complete treatment of fractured reservoir simulation

Oil & Gas Science and Technology Rev. IFP, Vol. 57 (2002), No. 5, pp. 499-514Copyright 2002, ditions Technip

Numerical Simulation of Three-Phase Flowin Deforming Fractured Reservoirs

R.W. Lewis1 and W.K.S. Pao1

1 School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2-8PP - United Kingdome-mail: [email protected] - [email protected]

Rsum Modlisation numrique dcoulements de fluides triphasiques et de la dformation dela roche pour des rservoirs fracturs Dans cet article, les auteurs, sappuyant sur la thorie de ladouble porosit, dveloppent une formulation mathmatique tridimensionnelle des coulements defluides triphasiques et de la dformation de la roche pour des rservoirs fracturs. La formulationactuelle, qui recouvre la fois les quations dquilibre mcanique et les quations de conservation de lamasse dans le cas dcoulements multiphasiques, rend compte de linfluence significative du couplageentre coulements fluides et dformations solides, gnralement non pris en compte ou ngligs dans lespublications traitant de la simulation des rservoirs. Une mthode par lment fini de type Galerkin estemploye pour discrtiser les quations principales dans lespace et un schma en diffrences finies entemps est utilis pour dterminer lvolution au cours du temps. Compar des modles similairesexistants, celui-ci identifie non seulement la dformation interne du squelette rocheux sous leffet de lapression diffrentielle entre le milieu poreux et le milieu fissur, mais les quations de conservation demasse qui en rsultent sont entirement couples. titre de comparaison, un exemple lchelle dunrservoir sert de test alpha de la robustesse de la mthodologie adopte. Les rsultats indiquent que lecomportement du rservoir est sensiblement diffrent lorsque leffet du couplage est reprsent.

Abstract Numerical Simulation of Three-Phase Flow in Deforming Fractured Reservoirs Themathematical formulation of a three-phase, three-dimensional fluid flow and rock deformation infractured reservoirs is presented in this paper. The present formulation accounts for the significantinfluence of coupling between the fluid flow and solid deformation, an aspect usually ignored in thereservoir simulation literature. A Galerkin-based finite element method is applied to discretise thegoverning equations in space and a finite difference scheme is used to march the solution in time. Thefinal set of equations, which contain the additional cross coupling terms as compared to similar existingmodels, are highly nonlinear and the elements of the coefficient matrices are updated implicitly duringeach iteration in terms of the independent variables. A field scale example is employed as an alpha caseto test the validity and robustness of the currently formulation and numerical scheme. The resultsillustrate a significantly different behavior for the case of a reservoir where the impact of coupling is alsoconsidered.

Oil & Gas Science and Technology Rev. IFP, Vol. 57 (2002), No. 5

INTRODUCTION

Productivity of a reservoir, to some extend, depends on themicroseismic fissures that exist in the oil formation. Themore intense the extend of fissuring, the more economic thereservoir is as fissures promote flow in the oil bearing strata.Due to the random statistical distribution of the fissures,direct numerical modeling of each individual fissures is aformidable task due to the large number of data sets involvedin one hand, and the exceedingly high computational cost onthe other. Hence, a continuum approach based on the doubleporosity model introduced by Barenblatt et al. (1960), andlater extended by Warren and Root (1963) is more suitable inthis scenario.

Over the past few decades, double porosity models withvarying degree of sophistication have been developed tosimulate flow through fractured porous media. Taking intoaccount the deformability of the oil formation, Aifantis andco-workers (Aifantis, 1985; Wilson and Aifantis, 1982;Khaled et al., 1984) extended Barenblatts model to what isnow known as deformable double porosity formulation.Alternative of Aifantis formulation were also given latter byothers in this area, e.g. Valliappan and Khalili (1990), Cho etal. (1991), Bai et al. (1993), Berryman and Wang (1995),Ghafouri and Lewis (1996), Chen and Teufel (1998), inchronological order. The majority of these works, however,concentrated on single-phase flow phenomena in adeformable geo-material, which has its special utility in civilengineering application like footing analysis etc. Directapplication of such models to a deformable oil reservoir is anidealization far from realistic as far as the petroleum industryis concerned. More recently, Lewis and Ghafouri (1997) andBai et al. (1998) have extended the double porosity conceptto include multiphase flow. To the authors knowledge, themodels developed by these two groups represent the onlyexisting simulators able to analyze multiphase flow in de-formable fractured geo-material, with particular emphasis onpetroleum reservoir mechanics. However, a few inadequaciesexist in these models. The formulation given in (Lewis andGhafouri, 1997) neglected the fracture deformation whilethose in (Bai et al., 1998) only considered a two-phase oil-water problem.

The objectives of the present paper are threefold. First, toeliminate the deficiencies in the present multiphase doubleporosity model. Second, to recast the governing equationsbased on a multiphase flow formulation and finally, topresent a numerical analysis of the proposed model.

1 PRELIMINARY CALCULATIONS

The model presented here conceptually consists of twoseparate models: the deformable skeleton; and the multiphase fluid flowing through the porous media.

The skeleton description is based on elastic theory, whilethe flow model is based on the double porosity concept. Twooverlapping flow regions are considered, one representing thefissured network and the other representing the porousblocks. It is postulated that an elementary scale exits, whichis sufficiently large for the formulation to be valid. In thefollowing derivation, the subscripts = 1 and 2, refer to theporous block and fissured block, respectively; and subscriptss, w, o and g stand for the skeleton, water, oil and gas phases.Other symbols and notations will be defined locally as theyappear in the script, and the assumptions stated explicitly aswe progress.

The porosities for both flow regions is defined as:

(1)

where is the pore volume, and b is the bulk volume. Weuse the notation:

(2)

to denote the total derivative of a quantity and vs

is thebarycentric velocity of that quantity. Small perturbation istacitly hypothesized assumed throughout the paper. In amulti-phasic domain containing oil, water and gas,conservation of mass required that:

(3)The interacting motion of each phase (not including the

skeleton constituents) can be linked to the evolution of thepartial pressures of each phase to their saturation values. Tothis end, a capillary pressure-saturation curve is used, whichis an approximation of both the drainage (drying) and theimbibitions (wetting) curves. In a saturated oil reservoir, thefluid pressure values at any point are related by their capillarypressure relationships. In general, capillary pressure isdefined as the difference between the nonwetting and thewetting phase pressures, i.e.:

(4)Therefore, for water-wet oil reservoirs, the following

expressions are used:For oil-water system:

(5)For oil-gas system:

(6)

From Equations (3), (5) and (6), the following saturation-pressure relationships can be derived:

(7) =

St

S pt

pt

ww

o w

'

p p pcg g o =

p p pcw o w =

p p pc nwet wet=

S S Sw o g + + = 1

ddt t s

=

+

v x

=b

500

RW Lewis and WKS Pao / Numerical Simulation of Three-Phase Flow in Deforming Fractured Reservoirs

(8)

(9)

where:

(10)

and:

(11)

are the slope of the S vs. pc plot for an oil-water and an oil-gas system, respectively.

