ICES REPORT Convergence and Error Analysis of Fully Discrete Iterative Coupling Schemes for Coupling Flow with Geomechanics

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1 ICES REPORT 16-4 October 016 Convergence and Error Analysis of Fully Discrete Iterative Coupling Scemes for Coupling Flow wit Geomecanics by Tameem Almani, Kundan Kumar, Mary F. Weeler Te Institute for Computational Engineering and Sciences Te University of Texas at Austin Austin, Texas 7871 Reference: Tameem Almani, Kundan Kumar, Mary F. Weeler, "Convergence and Error Analysis of Fully Discrete Iterative Coupling Scemes for Coupling Flow wit Geomecanics," ICES REPORT 16-4, Te Institute for Computational Engineering and Sciences, Te University of Texas at Austin, October 016.

2 Convergence and Error Analysis of Fully Discrete Iterative Coupling Scemes for Coupling Flow wit Geomecanics T. Almani 1,3, K. Kumar, M. F. Weeler 1 1 Center for Subsurface Modeling, ICES, UT-Austin, Austin, TX 7871, USA Matematics Institute, University of Bergen, Norway 3 Saudi Arabian Oil Company Aramco, Saudi Arabia {tameem,mfw}@ices.utexas.edu, kundan.kumar@uib.no October 15, 016 Abstract We consider single rate and multirate iterative coupling scemes for te Biot system, based on te fixed-stress split coupling algoritm. In contrast to te single rate sceme, in wic bot flow and mecanics sare te same time step, te multirate sceme exploits te different time scales of te two problems by allowing for multiple finer time steps for flow witin one coarser mecanics time step. For te single rate sceme considered in tis work, we derive error estimates for quantifying te error between te solution obtained at any iterate and te true solution. Our approac is based on studying te equations satisfied by te difference of iterates and utilizing a Banac contraction argument to sow tat te corresponding sceme is a fixed point iteration. Obtained contraction results are ten used to derive teoretical convergence error estimates for te single rate iterative coupling sceme. Numerically, we first validate te efficiency of multirate scemes versus single rate scemes for field-scale problems. Second, we compare teoretically derived contraction estimates against numerical computations, and conclude tat teoretical estimates can predict te contracting beavior, and subsequently, te linear rate of convergence of te iterative sceme wit ig accuracy. Keywords. poroelasticity; Biot system; fixed-stress split iterative coupling; multirate sceme; contraction mapping; a priori error estimates 1

3 1 Introduction Due to oil extraction, especially in stress-sensitive reservoirs, rock compaction may occur inducing a subsidence event. Suc a subsidence migt not only affect te surrounding environment adversely, but can also result in a dramatic impact on reservoir production [34]. Examples of oils fields tat experienced subsidence events in te past, due to oil extraction activities, include te Valall field, located in te central graben of te Nort Sea [9], and te Willmington field, located near te soutern edge of te Los Angeles sedimentary basin [1]. Surface subsidence is just one example of an induced environmental penomenon tat necessitates te development of accurate, and reliable ways to model subsurface fluid flow coupled wit mecanical interactions. Oter examples include well stability, sand production, waste deposition, ydraulic fracturing, and CO sequestration [15], []. Classical tecniques of incorporating te effects of mecanical deformations on fluid flow in porous media involve expressing te porosity of te reservoir as a function of a pore compressibility factor. Tis is found to be insufficient for structurally weak and stress-sensitive reservoirs. Te accurate modeling of suc effects can not be accomplised witout solving te flow and mecanics equations in a coupled manner. Tis can be performed in tree different ways, known as te fully implicit or simultaneous coupling, te loose or explicit coupling, and te iterative coupling metods. Te first approac solves te coupled equations simultaneously, and results in te most accurate, or reference solution. Altoug it poses several computational callenges to te underlying linear solver, it exibits excellent stability properties [17]. Te explicit coupling approac, on te oter and, enjoys a lower computational cost compared to te aforementioned fully implicit approac. However, it is only conditionally stable. Te iterative coupling approac comes in between tese two extremes and iterates between te fully decoupled flow and mecanics equations until a certain convergence criterion, wit an acceptable tolerance, is acieved for a particular time step [8, 1,, 35]. From an implementation point of view, tis approac is easy to implement as it allows te use of existing reservoir simulators, wile maintaining a relatively fast convergence rate. We sall focus our attention on studying te error analysis of te iterative coupling approac in tis work. Te problem of coupling flow wit geomecanics as been extensively explored in te past. Tis can be tracked down to te work of [33] and [6,7]. Terzagi predicted te settlement of differesnt types of soils, wic lead to te creation of te te science of soil mecanics, followed by Biot wo extended Terzagi s work to te generalized teory of consolidation [7]. Several years later, [9] presented te general teory of termoporoelastoplasiticy for saturated materials. A compreensive treatment of te teory of poromecanics can be found in [10]. Oter nonlinear extensions of te teory of poroelasticity can be found in [9, 1, 31]. Te existence, uniqueness, and regularity of te Biot system ave been studied by a number of autors including [3], [5], and et al [14]. Moreover, [30] presented explicit and iterative coupling scemes for coupling flow wit geomecanics involving fracture propagation. Tis paper addresses te error analysis and contracting beavior of iteratively coupled flow and mecanics problems. It sould be noted tat te rigorous matematical analysis of te devised coupling scemes as received relatively less attention compared to te proposed linear and nonlinear

