Z ZHU ET AL - ustc.edu.cnstaff.ustc.edu.cn/~zuojin/arts/CNME-147-02.pdf · Z ZHU ET AL Ltd key w...

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

Commun� Numer� Meth� Engng �� Prepared using cnmauth�cls �Version� �� v��

Numerical Simulation of Two Dimensional Transient Water

Driven Non�Newtonian Fluid Flow in Porous Media y

Zuojin Zhu� Qingsong Wu� Chunfu Gao

and Xiuyi Du

�� Department of Thermal Science and Energy EngineeringInstitute of Engineering Science

University of Science and Technology of ChinaAnhui� Hefei� �� P�R� China

� Exporation and Development Research Institue of Jiang HanOil Field� Hubei� �� P�R� China

SUMMARY

Numerical simulation of two dimensional transient water driven non�Newtonian �uid �ow in porous

media has been performed� The hyperbolic non�Newtonian �uid model was used to describe the

characteristics of non�Newtonian �uid �ow� Governing equations were �rst approximated by implicit

�nite di�erence� and then solved by a stabilized bi�conjugate gradient �Bi�CGSTAB approach� A

comparison of the numerical results for the case of water driven Newtonian �uid was made to validate

the numerical method� For water driven Newtonian �uid �ow� it was found that the numerical results

are satisfactorily consistent with those obtained by commercial softwareVIP which is the abbreviation

of Vector Implicit Procedure for numerical simulation of Newtonian �uid �ow in porous media� The

maximum deviation for average pressure is less than �� the distribution of water saturation is

almost the same as that obtained by VIP� For water driven non�Newtonian �uid �ow in porous

media� it was found that the limit of pressure gradient of the non�Newtonian �uid has signi�cant

e�ects on the process of oil recovery� The correction of numerical simulation based on the global mass

balance plays an important role in oil reservoir simulation� Copyright c� �� John Wiley � Sons�

�Correspondence to� Z� Zhu� Department of Thermal Science and Energy Engineering� Institute of EngineeringScience� University of Science and Technology of China� Anhui� Hefei� �� P� R� China� E�mail�zuojin�ustc�edu�cn

Received �� April ��

Copyright c� �� John Wiley � Sons� Ltd� Revised � July ��

� Z� ZHU ET AL�

Ltd�

key words� Transient two phase �ow in porous media� Hyperbolic non�Newtonian �uid model�

Stabilized bi�conjugate gradient algorithm�

�� INTRODUCTION

With the development of computer science and technology� numerical simulation of oil reservoir

has become an important tool in petroleum engineering� Thus the implementation of e�cient

numerical methods for this purpose is of great signi�cance�

For non�Newtonian �uid �ow in porous media� series models have been proposed� such as

bi�linear model Mirzadjanzade� �� hyperbolic modelMolokovich� � �� power law model

Bird� �� and Bingham model Entov et al�� Wu et al�� among which the

latter two models have been extensively used� For example� for one dimensional immiscible

displacement of the Newtonian �uid by a non�Newtonian one in porous media� an analytical

solution of Buckley�Leverett type was obtained and validated by the numerical results based

on power law model�Wu et al�� A general simulator�TOUGH� for multiphase �ow in

porous media has been developed by Wu and Pruess �� where both the power�law and

Bingham non�Newtonian �uid models were employed� An attempt to explain the non�Darcy

e�ects of non�Newtonian �uid �ow has been presented by Ma and Ruth � �� Recently� the

challenges and approaches for multiphase �ow and transport in heterogeneous porous media

have been reported in detail by Miller et al��

In this study� the hyperbolic non�Newtonian �uid model� which has been justi�ed by

Mychidiniv et al� �� is employed� Finite di�erence approximation was used to obtain the

Copyright c� �� John Wiley � Sons� Ltd� Commun� Numer� Meth� Engng ��

Prepared using cnmauth�cls

NUMERICAL SIMULATION OF NON�NEWTONIAN FLUID FLOW IN POROUS MEDIA �

discretised equations� which were solved by the Bi�CGSTAB algorithm developed by Von Der

Vorst �� For Newtonian �uid �ow� it was found that the numerical results are satisfactorily

consistent with those given by VIP� a commercial software developed by using Vector Implicit

Procedure has been widely used in oil reservoir simulation� For the problem on hand� it was

found that the limit of pressure gradient has pronounced in�uences on the water fraction in

the liquid of production and the amount of residual oil in the reservoir�

�� THE GOVERNING EQUATIONS

�� The Governing Equations

Consider water driven non�Newtonian �uid �ow in porous media� it is postulated that�

