porous media transport phenomena (civan/transport phenomena) || fluid transport through porous media

CHAPTER 7

FLUID TRANSPORT THROUGH POROUS MEDIA

7.1 INTRODUCTION

Single - and multiphase mass and momentum balance equations are coupled under isothermal conditions to derive a hydraulic diffusion equation, and its applications are demonstrated by several examples. *

First, a generalized leaky - tank reservoir model is developed, including the non - Darcy and generalized fl uid effects (Civan, 2002e ). The resulting model equa-tions are applied for a typical case. It is demonstrated that special plotting schemes can provide a practical technique for the determination of the thickness, permeabil-ity, and the water - drive strength of petroleum reservoirs from deliverability test data, including the non - Darcy effect due to converging fl ow pattern in the near - wellbore region, showing the effect of convective acceleration.

Next, convenient formulations of the immiscible displacement in porous media are presented and applied for waterfl ooding. The macroscopic equation of continuity for immiscible displacement is derived. Richardson ’ s (1961) approach and the fractional fl ow formulation are extended and generalized for anisotropic and heterogeneous porous media. The integral transformations according to Douglas et al. (1958) and the coordinate transformations lead to differential equations, which do not involve the variable fl uid and porous media properties explicitly in the dif-ferential operators. Fractional fl ow and unit end - point mobility ratio formulations are also derived for specifi c applications to reduce the computational requirements and to accomplish rapid simulation of waterfl ooding of petroleum reservoirs. The resulting equations can be discretized and solved more conveniently and accurately

Porous Media Transport Phenomena, First Edition. Faruk Civan.© 2011 John Wiley & Sons, Inc. Published 2011 by John Wiley & Sons, Inc.

177

* Parts of this chapter have been reproduced with modifi cations from the following:

Civan, F. 1996b. Convenient formulation for immiscible displacement in porous media. SPE 36701, Proceedings of the 71st SPE Annual Tech Conference and Exhibition (October 6 – 9, 1996), Denver, Colorado, pp. 223 – 236, © 1996 SPE, with permission from the Society of Petroleum Engineers;

Civan, F., 2000e. Leaky - tank reservoir model including the non - Darcy effect. Journal of Petroleum Science and Engineering, 28(3), pp. 87 – 93, with permission from Elsevier; and

Penuela, G. and Civan, F. 2001. Two - phase fl ow in porous media: Property identifi cation and model vali-dation, Letter to the Editor. AIChE Journal, 47(3), pp. 758 – 759, © 2001 AIChE, with permission from the American Institute of Chemical Engineers.

c07.indd 177c07.indd 177 5/27/2011 12:30:01 PM5/27/2011 12:30:01 PM

178 CHAPTER 7 FLUID TRANSPORT THROUGH POROUS MEDIA

than the conventional formulation, which requires cumbersome discretization for-mulae for mixed derivatives involving the fl uid and porous media properties. The convenient formulations offer potential advantages over the usual formulation used in the simulation of waterfl ooding, such as improved accuracy and reduced compu-tational effort.

Finally, models facilitating the streamline and stream tube or fl ow channel concepts, which are frequently used for description of fl ow in porous media, are described. It is shown that these models offer certain advantages such as conve-nience, practicality, reduced computational effort, and insight into fl ow patterns. Suitable fl ow problems involving porous media are formulated as potential fl ow problems and are solved analytically by means of the methods of superposition, images, and front tracking.

7.2 COUPLING SINGLE - PHASE MASS AND MOMENTUM BALANCE EQUATIONS

Consider the equation of mass conservation given by (Civan, 2002e )

∂( )

∂+ ∇ ⋅( ) = ∇ ⋅ ⋅∇( )[ ] +φρ ρ φρ

tru D , (7.1)

where r denotes the source term and D is the hydraulic dispersion coeffi cient tensor. Eq. (7.1) can be rearranged as

∂( )

∂+ − ∇ ⋅( ) ⋅∇( ) + ∇ ⋅ = ⋅∇ ( ) + =φρ φρ φρ φρ

φtrv D v D v

u2 , . (7.2)

If ρ = ct., then Eq. (7.2) becomes

∂∂

+ − ∇ ⋅( ) ⋅∇ + ∇ ⋅ = ⋅∇ + =φ φ φ φρ φt

rv D v D v

u2 , . (7.3)

If φ = ct., then Eq. (7.2) becomes

∂∂

+ − ∇ ⋅( ) ⋅∇ + ∇ ⋅ = ⋅∇ + =ρ ρ ρ ρφ φt

rv D v D v

u2 , . (7.4)

If φ = ct. and ρ = ct., then Eq. (7.2) becomes

∇ ⋅ = =v vur

φρ φ, . (7.5)

Now, neglect the dispersion term in Eq. (7.1) so that it can be written as

∂( )

∂+ ∇ ⋅( ) =φρ ρ

tru . (7.6)

Consider Darcy ’ s law for horizontal fl ow, given by

uK

p= − ∇μ

. (7.7)

c07.indd 178c07.indd 178 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM

7.3 CYLINDRICAL LEAKY-TANK RESERVOIR MODEL INCLUDING THE NON-DARCY EFFECT 179

The isothermal fl uid and pore compressibility factors are defi ned by

cp

cp

c p p c c co o Tρ φ ρ ρ φρρ

φφ ρ ρ= ∂

∂= − ∂

∂= −( )[ ] = +1 1

, , exp , . (7.8)

Combining Eqs. (7.6) – (7.8) yields

∂∂

= ∇ + ∇( ) + ∇ ⋅( )[ ]∇

+−( )[ ]

p

tp c p

cc p

r

c c p p

h hT

T

T o o

α αφ

φρ

φ ρ

ρ

ρ

2 2 1

exp, αα

φρhT

K

c= ,

(7.9)

where αh denotes the hydraulic diffusivity. If all the parameters are constants, there is no source, and ∇( ) ≅p 2 0, then Eq.

(7.9) simplifi es as

∂∂

= ∇p

tphα 2 . (7.10)

This equation is referred to as the hydraulic diffusion equation.

7.3 CYLINDRICAL LEAKY - TANK RESERVOIR MODEL INCLUDING THE NON - DARCY EFFECT

Frequently, the cylindrical tank model, as shown in Figure 7.1 , is resorted to describe the fl ow within the reservoir drainage area of wells, for convenience in the formula-tion of the governing equations of fl ow and to generate analytic solutions. The resulting simplifi ed radial fl ow model is assumed to approximately represent the reservoir fl uid conditions within the drainage area and leads to simplifi ed analytic

Figure 7.1 Schematic leaky - tank reservoir model (after Civan, 2000e ; reprinted by permission from Elsevier).

ww

ww

ww

pwe

K,f rw rer

pw

pe

h

c07.indd 179c07.indd 179 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


solutions, which reduce the complexity of the reservoir engineering analysis and are justifi able in view of the uncertainties involving in most reservoir conditions.

Kumar (1977a,b) formulated the radial fl ow equations necessary for the deter-mination of the water - drive strength from transient well test data using Darcy ’ s law. Civan and Tiab (1991) extended Kumar ’ s (1977a,b) formulation for the non - Darcy effect. Civan (2000e) generalized and extended their methodologies and formula-tions as described in the following. This leads to improved and practical infl ow deliverability equations, considering the non - Darcy effects associated with the converging/diverging fl ow around wellbores.

Here, the leaky - tank model by Civan (2000e) is described. Its defi ning equa-tions and the conditions of solutions are derived for generalized applications, irre-spective of the type of the reservoir fl uid, which may be classifi ed as incompressible, slightly compressible, or compressible. The assumptions and their implications are expressed quantitatively. The potential effects of the non - Darcy fl ow and partial water - drive conditions are also considered by the application of Forchheimer ’ s equa-tion of motion. The leaky - tank model provides a valuable tool for the determination of reservoir formation permeability and thickness, and the strength of partial water drive from deliverability test data.

Consider the cylindrical leaky - tank model shown schematically in Figure 7.1 . This model is intended to approximate the drainage area of a well completed in a reservoir, undergoing production by a partial water - drive mechanism. Although the following derivation is carried out in the radial coordinate, the results can be readily transformed to linear and elliptic fl ow conditions. The porosity, ϕ , permeability, K , and thickness, h , of the reservoir formation are assumed constant. A partially acting aquifer surrounding the reservoir is considered. A piston (or unit mobility ration) displacement of the reservoir fl uid by the incoming water from the surrounding aquifer is assumed. The reservoir formation and fl uid are assumed isothermal. The reservoir is producing at a constant terminal rate. The shrinkage of the radius of the outer reservoir boundary due to water infl ux is neglected because it is much greater compared with the wellbore radius.

The formulations are carried out according to Civan (2000e) in terms of the quantities expressed in mass units to avoid complications involving the varying fl uid properties and to maintain generality irrespective of the fl uid types.

The bulk cylindrical area normal to fl ow at a radius r from the well centerline is given by

A rh= 2π . (7.11)

The volumetric and mass fl ow rates can be expressed, respectively, as

q Au= (7.12)

and

w q= ρ , (7.13)

where ρ and u are the density and volumetric fl ux of the fl uid. Frequently, it is convenient to express the fl ow rates in terms of the volumes

expressed at appropriate base conditions, such as the standard, usually taken as

c07.indd 180c07.indd 180 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


14.7 psia (1 atm) and 60 ° F (15.5 ° C) in the petroleum industry, or appropriate refer-ence temperature and pressure conditions, denoted here by the subscript b . Hence, Eqs. (7.12) and (7.13) can be written in alternative forms, respectively, as

q Aub b= (7.14)

and

w qb b= ρ . (7.15)

Applying Eqs. (7.12) – (7.15) , the following derivations can be readily expressed in conventional terms, that is, in pressure and volumetric fl ow.

The mass balance of the fl uid for the cylindrical leaky - tank model shown in Figure 7.1 is given by

d

dtr r h w we w e wπ φρ2 2−( )⎡⎣ ⎤⎦= − , (7.16)

in which t is time, r w and r e denote the wellbore and drainage area radii, and ρ is the average density of the fl uid in the reservoir. w e denotes the mass fl ow rate of the reservoir fl uid displaced due to water infl ux at the outer reservoir boundary, and w w denotes the constant terminal mass production rate of the reservoir fl uid at the wellbore.

Kumar (1977a) defi ned the ratio of the reservoir infl ux and effl ux rates as the water - drive strength factor by

f w w q qe w eb wb= = , (7.17)

where q eb and q wb are the volumetric water infl ux and well production fl ow rates expressed at the base conditions.

