ICES REPORT February Tameem Almani, Kundan Kumar, Ali H. Dogru, Gurpreet Singh, Mary F. Wheeler

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1 ICES REPORT February 2016 Convergence Analysis of Multirate Fixed-Stress Split Iterative Scemes for Coupling Flow wit Geomecanics by Tameem Almani, Kundan Kumar, Ali H. Dogru, Gurpreet Sing, Mary F. Weeler Te Institute for Computational Engineering and Sciences Te University of Texas at Austin Austin, Texas Reference: Tameem Almani, Kundan Kumar, Ali H. Dogru, Gurpreet Sing, Mary F. Weeler, "Convergence Analysis of Multirate Fixed-Stress Split Iterative Scemes for Coupling Flow wit Geomecanics," ICES REPORT 16-07, Te Institute for Computational Engineering and Sciences, Te University of Texas at Austin, February 2016.

2 Convergence Analysis of Multirate Fixed-Stress Split Iterative Scemes for Coupling Flow wit Geomecanics T. Almani 1,3, K. Kumar 2, A. Dogru 3, G. Sing 1, M. F. Weeler 1 1 Center for Subsurface Modeling, ICES, UT Austin, USA 2 Matematics Institute, University of Bergen, Norway 3 Saudi Arabian Oil Company, Saudi Arabia {tameem,gurpreet,mfw}@ices.utexas.edu, kundan.kumar@uib.no, ali.dogru@aramco.com February 23, 2016 Abstract We consider multirate iterative scemes for te Biot system modelling coupled flow and geomecanics in a poro-elastic medium. Te multirate iterative coupling sceme exploits te different time scales for te mecanics and flow problems by taking multiple finer time steps for flow witin one coarse mecanics time step. We adapt te fixed stress split algoritm tat decouples te flow and mecanics equations for te multirate case and perform an iteration between te two problems until convergence. We provide a fully discrete sceme tat uses Backward Euler time discretization and mixed spaces for flow and conformal Galerkin for mecanics. Our analysis is based on studying te equations satisfied by te difference of iterates and using Banac contraction argument to prove tat te corresponding sceme is a fixed point contraction. Te analysis provides te value of an adjustable coefficient used in te proposed iterative coupling algoritms. Furtermore, we sow tat te converged quantities satisfy te variational weak form for te coupled discrete system. Keywords. poroelasticity; fixed-stress split iterative coupling; multirate sceme; contraction mapping 1 Introduction Te accurate modeling of coupled fluid flow and mecanical interactions as received more attention and importance for bot environmental and petroleum engineering applications. Accurate and reliable numerical metods for solving suc problems are needed for te accurate modeling of multiscale and multipysics penomena suc as reservoir deformation, surface subsidence, well stability, sand production, waste deposition, pore collapse, fault activation, ydraulic fracturing, CO 2 sequestration, and ydrocarbon recovery [13], [17]. Traditionally, canges in mecanical deformations are visible to fluid flow troug a pore compressibility factor, wic is insufficient for stress sensitive, and structurally weak reservoirs. In fact, it is only troug te coupling between flow and mecanics tat reliable 1

3 reservoir models can be obtained. In several of te applications listed above, te mecanics and flow equations ave different caracteristic time scales. Multirate scemes exploit te different time scales of tese two equations and allow taking different time steps for eac of tese two problems. Tis is naturally acieved by decoupling te two equations. Tere are typically tree different coupling approaces employed in modeling fluid flow coupled wit reservoir geomecanics. Tey are known as te fully implicit, te explicit or loose coupling, and te iterative coupling metods. Te fully implicit approac solves reservoir multipase flow and mecanics equations simultaneously and enjoys excellent stability properties [15] toug it poses certain computational callenges for te linear solver. On te oter and, te loosely coupled approac is less accurate, only conditionally stable but, contrary to te implicit coupling sceme, as a lower computational cost. Te iterative coupling approac lies in between te two extremes, and solves te two coupled subsystems iteratively by excanging te values of te sared state variables in an iterative manner. Te procedure is iterated at eac time step until te solution is obtained wit an acceptable tolerance [6, 16, 17, 26]. Te iterative coupling approac allows te use of existing reservoir simulators, is easy to implement, is robust and as fast convergence provided it as been designed appropriately. Our proposed numerical metod is based on suc an iterative approac. Tese iterative metods can be also used as a pre-conditioner for te fully implicit metod. Te work of Gai et al [8, 10] interpreted te fixed stress split iterative coupling sceme as an effective pysics-based preconditioning strategy applied to a Ricardson fixed point iteration. Te same preconditioning operator can be applied to te fully implicit coupled system, enancing te underlying Krylov subspace iteration as well [4, 5, 8]. Te coupled flow and geomecanics problem as been intensively investigated in te past. Starting from te pioneering work of Terzagi [24] and Biot [2, 3], several nonlinear extensions ave been proposed and investigated [7, 9, 21]. Te work of Settari and Mourits [20] proposed a robust iterative and explicit coupling scemes for coupling flow wit geomecanics along wit fracture propagation. Te existence, uniqueness, and regularity of te Biot system ave been investigated by a number of autors Sowalter [23], Pillips & Weeler [19], and Girault et al [12]. However, te development and analysis of teoretically convergent iterative coupling algoritms in poro-elastic media ave received relatively less attention. Recently, te work of Mikelić and Weeler [18] establises geometric convergence contraction wit respect to appropriately cosen metrics for different flow and geomecanics iterative coupling scemes. In addition, Kim et al. [14, 15] ave used von Neumann stability analysis to study te stability and convergence of similar scemes. Our work is inspired from te previous work of Mikelić and Weeler [18] see also [11] and extends teir results to cover te case of fully discrete multirate iterative coupling scemes. Convergence properties of multirate explicit coupling scemes ave been eavily investigated in [22, 29] for te non-stationary Stokes-Darcy model. In contrast, we consider multirate iteratively coupled flow and geomecanics problems in tis work. Figures 1.1a and 1.1b illustrate te differences between single rate versus multirate iterative coupling scemes. Figure 1.1a represents a typical single rate sceme, in wic te flow and mecanics problems sare te exact same time step, and te coupling iteration continues until convergence. In contrast, Figure 1.1b demonstrates a typical multirate sceme, in wic te flow problem takes multiple finer local time steps witin one coarser mecanics time step for eac iterative coupling iteration. Te process is iterated until convergence. In tis work, we 2