�� Water is Newtonian �uid� oil is visco�plastic non�Newtonian �uid�

�� The two�phase system is isothermal and under a pressure beyond the bubbling point of

pressure of oil thase�

�� Both �uids are micro�compressible� but the porous medium is heterogeneous�

Let q� denote the production or injection rate under standard storage condition� B� denote

volumetric coe�cient� uj� denote velocity of �th phase in the porous medium� From mass

conservation law� the continuity equation can be written as

�

�t

�S��

B�

�� q� �

�uj��xj

��

where the �ow velocity is given by

u� � �kijkr��

f��p� � �� jxj��p��

and � � �� is the unit vector in the vertical direction� and x is the positional vector� Here

� � � represents water phase and � � � represents oil phase� From hyperbolic non�Newtonian




model� the modi�cation factor is

f��p� � �� jxj� �

��

� for � � �

j�p��jxjj

��p��j�p��jxjj

�for � � �

��

Substituting for u� into the continuity equation�� we have the governing equation

�

�t

�S��

B�

�� q� �

�

�xj

�ij��

��p��xj

� ��kxk�

�xj

��

where � is the porosity of the porous medium� S� is the saturation of �th phase� For two

dimensional �ow� it is clear that ��x� � �� The transmissibility ij�� of phase � is given by

ij�� kijkr��B�

f��p��

�� The Supplementary Relations

The constrain condition for saturation is

�X��

S� � � ��

The relative permeability� the capillary pressure are assumed to be functions of water

saturation � � ��

kr� � kr�S�� pc � p� � p� � pcS��

Finally� the micro�compressible property for both �uids requires

B� � B�� C�p� � p��

�� C�p� � p��

�� C��p� � p��

�� pb�� C��p� � pb��

� � �� Crpav � p�av��




where the superscript � denotes the state at the pressure for reference point x� � x�� x�� x��

C� and C�� are the compressibility of �uid and the visco�pressure index of phase �� Cr is

the compressibility of porous medium� The subscript av indicates the arithmetic mean� e�g�

pav � p� � p�� Additionally� pb is the bubbling point of pressure for oil phase�

�� Initial and Boundary Conditions

Solutions of the governing equations �� must be sought which satisfy the initial and boundary

conditions described as follows�

�� Initial Conditions

p� jt�� p�x� �� S� jt�� S�x� ��

�� Boundary Conditions

The inner condition has the form�

Z ��

�

�ur��h�md� � �Q��m ��

where m � �� M denotes the well number in the considered oil reservoir� with M to be

the total well number� ur is the magnitude of radial velocity of �ow in the porous media at a

well with number m� � � � denotes the second phase� and h is the perforation thickness of oil

layer at the location of an oil well�

The outer condition is written as

��p��n

� ��

�n�jxj��

�

�

�

� � ��

where � is the boundary of the domain for simulation with n to be its unit normal vector on

the boundary of the domain�




�� THE NUMERICAL METHOD

�� The Discretisation of the Governing Equations

Since the choice of both pressures as the mandatory variables leads to a di�culty in the

determination of water saturation �eld in the latter stage of oil recovery� Thus� S� is taken as

the mandatory variable� Taking the pressure potential as an alternative of pressure p� gives

rises to a choice to simplify the governing equation for the non�Newtonian phase� Accordingly�

by de�ning � � �p��kxk� and after some algebraic operations� we obtain

�

�t

�S��

B�

�� q� �

�

�xjij��

�

�xj �

� �

�xjij��p

�c

�

�xjS��

�

�xj�ij��

�

�xj�kxk��

�

�t

�S��

B�

�� q� �

�

�xjij��

�

�xj � � �

which are discretised by a �nite di�erence approximation see� Aziz et al�� The change

of capillary pressure in a time interval is neglected� This implies that pc is very small as

compared with the pressure p� in porous media� To maintain the physical meaning of the

numerical solution� the relative permeability is upstream weighted�

�� The Bi�CGSTAB Algorithm

The discretised equations of the governing equations can be written as

AX � B ��

Since both the relative permeability and capillary pressure are closely related to the water

saturation� and the �ow velocity in the porous media is related to the pressure gradient� the

problem considered is strongly coupled with high non�linearity� The convergence history of

general conjugate gradient method is not better than that of Bi�CGSTAB which was used to