Although the radius of the leaky boundary shrinks during water infl ux, that is, r r te e= ( ), the variation of the outer radius is negligible at the actual fi eld conditions. Also, note that r rw e� , and h and ϕ are constants. Consequently, substituting Eq. (7.17) into Eq. (7.16) leads to the following simplifi ed mass balance equation:

π φ ρr h

d

dtf we w

2 1= − −( ) , (7.18)

subject to the initial condition given by

p p p ti i= = ( ) =, , .ρ ρ 0 (7.19)

Therefore, the analytic solution of Eqs. (7.18) and (7.19) yields

tr h

f wp pe

wi=

−( )( ) − ( )[ ]π φ ρ ρ

2

1, (7.20)

where p is the average reservoir fl uid pressure in the leaky - tank model given by (Dake, 1978 )

p p rh dr r r he w

r

r

w

e

= −( )⎡⎣ ⎤⎦∫ 2 2 2π φ π φ . (7.21)

c07.indd 181c07.indd 181 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


The equation of continuity in the radial coordinate is given by

∂( )∂

+∂( )∂

=φρ ρt r

r u

r

10. (7.22)

ϕ is constant and substituting Eqs. (7.11) – (7.13) into Eq. (7.22) results in

2 0π φ ρh r

t

w

r

∂∂

+ ∂∂

= . (7.23)

This equation can be solved subject to the following conditions. For all practical purposes, the reservoir fl uid conditions in a very large reser-

voir can be assumed stabilized under the infl uence of the outer boundary conditions of the reservoir, if the reservoir has been producing for a suffi ciently long time (Dake, 1978 ). Therefore, the stabilized state condition with respect to time can be approximated by Eq. (7.18) as

∂∂

≅ = −−( ) =ρ ρπ φt

d

dt

f w

r hw

e

12

constant, (7.24)

where (Kumar, 1977b )

f = 0 for semi - steady - state closed reservoir

zero water influ( ,

xx condition) , (7.25)

0 1< <f for intermediate - state leaky( external

,boundary partial watter influx condition) , (7.26)

f =1 for steady - state constant external boundary

pressure ful(

, ll water influx condition and) , (7.27)

f >1 for excess fluid injection condition. (7.28)

The conditions with respect to the radial distance or the boundary conditions at the wellbore and external radii are given, respectively, by

p p w w r rw w w= = − =, , (7.29)

and

, , .e e w ep p w w fw r r= = − = − = (7.30)

Eliminating ∂ ∂ρ t between Eqs. (7.23) and (7.24) yields

dw

drf w

r

rw

e

= −( )12

2, (7.31)

which, upon integration, subject to the leaky - boundary condition given by Eq. (7.30) , yields

w w f r rw e= − − −( )( )⎡⎣ ⎤⎦1 1 2 . (7.32)

The minus sign in Eqs. (7.19) , (7.20) , and (7.22) indicates that the fl ow direc-tion is opposite of the radial coordinate direction.

c07.indd 182c07.indd 182 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


The equation of motion in the radial coordinate is given by (Forchheimer, 1901, 1914 )

− ∂∂

= +p

r Ku u u

μ βρ . (7.33)

Following Civan and Evans (1998) , consider a pseudopressure function defi ned by

m p dpp

p

b

( ) = ∫ρμ

. (7.34)

Hence, substituting Eqs. (7.11) – (7.15) and (7.34) into Eq. (7.33) yields an alternative form of the Forchheimer equation as

−( ) = + ⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

dm p

dr

w

rh K

w

rh2

1

2πβμ π

. (7.35)

For all practical purposes, the effect of pressure on the fl uid viscosity is small. Thus, invoking Eq. (7.32) and the average fl uid viscosity defi ned by (Civan and Evans, 1998 )

μ μ μ≈ = −( )∫ dp p pp

p

2 1

1

2

, (7.36)

where p p p1 2< < is some representative range of pressure variation of the reservoir fl uid, such as p pw1 = and p pe2 = ; Eq. (7.35) can be solved, subject to the wellbore boundary condition given by Eq. (7.29) , to obtain an analytic solution as

m p m pw

hKn

r

rf

r r

r

w

h

ww

w

w

e

w

( ) − ( ) = ⎛⎝⎜

⎞⎠⎟− −( )⎡

⎣⎢− ⎤

⎦⎥

+

21

2

2

2 2

2π

βμ π

�

⎛⎛⎝⎜

⎞⎠⎟

− + − −( )⎡⎣⎢

− + −( ) − ⎤⎦⎥

2

2

23 3

4

1 12 1 1

3r rf

r r

rf

r r

rw

w

e

w

e

.

(7.37)

Applying Eq. (7.30) and rearranging Eq. (7.37) leads to the following linear form expressing the reciprocal pseudo - productivity index as a linear function of the production rate:

m p m p

wa bwe w

ww

( ) − ( ) = + , (7.38)

where, considering r rw e� ,

a n r r f hKe w≅ ( ) − ( ) −( )[ ] ( )� 1 2 1 2π (7.39)

and

br hw

≅( )β

μ π2 2. (7.40)

c07.indd 183c07.indd 183 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


Thus, f and ( Kh ) can be determined from a straight - line plot of

m p m p

wwe w

ww

( ) − ( )[ ]versus .

The preceding formulations are general, irrespective of the fl uid types. The present formulations can be readily applied for incompressible, slightly compress-ible, and compressible (gas) fl uids by substituting the following expressions for the fl uid density, respectively:

ρ ρ= =o constant, (7.41)

ρ ρ= −( )[ ]b bc p pexp , (7.42)

and

ρ = Mp

ZRT, (7.43)

where c is the compressibility coeffi cient, M is the molecular weight of the gas, Z is the real gas deviation factor, T is the absolute temperature, and R is the universal gas constant.

The data considered for the example are h = 10 m, K = 9.0 × 10 − 15 m 2 , β = 5.0 × 10 10 m − 1 , ϕ = 0.20, r w = 0.0508 m, r e = 9100 m, T = 410 ° K, p i = 4.83 × 10 7 Pa (or kg/m · s 2 ), and μ = × ⋅−1 25 10 5. /kg m s. The reservoir fl uid is a 0.65 - gravity natural gas. Using these data, the typical dimensionless average fl uid density versus dimensionless time, dimensionless mass fl ow rate versus dimension-less radial distance, and the reciprocal pseudo - productivity index versus mass pro-duction rate trends for f = 0, 0.5, 1.0, and 1.5 are plotted in Figures 7.2 – 7.4 , respectively.

When the actual fi eld measurements of fl owing bottom - hole pressure, p w , versus production rate, q wb , of a well and the fl uid properties data are available, a straight - line plot of the reciprocal pseudo - productivity index versus mass production rate similar to Figure 7.4 can be constructed by means of the least squares linear regression method. The values of the a and b parameters of Eq. (7.38) are determined

Figure 7.2 Dimensionless average density versus dimensionless time (after Civan, 2000e ; reprinted by permission from Elsevier).

0.00

0.50

1.00

1.50

0.00 0.25 0.50 0.75 1.00

Dimensionless time, wwt/(p re2hfri)

Dim

ensi

onle

ss a

vera

gede

nsity

, r/r

i

f = 1.5f = 1.0f = 0.5f = 0.0

c07.indd 184c07.indd 184 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


by the intercept and slope of this line, respectively. These values are substituted into Eqs. (7.39) and (7.40) , which are then expressed for h and Kh as

Kh r r f ae w= ( ) − ( ) −( )[ ] ( )ln 1 2 1 2π (7.44)

and

hr bw

= 1

2πβμ

. (7.45)

Thus, eliminating h between Eqs. (7.44) and (7.45) yields

f Kar b

r

rw

e

w

= + − ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥1 2

βμ

ln . (7.46)

The β value can be estimated using an appropriate empirical correlation, such as by Liu et al. (1995) (See Chapter 5 ).

Note that Eqs. (7.44) – (7.46) require an implicit solution approach for the determination of the h , Kh , and f values, because the Liu et al. (1995) or other available β correlations are given as functions of permeability K , as well as other parameters, such as porosity ϕ and tortuosity τ , whose values should be acquired by appropriate means.

Figure 7.3 Dimensionless mass fl ow rate versus dimensionless radial distance (after Civan, 2000e ; reprinted by permission from Elsevier).

Dim

ensi

onle

ss m

ass

flow

rat

e, |

w/w

w|

f = 1.5f = 1.0f = 0.5f = 0.0

0.00

0.50

1.00

1.50

0.00 0.25 0.50 0.75 1.00

Dimensionless radial distance, r/re

Figure 7.4 Reciprocal pseudo - productivity index versus mass production rate (after Civan, 2000e ; reprinted by permission from Elsevier).

Rec

ipro

cal p

seud

o-pr

oduc

tivity

inde

x,10

–13 [m

(pe)

–m(p

w)]

/ww(m

–3)

f = 1.5f = 1.0f = 0.5f = 0.0

2.00

2.10

2.20

2.30

2.40

0 0.025 0.05 0.075 0.1

Mass production rate, ww (kg/s)

c07.indd 185c07.indd 185 5/27/2011 12:30:02 PM5/27/2011 12:30:02 PM


7.4 COUPLING TWO - PHASE MASS AND MOMENTUM BALANCE EQUATIONS FOR IMMISCIBLE DISPLACEMENT

Formulations of the governing equations and boundary conditions for immiscible displacement in porous media are presented according to Civan (1996b) . Neglecting the capillary pressure effect, Buckley and Leverett (1942) obtained a simplifi ed formulation. This equation can be solved analytically using Welge ’ s (1952) method in an isotropic homogeneous media for one - dimensional linear, radial, and spherical fl ow. The formulation results in a convection - dispersion type of equation when the capillary pressure effect is included.

The formulations are applied to water/oil systems, and the generalized bound-ary conditions are derived. Convenient formulations are presented by transforming the governing equations by means of an integral transformation of the function and transformation of the coordinates. The fractional fl ow and end - point mobility ratio formulations are also presented for specifi c applications.

7.4.1 Macroscopic Equation of Continuity

The mass balance of phase j can be written as (Civan, 2002e )

∂∂( ) +∇ ⋅( ) −∇ ⋅ ⋅∇( ) + =

trj j j j j j j jε ρ ρ ε ρu D 0, (7.47)

where D denotes the hydraulic diffusivity tensor and ε j is the volume fraction of the fl owing phase j in porous media given by

ε φj j jiS S= −( ). (7.48)

ϕ is the porosity and Sj is the saturation of phase j and S ji is the immobile phase j saturation. rj denotes the loss of fl uid mass per unit bulk volume of porous media per unit time.

Adding Eq. (7.47) over all the fl uid phases yields the total mass balance equation as

∂( )∂

+∇ ⋅( ) −∇ ⋅ ⋅∇( ) + =φρ ρ φ ρt

ru D 0, (7.49)

in which the following average quantities have been used:

ρ ρ=∑Sj jj

, (7.50)

ρ ρu u=∑ j jj

, (7.51)

D D⋅∇ = ∇⋅∑ρ ρSj j jj

, (7.52)

and

r rjj

=∑ . (7.53)

c07.indd 186c07.indd 186 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM

7.4 COUPLING TWO-PHASE MASS AND MOMENTUM BALANCE EQUATIONS 187

For incompressible fl uids, invoking Eq. (7.49) into Eq. (7.47) yields a volu-metric balance equation as

∂( )∂

+∇ + =⋅φS

tqj

j ju 0, (7.54)

where

q rj j j= ρ . (7.55)

Thus, adding Eq. (7.54) over all the fl uid phases results in the following total volumetric balance equation:

∂∂+∇ ⋅ + =φ

tqu 0, (7.56)

in which

u u=∑ jj

(7.57)

and

q qjj

=∑ . (7.58)

7.4.2 Application to Oil/Water Systems

The macroscopic equation of continuity for the water phase can be obtained from Eqs. (7.47) and (7.48) as

∂∂( ) +∇ ⋅( ) −∇ ⋅ ⋅∇( ) + =

tS S rw w w w w w w wφ ρ ρ φ ρu D 0. (7.59)

In Eq. (7.59) , t is time, ∇ is the divergence operator, ϕ is the porosity, S w is the water saturation in the porous media, ρ w is the density of water, uw is the volu-metric fl ux of the water phase, Dw is the coeffi cient of hydraulic dispersion due to spatial variations in porous media, and r w is the mass rate of the water lost.

The volumetric fl uxes of the water and oil phases through porous media are given, respectively, by

u Kwr

ww w

kp g zw= − ⋅ ∇ + ∇( )

μρ (7.60)

and

u K Kor

oo o

r

ow c o

kp g z

kp p g zo o= − ⋅ ∇ + ∇( ) = − ⋅ ∇ +∇ + ∇( )

μρ

μρ , (7.61)

because the capillary pressure is defi ned as

p p pc o w= − . (7.62)

In Eqs. (7.60) and (7.61) , k rw and k ro , μ w and μ o , and ρ w and ρ o denote the relative permeability, viscosity, density, and pressure of the water and oil phases,

c07.indd 187c07.indd 187 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM


respectively. K, g , and z are the permeability tensor, the gravitational acceleration, and the distance in the gravity direction.