4 extensions currently available in literature. To te best of our knowledge, te first asymptotic error estimates for spatially discrete Galerkin approximations of te Biot s model were presented by [4]. Few years later, [13] considered finite difference metods for te Biot s model on staggered grids, derived stability estimates, and analyzed convergence for te discretized system. In a sequence of two papers, [6, 7] studied te continuous in time and fully-discrete Biot s model in wic mixed formulation is used for flow and continuous Galerkin is used for mecanics. A priori error estimates are derived in bot cases respectively. [11], on te oter and, derived a posteriori error estimates for te quasi-static Biot model, resulting in reliable error bounds wit all constants involved in te estimates are being specified. Suc error estimators can be used to perform adaptive simulations. Recently, [36] derived a priori error estimates for te quasi-static Biot model in wic flow is discretized by te multipoint flux mixed finite element metod, and elasticity uses continuous piecewise linear Galerkin finite elements. [8] considered finite element discretizations of te Biot s model based on MINI and stabilized P1-P1 elements, and derived error estimates of te fully discrete system accordingly. Te work of [0] considers a formulation of te Biot s system in four unknowns including pore pressure, fluid flux, stress tensor, and solid displacement, using a combination of two-mixed formulations for te flow and mecanics, and derived a priori error estimates of te fully coupled system accordingly. We note ere tat all previously derived error estimates consider simultaneous coupling of flow and mecanics. In tis work, we consider iterative coupling scemes instead, and drive error estimates for te fixedstress split iterative coupling sceme for te quasi-static Biot model. Te contracting beavior of bot scemes as been rigorously establised by [3]. In addition, [17,18] utilized von Neumann stability analysis tecniques to study te stability and convergence of oter iterative coupling scemes, including te fixed-strain and drained split tecniques. We sall consider bot te single rate and multirate iterative coupling scemes in tis work. Figures 1.1a and 1.1b demonstrate te coupling iteration in bot scemes respectively. In te single rate case, te flow and mecanics are solved for te same time step during eac flow-mecanics coupling iteration. In contrast, in te multirate case, during every iterative coupling iteration, te flow is solved for a number of fine time steps, ten mecanics is solved for a coarse time step. Te coupling iteration is continued until a certain convergence criterion is met, before advancing to te next coarse time step. We note ere tat te contracting beavior of te multirate iterative sceme for te Biot s system as been establised by [, 3] for te fixed-stress split coupling sceme, and by [5, 19] for te undrained-split coupling sceme. Moreover, te stability of te multirate explicit coupling sceme for te quasi-static Biot system as been establised by [4] under mild assumptions on material parameters. Te approac we follow in deriving our a priori error estimates utilizes previously establised results in a clever way, under te assumption tat te solution obtained by te iterative coupling sceme converges to te solution obtained by te simultaneously coupled sceme. Under suc assumption, te problem is simplified into estimating te error between te solution obtained by te iterative coupling sceme, and te one obtained by te simultaneously coupled sceme. In fact, we sow tat te former converges to te later geometrically by a Banac contraction argument. To te best of our knowledge, tis is te first rigorous derivation of a priori error estimates for te fixed-stress 3

5 coupling sceme for te Biot system. Te rest of te paper is structured as follows. Te model and discretizations are presented in Section 1. Banac fixed point contraction results for te single rate and multirate scemes are presented in Section. A priori error estimates for te fixed-stress split single rate iterative coupling sceme are derived in 3. Numerical results are presented in Section 4. We discuss te conclusions and future work in Section 5. Model equations & discretization Let be an open and connected bounded domain of IR d, were te dimension d = or 3, wit a Lipscitz continuous boundary. Let Γ denotes te part of te boundary wit positive measure. Wen d = 3, te boundary of Γ is also assumed to be Lipscitz continuous. Let Γ D denotes te part of te boundary associated wit Diriclet boundary conditions, and Γ N denotes te part associated wit Neumann boundary conditions, suc tat Γ D Γ N = Γ. Te model we study in tis work assumes a linear, omogeneous, and isotropic poro-elastic medium R d, wit a sligtly compressible viscous fluid saturating te reservoir. Te density of te fluid is a linear function of pressure, wit a constant viscosity µ f > 0. Te Lamé coefficients λ > 0 and G > 0, te Biot coefficient α, te reference density of te fluid ρ f > 0, and te pore volume ϕ are all assumed to be positive. Te absolute permeability tensor, K, is assumed to be bounded, symmetric, and uniformly positive definite in space and constant in time. Te quasi-static Biot model [7] we consider in tis work reads: Find u and p satisfying te equations below for all time t ]0, T [: Flow Equation: t 1 M + c f ϕ 0 p + α u 1 µ f K p ρ f,r g η = q in div σ por u, p = f in, Mecanics Equations: σ por u, p = σu α p I in, σu = λ ui + Gεu in Boundary Conditions: u = 0 on, K p ρ f,r g η n = 0 on Γ N, p = 0 on Γ D, Initial Conditions t=0: 1 M + c f ϕ 0 p + α u 0 = 1 M + c f ϕ 0 p0 + α u 0. were: η is te distance in te direction of gravity, wic is assumed to be constant in time, ρ f,r > 0 is a constant reference density wit respect to a reference pressure p r, g is te gravitational acceleration, M is te Biot modulus, ϕ 0 is te initial porosity, q = q ρ f,r were q is a mass source or sink term injection or production wells. Te above system is linear and coupled troug terms involving te Biot coefficient α. 4

6 t flow, t mec = 0 initial time = 0 k = 0 n = 0 iterative coupling index m = 1 flow iteration index t flow, t mec = 0 initial time = 0 k = 0 Fluid Flow: t flow = t flow + t Compute pore pressure: p n+1,k+m n = 0 iterative coupling index Fluid Flow: t flow = t flow + t Compute pore pressure: p n+1,k+1 Mecanics Biot Model: t mec = t mec + t Compute displacement: u n+1,k+1 Update pore volume m = m + 1 No m = Max flow iterations: q? Yes Mecanics Biot Model: t mec = t mec + q t Compute displacement: u n+1,k+q Update pore volume t flow = t flow t t mec = t mec t n = n + 1 No Yes Converged? k = k + 1 t flow = t flow q t t mec = t mec q t n = n + 1 No Converged? Yes k = k + q a Single Rate b Multirate Figure 1.1: Flowcart for iterative algoritm using single and multirate time stepping for coupled geomecanics and flow problems.1 Mixed variational formulation For te spatial discretization, we use a mixed finite element formulation for flow and a conformal Galerkin formulation for mecanics. For temporal discretization, we follow a backward-euler 5

