Advances in Coupled Geomechanical

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SummaryCoupled geomechanical and reservoir modeling is becoming feasi-ble on a full-field scale. This paper describes the advances of thecoupled model described previously, 1 its extensions for modelingcompaction, and the application of the model in full-field studies.

The advances of the theory and numerical implementation sincethe original work 1 make it practical to perform full-field coupledstudies with complex, realistic descriptions of the geomechanicalbehavior of the reservoir and shales, which allows prediction of stress changes, reservoir compaction, and surface subsidence.These capabilities are demonstrated in field examples that analyzeand predict classical pressure-induced compaction in a gas fieldand thermally induced compaction in a heavy-oil field. In bothcases, the methodology of interpreting the geomechanical labora-

tory and field data and their integration in the coupled modelingprocess was the key to obtaining a realistic predictive tool. Theexamples demonstrate that the technology is maturing to the pointthat conventional studies can be converted to coupled modeling ona fairly routine basis.

IntroductionThe geomechanical behavior of porous media has become increas-ingly important to hydrocarbon operations. Numerical modeling of such processes is complex and has been carried out historically inthree separate areas: geomechanical modeling (with the primarygoal of computing stress/strain behavior), reservoir simulation(essentially modeling multiphase flow and heat transfer in porousmedia), and fracture mechanics (dealing in detail with crack prop-agation and geometry). A modular system has been developed cou-pling these three modeling components in such a manner that thealready highly developed modeling techniques for each componentcan be used fully. 1 This model has been applied to several geome-chanical/reservoir problems assisting in reservoir development.

The paper will first discuss the theory of different degrees of coupling and its consequences for the formulation of the constitu-tive models and running efficiency of the software. Next, themodeling of compaction by rigorous means (plasticity) and itssimplifications, which lead to a considerable increase of computa-tional efficiency, will be presented. In addition to classicalpressure-depletion-induced compaction, the paper will describe thetheoretical and modeling aspects of thermal compaction phenomena,which have been observed in some applications.

Methods of CouplingThe key idea in the modular coupled system is the reformulation of

the stress-flow coupling so that the conventional stress-analysiscode can be used in conjuction with a standard reservoir simulator.This is termed a partially coupled approach because the stress andflow equations are solved separately for each time increment.However, the method solves the problem as rigorously as a fullycoupled (simultaneous) solution if iterated to full convergence.

The coupling takes place through the use of interface codedeveloped to allow communication between simulators. In ageomechanics/reservoir problem, for instance, the pressure andtemperature changes occurring in the reservoir simulator arepassed to the geomechanical simulator. The updated strains andstresses are passed back to the reservoir simulator and used to com-pute coupled parameters in the reservoir formulation (i.e., porosityand permeability). An iterative method then must be used to obtainconvergence. The interface is flexible enough to allow the user tochoose several degrees of coupling. The degree of coupling mayaffect the accuracy of the solution as well as the computationalefficiency; therefore, tradeoffs may be made to optimize run times.

To see the different degrees of coupling, consider first thegeneral formulation of the coupled problem in a finite-element set-

ting. After discretization in space and time, such a system can bewritten in matrix form as 2,3

where [K ] the stiffness matrix, the vector of displacements,[L ] the coupling matrix to flow unknowns, [E ] the flow matrix,and P the vector of reservoir unknowns (i.e., pressures, satura-tions, and temperatures). On the right side, F the vector of forceboundary conditions, and

R the right side of the flow equations.

The symbol t denotes the change over timestep; i.e.,

Note that in the conventional reservoir simulation notation,4[E ] [T ] [D], where [T ] the symmetric transmissibility matrix,[D] the accumulation (block diagonal) matrix, and

R Q [T ]

P

n,

where Q the vector of boundary conditions (well terms).

Decoupled. Consider now the flow part of the coupled systemonly, by assuming that t 0. This is the assumption made inreservoir simulation (i.e., stresses do not change), which gives thefamiliar matrix equation

Conversely, if we assume that t P 0, we obtain the classical elas-

ticity equations. In many stress analysis packages, pressure and/ortemperature can be imposed as external loads, which correspondsto assuming that t P is known. Then the top half of Eq. 1 can bedecoupled and written as

In practice,decoupledsimulations canbe carried out in several ways.With a reservoir model only. To account for at least zero-

dimensional effects of stress changes, the compressibility terms inthe [D] matrix must be modified to account for the expected typeof containment in terms of deformations and the fact that the reser-voir simulator uses a nondeforming grid (i.e., constant bulkvolumes of gridblocks). As a result, if the bulk compressibility cb is

. . . . . . . . . . . . . . . . . . . . . . . . . . . (4)t t F P = K L

.

. . . . . . . . . . . . . . . . . . . . . . (3)( ) nt P Q P = T D T

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (2b)1n nt P P P +

=

.and

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2a)1n n

t

+ =

. . . . . . . . . . . . . . . . . . . . . . . . (1)t

t

F

R P

=

K L

L E

T ,

Advances in Coupled Geomechanicaland Reservoir Modeling

With Applications to Reservoir CompactionA. Settari,* SPE and Dale A. Walters,* SPE, Duke Engineering and Services Inc.

* Now with Taurus Reservoir Services Ltd.

Copyright 2001 Society of Petroleum Engineers

This paper (SPE 74142) was revised for publication from paper SPE 51927, first presentedat the 1999 SPE Reservoir Simulation Symposium, Houston, 1417 April. Original manu-script received for review 16 December 1999. Revised manuscript received 14 March 2001.Manuscript peer approved 30 March 2001.

