485696 0.472847 0.455062 0.43234 0.404683 0.37209 0.33456 0.292094 0.244692

0.8 0.449329 0.446567 0.438258 0.424403 0.405 0.380052 0.349556 0.313514 0.271925 0.22479 0.172108

0.9 0.40657 0.403882 0.395027 0.380004 0.358814 0.331456 0.297931 0.258238 0.212377 0.160349 0.102154

1 0.367879 0.365155 0.355635 0.339317 0.316202 0.286291 0.249583 0.206078 0.155776 0.098677 0.034782

FIGURE 5.3: Graphical Representation of Versus Distance X for Different Time T.

65


.

FIGURE 5.4: Graphical Representation of Versus Time T for Distance X.

5.5 Conclusion:

The Homotopy Perturbation Method was used to derive the analytical solution of the

model.

We consider that the sides of basins are limited by rigid boundaries & bottom by a thick

layer of water flows only in positive -direction.

It is interpreted from graph that at particular time level, saturation of injected liquid has

decreasing approach with increase in value of and as time increases, rate of increase

of the saturation of injected liquid lessen at each layer. So, rate of increase of the saturation

slows down at each point as distance and time increases.

CHAPTER 6

COUNTER-CURRENT IMBIBITION

PHENOMENON WITH EFFECT OF

INCLINATION AND GRAVITATIONAL FORCE

SOLVED BY HOMOTOPY PERTURBATION

METHOD

67

CHAPTER 6

Counter-current Imbibition Phenomenon with Effect

of Inclination and Gravitational Force Solved by


6.1 Introduction:

In co-current flow, water and oil flow in the same direction and water pushes oil out of the

matrix. In counter – current flow, oil and water flow in opposite direction and oil escapes

back by flowing back in the same direction along which water has imbibed.

Many researchers have been researched this type of problem with different point of view.

Mehta [70] and Verma [16] studied the imbibition phenomenon in homogeneous porous

media, with appropriate conditions, by singular perturbation approach. Mishra [37] have

solved the imbibition phenomenon model in the double phase flow through porous media

of groundwater replacement. The numerical solution of imbibition phenomena has been

obtained by the Crank–Nicolson Scheme for finite differences by Pradhan and Verma [99].

Joshi and Mehta [74] applied group invariant method and Parikh [22] applied the

generalized separable solution of counter–current imbibition phenomena.

The counter current phenomenon with the effect of inclination and gravitational force has

been solved using Homotopy Perturbation Method in this chapter.

68

Counter-current Imbibition Phenomenon with Effect of Inclination and Gravitational Force

solved by Homotopy Perturbation Method

6.2 Mathematical Formulation of the problem:

FIGURE: 6.1 Schematic Diagram of the Problem

Here a cylindrical piece of a homogeneous porous matrix having length L having its three

sides are surrounded by an impermeable surface is considered whose one open end is

labelled as imbibition face x = 0 and it is inclined at an angle .

During imbibition, when water is injected into an oil saturated porous matrix at imbibition

face x = 0. The oil is displaced through a small distance x = l due to the difference in phase

viscosity. Since water is injected at common interface in inclined homogeneous porous

matrix contenting oil that will displace by injecting water.

The injected water and native oil both satisfy Darcy’s Law velocities of water and oil

respectively as

(

) (6.1)

(

) (6.2)

Where

K = Permeability of homogeneous medium,

69


Permeability of the injected liquid (water),

= Permeability of the native liquid (oil),

= Density of injected liquid (water),

Density of injected liquid (oil),

iw = Kinematic viscosity of the injected liquid (water),

no = Kinematic viscosity of the native liquid (oil),

= An inclination angle of the bed,

g = An acceleration due to gravity.

The x – co-ordinate is a measurement along the axis of the cylindrical medium, the origin

is located at the imbibition face x = 0.

The flow is counter-current of imbibition phenomenon so we take,

(6.3)

Now from equation (6.1) and (6.2),

(

) +

(

) = 0 (6.4)

Now by the definition of capillary pressure,

(6.5)

Form equations (6.4) and (6.5), we get

(

)

(

) (6.6)

Putting the value of

from (6.6) into (6.1), we got,

70



(

) (

) (

( ) )

(

) (6.7)

Injected water and displaced native oil that will satisfy the equation of continuity as,

(6.8)

Where is the porosity of medium and is the saturation of injected water (wetting

phase).

