Afully Coupled Hydro-Thermo-poro-mechanical Model for Black Oil Reservoir Simulation

*Correspondence to: R. W. Lewis, Mechanical Engineering Department, University of Wales Swansea,Swansea SA2-8PP, U.K.

E-mail: R.W. [email protected]

Contract/grant sponsor: Fractures Reservoir Project, University of Wales (under the control of Norwegian GeotechnicalInstitute) BP-AMOCO, ELF and Norwegian Research Council

Received 11 November 1999Copyright 2001 John Wiley & Sons, Ltd. Revised 2 March 2001

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2001; 25:1229}1256 (DOI: 10.1002/nag.174)

A fully coupled hydro-thermo-poro-mechanical model for blackoil reservoir simulation

W. K. S. Pao, R. W. Lewis* and I. Masters

Mechanical Engineering Department, University of Wales Swansea, Swansea SA2-8PP, U.K.

SUMMARY

A fully coupled formulation of a hydro-thermo-poro-mechanical model for a three-phase black oil reservoirmodel is presented. The model is based upon the approach proposed by one of the authors which fullycouples geomechanical e!ects to multiphase #ow. Their work is extended here to include non-isothermale!ects. The gas phase contribution to the energy equation has been neglected based on a set of assumptions.The coupled formulation given herein di!ers in several ways when compared to the earlier work and anattempt is made to link the #ow based formulation and mixture theory. The Finite Element Method isemployed for the numerical treatment and essential algorithmic implementation is discussed. Numericalexamples are presented to provide further understanding of the current methodology. Copyright 2001John Wiley & Sons, Ltd.

KEY WORDS: coupled model; petroleum reservoir; hydro-thermo-poro-mechanical; "nite element

1. INTRODUCTION

As recognised by Gutierrez and Lewis [1], the role of geomechanics in petroleum reservoirengineering is becoming increasingly important as deeper formations are detected and explored.The stress, #uid pressure and temperature conditions encountered at large depth give riseto a range of situations where conventional reservoir modelling fails to provide an accurateanalysis [2].Since the pioneering work to Biot [3] on isothermal consolidation theory, many advances has

been made by various researchers with applications in civil engineering and hydrology. Bydrawing the analogy between thermoelasticity and poroelasticity, the Biot equations wereextended by many workers to include thermal e!ect, e.g. References [4}7]. A direct application ofgeomechanics to hydrocarbon reservoirs is relatively rare but can be tracked in the earlier work of

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Geertsma [8,9], Gassman [10] and later, by other (see References [11}24]). A more exhaustiveliterature review on the subject can be found in Lewis and Schre#er [25].Based on the above literature study, we see that, although geomechanics aspects have been

applied to petroleum reservoirs since 1957, these models only make use of the pore com-pressibility as a sole coupling factor. The heuristicity of this approach cannot be justi"edanalytically or rationally. On the other hand, for coupled formulations, a discrepancy of opinionstill exists among reservoir engineers, who prefer #ow based formulation rather than those basedon the theory of mixtures, as discussed by Chen and Teufel [26]. We therefore feel that there isa need to settle this problematic issue and provide a consistent geomechanics integration intoso-called &traditional' petroleum reservoir simulators. Also, we are of the opinion that extendingthis rationale to take into account thermoelastic e!ects is highly desirable.

2. PRELIMINARY SUPPOSITION

In a multi-components model, we assume that water, an evolving original heavy crude oil in place(OOIP) and an organic gaseous phase co-exist in the porous arc. The gaseous phase consists ofboth the original organic gas in place ( OGIP) and the fragmentation of long hydrocarbon chainsinto shorter components below the bubble point pressure. As far as the gas phase is concerned, nodistinction is made in between these two gas components at the present stage. We further assumethat there are two distinct zones which exist in the porous continua, namely a dominant water}oilzone and a dominant oil}gas zone. We suppose that in the water}oil zone, the system iswater-wet, while in the oil}gas zone, the system is oil-wet.According to standard petrology for immiscible #uids, each phase will occupy a partial pore

volume S"d

/

such that

S#S

#S

"1 (1)

in which Sis termed the saturation of phase i. The subscript i"w, o and g denotes water, oil and

gaseous phase, respectively.