334 September 2001 SPE Journal


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known, one can calculate compressibility c R I (corresponding to free

deformation) or c R II (corresponding to laterally constrained defor-

mation) to use in the reservoir simulator. 1 While this implicitlyassumes stress changes in each block, the stresses are not calculated.

Using a reservoir solution (Eq. 3), one can compute the entiretime history of

P and use it to compute a transient stress solution

with Eq. 4. This approach is being employed frequently becauseconventional models can be used; 5,6 it essentially amounts to com-puting two independent time histories with no coupling (at best, theporosity or permeability relationships can be updated in specifictime intervals manually 7).

A significant improvement to the reservoir model only

approach can be obtained by employing the zero-dimensionalstress solution assuming zero lateral strain. In an idealized case, thechange of average horizontal stress havg can be computed fromchange of average reservoir pressure pavg by

Then, the coupling term can be formally included in Eq. 3.

In practice, because the reservoir model is usually finite-differenceand the stress model is finite-element, the coupling through the[L ]T matrix is reflected in porosity and permeability dependence on

stress. One would calculate the local effective stresses from block pressure and average stresses and define porosity and permeabilityas a function of the effective stress state. This approach accountsfor overall depletion effects and has been used quite early 8 tomodel stress-dependent permeability.

Explicitly Coupled. Explicit coupling can be achieved by laggingthe coupling terms one timestep behind. Starting with the reservoirsolution and known change of stress over the previous timestep,

t n, we first solve

Then, using the flow solution, t P n 1, the stress solution isadvanced by

Again, the implementation does not involve the use of the [L ]matrix because the discretizations are usually different. Instead,the porosity in Eq. 7 is expressed as a function of both pressureand mean stress. The discretization of the accumulation terms inEq. 7 can then be done in a mass conservative fashion. The stress-dependent terms in the [T ] matrix can be treated explicitly interms of stress. A variation of this approach, where only thecoupling through the [T ] matrix is considered (and the volumecoupling ignored), has been used for modeling cases where thedominant feature is the stress-dependent enhancement of perme-ability (i.e., waterflooding 9).

The explicit coupling is a special case of the iteratively coupledsystem, 1 in which only one iteration per timestep is performed (see

the next section).

Iteratively Coupled. The iterative method consists of a repeatedsolution of the flow and stress equations during the timestepaccording to

Obviously, when the iteration (Eq. 9) converges, P (v) P n 1, and

(v) n 1 the solution is identical to the fully coupled system, Eq.

1, provided that any iterative processes in either formulation havebeen converged. Again, including the coupling term in Eq. 9a isequivalent to expanding the porosity in the reservoir model as afunction of p, T, and mean stress, or I 1, as detailed in Ref. 1. Thus,the porosity in the reservoir model is determined directly by thevolumetric strain computed from the stress model, rather than bysome compressibility relation (this point has often been misunder-stood). The coupling through flow properties (i.e., effect of stresson matrix [T ]) can be explicit ( [T ] [T ]n) or implicit ( [T ] [T ]( )).

Convergence of the iteration on the volume coupling has beenestablished. 10 The iterative method as implemented in this work isthe most flexible. It includes the explicit coupling (one

iteration/timestep) and can be simplified further by specifying theporosity vs. effective stress relation directly in the reservoir model.This latter approach often increases computational efficiency with-out sacrificing accuracy, as discussed in the next section.

Fully Coupled. The fully coupled approach has the advantages of internal consistency, as the full system (Eq. 1) can be solved simul-taneously with the same discretization (usually finite-element).There are only a few models that currently treat multiphaseflow. 2,11,12 Large development efforts will be needed to bring theirflow-model capabilities on par with existing commercial (finite-difference) simulators.

Modeling of CompactionModeling reservoir deformation is of considerable importance in

soft and/or thick reservoirs, where the results of compaction mayprovide an important production mechanism, cause well failures,and/or cause ground subsidence or heave with environmental con-sequences. Field development of large compacting fields such asGroningen, Wilmington, the Bolivar coast of Venezuela, or Ekofisk led to the development of techniques for estimating compaction,starting with the work of Geertsma 1315 and followed by a numberof modified reservoir models. 1619 The common feature of suchreservoir-compaction models was that the compaction is treated asa 1D problem (uniaxial strain) by assuming that (a) only verticaldeformations take place and (b) each vertical column of blocksdeforms independently. Consequently, the porosity changes werecalculated by modifying the conventional compressibility c R basedon the results of uniaxial strain laboratory experiments, and thestress problem was not solved. The relation between reservoircompaction and surface subsidence was typically obtained by anindependent solution of a stress problem using the computed com-paction as a boundary condition.

In the context of the thermal coupled model presented here,both compaction and subsidence are obtained naturally as part of the solution. Typically, the stress part of the model would beextended upward to the surface as well as laterally so that any archingeffects of the over- and sideburden would be captured. Moreover,the laboratory-compaction data are used to calibrate thestress/strain (constitutive) model of the skeleton rather than todefine c R. Therefore, the coupled model can represent the materialbehavior under general triaxial loading paths and can includeeffects of shear and temperature. Finally, the model provides thestresses and displacements necessary to analyze casing failure,a frequent problem in compacting reservoirs. 20 In particular, thecasing-failure study in thermal operations 21 was performed with

the coupled system described here.The constitutive model is the key element of the compaction

model [analogous to the pressure/volume/temperature (PVT)model in compositional simulation, for example ]. Elastic andplastic deformations may occur in the reservoir, but we typicallyassociate compaction with the plastic deformations. Therefore, itsmodeling requires an elastoplastic constitutive model. However,useful approximations can also be obtained using a nonlinearelastic model with hysteresis. These two approaches will bedescribed and compared next.

