SPE-72095-PA

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Summary

This paper presents a discussion of the issues related to the interac-

tion between rock deformation and multiphase fluid flow behaviorin hydrocarbon reservoirs. Pore-pressure and temperature changesresulting from production and fluid injection can induce rock defor-mations, which should be accounted for in reservoir modeling.Deformation can affect the permeability and pore compressibility of the reservoir rock. In turn, the pore pressures will vary owing tochanges in the pore volume. This paper presents the formulation of Biot’s equations for multiphase fluid flow in deformable porousmedia. Based on this formulation, it is argued that rock deformationand multiphase fluid flow are fully coupled processes that should beaccounted for simultaneously, and can only be decoupled for prede-fined simple loading conditions. In general, it is shown that reser-voir simulators neglect or simplify important geomechanicalaspects that can impact reservoir productivity. This is attributed tothe fact that the only rock mechanical parameter involved in reser-

voir simulations is pore compressibility. This parameter is shown tobe insufficient in representing aspects of rock behavior such asstress-path dependency and dilatancy, which require a full tensorialconstitutive relation. Furthermore, the pore-pressure changescaused by the applied loads from nonpay rock and the influence of nonpay rock on reservoir deformability cannot be accounted forsimply by adjusting the pore compressibility.

Introduction

In the last two decades, there has been a strong emphasis on theimportance of geomechanics in several petroleum engineeringactivities such as drilling, borehole stability, hydraulic fracturing,and production-induced compaction and subsidence. In theseareas, in-situ stresses and rock deformations, in addition to fluid-flow behavior, are key parameters. The interaction between geo-mechanics and multiphase fluid flow is widely recognized in

hydraulic fracturing. For instance, Advani et al.1 and Settari et al.2

have shown the importance of fracture-induced in-situ stresschanges and deformations on reservoir behavior and how hydraulicfracturing can be coupled with reservoir simulators. However, inother applications, geomechanics, if not entirely neglected, is stilltreated as a separate aspect from multiphase fluid flow. By treatingthe two fields as separate issues, the tendency for each field is tosimplify and make approximate assumptions for the other field. Thisis expected because of the complexity of treating geomechanics andmultiphase fluid flow as coupled processes.

Recently, there has been a growing interest in the importance of geomechanics in reservoir simulation, particularly in the case of heavy oil or bituminous sand reservoirs,3,4 water injection in frac-tured and heterogeneous reservoirs,57 and compacting and sub-siding fields.8,9 Several approaches have been proposed to imple-

ment geomechanical effects into reservoir simulation. Theapproaches differ on the elements of geomechanics that should beimplemented and the degree to which these elements are coupledto multiphase fluid flow.

The objective of this paper is to illustrate the importance of geo-mechanics on multiphase flow behavior in hydrocarbon reservoirs.An extension of Biot’s theory10 for 3D consolidation in porousmedia to multiphase fluids, which was proposed by Lewis and

Sukirman,11 will be reviewed and used to clarify the issuesinvolved in coupling fluid flow and rock deformation in reservoir

simulators. It will be shown that for reservoirs with relativelydeformable rock, fluid flow and reservoir deformation are fullycoupled processes, and that such coupled behaviors cannot be rep-resented sufficiently by a pore-compressibility parameter alone, as isdone in reservoir simulators. The finite-element implementation of the fully coupled equations and the application of the finite-elementmodels to an example problem are presented to illustrate theimportance of coupling rock deformation and fluid flow.

Multiphase Fluid Flow in DeformablePorous Media

Fig. 1 illustrates the main parameters involved in the flow of multi-phase fluids in deformable porous media and how these parametersideally interact. The main quantities required to predict fluid move-ment and productivity in a reservoir are the fluid pressures (and

temperatures, in case of nonisothermal problems). Fluid pressuresalso partly carry the loads, which are transmitted by the surroundingrock (particularly the overburden) to the reservoir. A change in fluidpressure will change the effective stresses following Terzaghi’s12

effective stress principle and cause the reservoir rock to deform(additional deformations are induced by temperature changes innonisothermal problems). These interactions suggest two types of fluid flow and rock deformation coupling:

• Stress-permeability coupling, where the changes in pore struc-ture caused by rock deformation affect permeability and fluid flow.

• Deformation-fluid pressure coupling, where the rock defor-mation affects fluid pressure and vice versa.

The nature of these couplings, specifically the second type, arediscussed in detail in the next section.

Stress-Permeability Coupling

This type of coupling is one where stress changes modify the porestructure and the permeability of the reservoir rock. A commonapproach is to assume that the permeability is dependent onporosity, as in the Carman-Kozeny relation commonly used in basinsimulators. Because porosity is dependent on effective stresses, per-meability is effectively stress-dependent. Another important effect,in addition to the change in the magnitude of permeability, is on thechange in directionality of fluid flow. This is the case for rocks withanisotropic permeabilities, where the full permeability tensor can bemodified by the deformation of the rock.

Examples of stress-dependent reservoir modeling are given byKoutsabelouliset al.6 and Gutierrez and Makurat.7 In both examples,the main aim of the coupling is to account for the effects of in-situstress changes on fractured reservoir rock permeability, which inturn affect the fluid pressures and the stress field. The motivation for

the model comes from the field studies done by Heffer et al.5 show-ing that there is a strong correlation between the orientation of theprincipal in-situ stresses with the directionality of flow in fracturedreservoirs during water injection. There is also growing evidencethat the earth’s crust is generally in a metastable state, where mostfaults and fractures are critically stressed and are on the verge of fur-ther slip.13 This situation will broaden the range of cases with strongpotential for coupling of fluid flow and deformation.