To relate the skeleton motion due to the fluid, the effectiveaverage pore pressure is defined, where it is an average porepressure weighted by the saturations for both the matrix andfractures systems viz:

(12)

On incorporating Equations (7)-(9) into (12), the fol-lowing equation is obtained for the average pore pressure ina three-phase flow regime:

(13)

where:(14)

(15)

(16)

2 MATHEMATICAL FORMULATIONS

2.1 Skeleton Deformation

The linear momentum balance equation for the skeleton canbe written as:

(17)where is the total stress, and f is any arbitrary externaland/or body force per unit volume. The total stress canbe expressed in terms of the effective stress ' and theaverage pore pressure p, according to Biots effective stresslaw as:

(18)

In Equation (18), the phenomenological constants aretermed the modified Biots coefficients (or modified Biot-Willis parameter). As pointed out by Berryman and Wang(1995), the form given by Equation (18) is not seriously indoubt. An important characteristic of this equation is thateach point in space now has two weighted fluid pressuresassociated with it, and therefore these pressures are not thetrue microscopic pressures in the fluid, but the average oversome representative volume.

To obtain an explicit expression for the phenomenologicalconstants in terms of physically measurable parameters, onecan use Bettis reciprocity theorem. From the analysis carriedout in (Khalili and Valliappan, 1996) for an isotropic elasticmedium, it can be shown that:

(19)

in which K, Kp and Km are the drained bulk moduli of theskeleton, fissured block and the solid constituent, respec-tively. It is not difficult to show from Equation (19) that:

(20)

where is the well-known Biot coefficient for a non-fissured material. This shows that Equation (18) lies withinthe framework of Biot poroelasticity (Biot, 1941). It isworthwhile mentioning that if the volume of fissures isreduced to zero, i.e. Kp = K, then

Substituting Equation (18) into (17), one obtains:(21)

Here, the issue of whether Equation (21) should be castinto an incremental or finite form will not be discussed. For amore detail treatment of this, we refer the reader to (Schreflerand Gawin, 1996).

2.2 Flow Model

For mltiphase flow in porous continua, the followinggeneralized mass balance equation at reservoir conditionsholds (Pao, 1998; Pao et al., 2001):

(22)

in which the L() is the logical operator, defined as:

(23)L go w

( ) :: ,

=

=

=

10

+[ ]+

+( )

+ =

div

S L R S

Qt

S L R S

o so o o

STC STC o so o

v v( )

( ) ( )( )

1

1 0

div grad grad 00 = 1 1 2 2p p

2 10= = and .

= + = 1 2 1K

Km

, 1 2 1= = K

KK

KK

Kp m p

= ' 1 1 2 2p p

div 00+ =

S S p Sg g cg g '' '= +

S S p Sw w cw w '' '= +

S S p S p So o cg g cw w '' ' '=

=

+

+

pt

S pt

S pt

Sp

too

ww

gg

'' '' ''

p S p S p S pw w o o g g = + +

SSpg

g

cg'

=

S Spw

w

cw

'

=

=

+

St

S pt

S S pt

Sp

to

ww

g wo

gg

' ( ' ' ) '

=

St

Sp

t

pt

gg

g o

'

501


The subscript STC denotes the quantity evaluated at stocktank conditions, and v is the absolute velocity of the fluidphase in a medium . The density of each phase may beexpressed at the STC via the relation:

(24)

where B is the formation volume factor of the fluid phase .In Equation (22), R

so is the solution oil-gas ratio in themedium , and is the so-called leakage function. Itsexpression emanates from the fact that during someinfinitesimal time dt, a fluid of volume:

(25)

per unit of bulk volume is transferred, at reservoir conditions,between the porous block and the fissured network.Substituting Equation (24) into (22) yields:

(26)

The relative velocities of fluid phase in the medium can be described by:

(27)

On the other hand, the relative velocity of the fluid can belinked to Darcys law via:

(28)

in which K is termed the effective permeability of phase .In a multiphase system, the simultaneous flow of the fluidwill cause each fluid to interfere with the flow of the otherfluid components. This effective permeability value must beless than the single-fluid permeability of the medium. Then,one can define the relative permeability as:

(29)

Substituting Equations (2), (27), (28) and (29) into (26),one obtains:

From Equation (30), various alternative formulations existand one can recast the mass conservation equations in termsof pressure, capillary pressure or saturation (Aziz and Settari,1979). From a literature survey, it was found that Schreflerand Simoni (1991) made an extensive search on this issue andconcluded that, for a nonfissured two phase flow problem, thebest convergence was found using the combination ofdisplacements and pressures as primary unknowns. Based ontheir conclusion, Equation (30) will now be recast in terms offluid pressures as primary unknowns. Note that the timederivative of the formation volume factor and the solution oil-gas ratio can be rewritten as:

(31)

and: (32)

which pose no serious difficulty. Therefore, attention is nowfocused on the porosity term. Making use of the definition inEquation (1), we have:

(33)

Equation (33) states that the increase in porosity is due tothe increase in the pore volume minus the increase in thebulk volume. Note that the second expression on the RHS ofEquation (33) is actually the volumetric strain of theskeleton, i.e.:

(34)

Upon substitution of Equation (34) into (30), and aftersome manipulation, we have the following eight unknowns(p

w1, po1, pg1, pw2, po2, pg2, 1, 2) but only six equations.One way to obtain the two additional equations is by relatingthe unknowns 1 and 2 with the primary field variables p,and the skeleton displacement, u. In order to do so, weshould assume that the eight unknowns could be lumped as(p1, p2, 1, 2) according to the assumption made earlier inEquation (12). The reason for this is because it is impossibleto analyze, e.g. the effect of p

oonalone without involving

pw

and pg, and vice versa. By making this assumption, wecan now consider a representative volume of the fissuredporous medium subjected to the stress conditions as shown inFigures 1 to 3.

d bb

s

=

=v

xtr

d d db

b

b =

=

=

Rt

Rp

pt

Rp

tso so

o

oso

o

'

=

=

t B p B

pt

Bp

t

1 1

'

kr

= Kk

1

vK

r p= grad

v v v r sS= ( )

+

+

+

+ =

div

SB

LR S

BQ

t

SB

LR SB B

so o

oo STC

so o

o

v v( ) ( )( )

( ) ( )

1

1 0

dttt dt+

=

( )STCB

502

+

+ +

+ +

+

div grad gradk k

v

kB

p LR k

Bp

ddt

SB

LR SB

SB

LR SB x

Q

r so ro

o oo

so o

o

so o

o

s

( ) ( )

( ) ( )( )1 STCSTC B+ =( )1 0

(30)


Figure 1Stress state Case I corresponds to an external hydrostaticpressure of dM, an internal matrix pressure of dp

1 and aninternal fissure pressure of dp2 (adapted from Khalili andValliappan, 1996).