The total volumetric fl ux is given by

u u u= +w o. (7.63)

Substitution of Eqs. (7.60) and (7.61) into Eq. (7.63) yields

u K K= − +⎛⎝⎜

⎞⎠⎟

⋅∇ − ⋅∇ − +k kp

k dp

dSS

k kr

w

r

ow

r

o

c

ww

r

ww

r

oo

w o o w o

μ μ μ μρ

μρ⎛⎛

⎝⎜⎞⎠⎟

⋅∇g zK . (7.64)

Eliminating ∇pw between Eqs. (7.60) and (7.64) and rearranging leads to

u u K Kw wr

o

c

ww

r

o

Fk dp

dSS

kg zo o= + ⋅∇ − ⋅∇⎛

⎝⎜⎞⎠⎟μ μ

ρΔ , (7.65)

in which

Δρ ρ ρ= −w o (7.66)

and

Fk

kw

r w

o r

o

w

= +⎛⎝⎜

⎞⎠⎟−

11μ

μ, (7.67)

where M is the mobility ratio given by

Mk

kr w

r o

o

w

= μμ

. (7.68)

For formulation of the boundary conditions, the total mass fl ux of water is defi ned as the sum of the transport by bulk fl ow (convection) and hydraulic disper-sion according to Eq. (7.59) as

m u Dw w w w wS= − ⋅∇ρ φ ρ . (7.69)

Then, the water injection end boundary condition can be written as

ρ φ ρ ρw w w w w wSu D u− ⋅∇ = ( )in. (7.70)

Note that the right of Eq. (7.70) is prescribed by the conditions of the injected water. Similarly, the outlet (or production) end boundary condition is obtained as

φ ρSw wD ⋅∇ = 0. (7.71)

For incompressible fl uids, Eq. (7.70) simplifi es to

u uw w= ( )in, (7.72)

and Eq. (7.71) drops out.

7.4.2.1 Pressure and Saturation Formulation Consider incompressible fl uids and formation. The immiscible displacement of oil by water is described by the following differential equations, respectively, by invoking Eq. (7.64) into Eq. (7.56) and Eq. (7.60) into Eq. (7.59) :

c07.indd 188c07.indd 188 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM


∇⋅+⎛

⎝⎜⎞⎠⎟

⋅∇ + ⋅∇

+ +⎛

k kp

k dp

dSS

k k

rw

w

ro

ow

ro

o

c

ww

rw

ww

ro

oo

μ μ μ

μρ

μρ

K K

⎝⎝⎜⎞⎠⎟

⋅∇

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=g z

q

K

. (7.73)

and

φμ

ρ∂∂

−∇ ⋅ ⋅ ∇ + ∇( )⎡⎣⎢

⎤⎦⎥+ =S

t

kp g z qw rw

ww w wK 0. (7.74)

7.4.2.2 Saturation Formulation Assuming incompressible fl uids and substi-tuting Eq. (7.65) into Eq. (7.59) according to Richardson (1961) and Peters et al. (1993) yields

∂( )∂

+∇ ⋅ + ⋅∇ − ⋅∇⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥+

φμ μ

ρS

tF

k dp

dSS

kg z qw

wr

o

c

ww

r

o

o ou K KΔ ww = 0, (7.75)

where q w is defi ned according to Eq. (7.55) . Thus, substituting Eq. (7.56) into Eq. (7.75) (or eliminating ∇⋅( )u ) yields

φ φμ

∂∂

+ −( ) ∂∂+ ⋅∇ +∇ ⋅ ⋅∇⎡

⎣⎢⎤⎦⎥

−

S

tS F

t

dF

dSS F

k dp

dSSw

w ww

ww w

r

o

c

ww

ou K

∇∇⋅ ⋅∇⎡⎣⎢

⎤⎦⎥− + =F

kg z F q qw

r

ow w

o

μρΔ K 0.

(7.76)

When q = 0 , q w = 0 , and an isotropic and homogeneous porous media is considered, Eq. (7.76) simplifi es to Eq. (16.101) given by Richardson (1961) . Eq. (7.76) is interesting because it reveals a transient state, convection, dispersion, with source nature of the water phase transport. Eq. (7.76) can also be written in the following compact form:

φ φ∂∂

+ −( ) ∂∂+ ⋅∇ −∇ ⋅ ( ) ⋅∇[ ]

−∇ ⋅ ( ) ⋅∇

S

tS F

t

dF

dSS D S S

T S

ww w

w

ww w w w

w w

u K

K zz F q qw w[ ]− + = 0,

(7.77)

which, upon expansion, yields the following alternative form:

φ φ∂∂

+ −( ) ∂∂+ − ⋅ ⋅∇( ) − ⋅∇( )⎡⎣⎢

⎤S

tS F

t

dF

dS

dD

dSS

dT

dSzw

w ww

w

w

ww

w

w

u K K⎦⎦⎥⋅∇

− ∇ ⋅ ⋅∇( ) − ∇ ⋅ ⋅∇( ) − + =

S

D S T z F q q

w

w w w w wK K 0,

(7.78)

in which D w and T w can be referred to as the capillary and gravitational dispersion coeffi cients defi ned, respectively, as

D S Fk dp

dSw w w

r

o

c

w

o( ) = −⎛⎝⎜

⎞⎠⎟μ

(7.79)

and

T S Fk

gw w wr

o

o( ) =μ

ρΔ . (7.80)

c07.indd 189c07.indd 189 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM


7.4.2.3 Boundary Conditions The conditions at permeable boundaries can be formulated from the continuity of the water phase and total phase fl uxes across permeable boundaries, respectively, expressed by

u n u nw w⋅( ) = ⋅( )− + (7.81)

and

u n u n⋅( ) = ⋅( )− + . (7.82)

Note that − and + denote the inside and outside of the permeable boundary. The outside conditions are assumed to be prescribed. Therefore, by substituting Eqs. (7.63) and (7.65) into Eqs. (7.81) and (7.82) , the following general boundary condi-tion is obtained:

dp

dSS g z

k kc

ww

w

rw

o

w

K K n u n⋅∇ − ⋅∇⎛⎝⎜

⎞⎠⎟⋅⎡

⎣⎢⎤⎦⎥= ⎛⎝⎜

⎞⎠⎟ ⋅( ) −

− −+Δρ μ μ

rro

o

⎛⎝⎜

⎞⎠⎟ ⋅( )−

+u n . (7.83)

7.4.3 One - Dimensional Linear Displacement

For illustration purposes, consider the classical problem of one - dimensional fl ow of oil/water immiscible displacement in a homogeneous core (Civan, 1996b ). Assume that the core is initially saturated with oil up to the connate water saturation and the fl ow begins with injecting water at a constant rate to displace oil.

For this case, Eq. (7.76) simplifi es as

φμ

∂∂

+ ∂∂

+ ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟= ≤ ≤S

t

dF

dSu

S

x xF

k dp

dSK

S

xx L tw w

w

ww

r

o

c

w

wo 0 0, , >> 0. (7.84)

The initial condition is given by

S S x L tw wc= ≤ ≤ =, , .0 0 (7.85)

At the inlet boundary, because only water is injected, ( u o ) + = 0 and ( u w ) + ≡ ( u w ) in . Thus, Eq. (7.83) simplifi es to

dp

dS

S

x

u

Kkx tc

w

w w w in

rw

∂∂

=( )

= >μ

, , .0 0 (7.86)

At the outlet, three separate conditions need to be formulated.

1. Until the water break through time, t * , ( u w ) + = 0 and the overall volume balance dictates for incompressible fl uids that ( u o ) + = ( u o ) out = ( u w ) in . Thus,

dp

dS

S

x

u

Kkx L t tc

w

w o w

ro

∂∂

= −( )

= < ≤μ

in , , .*0 (7.87)

Note that Eqs. (7.84) , (7.86) , and (7.87) can be shown to be the same as the equations of Douglas et al. (1958) .

2. Between the breakthrough and infi nite water injection, both ( u w ) + = ( u w ) out ≠ 0 and ( u o ) + = ( u o ) out ≠ 0. Thus, Eq. (7.83) is written as

c07.indd 190c07.indd 190 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM


dp

dS

S

x

u

Kk

u

Kkx L t tc

w

w w w

r

o o

rw o

∂∂

=( )

−( )

= < < ∞μ μ

out out , , .* (7.88)

In addition, the overall balance for the core dictates that

u u uw w o( ) = ( ) + ( )in out out. (7.89)

3. During the infi nite water injection ( u o ) + = ( u o ) out = 0 and ( u w ) + = ( u w ) out = ( u w ) in . Thus, Eq. (7.83) simplifi es to

dp

dS

S

x

u

Kkx L tc

w

w w w

rw

∂∂

μ=

( )= →∞in , , . (7.90)

At later times, Eqs. (7.88) and (7.90) can be replaced by the following simpli-fi ed boundary condition:

S Sw or= −1 . (7.91)

7.4.4 Numerical Solution of Incompressible Two - Phase Fluid Displacement Including the Capillary Pressure Effect

For the description of a two - phase fl ow in porous core plugs, Penuela and Civan (2001) considered that the core, water, and oil properties remain constant; the core is initially saturated with water; the fl ow begins by injecting oil at a constant rate to displace the water; the pressure at the production outlet face of the core is constant; and the fl ow is one dimensional and horizontal. Under these conditions, the immis-cible displacement in a core plug can be described by the following equations (Richardson, 1961 ; Dullien, 1992 ; Civan, 1996b ):

φμ

∂∂

+ ∂∂

+ ∂∂

∂∂

⎡⎣⎢

⎤⎦⎥= ≤ ≤S

t

dF

dSu

S

x xF

k dP

dSK

S

xx L tw w

w

ww

ro

o

c

w

w 0 0, , >> 0, (7.92)

subject to the initial condition given by

S S x x L tw w= ≤ ≤ =* ( ), , ,0 0 (7.93)

and the injection and production face boundary conditions given, respectively, by

∂∂

= −( ) ⎡

⎣⎢⎤⎦⎥

= >−

S

x

u

Kk

dP

dSx tw o o inj

ro

c

w

μ 1

0 0, , (7.94)

and

S S x L tw w Pc= = >=0 0, , . (7.95)

The zero capillary pressure and zero gravity fractional water function is given by Eq. (7.67) .

In Eqs. (7.92) – (7.95) , x and t denote the distance and time; φ and K are the porosity and permeability of the core plug; u is the total volumetric fl ux of the pore fl uids, which is equal to the injection fl uid volumetric fl ux under the considered conditions; L is the length of the core plug; μo and k ro denote the viscosity and rela-tive permeability of the oil; μw and k rw are the viscosity and relative permeability of

c07.indd 191c07.indd 191 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM


the water; Pc denotes the capillary pressure; S w is the water phase saturation in the core; S xw

* ( ) represents the initial water saturation distribution in the core; and Sw Pc=0 defi nes the water saturation corresponding to zero capillary pressure. For the problem considered by Kulkarni et al. (1998) , S xw

* ( ) =1 and S Sw P orc== −0 1 , where the resid-

ual oil saturation is taken as zero, S or = 0. uo inj( ) is the volumetric fl ux of the oil injected into the core plug.

Figure 7.5 shows a comparison of the fully implicit numerical solutions of Eqs. (7.92) – (7.95) obtained by Penuela and Civan (2001) , using a fi rst - order accurate temporal and second - order accurate spatial fi nite difference approximations, with the solutions given by Kulkarni et al. (1998) . As can be seen, the Penuela and Civan (2001) solutions and those given by Kulkarni et al. (1998) are very close.

7.4.5 Fractional Flow Formulation

Although it is not necessary, it has been customary to express the water phase volu-metric fl ux, uw, in terms of the fractional fl ow of water, f w . For one - dimensional problems, however, this formulation provides some convenience. Civan (1994b) defi ned a scalar fractional water, f w , according to

u uw wf= . (7.96)

Thus, equating Eqs. (7.65) and (7.96) leads to the following expression:

f Fk

u

dp

dSS

k

ug zw w

r

o

c

ww

r

o

o o= + ⋅ ⋅∇ − ⋅ ⋅∇⎛⎝⎜

⎞⎠⎟

12 2μ μ

ρu K u KΔ , (7.97)

Figure 7.5 Comparison of the numerical solutions obtained by Kulkarni et al. (1998) and Penuela and Civan (2001) (after Penuela and Civan, 2001 ; © 2001 AIChE, reprinted by permission from the American Institute of Chemical Engineers).