Deformation-Fluid Pressure Coupling:Biot’s Theory for Multiphase Flow inDeformable Porous Media

The coupled deformation and fluid-flow problem was first ana-lyzed by Terzaghi12 in 1925 as a consolidation problem. Since then,

164 June 2001 SPE Reservoir Evaluation & Engineering

Petroleum Reservoir Simulation CouplingFluid Flow and Geomechanics

M. Gutierrez, SPE, Virginia Tech, and R.W. Lewis and I. Masters, SPE, U. of Wales, Swansea

Copyright © 2001 Society of Petroleum Engineers

This paper (SPE 72095) was revised for publication from paper SPE 50636, first presentedat the 1998 SPE European Petroleum Conference, The Hague, 20–22 October. Originalmanuscript received for review 15 February 1999. Revised manuscript received 11 April2001. Paper peer approved 16 April 2001.

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June 2001 SPE Reservoir Evaluation & Engineering 165

Terzaghi’s 1D consolidation theory has been used widely in settle-ment problems in saturated soils. Biot10 extended the theory into amore general 3D case, based on a linear stress-strain relation and asingle-phase fluid flow. Here, we present an extension of Biot’sequations for two-phase immiscible and isothermal flow.Equations for three-phase flow can be found in Lewis andSukirman.11 In the following, tensorial notation is used and sum-mation is implied for repeated indices.

For two-phase fluid flow, the generalized Darcy’s law is given as

In addition to Darcy’s law, Biot’s theory includes(a) Terzaghi’s effective stress principle,

(b) the stress-strain constitutive relation, including the com-pressibility of the solid grains,

(c) the strain-displacement compatibility relation,

and (d) the static-equilibrium equation,

Eqs. 2 through 5 constitute the geomechanical part of Biot’sequation. The equations relate the applied internal and externalloads F i from the static equilibrium condition (Eq. 5) and the porepressure p from the effective stress equation (Eq. 2) to the defor-mation of the rock (Eqs. 3 and 4). The final equation is for massbalance, which is written as

The fluid accumulation term on the right side of Eq. 6 consistsof the following contributions:

(a) the rate of change of fluid volume and saturation for eachphase ,

(b) the rate of rock volumetric change,

(c) the rate of change of solid particle volume owing to porepressure change,

and (d) the rate of solid-particle-volume change caused by the

change in mean effective stress,

Contributions (c) and (d) account for poroelastic effects byincluding the grain compressive modulus K s of the reservoir rock.For reservoir rocks, poroelastic effects can be significant when thematrix bulk modulus K has the same order of magnitude as K s.

The initial formulation of Biot’s theory emphasizes mechanicalissues over fluid-flow issues. Because of this, the theory is lesscompatible with conventional fluid-flow models in terms of theparameters involved. A reformulation of the theory along the lineof conventional fluid-flow modeling can be found in Chen et al.14

Dual-porosity coupled models also have been proposed for frac-

tured reservoirs by Chen and Teufel15 and Ghafouri.16 It shouldalso be noted that the theory is not restricted to elastic response of the rocks and has been extended to thermoporoelastoplasticity.17,18

Reservoir Simulation

The main purpose of reservoir simulation is to model multiphasefluid flow and heat transfer in porous media. The more advancedreservoir simulators can handle multicomponent three-phase fluidswith complicated pressure/volume/temperature (PVT) relationsand relative permeabilities.

The equations governing the behavior of two immiscible fluidsflowing in a porous medium can be obtained by combiningDarcy’s law (Eq. 1) with the mass-balance equations for each flow-ing phase. In contrast to Eq. 6, the mass-balance law in reservoirsimulators is written simply as

As in Biot’s equations, these two equations are supplementedby the equations of state for various fluid properties.

. . . . . . . . . . . . . . . . . . . . . . . . . (14)( , )c o w c o w P p p p S S = − = ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)( , )r r o wk k S S π π= ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)( ) pπ π πµ µ= ,

. . . . . . . . . . . . . . . . . . (11)1 1

i

i

v S q x B t B

π π π

π π

φ ∂ ∂

− = + ∂ ∂

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)3

ij ij

s K t

δ σ ′∂−

∂

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)(1 )

s

p

K t

φ− ∂

∂;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)ijvij

t t

εεδ

∂∂=

∂ ∂

;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)1

S t B

π

π

φ ∂

∂ ;

. . . . . . . . . . . . . . . . . . . . (6)

1 1i

i

v S x B t B

π π

π π

φ ∂ ∂

− = ∂ ∂

3

oi j i jkl kl

kl o

s

DS

B K t π

π

δ εδ

∂ + −

∂

( )2

1

3

oi j i jkl kl

o

s s

DS pq

B K t K

ππ

π

δ δ φ

− ∂ + − + ∂

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)0ij

i

j

F x

σ ∂+ =

∂

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)1

d2

jiij

i j

uu

x xε

∂∂= +

∂ ∂

;

. . . . . . . . . . . . . . . . . . . . . . . . . . (3)d

d d3

ij ijkl kl kl

s

p D

K σ ε δ ′ = +

;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)ij ij ij

pσ σ δ ′ = −

;

. . . . . . . . . . . . . . . . . . . . . . (1)ij r

i

j

k k v p gh

x

π

π π π

π

∂ ρ

µ ∂

= − +

.

Fig. 1—Schematic of the interaction between rock deformation,fluid flow, and temperature in a deformable reservoir.