7 sceme. Let T denote a regular family of conforming elements of te domain of consideration,. Using te lowest order Raviart-Tomas RT spaces, our discrete spaces are given as follows: Discrete Displacements: V = {v H 1 d ; T T, v T P 1 d, v = 0}, Discrete Pressures: Q = {p L ; T T, p T P 0 }, Discrete Fluxes: Z = {q Hdiv; d ; T T, q T P 1 d, q n = 0 on }. For te single rate sceme, te time step is given by t k = t k t k 1. For te multirate sceme, we assume tat t k denotes te fine flow time step, and q t k denotes te coarse mecanics time step, were q represents te number of flow fine time steps contained in one coarse mecanics time step. If we assume a uniform fine flow time step t and te total number of fine time steps is denoted by N, ten te total simulation time is given by T = t N, and t i = i t, 0 i N give te discrete time points. We note ere tat te proof we are about to present can be extended to oter mixed formulation approaces, for instance see [37], or Conformal Galerkin discretizations. Notation: For te single rate sceme, k denotes te flow/mecanics time step index. For te multirate sceme, k denotes te coarse mecanics time step index, m denotes te fine flow time step index, t denotes te unit fine time step size, wic is assumed to be uniform witin one coarse mecanics time step, and q denotes te number of flow time steps per coarse mecanics time step. We note tat te parameter q can vary across mecanics coarse time steps, and all our obtained results remain valid. 3 Previous results 3.1 Standard Fixed stress split algoritm Te fixed-stress split iterative coupling sceme assumes a constant volumetric mean total stress during te flow solve. In tis sceme, te flow problem is solved first followed by te elasticity problem. For te sake of clarity, we start by presenting te continuous strong form of te splitting sceme. We note ere tat n denotes te coupling iteration index between te flow and mecanics: Step a: Given u n, we solve for p n+1, z n+1 satisfying 1 + c M fϕ 0 + α λ t pn+1 z n+1 = α λ z n+1 = 1 µ f K p n+1 ρ f,r g η Step b: Given p n+1, z n+1, we solve for u n+1 satisfying div σ por u n+1, p n+1 = f σ por u n+1, p n+1 = σu n+1 α p n+1 σu n+1 = λ u n+1 I + Gεu n+1 t pn α t un + q 6

8 wit te initial condition 1 M + c fϕ 0 p n+1 + α u n+1 0 = 1 M + c fϕ 0 p0 + α u We first note tat te initial condition is independent of te coupling iteration index n. Second, te regularization terms α /λ t p n+1, α /λ t p n+1, added to te left and rigt and sides of te flow equation respectively, cancel eac oter upon convergence, and te original quasi-static Biot model is retrieved. 3. Fully discrete sceme for single rate Using a mixed formulation for te flow, continuous Galerkin for te mecanics, and te backward Euler finite difference metod in time, te weak formulation of te single-rate sceme reads as follows. Definition 3.1 Find p k Q, and z k Z suc tat, flow equation and mecanics equation find u k V suc tat, 1 1 θ Q, t M + c fϕ 0 p k p k 1, θ + 1 z k µ, θ = f α u k u k 1, θ + q, θ, 3. q t q Z, K 1 z k, q = p k, q + ρ f,r g η, q, 3.3 v V, G εu k, εv + λ u k, v α p k, v = f, v. 3.4 wit te initial condition for te first discrete time step, 3.3 Fully discrete sceme for multirate p 0 = p Similar to te single-rate sceme, te weak formulation of te multirate sceme reads as follows. Definition 3. For 1 m q, find p m+k Q, and z m+k Z suc tat, 7

9 flow equation θ Q, q Z, 1 1 t and mecanics equation find u k+q V suc tat, v V, G εu k+q α q t M + c fϕ 0 p m+k u k+q K 1 z m+k, q = p m 1+k, θ + 1 z m+k, θ = µ f, θ q, θ u k + p m+k, q + wit te initial condition for te first discrete time step,, 3.6 ρ f,r g η, q, 3.7, εv + λ u k+q, v α p k+q, v = f, v. 3.8 p 0 = p We note tat te pressure and flux unknowns are solved at fine time steps t k+m, m = 0,..., q. In contrast, te displacement unknowns are solved at coarse mecanics time steps t iq, i N. Terefore, tere are q flow solves for eac mecanics time step of size q t, wic justifies te nomenclature of multirate. 3.4 Weak formulation of te single rate sceme: Te weak formulation of te fully discrete single rate fixed-stress split iterative coupling sceme reads: Step a: Find p n+1,k Q, z n+1,k Z suc tat: θ Q, p k 1, θ + 1 z n+1,k, θ = µ f Step b: Given p n+1,k v V, G εu n+1,k 1 1 t M + c fϕ 0 + L p n+1,k 1 t L p n,k p k 1 α u n,k u k 1, θ + q, θ 3.10 q Z, K 1 z n+1,k, q = p n+1,k, q + ρ f,r gη, q 3.11, z n+1,k, find u n+1,k V suc tat,, εv + λ u n+1,k, v α p n+1,k, v = f, v, 3.1 We note tat in te above sceme, L is an adjustable coefficient tat will be determined by matematical analysis. In wat follows, we adopt te following notation: for a given time step t = t k, we define te difference between two iterative coupling iterates as: δξ n+1,k = ξ n+1,k ξ n,k. were ξ may stand for p, z, and u. 8

10 3.5 Banac fixed-point contraction result of te single rate sceme: Teorem 3.3 For a particular time step t k, L = α, c λ 0 = 4λ, and δσ n,k α v single rate fixed-stress iterative coupling sceme is a contraction given by t K 1/ βµ f δz n+1,k + Gc 0 εδu n+1,k + δσ n+1,k v = δp n,k α L δun,k, te Mα δσ n,k λ+mλc f ϕ 0 +Mα v Furtermore, te converged solution is a unique solution to te weak formulation Weak formulation of te multirate sceme: Te weak formulation of te fully discrete multirate fixed-stress split iterative coupling sceme reads: Step a For 1 m q, find p n+1,m+k θ Q, 1 1 t M + c fϕ 0 + L p n+1,m+k + 1 z n+1,m+k, θ = 1 µ f t L Q, and z n+1,m+k p n,m+k p n+1,m 1+k p n,m 1+k q Z, K 1 z n+1,m+k, q = p n+1,m+k, q + Z suc tat,, θ α q u n,k+q wit te initial condition, independent of n, for te first discrete time step, Step b Given p n+1,k+q u n,k, θ + q, θ, 3.13 ρ f,r g η, q, 3.14 p n+1,0 = p and, z n+1,k+q, find u n+1,k+q V suc tat, v V, G εu n+1,k+q, εv + λ u n+1,k+q, v α p n+1,k+q, v = f, v Similar to te single rate sceme, L is an adjustable coefficient wic will be determined appropriately by working out te contraction proof. In addition, q denotes a user-defined number of finer flow steps. 9