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.0

x , length (normalized)

Sw

, wat

er s

atur

atio

n (

frac

tion

)

Kulkarni et al. (1998)Present study

20 min

70 min

100 min

250 min

c07.indd 192c07.indd 192 5/27/2011 12:30:03 PM5/27/2011 12:30:03 PM


in which

u u u ux y z= = + +( )u 2 2 2 1 2, (7.98)

where u x , u y , and u z denote the volumetric fl ux components in the x - , y - , and z - direction. Consequently, assuming incompressible fl uids and invoking Eqs. (7.56) , (7.96) , and (7.98) into Eq. (7.54) for the water phase yields

φ φ∂∂

+ ⋅∇ − + + ∂∂

=S

tf qf q S

tw

w w w wu 0. (7.99)

7.4.6 The Buckley – Leverett Analytic Solution Neglecting the Capillary Pressure Effect

For this purpose, the Buckley and Leverett (1942) equation expressing the velocity of phase 1 at a given saturation can be expressed as the following (Marle, 1981 ). Consider

φ ∂∂

+ ∂∂

=S

tu

f

x1 1 0, (7.100)

where

S S x t1 1= ( ), . (7.101)

Thus, for a prescribed value of saturation S 1 ,

dSS

tdt

f

xdx1

1 10= = ∂∂

+ ∂∂

, (7.102)

we obtain

∂∂

= − ∂∂⎛⎝⎜

⎞⎠⎟

S

t

f

x

dx

dt S

1 1

1

. (7.103)

Consequently, substituting Eq. (7.103) into Eq. (7.100) yields

dx dt u df dSS S( ) = ( )( )

1 11 1φ . (7.104)

However, the superfi cial fl ow u and cumulative volume Q are related by

uA

dQ

dt= 1

, (7.105)

where A denotes the cross - sectional area of porous material normal to fl ow. Substituting Eq. (7.105) , Eq. (7.104) becomes

dx df dS dQ AS S1 11 1= ( ) ( )φ . (7.106)

Because f 1 = f 1 ( S 1 ) only, then ( / )df dS S1 1 1 is a fi xed value. Thus, integrating and applying the initial condition that

x Q tS1 0 0 0= = =, , . (7.107)

c07.indd 193c07.indd 193 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM


Eq. (7.106) leads to an expression for the location of point with a given saturation value as

xQ

A

df

dSS

S1

1

1

1

= ⎛⎝⎜

⎞⎠⎟φ

. (7.108)

This equation is used to generate a semianalytic solution, which provides the locations of prescribed fl uid saturations for various cumulative volumes of injected fl uid.

7.4.7 Convenient Formulation

As stated by Civan and Sliepcevich (1985a) , the transformations of transport equa-tions, in which “ the variable properties do not appear explicitly, ” may lead to numeri-cal solutions of “ better accuracy with less computational effort. ” Therefore, Eq. (7.77) will be converted to a form involving direct applications of the various math-ematical operations to the function.

For this purpose, fi rst, defi ne an integral transformation according to Douglas et al. (1958) as

r SZ

D S dSw w

S

S

w w

wc

w

( ) = ( )∫1

, (7.109)

where S wc and S or denote the connate water and residual oil saturations, respec-tively, and

Z D S dSw

S

S

w w

wc

or

= ( )−( )

∫1

. (7.110)

Thus, applying Eq. (7.109) , Eq. (7.77) can be transformed into

φD

r

t Z

dF

dr

dT

drz r r

T

Zz

w

w w

w

∂∂+ + ⋅∇( )⎡

⎣⎢⎤⎦⎥⋅∇ +∇ ⋅ ⋅∇( )

+ ∇ ⋅ ⋅∇( )

1u K K

K ++ −⎛⎝⎜

⎞⎠⎟+ −( ) ∂

∂=1

0Z

rF

rS F

tw

ww w wρ ρ

φ.

(7.111)

Next, consider coordinate transformation from ( x , y , z ) to ( ξ , η , ζ ) such that the mapping transformation is given by

∂∂∂∂∂∂

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

r

r

r

x y z

x y z

x

ξ

η

ζ

ξ ξ ξ

η η η

ζζ ζ ζ∂∂

∂∂

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

∂∂∂∂∂∂

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

y z

r

xr

y

r

z

(7.112)

or simply by

�∇ = ⋅∇r rJ , (7.113)

c07.indd 194c07.indd 194 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM


where J denotes the Jacobian matrix as shown in Eq. (7.112) and ∇ and �∇ are the gradient operators in the original and the transformed coordinates, respectively. To make Eq. (7.111) free of properties in the various operators, defi ne

�∇ = ⋅∇r rK . (7.114)

Thus, comparing Eqs. (7.113) and (7.114) leads to the following Jacobian matrix, which can be used to develop the mapping functions for the desired transformation:

J K= . (7.115)

This method of transformation can be readily extended to other coordinate systems. Consequently, applying Eq. (7.114) , Eq. (7.111) is transformed to the following convenient form:

φD

r

t Z

dF

dr

dT

drz r r

T

Z

w

w w

w

∂∂+ + ∇⎡

⎣⎢⎤⎦⎥⋅ ⋅∇ + ⋅∇ ⋅∇

+ ⋅

− −

−

1 1 1

1

u K K

K

� � � �

�� ∇⋅∇ + −( ) + −( ) ∂∂

=zZ

q F q S Ft

w w w w1

0φ

.

(7.116)

For example, for horizontal one - dimensional fl ow in an isotropic homoge-neous core, with no source terms, Eq. (7.116) simplifi es to

φ

ξ ξD

r

t Z

dF

druK

rK

r

w

w∂∂+ ∂

∂+ ∂

∂=− −1

01 12

2. (7.117)

Eq. (7.117) is more convenient for numerical discretization than Eq. (7.84) because it involves simple direct derivatives of the function.

7.4.8 Unit End - Point Mobility Ratio Formulation

Inferred by Collins (1961) , Craig (1980) , and Dake (1978) , the following simplifying assumptions are considered for the unit mobility ratio formulation (Civan, 1993, 1996a,b ). The fl uid properties (density and viscosity) are assumed constant. The capillary pressure effect is neglected:

p p p pc o w= = =0, . (7.118)

The relative permeabilities are approximated by linear functions of the normal-ized saturations (Yokoyama and Lake, 1981 ):

k k Srw rw w= ′ ˆ (7.119)

and

k k Sro ro w= ′ −( )1 ˆ , (7.120)

where the normalized mobile water saturation is given by

ˆ .SS S

S Sw

w wc

wc or

= −− −1

(7.121)

c07.indd 195c07.indd 195 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM


The fl uid densities are replaced by the arithmetic average value as

ρ ρ ρw o= = avg , (7.122)

where

ρ ρ ρavg = +( )w o 2. (7.123)

The end - point relative mobilities are replaced by the arithmetic average value as

′=

′= ⎛⎝⎜

⎞⎠⎟

k k krw

w

ro

o

r

μ μ μ avg

, (7.124)

where

k k kr rw

w

ro

oμ μ μ⎛⎝⎜

⎞⎠⎟

=′+

′⎛⎝⎜

⎞⎠⎟avg

1

2. (7.125)

A fl ow potential for incompressible fl uids is defi ned as

Ψ = +p gzρ . (7.126)

Based on these considerations, Eqs. (7.64) and (7.97) simplify, respectively as

u K= −( ) ⋅∇kr μ Ψ (7.127)

and

f Sw w= ˆ . (7.128)

Therefore, substituting Eq. (7.127) into Eq. (7.56) leads to the following fl ow potential equation:

−( )∇ ⋅ ⋅∇( ) + + ∂ ∂ =k q tr μ φK Ψ 0. (7.129)

Substituting Eqs. (7.121) and (7.118) into Eq. (7.99) results in the following water saturation equation:

1− −( ) ∂∂

+ ⋅∇ − + + − +( )⎡⎣ ⎤⎦∂ ∂S SS

tS S q q S S S S tor wc

ww w w wc wc or wφ φ

ˆˆ ˆ ˆu == 0. (7.130)

Note that Eq. (7.129) can be more conveniently solved after transforming into the following form by applying the transformation given by Eq. (7.114)

−( ) ⋅∇ ⋅∇ + + ∂ ∂ =−k q tr μ φK 1 0� �Ψ . (7.131)

Note that Eq. (7.131) is solved once if all the parameters are independent of time.

7.4.8.1 Example 1 This example deals with the time – space numerical solution of the following problem (Civan, 1996a,b ):

∂∂+ ∂

∂+ ∂

∂=S

t

u S

x

u S

yx y

φ φ0, (7.132)

c07.indd 196c07.indd 196 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM


subject to

S x y t= ≤ ≤ =0 0 1 0, , , , (7.133)

S x y t= = ≤ ≤ >1 0 0 1 0, , , , (7.134)

and

S x y t= ≤ ≤ = >1 0 1 0 0, , , . (7.135)

A fi nite difference discretization of Eq. (7.132) at n , n + 1, and n + 2 time levels according to Figure 7.6 yields, respectively:

− − + −( ) ( ) + ( ) −(− + +

−2 3 6 61 1 21S S S S t u S Sij

nijn

ijn

ijn

x ij

nijn

i jnΔ φ , ))

+ ( ) −( ) =−

Δ

Δ

x

u S S yy ij

nijn

i jnφ , ,1 0

(7.136)

S S S S t u S Sij

nijn

ijn

ijn

x ij

nijn

i− + + + +

−− + +( ) ( ) + ( ) −1 1 2 1 116 3 2 6Δ φ , jj

n

y ij

nijn

i jn

x

u S S y

+

+ +−+

( )+ ( ) −( ) =

1

1 11

1 0

Δ

Δφ , , (7.137)

and

− + − +( ) ( ) + ( ) −− + + + +2 9 18 11 61 1 2 2 2S S S S t u S Sij

nijn

ijn

ijn

x ij

nijnΔ φ ii j

n

y ij

nijn

i jn

x

u S S y

−+

+ +−+

( )+ ( ) −( ) =

12

2 21

2 0

,

, .

Δ

Δφ (7.138)

Eqs. (7.136) – (7.138) are solved analytically for S Sijn

ijn, ,+1 and Sij

n+2. Figure 7.7 a – c show typical solutions obtained using a 10 × 10 equally spaced spatial grid Δ Δx y=( ) over a square domain and Δ Δt x = 0 1. . The solutions using 0.5, 1.0, 5.0, and 10.0 for Δ Δt x have also been carried out. The accuracy of the numerical solutions degenerates by increasing values of Δt. This is due to the low - order discretization

Figure 7.6 Schematic of the computational molecule having three time steps and three space points (after Civan, 1996b ; © 1996 SPE, with permission from the Society of Petroleum Engineers).

Δt

Δt

Δt

n + 1

n + 2

n

n – 1

c07.indd 197c07.indd 197 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM


of the spatial derivatives used in this specifi c example and can be readily alleviated by using higher - order discretization according to Civan (1994a, 2009) .

7.4.8.2 Example 2 Consider the following problem describing a waterfl ooding in a homogeneous, anisotropic, and rectangular domain without any wells inside, as shown. Eq. (7.129) simplifi es as (Civan, 1996a )

kp

xk

p

yx y∂∂

+ ∂∂

=2

2

2

20. (7.139)

Assuming S wc = S or = 0, Eq. (7.130) simplifi es as

φ ∂∂

+ ∂∂

+ ∂∂

=S

tv

S

xv

S

yw

xw

yw 0, (7.140)

Figure 7.7 (a – c) Saturation profi les after 5, 15, and 30 time steps for Δ Δx t/ .= 0 1 (after Civan, 1996b ; © 1996 SPE, with permission from the Society of Petroleum Engineers).

(a)

(b)

(c)

1.0

14

710

0.8 – 1.00.6 – 0.80.4 – 0.60.2 – 0.40 – 0.2

0.80.6

Nor

mal

ized

satu

ratio

n

0.40.2

Grid points in x

Grid points in y

dt/dx = 0.1

0

1 4 7

10

0.8 – 1.00.6 – 0.80.4 – 0.60.2 – 0.40 – 0.2

0.8 – 1.00.6 – 0.80.4 – 0.60.2 – 0.40 – 0.2

1.0

14

710

0.80.6

Nor

mal

ized

satu

ratio

n

0.40.2

Grid points in x

Grid points in y

dt/dx = 0.1

0

1 4 7

10

1.0

14

710

0.80.6

Nor

mal

ized

satu

ratio

n

0.40.2

Grid points in x

Grid points in y

dt/dx = 0.1

0

1 4 7

10

c07.indd 198c07.indd 198 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM

where

vk p

xx

x= − ∂∂μ

(7.141)

and

vk p

yy

y= − ∂∂μ

. (7.142)

The initial condition is assumed as

S tw = =0 0, , .everywhere (7.143)

The boundary pressures are prescribed as shown in Figure 7.8 . The saturation boundary condition is prescribed as

S x yw =1 along the and -axes. (7.144)

This problem is solved for the pressure and saturation profi les as described in the following.

