GEOPHYSICS, VOL. 69, NO. 1 (JANUARY-FEBRUARY 2004); P. 200–211, 13 FIGS., 3 TABLES.10.1190/1.1649388

Coupled geomechanics and flow simulation for time-lapse seismic modeling

Susan E. Minkoff∗, C. Mike Stone‡, Steve Bryant∗∗, and Malgorzata Peszynska§

ABSTRACT

To accurately predict production in compactible reser-voirs, we must use coupled models of fluid flow and me-chanical deformation. Staggered-in-time loose couplingof flow and deformation via a high-level numerical inter-face that repeatedly calls first flow and then mechanics al-lows us to leverage the decades of work put into individ-ual flow and mechanics simulators while still capturingrealistic coupled physics. These two processes are oftennaturally modeled using different time stepping schemesand different spatial grids—flow should only model thereservoir, whereas mechanics requires a grid that extendsto the earth’s surface for overburden loading and may ex-tend further than the reservoir in the lateral directions.Although spatial and temporal variability between flowand mechanics can be difficult to accommodate with fullcoupling, it is easily handled via loose coupling. We cal-

culate the total stress by adding pore pressures to theeffective rock stress. In turn, changes in volume straininduce updates to porosity and permeability and, hence,dynamically alter the flow solution during simulation.Incorporating the resulting time-dependent pressures,saturations, and porosities (from coupled flow and me-chanics) into Gassmann’s equations results in seismicwave velocities and densities that can differ markedlyfrom those calculated from flow alone. In a synthetic nu-merical experiment based on Belridge field, California,incorporation of coupled flow and mechanical defor-mation into time-lapse calculations produces compres-sional wave velocities that differ markedly from thoseproduced by flow alone. In fact, it is the closing of thepores themselves (reduction in permeability) in this ex-ample which has the greatest impact on fluid pressuresand saturations and, hence, elastic wave parameters suchas velocity.

INTRODUCTION

Over the last decade, time-lapse seismic modeling has gainedtremendous ground as a technology of interest to companieswishing to explore producing reservoirs for trapped oil and gas.The basic methodology involves imaging a section of the sub-surface at periodic intervals and differencing these collecteddata sets to try to analyze the state of the field. As pointedout by Nur (1989), traditional production methods may miss70–90% of the fluids in a field. Therefore, to increase oil andgas reserves, we must explore carefully selected enhanced re-covery techniques. Time-lapse seismic imaging can highlightpockets of trapped oil and gas, but perhaps more significantly,it allows us to improve our overall understanding of the keyreservoir parameters—porosity, permeability, saturation and

Manuscript received by the Editor September 13, 2002; revised manuscript received May 16, 2003.∗University of Maryland, Baltimore County, Department of Mathematics and Statistics, 1000 Hilltop Circle, Baltimore, Maryland 21250. E-mail:sminkoff@math.umbc.edu.‡Sandia National Laboratories, Computational Solid Mechanics and Structural Dynamics Department 9142, Mail Stop 0847, Albuquerque, NewMexico 87185. E-mail: cmstone@sandia.gov.∗∗University of Texas at Austin, Center for Subsurface Modeling, Institute for Computational and Engineering Sciences (ICES), Austin, Texas78712. E-mail: sbryant@ices.utexas.edu.§Formerly University of Texas at Austin, Center for Subsurface Modeling, Institute for Computational and Engineering Sciences (ICES), Austin,Texas 78712; presently Oregon State University, Department of Mathematics, Corvallis, Oregon 97331-4605. E-mail: mpesz@math.orst.edu.c© 2004 Society of Exploration Geophysicists. All rights reserved.

temperature—which are used to assess the feasibility of differ-ent production scenarios.

Time-lapse seismic studies require careful collection andprocessing of the data [see Lumley et al. (1997) and Rickettand Lumley, (2001)]. In fact, interesting field studies are under-way to assess the usability of data recovered from permanentlydeployed sensors [see, for example, papers by Beasley et al.(1997) and Ebrom et al. (1998)]. Numerical time-lapse studiescarried out on computers can provide inexpensive but valu-able information about the visibility of subsurface features inthe face of inconsistencies in source and recording equipment,data acquisition geometry, and other types of noise.

One area in which time-lapse seismic analysis is just begin-ning to be used is in the modeling of compactible (or weak-formation) reservoirs. Despite the fact that these reservoirs

200

Flow and Mechanics for 4D Seismic 201

introduce a host of new data processing and modeling diffi-culties, such fields often have ideal characteristics for applyingtime-lapse analysis. As pointed out by Lumley et al. (1997), cer-tain reservoir conditions act as indicators that time-lapse anal-ysis is likely to be successful. These conditions include shallowreservoirs with low dry bulk modulus (high rock compressibil-ity) and high porosity—exactly the conditions which are foundin weak-formation fields. In one of the first papers to includeboth flow parameters and mechanical stress in time-lapse cal-culations, Olden et al. (2001) model a 2D gas-water reservoirusing a flow simulator. They calculate changes to elastic rockproperties from changes to pressure and saturation via fluidsubstitution and then condition the rock properties a secondtime (using empirical rock physics relationships derived fromlab measurements) to incorporate stress changes. They indicatethat their modeling could be improved by developing softwarethat incorporates the different types of data in a more straight-forward fashion. Pennington et al. (2001) describe a case studyin which faulting in the reservoir gives unexpected time-lapseresults. Xu and Nur (2001) integrate reservoir simulation andsatellite imaging to detect subsidence and to continuously mon-itor hydrocarbon production. Their model is again 2D and as-sumes single-phase flow. Their aim is to better estimate reser-voir permeability via continuous radar monitoring. Vidal et al.(2002) monitor an underground gas-storage reservoir. Theyuse flow simulation to estimate pore pressures and satura-tions, and then geomechanical stresses are calculated from thepore pressures. Again, they use empirical rock physics models(Hertz-Mindlin theory) to modify seismic velocity based onstress changes. Finally, Guilbot and Smith (2002) develop a setof equations that allow them to calculate changes to overbur-den and reservoir velocities from compaction. They apply theirmethod to data from the Ekofisk field in the North Sea.