Elastoplastic Constitutive Model. A variety of plasticity modelsmay be used when modeling compaction. For the example consid-ered here, a generalization of the Drucker-Prager model, including

. . . . . . . . . . . . (10)( ) ( ) ( ) ( ),n nt t P P P = =

.where

. . . . . . . . . . . . . . . . (9b)( 1) ( 1) , 1,t t F P + + = = K L

and

, . . . . . . . . . . (9a)( ) ( 1) ( )T nt t P Q P + = T D T L

. . . . . . . . . . . . . . . . . . . . . . . (8)1 1n nt t F P + + = K L

.

. . . . . . . . . . . . . . (7)( ) 1 T n n nt t P Q P + = T D T L

. . . . . . . . . . . . . (6)( ) avgT nt t P Q P = T D T L

.

. . . . . . . . . . . . . . . . . . . . . . . . . (5)avg avg1 21h p

=

.

September 2001 SPE Journal 335


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a compressive cap, will be used. The failure criterion is formulatedin terms of invariants of the effective stress tensor, with compres-sive stresses defined as negative values. The following standardstress invariants are used.

The Drucker-Prager yield surface is shown in Fig. 1. The yieldsurface has two sections. One is the standard Drucker-Prager fail-ure surface, denoted here as the cone. On the cone, the expressionfor the yield surface is

The expression for the cap portion of the yield surface is

The equation F 0 defines the yield surface. The cap and coneportions of the surface are constrained to meet at their commonpoint of tangency. If the friction angle is constant, the value of aand the location of the transition point can be computed from thefollowing expressions.

The Drucker-Prager model is a so-called two-invariant modelbecause the yield function depends only on two of the stress invari-ants. Hardening of the cap is an important part of the modelbecause this is the region of compaction. It is controlled in themodel by a user-defined relationship between the volumetric plasticstrain and the mean effective stress. The change of the porosityduring plastic hardening then corresponds to the observed com-paction compressibility.

Nonlinear Elastic Constitutive Model. The nonlinear elasticmodel used for the analyses of this paper is a modified hyperbolicmodel. 22 The model varies the Youngs modulus E and bulk mod-ulus B as a function of the mean stress as follows.

where E i the initial Youngs modulus, Bm the tangential bulk

modulus, r the reference stress (may be either the minimumeffective stress or the mean effective stress), K e , K b and ne , n b theconstant and exponent parameters for describing the magnitudeand shape of the Youngs and bulk moduli, and pa atmosphericpressure. The tangential Youngs modulus E t is then computed withthe following formulas.

where d the deviatoric stress ( 1 3), R f the failure ratio rep-

resenting the maximum deviator stress (calculated with a Mohr-Coulomb failure criterion) to the ultimate deviator stress predictedfrom the hyperbolic fit, and e1 and e2 the exponent parameters 23

that govern the behavior of the tangential Youngs modulus as thedeviator stress increases. The classical hyperbolic model 22 isobtained by setting e1 2 and e2 1. Thus, for the classical model,as the deviator stress increases, a reduction or softening of theYoungs modulus occurs according to the value of R f (normallybetween 0.5 and 0.9). The friction angle , used to calculate themaximum deviator stress according to a Mohr-Coulomb criterion,was allowed to decrease with increasing stress level as follows.

where 1 the value of the friction angle at a confining stress of pa(1 atm), the reduction in friction angle for a 10-fold increase inconfining stress (1 log cycle), and 3 the minimum effective stress.

Comparison. The two constitutive models described previouslyare both capable of predicting nonlinear stress/strain behavior. Theelastoplastic formulation is a much more rigorous approach indealing with the post-failure material behavior. The nonlinearelastic formulation was historically developed for prefailurebehavior. It is a good representation for the stress-strain responsefor many soils and soft rocks under standard triaxial loading at con-stant confining stress up to a shear-induced failure. 22 Once shearfailure occurs (i.e., the failure criterion is reached), the hyperbolicmodel is unable to implement post-failure phenomenon (i.e., strainhardening or softening); rather, the stress path is restricted to theelastic stress space.

This limitation in post-shear-failure behavior is not an issue in

the majority of reservoirs experiencing compaction effects becausethe mechanism is a cap failure. Although the deviatoric stress mayincrease during pressure depletion (considering uniaxial strain con-ditions), the shear stress developed is likely still below the shearfailure criterion. Therefore, the hardening behavior associated withcompaction may be captured by the hyperbolic formulation byusing the mean effective stress as the reference stress. Thus, as themean effective stress increases owing to pressure depletion, themoduli values will increase (Eqs. 18 and 19). Also, as the pressuredecreases, the deviatoric stress will increase (assuming uniaxialstrain conditions) according to

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)(1 )

(1 2 )d p

=

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)31 loga p

=

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)( )( )

max

ult

d f

d

R

=

, . . . . . . . . . . . . . . . . . . . . . . . (20)( )

12

max

1

ee

d t i f

d

E E R

=

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)bn

r m b a

a

B K p p

=

and

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)en

r i e a

a

E K p p

=

. . . . . . . . . . . . . . . . . . . . . . . (17)2

1transition 2 21 p

p

X a K R I

R

+ =

+.and

. . . . . (16)( ) ( )22 22 2 2 21

p p p pa K R R X R X K R = + + + + .