11 3.7 Banac fixed-point contraction result of te multirate sceme: Teorem 3.4 For a coarse mecanics time step t k+q, L = α, λ χ = L, c 0 = L, and δσ n,m+k qχ v = L χ δpn,m+k δp n,m 1+k α qχ δun,k+q, for 1 m q, te multirate iterative sceme is a contraction given by Gc 0 εδu n+1,k+q + q m=1 δσv n+1,m+k + t K 1/ βµ f δz n+1,k+q + t K 1/ δz n+1,m+k δz n+1,m 1+k q δσv n,m+k. βµ f q m=1 Mα λ+mλc f ϕ 0 +α M Furtermore, te sequences defined by tis sceme converge to te unique solution of te weak formulation Remark 3.5 Te contraction coefficient obtained in teorem 3.3 can be strengtened by taking advantage of te extra terms on te left and side of te contraction result. By triangle s inequality, te norm of te quantity of contraction can be written as δσ n+1,k v α L δun+1,k + δp n+1,k. By standard mixed metod tecniques of estimating te pressure by te flux using Poincare inequality, and bounding te volumetric strain term by Korn s inequality, we obtain δσ n+1,k v C n+1,k εδu + δz n+1,k for a constant C > 0. Now, denote te first two terms on te left and side of te result of teorem 3.3 by I n+1,k : I n+1,k = t K 1/ δz n+1,k + Gc 0εδu n+1,k βµ. f For a generic constant C > 0, inequality 3.17 can be written as: δσ n+1,k v CI n+1,k Te contraction result in teorem 3.3 takes te form: 1 δσ C + 1 n+1,k Mα v δσ n,k λ + Mλc f ϕ 0 + Mα v resulting in te improved contraction constant, δσ n+1,k v C C + 1 m=1 Mα δσ n,k λ + Mλc f ϕ 0 + Mα v

12 It is not difficult to see tat te constant C increases monotonically as te values of te Lamé coefficients, λ and G, increase assuming te Poisson ratio, ν, is fixed. However, estimating its exact value is difficult in practice. Te above computations sow tat te contraction coefficient C obtained earlier is damped by a factor strictly less tan one: < 1. Terefore, for C 1, C+1 te contraction estimate obtained in teorem 3.3 is relatively sarp. However, for small values of C C, te contraction coefficient obtained earlier can be severely affected by te damping factor. C+1 We will validate tese observations numerically by varying te values of te Lamé coefficients and computing contraction estimates numerically. We will sow tat for larger Lamé coefficients values, contraction estimates computed numerically approac te value of te teoretical estimate obtained in teorem 3.3. In oter words, our teoretical contraction estimates are sarper for larger Lamé coefficients. In fact, tese computations sow te impact of te extra positive terms on te left and side of te result in teorem 3.3 on te sarpness of te contraction coefficient. Remark 3.6 As in te single rate case, te contraction coefficient in te multirate case, derived in teorem 3.4, can be improved as well. Te improved contraction result reads: q q m=1 δσ n+1,m+k v For a constant C > 0. C 1 + C Mα λ + Mλc f ϕ 0 + α M m=1 δσ n+1,m+k v. 3.0 Remark 3.7 We can teoretically derive te value of te constant C given in te previous remark. For simplicity, consider te single rate case q = 1. For L = α, χ = L, and by Poincare s and λ Korn s inequalities, we write: δσ n+1,k α v δun+1,k L L + δp n+1,k L Tus, we ave: δσ n+1,k v 4λ C κ α 4λ C κ α λ α δun+1,k λc κ α λc κ α εδu n+1,k + λc κp ɛ α H 1 + δp n+1,k L εδu n+1,k εδu n+1,k + P L δp n+1,k L + P L K 1 δz n+1,k L + L P K 1 δz n+1,k L εδu n+1,k + 4λC κp α εδu n+1,k + L L K 1 δz n+1,k L P + λc κp αɛ K 1 δz n+1,k L

13 For ɛ > 0. Assuming tat te permeability tensor K is uniformly bounded and uniformly elliptic. Tere exits positive constants λ min, and λ max, suc tat We write: δσ n+1,k 4λ C κ v Terefore, we ave: δσ n+1,k v 4λ C κ Max α α λ min ξ ξ t Kxξ λ max ξ λc κp ɛ εδu n+1,k α + 1 L λ min + λc κp ɛ, α 1 λ min For te single rate case recall: c 0 = 4λ, I n+1,k α q I n+1,k q P + λc κp αɛ takes te form: εδu n+1,k = Gc 0 εδu n+1,k + t K 1/ δz n+1,k L βµ f 8Gλ εδu n+1,k α + 1 t δz n+1,k L λ max βµ f 8Gλ Min α, 1 λ max t Combining 3.4 and 3.5, we ave: δσ n+1,k v Max 4λ C κ α εδu n+1,k βµ f + λcκp ɛ α Min wit te constant C given by: 4λ Max C κ α C = Min + λcκp ɛ α, 1 λ min P + λc κp αɛ L L L + δz n+1,k 8Gλ 1 t, α λ max βµ f, 1 λ min P + λcκp αɛ P + λcκp αɛ δz n+1,k L L δzn+1,k L L I n+1,k q Gλ α, 1 λ max t βµ f 3.6 Clearly, C scales monotonically wit te values of Lame s parameters G and λ. Terefore, for larger Lame parameters, te value of te constant C increases, and in turn, te damping factor C start approacing te value of one. Tis results in reducing te gap between te teoretical 1+C Mα, value of te contraction coefficient δσv λ+mλc f ϕ 0 +α M and te ratio of n+1,k. Tis is validated δσv n,k numerically for te Frio field model in figure

14 4 Error analysis of te single rate iterative sceme For a given time step t = t k, and a given iterative coupling iteration n 0, we need to estimate ξ n,k ξt k, were ξ may stand for p, z, and u. By te triangle inequality, we can write: ξ n,k ξt k ξ n,k ξ k + ξ k ξt k were ξ k is te solution obtained by solving te coupled flow and mecanics equations simultaneously. Error estimates for te second term on te rigt and side ave been derived in te work of [6, 7]. It only remains to estimate te first term ξ n,k ξ k. Tis can be done in two steps: first we derive a Banac contraction argument on te difference between te solution obtained at a particular iterative coupling iteration ξ n,k, and te solution obtained by solving te coupled system simultaneously fully implicit sceme, ξ k. Ten, we derive stability estimates for te fully implicit sceme, and combine te two to bound te term ξ n,k ξ k. Te two steps are detailed below. Step 1: Banac Contraction estimate on ξ n,k ξ k : We first note tat te weak formulation of te fully discrete single-rate fixed-stress split iterative coupling sceme is given in equations In contrast, te weak formulation of te fully discrete implicit sceme reads: Find p k Q, z k Z, and u k V suc tat, 1 1 θ Q, t M + c fϕ 0 p k p k 1, θ + 1 z k, θ µ f = α u k u k 1, θ + t q Z, K 1 z k, q = p k, q + q, θ, 4.1 ρ f,r g η, q, 4. v V, G εu k, εv + λ u k, v α p k, v = f k, v. 4.3 Subtracting equations 4.3, 4., and 4.1, from 3.1, 3.11, and 3.10 respectively, and noting tat f n+1,k = f k, we get: 1 1 θ Q, t M + c fϕ 0 p n+1,k p k, θ + 1 z n+1,k z k µ, θ f q Z, = α t u n,k u k, θ L t p n+1,k p n,k, θ, 4.4 K 1 z n+1,k z k, q = p n+1,k p k, q, 4.5 v V, G εu n+1,k u k, εv + λ u n+1,k u k, v α p n+1,k p k, v