In this paper, we describe a very different type of study.We have coupled together a 3D finite-element multiphase flowcode with an equally complex 3D finite-element mechanicaldeformation code via a staggered-in-time algorithm. Thesetwo codes were chosen primarily because they model a va-riety of realistic production environments. The two simulatorsare loosely coupled via an interface which allows two-way pas-sage of information between the simulators (pressures to me-chanics, updated porosity and permeability to flow). In thisway, we are able to leverage the substantial development effortwhich has gone into the two codes (Minkoff et al., 2003). More-over, our algorithm serves as a prototype for coupling any twosuch independently-developed codes. Output pressures, satu-rations, and porosities from the coupled code are then fed di-rectly into Gassmann’s equations to calculate changes to elasticrock properties for seismic time-lapse analysis. Preliminary re-sults from this project were described in the paper by Minkoffet al. (1999). In that work, the authors analyzed a synthetic ex-periment based on a single layer of Belridge field, California.Porosity changed dynamically during flow simulation as a re-sult of stress effects. Permeability was held fixed. Time lapseanalysis indicated that flow and mechanics coupled togethercan impact seismic properties.

In the remainder of this paper, we describe the reservoirflow simulator, the mechanical deformation code, the couplingalgorithm, and the modifications made to the two simulatorsin order to link them together. We detail how time-lapse seis-mic modeling of compactible reservoirs differs from 4D anal-

ysis of fields in which the reservoir rock is assumed fixed. Fi-nally we describe two realistic numerical experiments based onBelridge field. The first experiment is a complex multilayer ex-ample which closely models materials found in the extensivediatomite reservoir of Belridge field. For this numerical ex-periment, porosity changes dynamically due to stress effects.The second numerical experiment we describe focuses on asingle layer of Belridge diatomite material, but both porosityand permeability change dynamically during coupled simula-tion. These numerical experiments clearly illustrate that seis-mic time-lapse rock properties determined from coupled flowand mechanical deformation modeling can differ markedlyfrom those derived from flow alone.

DESCRIPTION OF SIMULATORS

Reservoir simulator

The fluid flow simulator, IPARS (Integrated Parallel Accu-rate Reservoir Simulator) (Parashar et al., 1997; Wang et al.,1997; Wheeler et al., 1999) built at the University of Texas,is a subsurface simulator which includes multiple physical flowmodels. The IPARS simulator was chosen for this work becauseof its versatility and sophistication. As an example, IPARS isuniquely able to follow geologic structure (especially faults)via blocking, making it a good choice for a complex flow andmechanics simulator. [For further details on the multiblocktechnique in IPARS, see Arbogast et al. (1996), Yotov (1998),Peszynska et al. (1999), Wheeler et al. (2000), Lu et al. (2001),and Peszynska et al. (2002).]

In this paper, we focus on one of the flow models in IPARS—the black oil model—which is the simplest model that includesall three fluid phases (oil, gas, and water) needed for seismictime-lapse studies (see Lumley, 1995). We make the assump-tion here that the reservoir is isothermal and that the perme-ability tensor is diagonal. Although fluid viscosity generallydepends on pressure, in this paper we assume it is constantbecause it varies so little over the pressure range seen in ournumerical experiments.

In this work, we use the default no-flow Neumann boundaryconditions applied on the external boundary. Numerically, theblack oil model code is fully implicit, 3D, and uses an expandedmixed finite-element method to maintain local conservation ofmass (Lu, 2000; Lu et al., 2001).

Black oil formulation.—We assume that up to three phasesand three components are present. The uppercase subscriptsW, O, and G are used for the fluid components: water, heavyhydrocarbon or oil, and light hydrocarbon or gas, respectively.Lowercase subscripts w, o, and g are used for the fluid phases.The model is partially miscible. [For references to the black oilmodel, see Peaceman (1977) and Lake (1989)].

Given that NI is the stock tank volume of component I perunit pore volume, φ is porosity, Ro is the stock tank volume ofgas dissolved in a stock tank volume of oil, and qI is the totalstock tank rate of injection of component I , the mass balanceequations are

∂

∂t(φNG) = −∇ · (Ug + RoUo)+ qG,

∂

∂t(φNI ) = −∇ ·Ui + qI I = oil , water. (1)

202 Minkoff et al.

Darcy’s Law gives the mass velocity U of phase i (here i = oil,gas, or water) as

Ui = −Kkri

Biµi· (∇Pi − ρi g∇D). (2)

Here, K is the absolute permeability tensor, and kri is the rela-tive permeability of phase i . Bi is the formation volume factorfor phase i . Pi is pressure, µi is viscosity, and ρi is density.Gravity has magnitude g, and D is depth. Finally, the satura-tions must satisfy the constraint

Sg + So + Sw = 1, (3)

and capillary pressures are defined by

Pcow(Sw) = Po − Pw, Pcgo(Sg) = Pg − Po. (4)

The gas component is soluble in the oil phase. The maximumconcentration of gas that can be dissolved in the oil phase ata given pressure is given by Rs. Reducing the pressure of avolume of gas-saturated oil causes gas to come out of solu-tion and form a free gas phase. Thus two-phase conditions andthree-phase conditions may prevail simultaneously in differentregions of the same reservoir.