The fi nite difference solution can be obtained by discretizing Eqs. (7.139) – (7.144) respectively as the following:

kp p p

xk

p p p

yx

i j ij i jy

i j ij i j+ − + −− +( )

+− +( )

=1 1

2

1 1

2

2 20, , , , ,

Δ Δ (7.145)

vk p p

xx

x i j i jij = −

−+ −

μ1 1

2, , ,Δ

(7.146)

vk p p

yy

y i j i jij = −

−+ −

μ, , ,1 1

2Δ (7.147)

v v vx y= +2 2 , (7.148)

and

φ S S

tv

S S

xv

S S

yin

in

xi jn

i jn

yi jn

i jn

ij ij

− +−

+−

=−

− −1

1 1 0Δ Δ Δ

, , , , . (7.149)


Figure 7.8 Schematic of a rectangular computational region and a general fi ve - point computational molecule (prepared by the author).

i, j + 1

i, j – 1

i + 1, ji – 1, j i, j

150 atm

150 atm

200 atm

200 atm

c07.indd 199c07.indd 199 5/27/2011 12:30:04 PM5/27/2011 12:30:04 PM


First, the pressure equation, Eq. (7.145) , is solved. Then, the velocity compo-nents are calculated using Eqs. (7.146) and (7.147) . Finally, Eq. (7.149) is solved explicitly for the saturation.

The sweep effi ciency can be calculated by

ES S

Sdydx

S S

SdydxD

oi o

oiyx

w wc

wcyx

= − = −−∫∫ ∫∫ 1 (7.150)

where the double integration can be evaluated by an appropriate numerical proce-dure, such as the trapezoidal rule.

7.5 POTENTIAL FLOW PROBLEMS IN POROUS MEDIA

Irrotational fl ow in porous media can be formulated conveniently in terms of the fl ow potential, Ψ , defi ned by

Ψ = + −( )∫dp

g z zp

p

o

oρ

, (7.151)

and using the fl uid fl ux expressed by Darcy ’ s law, given by

u K= − ⋅∇ρμ

Ψ. (7.152)

The potential fl ow problems can be solved analytically under certain simplify-ing conditions. Analytic solutions can provide an accurate representation of intricate fl ow problems, such as preferential fl ow through fractures, cross - fl ow across dis-continues, and converging/diverging fl ow around wellbores, in heterogeneous reservoirs involving anisotropy and/or stratifi cation and wells and fractures. Hence, for such complicated fl ow situations, analytic solutions may be more advantageous than numerical solutions, the accuracy of which often suffers from insuffi cient levels of resolution of numerical grid systems (Shirman and Wojtanowicz, 1996 ). Frequently, the inherent grid point orientation effect and numerical dispersion and instability problems associated with numerical solution methods for differential equations render inaccurate numerical solutions. Under these conditions, well - conditioned analytic solutions with accuracies limited by simplifying assumptions may be favored over numerical solutions with inherent diffi culties.

7.5.1 Principle of Superposition

The superposition principle expresses the solution of linear equations as a weighted linear sum of their special solutions. For example, the following Laplace equation is a linear differential equation:

∇ =2 0Ψ . (7.153)

c07.indd 200c07.indd 200 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM

7.5 POTENTIAL FLOW PROBLEMS IN POROUS MEDIA 201

If N special solutions denoted by Ψ i : i = 1, 2, … , N can be found for this equation, then each will satisfy Eq. (7.153) . Thus,

∇ = = …2 0 1 2Ψ i i N: , , , . (7.154)

Consequently, multiplying Eq. (7.154) by some constant weighting coeffi -cients C i : i = 1, 2, … , N and then adding yields

∇ ==∑2

1

0Ci ii

N

Ψ . (7.155)

Therefore, by comparing Eqs. (7.153) and (7.155) , the solution of Eq. (7.153) can be expressed as a weighted linear sum of the special solutions, Ψ i i N: , , ,=1 2 … , as follows:

Ψ Ψ==∑Ci ii

N

1

. (7.156)

The weighting coeffi cients C i : i = 1, 2, … , N are determined in a manner to satisfy the prescribed conditions of specifi c problems, such as the consistency and compatibility conditions and the source/sink conditions.

As an example, consider the three - dimensional potential fl ow involving multiple - point sinks in an infi nite size homogeneous porous media. When only a single well i producing at a rate of q i is present, the fl ow fi eld around the well is spherical and the solution of the potential fl ow problem is given by the integration of Darcy ’ s law in radial distance for constant fl ow rate q i :

Ψ Ψiir

q

Kr( ) = −∞

μπ4

, (7.157)

where the radial distance r is given by

r x x y y z zi i i= −( ) + −( ) + −( )2 2 2 , (7.158)

where x , y , and z denote the general Cartesian coordinates, and x i , y i , and z i denote the Cartesian coordinates of well i . For convenience, the potential can be expressed relative to the potential at the infi nite distance as

Ψ Ψ Ψri iir r

q

Kr( ) ≡ ( ) − = −∞

μπ4

. (7.159)

When there are N wells present, producing at the rates of q i : i = 1, 2, … , N , then the analytic solution can be readily obtained by applying Eq. (7.156) as

Ψ Ψr i rii

N

ii

i

N

C Cq

Kr= = −

= =∑ ∑

1 1 4

μπ

. (7.160)

Note that Eq. (7.160) constitutes a solution regardless of the values of the weighting coeffi cients C i : i = 1, 2, … , N . For example, they all might as well be assigned a value of unity; that is, C i = 1 : i = 1, 2, … , N .

c07.indd 201c07.indd 201 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


7.5.2 Principle of Imaging

The method of images is a powerful technique used to generate analytic solutions for potential fl ow problems in intricate domains involving discontinuities by utiliz-ing analytic solutions of problems considering infi nite size domains. Here, the method of images is described according to Shirman and Wojtanowicz (1996) , who explained this method by analogy to an optical problem. The following presentation involves some modifi cations on the treatise presented by Shirman and Wojtanowicz (1996) for consistency with the materials covered in this chapter.

Shirman and Wojtanowicz (1996) explained the principle of imaging by means of two lamps and two photometers symmetrically placed in a box with respect to its center. Thus, the photometers indicate the same values when the strengths of the lamps are exactly the same. However, if the lamp on the right is turned off, the light available in the box will be less and will be unevenly distributed. Therefore, both photometers will read lower values than the previous case where both lamps are turned on. Further, the photometer closer to the lamp on the left will read a higher value than that on the right and farther away from the lamp. However, when a perfect mirror, capable of refl ecting with 100% effi ciency, is placed in the middle of the box, the photometer on the left side of the mirror now reads the full value because of the superposition of the lights coming from the lamp and its mirror image. However, the photometer on the right of the mirror reads zero because the mirror does not permit any light to pass to the other side on the right behind the mirror and therefore acts as a barrier to light transmission, or becomes an analog to a no - fl ow - type boundary.

As a next exercise, Shirman and Wojtanowicz (1996) assume that the 100% refl ecting mirror is now replaced by a 50% refl ecting and therefore 50% transparent mirror. Consequently, in contrast to the previous example, the photometer on the right of the mirror now indicates some value instead of zero and the photometer on the left of the mirror indicates a reading less than the previous 100% refl ecting mirror case. Hence, the 50% refl ecting mirror acts as an analog to a cross - fl ow boundary.

The lamps or light sources considered in the preceding example by Shirman and Wojtanowicz (1996) correspond to the wells or fl uid sinks in fl ow through porous media. The mirror corresponds to a discontinuity, no - fl ow, or cross - fl ow boundaries, depending on its transmissibility/refl ectivity conditions. The photometer readings correspond to the fl ow potential.

7.5.3 Basic Method of Images

By analogy to the above optical problem, Shirman and Wojtanowicz (1996) derived the following rules for applications on fl ow through porous media, as quoted below:

“ 1. The image well affects only the zone it was refl ected from, and its strength is equal to the product of the strength of a real well and a coeffi cient of refl ection C, and

2. the strength of the real well across the discontinuity is reduced by the refrac-tion coeffi cient D. ”

c07.indd 202c07.indd 202 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM

7.5 POTENTIAL FLOW PROBLEMS IN POROUS MEDIA 203

As an example of the application of these rules, Shirman and Wojtanowicz (1996) consider a single - point sink located in the upper layer near a planar discon-tinuity separating two layers of different permeability having K 1 permeability in the upper layer and K 2 permeability in the lower layer. A real well producing at a constant rate, q , acting as a point sink is placed in the upper layer at a distance, z R , from the planar cross - fl ow boundary. Thus, the image well is placed in the lower layer at an equal distance, z I = z R , from the planar cross - fl ow boundary.

If the real well existed alone in an infi nite - size porous medium of uniform permeability K , the potential distribution around the well would have been given by applying Eq. (7.157) :

Ψ Ψ= − −⎛⎝⎜

⎞⎠⎟

ee

q

K r r

μπ4

1 1, (7.161)

where

r x x y y z zR R R= −( ) + −( ) + −( )2 2 2 . (7.162)

Ψe denotes the fl ow potential at a radius of r e . When there is a porous media dis-continuity, however, the problem is analogous to the above - described partial refl ect-ing and therefore partial transparent mirror case. Thus, applying the superposition principle and rule 2, the potential distribution in the upper layer of permeability K 1 containing the real well is given by the sum of the full contribution of the real well and the partial contribution of its image with respect to the planar discontinuity according to

Ψ Ψ Ψ1 = +R IC , (7.163)

whereas the potential distribution in the lower layer of permeability K 2 is given by the partial contribution of the real well as the following, applying rule 2:

Ψ Ψ2 = D R . (7.164)

In Eqs. (7.163) and (7.164) , C and D denote the refl ection and refraction coef-fi cients of the planar discontinuity. The values of the coeffi cients are determined to satisfy the compatibility and consistency conditions at the planar discontinuity given, respectively, by

Ψ Ψ1 2 0= =, z (7.165)

and

u uK

z

K

zz1 2

1 1 2 2 0= − ∂∂

= − ∂∂

=orμ μ

Ψ Ψ, . (7.166)

Ψ ΨR Iand denote the fl ow potentials of the real and image wells, when they are present alone in a uniform permeability media K in general, given, respectively, by Eqs. (7.167) and (7.168)

Ψ ΨR e

R R R e

q

K x x y y z z r= −

−( ) + −( ) + −( )−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

μπ4

1 12 2 2

(7.167)

c07.indd 203c07.indd 203 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


and

Ψ ΨI e

R R R e

q

K x x y y z z r= −

−( ) + −( ) + −( )−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

μπ4

1 12 2 2

. (7.168)

As a result, substituting Eqs. (7.167) and (7.168) into Eqs. (7.163) – (7.166) yields, respectively,

Ψ Ψ11

2 2 2

1

4

1 1

4

1

= −−( ) + −( ) + −( )

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

−

e

R R R e

q

K x x y y z z r

Cq

K x

μπ

μπ −−( ) + −( ) + −( )

−⎛

⎝⎜⎜

⎞

⎠⎟⎟x y y z z r

R R R e2 2 2

1,

(7.169)

Ψ Ψ22

2 2 24

1 1= −−( ) + −( ) + −( )

−⎛

⎝⎜⎜

⎞

⎠⎟⎟e

R R R e

Dq

K x x y y z z r

μπ

, (7.170)

1

1 2

+ =C

K

D

K, (7.171)

and

1− =C D. (7.172)

Consequently, Eqs. (7.171) and (7.172) can be solved to obtain the coeffi cients of refl ection and refraction as functions of the permeability of the two layers as

CK K

K KC= −

+≤ ≤1 2

1 2

0 1, (7.173)

and

DK

K KD=

+≤ ≤2

0 12

1 2

, . (7.174)

Thus, the potential distributions in the K 1 and K 2 permeability layers can be determined by invoking Eqs. (7.173) and (7.174) into Eqs. (7.169) and (7.170) .