Figure 2

Stress state Case II corresponds to an equal external andinternal matrix pressure of dp1 (adapted from Khalili andValliappan, 1996).

Figure 3

Stress state Case III an equal external and internal fissurepressure of dp1 and a pore pressure of 0 (adapted from Khaliliand Valliappan, 1996).

Applying Bettys reciprocity theorem to Figures 1 and 2,Figures 1 and 3, and Figures 2 and 3, it can be shown that(Khalili and Valliappan, 1996):

(35)

(36)

According to Equations (35) and (36), the pore volumechange per unit of bulk volume in a fissured porous mediumis due to three components on the RHS. The first term is thevolumetric change due to a change in the overall bulkvolume of the solid skeleton. It depends on the overall bulkcompressibility of the material, which is dictated by themultiplying coefficient (modified Biots coefficients) in frontof the volumetric strain. In the case where the material isincompressible, and are equal to unity, otherwise,they are less than one. The second term is the volumetricchange due to a change in the fluid pressure occupyingthe pores or fissures. It depends on the matrix/graincompressibility of the solid constituent. The third term is thevolumetric change due to the pressure difference between thepores and fissures. It is responsible for the internaldeformation of the material. It should be noted that evenunder the situation where the matrix is incompressible, thereis still a pore volume change due to the first and third term.

The coupled formulation is obtained by substitutingEquations (35) and (36) into (30). Assuming that in mostpractical problems v

s 0, then:

(37)

the following compact form of the mass balance equationsfor each fluid phase is finally obtained:

For the water phase in the pore matrix:

(38)

+ +

+

+

+ +

+

div gradk

m

1 1

11 1 1 2

1 11

1 11

1 11

1 22

1 22

1 21 1

1

kB

p p p

pt

pt

pt

pt

pt

pt

SB

rw

w ww w w w

w ww

w oo

w gg

w ww

w oo

w gg w

w

( )

TTt

= 0

vsx t

ddt t


where:(39)

(40)

(41)

(42)

(43)

(44)

From Equations (38) and onwards, the four indexsubscripts which appear in the parameter indicate therelationship of each phase in either the matrix or the fracturedmedium to the other phases, i.e.

w1o2 should read lambda() relating water in matrix (w1) to oil in fracture (o2). InEquation (38), m is a hydrostatic vector defined as:

(45)

By the same argument, the water equation in the fracturescan be obtained as follows:

(46)

where:

(47)

(48)

(49)

(50)

(51)

In order not to bombard the reader with equations, we giveonly the expression for water equations here while thecomplete coupled equations can be found in the Appendix.Before proceeding further, it is important to make a fewremarks at this stage.

Remark 1The governing equations derived here are significantlydifferent from those obtained either by Lewis and Ghafouri(1997) or Bai et al. (1998). In their derivation, the crosscoupling terms

w1w2, w1o2, w1g2, w1w2, etc. are zero. Thisimplies that the multiphase flow fields in their equations areindependent of each other in the matrix and fractures, whilein the present formulation these are coupled. These crosscoupling terms are present because the internal deformationof the skeleton is a function of the pressure differentialbetween the matrix and the fissures.

Remark 2The meaning of these cross coupling terms and theirinfluence in Equations (38) and (46) can be explainedqualitatively by looking at the sign in front of thesecoefficients. For example, the quantity (volume of water perunit of bulk volume per unit of time):

(52)

in the matrix equation (see w1w1) implies that the liquid

water is attempting to drain from the matrix due to itssaturation derivatives S''

w1. This drainage is possible due tothe matrix compressibility , and the internaldeformation /K, due to the pressure increment p

w1. On theother hand, the quantity:

(53)

in the matrix equation (see w1w2) is trying to prevent this

drainage from happening in the matrix due to the saturationderivatives S''

w2 in the fractures. This interference arisesbecause the fractures are trying to prevent the matrix fromdeforming independently.

Remark 3The definition of the relative flow vector of each phase,defined by Equation (27) is essential in dictating the finalform of the coupled multiphase formulation. For example, inBai et al. (1998), they define the relative flow vector as:

(54)

Note the difference in the term vs in Equation (54) as

compared to Equation (27).

v v v r sS= ( )

SB

SK

pt

w

ww

w1

12

2''

( ) / 1 1 Km

+

SB

SK K

pt

w

ww

m

w1

11

1 1 1''

w g ww

gm

SB

SK K2 2

2

22

2 2=

+

''

w ow

ww

wo

mBS S

BS

K K2 22

22

2

22

2 2=

+

'

''

w w w ww

ww

ww

m

S BB

S SB

SK K2 2 2 2 2

2

22

2

22

2 2=

+

' '

''

w w ww

w w ow

wo w g

w

wg

SB K

S SB K

S SB K

S2 1 22

1 2 12

21 2 1

2

21= = =

'' '' '', ,

w rww w

kB2

2 2

2=

k

+ +

+ +

+

+

+ +

+

div gradk2 22

2 2 1 2

2 11

2 11

2 11

2 22

2 22

2 22 2

kB

p p p Q

pt

pt

pt

pt

pt

pt

rw

w ww w w w w STC

w ww

w oo

w gg

w ww

w oo

w gg

( ) ( )

SSB t

w

w

T2

20m =

mT = { , , , , , }1 1 1 0 0 0

= +

+

1 2 1 21 2

1 2

w w ww

w w ow

wo w g

w

wg

SB K

S SB K

S SB K

S1 2 11

2 1 21

12 1 2

1

12= = =

'' '' '', ,

w g ww

gm

SB

SK K1 1

1

11

1 1=

+

''

w ow

ww

wo

mBS S

BS

K K1 11

11

1

11

1 1=

+

'

''

w w w ww

ww

ww

m

S BB

S SB

SK K1 1 1 1 1

1

11

1

11

1 1=

+

' '

''

w rww w

kB1

1 1

1=

k

504


What vs actually implies is that:

(55)i.e. two relative flow vector are defined, one with respectto the porous block and the other one with respect tothe fissured block. By imposing the stress equilibriumcondition:

(56)and assuming that the strains are additively decomposable,i.e.:

(57)one will find, after following the outlined derivationprocedure, the following multiplying coefficient in front ofthe volumetric strain term:

(58)

in the matrix Equation (38) in which C1 and C2 are the elastictangent moduli of the matrix and fractured block,respectively. Notice that the quantity:

(59)

defines another compressibility coefficients. It is difficult toexplain the physical meaning of Equation (59) since already measures the compressibility of the porous block.The same argument also holds for the fissured block.Because of this confusion, we should use the relative flowvector of each phase as defined by Equation (27), whichdefines the relative velocity with respect to the solid skeleton,in a macroscopic sense.