15 Define e n+1 p = p n+1,k p k, en+1 u and 4.6 can be written as: = u n+1,k u k, and en+1 z = z n+1,k z k. Equations 4.4, 4.5, 1 1 θ Q, t M + c fϕ 0 + L e n+1 p, θ + 1 e n+1 z, θ = 1 α e nu + Le np, θ, µ f t 4.7 q Z, K 1 e n+1 z, q = e n+1 p, q, 4.8 v V, G εe n+1 u, εv + λ e n+1 u, v α e n+1 p, v = Let β = 1 + c M fϕ 0 + L. Testing 4.7 wit θ = e n+1 p, and 4.8 wit q = e n+1 z, we obtain: βe n+1 p + t e n+1 z, e n+1 p = α e n u + Le n µ p, e n+1 p f K 1 e n+1 z, e n+1 z = e n+1 p, e n+1 z Substituting 4.11 into 4.10, defining e n σ as χe n σ = Le n p α e n u, were χ is an adjustable parameter, and applying Young s inequality, we obtain: βe n+1 p + t K 1/ e n+1 z 1 e n σ + ɛ e n+1 p µ f ɛ. Te coice ɛ = β gives after multiplying by β : e n+1 p + t K 1/ e n+1 z βµ f 1 χe n β σ. 4.1 Multiplying te elasticity equation 4.9 by a free parameter c 0, and testing wit v = e n+1 u, we get: Gc 0εe n+1 u + λc 0 e n+1 u αc 0 e n+1 p, e n+1 u = Combining flow 4.1 wit elasticity 4.13, we obtain: Gc 0 εe n+1 u { e + n+1 p αc 0 e n+1 p, e n+1 u + λc } 0 e n+1 u + t K 1/ e n+1 z χ e n βµ f β σ Expanding te rigt and side to matc terms on te left and side to form a complete square: e n σ = L e n χ p αl e n χ p, e n u + α δe n χ u 14.

16 Te following inequalities sould be satisfied: 1 > L χ, αl χ = αc 0, and λc 0 = α χ. Te second and tird equalities lead to te following parameter assignments: c 0 = L χ, and L = α Te first inequality leads to te condition: χ > α λ. Now, 4.14 can be written as: Gc 0εe n+1 u + t K 1/ e n+1 z + 1 L e n+1 βµ f χ p + e n+1 σ λ. χ e n σ β For contraction to old, we require χ α < 1. Togeter wit te previous condition χ >, te β λ value of χ sould be cosen suc tat α λ < χ < 1 M + c fϕ 0 + α λ Tis imposes te following condition on our given parameters wic corresponds to te condition on te constrained specific storage coefficient in te work of [6, 7]: In general, for n 0, we can write: 1 M + c fϕ 0 γ 0 > 0. for some positive constant γ e n+1 σ χ e n σ β χ n+1 e 0 σ β. χ n+1 Le 0 p α e 0 u 4.17 β. Combining 4.15 wit 4.17, togeter wit Young s inequality, we can write: 1 L e n+1 χ p χ n+1 Le 0 p α e 0 u β χ n+1 L e 0p + α e 0 β u Lαe 0 p, e 0 u χ n+1 L e 0p + α e 0 β u + Lα 1 e 0 p ɛ χ n+1 L + Lα e β ɛ 0 p + α + Lαɛ e 0 u + ɛ e 0 u for ɛ >

17 Similarly, we can write: Gc 0εe n+1 u χ n+1 L + Lα e β ɛ 0 p t K 1/ e n+1 z χ n+1 L + Lα e βµ f β ɛ 0 p Combining 4.18, 4.19, and 4.0, we ave: e n+1 p [ were C 1 = u k 1 + εe n+1 u K + 1/ e n+1 z χ n+1c1 L + Lα e β ɛ 0 p ] χ χ L + 1 Gc 0 + βµ f t u k, 4.1 can be written as: p n+1,k p k χ n+1c1 β + εu n+1,k L + Lα ɛ + α + Lαɛ e 0 u + α + Lαɛ e 0 u + α + Lαɛ e 0 u Noting tat: e 0 p = p 0,k pk = pk 1 p k, and e0 u = u 0,k uk = u k K + 1/ z n+1,k p k p k 1 χ + β z k n+1c1 α + Lαɛ u k u k 1 Let η 1 = C 1 L + Lα, and η ɛ = C 1 α + Lαɛ for ɛ > 0, 4. reduces to: p n+1,k p k + εu n+1,k u k K + 1/ z n+1,k z k χ n+1 p k η 1 β p k 1 + η u k u k Step : Stability estimate on ξ k ξk 1 : We recall tat te weak formulation of te implicit sceme is given by equations Te derivation of te stability estimate for te implicit sceme is carried out in tree steps: by first considering te flow equations, followed by te mecanics equation and ten combining te two to derive te final estimate. For simplicity, we define c f = 1 + c M fϕ 0. 16

18 4.1 Flow equations Testing 4.1 wit θ = p k pk 1, and multiplying te wole equation by t, we obtain p k c f p k 1 + t z k µ, p k p k 1 = α u k u k 1, p k p k 1 + q, p k p k 1 f 4.4 Next, we consider te flux equation 4.. Taking te difference of two consecutive time steps t = t k and t = t k 1 and testing wit q = z k, we obtain: K 1 z k z k 1, z k = p k p k 1, z k 4.5 Substituting 4.5 into 4.4, wit some algebraic manipulations of te resulting term using te identity: aa b = 1 a b + a b, we derive p k c f p k 1 + t K 1/ z k K 1/ z k 1 + K 1/ z k z k 1 µ f = α u k u k 1, p k p k 1 + q, p k p k 1 4. Elasticity equation 4.6 Considering 4.3 for te difference of two consecutive time steps, t = t k and t = t k 1, and testing wit v = u k uk 1, we obtain Gεu k u k 1 + λ u k u k 1 α p k p k 1, u k u k 1 = f k f k 1, u k u k Combining flow and elasticity equations Combining 4.6 wit 4.7 yields p k c f p k 1 + t K 1/ z k K 1/ z k 1 + K 1/ z k z k 1 µ f +Gεu k u k 1 + λ u k u k 1 = q, p k p k 1 + f k f k 1, u k u k 1 }{{}}{{} R 1 R