. . . . . . . . . . . . . . . . . . . . . . (15)( )2 21 2

cap 2 1 I X a R J

F a

+=

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (14b)( )6 cos

3 3 sinc K =

and

, . . . . . . . . . . . . . . . . . . . . . . . . . . (14a)( )2sin

3 3 sin p

=

with

, . . . . . . . . . . . . . . . . . . . . . . . . . . (13)cone 2 1 p F J I K =

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12b)3 det( )ij J s=and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12a)12 2 i j i j J s s= ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11c)ij ij ij s = ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11b)13

I = ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11a)1 ii I = ,

Fig. 1The Drucker-Prager yield surface.

J 2

X= -I 1

a

b R =a

b

I 1transition

cone

cap

hardening



4/9

The negative sign indicates that the deviatoric stress will increasewith a decrease in pressure (negative p) and will soften the mod-ulus according to Eq. 20. Therefore, the two mechanisms are incompetition. The hyperbolic parameters may be adjusted so thatone or the other dominates, as the lab data dictate.

An example lab data set has been used to illustrate that both con-stitutive models may be adjusted to obtain a good representation of the material behavior. Fig. 2 compares the raw data and best fits of both models. The comparison shows that both constitutive modelsmay be used to match common compaction data. Tables 1 and 2contain constitutive model parameters used for the history match.

The hyperbolic model cannot match the elastic and the plastic

part of the typical compaction path simultaneously, as shown inFig. 3. Although in some reservoirs both elastic and plastic defor-mations occur (e.g., Fig. 4 of Ref. 7), sometimes the reservoirstress is already close to the cap [i.e., the original elastic zone up tothe preconsolidation stress, pre , is very small (see Fig. 15 of Ref.24) ]. In this case, the hyperbolic model can be matched to the plas-tic path, and the initial elastic loading can be ignored. The modelimplemented also includes hysteresis of mechanical properties,whereby a completely different set of parameters will be used (inan incremental fashion) when the effective stress starts to decrease.The hysteresis is used to model the elastic behavior along therebound curves, as shown in Fig. 3.

The hysteresis also offers a means to treat the general case inwhich the initial part of the loading is elastic. If the preconsolidationstress, pre, at which the cap is reached, is greater than the initial stressstate of the reservoir, it is sufficient to set the maximum historicalstress to pre and then initialize the model on the hysteresis curve.

The main advantage of using the nonlinear elastic model vs. theelastoplastic model is computing efficiency. For example, a simpleseven-spot element-of-symmetry model with a 6 10 16 grid wasrun through a 10-year depletion scenario with both constitutivemodels described earlier. Both models were run with the explicittimestep coupling. In addition, the elastic model was run iterative-ly coupled (converged with a pressure tolerance of 0.001). The

comparison of the surface displacement u z at the surface and runtimes is shown in Table 3.The elastoplastic model run times were approximately twice as

long as those of the nonlinear elastic model, and the fully coupledmodel was also 3.5 times slower, while all the results were within5%. Of course, run times will be problem-dependent, but for manycompaction problems, plastic failure begins at the onset of pressuredepletion. Thus, all elements are in plastic failure throughoutdepletion, resulting in significant computing costs if the elastoplas-tic model is used. There are some instances in which the nonlinearelastic model may not be applicable (i.e., more complicated stresspaths that may occur near the wellbore), but for the majority of thereservoir, a nonlinear elastic model can be used. Thus, significantgains in computational efficiency can be obtained with little loss inthe accuracy of the solution.

Thermal CompactionThe constitutive models presented so far have neglected thermaleffects on the stress/strain behavior. In certain reservoir materials,this may not be valid. Coussy 25 has presented a brief description of incorporating thermal effects in the elasticity domain for a general

TABLE 1ELASTOPLASTIC PARAMETERS FORHI STORY MATCH

Youngs modulus 1.5 10 5 psiPoissons ratio 0.3Grain bulk modulus 3 10 9 psiCohesion 14.5 psiFriction angle 30 X , initial intersection of cap with I 1 axis 45.0 psi

User-defined hardening function( vp vs. mean ) vpmean

(psi)

0 1.5 10 1

1.80 10 2 2.6 10 2

5.30 10 2 9.0 10 2

8.50 10 2 1.9 10 3

1.23 10 1 3.3 10 3

Fig. 2Compaction curve-example data.

27

28

29

30

31

32

33

34

35

36

37

0 500 1,000 1,500 2,000 2,500 3,000

mean' , psia

P o r o s

i t y ,

%

Raw dataNonlinear elastic model

Elasto plastic model

Fig. 3Compaction curve for nonlinear elastic model.

