A Coupled Flow-Geomechanics Model for Fluid and ... - OnePetro

ARMA 11-213A Coupled Flow-Geomechanics Model for Fluid and Heat Flow forEnhanced Geothermal ReservoirsPerapon Fakcharoenphol and Yu-Shu WuColorado School of Mines, Golden, CO, US.Copyright 2011 ARMA, American Rock Mechanics AssociationThis paper was prepared for presentation at the 45 th US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, June 26–29,2011.This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review ofthe paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, ormembers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMAis prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. Theabstract must contain conspicuous acknowledgement of where and by whom the paper was presented.ABSTRACT: This paper presents a numerical model for simulating coupled fluid flow, heat transfer, and geomechanics inenhanced geothermal systems (EGS). The developed simulator is built on the TOUGH2 [1], a well-established simulator for geohydrologicalanalysis with multiphase, multi-component fluid and heat flow [2]. The key capability for an EGS simulator is how tohandle fluid and heat flow in such multi-scale fractured systems. In the model, we represent different-scaled fractures or fracturedzones using hybrid fracture model. In handling the effect of rock deformation, we use the simplified model of the uniaxial strain.The numerical scheme is verified against the analytical solutions of the classical one-dimensional consolidation problem and onedimensionalheat conduction in deformable rock column. To demonstrate the capability of our model, a closed loop circulation ofan EGS reservoir with one injector and one producer is simulated. The results show that thermal-induced stresses are morepronounced than pressure-induced stresses, especially, close to the injector well. While the vertical stress is relatively constantthroughout the simulation run, the horizontal stresses reduce significantly. As a result, the stress changes may exceed rockstrength, creating new fractures or reactivating healed natural fractures.1. INTRODUCTIONMost geothermal reservoirs are situated in igneous andmetamorphic rocks that have low matrix permeability.Well-connected fractures provide high permeable fluidflowpath to un-fractured tight rock matrix. Under anexternal stimulus, the contrast of rates ofthermodynamics changes between high permeablefractures and tight matrix rock causes largethermodynamics variation between the two media. Inenhanced geothermal reservoirs, natural cracks andfractures are typically scarce. Artificial fractures orhydraulic fractures are required to provide highpermeable paths for fluid and heat flow to wells.Mechanical deformation process in porous and fracturedrock and their coupling to thermal and hydrologicalprocesses are important in enhanced geothermal systems(EGS) [3]. Stresses and strains changes due to externaldisturbances, e.g. cold-water injection, causesdeformation in rock frame. This deformation, in turn,alters hydraulic properties, such as porosity andpermeability, and affects fluid and heat flow. Thus, tosimulate this complex system, it is important to: 1)include the effect of stresses/strain changes on hydraulicproperties, especially fractured rock, and 2) account formulti-scaled fractures.2. MATHEMATICAL MODEL2.1. TOUGH2 mathematical frameworkThe general framework mathematical and numericalmodel solves mass and energy balance equations ofdescribing fluid and heat flow in general multiphase,multi-component systems. Fluid flow is described with amultiphase extension of Darcy’s law; in addition, thereis diffusive mass transport in all phases. Heat flow isgoverned by conduction and convection, also includingsensible as well as latent heat effects. Following [1], themass and heat balance equations in every model subdomainor REV of an EGS can be written in the form:d κκκ∫ M dVn= ∫ F • ndΓn+ ∫ q dVn(1)dtVnΓnκ= 1, ..., NK (total number of components) and n=1,…,NEL (total number of gridblocks).The integration in Eq. (1) is over an arbitrary subdomainor a control volume, V n , of the flow system under study,which is bounded by the closed surface Γ n . The quantityM appearing in the accumulation term (left hand side)Vn