4 propose different multirate iterative scemes and teir analyses and deduce te contracting caracter of eac sceme. Convergence immediately follows by applying Banac s fixed point teorem. Te presence of two different time steps for different equations in suc a system of PDEs introduces several complications. We define an appropriate expression of te volumetric mean stress for te multirate sceme and use te flow and mecanics estimates to derive a contraction for te difference of two successive coupling iterates. In addition, we employ matematical induction along wit a compactness argument to deduce strong convergence of te pressure and flux unknowns for flow finer time steps witin a coarser mecanics time step. Our analysis also reveals te optimal value of te fixed stress split regularization term in te mass conservation equation. Moreover, we introduce a modified multirate iterative coupling sceme tat successively corrects te fluxes in even coupling iterations so tat te resulting sceme as te same convergence properties as of single rate sceme. To te best of our knowledge, tis is te first analysis of multirate scemes for Biot equations. To summarize, our contributions in tis work are as follows: we formulate two multirate iterative coupling scemes for te Biot system tat can be viewed as te extensions of te classical fixed-stress split coupling algoritm see [18] to te multirate settings in wic flow takes finer time steps compared to te mecanics problem. Furtermore, we establis te contracting beavior of bot scemes leading to geometric speed of convergence wit an explicit expression for te contracting factor. In terms of numerical analysis, te novelty is in combining te compactness property wit an induction argument to sow tat te obtained solution converges to te unique solution of te original weak formulation given in Definition 2.2. Moreover, te numerical examples sow te sarpness of te teoretical estimates. Tis also reveals te CPU time savings as a result of te reduction in te number of mecanics linear iterations for te multirate sceme versus te single rate sceme. Finally, our proof outlines a general strategy tat is likely to be useful for obtaining similar estimates in oter contexts. For instance, a similar strategy as been applied to te multirate undrained-split coupling sceme in [1] improving te contraction obtained in [18]. Te paper is structured as follows. We present te model and discretization in Section 2. Te multirate sceme is introduced and analysed in Section 3. We also present a modified multirate sceme along wit its analysis in Section 4. We discuss te conclusions and outlook in Section Preliminaries Let Ω be a bounded domain open and connected of IR d, were te dimension d = 2 or 3, wit a Lipscitz continuous boundary Ω, and let Γ be a part of Ω wit positive measure. Wen d = 3, we assume tat te boundary of Γ is also Lipscitz continuous. In general, we assume tat te boundary is decomposed into Diriclet boundary Γ D, and Neumann boundary Γ N, associated wit Diriclet and Neumann boundary conditions respectively, suc tat Γ D Γ N = Ω. In addition, Let DΩ be te space of all functions tat are infinitely differentiable and wit compact support in Ω and let D Ω be its dual space, i.e. te space of distributions in Ω. As usual, we denote by H 1 Ω te classical Sobolev space H 1 Ω = {v L 2 Ω ; v L 2 Ω d }, 3

5 a Single Rate b Multirate Figure 1.1: Flowcart for iterative algoritm using single rate and multirate time stepping for coupled geomecanics and flow problem equipped wit te semi-norm and norm: v H 1 Ω = v L 2 Ω d, v H 1 Ω = v 2 L 2 Ω + v 2 H 1 Ω 1/2. More generally, for 1 p <, W 1,p Ω is te space normed by W 1,p Ω = {v L p Ω ; v L p Ω d }, v W 1,p Ω = v L p Ω, v W 1,p Ω = v p L p Ω + v p W 1,p Ω 1/p, wit te usual modification wen p =. We also define: H 1 0 Ω = {v H 1 Ω ; v Ω = 0}, and for te divergence operator, we sall use te space equipped wit te norm Hdiv; Ω = {v L 2 Ω d ; v L 2 Ω}, v Hdiv;Ω = v 2 L 2 Ω + 1/2 v 2 L 2 Ω 2 Model equations, discretization and splitting algoritm We assume a linear, elastic, omogeneous, and isotropic porous medium Ω R d, d = 2 or 3, in wic te reservoir is saturated wit a sligtly compressible viscous fluid. 4

6 2.1 Assumptions We ave te following assumptions on te model and data: 1. For mecanical modeling, te reservoir is assumed to be omogeneous, isotropic and saturated poro-elastic medium. Te reference density of te fluid ρ f > 0 is given and positive. 2. Te Lamé coefficients λ > 0 and G > 0, te dimensionless Biot coefficient α, and te pore volume ϕ are all positive. 3. Te fluid is assumed to be sligtly compressible and its density is a linear function of pressure. Te viscosity µ f > 0 is assumed to be constant. 4. Te absolute permeability tensor, K, is assumed to be symmetric, bounded, uniformly positive definite in space and constant in time. We will use ere a quasi-static Biot model wic is quite standard in literature [3,13]. Te model reads: Find u and p satisfying te equations below for all time t ]0, T [: t div σ por u, p = f in Ω, σ por u, p = σu α p I in Ω, σu = λ ui + 2 Gεu in Ω, 1 M + c f ϕ 0 p + α u 1 µ f K p ρ f,r g η = q in Ω, Boundary Conditions: u = 0 on Ω, K p ρ f,r g η n = 0 on Γ N, p = 0 on Γ D, Initial Condition t = 0 : 1 M + c f ϕ 0 p + α u 0 = 1 M + c f ϕ 0 p0 + α u 0. were: g is te gravitational constant, η is te distance in te vertical direction assumed to be constant in time, ρ f,r > 0 is a constant reference density relative to te reference pressure p r, ϕ 0 is te initial porosity, M is te Biot constant, q = q ρ f,r were q is a mass source or sink term taking into account injection into or out of te reservoir. We remark tat te first tree equations describe te mecanics wereas te fourt one is te flow equation. Note tat te above system is linear and coupled. 2.2 Mixed variational formulation We will use a mixed formulation for te flow and conformal Galerkin formulation for te mecanics equation. Te mixed metod defines flux as a separate unknown and rewrites te flow equation as a system of first order equations. Suc a formulation for te flow is standard and is preferred because it is locally mass conservative and as an explicit computation for te flux. Accordingly, for te fully discrete formulation discrete in time and space, we use a mixed finite element metod for space discretization and a backward- Euler time discretization. Let T denote a regular family of conforming triangular elements of te domain of interest, Ω. Using te lowest order Raviart-Tomas RT spaces, we ave 5