Fluid Pressure(Temperature)

In-Situ Stresses

Rock Deformation Permeability

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Expanding the time derivative on the right side of Eq. 11 gives

The first term on the right side of Eq. 16 is the change in thevolume factor B with pressure, giving the fluid compressibility of phase as

If the average fluid pressure p causing pore-volume change inthe effective stress law (Eq. 2) is calculated from11

then the second term on the right side of Eq. 16 may be rewritten as

where c ppore compressibility defined as

Substituting Eqs. 1, 17, and 19 into Eq. 11 yields the two-phasehydraulic diffusivity equation,

Reduction of the Coupled Equations to aHydraulic Diffusivity Equation

The hydraulic diffusivity equation from Biot’s theory is obtainedby introducing Darcy’s law (Eq. 1) into the mass-balance equation(Eq. 6). For the sake of simplicity, single-phase flow under isother-mal conditions will be considered.

where K f 1/ c f is the bulk modulus of the fluid. This equationshould be compared to the single-phase version of the fluid diffu-sivity equation (Eq. 21) used in reservoir simulation,

where ct c f c p is the total compressibility. Again, for the sake of simplicity, elastic stress-strain behavior will be considered, in

which case the constitutive tensor D

ijkl equals

where K and Gthe shear and bulk moduli, respectively, and arerelated to the Young’s modulus E and Poisson’s ratio as

Substituting Eq. 24 into Eq. 3 and the resulting equation into Eq. 2yields the poroelastic stress-strain relation, which relates the total

stress increment d

ij to the strain increment d

ij and pore-pressureincrement d p.

where Biot’s poro-elastic constant defined as

Substituting Eq. 24 in Eq. 22 yields the poroelastic hydraulicdiffusivity equation,18,19

where M Bthe Biot modulus defined as

Note that the fluid diffusivity equation (Eq. 28) is coupled tothe poroelastic stress-strain relation (Eq. 26) by the volumetricstrain increment d . Under specific conditions, these two equa-tions can be decoupled and the problem reduced to that commonlyused in reservoir engineering. To do this, two assumptions mustbe made to relate the volumetric strain d in Eq. 26 to the pore-pressure change d p:

• The changes in the total stresses.• The loading condition (called stress path in geomechanics).It should be noted that these assumptions are defined locally and

therefore applied to every point, as opposed to boundary conditions,

which are applied at the boundary of the domain of interest.Consequently, the strain-displacement compatibility relation (Eq. 4)and the equilibrium equation (Eq. 5) do not have to be invoked inmaking these assumptions.

Hydrostatic Compaction. To decouple Eqs. 26 and 28 locally, oneassumption that can be made is that the reservoir rock is subjectedto a hydrostatic loading, with equal horizontal and vertical defor-mations under constant total stresses. Applying the conditions

d x d y d zd and d

ij0 in Eq. 26 yields

Substituting this into Eq. 28 gives the decoupled flow equation

. . . . . . . . . . . . . . . . . . . . . . . . . . . (30)d d d 0 z v K pσ ε α= − = .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)1

B s f M K K

α φ φ−= + .

. . . . . . . . . . . . (28)1ij v

i j B

k p p gh q

x x t M t

ερ α

µ

∂ ∂ ∂ ∂

+ = + + ∂ ∂ ∂ ∂

,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)1 s

K

K α = − .

. . . . . . . . . . . . . (26)2d d 2 d d3

ij v ij ij ij K G G pσ ε δ ε α δ

= − + −

,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25b)( )2 1

E G

ν =

+.and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25a)( )3 1 2

E K

ν =

−

. . . . . . . . . . . . . (24)( )2

3ijkl ij kl ik jl il jk D K G Gδ δ δ δ δ δ

= − + +

,

. . . . . . . . . . . . . . . . . . . . (23)ij

t

j

k p p gh c q

x t ρ φ

µ

∂ ∂+ = +

∂ ∂

,

. . . . . . . . . . . . . . . . . . . . . . (22)

( )2

1

3

i j i jkl k l

f s s

D pq

K K t K

δ δ φ φ

− ∂ + + − + ∂

,

3

ij ij ijkl kl kl

i j s

k D p gh x x K t

δ ερ δ µ

∂ ∂ ∂ + = − ∂ ∂ ∂

. . . . . . . . . . . . . . (21)1

f p

S S pc S c q

B S p t

π π ππ π π

π π π

φ ∂ ∂= + + +

∂ ∂ .

ij r

j

k k p gh

x

π

π π

π

ρ µ

∂+

∂

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)1

pc p

φ

φ

∂=

∂.

. . . . . . . . . . . . . . . . . . . . . . . . (19)1 1

p

pS c

p p p π

π π

φ φ

φ φ

∂ ∂ ∂= =

∂ ∂ ∂,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)w w o o p p S p S = + ,

. . . . . . . . . . . . . . . . . . . . . . (17)1 1

f B c p B p

ππ π

π π π π

ρ

ρ

∂ ∂= =

∂ ∂ .

. . . . . . . . . . . . . . . . . . . . . . (16)

.

1 1 pS S

t B p B t π

π π

π π π

φ φ ∂ ∂ ∂

= ∂ ∂ ∂

1 1 1S S p B

B p B p S p t

π π ππ

π π π π π π

φ φ

φ

∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)1o wS S + = .and

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Comparing this equation with Eq. 23 gives the total compressibility as

The subscript hydro is used in the above equation to indicate thatthe pore compressibility was calculated assuming hydrostatic load-ing conditions. In the case of incompressible fluids, K f and

c f 0; hence, c pct gives

Neglecting further poroelastic effects by assuming that K sgives 1, and the pore-compressibility parameter for hydrostaticloading condition becomes

Oedometric Deformation. Another stress path commonlyassumed in determining the pore-compressibility parameter is uni-

axial strain compaction (also called K o compaction in geomechan-ics), in which the pore pressure is varied with constant total stressesand the horizontal displacements are blocked. Uniaxial straincompaction is usually assumed to be a good approximation of theconditions undergone by a reservoir during depletion.20 Applying

the conditions d x d y 0 and d

ij0 in Eq. 26 yields

Substituting Eq. 35 in Eq. 28 yields

Correspondingly, the oedometric total compressibility is

where M the constrained modulus defined as

Again, neglecting fluid-compressibility and poroelastic effectsyields the pore-compressibility parameter for oedometric condi-tion, which is

Consequences for Reservoir Simulation. The previous discus-sion has shown that the equations of poroelasticity can be reduced,under specific local assumptions, to a hydraulic-diffusion-typeequation. Whereas the compressibility of a fluid can be consideredan intrinsic property under constant temperature, however, the porecompressibility depends on the local conditions assumed. It wasshown that assuming local hydrostatic or oedometric conditionsgives two different values of pore compressibilities. The differ-ences in the pore compressibilities for these two conditions can besignificant. It can be shown that the ratio of the two compressibil-ity values is equal to

For instance, a Poisson’s ratio of 0.2 (a typical value for theelastic response of typical reservoir rocks), the hydrostatic porecompressibility is twice the oedometric pore compressibility.