epresents mass or energy per unit volume, F denotesmass or heat flux (see below), and q denotes sinks andsources. n is a normal vector on surface element dΓ n ,pointing inward into V n .F nmA nmwhere, x i,p represents the value of i th primary variable atthe p th iteration step.In evaluation of the terms in Eq. (2), mass accumulation,flux, source and sink must be calculated at each Newtoniteration step. The general form of the massaccumulation term is:∑κκM = φSβρβXβ(4)κ = 1, ..., NK, and β=1, …NPH (total number of phases).Fig. 1. Space discretization and flow-term determination in theintegral finite difference method [1].For numerical simulation the continuous space and timevariables must be discretized using the integral finitedifference method (IFDM, [4]), as shown in Fig. (1)IFDM avoids any reference to a global system ofcoordinates, and thus offers the advantage of beingapplicable to regular or irregular discretization in one,two, and three dimensions of porous and fracturedmedia. The IFDM also makes it possible, by means ofsimple preprocessing of geometric data, to implementdouble- and multiple-porosity methods for modelingflow in fractured media. Time will be discretized fullyimplicitly as a first-order backward finite difference.Time and space discretization of Eq. (1) results in a setof coupled non-linear equations, which can be written inresidual form as follows [1]:Rκn( xt + 1∆t− {Vn) = M∑mAnmκnF( xκnmt + 1( x) − Mt + 1κnn( x) + V qt)κ , t + 1nwhere, the vector x t consists of primary variables at timeκt, Rnis the residual of component κ for grid block n, Mdenotes mass or thermal energy per unit volume for acomponent, V n is the volume of the block n, q denotessinks and sources of mass or energy, ∆tdenotes thecurrent time step size, t+1 denotes the time level oft+ ∆ t , A nm is the interface area between neighboringblocks n and m, and F nm is the “flow” term (includingfluid flow, heat transfer, and advective and diffusivemass transport) between them. Eq. (2) is solved byNewton/Raphson iteration, leading to}(2)κ,t+1∂Rn κ,t+1− ∑ (xi,p+1− xi,p) = Rn(xi,p) (3)∂xiipwhere, φ is porosity,ρ βis density of phase β,Sβis theκsaturation of phase β, and Xβis the mass fraction ofcomponent κ in phase β. Before the calculation of massaccumulation, the parameters on the right hand side ofEq. (4) are calculated as a function of primary variables,which are directly solved from Eq. (3) for previousNewton iteration step, or from initial conditions at firstiteration.The heat accumulation term includes contributions fromthe rock matrix, gaseous and fluid phases and is given byequation:Mκ∑= ( 1−φ)ρ C T + φ S ρ u (5)κ = NK+1 (NK+heat component) and β=1,…, NPH.RHere ρRandC Rare grain density and specific heat of thehost rock respectively, T is temperature, and u βisspecific internal energy in phase β.The mass fluxes of aqueous and gaseous phases aredetermined by a multiphase version of Darcy’s law,written in the form:Fβb krβρ β= −k0 (1 + ) ( ∇Pβ− ρ β g)P µββ=1, ..., NMPH (total number of mobile phases).βRββββ(6)where, k 0 is absolute permeability, b is the Klinkenbergfactor [5] for gas slippage effect (b=0 when β=aqueousphase), krβis relative permeability to phase β, µ βisviscosity, Pβis pressure in the β phase, and g is thevector of gravitational acceleration.The diffusive fluxes are evaluated by the formulationprovided:κκ κJ = −φτρ d ∇X(7)βββββ

κwhere, dβis the molecular diffusion coefficient forcomponent κ in phase β, τ βis the tortuosity which is afunction of rock property and phase saturation, andX is mass fraction of component κ in phase β.κβstresses and strains. No additional primary variable isrequired. Thus, the computational workload required forcoupled geomechanics to a flow simulator is minimal.A reservoirThe heat flux term accounts for conduction, advectionand radiation heat transfer, and is given by:Fκ= −[(1−φ)K+ fσσ ∇T0R4+ φ+∑∑ββ = 1,2,3hββ = 1,2SFβKβ] ∇T(8)(a)where, KRis thermal conductivity of the rock, K βisthermal conductivity of phase β, T is temperature, hβisspecific enthalpy of phase β, fσis radiant emittancefactor, and σ0is the Stefan-BoltzmannA reservoir2.2. coupled geomechanicsA fully coupled flow-geomechanics model inheritsexpensive computation workload caused by additionalthree primary variables. For example, in a two-phaseblack-oil system, two primary variables (pressure and,saturation) are required. Coupling geomechanics-flowinvolves five primary variables and simultaneous solvingthis model can cause more than quadruple incomputational time of