7 te following discrete spaces V for discrete displacements, Q for discrete pressures, and Z for discrete velocities fluxes: V = {v H 1 Ω d ; T T, v T P 1 d, v Ω = 0} 2.1 Q = {p L 2 Ω ; T T, p T P 0 } 2.2 Z = {q Hdiv; Ω d ; T T, q T P 1 d, q n = 0 on Ω} 2.3 Te space of displacements, V, is equipped wit te norm: v V = d v i 2 1/2. Ω i=1 We also assume tat te finer time step is given by: t = t k t k 1. If we denote te total number of timesteps by N, ten te total simulation time is given by T = t N, and t i = i t, 0 i N denote te discrete time points. For te fully discrete sceme, we ave cosen te Raviart-Tomas spaces for te mixed finite element discretization. However, te proof extends to oter coices for te mixed spaces and we will state te results for Multipoint Flux Mixed Finite Element MFMFE spaces [26, 27] in Remark 4.4. Remark 2.1 Notation: In te following, tere will be two indices, one for te time step and te oter for te coupling between te flow and mecanics. To avoid any confusion, let us empasise te following notations, n denotes te coupling iteration index, k is te coarser time step iteration index for indexing mecanics coarse time steps, m is te finer local time step iteration index for indexing flow fine time steps, t stands for te time step, and q is te fixed number of local flow time steps per coarse mecanics time step. 2.3 Fully discrete sceme for multirate As discussed above, using te mixed finite element metod in space and te backward Euler finite difference metod in time, te weak formulation of a multirate sceme reads as follows. Definition 2.2 For 1 m q, find p m+k flow equation θ Q, q Z, 1 1 t α q t and mecanics equation M + c f ϕ 0 p m+k u k+q K 1 z m+k, q = Q, and z m+k p m 1+k, θ q, θ, θ + u k p m+k, q + Z suc tat, + 1 z m+k µ, θ = f, 2.4 ρ f,r g η, q, 2.5 6

8 find u k+q V suc tat, v V, 2G εu k+q, εv + λ u k+q, v α p k+q, v = f, v. 2.6 wit te initial condition for te first discrete time step, p 0 = p Note tat te pressure unknowns p and flux unknowns z are being solved at finer time steps t k+m, m = 0,..., q wereas te mecanics variables u are being solved at t iq, i N. Terefore, for eac mecanics time step of size q t, tere are q flow solves justifying te nomenclature of multirate. Moreover, te above system of PDEs is linear but coupled wit te coupling terms being computed at te coarse time steps. Instead of solving te problem in a coupled manner, as discussed before, we will apply a splitting algoritm to decouple te two equations and iterate between tem until te solutions satisfying te above system are obtained. In practice, tere are 4 major splitting algoritms drained, undrained, fixed strain and fixed stress used for studying te Biot system depending upon weter one solves te mecanics first or flow and te pysical variables wic are being lagged. We will use te fixed stress splitting algoritm ere because of its well establised stability and excellent convergence properties. 2.4 Standard Fixed stress split algoritm In te fixed stress split iterative coupling algoritm, we first solve te flow problem followed by te geomecanics problem. Even toug we use te splitting strategy at te discrete level, it is probably easier to see tis in te continuous strong form. Recalling tat n denotes te coupling iteration index between te flow and mecanics problems, te steps are as follows: Step a: Given u n, we solve for p n+1, z n+1 1 M + c f ϕ 0 + α2 λ t pn+1 z n+1 = α2 λ z n+1 = 1 µ f K p n+1 ρ f,r g η Once te flow is computed, we update te displacement solution. 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 wit te initial condition, independent of n, σ por u n+1, p n+1 = σu n+1 α p n+1 σu n+1 = λ u n+1 I + 2 Gεu n+1 t pn α t un + q 1 M + c f ϕ 0 p n+1 + α u n+1 0 = 1 M + c f ϕ 0 p0 + α u Note tat te flow equation as a regularization term α 2 /λ t p n+1 added to te left and side and a similar term added to te rigt and side for consistence wile te mecanics 7

9 equation remains uncanged. In te case of convergence, tis term vanises retrieving te original equation. Indeed, tis as been analyzed in literature and we simply state te results to elucidate our approac. Following result is obtained in Mikelić and Weeler [18], and adapted to our model equations. Teorem 2.3 [Mikelić & Weeler [18]] Let Ω t := Ω 0, t, σ n v := σ v,0 + λ u n αp n, and δ denoting te difference of two successive iterates, te fixed stress split sceme as given in Section 2.4 is a contraction given by t δσv n λmα 2 Ω t µ f Mα 2 +λ1+mc f ϕ 0 K 1/2 δ p n+1 t 2 + 4Gλ ε t δu n+1 2 Ω Ω t +λ 2 δu n+1,k 2 2 Mα 2 t Ω t λ+mλc f ϕ 0 δσ n +Mα 2 v 2 Ω t. Te proof of te above results can be adapted to te fully discrete case in wic a mixed formulation is used for space discretization see section 4 and Teorem 4.3. Moreover, in te Teorem 2.3, te contraction is obtained on te volumetric mean stress, σ v, involving bot pressure flow and displacement mecanics unknowns. A relatively straigtforward compactness argument sows tat te converged quantities solve te original coupled equations in a weak form. In wat follows, we will derive similar estimates for te case of te multirate iterative coupling sceme. Two different multirate iterative coupling algoritms are discussed and analyzed below. Even toug our approac is similar to te one in [18], te fact tat we solve for multiple flow finer time steps witin one coarser mecanics time step leads to several complications. Te adaptation of te fixed stress algoritm requires defining an appropriate mean stress quantity and te analysis introduces two adjustable parameters. Careful algebraic manipulations are required to sow te contraction. Even after te contraction is acieved, te presence of te two different time scales in te coupled problem requires non-trivial arguments involving te matematical induction to sow convergence to te weak formulation Multirate iterative coupling scemes 3.1 Multirate iterative sceme Here, we provide a multirate formulation of te fixed stress split iterative coupling algoritm and analyze its convergence properties in te next section. Recall tat n denotes te coupling iteration index, k te coarser time step iteration index for indexing mecanics time steps, m te finer local time step iteration index for indexing flow finer time steps, t te unit time step, and q denote fixed number of local flow time steps witin one coarse mecanics time step. We begin by describing te algoritm. Te weak formulation of equations reads: 8