Obviously, depending on the loading condition, a wide range of pore-compressibility values is possible even for idealized elasticmaterials. The deformation of reservoir rocks is, however, nonlin-ear and elastoplastic. Consequently, an even wider range of porecompressibilities can be expected for reservoir rocks than from

elastic behavior. Pore compressibility can be infinite for nonstrain-hardening perfectly plastic rocks. In the extreme case, pore com-pressibility can be negative when shear loading causes the rock volume to increase under constant or increasing pore pressure.

In practice, rock compressibility will have an impact when itsabsolute value is comparable to or greater than that of the fluid. Themagnitude of pore compressibility in relation to fluid compressibil-ity should provide an initial indication of the possible importance of fluid flow and deformation coupling on pore-pressure distribu-tion and productivity. The bulk modulus of many reservoir rocksrarely exceeds 10 GPa. In comparison, the bulk modulus of wateris about 2.25 GPa at standard conditions. It is expected, therefore,that reservoir rock pore compressibility will generally be greaterthan, or at least of the same order of magnitude as, the compress-ibility of reservoir fluids in oil/water systems. However, for cases

involving pressure maintenance by gasflooding or in case of asolution-gas drive, the gas can increase the total system com-pressibility considerably, making the effects of pore compress-ibility less important.

Two remarks can be made at this point. First, it should be notedthat pore compressibility as defined in reservoir engineering is meas-ured under constant total stresses. Also, from the previous discus-sion, the measured value of the compressibility pertains only to aspecific stress path and cannot be used for other stress paths.Boutéca19 emphasized the need to measure the pore compressibilityunder oedometric conditions, on the basis that most of the reservoirdeforms under oedometric conditions during depletion.

Second, the assumption of oedometric or uniaxial strain condi-tion for every point in a reservoir is only valid, strictly speaking,for a horizontally infinite reservoir under uniform pressure draw-down. Reservoirs are, however, bounded laterally and do not

deform uniformly even under uniform pressure drawdown. This isillustrated by the results of a stress analysis of an axisymmetric(disk-shaped) thin reservoir, whose radius is the approximately thesame as its depth, subjected to a uniform pore-pressure reduction(Fig. 2). Because of the stiffness and bending of the overburden,the reservoir deforms nonuniformly, as shown by the displacementvectors. Close to the reservoir centerline, the displacement vectorsare vertical; hence, the rocks are under uniaxial strain conditions.Near the flanks, the horizontal displacements are comparable to thevertical components depicting hydrostatic loading of the rock. Ingeneral, the displacement and stress fields in a reservoir willdepend on the reservoir geometry, boundary conditions, and pore-pressure distribution, and will be different from the idealized uni-axial strain condition. The presence of discontinuities (e.g., faultsand fractures) and material inhomogeneities (e.g., layering of the

reservoir) will also affect the stress distribution.

Petroleum Reservoir Simulation CouplingMultiphase Flow and Deformation

Biot’s formulation constitutes a fully coupled system of equationsfollowing the definition of Zienkiewicz.21 Based on this definition,fluid flow (Eqs. 1 and 6) and geomechanics (Eqs. 2 through 5)form a set of separate domains that cannot be analyzed separately.Consequently, the dependent variables (e.g., fluid pressures androck displacements) cannot be explicitly eliminated except, as seenpreviously, when assuming specific local conditions. By compar-ing the governing equations for reservoir simulation (Eq. 21) andBiot’s theory (Eqs. 2 through 6 and 22), the following differencesmay be observed:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (40)( )

( )( )hydro

oedo

3 1

1

p

p

c

c

ν

ν

−=

+.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (39)

( )oedo1

pc

M φ= .

. . . . . . . . . . . . . . . . . . . . (38)( )

( )( )

14

3 1 2 1

E M K G

ν

ν ν

−= + =

− +.

. . . . . . . . . . . . . . . . . . . . . . . . . (37)( )2

oedo

1 1t

B

cM M

α

φ

= +

,

. . . . . . . . . . . . (36)2 1ij

i j B

k p p gh q

x x M M t

αρ

µ

∂ ∂ ∂ + = + +

∂ ∂ ∂

.

. . . . . . . . . . . . . . . . . . . . (35)4

d d d 03

z v K G pσ ε α

= + − =

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (34)( )hydro

1 pc

K φ= .

. . . . . . . . . . . . . . . . . . . . . . . . (33)( )2

hydro

1 p

s

c K K

α α φ

φ

−= +

.

. . . . . . . . . . . . . . . . . . . . . . . . . (32)( )2

hydro

1 1t

B

c K M

α

φ

= +

.

. . . . . . . . . . . . (31)2 1ij

i j B

k p p gh q

x x K M t

αρ

µ

∂ ∂ ∂ + = + + ∂ ∂ ∂

.


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• Reservoir simulation does not include any of the equationsfrom the geomechanical relations. This means, for instance, thatthe calculated pore pressures may not be in equilibrium with theoverburden loads because the effective stress principle and theequilibrium condition are not accounted for.