the flow model. Many researchers[2, 6, 7], thus, focus on sequential-coupling technique toreduce computational time. However, the sequentialtechnique does not guarantee mass or energyconservation. Iteration between flow and geomechanicsis required to minimize the mass balance error.We formulated a boundary-based simplified coupledflow-geomechanics model which simultaneously solvingfluid, heat flow and geomechanics equations. Thus, massand energy conservations are guaranteed. A typicalcoupled flow-geomechanics model assumes the undeformableboundary at the lateral and bottom modelboundaries, and the constant stress boundary at the topof the model, see Fig. 2a. Rock deformations between anelement and its neighboring elements follow forcesbalance equations and a stress-strain relationship. In ourformulation, we eliminate force interaction of eachelement and impose boundary conditions, used formodel outer boundaries, on each element interface. Forexample, we can impose the uniaxial strain boundary,un-deformable boundary at the lateral and bottominterfaces and constant stress boundary at the top, toevery element, see Fig. 2b. Using this formulation, wecan directly calculate pressure and temperature induced(b)Fig. 2. Mechanical boundary condition of (a) a typical coupledflow-geomechanics model and (b) the proposed coupled flowgeomechanicsmodel.Under the thermo-poro-elastic and the small strainassumptions, pressure and temperature induced stressesand strains can be expressed as follows:(i)Uniaxial strain: un-deformable boundary at thelateral and bottom interfaces, and constant stressboundary at the top interface of an element⎛1−2ν⎞ ⎛ E ⎞∆σh= ⎜ ⎟αP∆P+ ⎜ ⎟αT∆T⎝ 1−ν⎠ ⎝1−ν⎠(9)ν ⎛ E ⎞∆σ' h= − αP∆P+ ⎜ ⎟αT∆T1−ν⎝1−ν⎠(10)σ = 0(11)∆ v∆σ ' = −α∆P(12)vP( 1 + ν )( 1 − 2ν)E( 1 −ν)1 + νεv= − αT∆T− αP∆P(13)1 −ν

(ii)(iii)where,∆ σh, ∆σvare total stress changes inhorizontal and vertical directions, respectively,∆ σ ' h, ∆ σ ' vare effective stress changes inhorizontal and vertical directions, ∆ P is pressurechange, ∆ T is temperature change, ν is Poissonratio, E is Young modulus, αPis Biot constant,αTis linear thermal expansion, and εvis volumemetric strain, where positive sign indicatesexpansion and negative sign indicates contraction.Un-deformable boundary: un-deformableboundary at all interfaces of an elementE∆σh= ∆σv= αP∆P+ αT∆T1−2ν(14)E∆σ'h= ∆σ'v= αT∆T(15)1−2νεv= 0(16)Constant stress boundary: constant stress boundaryat all interfaces of an element∆σ = ∆σ= 0(17)hv∆σ' = ∆σ' = −α∆P(18)hvP( 1− 2ν)εv= −αT∆T−αP∆P(19)E2.3. Stress dependent hydraulic propertiesIt has been recognized for over half a decade thatpermeability of sand stone cores is reduced by anapplied, external stress [9]. Under a reservoir condition,fluid production and injection causes stress changes anddeformation in a reservoir. Rock porosity andpermeability are altered as the consequence. Manyresearch efforts have been made to quantify thesechanges. Number of stress/strain induced porosity andpermeability change correlations are available inliterature [10 – 14]. The selected numerical schememake possible to incorporate these correlations as userselectable options. The general form of porosity andpermeability change is as follows:φ = φ( σ ', ε )(20)k = k( σ ', ε )(21)where, φ is porosity, k is permeability, σ ' is effectivestress, and ε is strain.Capillary pressure can be scaled according to theLeverett J-function [2].pc=pc0⎛⎜⎜⎝k ⎞0/ φ0⎟k / φ ⎟⎠(22)where, k0is reference permeability, φ0is referenceporosity, pcis capillary pressure, and pc0is capillarypressure at reference porosity and permeability.Emphasis should be made for the hydraulic properties ofthe fractured medium. Rutqvist et al. [2] developed anempirical correlation for fracture hydraulic propertieschanges caused by stress alteration. They correlatedfracture width using an exponentially decaying pathbetween zero-effective-stress fracture width andresidual-high-stress fracture width. Porosity andpermeability, then, can be calculated from the computedfracture width.' '[ d( σ −σ)]w = wi+ ( w0 −wr) expn ni(23)where, w is fracture width,wi, w0,zero-stress, and residue fracture width,w are the initial,rσ , σ arenormal effective stress and the initial normal stress, andd is experimental fitting curve.w1+ w2+ w3φ = φi(24)w + w + w1iwhere, φiis the initial porosity, w is fracture width ofthe fracture set 1, 2, and 3 where they are