10 Algoritm 1: Multirate Iterative Coupling Algoritm 1 for k = 0, q, 2q, 3q,.. do /* mecanics time step iteration index */ 2 for n = 1, 2,.. do /* coupling iteration index */ 3 First Step: Flow equations 4 Given u n,k+q assuming an initial value is given for te first iteration: u 0,k+q 5 for m = 1, 2,.., q do /* flow finer time steps iteration index */ 6 Solve for p n+1,m+k and z n+1,m+k satisfying: 1 M + c f ϕ 0 + L p n+1,m+k z n+1,m+k p n,m+k L p n+1,m 1+k t p n,m 1+k t α + 1 z n+1,m+k µ f u n,k+q = u n,k q t + q 3.1 = K p n+1,m+k ρ f,r g η Second Step: Mecanics equations 8 Given p n+1,k+q and, z n+1,k+q, solve for u n+1,k+q satisfying: div σ por u n+1,k+q, p n+1,k+q = f 3.3 σ por u n+1,k+q σu n+1,k+q, p n+1,k+q = σu n+1,k+q α p n+1,k+q I 3.4 = λ u n+1,k+q I + 2 Gεu n+1,k+q 3.5 9

11 Step a For 1 m q, find p n+1,m+k θ Q, 1 1 t Q, and z n+1,m+k Z suc tat, M + c f ϕ 0 + L p n+1,m+k p n+1,m 1+k, θ L p n,m+k p n,m 1+k α q u n,k+q + 1 z n+1,m+k µ, θ = 1 f t + u n,k, θ q, θ, 3.6 q Z, K 1 z n+1,m+k, q = p n+1,m+k, q + ρ f,r g η, q, 3.7 wit te initial condition, independent of n, for te first discrete time step, Step b Given p n+1,k+q p n+1,0 = p and, z n+1,k+q, find u n+1,k+q V suc tat, v V, 2G εu n+1,k+q, εv + λ u n+1,k+q, v α p n+1,k+q, v = f, v. 3.9 In te above sceme, L is te adjustable coefficient tat will be cosen appropriately later tis coice completely determines te sceme and q is a user-defined number of finer flow steps. Below we analyze te above weak formulation and deduce te contracting caracter of te iterative sceme. Te proof relies on studying te difference of two successive iterates and uses Banac s fixed point teorem. Te final step is to sow tat te converged quantities satisfy te weak formulation Proof of contraction For a given time step t = t k, we define te difference between two coupling iterates as: δξ n+1,k = ξ n+1,k ξ n,k, were ξ may stand for p, z, and u. In addition, for notational convenience, we define, β = 1 M + c f ϕ 0 + L Step 1: Flow equations For n 1, by taking te difference of two successive iterates of 3.6, wic corresponds to one local flow iteration and its corresponding local flow iteration in te previous flow and geomecanics iterative coupling iteration, testing wit θ = δp n+1,m+k β δp n+1,m+k L δp n,m+k δp n,m 1+k, we obtain 2 + t δz n+1,m+k µ f α δu n,k+q q 10 δu n,k, δp n+1,m+k, δp n+1,m+k = 3.11.

12 Similarly, for te flux equation 3.7, by taking te difference of two successive iterates, followed by taking te difference at two consecutive finer time steps, t = t m+k, and t = t m 1+k, and testing wit q = δz n+1,m+k, we obtain K 1 δz n+1,m+k δz n+1,m 1+k, δz n+1,m+k = δp n+1,m+k, δz n+1,m+k We combine 3.11 wit 3.12, apply Young s inequality and use δu n,k = 0 to obtain βδp n+1,m+k 2 + t K 1 δz n+1,m+k µ δz n+1,m 1+k, δz n+1,m+k f 1 L δp n,m+k 2ε δp n,m 1+k α q δun,k+q 2 + ε δp n+1,m+k 2 2. Te coice ε = β absorbs te pressure term on te rigt and side. Togeter wit a simple expansion of te flux product, we derive β δp n+1,m+k t { K 1/2 δz n+1,m+k 2µ 2 K 1/2 δz n+1,m 1+k 2 f + K 1/2 δz n+1,m+k δz n+1,m 1+k 2} 1 L δp n,m+k 2β δp n,m 1+k α q δun,k+q Te rigt and side constitutes an expression for a quantity to be contracted on. Introducing a new parameter χ, we define te volumetric mean stress for 1 m q as χδσ n,m+k v = Lδp n,m+k δp n,m 1+k α q δun,k+q Te value of χ will be cosen suc tat contraction can be acieved on te norm of σv n,m+k, summed over q flow finer time steps, witin one coarser mecanics time step. Multiplying 3.13 by 2 β, summing up for 1 m q, substituting te new definition of te volumetric mean stress 3.14, and noting tat δz n+1,k = 0, we obtain δp n+1,m+k 2 + t K 1/2 δz n+1,k+q βµ 2 f + t βµ f K 1/2 δz n+1,m+k δz n+1,m 1+k 2 1 β 2 χδσv n,m+k Step 2: Elasticity equation For n 1, we take te difference of successive iterates of te mecanics equation 3.9, multiply by a newly introduced parameter, c 0, and test wit v = δu n+1,k+q to get 2Gc 0 εδu n+1,k+q 2 + λc 0 δu n+1,k+q 2 αc 0 δp n+1,k+q, δu n+1,k+q =

13 For te iterative sceme to be contractive, a quantity similar to te rigt and side of 3.15, for te next iterative coupling iteration, n + 1, as to be formed. To acieve tat, we introduce a term involving a summation over all flow finer time steps in 3.16 by noticing tat δp n+1,m+k Substituting 3.17 into 3.16 leads to 2Gc 0 εδu n+1,k+q αc 0 q = δp n+1,k+q λc 0 δu n+1,k+q 2 δp n+1,m+k, δu n+1,k+q = Step 3: Combining flow and elasticity equations By combining 3.18 wit 3.15, and rearranging terms, we form a square term, in expanded form, summed over flow finer time steps witin one coarser mecanics time step, 2Gc 0 εδu n+1,k+q 2 + αc 0 δp n+1,m+k + t βµ f { δp n+1,m+k K 1/2 δz n+1,m+k }, δu n+1,k+q + t δz n+1,m 1+k 2 χ 2 β λc 0 δu n+1,k+q q 2 K 1/2 δz n+1,k+q βµ 2 f δσ n,m+k It remains to coose te values of our newly introduced parameters, χ, L, and c 0, suc tat te coefficients of te expanded square contributes only positive terms to te left and side of Terefore, we expand te rigt and side of 3.19 as δσv n,m+k 2 = L2 δp n,m+k χ 2 + α2 δp n,m 1+k 2 2αL qχ 2 δp n,m+k v δp n,m 1+k, δu n,k+q δu n,k+q χ 2 q Now, we matc te coefficients of te expansion in 3.20 to te coefficients of te expanded square on te rigt and side of For te left and side of 3.19 to remain positive, te following inequalities sould be satisfied 1 L2 χ 2, 2αL qχ 2 = αc 0, λc 0 q α2 χ 2 q 2. Te second inequality gives rise to c 0 = 2L. Te tird inequality gives L α2 qχ 2 2λ. Since te contraction factor is monotone wit respect to L, its minimum is acieved wen 12