Remark 4The fissured block is modeled as blocks of orthogonalfractures, e.g. parallel plates and as a sugar cube model. Thegeometric parameter, can be written for quasi steady-statecondition as (Warren and Root, 1963):

(60)

and:

(61)

n = 1,2,3 is the number of normal sets of fractures and d1, d2,d3 are the fracture intervals in each direction.

3 INITIAL AND BOUNDARY CONDITIONS

Normally, the initial conditions of a reservoir system can bedefined by specifying the initial distribution of fluid pressurewithin the reservoir and/or its saturation, depending on theunknowns used in the formulation and the updatingprocedure. Therefore, the initial conditions for a three-dimensional reservoir system can be defined as follows:

(62)

(63)where p0 is the fluid pressure at position vector x at timezero and S0 is the corresponding saturation. For theequilibrium equations, the initial conditions can either be aprescribed initial displacement or a prescribed initial stress,i.e.:

(64)For the sake of simplicity in this paper, we should assume

that u0 = 0 = 0. In the case where the initial stress is non-zero, the above procedure can also be applied, but now thecalculated Cauchy stress has to be added to the initial stressvalue to obtain the in situ stress. This additivity of thestresses is reasonable in this case because we have assumedan elastic constitutive equation for the skeleton and thus thetheory of superposition can be applied.

Generally, the flow boundary condition may be writtenbased on Darcys equation where the flow rate q of the-phase, which can be either flow in or out of the system,must satisfy the following condition:

(65)

In practice, two different types of flow boundary condi-tions are usually applied, firstly by prescribing the flow rate:

(66)or secondly, by specifying the value of pressure at theboundary as:

(67)The boundary segments q and p have to satisfy the

conditions such that:(68)

For the equilibrium equations, one requires the displace-ment boundary conditions, i.e.:

(69)and the traction boundary conditions. For the sake ofsimplicity, the traction boundary condition is set equal tozero, i.e.:

(70) = = n 0 on t

u u= on u

= = q p q p and

p p p = on

q q q = on

= nk

kB

p qr qgrad on

u u0 0= =i i;

S S i 0 ( ) ( )x x=

p p i 0 ( ) ( )x x=

l

d nd d

d dn

d d dd d d d d d

n

=

=

+ =

+ + =

1

1 2

1 2

1 2 3

1 2 2 3 3 1

12 2

3 3

= +4 22

n n

l( )

1

1 12

1+

CC C1

SB t

w

w

T1

11 1

1mC

C C +

+

1

1 2

= +1 2

= =1 2

v vs s1 1 2 2= [ ] [ ] = [ ] [ ]tr tr tr tr ,

505


Again, the prescribed boundary segments u

and thave to satisfy the conditions such that:

(71)

4 NUMERICAL DISCRETISATION

A Galerkin-based Finite Element Method was applied to thegoverning equations obtained in the Section 2, where thedisplacements, u, and the fluid pressures, p, in the twooverlapping continua, namely the matrix and the fracturednetwork, are the primary unknowns. By first recasting thegoverning equations in a weak form and approximating theunknown vector as:

(72)where , the following equation is obtained:

(73)

Applying the finite difference implicit -time steppingscheme to Equation (73) and letting:

(74)and:

(75)the temporally discretised form of Equation (73) willultimately be given as:

(76)where n is the iteration counter. The details of the FiniteElement Method will not be elaborated here but can be found

elsewhere, e.g. Lewis and Schrefler (1998), Pao et al. (1999),Pao (2000), Hughes (2000). All the coefficients matricesappearing in A and B are nonlinear and are dependent onthe values of the sought unknowns. Therefore, iterativeprocedures are performed within each time step to obtain thefinal solution. In fact, Equation (76) has been implemented inthe code CORES (COupled REservoir Simulator) developedat the University of Wales Swansea (Pao, 1998; Pao et al.,1999; Pao, 2000). In order to test the validity of theformulation, the alpha test suite for the implementation nowfollows.

5 NUMERICAL EXAMPLE

The selected numerical example is a pseudo three-dimensional, field scale case, which was a part of the sixthSPE comparative solution project for double porositysimulators (Firoozabadi and Thomas, 1990). This examplewas chosen not only because the information required for thesimulator is complete, but also a comparison of the solutiontechnique with other simulators is also possible. A linearsection of reservoir was modeled (Fig. 4). Vertically, fivelayers each having a height of 50 ft [15.2 m] was used.Horizontally, the reservoir was divided into ten 200 ft [61 m]grid blocks. A uniform thickness of 1000 ft [305 m] was usedin the y-direction. The layer description for the cross-sectionis given in Table 1. Whereas both the shape factors andfracture permeabilities may be determined in terms of thematrix block size, the corresponding values given in Thomaset al. (1983), and presented in Table 1, were used directly.The oil production rate was calculated as follows:

(77)Q I po p o2 2=

B A X B A X F+[ ] = [ ] ++ ++ + + + t t tnt nt nt nt nt ) )11 11( )

d t t t) ) )X X X= +1

) ) )X X Xt t t+ += + 1 1( )

AX B X F

) )+ =

ddt

) ) ) )X u p p= , 1 2 ,{ } u Nu p Np p Np = =

) ) ) ) ), , 1 1 2 2

)X

= = u t u tand

506

1000 ft

10 x 200 ft

Fra

cturedre

servoir

Ove

rburden

strata

5 x

50 ft

Figure 4

Finite element mesh used for the SPE example (not in scale).


where po2 is the well drawdown in psi and Ip is the

productivity index, defined as:

(78)

in which kro2 is the oil relative permeability and o2 (cp) and

Bo2 (ResB/STB) are the viscosity and formation volume

factor of the produced oil, respectively, at well bottom hole

pressure. The coefficient J has dimension of ResB-cp/day-psi, or Darcy-ft, and the corresponding values for each layerare given in Table 1.