Equation (6.8) becomes (6.9) by putting the value of from Equation (6.7),

*

(

( ) )

+ (6.9)

Equation (6.9) is a non-linear differential equation, which described the linear counter-

current imbibition phenomenon of two immiscible fluid flows through homogeneous

porous cylindrical medium with impervious boundary surface on three sides.

Some standard relationship among the relative permeability, phase saturation and capillary

pressure phase saturation as [15],

(6.10)

Where proportionality constant.

Since the present investigation includes water and viscous oil, so from the scheidegger [3]

we have,

*

+

(6.11)

71


Putting the values of and from Equation (6.10) and (6.11) into equation (6.9), we

get,

*

+

* ( ) + (6.12)

Using dimensionless form

We get,

*

+

( )

= 0 (6.13)

*

+

= 0 (6.14)

Where B ( )

Now, choosing an appropriate initial and the boundary conditions to solve above equation

using Homotopy Perturbation Method.

6.3 Solution by Homotopy Perturbation Method

We take initial condition,

( )

And boundary conditions are,

( ) ( )

Now using Homotopy Perturbation Method.

Homotopy ( ) [ ] for equation (6.14) is define as,

72



( ) ( ) *

+ [

(

)

]

[

(

)

]

[ (

)

(

)

(

)

(

)]


(

) (

)

( ) ( ) (

)

( )

( ) ( ) (

)

= 0

( ) ( ) (

) ( )

(

)

73


Solving the above equations we get,

( )

( ) (

(

) (

)

(

) )

( ) ((

) *

( ) ( )

( ) (

) +

[ (

) [

(

) (

)

(

)]

(

( ) ( )

( ) (

))]

+

(

) (

( ) ( ) (

) (

))

*

( ) ( ) (

) (

)+ )

………………………………………. (6.15)

Solution of the equation (6.15) can be written as,

( ) ( ) ( ) + ( ) ( )


An approximate analytical solution has been obtained for equation (6.14) with appropriate

conditions by using Homotopy Perturbation Method. Here, we studied the saturation rate

of wetting phase in homogenous porous media with different inclination and gravitational

force. From the tables and the graphs, we can see the saturation of water is high at

and low at for distance and Time

.