To complete the formulation, we must specify the state equa-tions. We denote the stock tank density of component I by ρI S.Then, the water, gas, and oil phase densities are given, respec-tively, by

ρw = ρW S/Bw,

ρg = ρGS/Bg,

ρo = (ρOS+ RoρGS)/Bo. (5)

Gas and oil phase saturations are defined by

Sg = Bg(NG − RsNO),

So = BoNO, (6)

and the water phase saturation is defined by the following twoequations:

Sw = BwNw, Sw = 1− So − Sg. (7)

Geomechanical deformation modeling

The geomechanics code from Sandia National Laboratories,JAS3D, is a 3D, quasi-static finite-element code. Like IPARSfor fluid flow, JAS3D is a sophisticated mechanical deformationcode which has several unique capabilities. Specifically, JAS3Dcan accurately model large deformations, sliding contact sur-faces, and both elastic and inelastic material responses from awide array of constitutive models (Arguello et al., 1998). Thefinite element technology which is used in JAS3D is based oniterative quasi-static solvers which allow problems with largenumbers of unknowns to be efficiently solved. Two iterativetechniques are currently used in JAS3D: a preconditioned con-jugate gradient technique (Biffle, 1993) and an adaptive dy-namic relaxation technique (Stone, 1997).

We consider the following field equation governing the de-formation of a body occupying a volume V :

∂σi j /∂xj + ρbi = 0 on V. (8)

Equation (8) is the quasi-static equation of motion, where σis the Cauchy stress tensor, x is the position vector, ρ is themass per unit volume, and b is a specific body force vector. Thesolution to Equation (8) is sought subject to the kinematic andtraction boundary conditions:

ui (x, t) = Ui (x, t) on SU , (9)

σi j n j = si (x, t) on ST , (10)

where SU represents the portion of the boundary on whichkinematic quantities are specified (i.e., displacements ui ), nj

is a unit normal vector, and ST represents the portion of theboundary on which tractions are specified. The boundary ofthe body is given by the union of SU and ST .

MODIFICATIONS TO FLUID FLOW

Traditional flow simulators initialize porosity and permeabil-ity at the start of the computation and leave them essentiallyfixed throughout the simulation. To dynamically incorporateoutput from mechanics into flow during simulation, quantitiesinvolving porosity and permeability need to be updated eachtime the code completes a JAS3D step.

In traditional flow simulators, small changes to reservoirporosityφ are accounted for by the following linear expression:

φ = φ∗(1+ cr (P − P∗)), (11)

where φ∗ is the initial porosity at initial pressure P∗, P is cur-rent pore pressure, and the rock compressibility cr is a constant(typically on the order of 10−7 to 10−10 Pa−1). Such small valuesfor cr accurately model linear elastic deformation but are notable to accommodate more complex models (i.e., nonlinearityor inelasticity). In our coupled simulator, we do not rely onequation (11) to determine changes to porosity from deforma-tion. Porosity values at the beginning of each set of simulationtime steps are calculated for the flow simulator (IPARS) frommechanics (JAS3D), as depicted in Figure 1. We are able totake advantage of the range of complex constitutive relationsdescribing different materials available in JAS3D. (The numer-ical experiments described in this paper use a plastic cap model,for instance.) The nonlinear system resulting from the finite-element discretization of equations (1) and (2) is solved viaNewton’s method. Porosity updates at time step n+ 1 (frommechanics) are directly incorporated into the Newton methodresidual for the flow equations. Mass is conserved, and theNewton update accounts for a decrease in pore space (for ex-ample) by increasing the well rates and pressures at the nexttime step.

Nonetheless, initial numerical experiments indicate that theflow solver may have difficulty accommodating large jumpsin reservoir parameters produced by infrequent calls to themechanics code. (Recall that one of the advantages of loosecoupling is that displacements, stresses, and strains need notbe recalculated each time a flow step is taken as in full cou-pling.) We found that using an estimate of rock compressibil-ity as a preconditioner to Newton’s method helps the flowsolver reach convergence after a return from the mechanicscode. An approximate compressibility value is incorporatedinto the Jacobian calculation for the Newton system to guidethe flow solver in the direction of porosity changes with pres-sure (Minkoff et al., 2003). In the second numerical experiment

Flow and Mechanics for 4D Seismic 203

described in this paper (single layer Belridge), we improveupon this idea. Equation (11), used in traditional flow simu-lators to update porosity, relates the current porosity to thechange in fluid pressure from the initial state. In order to im-prove the computational performance of the flow simulator be-tween geomechanics updates (explicit calls to JAS3D), we esti-mate small changes in porosity coming from pressure changes.We recast equation (11) into incremental form. The expressionfor porosity at time step n becomes

φn = φm(1+ cr (Pn − Pm)), (12)

where the superscript m denotes porosity calculated from themost recent output of the mechanics code (JAS3D) at pressurePm.

Dynamic updates to absolute permeability during flow sim-ulation impact large portions of the code. The values of trans-missibility are generally assumed fixed in uncoupled flow sim-ulators. In our code, both the transmissibilities and the wellmodel must be updated after each mechanics step to accountfor the permeability changes induced by strain. [For more de-tail on these modifications to the fluid flow code, see Minkoffet al. (2003)].