The water and gas rates were obtained in terms of themobility ratio values as follows:

(79)

(80)Q M Q R Qo go o so o2 2 2 2 2= +

Q M Qw wo o2 2 2=

Ik J

Bpro

o o

=2

2 2

507

TABLE 1

Basic data for the sixth SPE comparative solution project

Matrix block Fracture Shape JLayer size, ft Permeability, mD Factor, ft2 Ft2[m2]

[ 0.3048 m] [ 9.87 1010 m2] [ 0.0929 m2] RB-cp/D psi [m3]1 25 [7.62] 10 [9.89 109] 0.040 [0.431] 1 [0.0283]2 25 [7.62] 10 [9.89 109] 0.040 [0.431] 1 [0.0283]3 5 [1.524] 90 [8.88 108] 1.000 [10.76] 9 [0.2549]4 10 [3.048] 20 [1.97 108] 0.025 [0.269] 2 [0.0566]5 10 [3.048] 20 [1.97 108] 0.025 [0.269] 2 [0.0566]

Matrix permeability, k1, mD [m2] 1[9.89 1010]Matrix porosity, 1 0.29Fissured block porosity, 2 0.01Initial oil pressure, psi [MPa] 6000 [41.37]Initial water-oil capillary pressure, psi [Pa] 0.87 [6000]Initial gas-oil capillary pressure, psi [Pa] 0.02 [500]Matrix compressibility, vol/vol psi [vol/vol Pa] 3.5 106 [5.075 1010]Z-direction transmissibilities Multiply calculated values by 0.1

TABLE 2

PVT data for the Sixth SPE comparative solution project

Pressure Bo

Bg Rso o g (psi) (RB/STB) (RB/SCF) (SCF/STB) (cp) (cp) (dyne/cm)

1674.0 1.3001 0.00198 367.0 0.529 0.0162 6.02031.0 1.3359 0.00162 447.0 0.487 0.0171 4.82530.0 1.3891 0.00130 564.0 0.436 0.0184 3.32991.0 1.4425 0.00111 679.0 0.397 0.0197 2.23553.0 1.5141 0.000959 832.0 0.351 0.0213 1.284110.0 1.5938 0.000855 1000.0 0.310 0.0230 0.0724544.0 1.6630 0.000795 1143.0 0.278 0.0244 0.4444935.0 1.7315 0.000751 1285.0 0.248 0.0255 0.2555255.0 1.7953 0.000720 1413.0 0.229 0.0265 0.1555455.0 1.8540 0.000696 1530.0 0.210 0.0274 0.0907000.0 2.1978 0.000600 2259.0 0.109 0.0330 0.050

Original bubble point, psi [MPa] 5545 [38.23]Density of stock-tank oil, lbm/cu ft [kg/m3] 51.14 [810.04]Gas density at standard condition, lbm/cu ft [kg/m3] 0.058 [0.91870]Water formation volume factor 1.07Water compressibility, vol/vol psi [vol/vol Pa] 3.5 106 [5.075 1010]Water viscosity, cp [Pa s] 0.35 [0.35 103]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Krog

Krg

Gas saturation, Sg

Rel

ative

per

meabi

lity, K r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gas saturation, Sw

Rel

ative

per

meabi

lity, K r

Krow

Krw

Figure 5

Saturation versus relative permeabilities in the oil-gas systemfor the matrix continuum.

Figure 6

Saturation versus relative permeabilities in the oil-watersystem for the matrix continuum.

0.1 0.2 0.3 0.4 0.5 0.6 0.7500

1000

1500

2000

2500

3000

Gas saturation, Sg

Capi

llary

pre

ssur

e, P c

g (P

a)

0.20.10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-300

-150

-200

-250

-100

-50

0

50

Water saturation, Sw

Capi

llary

pre

ssur

e, P c

w

(Pa)

Figure 7

Capillary pressure versus saturation in the oil-gas system forthe matrix continuum.

Figure 8

Capillary pressure versus saturation in the oil-water systemfor the matrix continuum.

where Mwo2 and Mgo2 are the mobility ratios of water to oil

and mobility ratio of gas to oil, respectively, and arecalculated as follows:

(81)

(82)

The production rates are calculated using data evaluated atthe beginning of each time step, i.e. an explicit loading is

assumed. Depletion runs were carried out to a maximumof 10 years, or whenever production declined to less than1 STB/D [0.16 STM/D]. The production well has a maximumrate of 500 STB/D [80 STM/D] and was limited by amaximum drawdown of 100 psi [6894.757 KPa]. This wellwas located at the far right column and perforated only in thebottom layer. Basic pressure-volume-temperature (PVT) datafor the example are given in Table 2 (Firoozabadi andThomas, 1990). Also, Figures 5 to 8 illustrate the saturation-relative permeability and capillary pressure-saturation curvesfor gas-oil and water-oil systems, respectively.

Mk Bk Bgorg g g

ro o o2

2 2 2

2 2 2=

( / )( / )

Mk Bk Bworw w w

ro o o2

2 2 2

2 2 2=

( / )( / )


These curves apply to the matrix continuum only. Becauseof the limitation associated with the formulation, a pseudo-capillary pressure curve, having a negligible effect on thedepletion behavior, was assumed. The variation of gas/oilinterfacial tension (IFT), , with pressure, as given in Table 2has also been incorporated into the problem. For thispurpose, the gas/oil capillary pressure is directly related tothe IFT and therefore the gas/oil capillary pressure should beadjusted according to the ratio of IFT at reservoir pressure,divided by the value of the IFT at which the capillarypressure-saturation curve is specified, i.e. bubble pointpressure, or in a mathematical form:

(83)

where the term pcgI corresponds to input capillary pressure

values calculated using a surface tension 1. Instead of anadaptive time step value, an equally spaced time step sizes of0.1 year were used, and no attempt was made to optimize thetime step size. The domain was assumed to be sealed and noflux boundary condition was prescribed at the surfaces. Thereservoir was assumed to deform vertically and alldisplacements perpendicular to the lateral surfaces were fixedat zero. Also, the base of the reservoir was assumed to haveno movement in all directions. The Youngs modulus ofelasticity for the solid skeleton was estimated in terms of thecompressibility values of the porous skeleton using therelationship given by equation K = E/(3(1 2v)) and byassuming a constant Poissons ratio v = 0.2. Due to the lackof data on matrix bulk modulus, K

mand pore fissured bulk

modulus, Kp, it was assumed that and such that the equivalent Biot-Willis parameter is equal to 0.8. Based on this, the values of K

mand Kp can be calculated

using Equation (19).