M e a n E f f e c t i v e S t r e s s , i

p r e

Rebound

Plastic compaction

TABLE 3COMPARISON OF DIFFERENTSOLUTION METHODS

Model u z (ft) Run time (minutes)

Elastic, expl. coupling 3.64 6.75Elastoplastic, expl. coupling 3.48 14.59Elastic, fully coupled 3.45 24.37

TABLE 2HYPERBOLIC PAR A METERS FOR HISTORY MATCH

Hyperbol ic parameter Pr imary Curve Hysteres is Curve

K e 485.0 700.0ne 0.3 0.6K b 404.2 583.3nb 0.3 0.6

R f 0.85 0.85Cohesion (psi) 14.5 14.5

1 (degrees) 40 40 (degrees) 12 12

pa (psi) 14.7 14.7



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thermoporoelastoplastic formulation. In short, the equation gov-erning the free energy of a system was modified to include thermaleffects in the hardening parameter governing the evolution of theelastic domain. Therefore, temperature changes are incorporatedinto a general loading function, which may cause thermal hardeningand plastic strain (depending on the direction of the temperaturechange). This rigorous formulation was simplified and applied tothe nonlinear elastic model described earlier. The modifications tothe nonlinear elastic formulation are described next.

Modified Formulation of Thermoporoelasticity. The generalequation of thermoporoelasticity (compression is negative) may be

formulated as

where ij and ij total stress and strain components, p and T porepressure and temperature; other symbols are defined inthe Nomenclature.

Eq. 24 was modified to include terms associated with thermalcompaction and hardening. The following equation was used torepresent the strain for an increment in temperature.

Here, K and n describe the shape of the thermal compactioncurve (as a function of temperature), and T ref a reference temper-ature (usually ambient or initial reservoir temperature). The vari-able comp represents essentially the same material behavior as thecoefficient of linear thermal expansion, although comp may have adifferent sign and is a nonlinear function of temperature. The ther-moporoelastic formulation (Eq. 24) is modified so that L isreplaced with the sum of the thermal expansion and compactioneffects of the material.

It is expected that any thermal compaction occurring (a nonrecov-erable plastic compression of the matrix) would induce hardeningof the Youngs and bulk moduli of the material. Thus, the nonlin-ear elastic constitutive model discussed previously was modifiedas follows.

A modification of the moduli will occur only when the reservoir isheated or cooled from the initial reference temperature. The expo-nents me and mb describe the shape of the modulus multiplier.Hysteretic behavior may be used for these modified variables andis based on the direction of temperature change as opposed to pres-sure or effective stress.

Calibration With Lab Data. A hypothetical example of test datawill be used to demonstrate how to calibrate the model. The test(similar to actual tests that led to the development of the method)consists of a series of temperature loads applied to a sample underisotropic confining stress conditions. The volumetric strain ismeasured throughout the test. To calibrate the hardening parame-ters, mechanical loading stages should follow each temperatureload. Fig. 4 presents a comparison of the hypothetical raw data anda best fit using the nonlinear elastic model. The thermal com-paction parameters required to history match the lab data are listedin Table 4. The values for the thermal hardening parameters werenot calibrated using the data presented in Fig. 4 because alternatingmechanical and thermal loading is needed. All other parametersrequired for the nonlinear elastic model are contained in Table 3.

The loading data shown in Fig. 4 are contrary to what would beexpected based only on the thermal expansion of the solid (whichwould follow the unloading path of Fig. 4). A number of mecha-nisms can be responsible for the compaction, including weakeningof bonds, grain rearrangement, and shrinkage of some mineral com-ponents. The plastic hardening is evident from the curvature on Fig.4. Finally, it should be noted that the amount of thermal compactionwill be dependent on the loading path and type of material, and thecase shown here should not be regarded as typical.

Field ExamplesThermally Sensitive Heavy-Oil Field. The following exampleillustrates the thermal effects caused by steamflooding a heavy-oilreservoir. The field consists of a flat, five-layer reservoir systemwith 32 production wells interspersed with 17 water and steaminjectors. Table 5 lists the general reservoir properties used in theexample. The production wells were perforated in Layers 2through 4, while the injectors were only perforated in Layer 4.

Lab Data Match. The lab data used for the example are pre-sented in Figs. 3 and 4. As discussed, the nonlinear elastic modelprovides a good match of the data and was used for this example.The material shows sensitivity to both pressure decreases andtemperature increases, indicating that compaction may be an issuedepending on the magnitude of pressure depletion and heating of the reservoir.

Full-Field Compaction. The full-field simulation consisted of

a 10-year production/injection period. All wells in the full-fieldmodel were set to constant fluid-rate constraints for the full simu-lation period. The production wells were set to produce at 450BOPD with a minimum bottomhole pressure (BHP) of 15 psia. Thewater injectors were set to 650 BWPD (at T inj T res), and the steaminjectors were set to 1,000 BWPD (0.8 steam quality atT inj 550F).

The results of the simulation are illustrated in Figs. 5 through 7.It is apparent that the thermal component of the compaction isdominant, as the total compaction pattern follows the temperaturepattern. This is, of course, a function of the material behavior asexpressed by Fig. 4. For comparison, Fig. 8 shows the case inwhich the thermal compaction component was removed and thenormal thermal expansion coefficient was used. This case shows

. . . . . . . . . . . . . . . . . . . . . . . . (28)ref

b bn m

mm b a

a

T B K p

p T

=

. . . . . . . . . . . . . . . . . . . . . . . . . (27)ref

e en m

mi e a

a

T E K p

p T

=

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)tot comp L = +

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)compref

nT

K T

=

, . . . . . . . (24)1

21 2ij ij v L ij ij

G T p

+ = +

TABLE 4THERMAL PARAMETERS FOR HISTORY MATCH

P ar amet er Primar y C ur ve H yst er es is Cur ve

L (1/F) 5.6 10 6 5.6 10 6

K (1/F) 8.36 10 5 0.0n 0.33 0.0m e 1.85 1.5m b 1.85 1.5

Fig. 4Thermal compaction curve-example data.