MODIFICATIONS TO MECHANICS

The mechanics code is capable of calculating point-wisechanges in pore volume in the reservoir, coming (for example)from field production, pressure decrease, and then compaction.To calculate these field property changes in the coupled code,the reservoir simulator sends the pore pressure field to the ge-

FIG. 1. Flow chart for loose coupling algorithm.

omechanics part of the code where it is used in the calculationof total stress:

σ Ti j = σi j + pδi j . (13)

Here, p the pore fluid pressure, δi j is the Kronecker delta func-tion, and σi j is the “effective” stress which is used in the con-stitutive model. The total stress is used in the determination ofthe equilibrium state for the reservoir subject to the overbur-den loads, kinematic boundary conditions, and changing porepressure field. On output, the geromechanics fcode provides anupdated porosity φ at the current time step via the followingexpression:

φ = 1− (1− φ0)eεv

, (14)

where φ0 is the initial porosity, and εv is the total volume strain(the sum of the elastic and inelastic components of strain). Wenote that although JAS3D can handle thermal strains gener-ated in the rock (temperature changes could be computed bythe flow simulator and then passed to JAS3D in a similar fash-ion as the pore pressures), in this work we assume isothermalconditions.

For this work, we have chosen to use an exponential modelrelating permeability to volume strain (although any otherfunctional form for permeability could be substituted into thecode). This model was based on lab measurements made ofpermeability versus differential stress for several different rocktypes (A. Fossum, personal communication, 1999):

K = AeBεv . (15)

204 Minkoff et al.

Here A and B are material constants and εv is again the totalvolume strain. This empirical relationship assumes permeabil-ity changes will be isotropic.

THE LOOSE COUPLING ALGORITHM

Various algorithms have been investigated for coupling to-gether two or more distinct physical simulators. These algo-rithms range from the tightest coupling of physics (full cou-pling) to the weakest (one-way coupling). In this work, we wishto model realistic physics while making use of the two sophis-ticated simulators described earlier in this paper (IPARS andJAS3D). Our algorithm passes information between the twosimulators via a high-level interface. Related two-way loosecoupling algorithms have been investigated by Fung et al (1994)Settari and Mourits (1994), and Settari and Walters (1999).

In our algorithm, the user designates a mechanics time stepand a separate flow time step. (We will call the time intervalbetween mechanics time steps1t , where1t1= t1− t0; a typicalvalue of 1t being anywhere from one month to one or twoyears.) The flow simulator runs first and will likely break the1t1 time interval into multiple flow time steps (typically onthe order of a fraction of a day to a few days). At the end of thetime interval1t1, the pore pressure is passed to the mechanicscode, and the mechanics code then runs the simulation for thatsame (prior) 1t1 time interval. The geomechanics code maytake only one time step for this time interval (or at least it willlikely take different subtime steps relative to the flow simulatortime step sequence). The mechanics calculation is quasi-static,nonlinear, and time independent. In these calculations, we haveno time-dependent material response such as creep. However,there may be a time lag in subsidence relative to productionchanges due to the flow properties of the rock. For example,we might stop production of the well, but subsidence mightcontinue to occur as reservoir pressure gradients equilibratewith the overburden.

The mechanics code uses the pore pressures to calculatestrains and ultimately updates to porosity and permeability.Using the updated values of the flow parameters from me-chanics, the coupled code then simulates flow for the next timeinterval 1t2= t2− t1. The two-way staggered-in-time couplingalgorithm proceeds until the both simulators reach the finaltime tend. The loose-coupling algorithm is shown as a flow chartin Figure 1.

Between the flow and mechanics simulations in Figure 1 liesa column devoted to mapping output and input quantities fromone simulator to the other. This column is necessary becausethe loose coupling algorithm does not require the two simu-lators to use the same computational grid. (In fact, this is anadvantage of loose coupling!) The flow domain is typically asubset of the mechanics domain. The flow simulator will onlymodel the reservoir, whereas the mechanics code may need toextend further in the lateral directions than the reservoir, willsurely need to extend up to the earth’s surface (for overbur-den loading), and may cover an area below the reservoir aswell. Note that fully coupled simulators typically assume a sin-gle computational domain requiring unnecessary calculationson the reservoir side. The current algorithm also does not re-quire the two codes to have the same grid spacing in the partsof the domain where there is overlap (the reservoir). Physi-cally, there may be a sound reson why the spatial discretization

on the flow side will differ from that for mechanics (exam-ples: multiple material layers in the reservoir or a need forfiner resolution to capture complex physics). We have chosenhere to use a software package from Sandia’s engineering suite,MAPVAR (Wellman, 1999) for the upscaling. Different spa-tial domains for flow and mechanics are sent to MAPVAR,and variables are mapped on the overlapping portions of thesedomains. MAPVAR uses either nodal averaging, constrainedleast squares, or direct transfer for data interpolation. In thiswork, we are using constrained least squares, which is the mostaccurate of the three options because it is able to capturegradients in the solution. The interpolation within MAPVARcan be done on either undeformed or deformed geometries.Computation time for the data transfer is insignificant com-pared to the computation times for the flow and geomechanicscodes.

TIME-LAPSE SEISMIC MODELING

Time-lapse seismic modeling fundamentally relies on be-ing able to incorporate reservoir (fluid flow) effects into elas-tic wave propagation. We use Gassmann’s equations to con-vert the time-dependent reservoir pressures and saturationsresulting from the flow simulation (or flow simulation and ge-omechanics) into elastic rock parameter changes (Gassmann,1951). Gassmann’s equations provide a way to compute theeffective bulk and shear moduli (K and µ, respectively) ofreservoir rock saturated with a given composition of pore flu-ids from the elastic moduli of the dry rock. These equationsare

µsat = µdry, (16)

Ksat

Ksolid− Ksat= Kdry

Ksolid− Kdry+ K f luid

φ(Ksolid− K f luid). (17)

Gassmann’s equations require knowledge of (1) the effectivebulk modulus of the pore fluid K f luid , (2) the porosity φ, (3) thebulk and shear moduli of the dry rock with empty pores Kdry,µdry, and (4) the bulk modulus of the mineral material makingup the rock Ksolid.