74



TABLE 6.1: Numerical values of saturation of injected water at

X/T 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 0 0 0 0 0 0 0 0 0 0 0

0.1 0.0612 0.0617 0.0622 0.0627 0.0631 0.0636 0.0641 0.0646 0.0651 0.0655 0.0660

0.2 0.1289 0.1295 0.1302 0.1308 0.1315 0.1322 0.1328 0.1335 0.1342 0.1348 0.1355

0.3 0.2036 0.2045 0.2054 0.2063 0.2072 0.2081 0.2090 0.2099 0.2107 0.2116 0.2125

0.4 0.2862 0.2874 0.2886 0.2898 0.2910 0.2921 0.2933 0.2945 0.2957 0.2969 0.2980

0.5 0.3775 0.3791 0.3806 0.3822 0.3837 0.3853 0.3868 0.3883 0.3899 0.3914 0.3930

0.6 0.4785 0.4805 0.4824 0.4844 0.4864 0.4884 0.4904 0.4924 0.4944 0.4964 0.4984

0.7 0.5900 0.5925 0.5951 0.5977 0.6002 0.6028 0.6053 0.6079 0.6105 0.6130 0.6156

0.8 0.7132 0.7165 0.7198 0.7230 0.7263 0.7296 0.7328 0.7361 0.7394 0.7426 0.7459

0.9 0.8495 0.8536 0.8577 0.8619 0.8660 0.8701 0.8743 0.8784 0.8826 0.8867 0.8908

1 1 1 1 1 1 1 1 1 1 1 1

FIGURE 6.2: Saturation versus X and Time T for

75




X/T 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 0 0 0 0 0 0 0 0 0 0 0

0.1 0.0612 0.0616 0.0620 0.0624 0.0627 0.0631 0.0635 0.0639 0.0643 0.0646 0.0650

0.2 0.1289 0.1294 0.1300 0.1305 0.1311 0.1316 0.1322 0.1327 0.1333 0.1338 0.1344

0.3 0.2036 0.2044 0.2052 0.2059 0.2067 0.2075 0.2082 0.2090 0.2098 0.2106 0.2113

0.4 0.2862 0.2873 0.2883 0.2894 0.2904 0.2915 0.2925 0.2936 0.2946 0.2957 0.2967

0.5 0.3775 0.3789 0.3803 0.3817 0.3831 0.3845 0.3859 0.3873 0.3887 0.3901 0.3915

0.6 0.4785 0.4803 0.4821 0.4840 0.4858 0.4876 0.4895 0.4913 0.4931 0.4950 0.4968

0.7 0.5900 0.5924 0.5947 0.5971 0.5995 0.6019 0.6043 0.6067 0.6090 0.6114 0.6138

0.8 0.7132 0.7163 0.7194 0.7224 0.7255 0.7286 0.7316 0.7347 0.7378 0.7408 0.7439

0.9 0.8495 0.8534 0.8573 0.8612 0.8651 0.8690 0.8730 0.8769 0.8808 0.8847 0.8886

1 1 1 1 1 1 1 1 1 1 1 1

76



FIGURE 6.4 : Saturation Versus X and Time T for

FIGURE 6.5: Saturation Versus X and Time T for

77



FIGURE 6.6 : Saturation versus X and Time T for

X/T 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 0 0 0 0 0 0 0 0 0 0 0

0.1 0.0612 0.0615 0.0618 0.0621 0.0623 0.0626 0.0629 0.0632 0.0635 0.0638 0.0641

0.2 0.1289 0.1293 0.1297 0.1302 0.1306 0.1311 0.1315 0.1320 0.1324 0.1329 0.1333

0.3 0.2036 0.2043 0.2049 0.2056 0.2062 0.2069 0.2075 0.2082 0.2088 0.2095 0.2101

0.4 0.2862 0.2871 0.2881 0.2890 0.2899 0.2908 0.2917 0.2926 0.2935 0.2945 0.2954

0.5 0.3775 0.3788 0.3800 0.3813 0.3825 0.3838 0.3850 0.3863 0.3875 0.3888 0.3900

0.6 0.4785 0.4801 0.4818 0.4835 0.4851 0.4868 0.4885 0.4902 0.4918 0.4935 0.4952

0.7 0.5900 0.5922 0.5944 0.5966 0.5988 0.6010 0.6032 0.6054 0.6076 0.6098 0.6120

0.8 0.7132 0.7161 0.7190 0.7218 0.7247 0.7276 0.7304 0.7333 0.7362 0.7391 0.7419

0.9 0.8495 0.8532 0.8569 0.8606 0.8643 0.8680 0.8717 0.8754 0.8791 0.8828 0.8865

1 1 1 1 1 1 1 1 1 1 1 1

78





X/T 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 0 0 0 0 0 0 0 0 0 0 0

0.1 0.0612 0.0614 0.0616 0.0618 0.0620 0.0621 0.0623 0.0625 0.0627 0.0629 0.0631

0.2 0.1289 0.1292 0.1295 0.1299 0.1302 0.1305 0.1309 0.1312 0.1316 0.1319 0.1322

0.3 0.2036 0.2041 0.2047 0.2052 0.2057 0.2063 0.2068 0.2073 0.2079 0.2084 0.2089

0.4 0.2862 0.2870 0.2878 0.2886 0.2894 0.2902 0.2909 0.2917 0.2925 0.2933 0.2941

0.5 0.3775 0.3786 0.3798 0.3809 0.3820 0.3831 0.3842 0.3853 0.3864 0.3875 0.3886

0.6 0.4785 0.4800 0.4815 0.4830 0.4845 0.4860 0.4875 0.4890 0.4906 0.4921 0.4936

0.7 0.5900 0.5920 0.5940 0.5961 0.5981 0.6001 0.6021 0.6042 0.6062 0.6082 0.6102

0.8 0.7132 0.7159 0.7186 0.7213 0.7239 0.7266 0.7293 0.7320 0.7346 0.7373 0.7400

0.9 0.8495 0.8529 0.8564 0.8599 0.8634 0.8669 0.8704 0.8738 0.8773 0.8808 0.8843

1 1 1 1 1 1 1 1 1 1 1 1

79

Conclusion



80



6.5 Conclusion:

Here we studied the initial water saturation as well as the recovery rate for counter-current

imbibition phenomenon in an inclined homogenous porous media.