14 L = α2 2λ. Te first inequality gives χ2 L 2. Te minimum value of te contraction factor is acieved wen χ 2 = L 2. Terefore, wit L = α2 2λ, χ2 = L 2, c 0 = 2L qχ 2, we group te terms of te expanded square on te left and side of 3.19 to form te quantity of contraction for te next iterative coupling iteration, n + 1, as 2Gc 0 εδu n+1,k+q t βµ f K 1/2 δz n+1,m+k δσv n+1,m+k Clearly, te contraction coefficient: independent of q. 2 + t K 1/2 δz n+1,k+q βµ f δz n+1,m 1+k 2 L 1 M +c f ϕ 0 +L 2 = 2 L 1 M + c f ϕ 0 + L 2 δσv n,m+k 3.21 Mα 2 2λ+2Mλc f ϕ 0 +α 2 M 2 < 1, and Convergence to discrete multirate formulation From te derivation above, we establis convergence of te sequences generated by te multirate fixed stress split algoritm and sow tat te converged quantities satisfy te weak formulation Te proof uses te matematical induction for te finer flow equations combined wit te contraction estimates obtained above. Lemma 3.1 For every coarser mecanics time step, t = t k, tere exist a limit function u k suc tat u n,k u k strongly in H 1 Ω d. Proof. Te contraction result in 3.21 implies tat for a coarser time step t = t k, εδu n+1,k converges geometrically to zero. Tis implies tat εu n+1,k is a Caucy sequence converging geometrically to a unique limit in L 2 Ω. It follows immediately tat u n+1,k is a Caucy sequence converging geometrically to a unique limit in H 1 Ω d, being a Hilbert space. Lemma 3.2 For every two consecutive coarser mecanics time steps, t = t k, and t = t k+q, and for every 1 m q, tere exist limit functions p m+k, z m+k suc tat p n,m+k p m+k in L 2 Ω, z n,m+k z m+k in Hdiv, Ω d, wit strong convergence in te norms of te above spaces. 13

15 Proof. Te contraction result in 3.21 implies tat te quantities q K 1/2 δz n+1,m+k δz n+1,m 1+k 2, and q δσv n+1,m+k 2 converge geometrically to zero. It follows tat for 1 m q, K 1/2 δz n+1,m+k δz n+1,m 1+k 2, and δσv n+1,m+k 2 converge geometrically to zero. Moreover, by 3.2, and Poincaré inequality, K 1/2 δp n+1,m+k δp n+1,m 1+k 2 and δp n+1,m+k 2 converge geometrically to zero, respectively. Tis implies tat for every 1 m q, te finer time step differences p n,m+k p n,m 1+k, z n,m+k z n,m 1+k, and te volumetric mean stress defined by are Caucy sequences in L 2 Ω. σ n,m+k v We will sow strong convergence of te pressure sequence by induction. Te proof of strong convergence of te flux sequence follows in te same way. Given an initial pressure value for t = t 0 : p n,0 = p 0, from te above discussion, p n,1 p 0 is a Caucy sequence in L 2 Ω, and, in turn, p n,1 is a Caucy sequence in te complete space L 2 Ω, and tus as a unique limit. Tis completes te base case for induction. For te inductive ypotesis, we assume tat for any coarser mecanics time step t = t k, and for any 1 m q, p n,m+k is a Caucy sequence converging to a unique limit in L 2 Ω: p n,m+k p m+k in L 2 Ω. We will sow tat p n,m+k+2 is also a Caucy sequence converging to a unique limit in L 2 Ω. Consider te two Caucy sequences in L 2 Ω: p n,m+k+2 p n,m+k+1 and p n,m+k+1 p n,m+k. Let p n,m+k+2 p n,m+k+1 for a, b L 2 Ω. It follows tat Tus, p n,m+k+2 p n,m+k+1 p n,m+k+2 p n,m+k+1 a in L 2 Ω, p n,m+k b in L 2 Ω, + p n,m+k+1 p n,m+k a + b + p m+k a + b in L 2 Ω. in L 2 Ω, by te inductive ypotesis. Tis completes te inductive step. Terefore, we obtain tat for all coarser mecanics time steps t = t k, and for 1 m q, p n,m+k, z n,m+k are Caucy sequences converging geometrically to unique limits in L 2 Ω. For te divergence of te flux, we note tat 3.6 amounts to te following equality a.e. in L 2 Ω: δz n+1,m+k Te convergence of z n,m+k in L 2 Ω follows from te convergence of te differ- and σv n,m+k in L 2 Ω, establised above. Tus, we ave bot converging geometrically to unique limits in L 2 Ω, and ence ence p n,m+k z n,m+k p n,m+k and z n,m+k = βµ f t δpn+1,m+k µ f χ t δσn,m+k v. 14

16 z n+1,k converges to a unique limit in Hdiv, Ω d. It remains to pass to te limit in Tis is straigtforward since te equations are linear and all operators involved are continuous in te spaces invoked in te statements of Lemmas 3.1 and 3.2. Moreover te convergences are strong. Terefore, we easily retrieve te discrete in time formulation. Te above discussions are summarized in te following main result: Teorem 3.3 [Multirate 1] For L = α2 2λ, χ2 = L 2, and c 0 = 2L, te multirate qχ 2 iterative sceme is a contraction given by 2Gc 0 εδu n+1,k+q 2 + q δσv n+1,m+k 2 + t K 1/2 βµ f δz n+1,k+q + t q βµ f K 1/2 δz n+1,m+k δz n+1,m 1+k 2 2 Mα 2 q 2λ+2Mλc f ϕ 0 +α 2 M 2 δσv n,m+k Furtermore, te sequences defined by tis sceme converge to te unique solution of te weak formulation Remark 3.4 We note tat te contraction coefficient obtained in Teorem 3.3 exactly matces te contraction coefficient of te single rate optimized fixed stress split iterative metod in te work of Mikelić and Weeler [18]. 4 Modified Multirate Sceme We introduce te modified multirate iterative coupling algoritm wic results in Banac contraction on te volumetric mean total stress as defined by Mikelić and Weeler [18] for te single rate fixed stress split iterative metod. Te algoritm involves a sligt modification in te iterative coupling algoritm, in wic we employ successive corrections in te flow problem te corrections cancel out in te limit. We split te iterative coupling iteration into an even and odd iterations: in odd coupling iterations, we solve exactly te same mass balance equation solved in te single rate case, in contrast, for even coupling iterations, we add flux correction terms to te left and rigt and sides of te mass balance equation. Te idea is to correct for te flux, as we take finer time steps witin one coarser mecanics time step, so tat te summation of te finer flow equations over one coarser mecanics time step retrieves te weak formation of te single rate case, ence, deduce a contraction result similar to te one obtained by Mikelić and Weeler [18] but for a fully discrete setting