In the examples described in this paper, we assume a fixedoil and water phase bulk modulus but used a modified idealgas law to calculate the gas modulus [see Dake (1978), Lumley(1995)]. The modified ideal gas law includes a “Z factor” tocorrect for the difference between reservoir and atmosphericpressures. The total fluid bulk modulus is then calculated as

1K f luid

= SwKw

+ So

Ko+ Sg

Kg. (18)

The dry moduli are determined entirely by rock physics mea-surements made on core samples at differing differential pres-sures (where differential pressure is the difference betweenoverburden and pore pressures). To calculate the total fluiddensity, we calculate the individual phase densities via equa-tion (5) and then take a saturation-weighted average of thethree phase densities as follows:

ρ f luid = Swρw + Soρo + Sgρg. (19)

Flow and Mechanics for 4D Seismic 205

Finally, from Gassmann’s equations and relationships be-tween elastic parameters, we can determine the density ρ, com-pressional wave velocity Vp, and shear wave velocity Vs at agiven fluid saturation, pressure, and porosity:

ρsat = (1− φ)ρsolid + φρ f luid , (20)

V psat =

√√√√√Ksat+ 43µsat

ρsat, (21)

V ssat =õsat

ρsat. (22)

There are a number of places where output from coupledflow and mechanics used as input to Gassmann’s equationswill strongly influence the results. When pore fluid pressuresare used as input loads for the mechanics calculation, new val-ues of reservoir porosity and permeability resulting from stresschanges in turn affect future pressure and saturation calcula-tions from flow. Items (1) and (3) above (K f luid and Kdry, µdry)are affected by pressure changes [as is item (2) porosity indi-rectly via geomechanics]. The fluid modulus is also affected bysaturations. Porosity [item (2)] is directly updated in the defor-mation code and will, therefore, also differ from flow simulationalone. In fact, one traditionally assumes porosity is constantwhen applying Gassmann’s equations. In our case, porosity willchange (dramatically in places) with time. The level of impactcoupled simulation can have on seismic properties is illustratedin the Belridge field examples described below.

NUMERICAL EXAMPLES

The examples we describe here are based on data fromBelridge field. Belridge is located west of Bakersfield,California, and was the fifth most productive field in the U.S.in 1996. In fact, the field was estimated to have more than5× 108 m3 of original oil-in-place. The field contains two reser-voirs: a small reservoir located in the shallow Tulare sand, anda slightly deeper reservoir located in diatomite, which extendsnearly 305 m in depth. Despite the fact that this large diatomitereservoir was discovered in the early 1900s, by the mid 1990s,most of the production was still focused on the Tulare reser-voir (Fredrich et al., 1996). The diatomite has unusual geologicproperties which make production difficult. This reservoir hasa large pore space (porosities range from 45% to 70%), butvery low permeability (∼0.1 md). In the mid 1970s, hydraulicfracturing (and hence the introduction of new flow paths) al-lowed production in this field for the first time. However, withincreased production came substantial subsidence (up to 6 mat the earth’s surface) and hence well failures, which by themid 1990s amounted to a well failure rate of 2–5% annually(Fredrich et al., 1996).

Minkoff et al. (1999) describe a synthetic experiment basedon Belridge field which focuses on a single layer of diatomitematerial (layer J). In that paper, porosity was modified dynam-ically during flow simulation, but permeability remained fixed.In this paper, we describe two related numerical experiments.The first is a more complete Belridge simulation in which wemodel all 18 material layers (in depth) which describe the ac-tual reservoir. Again, we assume permeability does not changeduring simulation but modify porosity based on the output of

the mechanical stress calculation. The second example is a sin-gle layer problem focused on the weakest layer of diatomite(layer J). The computational domains for mechanics and flowused in the multilayer numerical experiment and the relativeposition of layer J in the overall reservoir are shown in Figure 2.In the single layer synthetic experiment presented in this pa-per, both permeability and porosity change substantially dur-ing flow simulation as a result of mechanical compaction. Wenote that diatomite is an unusual material with a very largepore space that dramatically deforms due to changes in pres-sure and fluid saturation. While we do not claim that stud-ies done on diatomite produce typical oil field behavior, wedo feel the numerical experiments illustrate the capabilitiesof the coupled code for fields in which large displacementsoccur.

Multilayer Belridge experiment

The first example we describe closely follows the geologicstructure of Belridge field in depth and field properties. Thereservoir is assumed to be rectangular with four productionwells in the corners of the domain. Each well extends to aslightly different depth in the reservoir. The computational do-main corresponds to a single section of the total reservoir (inthe horizontal directions), and hence is unusual because it ex-tends noticeably further in z than in x and y. The reservoir do-main is broken into 21 grid blocks in x and in y, and 22 blocks inz. Grid spacing is fixed at 5.03 m in each of the lateral directions,but varies in z. The top of the reservoir (located at a depth of210 m from the surface) has an initial pore pressure of 2.28 MPa.Initial water saturation is 31.4% with minimal free gas of 5%.Porosity is initially uniform throughout the domain at 50%,and permeability is heterogeneous and anisotropic with valuesranging from 0.002 to 0.1385 md. Permeability (based on fieldmeasurements) is consistently ten times greater in the horizon-tal directions than in depth. We simulate 10 years of coupled

FIG. 2. Simulation domains for the multilayer Belridge fieldsynthetic experiment. Leftmost figure is the computational do-main for mechanical deformation. The figure on the right is thecomputational domain for reservoir simulation. Numerical ex-periment 2 focuses on layer J, which is labeled in the figure aswell.