From the tables and graphs, It is shown that saturation of water increases as time and

distance increases for and subsequently decreases

with time and distance as inclination of the bed increases results decreases in saturation

and recovery rate also decrease. Graphs and tables show that the saturation of wetting

phase be maximum for zero inclination.

Hence it may conclude that the saturation of wetting phase increases with distance and

time for zero inclination and small inclination results increase the recovery rate of the oil

reservoir but as the inclination increases it results lower the saturation rate implies less

recovery rate of the oil reservoir.

CHAPTER 7

GROUND WATER RECHARGE IN VERTICAL

DIRECTION IN UNI DIMENSIONAL THROUGH

POROUS MEDIA SOLVED BY HOMOTOPY

PERTURBATION METHOD

81

CHAPTER 7

Ground Water Recharge in Vertical Direction in Uni

Dimensional through Porous Media solved by Homotopy

Perturbation Method

7.1 Introduction:

The soil is a very important part of the hydrological cycle. Soil moisture is a moisture

content that is the quantity of water contained in a material. The saturated zone means the

void space is occupied by water. The imaginary surface which bounds the saturated zone is

a phreatic surface. There is no moisture in dry soil so the moisture content is zero (0) and

the unsaturated porous medium, its value is one (1). When the porous medium is fully

saturated by injected fluid water, the range of moisture content is [0, 1]. The region of the

soil is called the unsaturated zone. This is only the region where the most behaviour

observed.

The flow of water through the soil in many situations is slightly saturated and unsteady too.

The flow of water is unsteady because of the moisture content differs as a function of time

and slightly saturated because all the void space are not fully filed by flowing fluid. It is

very important that the water content of the soil in these flows, the solution of the equation

describing such flows are very useful in many branches of engineering like civil,

hydrologic, irrigation etc.

The phenomenon of one dimensional vertical groundwater recharge is very important for

who is in the field of hydrologists, agriculturists and for the people related to water

resources sciences.

Many authors have focused on these phenomena from a different point of views; here are

some examples. Klute [9], A numerical method for solving the flow equation for water in

unsaturated materials which is reduced diffusion equation to an ordinary differential

equation and employed a forward integration and iteration method. Verma [11] used the

Laplace transformation technique to solve this problem; an approximate solution

82

Ground Water Recharge in Vertical Direction in Uni Dimensional Through Porous Media


considering the average diffusivity coefficient of the whole range of moisture content and

treated as small constant by the method of singular perturbation technique has been

obtained by Mehta [69]. Prasad [53] made a mathematical model of water flow in the

unsaturated area and observed the effect of unsaturated soil parameters on water movement

during various processes such as gravity drainage and infiltration. Mishra and Verma [18]

solved one-dimensional vertical groundwater recharges through porous media. Mehta [70]

have considered aqueous conductivity directly proportional to depth, moisture content and

inversely proportional to time. They obtained an approximate solution for the vertical

groundwater recharge problem in slightly saturated porous media by using small parameter

method.

FIGURE 7.1: Representation of Groundwater Recharge Phenomenon


It is investigated mathematical model; we consider that the ground water recharge takes

place over a large basin of such geological location that sides are limited by rigid

boundaries and the bottom by a thick layer of water table. In this case, we assumed that the

83


flow is vertically downward through unsaturated porous media.

Here, the average diffusivity coefficient of the whole range of moisture content is regarded

as constant (Mehta)[70] and the permeability of the moisture content is assumed to a linear

function of moisture content. The theoretical formulation of the problem gives a nonlinear

partial differential equation for the moisture content.