19 To bound te terms R 1 and R, we will use Poincaré s and Korn s inequalities. Poincaré s inequality in H 1 0 reads: tere exists a constant P depending only on suc tat v H 1 0, v L P v H Korn s first inequality in H 1 0 d reads: tere exists a constant C κ depending only on suc tat v H 1 0 d, v H 1 C κ εv L By Poincaré, Korn, and Young inequalities, we bound R 1 & R as: R 1 1 ɛ 1 q + p k p k 1 ɛ 1 R 1 f k f k 1 + ɛ u k u k 1 ɛ 1 f k f k 1 ɛ + ɛ P C κ εu k u k 1. for ɛ 1, and ɛ > 0. Coosing ɛ 1 = c f, and ɛ = G, and summing for 1 k N, were N P C κ denotes te total number of time steps note telescopic cancellations, we derive c f p k p k 1 + t K 1/ z N + K 1/ z k z k 1 + G εu k u k 1 µ f +λ u k u k 1 t K 1/ z µ f c f Terefore, we can write: p k p k 1 u k u k 1 t K 1/ z c µ f c f f t K 1/ z µ f λ c f λ P q + C κ 4G P q + C κ G c f P q + C κ 4Gλ Combining 4.3 wit 4.33, we ave: p k p k 1 + u k u k 1 K t η 1/ 3 z 0 + η 4 18 f k f k 1, 4.31 f k f k 1, 4.3 f k f k 1. N q 4.33 N + η 5 f k f k

20 were η 3 = 1 µ f C, η 4 = 1 c f C, η 5 = P C κ C 1 G, and C = c f + 1. Combining 4.3 wit λ 4.34, for a generic constant C 3 > 0 wic will be revealed by te end of te derivation but we suppress its value now for te sake of simplicity, we can derive: p n+1,k p k + εu n+1,k u k K + 1/ z n+1,k z k χ n+1 p k η 1 β p k 1 + η u k u k 1 χ n+1c3 [ p k p k 1 + u k u k 1 ] β χ n+1c3 [ N p k p k 1 + u k u k 1 β χ n+1c3 [ K t η 1/ 3 z 0 + η 4 β χ n+1c3 [ tk 1/ z 0 + β N q + η5 Terefore, we can write: p n+1,k p k + εu n+1,k u k K + 1/ L z n+1,k z k L N ] f k f k 1 ] N q + f k f k 1 ] χ n+1 [ C 3 tk 1/ z 0 + β L q L L + f k f k 1 L 4.35 Now, we assume tat te permeability tensor K is uniformly bounded and uniformly elliptic. Tere exits positive constants λ min, and λ max, suc tat λ min ξ ξ t Kxξ λ max ξ We can write K 1/ z n+1,k z k L 1 z n+1,k λmax 1/ z k L. In addition, by Poincaré s inequality and Korn s first inequality, we ave for C k1 > 0: εu n+1,k u k L 1 u n+1,k u k C H 1. k1 ] 1/ 19

21 Terefore, 4.37 can be written as: p n+1,k p k + u n+1,k L u k + z n+1,k H 1 z k χ n+1 [ C 3 tk 1/ z 0 + β L q L + L f k f k 1 1/ 4.37 L ] We conclude tat for every coupling iteration n 0, p n+1,k pt k + u n+1,k l L ut k + z n+1,k l H 1 zt k l L p n+1,k p k + u n+1,k l L u k + z n+1,k l H 1 z k l L + p k pt k + u k l L ut k + z k l H 1 zt k χ n+1 [ C 3 tk 1/ z 0 + q + f k β L L f k 1 L + p k pt k + u k l L ut k + z k l H 1 zt k l L ] 1/ l L By [6, 7], we ave: p k pt k + u k l L ut k + z k l H 1 zt k C r1+ + r + O t l L for a positive constant C > 0 and mes size. We note tat r 1 denotes te degree of te polynomials used in te mixed space Q, Z, and r denotes te degree of te polynomials used in te displacement space V. In our case, r 1 = 0, and r = 1. Terefore, our final estimate takes te form: p n+1,k pt k + l L u n+1,k χ n+1 [ C 3 tk 1/ z 0 + β L ut k + l H 1 q L z n+1,k zt k + l L f k f k 1 L ] 1/ 1/ + 3 C + O t. were C 3 = C k1 + λ 1/ max Max η 1, η Max η 3, η 4, η 5 Te above discussions are summarized in te following teorem: Teorem 4.1 For a particular time step t k, and a particular flow-mecanics coupling iteration n 1, and assuming te lowest order Raviart-Tomas spaces for flow, and continuous piecewise linear approximations for mecanics, and assuming equations 4.16 and 4.36, and sufficient 0

22 regularity in te true solution, te following finite element error estimate, to te leading order in time, for te single rate fixed-stress split iterative coupling sceme olds: p n,k pt k + u n,k l L ut k + z n,k l H 1 zt k l L n [ χ C 1 tk 1/ z 0 + N q + ] N f k f k 1 1/ 1/ + C + O t were β L L L C 1 = C k1 + λmax 1/ Max η 1, η Max η 3, η 4, η 5 C = C T, K, M, c f, ϕ 0, C k1, p k, p k,t, z k, u k,t. Remark 4. We note tat te contracting beavior of te undrained split iterative coupling sceme as been establised in [3] wen Continuous Galerkin CG is used to discretize bot flow and mecanics. Te work of [5, 19] extends tis result to te case wen a mixed form is used for flow and CG is used for mecanics for bot te single rate and multirate scemes. Based on tat, a priori error estimates for te single rate undrained split sceme can be derived in te same way as in te fixed stress split sceme. A Banac contraction estimate on te difference ξ n,k ξ k can be obtained in a similar way as described in [5, 19]. Tis complete te first step of te error analysis. Te second step, involving te derivation of stability estimates on te difference ξ k ξ k 1 remains uncanged, as bot te fixed-stress split and undrained split iterative scemes converge to te solution obtained by te simultaneously coupled sceme. 5 Numerical Results Te single rate and multirate fixed-stress split iterative coupling scemes ave been implemented in te Integrated Parallel Accurate Reservoir Simulator IPARS for single-pase and two-pase flow models coupled wit a linear poroelasticity model. Conformal Galerkin is used for elasticity discretization and te Multipoint Flux Mixed Finite Element Metod MFMFE is used for flow discretization. Te two-pase model assumes an IMPES implicit pressure explicit saturation sceme. We first igligt efficiency gains of te multirate sceme over te single rate sceme by considering te Frio field model a field-scale problem run as a two-pase oil and water flow problem coupled wit mecanics. Ten, we te investigate te effects of Lame parameters on te sarpness of te predicted teoretical estimate for te Frio field model run as a single-pase flow problem coupled wit mecanics. 5.1 Frio Field Model Te Frio field model is a field-scale problem representing a reservoir model located at Sout Liberty oil field on te Gulf Coast, near Dayton, Texas. Te field is curved in te dept direction, and 1