206 Minkoff et al.

flow and mechanics. IPARS takes time steps of 0.2 to 10 days,and the JAS3D time steps occur every 30 days.

JAS3D contains more than 50 different constitutive relation-ships which can be used to simulate deformation of a rangeof materials. We use a modified Sandler-Rubin cap-plasticityconstitutive relationship (Sandler and Rubin, 1979) devel-oped specifically to model compaction of Belridge diatomite(Fossum and Fredrich, 2000). It provides a more realistic rep-resentation of the mechanical behavior of porous soil or rockthan the basic Drucker-Prager model (Drucker and Prager,1952) by incorporating lode-angle dependence of yield in thedeviatoric plane and nonassociativity in the meridional planeon the shear failure surface. The loading function is assumed tobe isotropic and to consist of two parts: an ultimate shear fail-ure surface which serves to limit the maximum shear stressesattainable by the material, and an elliptically shaped strain-hardening surface, or “cap,” that produces plastic volume andshear strains as it moves. This model is described more thor-oughly in papers by Fossum et al. (1995), Arguello et al. (1998),and Minkoff et al. (2003). In both the examples described in thispaper which involve diatomite, the vertical sides of the modelare constrained from movement normal to the boundary. Theprescribed in-situ stress state at the initial time is due to gravityloading and calculated from the density of the overlying mate-rials. The horizontal principal stresses (x and y) were calculatedby multiplying the vertical principal stress (z) by factors of 0.65and 1.20, respectively (Fredrich et al., 2000).

In the first numerical example, the JAS3D model contains18 different material layers corresponding to 12 different setsof material properties. Layer depths and material moduli usedfor the geomechanics calculation are given in Table 1. Only thediatomite reservoir is included in the flow grid. The mechanicsgrid extends outside the reservoir and includes the overlyingair sands and Tulare reservoir and underlying porcellanite. Infact, the flow domain was discretized into 9702 total grid blocks.In contrast, there were 37 548 total grid blocks used for themechanics calculation. All diatomite materials are modeledusing the cap-plasticity constitutive model. The more standardDrucker-Prager constitutive relationship is used for modelingstress and strain in the remaining materials.

In this example, the deformation portion of the coupled codeproduces dynamic changes to porosity, but permeability is heldfixed. At the start of simulation, the pore pressure decreasesdue to production are fairly uniform throughout the field (see

Table 1. Material layer properties used in Belridge experi-ment 1 for mechanical deformation calculation.

Layer Young’s In reservoirType of material depth (m) modulus (MPa) grid?

Air sands 46 124 noUpper Tulare 91 241 noLower Tulare 210 241 noDiatomite layer G 261 329 yesDiatomite layer H 311 434 yesDiatomite layer I 361 546 yesDiatomite layer J 404 414 yesDiatomite layer K 421 494 yesDiatomite layer L 479 775 yesDiatomite layer M 514 1121 yesDiatomite layer N 690 1121 yesPorcellanite 1158 469 no

Figure 3). But by year 7, most of the change in pore pressurehas accumulated in layer J, the portion of the reservoir whichundergoes the majority of compaction (see Figure 4). Layer Jis the weakest of the diatomite layers, with an elastic modulusthat is 15 times weaker than sandstone (Fredrich et al., 2000;J. Fredrich, personal communication, 1999). One can clearlysee in Figure 5 that by year 7, porosity has dropped from itsinitial value of 50% down to 48.5% in layer J. The pore volumedecrease is calculated from the change in strain as shown inFigure 6. Vertical displacement is approximately 3 m at theearth’s surface by year 7. The displacement profile is shownin Figure 7. The low permeability of the reservoir rocks and

FIG. 3. Pore pressure contours (in MPa) for the multilayer Bel-ridge field numerical experiment after one year of coupled flowand mechanical deformation modeling. Axis units shown arein meters.

FIG. 4. Pore pressure contours (in MPa) in the multilayerBelridge field reservoir after seven years of coupled flow andmechanical deformation modeling. Axes are in meters.

Flow and Mechanics for 4D Seismic 207

heavy oil in this field make movement of fluids to the wellsdifficult. However, to conserve mass while simulating flow in acompacting reservoir, the fluids must move out the wells. Thisphenomenon explains the large decrease in pore pressure seenin layer J (Figure 4).

In order to determine seismic rock property changes in thesimulated reservoir with time, we need the moduli of the soliddiatomite rock as well as rock physics curves which correlatechanges in modulus to pressure. For the solid matrix diatomite,we used modulus values for opal (Birch, 1966), namely thatKsolid is 15 457 MPa and µsolid is 18 000 MPa. The solid rockdensity ρsolid was taken to be 2.25 g/cm3. Pressure dependentdry rock values were taken from clean diatomite measurementsof the Lost Hills formation. These values were provided to us

FIG. 5. Porosity (dimensionless) after seven years of coupledflow and mechanical deformation modeling for multilayer Bel-ridge field numerical experiment. Axes are in meters.