2.1. Capillary pressure relationship

In order to specify the interacting motion of each phase (not including skeleton constituents), onerequires equations which link the evolution of the partial pressures of each phase to theirsaturation values. According to Hassanizadeh and Gray [27], the most practical method incharacterizing this interacting motion is to use an empirical correlation relating the capillarypressure, p

and the saturation, S. In this sense, the correlation may be thought of as being

representative equations which account for all e!ects and processes that in#uence the #uiddistribution. This argument is strongly supported via the Buckingham analysis carried out bySlattery [28], and is favourably viewed amongst practicing reservoir engineers [29]. It should benoted that at the macroscopic level, the capillary pressure is the di!erence of averaged pressuresand not the average of their di!erence.Following this, we can write

S"F (P

,) (2)

1230 W. K. S. PAO, R. W. LEWIS AND I. MASTERS

Copyright 2001 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 2001; 25:1229}1256

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Partially di!erentiating Equation (2) one obtains, for an oil}water system

S

t" S

p

p

t!p

t #S

t

(3)

Similarly, for an oil}gas system,

S

t" S

p

p

t!p

t #S

t

(4)

From Equation (1)

dS

dt"!dS

dt!dS

dt(5)

Using Equations (3) and (4), and substituting into Equation (5) results in

S

t"! S

p

p

t!p

t !S

p

p

t!p

t !S

#

S

t

(6)

Simplifying, and letting S"S

/p

, S

"S

/p

, S

"S

/ and S

"S

/ one

obtains

S

t"S

p

t!p

t #St

(7)

S

t"S

p

t!p

t #St

(8)

S

t"!S

p

t!p

t !S p

t!p

t !(S#S)t

(9)

2.2. Average pore pressure relationship

The e!ective pore pressure, pN , has been de"ned in di!erent ways by many authors based on a setof arguments. Tortike and Farouq-Ali [30], for example, argued that in a tar sand, the grains areentirely encapsulated by water and therefore, the water phase pressure is the only in#uencingmechanism on the skeleton. This argument has been adopted by many researchers, particularlythose in the area of hydrology. Here, we attempt to show a derivation which will be consistentwith those used by petroleum reservoir engineers.Let p

be de"ned as the molar pressure of the #uid in an in"nitesimal pore volume

. Then, the

volume averaged pore pressure pN , can be written as

pN" 1

pnd (10)

in which nis the total mole number and c is the number of components. The total mode number

is related to the mole fraction via

x"n

n

(11)

A FULLY COUPLED HYDRO-THERMO-PORO-MECHANICALMODEL 1231


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where nis the mole number of species i. Substituting Equation (11) into (10), one obtains

pN" 1

p

nx

d (12)

Equation (12) can be written as

pN" 1

p

nx

d (13)

But

d"

(14)

Substituting Equation (14) into (13) yields

pN" 1

p

nx

(15)

However S"

/

. Realising that n

/x

"1, and replacing i"w, o, g, one obtains

pN"Sp#S

p#S

p

(16)

This derivation is consistent with the thermodynamic pressure usually employed in physicalchemistry. On totally di!erentiating Equation (16), we have

dpN"Sdp

#p

dS

#S

dp

#p

dS

#S

dp

#p

S

(17)

Making use of the relationships from Equations (7)}(9) and simplifying by de"ning

S"S

!p

S!p

S

(18)

S"S

#p

S

(19)

S"S

#p

S

(20)

S"p

S!p

S

(21)

one obtains

pN"Sp

#S

p

#S

p

#S

(22)

Equation (22) has been obtained by considering small perturbation hypothesis in which theconvective component is assumed to be zero.