parallel to x, y,and z direction, subscript i indicates the initial condition.k13232i2i3333i3i'n'niw + w= k1i(24)w + wwhere, k 1 iis the initial permeability of direction 1, w 2 ,w 3 are fracture width of the fracture set normal todirection 1.2.4. Hybrid fracture modelingA typical EGS reservoir required massive hydraulicstimulation to achieve economic production [9]. Thefracture created by the stimulation can be considerablylarge. As such, it raises the challenge on how to modelthe system where there co-exist multi-scale fractures, alarge hydraulic fracture and small pre-existing naturalfractures. Two modeling approaches, explicitdiscretization and conceptualization, such as dualporosity,dual-permeability, and multi-interactioncontinuum (MINC) are discussed in literatures [15 – 19].The former approach deals with actual fracture geometry

and models them accordingly. It is well suited formodeling a hydraulic fracture where the fracturegeometry is known. The conceptual approach, on theother hand, captures heat and fluid flows betweenfracture and matrix media using idealized fracturematrixnetworks. It is appropriate for modeling naturalfractures where the geologic detail is complex andavailable data is rather incomplete [16].The integral finite difference method discussed in theprevious section, makes possible to implement allfractured modeling approaches aforementioned in thesame framework. Each fractured modeling can be treatedas a geometric preprocessing input while allcomputational procedures are all the same. The “flow”term ( F ) in Eq. (6) for multiphase flow is described byβa discrete version of Darcy’s law. This is the mass fluxof fluid phase β along the connection of element i and j,given byFβ, i j= λβ, ij+ 1/ 2γi j( ψβ j−ψβ i) (25)where λ β,i j+1/2 is the mobility term to phase β, defined as⎛⎛ ⎞, + 1/ 21⎟ ⎟ ⎞⎜⎜b ρβkrβλ + ⎟β i j(26)⎜⎝⎝Pβ⎠ µβ ⎠=ij+1/ 2and subscript ij+1/2 denotes a proper averaging orweighting of properties at the interface between twoelements i and j, γijis transmissivity calculated as:whereγi jijAijk=d + dii j+1/ 2j(27)A is the common interface area betweenconnected blocks or nodes i and j; d i is the distance fromthe center of block i to the interface between blocks i andj; and k ij+1/2 is an averaged (such as harmonic weighted)absolute permeability along the connection betweenelements i and j.The flow potential term in Equation (25) is defined asψ = − ρ g D(28)β iPβ i β,ij+1/ 2where D i is the depth to the center of block i from areference datum.To implement all fractured modeling approaches, specialattention is needed to calculate fracture-matrix masstransfer where a quasi-steady state is assumed, e.g. inDouble-porosity and Dual-permeability. The flowbetween fractures and matrix is still evaluated using Eq.(25); however, the transmissibility for the fracturematrixflow is given byiAFMkγij=lFMM(29)where A FM is the total interface area between fracturesand matrix of element i and j (one of them is a fractureand the other is a matrix blocks); k M is a matrix absolutepermeability; and l FM is a characteristic distancerepresenting an equivalent length of average matrixblock pressure for a quasi-steady state flow. It can becalculated for idealized 1-D, 2-D and 3-D dimensionalrectangular matrix blocks when using the doubleporositymodel [4, 17].Fig. 3 shows the example model for an EGS reservoir.The hydraulic fracture is modeled by discrete fractureapproach and preexisting natural fractures are capturedby the multi-continuum conceptualization.WellDiscrete fractureMINC , Dual-porosity,or dual-permeabilitySingle porosityFig. 3. An EGS model where a hydraulic fracture is modeledby discrete fracture approach and preexisting natural fracturesare captured by the multi-continuum modeling approach.3. MODEL VERIFICATIONSTo verify our numerical model two classical problemswere simulated and the results were compared toanalytical solution.3.1. One-dimensional heat conduction in thedeformable columnThe problem is heat conduction a solid rock column thatundergoes uniaxial strain in the vertical direction only.The column is subjected to a low temperature boundaryon the top, see Fig. 6. Only heat conduction is assumed.The cooling effect caused contraction. As heatconduction is a slow process, the semi-infinite space