23 contains several tin curved faults, wit a geometrically callenging geological formation [16]. In tis work, we consider te callenging geometry of te field, and its permeability distribution. Gravity effects are included in te simulation model. Oter input parameters are listed in Table 1. We recall tat q denotes te number of fine flow time steps of size t witin one coarse mecanics time step of size q t. We run te simulation for q = 1, wic corresponds to te single rate case, and for q =, 4, and 8, wic correspond to tree different multirate scemes, in wic we take two, four, and eigt flow fine time steps of size t witin one coarse mecanics time step of size t, 4 t, and 8 t, respectively. 5. Results Figures 5.a, 5.b, 5.c, and 5.d sow water pressure profile and mecanical displacements in te x,y, and z directions respectively for te frio field model after 18.0 simulation days. We clearly see tat te results for bot single rate and multirate implementations are identical. It sould be noted tat te pore pressure cange in tis case is relatively small compared to te value of te Young s modulus of te porous medium, wic leads to small variations in te displacement vectors. Figure 5.1a sows te accumulated CPU run time for te single rate case q = 1, and for multirate cases: q =, 4, and 8. Te multirate iterative coupling algoritm wit two flow finer time steps witin one coarser mecanics time step q = results in 1.5% reduction in CPU run time compared to te single rate case. Multirate couplings q = 4 and q = 8 result in 18.18% and 0.05% reductions in CPU run times respectively. Figure 5.1b explains te reduction in CPU run time observed in te multirate case. By just solving for two flow finer time steps witin one coarse mecanics time step q =, te total number of mecanics linear iterations was reduced by 47.78% wit reference to te single rate case. Multirate couplings q = 4 and q = 8 result in 73.07% and 85.75% reductions in te number of mecanics linear iterations respectively, wic in turn, reduce te CPU run time as well. Figure 5.1c sows te total number of flow linear iterations in te four cases. We see a sligt increase in te total number of flow linear iterations for multirate iterative coupling scemes. Te case q = results in 1.5% increase in te total number of flow linear iterations. Multirate couplings q = 4 and q = 8 result in.89% and 4.8% increase in te total number of flow linear iterations respectively. We conclude tat te uge reduction in te number of mecanics linear iterations outperform te overead introduced by te increase in te number of flow linear iterations. Tis is a key factor to te success of te iterative multirate coupling sceme in reducing te overall CPU run time. We note ere tat te observed increase in te number of flow linear iterations in te multirate case is a direct consequence of te increase in te number of iterative flow coupling-mecanics iterations in te multirate case, compared to te single rate case, as sown in figure 5.1d.

24 Wells: 3 production wells, 6 injection well Injection well 1: Pressure specified, psi Injection well : Pressure specified, psi Injection well 3: Pressure specified, psi Injection well 4: Pressure specified, psi Injection well 5: Pressure specified, psi Injection well 6: Pressure specified, psi Production well 1: Pressure specified, psi Production well : Pressure specified, psi Production well 3: Pressure specified, psi Total Simulation time: 18.0 days Finer Unit time step: 0.05 days Number of grids: 891 grids Absolute Permeabilities: k xx, k yy, k zz igly varying, range: 5.7E-10, 3.10E+3 md Initial porosity, ϕ 0 0. Water viscosity, µ w 1.0 cp Oil viscosity, µ o.0 cp Initial oil concentration, c o 44.8 lb m /ft 3 Initial oil pressure, p o psi Water compressibility c fw : 1.E-6 1/psi Oil compressibility c fo : 1.E-4 1/psi Rock compressibility: 1.E-6 1/psi Rock density: lb m /ft 3 Initial water density, ρ w : 56.0 lb m /ft 3 Initial oil density, ρ o 6.34 lb m /ft 3 Young s Modulus E 1.E6 psi Possion Ratio, ν 0.35 Biot s constant, α 1.0 Biot Modulus, M 1.E8 psi λ = 1.037E6 psi Eν 1+ν1 ν α L introduced fixed stress parameter λ Flow Boundary Conditions: no flow boundary condition on all 6 boundaries Mecanics B.C.: X+ boundary EBCXX1 σ xx = σ n x = 10, 000psi, overburden pressure X- - boundary EBCXXN1 u = 0, zero displacement Y+ - boundary EBCYY1 u = 0, zero displacement Y- - boundary EBCYYN1 σ yy = σ n y = 000psi Z+ - boundary EBCZZ1 u = 0, zero displacement Z- - boundary EBCZZN1 σ zz = σ n z = 1000psi Table 1: Input Parameters for Frio Field Model 3

25 Accumulated CPU Time minutes Accumulated CPU Run Time vs Simulation Period Single Rate Multirate q = Multirate q = 4 Multirate q = 8 Accumulated Mecanics Linear Iterations Accumulated # of Mecanics Linear Itrns vs Simulation Period Single Rate Multirate q = Multirate q = 4 Multirate q = Simulation Period days a CPU Run Time vs Simulation Days Simulation Time days b Total Number of Mecanics Linear Iterations vs Simulation Days Accumulated Flow Linear Iterations Accumulated # of Flow Linear Itrns vs Simulation Period Single Rate Multirate q = Multirate q = 4 Multirate q = 8 Number of Coupling Iterations Per Time Step Coupling Iterations per Coarse Time Step Single Rate q = 1 Multirate q = Multirate q = 4 Multirate q = Simulation Time days c Total Number of Flow Linear Iterations vs Simulation Days Time Steps days d Number of Iterative Coupling Iterations Per Coarser Time Step Figure 5.1: Frio Field Model Simulation Results: te top left plot sows te CPU time savings of different multirate scemes q =, 4, and 8 over te single rate sceme q = 1. Te top rigt plot sows te reduction in te number of mecanics linear iterations observed for te corresponding multirate sceme. Te bottom left plot sows te increase in te number of flow linear iterations induced by te sligt increase in te number of flow-mecanics coupling iterations, as sown in te bottom rigt plot, for multirate scemes q =, 4, and 8 compared to te single rate sceme q = 1. 4