NUMERICAL SIMULATION OF NON�NEWTONIAN FLUID FLOW IN POROUS MEDIA

perform the inner iteration� Due to the non�linearity of the problem� the outer iteration is

required� Assuming X�� let � equal a positive small number� say �� we can write the

outer iteration procedure as the following pseudo code�

�� Evaluate X�� by solving equation �� in term of Bi�CGSTAB�

�� Update A and B based on X��

�� Check k AX�� B k k B k� ��

If CONVER�TRUE��

Terminate the outer iteration�

Else

let � � � � � and return to step ��

Endif�

The inner iteration based on the Bi�CGSTAB algorithm can be written as�

�� Select E � diagfAg as a pre�conditioner� let iteration level s � �� and Xs� � X� � then

calculate residual rs� � B� AXs�� and let �r � rs��

�� For s � ��

�s�� rT rs��

if �s�� method fails�

if s � �

vs�� rs��

else

�s�� s��

��s��s��

�s��

vs�� rs�� s��v

s�� s��v

s��

endif�




�� solve E �v� � vs��

vs�� A �v�

�s� � ��s��

�rT v�

v� � rs�� s�vs��

Check if j v� j� �� if hold� Xs� � Xs�� v�� iteration terminated� otherwise� continue

step ��

�� Solve E �v� � v�

v � A �v�

�s� � vT� v�

vT� v�

Xs� � Xs�� s� �v� � �s� �v�

rs� � v� � �s�v

Check the convergence� continue to step � if necessary�

The evaluated X must satisfy the global mass balance equation� based on which a correction

term �X can be obtained to improve the numerical results�

�� NUMERICAL VALIDATION

To validate the numerical method described above� an isolate reservoir in Jiang Han Oil Field

was selected� Exploration of fossil resources in this region began at the end of �� Since then�

it has produced petroleum for about forty years by using water injection� It was found that

the oil viscosity in this region is low� and the thermal e�ects can be neglected� Depth of oil

layer is about ��m� and the original pressure is about � atm� The reason to do this choice

is that the production data has been �tted by the commercial software VIP which has been



NUMERICAL SIMULATION OF NON�NEWTONIAN FLUID FLOW IN POROUS MEDIA

widely used in petroleum engineering� where Newtonian �uid model and conjugate gradient

method were incorporated� The evolution of average pressure during the time range from the

end of �� to the end of June in �� are illustrated in Figure �� It is found that the curve

for pmean obtained by present method coincides completely with the results given by VIP in

the �rst three years of oil recovery� deviation appears at the latter stage� The primary reason

arises from the di�erent treatment of wells� It is seen that the largest deviation is about ��

atm� which occurs at t � �� Days� This is less than �� On the other hand� a satisfactory

agreement is also observed from the distribution of water saturation shown in Figure � a� and

b�� at the instant of t � �� Days�

�� THE APPLICATION TO SIMULATE THE RESERVOIR AT BA� MIAN� HE

The numerical method including Bi�CGSTAB was used for the simulation of water driven non�

Newtonian �uid �ow in porous media� The rock parameters and �uid properties were chosen

with respect to the reservoir at Ba Mian He�

The parameters required for the numerical simulation of non�Newtonian reservoir are

illustrated in Table �� where the over�bar of a parameter implies the volumetric average over

the whole reservoir� while S�max and S�min are respectively the connate and the irreducible

saturation for oil phase�

In accordance with the balance of vertical forces� the initial pressure and water saturation

�eld can be obtained� as seen in Figure �� where the values of pressure and water saturation in

the non�porous region are assigned to be zero� and water saturation in the pure water region

is of course unity�

Figure � a� and b� show the dependence of relative permeability capillary pressure pc on



�� Z� ZHU ET AL�

water saturation� These curves are important in the numerical simulation of an oil reservoir�

The capillary pressure in the sub�surface system is much smaller than the initial pressure�

indicating that it is permissible to neglect the time variation of pc during a time interval�

The variations of the comprehensive water fraction and the amount of residual oil are shown

in Figure � a� and � b�� It is found that a relatively large value of fw occurs when the limit

of pressure gradient � is large� corresponding to more residual oil in the reservoir�

Figure � a��b�� c�� and d� show respectively the time evolutions of water injection rate

Qw� oil production rate Qo� liquid production rate Ql � Qo � Qw� and the average pressure

pmean for four cases when � � �� Pam� Note that � � � means that the oil is

Newtonian �uid� The evolution of average pressure is dominated by the operating conditions

of oil recovery and the �ow performances of the two�phase system� It is observed from Figure