3. DEVELOPMENT OF THE GOVERNING EQUATIONS

3.1. Equilibrium equation

The starting point of the equilibrium equation is the skeleton momemtum balance, which reads

div !f"0 (23)



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in which is the total stress and f represents the external forces. The total stress is related to thee!ective stress via the following state law:

"!IpN (24)

where the #uid is assumed to be mechanically perfect as indicated by a second order identitytensor, I, in front of the volume averaged pore pressure, pN . From Equations (23) and (24), twooptions exist. One can either formulate the equilibrium equation in terms of the total or e!ectivestress. Here, we shall choose the e!ective stress formulation. The constitutive equation relatingthe e!ective stress to the strain is given by Lewis and Schre#er [25] as

d"C : (d!d!d

) (25)

when C is the drained consistent tangent of the skeleton and is the total strain of the skeleton.The quantity

represents the volumetric strains caused by uniform compression of the skeleton

matrix due to a con"ning pressure, which can be given as

d"! I

3K

dpN (26)

whereKis the bulk modulus of the matrix, which may well be di!erent from the bulk modulus of

the skeleton. The quantity is the volumetric thermal strain of the skeleton due to temperature

changes, i.e.,

d"I

3d (27)

Here, is the thermal coe$cient of expansion of the skeleton and d is de"ned w.r.t. a reference

temperature at a reference con"guration. Making use of the state equations given by (22),(24)}(27), Equation (23) reads

div C : !S ) grad p

!S

) grad p

!S ) grad p

!

3C : I ) grad !f"0 (28)

where is the usual Biot coe$cient. If C is isotropic, we have

"1! KK

(29)

where K is the skeleton bulk modulus. However, if C is anisotropic, then

"I!C : I3K

(30)

3.2. Continuity equations

There are many ways to derive the volumetric balance equations for the #uids. One of those hasbeen previously described by Pao et al. [31]. Here, we cast the volumetric equations ina terminology which is familiar to petroleum reservoir engineers.



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Consider a region of rock with bulk volume

, whose pore space contains an amount of #uid

massM. If the density of this particular #uid is denoted by

, the volume occupied by this #uid is

M/

. If the pore volume is partially saturated with the #uid phase i, the pore volume occupied by

this #uid is therefore,

d (S

)"d

M

(31)

where Equation (31) is expressed at reservoir conditions. In order to express Equation (31) intothe stock tank conditions, stc, w.r.t. the hydrocarbon component, we write

d (S

)"d

MB

(32)

where Bis the formation volume factor and

is the density of phase i at stc. Totally

di!erentiating Equation (32), and multiplying both sides by gives

Sd

#

dS

"B

dM

#M

dB

(33)

Dividing Equation (33) by

, we obtain

S

d

"BdM

!

dS#M

dB. (34)

where

"

(35)

and

M

"S

(36)

where is the porosity. It should be noted that in Equation (36), the #uid mass is expressed at stc,and not at the reservoir condition. Equation (34) describes a very interesting relationship, wherethe partial pore volumetric change per unit of bulk volume can be broken into three parts on theRHS. The "rst term is due to the additional #uid mass dM

that moves into the pore volume per

unit of bulk volume. The second term is due to the extraction of the #uid mass from the porevolume due to the change in saturation dS

, a phenomena known as drainage. Finally, the third

term is due to the compression of the #uid phase i that is already in that region.Following Biot's methodology, we introduce

d

"dM

(37)

Substituting expressions (35), (36) and (37) into Equation (34) and rearranging will yield thefollowing:

d

"

B

S

d

#B

dS#

S

d

1

B (38)



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Now, the quantity dShas been de"ned previously and the derivative of B

is easily determined.

However, we need an expression for d/

. Let us write

d

"d!

d

(39)

According to Charlez [2]

d

"1

1

K! 1

K (dN#dpN )!

dpNK

(40)

in which K is the bulk modulus of the skeleton whose expression can be obtained from theconsistent tangent matrix C. The value of K

has to be determined from the jacketted test of the

core samples. In Equation (40), is the so-called con"ning total stress, whose expression can beobtained via

"13

tr (41)

Multiplying (40) by /

, one gets

d

"1

K! 1

K (dN#dpN )!

dpNK

(42)

We now make use of the mean total stress equation, i.e.

d"d !1!K

K dpN (43)

Substituting Equation (43) into (42), one "nally "nds the following relationship:

d"1

K! 1

K d #

1

K

! KK

dpN (44)

and

d

"dpNK

(45)

It should be noted that Equation (44) is signi"cantly di!erent from the commonly acceptedrelationship used in an uncoupled petroleum reservoir simulation, (see e.g. References [16, 30]),which is given by

d"cdpN (46)

where cis commonly known as the pore compressibility. The validity of equation (46) is doubtful

since it is in fact a linear equation relating to pN . Experimental evidence, however, suggests thatno such relationship exists, see Figure 1. Equation (44) can also be cast, using the terminology ofLewis and Schre#er [25], as

d"d(tr )! 13K

I : C : d! 19K

I :C : I dpN# 1K

dpN (47)



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Figure 1. Evolution of con"ning yield pressure versus porosity, adapted from Reference [32].