31

32

33

34

35

36

0 100 200 300 400 500 600

Temperature, o F

P o r o s

i t y , %

Raw data

Nonlinear elastic model



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general compaction resulting from pressure reduction and localthermally induced expansion around steam injectors. In relativeterms scaled to the maximum compaction in the injector locationsof Fig. 7, the case of Fig. 7 shows general compaction of 0.08 to0.1 units, while the case of Fig. 8 shows general compaction of 0.04 to 0.06 and heave of 0.06 to 0.09 around the injectors.

Gas Field. The model described here has been used in develop-ment planning of a large offshore gas-condensate field. The modelhas been used directly to forecast both reservoir compaction andseafloor subsidence. The work to date has shown the importance of obtaining correct laboratory data, as well as detailed simulation of

the reservoir surroundings.While the complexity of the model does not allow detaileddescription here, selected results are shown to illustrate these points.

Lab Data Match. Owing to considerable reservoir heterogeneity,lab samples were obtained and tested for a wide range of porosi-ties. The results were grouped for modeling into three materialtypes based on permeability ranges. An example of the stress/strainand volumetric-strain match is shown in Fig. 9.

Example of Predictions. The reservoir model consists of a52 12 12 block grid with heterogeneous properties and consid-erable structure. This model represents one of the fault blocks inthe field. Initially, a stress model was built with the same52 12 12 areal grid, assuming free deformations at the top of thereservoir. This is the base-case scenario.

A more rigorous model was created by extending the finite ele-ment (FEM) grid for the stress solution above and below the reser-voir grid, modeling compaction transferred to the seafloor as wellas any rebound below the reservoir. The resulting grid consisted of 52 12 18 elements. This model (both the reservoir and FEMgrids) is shown in Fig. 10. Finally, a model was extended to theflanks of the reservoir (parallel to the faulting), resulting in a60 12 18 grid.

The fault block is produced by six vertical wells completed oversix layers so that the reservoir is depleted from the initial pressure of 3,400 psi to approximately 1,800 psi after 20 years of production.

The time history of predicted compaction from these three

cases [at the areal location (27,3) for the sum of all reservoir lay-ers] is shown in Fig. 11, normalized to the base case. It can be seenthat the compaction is not sensitive to the inclusion of the over-burden in this case. This is because of the soft properties of theoverburden; in cases in which the reservoir is surrounded by hard-er rock, significant arching may occur. To demonstrate this, anoth-er case was run with stiffer overburden, which is also shown in Fig.11 and results in decreased compaction.

The 20-year simulation took 176 minutes for the base case, 183minutes for the overburden case, and 255 minutes for the overbur-den+flanks case, on a 450 MHz Pentium with 500 MB of randomaccess memory (RAM). This compares with 13 minutes for anuncoupled run with the reservoir model only. Although the coupledmodeling requires an order of magnitude more time, the run times

Fig. 5Pressure distribution after 10 years.

INJ1

INJ10

INJ11

INJ12

INJ13

INJ14 INJ15

INJ16

INJ17

INJ2

INJ3

INJ4

INJ5

INJ6 INJ7

INJ8 INJ9

PROD1

PROD10 PROD11

PROD12 PROD13

PROD15

PROD16 PROD17

PROD18

PROD19

PROD2

PROD20

PROD21 PROD22

PROD23

PROD24

PROD25 PROD26

PROD27

PROD28 PROD29

PROD3 PROD30

PROD31

PROD32 PROD4 PROD5

PROD6

PROD7

PROD8 PROD9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0 2,000 4,000 6,000 0

2 , 0

0 0

4 , 0

0 0

6 , 0

0 0

1 2 3 4 5 6 7 89 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32

200 250 300 350 400 450 500 550 600 650 700 750

Layer 4, time = 3,650 days

TABLE 5RESERVOIR PROPERTIESEXAMPLE 1

Initial average reservoir pressure 559.5 psiDepth to reservoir top 1300 ftReservoir temperature 123FHorizontal stress gradient 0.665 psia/ftVertical stress gradient 0.95 psia/ftGrid size: 32 32 5

Layer K h (md) K v (md) z (ft) h (psia) z (psia)1. Shale .001 .001 0.15 25 334.8 708.92. Reservoir 1,300 130 0.28 37.5 341.6 724.63. Reservoir 800 80 0.34 12.5 347.2 737.3

4. Reservoir 1,200 120 0.25 18.75 351.6 746.25. Shale .001 .001 0.15 31.25 357.2 758.9



7/9

(under 5 hours) are manageable and show that full-field coupledsimulations are quite feasible.

Conclusions1. A coupled thermal reservoir/geomechanical model with modu-

lar coupling and elastic or elastoplastic rock behavior wasdeveloped and applied to several field studies. The system isnow sufficiently advanced so that geomechanical modeling canbe used for reservoir planning in the same manner as conven-tional reservoir simulators.

2. The coupling method developed offers different degrees of cou-pling from independent flow and stress solutions to a fully cou-pled system, with corresponding differences in run times. Thetimestep explicit coupling provides the best compromise betweencomputing efficiency and accuracy for compaction problems.

3. Correct representation of the stress/strain (constitutive) modelof the skeleton is the key for realistic predictions. A nonlinearelastic model with hysteresis can be used to approximate anelastoplastic model for modeling compaction at a fraction of itscomputer requirements.