17 Algoritm 2: Modified Multirate Iterative Coupling Algoritm 1 for k = 0, q, 2q, 3q,.. do /* mecanics time step iteration index */ 2 for n = 1, 2,.. do /* coupling iteration index */ 3 First Step: Flow equations 4 Given u n,k+q assuming an initial value is given for te first iteration: u 0,k+q 5 For te first local flow timestep iteration, solve for p n+1,1+k satisfying: 1 M + c f ϕ 0 + α2 p n+1,1+k λ p n+1,k t z n+1,1+k + 1 µ f z n+1,1+k = α 2 λ p n,1+k p n,k t u n,k+q α and z n+1,1+k u n,k q t + q 4.1 = K p n+1,1+k ρ f,r g η if modn,2 = 1 ten /* coupling iteration index n is odd */ 7 for m = 2,.., q do /* flow finer time steps iteration index */ 8 Solve for p n+1,m+k and z n+1,m+k satisfying: 1 M + c f ϕ 0 + α2 λ p n+1,m+k α 2 p n,m+k λ p n+1,m 1+k t z n+1,m+k p n,m 1+k t + 1 µ f z n+1,m+k α = u n,k+q u n,k q t + q 4.3 = K p n+1,m+k ρ f,r g η else /* coupling iteration index n is even */ 10 for m = 2,.., q do /* flow finer time steps iteration index */ 11 Solve for p n+1,m+k and z n+1,m+k satisfying: 12 1 M + c f ϕ 0 + α2 λ = α2 λ p n,m+k p n+1,m+k p m 1+k t p n+1,m 1+k t α z n+1,m+k + 1 u n,k+q z n+1,m+k 1 z n+1,m 1+k µ f µ f + q 1 z n,m 1+k µ f u n,k q t 4.5 = K p n+1,m+k ρ f,r g η Second Step: Mecanics equations 14 Given p n+1,k+q and, z n+1,k+q, solve for u n+1,k+q div σ por u n+1,k+q satisfying:, p n+1,k+q = f

18 Remark 4.1 As indicated earlier, in contrast to te original multirate iterative coupling algoritm Algoritm 1, in Algoritm 2, we split te iterative coupling iterations into even and odd iterations. For te first finer flow time step, we solve exactly te same set of equations for bot cases, as sown in line 5. For subsequent finer flow iterations, in te 1 case of an even coupling iteration, we subtract flux correction terms, µ f z n+1,m 1+k, and 1 µ f z n,m 1+k, from te left and rigt and sides of te mass balance equation respectively, as sown in line 11. Upon convergence, z n+1,m 1+k = z n,m 1+k and bot terms cancel eac oter. In te case of an odd coupling iteration, we solve te same set of equations as in te single rate case, as sown in Line 8. Wit te newly introduced flux correction terms, a summation over finer time steps result in reducing te weak formulation of te multirate sceme to tat of te single rate sceme. Tis allows us to obtain exactly te same contraction coefficient as te one obtained in te single rate case, Teorem 2.3. In addition, te modified sceme contracts on te volumetric mean total stress as defined in [18] for te single rate sceme. 4.1 Proof of contraction of modified multirate sceme Step 1: Reduction to single rate weak formulation Extending te work of [18] to te fully discrete formulation, we define te volumetric mean stress, constituting te quantity to be contracted on, for n 1, as: σ n,m+k v σ n,k+q v = σ n,k v = σ n,k v + λ u n,k + λ u n,k+q αp n,m+k αp n,k+q p n,k for 1 m q 1, 4.8 p n,k for m = q. 4.9 In terms of te difference between two coupling iterates, we ave δσ n+1,m+k v δσ n+1,k+q v = αδp n+1,m+k for 1 m q 1, 4.10 = λ δu n+1,k+q αδp n+1,k+q for m = q In order to obtain te single rate weak formulation, we sum up local flow iterations across one coarser mecanics time step. As we solve different mass balance equations in even versus odd coupling iterations, we need to consider eac case seperately: Coupling iteration index, n, is odd: 1 M + c f ϕ 0 + α2 1 λ t z n+1,m+k α 2 λ t p n,m+k q = K p n+1,m+k p m 1+k p n+1,m 1+k α q t + 1 µ f u n,k+q z n+1,m+k = u n,k + q q 4.12 p n+1,m+k + 1 Kqρ f,r g η 4.13 µ f 17

19 Coupling iteration index, n, is even: Equation 4.13 remains uncanged. For 4.5, we ave: 1 M + c f ϕ 0 + α2 1 λ t 1 µ f q 1 w=1 + q q 1 µ f z n+1,w+k q 1 w=1 p n+1,m+k = α2 λ t p n+1,m 1+k p n,m+k + 1 µ f p m 1+k α q t z n+1,m+k u n,k+q z n,w+k 4.14 Assuming, witout loss of generality, tat n+1 represents an even coupling iteration, and n represents an odd coupling iteration, subtracting 4.12 from 4.14 to form te difference between two consecutive coupling iterates, and taking advantage of 3.17, we derive 1 M + c f ϕ 0 + α2 1 λ t δpn+1,k+q + 1 δz n+1,q+k µ = α f λ t δσn,k+q v Equation 4.15 involves only coarser time step variables. Considering te modified multriate iterative coupling sceme as a single rate sceme, in wic bot te flow and mecanics problems sare te coarser time step, te weak formulation in terms of te differences between coupling iterates reads θ Q, 1 1 t M + c f ϕ 0 + α2 δp n+1,k+q λ, θ + 1 δz n+1,k+q µ, θ f = α λ t δσn,k+q v, θ, 4.16 q Z, K 1 δz n+1,k+q, q = δp n+1,k+q, q, 4.17 v V, 2G εδu n+1,k+q, εv + λ δu n+1,k+q, v α δp n+1,k+q, v = Step 2: Flow equations Recall β = 1 + c f ϕ Mα 2 α λ ; testing 4.16 wit θ = δp n+1,k+q and applying Young s inequality, we obtain β αδp n+1,k+q t δz n+1,k+q µ,δp n+1,k+q = 1 αεδp n+1,k+q f t, 1 ελ σn,k+q v 1 ε 2 αδp n+1,k+q t δσ n,k+q 2ε 2 λ 2 v 2. Te coice ε 2 = β absorbs te pressure term on te rigt and side by its corresponding term on te left and side, leading to βαδp n+1,k+q t δz n+1,k+q µ, δp n+1,k+q 1 δσ n,k+q f βλ 2 v u n,k 18