• The only rock mechanical parameter involved in reservoir

simulation is the pore compressibility c p. This is a scalar quantityand cannot sufficiently represent the behavior of rocks. Rock stress-strain behavior is not only nonlinear but also stress-pathdependent and requires a full tensorial relation. As shown earlier,even for linearly elastic and isotropic materials, different com-pressibility parameters are obtained depending on the loading path.

Gutierrez8 has analyzed numerically and theoretically the valid-ity of the uncoupled approach. With a finite-element formulation,the single-phase fluid flow diffusion equation was discretized as

where [p]the matrix of pore-pressure change, [p]the cur-rent pressure matrix, [C]the compressibility matrix, []thepermeability matrix, [q]the matrix of fluid fluxes, andt the time increment. From this equation, the pore-pressurechanges can be solved as

For the single-phase, fully coupled formulation, the finite-elementdiscretization is given as

where [K]the stiffness matrix, [L]the coupling matrix, and[F]the matrix of boundary and self-weight loads. As shown, thedisplacement increment matrix [u] and the pore-pressure-change

matrix [p] can be solved simultaneously from this equation.Solving the displacement field [u] from the first equation of Eq.43, substituting [u] in the second, and solving for [p] gives

The main differences between Eqs. 42 and 43 are:• Pore-pressure change is a function of the full-rock-stiffness

matrix [K] in the fully coupled formulation, while it is a functiononly of the rock-compressibility matrix [C] in the uncoupled for-mulation. The former accounts for the full constitutive behavior of

the reservoir and nonpay rock system, while the latter is a diagonalmatrix that can only be made dependent on the pore pressure.

• Pore-pressure change is also a function of the applied load[L]t [K]1[F], caused by the total stress changes in the fully cou-pled analysis. Such total stress changes come from the weight of the overburden, which is transferred nonuniformly to the reservoiragain, according to the pore-pressure distribution in the reservoir.

Application to a Field Case

The extended Biot equations for three-phase fluid flow indeformable porous media were discretized by Lewis andSukirman,11 and the discretized equations were implemented in thefinite-element code CORES (COupled REservoir Simulator).

CORES is a 3D black-oil (three-phase compressible and immisci-ble fluid flow) simulator. The reservoir rock is modeled by elasticand/or elastoplastic constitutive models, and the physical proper-ties of the fluids depend on fluid pressures and saturations. In thefinite-element implementation, implicit procedures are used tosolve the fully coupled governing equations where the rock dis-placements and fluid pressures are the primary unknowns.

To illustrate the importance of analyzing fluid-flow and geo-mechanical behavior as fully coupled processes, CORES isapplied to the simulation of an idealized North Sea reservoir. Thereservoir has an area of about 6.5 by 11 km and a thickness of 300m. The complete model, which includes the nonpay rock, has anarea of about 23 by 38.5 km and a thickness of 4 km (of which 3km is the overburden, 0.3 km the reservoir, and 0.7 km the under-burden). A simplified view of the whole model is shown in Fig.

3. A rough mesh with 8 by 20 by 14 eight-noded brick elementswas used in the simulation. However, the results of the 3D modelwere also verified by a 2D model with a much finer discretiza-tion. No vertical displacements are allowed at the base of themodel, and no lateral displacements are allowed at the four sidesof the model. It is a common practice in geomechanical modelingto extend the lateral boundaries as far away as computationallypossible from the main loaded region to simulate infinitely hori-zontal boundaries and minimize local boundary effects. The topof the model, which corresponds to the seabed, is allowed todeform freely.

The initial reservoir pressure is assumed to be 48 MPa, which isuniformly distributed in the reservoir. The initial effective verticalstress distribution vs. depth is integrated from the self-weights of

. . . . . . . . (44) ( ) p q L K Ft

t −

−∆ + ∆ + ∆ .

( )

p L K L t

t −

−

∆ = + ∆

. . . . . . . (43)( )

FK L u

p p + qL t

t t

=

,

. . . . . . . . . . (42)( ) ( )

p C p qt t −

∆ = + ∆ ∆ + ∆ .

. . . . . . . . . . . . (41)( ) ( )C [] [ p] [][p] [ q]t t + ∆ ∆ = ∆ + ∆ ,


Fig. 2—Rock displacements caused by a uniform pressuredrawdown in a disk-shaped reservoir.

Reservoir centerline →

Fig. 3—Fully coupled model of an idealized reservoir.

3 km

0.3 km

0.7 km

11 km

38.5 km

23 km

4 km

6.5 km

Hydrocarbonreservoir

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the different rock layers, while the initial effective horizontal stressesare assumed to be one-half of the total vertical stresses. Only thewater and oil phases are considered in the reservoir to simplify theanalysis. Undrained conditions (i.e., no fluid flow or zero perme-

ability) are used for the surrounding nonpay rocks. Realistic relativepermeability and capillary pressure curves, and standard oil andwater formation volume factor curves, were used for the fluid-flowpart of the flow simulation. Elastic rock properties were used in thegeomechanical simulation. The rock properties used are given inTable 1. The model is analyzed for a 14-year production scenarioby specifying production rates in the finite-element nodes corre-sponding to production wells within the reservoir. The productionrates applied in the production wells are based on recorded produc-tion data in the field.

The calculated pore-pressure distribution at the end of simula-tion in the top reservoir layer is shown in Fig. 4. After 14 years of production, the reservoir pore pressure has been reduced to approx-imately 25 MPa in much of the reservoir. However, despite thecontinuous production, the pore pressures have increased to

approximately 55 MPa in areas close to the reservoir flanks, whichis higher than the initial pressure of 48 MPa.