FIG. 6. Inelastic strain (dimensionless) after seven years offlow/mechanics simulation for the multilayer Belridge field nu-merical experiment.

by Joanne Fredrich of Sandia National Labs (personal com-munication, 1999) and are shown in Table 2. Figure 8 showsthe time-lapse difference between the start of simulation andtime 5 years for the saturated rock density calculated from flowalone and from coupled flow and mechanics. In the coupledsimulation, changes to the density in layer J due to compactionare evident. The change in layer density is not evident fromthe time-lapse calculation based on flow alone. While the seis-mic velocities calculated via fluid substitution for this exampleshowed some time lapse effects, they are not as striking as thechanges in density.

Single-layer Belridge experiment

The second numerical example we describe is a syntheticexperiment in which we model only layer J of the diatomitereservoir (see Figure 2). Like the multilayer Belridge exam-ple described above, this example has four production wells inthe corners of the rectangular domain. The single-layer reser-voir extends 44 m in depth. For flow simulation, the grid blockspacing and number of points is the same in x and y as in themultilayer case. However, we now use 18 blocks in z, each of

Table 2. Diatomite modulus values used in rock physics anal-ysis for the first numerical experiment (multilayer Belridge syn-thetic example). Lost Hills data courtesy of J. Fredrich, SandiaNational Labs.

Pressure Modulus(MPa) (MPa)

Young’s modulus 1.4 5236.9 1060

Shear modulus 1.4 2176.9 374

FIG. 7. Vertical displacement (in meters) after seven years ofcoupled simulation for the multilayer Belridge field numericalexperiment.

208 Minkoff et al.

width 2.4 m (a finer spacing than for the multilayer case). Inthis numerical experiment, the mechanics grid contains onlyone material and extends from the reservoir up to the earth’ssurface for overburden loading. The top of the reservoir is lo-cated at a depth of 361 m, and the corresponding initial porepressure is 3.76 MPa. Initial fluid saturations are the same asin the multilayer case, and initial porosity is again taken to beuniform throughout the domain at 50%. However, for this nu-merical experiment we assume the permeability is isotropic andhomogeneous everywhere at 0.1 md at the start of simulation.The calculation simulated five years of reservoir productionwith IPARS time steps ranging from 0.5 days to over 100 days,and calls to JAS3D performed initially every five days and grad-ually increasing to once a year. While adaptive time steppingwas not used for the numerical experiments described in thispaper, we have recently tested a local error adaptive time-stepalgorithm (Gear, 1971) to help optimize the number of calls tomechanics for the loose coupling algorithm. We believe com-putational savings over full coupling will be substantial. [For acomparison of coupling methods, see Dean et al. (2003).]

Besides the obvious difference in the simulation domain, thisproblem differs from the previous multilayer case in that bothporosity and permeability change dynamically during simula-tion. As described in the section on modifications to fluid flow,large reservoir property changes produced by the mechanicscode can place a burden on the flow solver. To ensure improvedconvergence for flow, we estimate porosity changes from incre-mental rock compressibility at flow time steps intermediate toexplicit calls to JAS3D [explicit calls to JAS3D are shown inthe flow chart,(Figure 1, right column)].

We contrast the results of flow simulation (IPARS) aloneon this data with coupled flow simulation and mechanical de-

FIG. 8. Time-lapse change in saturated rock density (in kg/m3)between start and end of five years of simulation for the mul-tilayer Belridge field numerical experiment. Top figure showsdensity calculated from flow simulation alone. Bottom figurecorresponds to density changes calculated from coupled flowand mechanics. Axes are in meters.

formation (IPARS/JAS3D). At the end of the five-year flowsimulation the pressures had decreased by 40% from theirinitial values. At the end of the coupled flow and mechanicssimulation, the pressures had decreased by 50%. A maximumdifference in pressure between the two simulations occurs atthe production wells (about 0.69 MPa).

The total strain at the end of the run is 2.5%. Figure 9 showsvon Mises stress for the coupled simulation, and one can see3D variations in stress at the wells due to different well sched-ules and different well completion depths. After five years ofcoupled simulation, a maximum subsidence of 0.41 m is calcu-lated at the surface in the region of the wells. During coupledsimulation, porosity decreases by 4%, but the permeabilitiesshow the biggest change, decreasing from 0.1 to 0.001 md atthe wells (see Figure 10 for the final permeability at the end ofthe coupled run).

For the time-lapse calculations, we used the rock physicsdata shown in Table 3 (courtesy of Z. Wang, ChevronTexaco,personal communication, 2003). The bulk and shear moduliwere not directly measured but were backed out from othermeasured material properties. As shown in Tables 2 and 3,modulus values for diatomite can vary substantially from onelocation to another.

FIG. 9. Von Mises stress (in MPa) for the single-layer Belridgefield numerical experiment at the end of five years of coupledflow and mechanical deformation modeling.

FIG. 10. Permeability (in md) at the end of five years of coupledflow and mechanical deformation modeling of the single-layerBelridge field data. At the start of simulation, the permeabili-ties are 0.1 md uniformly throughout the reservoir.

Flow and Mechanics for 4D Seismic 209

Table 3. Diatomite modulus values used in rock physicsanalysis for second numerical experiment (single layer Bel-ridge synthetic example). Lost Hills data courtesy of Z. Wang,Chevron Texaco.

Pressure Modulus(MPa) (MPa)

Bulk modulus 1.9 15806.2 1070

Shear modulus 1.9 13306.2 1270

FIG. 11. Time-lapse calculation of the difference in saturatedrock compressional wave velocity (in m/s) between the startand end of five years of coupled flow and mechanical deforma-tion modeling for single-layer Belridge data.

FIG. 12. Input and output parameters for three types of physics discussed in this work: fluid flow, mechanicaldeformation, and elastic wave propagation.