26 a Pressure Profiles after 18.0 simulation days b Displacement in x direction after 18.0 simulation days c Displacement in y direction after 18.0 simulation days d Displacement in z direction after 18.0 simulation days Figure 5.: Frio Field Model Pressure and Displacement Fields at Te End of Te Simulation 5

27 5.3 Teoretical Vs. Numerical Contraction Coefficients In tis section, we compare our teoretically derived contraction estimate against numerically observed values for te single rate case. We consider te Frio field model run as a single pase flow problem coupled wit mecanics. We assume te same permeability distribution as in te previous case. Te values of te Biot Modulus M, Biot s constant α, Poisson Ratio ν, fluid compressibility c f, and initial porosity ϕ 0 are 10 8 psi, 0.9, 0.35, /psi, and 0. respectively. We varied te value of te Young s Modulus E, and computed te numerical contraction estimates for first 1 simulation days in eac case. Numerical contraction estimates are computed by taking te ratio of te quantity of contraction between two consecutive iterative coupling iterations δσ n+1,m+k v / δσ. For eac time step, te maximum value of tis ratio across n,m+k v all coupling iterations is considered. Teoretical contraction estimates are given by te expression Mα. λ+mλc f ϕ 0 +Mα Results are sown in figure 5.3. We clearly see tat teoretical estimates act as upper bounds for numerically computed contraction estimates. Moreover, numerical contraction estimates are larger for early time steps, wic is expected, as te coupled problem as not reaced a steady-state yet. We also recall tat te improved contraction estimate for te single rate case derived in remark 3.5, is given by δσ n+1,k C Mα v δσ n,k C+1 λ+mλc f ϕ 0 +Mα v for a constant C > 0, wic is difficult to compute in practice. However, we anticipate tat it scales monotonically wit te values of Lamé s first parameter λ, and Young s modulus E wen fixing te value of te Poisson ratio. Terefore, teoretically, we expect te value of te damping factor to approac one, as te value of Young s modulus increases assuming ν is fixed, wic means tat our derived contraction estimates are sarper for larger Young s modulus values. Te results sown in figure 5.3 validate tis teoretical discussion numerically. We see tat as te value of te Young s modulus increases, te gap between teoretical contraction estimates, and numerically computed values srinks. For E = 10 6, teoretical estimates and numerical ones are bot of te order of Conclusions Tis paper considers te contracting beavior of te single rate and multirate fixed-stress split iterative coupling scemes bot teoretically and numerically. In te single rate case, te flow and mecanics problems sare te same time step, wile in te multirate case, te flow takes multiple fine time steps witin eac coarse mecanics time step. A priori error estimates for te single rate fixed-stress split iterative coupling sceme are derived in tis work. Te novelty of te approac used in deriving our error estimates lies in its te ability to utilize previously establised results for te simultaneously coupled sceme. We anticipate tat tis approac can be used to derive error estimates for oter contractive iterative scemes, including te undrained-split coupling sceme. C C+1 6

28 Figure 5.3: Numerical contraction estimates for different values of Young s modulus E for te single rate sceme for te first 1 simulation days. As te value of Young s modulus increases, te gap between teoretically predicted contraction estimates and numerically observed values srinks, validating our teoretical derivations sown in remarks 3.5 and 3.6. Numerical Contraction Estimates Numerical Contraction Estimates For Different Values of Young's Modulus Young's Modulus E = 1.E+4, Teoretical. Est. = Young's Modulus E = 5.E+4, Teoretical. Est. = Young's Modulus E = 1.E+5, Teoretical. Est. = Young's Modulus E = 5.E+5, Teoretical. Est. = Young's Modulus E = 1.E+6, Teoretical. Est. = 5.E Simulation Period days Our numerical results igligt te ability of te multirate sceme in reducing te number of mecanics linear iterations, and in turn, te CPU run time, efficiently wile maintaining te same level of accuracy compared to te results obtained by te single rate sceme. In addition, subject to te value of te damping factor obtained in remarks 3.5, and 3.6, by comparing our teoretical contraction estimates against numerical computations, we conclude tat te teoretical estimates can predict te contracting beavior, and subsequently, te rate of convergence of te corresponding iterative sceme wit ig accuracy. Acknowledgements TA is funded by Saudi Aramco. We tank Paulo Zunino and Ivan Yotov for elpful discussions. KK would like to acknowledge te support of StatOil Akademia Grant Bergen. Te autors would like 7

29 to acknowledge te CSM Industrial Affiliates program, DOE grant ER5617, and ConocoPillips grant UTA Moreover, we tank Gurpreet Sing for is tremendous elp and support wit IPARS. References [1] D. R. Allen. Pysical canges of reservoir properties caused by subsidence and repressurizing operations. Society of Petroleum Engineers, SPE [] T. Almani, A. H. Dogru, K. Kumar, G. Sing, and M. F. Weeler. Convergence of multirate iterative coupling of geomecanics wit flow in a poroelastic medium. Saudi Aramco Journal of Tecnology, Spring 016, 016. [3] T. Almani, K. Kumar, A. H. Dogru, G. Sing, and M. F. Weeler. Convergence analysis of multirate fixed-stress split iterative scemes for coupling flow wit geomecanics. Ices report 16-07, Institute for Computational Engineering and Sciences, Te University of Texas at Austin, Austin, Texas, 016. [4] T. Almani, K. Kumar, G. Sing, and M. F. Weeler. Stability of multirate explicit coupled of geomecanics wit flow in a poroelastic medium. Ices report 16-1, Institute for Computational Engineering and Sciences, Te University of Texas at Austin, Austin, Texas, 016. [5] T. Almani, K. Kumar, and M. F. Weeler. Multirate undrained splitting for coupled flow and geomecanics in porous media. Ices report 16-13, Institute for Computational Engineering and Sciences, Te University of Texas at Austin, Austin, Texas, 016. [6] M. A. Biot. Consolidation settlement under a rectangular load distribution. J. Appl. Pys., 15:46 430, [7] M. A. Biot. General teory of tree-dimensional consolidation. J. Appl. Pys., 1: , [8] L. Y. Cin, L. K. Tomas, J. E. Sylte, and R. G. Pierson. Iterative coupled analysis of geomecanics and fluid flow for rock compaction in reservoir simulation. Oil and Gas Science and Tecnology, 575: , 00. [9] O. Coussy. A general teory of termoporoelastoplasticity for saturated porous materials. Transport in Porous Media, 4:81 93, June [10] O. Coussy. Mecanics of Porous Continua. Wiley, West Sussex PO19 1UD, England,

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