The predicted seabed subsidence and reservoir compactionafter 14 years of production are shown in Fig. 5. The maximumcalculated subsidence is about 6 m, which is approximately 85% of the maximum reservoir compaction. An interesting result is theexpansion of the reservoir and the corresponding heave of theseabed around the reservoir flanks. This result is an outcome of the

increased pore pressures shown in Fig. 4.The increase in pore pressure close to the reservoir flanks isalso observed in the case of pressure-driven production. This canbe seen in Fig. 6, which shows the pressure distribution along anorth/south section of the reservoir caused by a pressure drawdownin a single well at the center of the reservoir. In this figure, the bot-tomhole pressure history used in the simulation to drive the pro-duction is based on the measured bottomhole pressures in the field.The prescribed bottomhole pressure decreased almost linearly withtime from 48 MPa at the start of production to about 24 MPa after14 years of production. The results of two analyses are shown inthis figure: one for a soft reservoir with a Young’s modulus of

E 50 MPa, and another for a stiff reservoir with E 850 MPa( 0.25 for both cases). The Young’s modulus of 50 MPa for thesoft reservoir corresponds to the post-yield elastoplastic responseof the reservoir rock, while the value for the stiff reservoir corre-

sponds to the elastic response of the reservoir rock.For the case of the coupled simulation of the soft reservoir, the

pore pressures have increased to about 52 MPa (which is higherthan the initial reservoir pore pressure of 48 MPa) from a distanceof about 1.7 km from the central producing well, despite the con-tinuous drawdown in the producing well. The pore-pressure distri-butions obtained from uncoupled reservoir simulations are alsoshown in the same figure. In the uncoupled reservoir simulation,the pore compressibilities used (see Table 1) correspond to a uni-axial strain loading condition and were calculated from Eq. 39 withthe same Young’s modulus and Poisson’s ratio used in the fullycoupled simulation. Again, this is based on the assumption that theexpected horizontal displacements in the reservoir will be negligi-ble in comparison to the horizontal dimensions of the reservoir. Assuch, these values are the best a priori estimates of the pore com-

pressibilities for the uncoupled simulations.As shown, the predicted pore pressures never exceeded the ini-

tial pore pressure of 48 MPa. For the case of the stiff reservoir, noincrease in pore pressure above the initial pressure is observed, asin the case of the soft reservoir, for both the coupled and uncoupledsimulations. However, there are still significant differences in thepredicted pore-pressure distributions from the two simulations.

The differences in the predicted pore pressures shown in Fig. 6,in the case of both soft and stiff reservoirs, are caused by the lack of geomechanical terms and the deficiency of using rock com-pressibility to account for the geomechanical effects in the reser-voir simulation. In the reservoir simulation, the rock deformationwas constrained and assumed to be a priori uniaxial. However, thestress paths followed by the different points in the reservoir are


TABLE 1—MATERIAL PROPERTIES USED IN MODELING

Overburden/Sideburden

Young’s modulus, E (GPa)

Poisson’s ratio,Absolute permeability, k (md)

2.50.45

0

Underburden

Young’s modulus, E (GPa)

Poisson's ratio,Absolute permeability, k (md)

13.50.45

0

Reservoir

Young’s modulus, E (GPa)Poisson’s ratio,

Porosity, φPore compressibility, c p * (/MPa)Permeability, k (md)

Stiff

0.850.2535

0.0028150

Soft

0.050.2535

0.0476150

* Used only for the uncoupled flow model and calculated from Eq. 39 with

the same E and ν used in the fully coupled models. For comparison,c f =4.4×10

–4 /MPa at initial reservoir conditions (used for both uncoupled

and fully coupled simulation).

ν

ν

ν

Fig. 4—Pore pressure (in MPa) distribution in the reservoir after14 years of production.

Fig. 5—Reservoir compaction and seabed subsidence at theend of 14 years of production.

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controlled by the interaction between the reservoir and the sur-rounding nonpay rock, and by the constitutive behavior of both thereservoir and the nonpay rocks.

The increase in pore pressure above initial value during pro-duction is an important effect that cannot be predicted by existingreservoir simulators. This increase in pore pressure is caused by theload-term load [L]t [K]1[F] from the total stress changes and theinteraction of reservoir and overburden in the fully coupled analy-sis. The pore-pressure reduction resulted in the compaction of the

central part of the reservoir. In turn, the compaction resulted in thedownward movement and bending of the overburden, causing thereservoir fluids to be squeezed and the pore pressures to increasetoward the reservoir flanks. This coupled response, which comesfrom the interaction between the reservoir and the overburden, is acomplicated process. The deformation of the overburden isdependent on the pore-pressure distribution in the reservoir; on theother hand, the pore-pressure distribution in the reservoir is alsocontrolled by deformation of the overburden. This structural inde-terminacy in rock deformation is one of the main reasons why it isnot always easy to decouple rock deformation from fluid flow.22

The increase in pore pressure above initial value during fluidextraction is analogous to the so-called Mandel-Cryer effectobserved in one of the first applications of Biot’s 3D consolidation

theory. Cryer23 showed that on withdrawal of fluid in a consolidat-ing layer of a fluid-saturated medium, the pore pressure instanta-neously jumped, then continued to increase for some time, beforepressure dissipation commenced. The pore-pressure increase isattributed to the downward movement of the layer above the con-solidating layer and the increase in total stresses as the weight of the layer above the consolidating layer is transferred to the fluids,causing the pore pressure to increase.

The increase in pore pressure owing to rock deformation hasalso been referred to as compaction drive in reservoir engineering.In standard reservoir simulation, the main mechanism accountingfor the compaction drive is the pore-volume reduction of the

reservoir rock. In fully coupled simulation, the downward move-ment of the overburden also contributes to the compaction drive.This contribution, particularly when the pore pressures increaseabove the initial reservoir pressure, cannot be accounted for sim-ply by adjusting the pore compressibility in reservoir simulations.The compaction drive will be very pronounced for soft reservoirs,but it can also be significant for the case of relative stiff reservoirs,as shown in Fig. 6. Note that this increase is only to be expectedfor reservoirs with low-permeability aquifers. Otherwise, theincrease in reservoir pressure from compaction drive will be dis-sipated into the aquifer. On the other hand, a compressible aquifermight also contribute to the increase in pore pressure from thecompaction drive.