Fig. 6Temperature distribution after 10 years.

INJ1

INJ10

INJ11

INJ12

INJ13

INJ14 INJ15

INJ16

INJ17

INJ2

INJ3

INJ4

INJ5

INJ6 INJ7

INJ8 INJ9

PROD1

PROD10 PROD11

PROD12 PROD13

PROD15

PROD16 PROD17

PROD18

PROD19

PROD2

PROD20

PROD21 PROD22

PROD23

PROD24

PROD25

PROD26

PROD27

PROD28 PROD29

PROD3 PROD30

PROD31

PROD32 PROD4 PROD5

PROD6

PROD7

PROD8 PROD9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0 2,000 4,000 6,000 0

2 , 0

0 0

4 , 0

0 0

6 , 0

0 0

1

2 3 4 5 6 7 89 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500


Fig. 7Compaction distribution-case with thermal compactioneffects.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0 2,000 4,000 6,000 0

2 , 0

0 0

4 , 0

0 0

6 , 0

0 0

1

2 3 4 5 6 7 8910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32


Fig. 8Compaction distribution-case without thermal com-paction effects.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0 2,000 4,000 6,000

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0 0

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0 0

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2

5

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32




8/9

4. Thermally induced compaction has been formulated and imple-mented in the model. This effect is potentially important inthermal projects.

5. Proper modeling of reservoir surroundings in terms of stresscan be important for the accuracy of compaction and/or subsi-

dence predictions.Nomenclature

a coefficient in the cap yield surface B bulk modulus

Bm tangential bulk moduluscb bulk compressibilityc R conventional rock compressibilityc R

I equivalent c R in unconstrained systemc R

II equivalent c R in laterally constrained system[D] flow accumulation matrix[E ] flow equations matrix

e1, e2 exponents in the generalized Youngs modulusequation (hyperbolic model)

E Youngs modulus E i Initial Youngs modulus (Eq. 27) E t tangential Youngs modulus

F cap yield surface functionF cone yield surface function

F vector of force/displacement boundary conditionsG shear modulus I 1 1st stress invariant

J 1, J 2, 2nd and 3rd stress invariants[K ] stiffness matrixK b constant for bulk modulus (hyperbolic model)K e constant for Youngs modulus (hyperbolic model)

[L ] coupling matrixme thermal hardening exponent for modulus

mb thermal hardening exponent for bulk modulusne ,n b exponents for E i and Bm (hyperbolic model) p pore-fluid pressure

pa atmospheric pressure (scaling constant)P vector of flow unknownsQ vector of source terms (wells)

[R ] residual of the flow equations R right side of the flow equations (Eq. 1) R f failure ratio (hyperbolic model)sij total deviatoric stress components, sij ij ij

[T ] transmissibility matrixT temperature

T ini initial reservoir temperatureT ref reference temperature

u x,y,z displacements X intersect of the cap surface with I 1 axis L coefficient of linear thermal expansion Biot constant, 1 cs / cb

comp thermal compaction coefficient p plastic parameter, Eq. 14

t change over time interval (timestep) h avg change of average horizontal stress pavg change of average reservoir pressure

vector of displacement unknownsij Kronecker deltaij strain componentsv volumetric strain

vp volumetric plastic strain

Poissons ratiofriction angle for shear failure

d deviatoric stress, 1 3

Fig. 9Example of stress/strain lab data match.

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

0 0.01 0.02 0.03 0.04 0.05

Uniaxial strain (=volumetric strain)

D e v

i a t o r a n

d M e a n

E f f e c

t i v e

S t r e s s ,

p s

i a Observed

Calculated

Deviator Mean

Fig. 10Reservoir and FEM grid for the gas-field-compaction example.



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ij total stress components pre preconsolidation stress

r reference stress

SuperscriptsT transpose (matrix)

Subscriptsmax maximum deviator stress

ult ultimate deviator stress

AcknowledgmentsThe authors want to acknowledge a number of people who con-tributed, directly or indirectly, to this work, particularly WashWawrzynek of FAC, Alda Behie, Peter Puchyr, and VladimirZhitomirsky. The plasticity model development was also support-ed by Imperial Oil Research Ltd.

References1. Settari, A. and Mourits, F.M.: A Coupled Reservoir and

Geomechanical Simulation System, SPEJ (September 1998) 219.2. Lewis, R.W. and Schrefler, B.A.: The Finite Element Method in the

Deformation and Consolidation of Porous Media, John Wiley & Sons,New York City (1987) 1344.

3. Vaziri, H.H.: Nonlinear Temperature and Consolidation Analysis of Gassy Soils, PhD dissertation, U. of British Columbia, Vancouver,British Columbia, Canada (1986).

4. Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, AppliedScience Publishers, New York City (1979) 1476.

5. Chin, L.Y. and Boade, R.R.: Full-Field, 3-D Finite ElementSubsidence Model for Ekofisk, 1990 North Sea Chalk Symposium,Copenhagen, Denmark, 1112 June.

6. Fredrich, J.T. et al.: Three-Dimensional Geomechanical Simulation of Reservoir Compaction and Implications for Well Failures in theBelridge Diatomite, paper SPE 36698 presented at the 1996 SPE

Annual Technical Conference and Exhibition, Denver, Colorado,69 October.7. Sulak, R.M., Thomas, L.K., and Boade, R.R.: 3D Reservoir

Simulation of Ekofisk Compaction Drive, JPT (October 1991) 1272;Trans., AIME, 291.