Figure 9 shows the plot of oil production versus elapsedtime. The results obtained by the present study show asignificantly delayed reduction of the production rate whencompared to the coupled model of Lewis and Ghafouri(1997). A negligibly small oscillation of the production ratewas observed when the production rate was still high. Theoscillation soon damped out after a few time steps. It isimportant to mention here that most of the previousuncoupled models predict a sharp decline of the productionrate after nearly 2-6 years, the present coupled model predictsa much longer period with a smooth decline in total oilthroughput, which finally ends after 9.2 years.

The same trend may be observed for the gas/oil ratio,GOR, in Figure 10. The results obtained by the SPEparticipants suggests a sharp increase in GOR after a fewyears due to the decline in the reservoir pressure, the presentmodel, where coupling effects are also considered, this waspostponed until the ninth year, see Figure 10. Note that therate of GOR increases abruptly after nine years. We believethat the reason why the solution procedure halted at 9.3 yearswas mainly due to the extremely high GOR ratio calculatedusing Equation (80) during that period, thus causing thesolution to diverge. Again, oscillations are observed in theproducing GOR during the initial period due to thefluctuation in the oil production rate.

In order to investigate the influence of the reservoirpressure drawdown, the average pressure values in gridblock(5,1,1), are compared in Figure 11. The rate of pressuredecline is significantly slower when the coupling effect isincorporated (Fig. 11), whereas all the SPE uncoupledmodels predict a similar trend with a rapidly decreasingpressure. This confirms the considerable impact of rockdeformation on maintaining the initial reservoir pressure for alonger time, which can improve long term economical

. 2 0 2= .1 0 6=

p pcgI

cgI2 =

509

20 4 6 8 100

400

300

200

100

500

Time (years)

OIl

rate

(STB

/day)

Solutionhalted

Present model

Lewis and Ghafouri

20 4 6 8 100

2500

5000

7500

10 000

12 500

15 000

17 500

20 000

Time (years)

GO

R (S

CF/S

TB)

Solutionhalted

Present model

Lewis and Ghafouri

Figure 9

Oil rate versus time for the SPE example.

Figure 10

Gas/oil ratio versus time for the SPE example.


Figure 11

Pressure at Grid block (5,1,1) versus time for the SPEexample (1 psi = 6894.8 Pa).

productivity of a reservoir. This effect, which is ignored inconventional uncoupled models, is particularly of great significance when the oil-bearing strata are quite deformable.An interesting comparison between the coupled models inFigure 11 shows that the present model predicted slowerpressure dissipation in the reservoir. This observation is inaccordance with the results reported in the literature (Khaledet al., 1984; Bai et al., 1993) where the matrix block pressuretends to dissipate much slower due to the presence offissures. Another reason that leads to this result is the factthat in the present case, both the matrix and fissured blocksare assumed to be compressible (dictated by the value of K

m

and Kp), while in (Lewis and Ghafouri, 1997), these valueswere assumed to be zero due to their formulation. In additionto that, the present formulation which include the crosscoupling terms can be another factor that slows down thepressure dissipation. Isobaric contours in the cross-section ofthe reservoir are shown in Figures 12-14 for 1, 5 and 9 years,respectively. The pressure declines towards the producingwell with a relatively high-pressure gradient near the wellbore. The pressure gradient in the region adjacent to thewellbore is usually very high compared to the far field. Thisis firstly due to a very high flow rate per unit area in thisregion. Moreover, in many producing wells the value ofabsolute permeability of the formation close to the well boreis greater than the values further back in the reservoir. Thiscauses even more pressure drawdown around the well bore,the so-called skin effect. Due to the combined effect of thetwo foregoing factors, usually a very rapid drawdown occursaround the well. However, this effect is not predicted in thepresent simulation. The reason is that the size of elementsused in this problem is larger than that of the region affectedby the phenomenon, which is usually of the order of a fewmeters. This implies that the pressure value around the well

Figure 12

Pressure distribution (psi) contours in the cross-section of thereservoir at 1 year (1 psi = 6894.8 Pa).

Figure 13

Pressure distribution (psi) contours in the cross-section of thereservoir at 5 years (1 psi = 6894.8 Pa).

Figure 14

Pressure distribution (psi) contours in the cross-section of thereservoir at 9 years (1 psi = 6894.8 Pa).

obtained in this study may be significantly different from thereal values. More accurate results may be obtained byrefining the mesh in this region. However, the pressurevalues in the other regions are by no means dramaticallyaffected.

In all the solutions obtained by the participants, includingLewis and Ghafouri (1997), no solution for the reservoirdisplacements was shown. Figure 15 illustrates the verticalreservoir subsidence profile at the top surface for differenttime intervals. The reservoir subsides almost 1.0 ft overa period of 9.2 years, giving a subsidence rate ofapproximately 0.1 ft/year. This seemingly small value canhowever do considerable damage to the oil platform, well

4255 4250

4255 4250 4244

4239

4260

4260

4842 4838

4834

4823 4815

4830

48274842 4838

4815

5728 5723 5719

5711

5715

5728 5723 5719

20 4 6 8 100

1000

2000

3000

4000

5000

6000

7000

Time (years)

Pres

sure

(psi)

Present model

Lewis and Ghafouri

510


casing and pipeline, e.g. some old fields in the United States,i.e. Buena Vista (0.27 m), Fruitvale (0.04 m) and Santa FeSpring (0.66 m). Figure 16 illustrates the displacement vs.time curve at the gridblock (5,1,1). Note that thedisplacement is almost linearly proportional to the oil rateand the pressure drawdown.

CONCLUSIONS

The mathematical formulation of a three-phase three-dimensional fluid flow and rock deformation in fracturedreservoir was introduced in this paper. The present for-mulation, consisting of both the equilibrium and multiphasemass conservation equations, accounts for the significantinfluence of coupling between the fluid flow and soliddeformation. The final set of equations, which contain theadditional cross coupling terms as compared to similarexisting models, accounts for the displacements compatibilitycondition between the porous rock and the fissured block. Afield scale example was employed as an alpha case to testthe soundness of the current formulation and the robustnessof the numerical scheme. The results illustrate a significantlydifferent behavior for the case of a reservoir where theimpact of coupling was also considered.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. Fabrice Cuisiat ofNorwegian Geotechnical Institute and Dr. Nasser Khalili ofCivil Engineering Department, University of New SouthWales, Australia, for reading the original manuscript andproviding many constructive comments.