Note that if K"R, then the evolution of is simply the volumetric strain of the skeleton.

Therefore, Equation (39) will not, after taking into account the volumetric thermal straincomponent, read as

d

"dtr #!K

dpN!(!)

3d (48)

Equation (48) can now be substituted into (38) to give

d

"

B

S dtr #

!K

dpN!(!)

3d#

B

dS#

S

d

1

B (49)

To obtain the volumetric balance equation, we use

div M#mR

"q

(50)

where Mis the mass #ux vector of phase i, q

is the volumetric source/sink and the overdot

denotes partial time derivative. We now de"ne dmas

dm"dM

(51)



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Utilizing the fact that for a small perturbation, d(*)/dt+ (*)/t, and substituting Equation (49)into (51), one obtains

div M#

B

S tr #

!K

pNQ!(!)

3Q

#B

SQ#

S

1B"q (52)

By assuming Darcy's law, the mass #ux vector takes the form

M"Kk

B

grad p

(53)

in which K is the intrinsic permeability and kis the relative permeability. SinceM

, pNQ , SQ

and etc.

are all known, the derivation of the #uid phase equation is now straight forward. For the waterphase, we obtain

div Kk

B

grad p#Q#

SS

B

!K

#SB!

B

S

p

t

#SS

B

!K

#

B

S

p

t#

S

B!K

p

t

#SS

B

!K

!

S

B

(!)#S

B

t#S

B

t"0 (54)

in which

B"

p1

B (55)

and

B"

1

B (56)

The source term is de"ned as

Q" q

(57)

Similarly, the oil equation can be derived as

div Kk

B

grad p#Q#

SS

B!K

#

B

S

p

t

#SS

B!K

!

B

(S!S

)#S

B

p

t



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#S

B!K

!

S

Bp

t

#S

B!K

!(!)#SB

t#S

B

t"0 (58)

in which

B"

p1

B (59)

and

B"

1

B (60)

Similarly, the source term is de"ned as

Q" q

(61)

The gas equation can be given as

div Kk

B

grad p#RKk

B

grad p#Q

#S !K

#

RS

B

p

t

#S !K

!

S

B

!RB

(S!S

)#S

B

R#R

SB

p

t

#S !K

!

RS

B

#B

S#S

B

p

t

# !K

S!(!)

#SB#S

B

R#R

SB

t

# !K

t"0 (62)

in which

"S

B

#RSB (63)

B"

p1

B , B"

1

B (64)

R"R

p

, R"R

(65)



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The source term for the gas reads

Q" q

#RQ

(66)

In Equations (62)}(65), Ris the volume of the dissolved gas in the oil phase per unit volume of oil

at stc.Before we proceed, it is important to observe that the formulation provided by Equations (54),

(58) and (62) is valid irrespective of whether the "nal consideration requires a coupled oruncoupled solution. For an uncoupled model, we need to eliminate the partial time derivativeterm of the volumetric strain, i.e. /t. Under the assumption that the rate of change of totalstress is zero, i.e. "constant, we can write, making use of Equation (24)

div v" K"IpNQ

K(67)

where is the skeleton velocity and an overdot denotes time derivative. Equation (67) can be usedto eliminate the time derivative of the volumetric strain term in Equations (54), (58) and (62)resulting in a decoupling of the #ow equations from the equilibrium equations.