Following Klute [9],

We may write fundamental equation as below. The equation of continuity for unsaturated

medium is given by,

( ) (7.1)

Here,

= Bulk density of the medium,

θ = Moisture content of the dry weight,

M = The mass flux of moisture.

Now, we take the water motion from the basic Darcy’s law,

(7.2)

Where,

∇ = The gradient of whole moisture potential,

V = The volume flux of moisture potential,

K = The coefficient of aqueous conductivity.

From equation (7.1) & (7.2) we got,

( ) (7.3)

Where, is the flux density.

Since in the present case we consider that the flow takes place only in the vertical

direction, equation (7.3) becomes,

84

Ground Water Recharge in Vertical Direction in Uni Dimensional through Porous Media


(

)

(7.4)

Here,

𝜓 = Capillary pressure potential,

g = The gravitational constant, and = ψ − gz.

The positive direction of the z-axis is the same as that of the gravity.

Considering θ and ψ to be conducted by a single valued function, we may write (7.4),

(

)

(7.5)

Where

and is known as diffusivity coefficient.

Replacing D by its average value and assuming, = 0.232, we get

(7.6)

Taking water table is situated at a depth L, and putting,

(7.7)

Uni-dimensional groundwater recharge through porous media with linear permeability is,

(7.8)

Where Z = Penetration depth (dimensionless)

T = Time (dimensionless)

= Flow parameter ( )

85


With appropriate boundary conditions are,

, ( ) (7.9)

Where the moisture content throughout the region is zero, at the layer Z = 0 it is , and at

the bottom means the water table Z = L it is assumed to remain 100% throughout the

process of investigation.


We choose initial condition and boundary conditions are,

, ( ) (7.10)

Using Homotopy perturbation Transform method,

Homotopy [ ] for equation (7.8) is define as

[

] + *

+ (7.11)

*

+

(

*

+

*

+

*

+)

86

Ground Water Recharge in Vertical Direction in Uni Dimensional Through Porous Media


*

+

[

]

*

+

*

+

*

+


–

(

)

(

)

(

)

………………………………….

Solving all above partial differential equation we get,

[ ]

[ ]

[ ]

…………………………

87

Solution by the Homotopy Perturbation

Solution of equation (7.11) can be written as,

[ ] [

]

[ ]

(7.12)

Equation (7.12) is a solution of Equation (7.8)

7.4 Numerical Values and the Graphical Presentation:

The nonlinear partial differential equation arising in ground water porous media has been

solved using Homotopy Perturbation Method. Table values and graph have been obtained

using Microsoft excel 2010 and Origin Lab. Table 7.1 shows the numerical values of depth

Z and time T and for and Figure 7.2 and 7.3 show the moisture content with

respect to time and depth by HMP and Figure 7.4 Shows the moisture content with respect

to time and depth by VIM.

TABLE 7.1: Numerical values table of moisture content Z at different Time T and

Z/T 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0 0 0 0 0 0 0 0 0 0 0

0.1 0.04066 0.0413446 0.04203228 0.04272 0.0434076 0.0440953 0.044782 0.0454705 0.046158 0.0468459

0.2 0.08987 0.0906887 0.09151168 0.092335 0.0931576 0.0939805 0.094804 0.0956264 0.096449 0.0972723

0.3 0.14898 0.1499546 0.15093375 0.151913 0.1528919 0.153871 0.15485 0.1558291 0.156808 0.1577873

0.4 0.21953 0.2206836 0.22184256 0.223002 0.2241605 0.2253194 0.226478 0.2276373 0.228796 0.2299552

0.5 0.30327 0.3046311 0.30599699 0.307363 0.3087287 0.3100945 0.31146 0.3128261 0.314192 0.3155578

0.6 0.40219 0.4037954 0.40539883 0.407002 0.4086057 0.4102091 0.411813 0.4134158 0.415019 0.4166226

0.7 0.51857 0.5204485 0.52232444 0.5242 0.5260761 0.527952 0.529828 0.5317036 0.533579 0.5354553

0.8 0.65499 0.6571724 0.6593603 0.661548 0.663736 0.6659239 0.668112 0.6702995 0.672487 0.6746752

0.9 0.81435 0.8168984 0.81944316 0.821988 0.8245326 0.8270774 0.829622 0.8321668 0.834711 0.8372563