REFERENCES

Aifantis, E.C. (1985) Introducing a Multi-Porous Medium.Developments in Mech., 9, 46-69.Aziz, K. and Settari, A. (1979) Petroleum Reservoir Simulation,Applied Science Publisher, London.Bai, M., Elsworth, D. and Roegiers, J.C. (1993) Modelling ofNaturally Fractured Reservoir Using Deformation DependentFlow Mechanism. Int. J. Rock. Mech. Min. Sci. Geomech., 30,1185-1191.Bai, M., Meng, F., Roegiers, J.C. and Abousleiman, Y. (1998)Modelling Two-Phase Fluid Flow and Rock Deformation inFractured Porous Media. In Poromechanics, Thimus et al. (eds),Balkema, Rotterdam.Barenblatt, G.I., Zheltov, I.P and Kochina, I.N. (1960) BasicConcepts in the Theory of Seepage of Homogeneous Liquids inFissured Rocks. J. Appl. Math. Mech., USSR, 24, 1286-1303.Berryman, J.G. and Wang, H.F. (1995) The Elastic Coefficientsof Double-Porosity Models for Fluid Transport in Jointed Rock.UCRL-JC-119722.Biot, M.A. (1941) General Theory of Three-DimensionalConsolidation. J. App. Phys., 12, 155-163.Chen, H.Y. and Teufel, L.W. (1998) Coupling Fluid-Flow andGeomechanics in Dual-Porosity Modeling of Naturally FracturedReservoirs. Paper SPE 38884 presented at SPE Annual TechnicalConference and Exhibition, San Antonio, Texas.Cho, T.F., Plesha, M.E. and Haimson, B.C. (1991) ContinuumModelling of Jointed Porous Rock. Int. J. Num. Anal. Meth.Geom., 15, 333-353.Firoozabadi, A. and Thomas, L.K. (1990) Sixth SPE ComparativeSolution Project: Dual-Porosity Simulators. JPT, 42,710-715.Ghafouri, H.R. and Lewis, R.W. (1996) A Finite Element DoublePorosity Model for Heterogeneous Deformable Porous Media.Int. J. Num. Analy. Meth. Geom., 20, 831-844.Hughes, T.J.R. (2000) The Finite Element Method: Linear Staticand Dynamic Finite Element Analysis, Dover Reprinted Series,New York.

511

5000 1000 1500 2000-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Distance (ft)

Verti

cal d

ispla

cem

ent (f

t)

year 1

year 3

year 5

year 7

year 9.2

20 4 6 8 10-1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

-0.1

0

Time (years)

Dis

plac

emen

ts (ft

)

Figure 15

Subsidence versus reservoir distance at various times(1 ft = 0.3048 m).

Figure 16

Displacement and gridblock (5,1,1) versus time for the SPEexample (1 ft = 0.3048 m).


Khaled, M.Y., Beskos, D.E. and Aifantis, E.C. (1984) On theTheory of Consolidation with Double Porosity-III a FiniteElement Formulation. Int. J. Num. Anal. Meth. Geom., 8, 101-123.Khalili, N. and Valliappan, S. (1996) Unified Theory of Flow andDeformation in Double Porous Media. European Journal ofMechanics, A/Solid, 15, 321-336.Lewis, R.W. and Ghafouri, H.R. (1997) A Novel Finite ElementDouble Porosity Model for Multiphase Flow through DeformableFractured Porous Media. Int. J. Num. Anal. Meth. Geom., 21,789-816.Lewis, R.W. and Schrefler, B.A. (1998) The Finite ElementMethod in the Static and Dynamics Deformation andConsolidation of Porous Media, 2nd Ed., John Wiley & Son,England.Pao, W.K.S. (1998) Coupling Flow and Subsidence Model forPetroleum Reservoir. Fractured Reservoir Project. ConfidentialReport submitted to UWS, NGI, BP-AMOCO, TotalFinaElf andNorwegian Research Council.Pao, W.K.S. (2000) Aspects of Continuum Modeling andNumerical Simulations of Coupled Multiphase Deformable Non-Isothermal Porous Continua. PhD Thesis, University of WalesSwansea, UK.Pao, W.K.S., Lewis, R.W. and Masters, I. (2001) A FullyCoupled Hydro-Thermo-Poro-Mechanical Model for Black Oil

Reservoir Simulation. Int. J. Num. Analy. Meth. Geom., 25, 1229-1256.Pao, W.K.S, Masters, I. and Lewis, R.W. (1999) Integrated Flowand Subsidence Simulation for Hydrocarbon Reservoir. 7thAnnual Conf. of ACME99, 175-178.Schrefler, B.A. and Gawin, D. (1996) The Effective StressPrinciple: Incremental or Finite Form? Int. J. Numer. Analy.Meth. Geomch., 20, 785-815.Schrefler, B.A. and Simoni, L. (1991) Comparison betweenDifferent Finite Element Solutions for Immiscible Two-PhaseFlow in Deforming Porous Media. In Comp. Meth. & Adv.Geomech., G. Beer et al. (eds.), 2, 1215-1220.Thomas, L.K., Dixon, T.N. and Pierson, R.G. (1983) FracturedReservoir Simulation. SPERE, 42-54.Valliappan, S. and Khalili, N.S. (1990) Flow through FissuredPorous Media with Deformable Matrix. Int J. Num. Meth. Eng.,29, 1079-1094.Warren, J.E. and Root, P.J. (1963) The Behaviour of NaturallyFractured Reservoir. Trans. AIME, SPEJ, 228, 244-255.Wilson, R.K. and Aifantis, E.C. (1982) On the Theory ofConsolidation with Double Porosity. Int. J. Engng. Sci., 20, 1009-1035.

Final manuscript received in June 2002

512


APPENDIX

1 EQUILIBRIUM EQUATIONS FOR SOLID

The solid equilibrium equations is derived from the principleof virtual work, which, in a convenient FE form read:

(A.1)The integral sign is implied, and K is the

solid stiffness matrix. N and B are the shape function andspatial derivative of the shape function, respectively. In thispaper, equal order of interpolation function is used for both uand p.