3.3. Energy balance equation

By enforcing the local postulate that the #uid reaches thermal equilibrium instantaneously withthe reservoir rock, then the only relevant mechanisms of energy transport are conduction andconvection. Neglecting kinetic energy, viscous and intrinsic dissipation, the energy balance on anin"nitesimal reservoir element can be written as

!div (grad )#cv

) grad#Q

!Q

c"

t( c) (68)

in which v"K (k

/

)grad p

, is the velocity of the #uid phase i. In Equation (68), Q

is the

external volumetric heat input into the system, e.g. via wellbore heating, etc. The quantityQc is

the energy density of the #uid phase i entering/leaving the system via a production or injectionwell. On the RHS of Equation (68), the quantity cdtd represents the increase, or decrease, ofthe internal energy of the system due to a temperature evolution "!

. Note that the

intensive quantity has been de"ned with respect to a reference temperature, and in Equation(68),

is taken to be zero. The bulk quantity c in Equation (68) represents an averaged heat

storage capacity of the reservoir system. For a microscale problem, unlike a petroleum reservoir,e.g. brick drying, the determination of the bulk value of c is possible via experimentation. Fora reservoir, this is impossible as our only window into the reservoir formation is via the outcropsand cores, drilled via a tiny hole of the wellbore. In addition, the sample size obtained from a "eldinvestigation may not be representative due to the chaotic distribution of reservoir material andthe heterogeneity of the formation. In this case, the bulk heat storage capacity has to be expressedin terms of the heat capacity of the pure substances. The time derivative of the heat capacity cannow be expressed as

t( c )"

t[(1!)

c#S

c

#S

c

#S

c

] (69)



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In Equation (69), instantaneous local thermal equilibrium has been enforced. If the time deriva-tive on the RHS of Equation (69) is taken into account, the resulting equation system will beprohibitively large. A simpli"cation can be made w.r.t. Equation (69), by "rst observing that;

in which the subscript l denotes the liquid phase. For most organic gases, c

+c

up to

5003F, from 15 to 3000 psia, see Reference [33]. Hence, the heat capacity of the gas phase can beneglected (the ratio of the heat contributed by gas to the liquid phase is approximately 1 : 1000)and Equation (69) now reads

t(N c)"

t[(1!)

c#S

c

#S

c

] (70)

A consequence of this assumption is that the application of Equation (70) is limited to a speci"crange of material and conditions. Provided that the saturation of the gas phase is small relative tothe liquid phase (as in most of the cases in black/green oil reserves), this assumption is justi"able.Following this, the energy equation can be derived as, after taking the time derivative of Equation(70) and neglecting /t, the mechanical heating term

!div ( ) grad )#(cv#

cv) grad

# (1!)c

K

S#S

c

K

!cS#

cS

p

t

# (1!)c

K

S#S

c

K

#cS!

cS

p

t

#!Sc!Sc#cS

!c

S

#(1!)c#S

c#S

c!(1!)c

# (1!)c

K

S

t#Q

!Q

c!Q

c"0 (71)

We may safely ignore the work contribution due to pore evolution and this can be justi"ed bystudying Equation (47), in which if K

"R, the magnitude of is of the same order of

magnitude as tr .Equations (28), (54), (58), (62) and (71) represent a set of highly non-linear partial di!erential

equations for three-phase #ow coupled with the consolidation behaviour occurring in a deform-able petroleum reservoir. The major non-linearities, i.e. the phase saturation S

, relative permeab-

ility k, formation volume factor B

, viscosities

and porosity , are strongly dependent on the

primary unknowns and therefore should be updated at appropriate time intervals. In order tocomplete the descriptions of the above governing equations, we need the initial and boundaryconditions. The initial conditions is given by

u"u; p"P

; " ; x at t"0 (72)



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on the domain . The boundary conditions can be prescribed as, t;

u"u x3, n;" x3 (73)

p"pN x3p, n; Kk

B

grad p"Q

(x, t) x3

(74)

"M x3, (

cv!grad) ) n;"q

(x, t) x3

(75)

with

", ", " (76)

in which L.

4. SPATIAL AND TEMPORAL DISCRETISATION

The "nite element discretisation of the balance equations may now be expressed in terms of thenodal displacements, u; , nodal #uid pressure, i.e. P

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Afully Coupled Hydro-Thermo-poro-mechanical Model for Black Oil Reservoir Simulation