20 Testing 4.17 wit q = δz n+1,k, we obtain K 1 δz n+1,k, δz n+1,k = δp n+1,k, δz n+1,k Combining 4.20 wit 4.19 leads to a sum of two positive squared norms on te rigt and side of 4.19, in wic te rigt and side constitutes te quantity to be contracted on, β αδp n+1,k+q Step 3: Elasticity equation Testing 4.18 wit v = δu n+1,k, we obtain 2G εδu n+1,k+q 2 + λ δu n+1,k+q Combining 4.21 wit 4.22, we infer { αδp n+1,k+q t K 1/2 δz n+1,k+q µ 2 1 δσ n,k+q f βλ 2 v αδp n+1,k+q + 4Gλ εδu n+1,k+q λ 2 δu n+1,k+q 2 α δp n+1,k+q, λ δu n+1,k+q, δu n+1,k+q = λ 2 δu n+1,k+q 2} 2 tλmα 2 µ f Mα 2 K 1/2 δz n+1,k+q + λ1 + Mc f ϕ Mα 2 2 δσ n,k+q λ + Mλc f ϕ 0 + Mα 2 v Te first tree terms form a square of te volumetric mean stress defined in 4.11, establising te quantity of contraction for te next iterative coupling iteration, n + 1, on te rigt and side of 4.23, as δσv n+1,k+q + λ 2 δu n+1,k+q 2 2 tλmα 2 + µ f Mα 2 K 1/2 δz n+1,k+q + λ1 + Mc f ϕ Gλ εδu n+1,k+q 2 2 Mα 2 2 δσ n,k+q λ + Mλc f ϕ 0 + Mα 2 v 2, Mα wit a contraction coefficient 2 λ+mλc f ϕ 0 +Mα < Convergence to te discrete form In te next lemma, we establis convergence of te sequences generated by te modified multirate iterative coupling sceme for coarser mecanics time steps. Lemma 4.2 For k = 0, q, 2q,.., tere exist limit functions p k, uk, zk suc tat p n,k p k in L 2 Ω, u n,k u k in H 1 Ω d, z n,k z k in Hdiv, Ω d, wit strong convergence in te norms of te above spaces. 19

21 Proof. Te contraction result in 4.24 implies tat δσv n+1,k Ω, δu n+1,k Ω, and K 1/2 δz n+1,k Ω converge geometrically to zero. Tis implies tat σv n+1,k, u n+1,k, and z n+1,k are Caucy sequences converging to unique limits in L 2 Ω. By 4.11, we conclude tat p n,k is a Caucy sequence converging geometrically to a unique limit in L 2 Ω, being a Hilbert space. For te displacements, 4.24 implies tat εδu n+1,k converges geometrically to 0 in L 2 Ω. It follows immediately tat u n+1,k converges geometrically to a unique limit in te Hilbert space H 1 Ω d. For te divergence of te flux, we note tat 4.16 amounts to te following equality a.e. in L 2 Ω: δz n+1,k Te convergence of z n+1,k L 2 Ω. Terefore, we ave bot z n+1,k ence z n+1,k = µ f 1 t M + c f ϕ 0 + α2 δp n+1,k λ µ f α λ t δσn,k v. and σv n,k in converging geometrically in L 2 Ω, in L 2 Ω follows from te convergences of p n+1,k and z n+1,k converges in Hdiv, Ω d. Te existence of te limiting function in Hdiv, Ω d follows from te completeness of te space. It remains to pass to te limit in te weak formulation of Tis is straigtforward in view of te linearity of equations and strong convergences obtained. Teorem 4.3 For coarser mecanics time steps, k = 0, q, 2q,.., te modified multirate iterative sceme is a contraction given by δσv n+1,k tλmα 2 Ω 2 Mα 2 δσ n,k λ+mλc f ϕ 0 +Mα 2 v 2. Ω µ f Mα 2 +λ1+mc f ϕ 0 K 1/2 δz n+1,k 2 + 4Gλ εδu n+1,k 2 Ω Ω + λ2 δu n+1,k 2 Ω Furtermore, te sequences defined by tis sceme converge to te unique solution of te weak formulation of Remark 4.4 All our obtained results remain valid wen te multipoint flux mixed finite element metod MFMFE [27, 28] is used for flow discretization. For clarification, consider te modified multirate sceme. Using te MFMFE metod for flow discretization, 4.24 translate to δσv n+1,k+q + λ 2 δu n+1,k+q 2 2 tλmα 2 + µ f Mα 2 K 1 δz n+1,k+q + λ1 + Mc f ϕ 0, δz n+1,k+q + 4Gλ εδu n+1,k+q 2 Q 2 Mα 2 2 δσ n,k+q λ + Mλc f ϕ 0 + Mα 2 v were K 1.,. Q is te quadrature rule defined in [27] for te MFMFE corresponding spaces. It was sown by Weeler and Yotov in [27], and ten extended to distorted quadrilaterals 20