Several schemes have been proposed in the literature to couplethe stress-strain behavior of rock and multiphase fluid flow.3,4

Settari and Mourits,

24

for instance, present an approach where theporosity is used as a coupling parameter between a finite-elementstress-analysis code and a reservoir simulator. The geomechanicaland reservoir simulators are used in a staggered manner. Pore-pressure changes are calculated from the reservoir simulation andconverted to nodal loads. From these nodal loads, the in-situ stresschanges and rock displacements are calculated in the geomechan-ical simulation. An iterative algorithm is used to ensure that theporosities calculated from the geomechanical simulator are thesame as those calculated from the reservoir simulator.

The iterative approach, however, does not rigorously addressthe coupling of geomechanics and reservoir simulations. One pos-sible drawback of such an iterative approach is that there appearsto be no proof that the approach will converge to a unique solution.For instance, it is not clear whether the approach can be used in thecase where the rock tends to increase in volume with a reduction in

pore pressure (e.g., owing to dilation during shearing). Such a vol-ume increase would require a negative pore compressibility in thereservoir simulation and may cause numerical instability.

Conclusions

The issues related to the interaction between fluid flow and rock deformation in reservoir simulation have been discussed in thispaper. A primary type of interaction concerns stress- induced per-meability changes, which in turn affect the fluid-pressure distribu-tion. This type of coupling is particularly important in fracturedand faulted reservoirs, where fracture- and fault-permeabilitychanges can be orders of magnitude greater than those of the bulk matrix. Moreover, fracture- and fault-permeability changes canalso influence fluid-flow directionality and sweep efficiency.

A comparison of the governing equations used for reservoir

simulations and Biot’s theory for multiphase fluid flow indeformable porous media was made. Based on this comparison andon the results of a simple case study, it was shown that geome-chanics and multiphase fluid flow in hydrocarbon reservoirsshould be analyzed as fully coupled processes. As fully coupledprocesses, fluid flow and geomechanics form a set of separatedomains that cannot be analyzed separately. The dependent vari-ables in each domain (e.g., fluid pressures and rock displacements)cannot be eliminated explicitly, except for simple cases correspon-ding to specific stress paths. However, in general, the deformationof the reservoir during depletion and recovery is a complicatedprocess for which a simple stress path cannot be assumed.

It is shown that reservoir simulators, by not taking these couplingphenomena into consideration, simplify important geomechanical


Fig. 6—Comparisons of reservoir pressures from fully coupledand standard reservoir simulations. Pressure-controlled pro-duction. Top: soft reservoir; bottom: stiff reservoir.

0 1000 2000 3000 Distance from production well, m

30

40

50

R e

s e r v o i r p r e s s u r e

Fully coupled modeling

Reservoir simulation

0 1000 2000 3000

Distance from production well, m

30

40

50

R e s e r v o i r p r e s s u r e

Fully coupled modeling

Reservoir simulation

Initial reservoir

pressure

Initial reservoirpressure

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aspects that can impact reservoir productivity. This is attributed to thefact that reservoir simulation does not include the governing equa-tions for geomechanics. Moreover, the only rock mechanical param-eter involved in reservoir simulations is pore compressibility. Thisparameter is not sufficient in representing aspects of rock behaviorsuch as stress-path dependency and dilatancy, which require a fullconstitutive relation, and the influence of the surrounding nonpayrock on reservoir deformability. Furthermore, the pore-pressurechanges caused by the applied loads from the nonpay rock cannot beaccounted for by simply adjusting the pore compressibility.

Nomenclature

B volume factor for phase c f fluid compressibility

c f fluid compressibility of phase c p pore compressibility

(c p)hydro pore compressibility under hydrostatic loading(c p)oedo pore compressibility under oedometric loading

ct total compressibility

[C] compressibility matrix

D

ijkl constitutive tensor E Young’s modulus

[F] matrix of incremental boundary or self-weight loadsF i boundary or self-weight loads

g acceleration owing to gravity

G shear modulus

h fluid heightk absolute permeability

k

ij permeability tensor

k r relative permeability for phase K bulk modulusK f fluid modulusK s grain compressive modulus

[K] stiffness matrix[L] coupling matrix M constrained modulus

M B Biot’s modulus p pore pressure

p fluid pressure for phase Pc capillary pressure

[p] incremental pore-pressure matrixq fluid source or sink

[q] matrix of incremental fluid sources or sinksS saturation for phase

t time

ui rock displacement[u] displacement increment matrix

i fluid velocity for phase x i spatial coordinate Biot’s coefficient (1-K / K s)ij Kronecker delta function

ij strain tensor

volumetric strain[] permeability matrix

porosity

fluid viscosity

viscosity for phase density

density for phase

ij total stress tensor

ij effective stress tensor

Poisson’s ratio

Subscripts

i, j,k ,l spatial coordinates x , y, zo oilw water

z vertical axis

fluid phase (o,w)

Acknowledgment

The assistance of Dr. Hamid Ghafouri in carrying out the fullycoupled modeling of the idealized North Sea reservoir is grate-fully acknowledged.

References

1. Advani, S.H. et al.: “Fluid Flow and Structural Response Modeling

Associated with the Mechanics of Hydraulic Fracturing,” SPEFE (June

1986) 309.

2. Settari, A., Ito, Y., and Jha, K.N.: “Coupling of a Fracture MechanicsModel and a Thermal Reservoir Simulator for Tar Sands,” J. Cdn. Pet.

Tech. (November 1992) 31, No. 9, 20.