In the first five years of coupled simulation, compressionalwave velocity at the wells was reduced by 350 m/s, primarilydue to gas coming out of solution (see Figure 11). Becausethe pore space is so large for the Belridge Field, seismic ve-locities tend to be very low. So this time-lapse velocity de-crease is substantial. Figure 12 lists the inputs, outputs, andtypical parameter units for fluid flow, mechanical deformation,and seismic modeling. We were surprised to find that the re-duction in permeability had such a noticeable impact on thecompressional wave velocities of the saturated reservoir rocks.Note that 1 darcy is equivilent to approximately 1× 10−12 m2

with typical seismic wavelengths being on the order of a fewkilometers/second. Yet when we compared three numerical ex-periments (flow alone, flow and mechanics where only porositychanges during simulation, and coupled flow and mechanicswhere both porosity and permeability change during simu-lation), it was the third numerical experiment which showedthe only substantial change in seismic wave velocity. Figure 13shows the difference in compressional wave velocity computedat the end of simulation contrasted for these numerical ex-periments. The permeability change causes a seismic wavevelocity discrepancy that would be detectable in a real fieldexperiment.

We emphasize again that diatomite is an unusual material(with a pore space of 45–70% in places). For these syntheticnumerical experiments, we assumed an initial porosity of 50%.Therefore, 50% of the time waves are traveling through flu-ids rather than rock. Sensitivity analyses show that some ofthe most striking changes we see in these numerical examplesare due to changes in fluid modulus, not movement from onepressure point to another on rock moduli curves. Whether oneuses the rock physics data in Table 2 or Table 3, the magnitudeof change in Vp is essentially the same due to the significantvariation in fluid saturation in this field with time. Couplingflow to mechanics can have a tremendous influence on reser-voir properties and hence on output pressures and saturations.

210 Minkoff et al.

FIG. 13. (Top) Difference in compressional wave velocity (inm/s) between flow simulation alone and coupled flow and me-chanical deformation modeling with only porosity changingdynamically. (Bottom) Difference in compressional wave ve-locity (in m/s) between flow simulation alone and coupled flowand mechanical deformation modeling with both porosity andpermeability changing dynamically during simulation. Bothfigures correspond to single-layer Belridge numerical exper-iments at five years.

Pressures and saturations coming from coupled flow and me-chanical deformation in turn affect seismic properties.

CONCLUSIONS

In this paper, we have described results of loosely couplingtogether two advanced 3D finite element simulators: IPARSfrom the University of Texas at Austin for flow and JAS3Dfrom Sandia National Labs for mechanics. A high-level inter-face calls first the reservoir simulator for a fixed set of time stepsand then the deformation code for that same (prior) set of timesteps. Pore pressures from flow are used in the total stress calcu-lation to determine mechanical property changes. The mechan-ics code then outputs updated flow parameters (porosity andpermeability) for the next set of time steps. The loose-couplingalgorithm is, therefore, staggered in time and includes two-waypassage of information. Mass is conserved despite updates toporosity during flow by forcing fluids out the wells. The two sim-ulators can take different time steps and have different spatialgrids both inside and outside the reservoir interval.

When this coupled flow and mechanics simulator is used asinput to a fluid substitution calculation, we can more accuratelymodel reservoir changes in compactible fields. In our time-lapse analyses, porosity values used in Gassmann’s equationsare no longer fixed in time. Pressure and saturation-dependent

fluid and dry rock moduli are also affected by the stress changesincorporated into flow.

We have described two synthetic numerical examples basedon data from the Belridge field diatomite reservoir. The first isa full-scale simulation (in depth) with 18 layers and 12 differentmaterials. The diatomite reservoir is modeled using a modifiedSandler-Rubin cap-plasticity constitutive model for nonrecov-erable deformation. In this example, porosity changes, but per-meability is held fixed. Strain accumulates in the weakest layerof the diatomite (layer J) causing 3–4 m of compaction at thesurface. A pore pressure gradient gradually localizes in thisweak layer as well. A fluid substitution calculation using thecoupled data is able to identify the change in saturated rockdensity in the reservoir at layer J. By contrast, fluid substitu-tion performed using flow simulation alone does not pick upthis change in rock density.

The second example includes only the weakest diatomitelayer of the reservoir. However, both porosity and permeabil-ity change dynamically due to stress. Permeability shows thebiggest change, dropping two orders of magnitude during thefive-year simulation. Seismic velocities are directly affected bythese flow property changes. In fact, permeability changes areshown to greatly impact compressional wave velocity despitethe vast difference in scale between these two rock properties.

We see little sensitivity in saturated rock parameters dueto the size of the mechanics time steps taken, indicating thatloose coupling is ideal for time-lapse seismic modeling of weak-formation reservoirs. In the future, we intend to incorporateadaptive time stepping based on the relative magnitude of dis-placement changes occurring in the reservoir. We also hope tovalidate our time-lapse calculations against well logs for realdata examples.

ACKNOWLEDGMENTS

We thank Mary Wheeler from UT Austin for her support ofthis project. We are indebted to Joanne Fredrich of Sandia Na-tional Labs and Zee Wang of ChevronTexaco, who contributedthe rock physics data for diatomite. The authors gratefully ac-knowledge support for this work from the U.S. Departmentof Energy’s Natural Gas and Oil Technology Partnership Pro-gram (NGOTP). Oil industry partners for this project includeBP, ChevronTexaco, ExxonMobil, Halliburton, and Schlum-berger. The second author is employed at Sandia NationalLaboratories. Sandia is a multiprogram laboratory operatedby Sandia Corporation, a Lockheed Martin Company, forthe United States Department of Energy under contract DE-ACO4-94AL85000.

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