solution is used to compare with the numerical results.Table 1 shows detail input parameters.Fig. 6a and 6b show the comparison between numericalresults and analytical solution. The excellent match isachieved which lend creditability to our mathematicalformulation.Constant temperature (T b ) at 10 °CT i =60 °CTable 1. Input parameters for one-dimensional consolidationproblemParameter Value UnitModel 1 ×1×100Grid size ∆x = 1, ∆y = 1, m.∆z = 0.5Heat conductivity (k T ) 2.34 W/m°KHeat capacity of rock (c s ) 690 J/kg°KRock density (ρ s ) 2,550 Kg/m 3Linear thermal expansion (α T ) 1.5×10 -6 K -1Young modulus (E) 14.4×10 9 PaPoisson ratio(ν) 0.20Fig. 6. Configuration of one-dimensional heat conduction indeformable media problem0.01.6E-03Depth, m10.020.030.040.0t=1.0e6 sec.t=1.0e7 sec.t=1.0e8 sec.Analytical solutionNumerical ResultsVertical Displacement, m1.4E-031.2E-031.0E-038.0E-046.0E-044.0E-042.0E-04Analytical SolutionNumerical Results50.00.0 20.0 40.0 60.0Temperature, °C(a)0.0E+001.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09Time, sec(b)Fig. 6. The comparisons between numerical and analytical solutions (a) temperature profile and (b) the displacement at the top ofthe column.3.2. One-dimensional consolidationThe problem concerns a porous permeable column thatundergoes uniaxial strain in the vertical direction only.The column is subjected to a constant load on the top,the fluid boundary pressure is set to zero gauge rightafter the load is imposed, and only vertical displacementtakes place, see Fig. 4. The numerical simulation wasinitialized at the undrained condition, as all subsurfacereservoirs is in the condition when they are discoveredand our numerical model cannot capture forceinteraction and deformation of each element interface ofthe undrained process. Thus, only drained process withstresses and strains changes is simulated. Detail inputparameters can be found in Table 2. The results arecompared with Jaeger's analytical solution [20]. Theanalytical solution can also be found in [21]. Thepressure profiles and the deformation at the of rockcolumn are shown in Fig 5a and 5b, respectively. Themodel can capture pressure change in various times andthe top model deformation profile with time.

σ ex =14.2 MPaFig. 4. Configuration of one-dimensional consolidationproblem.Table 2. Input parameters for one-dimensional consolidationproblemParameter Value UnitModel 1 ×1×100Grid size ∆x = 1, ∆y = 1, m.∆z = 0.5Porosity (φ) 0.19Permeability (k) 10 -13 m 2Rock compressibility (c φ ) 4.4×10 -10 Pa -1Fluid Density (ρ) 1000 Kg/m 3Fluid Viscosity (µ) 0.89×10 -3 Pa.sFluid compressibility (c f ) 6.0×10 -10 Pa -1Biot coefficient (α P ) 0.8Young modulus (E) 14.4×10 9 PaPoisson ratio(ν) 0.20Initial pressure (P i ) 3×10 6 Pa0.00.006Normalized Depth0.20.40.6t=3000 sect=1000 sect=100 secVertical Displacement, m0.0050.0040.0030.0020.8AnalyticalsolutionNumerical Results0.001Analytical SolutionNumerical Results1.00.0 0.2 0.4 0.6 0.8 1.0Normalized Excess Pressure(a)0.0001.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05Time, sec(b)Fig. 5.The comparisons between numerical and analytical solutions (a) pressure profile and (b) the displacement at the top of thecolumn.4. EXAMPLE APPLICATIONAn example geothermal reservoir model was run toshow capability of our model. A pair of injectionand production wells is placed in a deep EGSbasement reservoir. Three large fractures/faults inhorizontal plain are encountered from both wells.Hydraulic stimulation was performed. The seismicdata indicate micro-seismic activity in large volume[22].A three-dimensional model is created to simulateeffect of cool water injection. The model consists ofthree types of rock, 1) tight matrix block wheresingle-porosity is used, 2) large fracture zonemodeled by discrete fracture model, and 3) crushedzone where small fractures exist, but the no fracturegeometry data is available, thus dual-porosity modelis used, see Fig.7.After 10 years of circulation, temperature aroundthe injector is reduced significantly where aspressure change is minimal see Fig. 8. This coolingeffect cause contraction of rock and the effectivehorizontal stress is decreased substantially. On theother hand, the vertical stress is relatively constantthroughout the run. The Mohr-circle plot shows inFig. 9.indicated that the stress state is approachingrock failure line as time goes by. Eventually thestress changes may exceed rock strength, creatingnew fractures or reactivating healed naturalfractures.