2 WATER BALANCE EQUATIONSFOR MATRIX BLOCK

(A.2)

where:

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

3 WATER BALANCE EQUATIONSFOR FRACTURED BLOCK

(A.9)

where:

(A.10)

(A.11)

(A.12)

(A.13)

(A.14)

4 OIL BALANCE EQUATIONS FOR MATRIX BLOCK

(A.15)

where:

(A.16)

(A.17)

(A.18)

(A.19) o go

go

og

mBS S

BS

K K1 11

11

1

11

1 1=

+

'

''

o o o oo

g wo

oo

m

S BB

S S SB

SK K1 1 1 1 1

1

11 1

1

11

1 1= +

+

' ( ' ' )

''

o wo

oo

ow

mBS S

BS

K K1 11

11

1

11

1 1=

+

'

''

o roo o

kB1

1 1

1=

k

+

+ +

+

+

+ +

div gradk1 11

1 1 1 2 1 11

1 11

1 11

1 22

1 22

1 22 1 1

1

kB

p p ppt

pt

pt

pt

pt

pt

SB

ro

o oo o ow o o w

w

o oo

o gg

o ww

o oo

o gg o

o

( )

mmTt

= 0

w g ww

gm

SB

SK K2 2

2

22

2 2=

+

''

w ow

ww

wo

mBS S

BS

K K2 22

22

2

22

2 2=

+

'

''

w w w ww

ww

ww

m

S BB

S SB

SK K2 2 2 2 2

2

22

2

22

2 2=

+

' '

''

w w ww

w w ow

wo w g

w

wg

SB K

S SB K

S SB K

S2 1 22

1 2 12

21 2 1

2

21= = =

'' '' '', ,

w rww w

kB2

2 2

2=

k

+ +

+ +

+

+

+ +

+

div gradk2 22

2 2 1 2

2 11

2 11

2 11

2 22

2 22

2 22 2

kB

p p p Q

pt

pt

pt

pt

pt

pt

rw

w ww w w w w STC

w ww

w oo

w gg

w ww

w oo

w gg

( ) ( )

SSB t

w

w

T2

20m =

= +

+

1 2 1 21 2

1 2

w w ww

w w ow

wo w g

w

wg

SB K

S SB K

S SB K

S1 2 11

2 1 21

12 1 2

1

12= = =

'' '' '', ,

w g ww

gm

SB

SK K1 1

1

11

1 1=

+

''

w ow

ww

wo

mBS S

BS

K K1 11

11

1

11

1 1=

+

'

''

w w w ww

ww

ww

m

S BB

S SB

SK K1 1 1 1 1

1

11

1

11

1 1=

+

' '

''

w rww w

kB1

1 1

1=

k

+

+ +

+

+

+ +

div gradk

m

1 1

11 1 1 2 1 1

1

1 11

1 11

1 22

1 22

1 21 1

1

kB

p p ppt

pt

pt

pt

pt

pt

SB

rw

w ww w w w w w

w

w oo

w gg

w ww

w oo

w gg w

w

( )

TTt

= 0

(*) (*)=

d

K u B mN B mN

B mN B mN

B mN B mN

+

+

+

+ +

tS p

tS p

t

Sp

tS p

t

S pt

Sp

Tw

w To

o

Tg

g Tw

w

To

o Tg

g

'' ''

'' ''

'' ''

1 11

1 11

1 11

2 22

2 22

2 22

tt+ = f 0

513


(A.20)

5 OIL BALANCE EQUATIONSFOR FRACTURED BLOCK

(A.21)

where:

(A.22)

(A.23)

(A.24)

(A.25)

(A.26)

6 GAS BALANCE EQUATIONS FOR MATRIX BLOCK

(A.27)

where:

(A.28)

(A.29)

(A.30)

(A.31)

(A.32)

(A.33)

7 GAS BALANCE EQUATIONSFOR FRACTURED BLOCK

(A.34)

where:

(A.35)

(A.36)

(A.37)

(A.38)

(A.39)

(A.40)

g g g g

gg

so

og

g gm

S BB

S RB

S

c SK K

2 2 2 2 22

22

2 2

22

2 22 2

= +

+

' ' '

''

g o so o o

oo o

gg

so

og w g o

m

R S BB

S RB

S

RB

S S c SK K

2 2 2 2 2 22

22 2

2

22

2 2

22 2 2 2

2 2

= +

+

+

' ' '

( ' ' ) ''

g w soo

w g wm

RB

S c SK K2 2

2 2

22 2 2

2 2=

+

'

''

g w g w g o g o g g g gc K S c K S c K S2 1 2 1 2 1 2 1 2 1 2 1= = ='' '' ''

, ,

cSB

R SBg

g

g

so o

o2

2

2

2 2

2= +

grg

g g

kB2

2 2

2=

k

+ +

+ +

+

+

+ +

+

div gradk2 2

22 2 1 2

2 11

2 11

2 11

2 22

2 22

2 22 2

kB

p p p Q

pt

pt

pt

pt

pt

pt

rg

g gg g g g g STC

g ww

g oo

g gg

g ww

g oo

g gg

( ) ( )

SSB t

g

g

T2

20m =

g w g w g o g o g g g gc K S c K S c K S1 2 1 2 1 2 1 2 1 2 1 2= = ='' '' ''

, ,

g g g g

og

so

og

g gm

S BB

S RB

S

c SK K

1 1 1 1 11

11

1 1

11

1 11 1

= +

+

' ' '

''

g o

o

oso so o o

so

og w

gg g o

m

SB

R R S B RB

S S

BS c S

K K

1 11 1

11 1 1 1 1

1 1

11 1

1

11 1 1

1 1

= + +

+

' ' ( ' ' )

'

''

g w sow

w g wm

RB

S c SK K1 1

1 1

11 1 1

1 1=

+

'

''

cSB

R SBg

g

g

so o

o1

1

1

1 1

1= +

grg

g g

kB1

1 1

1=

k

+

+ +

+

+

+ +

div gradk1 1

11 1 1 2 1 1

1

1 11

1 11

1 22

1 22

1 22 1 1

1

kB

p p ppt

pt

pt

pt

pt

pt

SB

rg

g gg g g g g w

w

g oo

g gg

g ww

g oo

g gg w

w

( )

mmTt

= 0

o go

go

og

mBS S

BS

K K2 22

22

2

22

2 2=

+

'

''

o o o o

og w

o

oo

m

S BB

S S

SB

SK K

2 2 2 2 22

22 2

2

22

2 2

= +

+

' ( ' ' )

''

o wo

wo

ow

mBS S

BS

K K2 22

22

2

22

2 2=

+

'

''

o w oo

w o oo

oo o g

o

og

SB K

S SB K

S SB K

S2 1 22

1 2 12

21 2 1

2

21= = =

'' '' '', ,

o roo o

kB2

2 2

2=

k

+ +

+ +

+

+

+ +

+

div gradk2 22

2 2 1 2

2 11

2 11

2 11

2 22

2 22

2 22

kB

p p p Q

pt

pt

pt

pt

pt

pt

S

ro

o oo o o o o STC

o ww

o oo

o gg

o ww

o oo

o gg

( ) ( )

oo

o

TB t

2

20m =

o w oo

w o oo

oo o g

o

og

SB K

S SB K

S SB K

S1 2 11

2 1 21

12 1 2

1

12= = =

'' '' '', ,

complete treatment of fractured reservoir simulation

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