� that the dependence of evolutions on � is signi�cant�

The pressure �elds for � � �� Pam in the reservoir at Ba�Mian�He at the instant

of t � �� Day are illustrated in Figure a� and b�� From the comparison of shaded

areas between Figure a� and b�� it is observed that there is a pronounced di�erence

between the pressure �eld in a Newtonian reservoir and that in a non�Newtonian reservoir�

More dense contours occurs in the non�Newtonian reservoir indicating there exists a larger

pressure gradient �eld�

�� CONCLUSIONS

The Bi�CGSTAB method developed by Von Der Vorst was used to simulate the two

dimensional transient two�phase �ow of water driven non�Newtonian �uid �ow in porous

media� The role of the global mass balance was stressed and used to correct the solution of



NUMERICAL SIMULATION OF NON�NEWTONIAN FLUID FLOW IN POROUS MEDIA ��

discretised equations� For a de�nite small reservoir with Newtonian oil� the results obtained by

the method introduced were compared with that given byVIP i�e� Vector Implicit Procedure��

A satisfactory agreement was obtained� The non�Newtonian property shows signi�cant e�ects

on water driven non�Newtonian �uid �ow in porous media� The application of the numerical

method indicates that it has a potentiality in oil reservoir simulation�

ACKNOWLEDGEMENTS

This work is �nancially supported by Exploration � Development Institute of Jiang Han Oil Field�

SINOPEC� and supported by State Key Laboratory of Oil� Gas Reservoir Geology and Exploitation

with Grant No� PLN�� and Chinese National Science Foundation with Grant No� ��

Special gratitude is devoted to the anonymous reviewers for those useful comments�

REFERENCES

�� Aziz K�� and Serrari A�� Petroleum reservoir simulation� Applied Science Publication Ltd� London� ��

�� Bird R�B�� Stewart W�E�� and Lightfoot� E�N�� Transport phenomena� Wiley� New York� ��

� Bernadiner M�T�� Entov B�M�� Fluid dynamic theory of abnormal liquid �ltration� Science Publisher�

Moscow� �� In Russian��

� Ma H�� and Ruth D�� Physical explanations of non�darcy e�ects for �uid �ow in porous media� SPE�

Formation Evaluation� ��

�� Miller C� T�� Christakos G�� Imho� P� T�� Mcbride J� F�� Pedit J� A�� and Trangenstein J� A�� Multiphase

�ow and transport modeling in heterogeneous porous media� challenges and approaches� Advances in

Water Resources � �� No�� pp��

� Mirzadjanzade A�KH�� Fluid dynamic problems of visco�plastic and visco liquids in petroleum recovery�

Aznefteizdat� �� In Russian��

�� Molokovich U� M�� Ckworshov E� W� A measurement of �ltration of compressible visco�plastic liquid�

Kasan� �� In Russian��




�� Mychidinov N�� Mykimov N� and Cadeikov M� K�� Numerical simulation of non�linear �ltration� Tashkent�

Fan publisher� pp�� In Russian�

�� Von Der Vorst H�� Bi�CGSTAB� A fast and smoothly converging variant of BICG for the solution of

non�symmetric linear systems� SIAM� Journal on Scienti�c and Statistical Computing� �� pp ��

��

�� Wu Y�S�� Pruess K�� and Witherspoon P�A�� Flow and displacement of bingham non�Newtonian �uids in

porous media� SPE ��

�� Wu Y�S�� Pruess K�� and Witherspoon P�A�� Displacement of a Newtonian �uid by a non�Newtonian �uid

in a porous medium� Transport� in Porous Media �� pp��

�� Wu Y�S�� and Pruess K�� A numerical method for simulating non�Newtonian �uid �ow and displacement

in porous media� Advances in Water Resources � �� pp��




Nomenclature

B� � volumetric coe�cient Greek Symbols

C� � compressibility of �uid � � limit of pressure gradient

C�� visco�pressure index �� g � speci�c gravity

f� � factor of nonlinearity ��jxj � gradient of depth

fw � water fraction �� viscosity of �th phase �uid

h layer thickness � � pressure potential for oil

k � � temsor of permeability � � �uid density

kr� � relative permeability �� initial �uid density

n � unit normal vector on ij�� tensor of transmissibility

the boundary of the domain Superscript

pb � bubbling pressure of oil � � referred point� initial

p� � pressure of �th phase b � bubbling point

q� � source term l � time level� or liquid

pc � capillary pressure s � iteration level

�Q��m � production or injection rate � � derivative

for �th phase by a mth well Subscripts

Qo � oil production rate b well�bore

Qw � water production rate � �th phase

Ql � production rate of liquid ij grid node number

Qmro � amount of residual oil m well number

S� � saturation for �th phase o � oil

u� � velocity vector r remained� radial

w water




Table I� Parameters Used for Reservior Simulation

ky� �� m� � ��

h � �� m S�max��

pb � �� Pa S�min��

�� Pas rw�� m

C�� Pa C�� Pa

C� � �� Pa C� � �� Pa

Cr � �� Pa

y Here the over bar for k � and h means the average over

the whole volume of the reservoir considered�

t (Day)