Shirman and Wojtanowicz (1996) point out several interesting features of this solution. First, Eq. (7.172) confi rms that the fractional energies passing through and refl ected off the planar discontinuity surface add up to the unity, C + D = 1. Second, when the planar discontinuity is a full no - fl ow boundary or 100% refl ector, then C = 1 and D = 0. Third, if a uniform media having no planar discontinuity is con-sidered, then K 1 = K 2 = K and C = 0 and D = 1. Fourth, K 2 → ∞ , then the planar discontinuity represents a constant potential boundary and Eqs. (7.166) – (7.168) yield C = − 1, D = 2, and Ψ Ψ2 = e.

Shirman and Wojtanowicz (1996) present solution for another interesting case dealing with a single well dissected by the planar cross - fl ow boundary located between the porous layers of K 1 and K 2 permeability. In this case, both the real and image wells are located at the same point ( x I = x R , y I = y R , z I = z R = 0) on the cross - fl ow boundary, Hence, Eqs. (7.169) and (7.170) simplify, respectively, as

c07.indd 204c07.indd 204 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM

7.6 STREAMLINE/STREAM TUBE FORMULATION AND FRONT TRACKING 205

Ψ Ψ11

2 2 2

12 2 2

4

1 1

4

1 1

= −+ +

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

−+ +

−⎛

⎝⎜⎜

ee

e

q

K x y z r

Cq

K x y z r

μπ

μπ

⎞⎞

⎠⎟⎟

(7.175)

and

Ψ Ψ22

2 2 24

1 1= −+ +

−⎛

⎝⎜⎜

⎞

⎠⎟⎟e

e

Dq

K x y z r

μπ

, (7.176)

where C and D are given by Eqs. (7.173) and (7.174) .

7.5.4 Expanded Method of Images

As described by Shirman and Wojtanowicz (1996) , when a single - point sink is located between two parallel cross - fl ow boundaries, an infi nite number of images with respect to both boundaries need to be considered. Shirman and Wojtanowicz (1996) explain that the contribution of each new image well due to the presence of a refl ecting boundary is determined by a product of its rate of production and coef-fi cient of refl ection C . Therefore, the contribution of the higher - generation images rapidly diminishes, especially after the third image. When one or both boundaries are the no - fl ow types, then the coeffi cient of perfect refl ection of C = 1 is applied for all images with respect to the no - fl ow boundary.

Shirman and Wojtanowicz (1996) described that when a single - point sink is placed in a layer of a multilayer porous media, the strengths of the signals approach-ing the parallel cross - fl ow boundaries, having the refl ection coeffi cients of C i : i = 1, 2, … , N , will split into the refl ecting and refracting fractions consecutively. For instance, the strength of the image source resulting from the refl ection of a signal approaching the fi rst cross - fl ow boundary is qC 1 and the strength of the refracting signal crossing the fi rst cross - fl ow boundary will be q (1 − C 1 ). Similarly, the signal of strength q (1 − C 1 ) approaching the second cross - fl ow boundary will split into the refl ected strength [ q (1 − C 1 )] C 2 for the second image and the refracted strength [ q (1 − C 1 )](1 − C 2 ) crossing the second cross - fl ow boundary. This procedure is repeated for an infi nite number of images, each having a rapidly diminishing contribution.

7.6 STREAMLINE/STREAM TUBE FORMULATION AND FRONT TRACKING

The development and implementation of models based on streamline/stream tube formulation and front tracking have been attempted for variety of applications. The fundamentals of the streamline/stream tube formulation are presented according to Martin and Wegner (1979) , and Akai (1994) . Consider Figure 7.9 showing a porous medium confi ned inside a region with sealed boundaries. A fl uid is fl owing through this medium because of the injection and production of the fl uid at diagonally

c07.indd 205c07.indd 205 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


opposite corners. The paths of the fl uid particles are depicted by continuous lines extending from the injection to the production corners. These are called the stream-lines. The regions present in between the consecutive streamlines can be viewed as stream tubes or fl ow channels. Streamlines form no - fl ow boundaries for the stream tubes because the streamlines are tangent or parallel curves to fl ow directions. Therefore, stream tubes can be viewed as a nonleaky hose whose width is determined by the fl ow pattern. The volumetric rate of a fl uid fl owing through a stream tube can be expressed by a scalar function called the stream function. The value of this func-tion remains constant along each streamline, and the difference between the stream function values of the two streamlines forming a stream tube is equal to the volu-metric rate of fl uid fl owing through that stream tube.

7.6.1 Basic Formulation

Consider the pair of streamlines as shown in Figure 7.9 in an x - y domain. The fl uid is assumed incompressible and the fl ow is two dimensional for this case. The stream function in a two - dimensional domain is a prescribed scalar function, represented by

Ψ Ψ= ( )x y, . (7.177)

The exact differential of the stream function is given by

dx

dxy

dyΨ Ψ Ψ= ∂∂

+ ∂∂

. (7.178)

On the other hand, a volumetric balance over the unit - thick triangular element shown in Figure 7.9 leads to (Martin and Wegner, 1979 )

dq u x u yy x≡ = −ΔΨ Δ Δ . (7.179)

Figure 7.9 Schematic of two streamlines and a stream tube (prepared by the author).

Δx

Production well

uxΔy

ΔyΔs

uyΔx

Δψ

B

A C

ψ2= ψ + Δψ

ψ1 = ψ

Stream lines

Stream tube

Injection well

y

x

Impermeable region

c07.indd 206c07.indd 206 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


The limit of this equation as Δ Δx yand approach zero is

d u dx u dyy xΨ = − . (7.180)

Thus, comparing Eqs. (7.178) and (7.180) yields

uy

x = −∂∂Ψ

(7.181)

and

ux

y =∂∂Ψ

. (7.182)

Substituting Eqs. (7.181) and (7.182) into Eq. (7.180) yields

dΨ = 0. (7.183)

This indicates that the stream function value remains constant along a stream-line. Further, for a fl uid fl owing through a stream tube confi ned between two stream-lines, for which the stream function values are denoted by Ψ1 and Ψ2; the volume fl ow rate is given by (Akai, 1994 )

q = −Ψ Ψ2 1 . (7.184)

In addition, Eqs. (7.181) and (7.182) , and therefore the stream function, satisfy the incompressible fl uid equation of continuity in an incompressible porous media, given by

∂∂

+∂∂

=u

x

u

yx y 0. (7.185)

If s denotes the length along a streamline, the parametric representation of a streamline in two dimensions is given by

x x s= ( ) (7.186)

and

y y s= ( ), (7.187)

whereas the velocity components in the x - , y - , and s - directions are defi ned, respec-tively, by

udx

dtx = , (7.188)

udy

dty = , (7.189)

and

uds

dt= . (7.190)

Then, the equation of a streamline can be obtained by eliminating the time variable between Eqs. (7.188) and (7.189) as

c07.indd 207c07.indd 207 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


dy

dx

u

uy

xΨ==

ct.

. (7.191)

The same can be obtained by applying Eq. (7.183) to Eq. (7.180) (Akai, 1994 ). It is also true from Eqs. (7.188) – (7.190) that

dx

ds

u

ux

Ψ==

ct.

(7.192)

and

dy

ds

u

uy

Ψ==

ct.

. (7.193)

For horizontal two - dimensional fl ow, Darcy ’ s law expresses the components of the fl uid fl uxes in the x - and y - directions by

up

xx x= − ∂

∂λ (7.194)

and

up

yy y= − ∂

∂λ , (7.195)

where the directional fl uid mobility are given by

λ μx xK= (7.196)

and

λ μy yK= , (7.197)

where μ is the viscosity of the fl uid and K x and K y are the directional permeability of porous media.

Substituting Eqs. (7.194) and (7.195) into Eq. (7.185) yields

∂∂

∂∂

⎛⎝⎜

⎞⎠⎟+ ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟=

x

p

x y

p

yx yλ λ 0. (7.198)

Similarly, substituting Eqs. (7.181) and (7.182) into Eq. (7.185) yields

∂∂

∂∂

⎛⎝⎜

⎞⎠⎟= ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟x y y x

Ψ Ψ. (7.199)

This conforms to Schwartz ’ s rule of exactness and indicates that the stream function is exact.

Previously, the fl uid was assumed incompressible. Further assume inviscid fl ow, that is, fl ow involving no viscosity or frictional effects. Then, the fl ow obeys the following irrotationality condition (Bertin, 1987 ; Akai, 1994 ):

∂∂

− ∂∂

=u

x

u

yy x 0. (7.200)

c07.indd 208c07.indd 208 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


Thus, substituting Eqs. (7.194) and (7.195) into Eq. (7.200) yields

∂∂

∂∂

⎛⎝⎜

⎞⎠⎟= ∂∂

∂∂

⎛⎝⎜

⎞⎠⎟x

p

y y

p

xy xλ λ . (7.201)

Similarly, substituting Eqs. (7.181) and (7.182) into Eq. (7.200) yields the Laplace equation:

∂∂

+ ∂∂

=2

2

2

20

Ψ Ψx y

. (7.202)

Consider a boundary of the system as shown in Figure 7.10 . The volumetric fl ux vector can be decomposed into the components in the x - and y - directions, denoted as u x and u y , respectively. It is also possible to decompose the volume fl ux vector into the tangential and normal components relative to the boundary position. The latter is convenient for prescription of the boundary conditions for Eq. (7.202) and can be related to the former by (Marathe et al., 1995 )

u u un x y= −sin cosθ θ (7.203)

and

u u ut x x= +cos sin ,θ θ (7.204)

where θ is the angle between the tangent and the horizontal direction in x . Typical boundary conditions for Eq. (7.202) can be specifi ed as follows

(Shikhov and Yakushin, 1987 ; Marathe et al., 1995 ):

1. Impermeable, no - fl ow, or sealed boundaries. The normal fl ows are zero along sealed boundaries Γ:

ut

n ΓΓ

ΓΨ Ψ= ∂∂

= =0 0, , .ct (7.205)

For example, for a rectangular domain,

∂∂

= − =Ψ Γy

u yx y0 along the -boundary (7.206)

Figure 7.10 Decomposition of the volume fl ux at a boundary (prepared by the author).

System

uy

un

ux

ut

u

Γ - boundary

q

c07.indd 209c07.indd 209 5/27/2011 12:30:05 PM5/27/2011 12:30:05 PM


and

∂∂

= =Ψ Γx

u xy x0 along the -boundary . (7.207)

Therefore, Eqs. (7.206) and (7.207) imply, respectively, that

Ψ = ct. along the -directiony (7.208)

and

Ψ = ct. along the -direction.x (7.209)

For simplicity, the constant can be chosen as zero for reference purposes.

2. Equipotential boundaries. The pressures are constant and the tangential fl ows are zero along the equipotential boundaries. Thus,

p un

t= = ∂∂

=ct., , .ΓΓ

Ψ0 0 (7.210)

For example, for a rectangular domain,

up

x yxx x x= − ∂

∂= − ∂

∂=λ Ψ Γ0 along the -boundary (7.211)

and

up

y xyy y y= − ∂

∂= ∂∂

=λ Ψ Γ0 along the -boundary . (7.212)

3. Sources and sinks. Along the boundaries across which fl uid fl ows into and out of the system, such as wellbores and leaky boundaries, the difference between the stream function values of two streamlines is equal to the rate of fl uid fl owing into or out of the system. Hence, the strength of the source/sink is equal to the discontinuity in the stream function value. Thus,

Ψ Ψ1 2 0

0 0

− = >= <

q q

q q

, ,where for production

for noflow,and for injectiion. (7.213)

For example, consider Figure 7.11 showing several streamlines emanating from an injection well. The normal volume fl ux for axisymmetric fl ow is given by

uq

rhn =

∂∂

=ΨΓ 2π

, (7.214)

where Γ denotes distance along the wellbore boundary. An integration of Eq. (7.214) between any pair of i and j streamlines yields

Ψ Ψ Γ Γj i ij j iqq

rh− = = −( )

2π (7.215)

and

q qij=∑ . (7.216)

c07.indd 210c07.indd 210 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


7.6.2 Finite Analytic Representation of Wells in Porous Media

The fi nite analytic method generates fi nite difference - type numerical schemes for governing differential equations by using local analytic solutions (Civan, 1995, 2009 ).