InjectorProducerTight matrixCrushed zoneLarge fractureCrushed zone0.6 kmθ=7°3.0 kmFig. 7. Configuration of reservoir model in Cooper basin where one cold water injector and one producer in the reservoir.T (°C) : T i =250 °Ca)60 100 140 180 220P (Pa) : P i =7.40 E+7b)7.51E+077.535E+07σ‘ H (Pa) : σ‘ H,i =4.88 E+8c)2.8E+08 3.8E+08 4.8E+08σ‘ v (Pa) : σ‘ v,i =6.65 E+8d)6.6452E+086.6466E+08ε v (dimensionless)e)0.001 0.004 0.007 0.01Fig. 9. Simulation results after 10 years of 20 kg/s injection and production using Uniaxial-strain assumption: a) temperature, b)pressure, c) effective horizontal stress, d) effective vertical stress, and d) volume metric strain.

Shear stressσ' 3,t2 σ' 3,t1 σ' 3,t0Effective normal stressσ' 1Fig. 10. General stress change in Mohr-Coulomb plot5. CONCLUSSIONS• We developed a coupled flow-geomechanicsmodel based on simplified assumptions wherestresses and strains can be directly calculatedfrom pressure and temperature changes, noadditional primary variable is required innumerical solutions. Thus, minimal additionalcomputational workload is needed to solvecoupled flow-geomechanics comparing to that offlow simulator.• The numerical scheme is successfully verifiedagainst analytical solutions of one-dimensionalconsolidation and one-dimensional heatconduction in deformable media.• Hybrid fracture model is used to capture multiscalefractures in an EGS reservoir. Hydraulicfractures can be modeled using discretemodeling approach whereas multi-continuumfracture model can apply to pre-existing naturalfractures.• In an EGS reservoir, temperature induced stressis much more pronounce than that of pressureinduced stresses, especially at vicinity to a coldwaterinjection well. The stress change cancause rock failure and consequently, create orreactivate new fractures.ACKNOWLEDGMENTThis work is supported by the U.S. Department ofEnergy under Contract No. DE-EE0002762,“Development of Advanced Thermal-Hydrological-Mechanical-Chemical (THMC) Modeling Capabilitiesfor Enhanced Geothermal Systems“REFERENCES1. Pruess, K, C. Oldenburg, and G. Moridis, 1999,TOUGH2 user’s guide, V2.0. Lawrence BerkeleyNational Laboratory Report LBNL-43134, Berkeley,CA.2. Rutqvist, J., Y. S. Wu, C. F. Tsang, and G. Bodvarsson,2002, A modeling approach for analysis of coupledmultiphases fluid flow heat transfer, and deformation infractured porous rock., Int. J. Rock Mech. & Min. Sci.,39: 429-442.3. Liu, H. H., J. Rutqvist, J. G. Berryman, 2009, On therelationship between stress and elastic strain for porousand fractured rock, Int. J. Rock Mech. & Min. Sci.,46:289-296.4. Narasimhan, T., and P. Witherspoon, 1976, Anintegrated finite difference method for analyzing fluidflow in porous media, Water Resour. Res., 12(1): 57-64.5. Klinkenberg, L. J., 1941, The permeability of porousmedia to liquids and gases, Drilling and ProductionPractice, American Petroleum Inst.,:200–213.6. Settari, A. and F. M. Mouritis, 1998, A coupledreservoir and Geomechanical simulation system, SPEJournal, Sep:219-226, SPE 50939.

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