3. Fung, L.S.K., Buchanan, L., and Wan, R.G.: “Coupled Geomechanical-

Thermal Simulation of Deforming Heavy-Oil Reservoirs,” paper 36–1 pre-

sented at the 1992 CIM Annual Technical Conference, Calgary, June 1992.

4. Tortike, W.S. and Ali, S.M.: “Reservoir Simulation Integrated with

Geomechanics,” paper 39–1 presented at the 1992 CIM Annual

Technical Conference., Calgary, June 1992.

5. Heffer, K.J. et al.: “The Influence of Natural Fractures, Faults and Earth

Stresses on Reservoir Performance—Analysis by Numerical

Modelling,” Proc., 3rd Intl. Conference on North Sea Oil and Gas

Reservoirs, Trondheim, Norway (December 1992) 201.

6. Koutsabeloulis, N.C., Heffer, K.J., and Wong, S.: “Numerical

Geomechanics in Reservoir Engineering,” Proc., Intl. Conference on

Computer Methods and Advances in Geomechanics (1994) 2097.7. Gutierrez, M. and Makurat, A.: “Coupled HTM Modelling of Cold

Water Injection in Fractured Hydrocarbon Reservoirs,” Intl. J. Rock

Mech. Min. Sci. (1997) 34, 429.

8. Gutierrez, M.: “Fully Coupled Analysis of Reservoir Compaction and

Subsidence,” paper SPE 28900 presented at the 1994 SPE European

Petroleum Conference, London, 25–27 October.

9. Lewis, R.W. and Sukirman, Y.: “Finite Element Modelling for

Simulating the Surface Subsidence Above a Compacting Hydrocarbon

Reservoir,” Intl. J. Num. Analy. Meth. Geomech. (1993) 18, No. 9, 618.

10. Biot, M.A: “General Theory of Three-Dimensional Consolidation,” J.

Appl. Phys. (1941) 12, 155.

11. Lewis, R.W. and Sukirman, Y.: “Finite Element Modelling of Three-

Phase Flow in Deforming Saturated Oil Reservoirs,” Intl. J. Num.

Analy. Meth. Geomech. (1993) 17, No. 8, 577.

12. Terzaghi, K.: Erdbaumechanik auf Bodenphysikalicher Grundlage,

Franz Deutike, Vienna (1925).

13. Sornette, D., Davy, P., and Sornette, A.: “Structuration of the

Lithosphere in Plate Tectonics as a Self-Organized Critical

Phenomenon,” J. Geophys. Res. (1990) 95, No. 11, 353.

14. Chen, H.-Y., Teufel, L.W., and Lee, R.L.: “Coupled Fluid Flow and

Geomechanics in Reservoir Study—I. Theory and Governing

Equations,” paper SPE 30752 presented at the 1995 SPE Annual

Technical Conference and Exhibition, Dallas, 22–25 October.

15. Chen, H.-Y. and Teufel, L.W.: “Coupling Fluid-Flow and Geomechanics

in Dual-Porosity Modeling of Naturally Fractured Reservoirs,” paper

SPE 38884 presented at the 1997 SPE Annual Technical Conference and

Exhibition, San Antonio, Texas, 5–8 October.

16. Ghafouri, H.R.: “Finite element modelling of multi-phase flow through

deformable fractured porous media,” PhD thesis, U. of Wales, Swansea,

Wales (1996).

17. Coussy, O.: A General Theory of Thermoporo-elastoplasticity .Transport in Porous Media (1989) 4, 281–293.

18. Charlez, Ph.: Rock Mechanics, Vol. 2 Petroleum Applications, Technip

Ed., Paris (1997).

19. Boutéca, M.: “Elements of Poro-elasticity for Reservoir Engineering,”

Revue de l’Institut Français du Pétrole, (1992) 47, No. 4, 479.

20. Morita, N. et al.: “A Quick Method To Determine Subsidence,

Reservoir Compaction, and In-Situ Stress Induced by Reservoir

Depletion,” JPT (January 1989) 71.

21. Zienkiewicz, O.C.: “Coupled Problems and Their Numerical Solutions,”

Numerical Methods in Coupled Systems, Wiley, New York City (1984) 35.

22. Lewis, R.W., Schrefler, B.A., and Simoni, L.: “Coupling vs.

Uncoupling in Soil Consolidation,” Intl. J. Num. and Anal. Meth.

Geomech. (1991) 15, No. 8, 533.


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23. Cryer, C.W.: “A Comparison of the Three-Dimensional Consolidation

Theories of Biot and Terzaghi,” Q.J. Mech. Appl. Math. (1963) 16, 401.

24. Settari, A. and Mourits, F.M.: “Coupling of Geomechanics and

Reservoir Simulation Models,” Proc., Intl. Conference on Computer

Methods and Advances in Geomechanics, Morgantown, West Virginia

(1994) 2151.

SI Metric Conversion Factors

bar 1.0* E 05 Paft 3.048* E 01 m

mile 1.609 344* E 00 km

*Conversion factor is exact.

M. Gutierrez is Associate Professor in the Dept. of Civil andEnvironmental Engineering at the Virginia Polytechnic Inst.and State U. in Blacksburg, Virginia. e-mail: [email protected]. He was formerly a senior engineer responsible forreservoir mechanics at the Norwegian Geotechnical Inst. inOslo. Gutierrez holds a PhD degree in civil engineering fromthe U. of Tokyo. R.W. Lewis is Professor and Head ofMechanical Engineering at the U. of Wales, Swansea, U.K.An editor of three international journals on numerical mod-eling, he has cowritten a book on the finite-elementmethod in the deformation and consolidation of porousmedia and has written or collaborated on more than 250papers. Lewis holds PhD and DSc degrees from the U. of Wales.I. Masters is Lecturer in the Dept. of Mechanical Engineering at

the U. of Wales, Swansea. He holds a PhD degree from the U.of Wales.

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