Consider the schematic of a wellbore contained within a grid block as shown in Figure 7.12 . Because a wellbore diameter is generally much smaller than a typical reservoir grid block size considered for numerical solution in reservoir simulation, accurate representation of wells requires special attention in the numerical modeling of such wells. Among others, Shikhov and Yakushin (1987) , and Peaceman (1990) offered effective procedures that incorporate near - wellbore radial fl ow analytic solu-tions into numerical solution schemes. Here, the method of Shikhov and Yakushin (1987) is presented.

Shikhov and Yakushin (1987) developed a fi nite analytic numerical scheme for isotropic fl ow in the vicinity of a well. For this purpose, by substituting λ x = λ y = λ = ct. into Eq. (7.198) , the following Laplace equation is obtained:

Figure 7.11 Streamlines emanating from an injection well (prepared by the author).

ψ3 ψ2

ψ1ψ4

ψ6ψ5

Injectionwell

q61

q12

q56

q23

q45

q34

Γ

Figure 7.12 Schematic of a wellbore contained within a grid block (prepared by the author).

i, j + 2

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

i + 1, j i + 2, ji, jx

Δy

Δx

y

Well

(xo, yo)

c07.indd 211c07.indd 211 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


∂∂

+ ∂∂

= ⎛⎝⎜

⎞⎠⎟=

2

2

2

2

10

p

x

p

y r

d

drr

dp

dr. (7.217)

The analytic solution of this equation is given by

p pq

Kr p

q

Kx x y yo o o o= + = + −( ) + −( )μ μ

ln ln .2 2 (7.218)

Therefore,

uK p

x

q

h

x x

x x y yx

o

o o

= − ∂∂

= −−( ) + −( )μ π2 2 2 (7.219)

and

uK p

y

q

h

y y

x x y yy

o

o o

= − ∂∂

= −−( ) + −( )μ π2 2 2

. (7.220)

On the other hand,

uy

x = −∂∂Ψ

(7.221)

and

ux

y =∂∂Ψ

. (7.222)

Now, consider the four grid points associated with the grid block containing the well as shown in Figure 7.12 . The central fi nite difference approximations of Eqs. (7.221) and (7.222) are given, respectively, by

uy

x i j

i j i j( ) = −−

++

,

, ,

1

2

2

Ψ ΨΔ

(7.223)

and

ux

y i j

i j i j( ) =−

++

1

2

2,

, , .Ψ Ψ

Δ (7.224)

These equations can be solved for Ψ i j, , respectively, as

Ψ Ψ Δi j i j x i jy u, , ,

= − ( )+ +2 12 (7.225)

and

Ψ Ψ Δi j i j y i jx u, , ,

.= + ( )+ +2 12 (7.226)

Then, the above two expressions can be averaged arithmetically as

Ψ Ψ Ψ Δ Δi j i j i j x i j y i jy u x u, , , , ,

.= + − ( ) + ( )⎡⎣ ⎤⎦+ + + +

1

22 22 2 1 1

(7.227)

In this equation, u x and u y are calculated using Eqs. (7.219) and (7.220) . A formula can be generated for Ψ i j+1, also from Eq. (7.227) by progressing the index. Note that Eq. (7.227) is slightly different from the expression given by Shikhov and Yakushin (1987) .

c07.indd 212c07.indd 212 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


The central fi nite difference approximations of Eqs. (7.221) and (7.222) at the other two points are given, respectively, by

uy

x i j

i j i j( ) = −−Δ

+ −,

, ,Ψ Ψ1 1

2 (7.228)

and

ux

y i j

i j i j( ) =−

+ ++ + +

1 1

2 1 1

2,

, , .Ψ Ψ

Δ (7.229)

These equations can be solved for Ψ i j, +1 and then averaged arithmetically to obtain

Ψ Ψ Ψ Δ Δi j i j i j x i j y i jy u x u, , , , ,

.+ − + + + += + + ( ) − ( )⎡⎣ ⎤⎦1 1 2 1 1 1

1

22 2 (7.230)

In this equation, u x and u y are calculated using Eqs. (7.219) and (7.220) . A formula can be generated for Ψ i j+ +1 1, also from Eq. (7.230) by progressing the index.

7.6.3 Streamline Formulation of Immiscible Displacement in Unconfi ned Reservoirs

An unconfi ned reservoir is an infi nitely large theoretical system in which the fl ow potential in the vicinity of the production/injection wells is only infl uenced by the fl ow rates of these wells, while the potential at infi nite distance remains constant because there are no acting boundary effects.

Consider i = 1, 2, … , N wells in an unconfi ned reservoir that is an areal, uniform - thick, and isotropic porous medium. A unit mobility ratio displacement of the resident fl uid by the injected fl uids under steady - state fl ow conditions is consid-ered for simplicity.

The potential distribution around the i well, when it is present alone in an unconfi ned medium, is given by the following analytic solution:

Ψ Ψi wi

w

q

Kh

r

r= − ⎛

⎝⎜⎞⎠⎟

μπ2

ln , (7.231)

where Ψw denotes the fl ow potential at the wellbore. The radial distance from the well center is given by

r x x y yi i= −( ) + −( )2 2 , (7.232)

where x i and y i are the Cartesian coordinates of the center point of the i well. Applying the superposition principle of potential fl ow, the potential distribution incorporating the infl uence of all wells present in an unconfi ned medium can be expressed by

Ψ Ψ Ψx y x yKh

q x x y yii

N

w i i ii

N

, , ln .( ) = ( ) = − −( ) + −( )⎡⎣ ⎤⎦= =∑ ∑

1

2 2

14

μπ

(7.233)

c07.indd 213c07.indd 213 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


Applying Darcy ’ s law, the directional volume fl uxes (superfi cial velocities) are given by

uK

x h

q x x

x x y yx

i i

i ii

N

= − ∂∂

=−( )

−( ) + −( )=∑μ π

Ψ 1

2 2 21

(7.234)

and

uK

y h

q y y

x x y yy

i i

i ii

N

= − ∂∂

=−( )

−( ) + −( )=∑μ π

Ψ 1

2 2 21

. (7.235)

The pore fl uid directional velocities are given by

v

u

Sx

x

rr

=−⎛

⎝⎜⎞⎠⎟

∑φ 1 (7.236)

and

v

u

Sy

y

rr

=−⎛

⎝⎜⎞⎠⎟

∑φ 1, (7.237)

where S r denotes the immobile fl uid saturations, such as the connate water and residual oil saturations involved in the waterfl ooding of oil reservoirs.

Figure 7.13 of Marathe et al. (1995) shows the typical streamline patterns involving an injection well surrounded by four equally spaced production wells present in an unconfi ned reservoir.

7.6.4 Streamline Formulation of Immiscible Displacement Neglecting Capillary Pressure Effects in Confi ned Reservoirs

Marathe et al. (1995) developed a calculation scheme for the determination of fl ood patterns resulting from multiple injection and production wells present in irregularly

Figure 7.13 Typical streamline patterns involving an injection well surrounded by four equally spaced production wells present in an unconfi ned reservoir (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers).

D C

A B

P2P1

P3P4

c07.indd 214c07.indd 214 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


bounded reservoirs. They approximated the immiscible fl uid displacement process as a unit mobility ratio displacement, leading to a homogeneous reservoir treatment. Their approach for modeling immiscible displacement to determine the potential and streamline distributions and for displacing fl uid front positions is presented here with some modifi cations for consistency with the rest of the presentation of this chapter.

Consider Figure 7.14 by Marathe et al. (1995) , showing an irregularly bounded, uniform - thick areal reservoir containing a number of injection and production wells. This reservoir may be referred to as the “ actual ” system. For simplifi cation, Marathe et al. (1995) fi rst approximated the reservoir boundaries by several straight - line segments as depicted in Figure 7.15 . This reservoir may be referred to as the “ model ” system. Second, place N image wells outside, a distance away from and along the reservoir boundary as shown in Figure 7.16 to represent the effect of the boundaries

Figure 7.14 An irregularly bounded, uniform - thick areal reservoir containing a number of injection and production wells (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers).

P3

I2

P2

P1

PF

Domain Ω

P – Producer I – Injector

Boundary Γ

I1

Figure 7.15 Reservoir under pressure - maintenance no - fl ow boundaries (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers). P – Producer I – Injector

D

C

BA

E

P2

P3 I2 P1

P4

I1

c07.indd 215c07.indd 215 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


in a confi ned reservoir. Third, sample a suffi ciently large number of boundary points M ( M > N ) along the boundary segments and identify them with associated boundary conditions, such as constant potential or no - fl ow.

Applying the superposition principle for potential fl ow, the directional volume fl ux components are expressed as a sum of the contributions by the N R - real and N I - image wells as

uh

q x x

x x y yx

i i

i ii

NT

=−( )

−( ) + −( )=∑1

2 2 21π (7.238)

and

uh

q y y

x x y yy

i i

i ii

NT

=−( )

−( ) + −( )=∑1

2 2 21π

, (7.239)

where the total number of real and image wells is given by

N N NT R I= + . (7.240)

Because the streamlines are parallel to no - fl ow boundaries, the normal fl uid fl ux vanishes. Thus,

u u un x y= − =sin cos .θ θ 0 (7.241)

Substituting Eqs. (7.238) and (7.239) into Eq. (7.241) and then applying for the sample points located along the no - fl ow boundaries, and rearranging and com-posing the resulting system of M linear equations lead to a matrix equation as

Aq bI = , (7.242)

Figure 7.16 Schematic of N image wells and M boundary points (prepared by the author).

c07.indd 216c07.indd 216 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


where the elements of the q I column vector are the unknown fl ow rates q Ii of i = 1, 2, … , N I image wells placed outside the reservoir boundaries; the elements of the b column vector are given as follows in terms of the known fl ow rates q Ri of i = 1, 2, … , N R real wells present inside the reservoir boundaries:

bq x x y y

x x y yjj

Ri j k j k

j k j kk

NR

=−( ) − −( )[ ]−( ) + −( )

==∑ sin cos

,θ θ

2 21

1,, , ... , .2 M (7.243)

The elements of the coeffi cient matrix A are given by

ax x y y

x x y yj Mji

j i j i

j i j i

=−( ) − −( )

−( ) + −( )=

sin cos, , , ... ,

θ θ2 2

1 2 andd i N= 1 2, , ... , . (7.244)

Because the streamlines are normal to the equipotential boundaries, the tan-gential fl uid fl ux vanishes. Thus,

u u ut x y= + =cos sin .θ θ 0 (7.245)

An analysis similar to no - fl ow boundaries can be carried out for equipotential boundaries to determine the fl ow rates for the image wells placed along and outside the equipotential boundaries. The resulting linear equations are also incorporated into Eq. (7.242) , which can then be solved by an appropriate numerical method, such as the singular value decomposition method (Forsythe et al, 1977 ), for the unknown fl ow rates q Ii of i = 1, 2, … , N I image wells.

Now that the fl ow rates of the real and image wells and their positions are known, then Eqs. (7.238) and (7.239) can be used to determine the front positions and streamlines according to the procedure described in this chapter.

Figures 7.17 and 7.18 by Marathe et al. (1995) show the typical streamline patterns, and areal sweep effi ciency and water cut versus pore volume involving an

Figure 7.17 Typical streamline patterns for the fi ve - spot pattern confi ned case with image wells (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers).

X X X X X X

— Producer

— Injector

— Image well

X X X

X X X X X X X X X

X X

X X

X X

X X

X X

X X

XD C

A B

X

c07.indd 217c07.indd 217 5/27/2011 12:30:06 PM5/27/2011 12:30:06 PM


injection well surrounded by four equally spaced production wells in a confi ned reservoir. Figure 7.18 indicates that the streamline solution matches well with the analytic solution of the same problem, confi rming the validity of the streamline solution method. Figure 7.19 a – c by Marathe et al. (1995) show the streamline pat-terns for two injection and four production wells in an unconfi ned reservoir, a confi ned reservoir with no - fl ow boundaries, and a confi ned reservoir with mixed equipotential and no - fl ow boundaries, respectively. Figure 7.20 by Marathe et al. (1995) depicts the areal sweep effi ciency and water cut versus pore volume calcu-lated for the bounded reservoir case.