1 1 1 1 1 1 1 1 1 1 1

88

Ground Water Recharge in Vertical Direction in Uni Dimensional through Porous Media


FIGURE 7.2: Moisture Content vs. Time T and Depth Z

FIGURE 7.3: Moisture Content vs. Time T and Depth Z

89

Conclusion

FIGURE 7.4: Moisture Content vs. Time T and Depth ξ

(Graph - Variational Iteration Method)

7.5 Conclusion :

A specific problem of one dimensional flow in unsaturated porous media under certain

assumptions is discussed and its solution is obtained by Homotopy perturbation method.

The equations (7.12) represent an approximate solution of moisture content in terms of

and T.

The graphical behavior of solutions obtained by Homotopy Perturbation Method is

compared with the behavior of graphical presentation which is obtained by the Variational

iteration method and as a result the behavior is approximately same in both methods.

Figure (7.2), (7.3) and (7.4) present the graphical behavior of the solutions obtained by

HMP and VIM.

It is interpreted from the both graph time increases, the moisture content also increases at

each point in the basin and after some time, it becomes constant. After some particular

time, optimum moisture content rises with an increase in length.

90

7.6 Utilities of the Problem:

Because of the ground water recharge the salinity of the soil can be reduced because of the

increase of moisture content. Due to the increase in moisture content the fertility of soil

increases which helps the farmer in growing up a qualitative crop and in this case

production of the crop will also increase and quality of the ground water also increase.

91

CONCLUSION

The various mathematical models of different phenomena like imbibition phenomenon, the

counter-current imbibition phenomenon, the instability (fingering) phenomenon, the finger-

imbibition phenomenon have been solved successfully using Homotopy Perturbation

Method. The phenomenon of uni dimensional vertical groundwater recharge through

unsaturated porous medium has been studied too. The approximate analytical solution and

exact solution have been solved using appropriate initial and boundary conditions.

It is found that for fixed time, the saturation of the injected water increases as the

distance increases. When the saturation of entering water will increase, oil will be

displaced towards the production well. The solutions can be useful to determine the

amount of water required for injection and for the prediction of oil recovered.

It is found, the moisture content increases at each point in the basin and after some

time, it becomes constant. After some particular time, optimum moisture content

rises with an increase in length in ground water recharge process.

Homotopy Perturbation Method has been used to solve the problems. The solutions are

explained graphically as well as numerically using Microsoft excel 2010, Mat lab and

Origin Lab and hence get the exact idea of saturation of water in oil recovery process.

Through the study, the researcher came to the conclusion that the saturation of water level

increases with time and space as we inject the water and saturation of water decreased with

time and space in some examples as per our assumptions. At the same time, the moisture of

soil also gets increased when the saturation of water gets increased.

92

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100

LIST OF PUBLICATIONS

List of Publications

1. A Solution of Fluid flow through Porous medium Equation by Homotopy Perturbation

Transform Method, International Research Journal of Engineering and Technology,

3(7), (2016), 1848-1852.

2. An Analytical Solution of Non – Linear One dimensional Diffusion Equation By

Homotopy Perturbation Method, Journal of Applied Science & Computations,5(6),

(2018),328-333.

3. Solving Burger’s Equation Arising in Fingering Phenomenon by Laplace Transform

Method, International Journal of Management Technology and Engineering,10(11),

(2018),1313-1317.

4. A study of linear and nonlinear diffusion equations arising in fluid flow through porous

Media by Homotopy Perturbation Method, Journal of Emerging Technologies and

Innovative Research, 5(12), (2018),650-659.

Details of work presented in conference

1. The paper entitled as “A Study of nonlinear diffusion equation arising in fluid flow

thorough porous media by Homotopy Perturbation Method”, Advance in Pure and

Applied Mathematics, Mehsana Urben Institute of Science, Ganpat University, Kherva,

(2017).

MATHEMATICAL APPROACH IN DIFFERENT... · A Thesis submitted to Gujarat Technological University For...

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Transcript of MATHEMATICAL APPROACH IN DIFFERENT... · A Thesis submitted to Gujarat Technological University For...