8. Settari, A. and Price, H.S.: Simulation of Hydraulic Fracturing inLow-Permeability Reservoirs, SPEJ (April 1984) 141.

9. Koutsabeloulis, N.C., Heffer, K.J., and Wong, S.: Numerical geome-chanics in reservoir engineering, Computational Methods and

Advances in Geomechanics, Siriwardane and Zeman (eds.), Balkema,Rotterdam, The Netherlands (1994) 2097.

10. Settari, A. and Mourits, F.M.: Coupling of geomechanics and reservoirsimulation models, Computational Methods and Advances in

Geomechanics, Siriwardane and Zeman (eds.), Balkema, Rotterdam,The Netherlands (1994) 2151.

11. Gutierrez, M. and Lewis, R.W.: The Role of Geomechanics inReservoir Simulation, paper SPE 47392 presented at the 1998SPE/ISRM Eurock, Trondheim, Norway, 810 July.

12. Koutsabeloulis, N.C. and Hope, S.A.: Coupled Stress/Fluid/ThermalMulti-Phase Reservoir Simulation Studies Incorporating Rock Mechanics, paper SPE 47393 presented at the 1998 SPE/ISRMEurock, Trondheim, Norway, 810 July.

13. Geertsma, J.: Problems of rock mechanics in petroleum productionengineering, Proc., First Congress of the Intl. Soc. of Rock Mechanics, Lisbon, Portugal (1966) 1, 585594.

14. Geertsma, J.: A basic theory of subsidence due to reservoir com-paction: the homogeneous case, Verhandelingen Kon. Ned. Geol. Mijnbouwk. Gen. (1973) 28, 43.

15. Geertsma, J. and van Opstal, G.: A numerical technique for predicting sub-sidence above compacting reservoirs, based on the nucleus of strain con-cept, Verhandelingen Kon. Ned. Geol. Mijnbouwk. Gen. (1973) 28, 63.

16. Finol, A. and Farouq Ali, S.M.: Numerical Simulation of OilProduction With Simultaneous Ground Subsidence, SPEJ (October1975) 411; Trans., AIME, 259.

17. Merle, H.A. et al.: The Bachaquero StudyA Composite Analysis of the Behavior of a Compaction Drive/Solution Gas Drive Reservoir,

JPT (September 1976) 1107; Trans., AIME, 261.18. Rattia, A.J. and Farouq Ali, S.M.: Effect of Formation Compaction on

Steam Injection Response, paper SPE 10323 presented at the 1981SPE Annual Technical Conference and Exhibition, San Antonio, Texas,57 October.

19. Chase, C.A. Jr. and Dietrich, J.K.: Compaction Within the SouthBelridge Diatomite, SPERE (November 1989) 422.

20. Bruno,M.S.:Subsidence-InducedWellFailure, SPEDE (June1992)148.21. Smith, R.J.: Geomechanical Effects of Cyclic Steam Stimulation on

Casing Integrity, MS thesis, U. of Calgary, Calgary (1997).22. Duncan, J.M. and Chang, C.Y.: Nonlinear Analysis of Stress and

Strains in Soils, J. Soil Mechanics and Foundation Division, ASCE(1970) 96 (SM5) 1629.

23. Settari, A. et al.: Geotechnical Aspects of Recovery Processes in OilSands, Cdn. Geotech. J. (1993) 30, 22.

24. Colazas, X.C.: Subsidence, compaction of sediments and effects of water injection, Wilmington and Long Beach offshore fields, MS the-sis, U. of Southern California, Los Angeles (1971).

25. Coussy, O.: Mechanics of Porous Continua, John Wiley & Sons,Chichester, U.K. (1995).

SI Metric Conversion Factorsbbl 1.589 873 E 01 m3

ft 3.048* E 01 mF (F 32)/1.8 Cpsi 6.894 757 E 00 kPa

*Conversion factor is exact.

A. (Tony) Settari is president of Taurus Reservoir Solutions Ltd., apetroleum and geomechanics engineering firm based inCalgary, and holds the Petroleum Engineering chair at the U.of Calgary. e-mail: [email protected]. He is a leadingexpert in the area of reservoir engineering and computer sim-ulation of petroleum reservoirs, and in the analysis of fracturingand geomechanical processes in reservoirs. He has beeninvolved in a wide range of simulation applications, includingnaturally fractured reservoirs, enhanced recovery projects,hydraulic fracturing and acidizing, in-situ thermal processes inoil sands, perforation mechanics, and geomechanics. Settariholds a BS degree from the Technical U. of Brno,Czechoslovakia and a PhD degree in mechanical engineer-ing from the U. of Calgary. Dale Walters is a senior engineerwith Taurus Reservoir Solutions Ltd. in Calgary. e-mail:[email protected] He has been involved in a wide rangeof simulation applications, including reservoir compaction,hydraulic fracturing, thermal processes in oil sands, andenhanced recovery projects. Walters holds BS and MS degreesfrom the U. of Calgary, both in civil engineering.

SPEJ

Fig. 11Total reservoir compaction predicted with differentstress-boundary conditions.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000

Time, days

N o r m a

l i z e

d R e s e r v o

i r C o m p a c

t i o n Base case

OverburdenOver + sideburdenStiffer overburden


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Advances in Coupled Geomechanical