7.7 EXERCISES

1. Construct a plot of the dimensionless pressure versus dimensionless time for a real gas to replace the plot given in Figure 7.2 .

2. What form would Eq. (7.9) take if the derivation was carried out by including the dispersion term?

3. Obtain a Buckley – Leverett solution of saturation S 1 versus distance x S 1 at various times using Eq. (7.108) for typical assumed values.

4. Repeat the Penuela and Civan (2001) numerical solution approach using the relative permeability and capillary pressure data given by Kulkarni et al. (1998) after replacing the outlet boundary condition given by Eq. (7.95) by the general outlet boundary condi-tion formulated by Civan (1996b) given by Eq. (7.83) so that a numerical solution can be obtained until infi nite fl uid throughput at which condition all the mobile fl uid will have been displaced by the injection fl uid.

Figure 7.18 Areal sweep effi ciency and water cut versus pore volume involving an injection well surrounded by four equally spaced production wells in a confi ned reservoir. The streamline solution matches well with the analytic solution (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers).

AnalyticalStreamline

1.0

0.8

0.6

Are

al s

wee

p ef

ficie

ncy

Pat

tern

wat

er c

ut (

%)

0.4

0.2

0.00.30 0.60 0.90

Pore volume injected1.20 1.50

100

80

60

40

20

c07.indd 218c07.indd 218 5/27/2011 12:30:07 PM5/27/2011 12:30:07 PM

7.7 EXERCISES 219

Figure 7.19 Streamline patterns for two injection and four production wells in (a) an unconfi ned reservoir, (b) a confi ned reservoir with no - fl ow boundaries, and (c) a confi ned reservoir with mixed equipotential and no - fl ow boundaries (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers).

(a)

(b)

(c)

XX

XX X

XX

X X X

XX

X

X

X

X

X

XXXXXXXX

X

X

X

XX

XX

XX

XXXXXX

X

X

X

X

X

XX

XX

XX

X

X

X

X

X

X

X

X

— Producer

— Injector

D

C

A B

E

E

A B

C

D

— Producer— Injector— Image well

— Producer

No-flow boundary

— Image well

E

A B

C

D

c07.indd 219c07.indd 219 5/27/2011 12:30:07 PM5/27/2011 12:30:07 PM


5. Obtain a numerical solution of the problem given in Example 1 using the numerical method described there.

6. Obtain a numerical solution of the problem given in Example 2 using the numerical method described there.

7. Guevara - Jordan and Rodriguez - Hernandez (2001) derived the analytic solutions for pressure distribution over a model areal reservoir of a unit square shape and containing equal size, contrasting permeability porous formations, one having an injection well and the other having a production well with unit strengths, as illustrated in Figure 7.21 .

(a) Derive and confi rm their analytic solution given next for the special case of an infi nite size reservoir:

Figure 7.20 Areal sweep effi ciency and water cut versus pore volume for the bounded reservoir (after Marathe et al., 1995 ; © 1995 SPE, with permission from the Society of Petroleum Engineers).

1.0

0.8

0.6

Are

al s

wee

p ef

ficie

ncy

Pat

tern

wat

er c

ut (

%)

0.4

0.2

0.0

100

80

60

40

20

0.30 0.60 0.90Pore volume injected

1.20 1.50

Figure 7.21 Reservoir ’ s geometry for the validation of the numerical method (after Guevara - Jordan and Rodriguez - Hernandez, 2001 ; © 2001 SPE, with permission from the Society of Petroleum Engineers).

Production well

Interface

k2 = 1k1 = 10

Injection well

c07.indd 220c07.indd 220 5/27/2011 12:30:07 PM5/27/2011 12:30:07 PM

7.7 EXERCISES 221

p

K

K K

K K

K K

R Ix x x x x( ) = − + −+

⎛⎝⎜

⎞⎠⎟

−⎡⎣⎢

⎤⎦⎥

−+

1

4

1

2

11

12 1 2

1 21

2

1

π

π

ln ln

222

21

⎛⎝⎜

⎞⎠⎟

− −ln x x I Kin region,

(7.246)

and

p

K

K K

K K

K

R Ix x x x x( ) = − − − −+

⎛⎝⎜

⎞⎠⎟

−⎡⎣⎢

⎤⎦⎥

++

1

4

1

2

12

22 1 2

1 22

2

1

π

π

ln ln

KKKI

21

22

⎛⎝⎜

⎞⎠⎟

− −ln x x in region,

(7.247)

where the subscripts R and I denote the real and image wells, and 1 and 2 denote the production and injection wells, shown in Figure 7.21. K 1 = 1 and K 2 = 10 md, although no units were provided by the authors.

(b) Using the analytic solution given in (a), determine the streamline distribution for the infi nite - size reservoir and compare this with those given in Figure 7.22 .

(c) Applying the method of images and superposition with the analytic solutions given in (a) according to the mirror imaging scheme depicted in Figure 7.23 , derive the corresponding analytic solutions for the unit square reservoir with no - fl ow boundar-ies, shown in Figure 7.24 .

(d) Using the analytic solution derived in part c, determine the streamline distribution for the unit square reservoir with no - fl ow boundaries and compare this with those given in Figure 7.25 .

8. Consider Figure 7.26 showing a horizontal well completed inside a slab - shaped homo-geneous and isotropic reservoir containing a plane discontinuity. Determine the transient - state analytic solution and plot the isopotential lines for the fl ow potential distribution in this reservoir if this well begins production at a constant rate at a given time.

Figure 7.22 Streamline distribution without imaginary wells around the reservoir (after Guevara - Jordan and Rodriguez - Hernandez, 2001 ; © 2001 SPE, with permission from the Society of Petroleum Engineers).

c07.indd 221c07.indd 221 5/27/2011 12:30:07 PM5/27/2011 12:30:07 PM


9. Carry out an analysis similar to Shikhov and Yakushin (1987) for a well in an anisotropic porous media.

10. Carry out an analysis similar to Marathe et al. (1995) for an anisotropic reservoir.

11. The analytic solution of the leaky - tank model was plotted as straight lines for the dimensionless average fl uid density versus the dimensionless time by Civan (2000e) . Construct a plot of the average reservoir fl uid pressure versus the dimensionless time corresponding to the dimensionless density and dimensionless time range indicated by

Figure 7.23 Mirror imaging of the reservoir in Figure 7.24 used in the analytic solution by the method of images (after Guevara - Jordan and Rodriguez - Hernandez, 2001 ; © 2001 SPE, with permission from the Society of Petroleum Engineers).

2

1

0

–1

–1 0 1 2

k2

k2

k2

k2

k2

k2

k2

k2

k2

k1

k1

k1

k1

k1

k1

k1

k1

k1

+

++

++

++

––

––

––

+ +

– – –

Figure 7.24 Example of imaginary wells and boundary point distribution for the reservoir considered in the validation (after Guevara - Jordan and Rodriguez - Hernandez, 2001 ; © 2001 SPE, with permission of the Society of Petroleum Engineers).

Boundary points, interfaceImaginary well, region 2Imaginary well, region 1Boundary points

Injection wellProduction well

c07.indd 222c07.indd 222 5/27/2011 12:30:08 PM5/27/2011 12:30:08 PM

7.7 EXERCISES 223

Figure 7.2 for typical slightly compressible and compressible fl uids using the data presented by Civan (2000e) .

12. Solve the following problem using the data given as porosity ϕ = 0.25, K x = 250 mD = 250 × l0 − 15 m 2 , K y = 150 mD = 150 × l0 − 15 m 2 , formation thickness = 30 ft = 9.14 m, r w = 0.25 = 0.0762 m, oil production rate q w = 200 stb/day = 4.4 × l0 − 4 cm 3 /s, B o = 1.2 bbl/stb, oil gravity = 35 API, viscosity = 3.0 cp, pressure at the external bound-ary = 3000 psia = 204 atm, reservoir half - length ( x e ) = 4000 ft = 1219 m, reservoir half - width ( y e ) = 2500 ft = 762 m, number of grid blocks in the x - direction ( N ) = 40, and number of grid blocks in the y - direction ( M ) = 25. Prepare the two - dimensional ( x , y ) contour plots of the reservoir fl uid pressure, Darcy velocities in the x - and y - Cartesian directions, the resultant Darcy velocity, and its direction angle with respect to the x - axis over the quadrant of a rectangular - shaped reservoir of unit thickness based on the fi nite difference numerical solution method (Fig. 7.27 ). Make reasonable assumptions for any missing data. Explore ways of improving the solution method and accuracy. You may use any computing tool of your preference.

13. Consider the quadrant of a rectangular - shaped reservoir having a uniform thickness in which a well in the center is hydraulically fractured as shown in Figure 7.28 . Half - fracture length ( x f ) = 500 ft, fracture width ( W f ) = 1.0 in., half - reservoir length ( x e ) = 2500 ft, half - reservoir width ( y e ) = 1500 ft, reservoir permeability ( K ) = 0.1 darcy, fracture permeability ( K f ) = 1.0 darcy, wellbore pressure ( p w ) = 1500 psia, fl uid viscos-ity μ( ) = 0 7. cP, number of grid points in the x - direction N = 30, and number of grid

Figure 7.25 Streamline distribution computed with the new method for the validation problem (after Guevara - Jordan and Rodriguez - Hernandez, 2001 ; © 2001 SPE, with permission from the Society of Petroleum Engineers).

Figure 7.26 Cross - sectional view of a semi - infi nite slab reservoir containing a horizontal well (prepared by the author). x

yz

H

ZReservoir

Well

→∞

X

c07.indd 223c07.indd 223 5/27/2011 12:30:08 PM5/27/2011 12:30:08 PM


points in the y - direction M = 20. Assume a no - fl ow condition at the fracture tip. Repeat the solution using exactly the same data but assuming (a) constant pressure Dirichlet boundary conditions along the external reservoir boundaries prescribed by p = p e = 3000 psia, (b) constant fl ow Neumann boundary conditions along the external reservoir boundaries specifi ed by ∂ ∂ =p x 3 0. / psia ft, and ∂ ∂ =p y 3 0. / psia ft. (c) Compare the pressure profi les along the fracture obtained in parts a and b. (d) Compare the pressure profi les (isobar contours) over the reservoir domain obtained in parts a and b.

14. Write down the equation of continuity with a constant source term for one - dimensional transient - state radial fl ow of a single - component gas through a homogeneous and iso-tropic porous media. Substitute Darcy ’ s law to obtain a pressure equation. Solve this equation for fl ow of an ideal gas in the radial domain of r w < r < r e under the steady - state conditions, where r w and r e denote the wellbore and external reservoir boundaries along which the constant pressures are specifi ed as p w and p e , respectively.

Figure 7.27 Production well in the quadrant of a rectangular - shaped reservoir having a uniform thickness (prepared by the author).

(0, 0)

∂p/∂y = 0

∂p/∂x = 0

qw

P = Pe = constant

P = Pe =constant

xe

ye

x

y

Reservoir

qw/4

Figure 7.28 Hydraulically fractured well in the quadrant of a rectangular - shaped reservoir having a uniform thickness (prepared by the author).

(0, 0)

∂p/∂y = 0

∂p/∂x = 0

Pw

P = Pe or ∂p/∂y = constant

P = Pe

or ∂p/∂x = constant

xf

xe

ye

x

y

Reservoir

Fracture ∂p/∂x = 0

c07.indd 224c07.indd 224 5/27/2011 12:30:08 PM5/27/2011 12:30:08 PM

7.7 EXERCISES 225

15. The water - to - oil viscosity ratio is

μμ

w

o

=1 0. .

The water - to - oil relative permeability ratio correlation is given by

k

ka bSro

rww= −( )exp ,

where a = 1.0 and b = 2.0. At what water saturation value does the fractional water versus water saturation curve has an infl ection point?

16. If the water - drive strength factor f of the leaky - tank boundary is greater than 1.0, what does it indicate about the condition of the leaky external boundary condition?

c07.indd 225c07.indd 225 5/27/2011 12:30:08 PM5/27/2011 12:30:08 PM

porous media transport phenomena (civan/transport phenomena) || fluid transport through porous media

Documents