pm

ean

(10

5 Pa)

0 2500 5000 750060

80

100

120

140

160

180

200

220

VIPBi-CGSTAB

Figure � A comparison of average pressure where dashed curve is given by

VIP� solid curve is obtained by making use of the present method�




0

5

10

15

20

25

j

0 10 20 30 40 50i

1

1

1

1

1

2

22

2

23

2

22

3

4

3

2 4

3

4

2

2

4

33

43

4 5

5 5

4

5

3

3

45

4

4

5

5

5 5

5 4

77

4

3375

5

54

46 5

5

6

6

5

5

6

6

57

6

7

6

67

4

6

7 66

5

6

86

7

8

6

7

8

8

8

8

(b)

0

5

10

15

20

25

j

0 10 20 30 40 50i

111

1 11

3

12

2

33

3

23

34

2

22

2

3

3

3

3

4 43

4

3

3

53

6243

5

4

4

56

6

5

5

5

5

6

6

66

7

7

5

76

7

57

68

5

7

7

6

6

5

8

7 6

8

7

7

7

6

8

788

7

7

7

(a)

Figure �� A comparison of water saturation at the moment of t � ��

�Days from the beginning of oil recovery� �a Obtained by VIP� �b Obtained

by Bi�CGSTAB� The curves labelled �� are for the values of S� �

��




0

50

100

p 20(a

tm)

0

50

100

i0

25

50 j(a)

00.250.50.751

S10

0

50

100

i0

25

50 j(b)

Figure �� The initial �elds of �a pressure� and �b water saturation�

Saturation of Water

Rel

ativ

eP

erm

eabi

lity

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

kr1

kr2

(a) Saturation of Water

Cap

illar

yP

ress

ure

(atm

)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Pc(S1)

(b)

Figure �� The dependence of �a the relative permeability for each phase� and

�b the capillary pressure pc on water saturation�




t (Day)

Qm

ro(1

07kg

)

0 1000 2000 3000 4000

158

160

162

164

166

168

170

(b)

1

23

4

t (Day)

f w(%

)

0 1000 2000 3000 40000

10

20

30

40

50

60

70

80

90

100

(a)

1

2

3

4

Figure �� The time variations of �a water fraction �b residual oil� The curves

labelled �� and � are appropriate for values of the limit of pressure gradient

� � �� Pa�m� respectively�




t(Day)

Ql(m

3/D

ay)

1000 2000 3000 40000

50

100

150

200

250

300

350

400

450

(c)

1

2

3

4

t(Day)

Pm

ean(a

tm)

1000 2000 3000 4000100

110

120

130

140

150

160

170

180

190

200

210

220

(d)

1

2

3

4

t(Day)

Qo(

m3 /D

ay)

1000 2000 3000 4000

10

20

30

40

50

60

70

(b)

1

2

3

4

t(Day)

Qin

j(m3 /D

ay)

1000 2000 3000 40000

50

100

150

200

250

300

350

400

(a)

1

2

3

4

Figure �� The evolutions of �a the water injection rate Qw for the case of

� � � �� Pa m �b oil production rate Qo �c liquid production

Ql �d the average pressure in the porous media� The curves labelled ��

and � are appropriate for the cases of � � � �� Pa m� respectively�




020406080100120i

0

25

50

j

(a)

020406080100120

i

0

25

50

j

(b)

Figure �� The predicted contours of pressure at the moment of �ve years�

simulation� �a for Newtonian oil �b for non�Newtonian oil when � �

��Pa m� The curves labelled �� are for values of p� � �� p ��

��p �� p ��atm� where �p � ��atm�



Date post:	07-Sep-2019
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Z ZHU ET AL - ustc.edu.cnstaff.ustc.edu.cn/~zuojin/arts/CNME-147-02.pdf · Z ZHU ET AL Ltd key w...

Documents