22 and exaedra in [28], tat for any z Z, K 1 z, z C z 2, for a constant C > 0. Tis immediately leads to a similar contraction result. Te same argument olds for previously derived results in te first multirate sceme described earlier. 5 Numerical Results Te first multirate iterative coupling algoritm Algoritm 1 is implemented in te Integrated Parallel Accurate Reservoir Simulator IPARS on top of a two-pase oil-water flow model coupled wit a linear poroelasticity model. In IPARS, te two-pase oil-water flow model is solved by an iterative implicit pressure explicit saturation IMPES sceme, in wic te Multipoint Flux Mixed Finite Element Metod MFMFE is used for flow discretization. Conformal Galerkin Finite Element Metod is used for discretizing and solving te linear elasticity model. We refer te reader to [25] for a detailed description of te model equations. 5.1 Quarter Wellbore Model In order to assess te applicability of our devised multirate iterative coupling algoritms, we consider a simple test case, consisting of a quarter 3D wellbore model. Te model domain is a 25.0 ft 25.0 ft 25.0 ft cube wit a quarter of a cylindrical wellbore centered along one of its edges. Te mes contains 4200 grid elements, wit 30 elements in te radial direction, 20 elements in te oop direction, and 7 elements in te vertical direction. Finer grids are used near te wellbore, and tey coarsen as tey distance apart from te wellbore. A constant wellbore pressure of 300 psi is enforced on te wellbore surface. No flow boundary conditions are enforced on te rest of te boundary faces. For te mecanics model, we apply a zero displacement boundary condition on top of te cube. For te remaining boundaries, we apply zero normal and zero sear traction boundary conditions. Gravity is neglected in tis model. In addition, altoug te code can andle two-pase flow, we run it as a single pase model by assuming te initial oil concentaion to be zero trougout te wole domain. Detailed specifications of te input parameters can be found in Table Convergence Stopping Critera Te stopping criteria are based on te difference of two successive iterates of porosity. We define, δϕ n,k+q flow δϕ n,k+q mec 1 = M + α2 δp n,k+q λ 5.1 = α δun,k+q + 1 M δpn,k+q 5.2 Te expression 5.2 is te standard definition of te fluid content of te medium [13]. Upon convergence, 4.11 leads to δσv n,k+q = λ δu n,k+q αδp n,k+q = 0. Accordingly, we define mec two convergence stopping criteria as follows: ϕn,k+q flow = α u n,k+q α2 λ pn,k+q < L L ϕn,k+q ϕ n,k+q mec ϕ n,k+q mec 21

23 Total Simulation time: days Finer Unit time step: days Number of grids: 4200 grids Permeabilities: k xx, k yy, k zz 5, 20, 20 md Capillary Pressure: 0 Initial porosity, ϕ Water viscosity, µ w 1.0 cp Oil viscosity, µ o 2.0 cp Initial oil concentration, c o 0.0 lb m/ft 3 running as a single pase Initial pressure, p 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: lb m/ft 3 Initial oil density, ρ o 56.0 lb m/ft 3 Young s Modulus E 1.1E7 psi Possion Ratio, ν 0.4 Biot s constant, α 0.75 Biot Modulus, M 0.5E14 L introduced fixed stress parameter α 2 2λ Table 1: Input Parameters for te Quarter Wellbore Model TOL1, and R n,k+q+1 flow < TOL2, were te latter is te residual of te flow volume conservation equation using te last computed pressure and displacement values p n,k+q and u n,k+q. For te quarter wellbore model, we set TOL1 = TOL2 = Results & Discussion Figure 5.1a sows te accumulated CPU run time for te single rate case q = 1, and for multirate cases: q = 2, 4, and 8. Te case q = 2 results in 14.28% reduction in CPU run time compared to te single rate. q = 4, and q = 8 result in 20.97% and 25.09% 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 coarser mecanics time step q = 2, te total number of mecanics linear iterations was reduced by 45.21% wit reference to te single rate case. Multirate couplings q = 4, and q = 8 result in 70.46% and 84.36% reductions in te number of mecanics linear iterations respectively, wic in turn, reduce te CPU run time as well. For tis problem, te total number of flow iterations for bot te single rate and multirate coupling algoritms are found to be te same. In addition, all four cases perform te same number of flow/mecanics coupling iterations for eac coarse mecanics time step, reducing te number of accumulated mecanics linear iterations for multirate scemes, witout affecting te total number of flow linear iterations. Tis results in multirate coupling scemes to outperform te single rate sceme. We also compare te value of our teoretically driven contraction coefficient against numerically observed contraction coefficient values. Teorem 3.3 gives an expression of te 22

24 a CPU Run Time vs Simulation Days b Total Number of Mecanics Linear Iterations vs Simulation Days Figure 5.1: Quarter Wellbore Model 2 Mα contraction coefficient 2 2λ+2Mλc f ϕ 0 +Mα for te multirate algoritm considered in tis 2 case L = α2 2λ, leading to linear convergence of te multirate sceme. For tis test case, we 2 Mα ave 2 2λ+2Mλc f ϕ 0 +Mα = Table 2 lists te values of contraction coefficients 2 obtained numerically for q = 1, 2, 4, and 8. We consider te iterative coupling iteration for te first coarse mecanics time step, wic takes four coupling iterations to converge, according to te stopping criteria described earlier. We compute te values of te volumetric mean stress defined in 3.14 for te last two coupling iterations. Ratios of tose computed values give estimates of contraction coefficients, obtained numerically, as sown in Table 2. We notice tat contraction coefficients computed numerically are smaller tan te predicted teoretical estimate. Tis is expected since te extra terms on te left and side of te contraction result listed in teorem 3.3 are not included wen computing numerical estimates. Tis is due to te fact tat te main objective ere is to investigate te accuracy of our derived teoretical contraction estimates in capturing te contracting beavior of te sceme. In addition, we notice tat as te number of flow finer time steps solved witin one coarser mecanics time step increases, te values of te computed numerical contraction coefficient estimates decrease. We identify tree factors tat determine te efficiency of multirate scemes: 1. Te relative computational cost of te flow solve versus te mecanics solve: if te computational cost of solving te coupled problem is dominated by te mecanics solve, ten reducing te number of mecanics solve will substantially reduce te overall running time compared to single rate scemes. Te multirate scemes are expected to be more useful in tis case. 2. Longer simulation periods lead to larger time savings. During early time steps in te simulation, relatively larger numbers of coupling iterations are observed. As te model reaces me- 23

25 q δσv 3,m 2 / q δσv 2,m q = 1 q = 2 q = 4 q = Table 2: Numerical Contraction Estimates: Contraction estimates observed numerically are sown for different values of q te number of flow finer time steps witin one coarser mecanics time step. Tese are obtained by taking te ratio of te norms of σ v computed at te last two iterative coupling iterations during te first coarse time step: t, 2 t,4 t, and 8 t for q = 1, 2, 4, and 8 respectively. Te first coarse time step involves four iterative coupling iterations for all te four cases. canics equilibrium, te number of iterative coupling iterations per coarse mecanics time step gets reduced. Tis suggests a dynamic iterative coupling sceme, in wic a single rate sceme is employed during early time steps in te simulation, and as te problem approaces mecanics equilibrium, multirate sceme sould be employed wit adaptive q. 3. Tolerance values used in te convergence stopping criteria affect te efficiency of multirate coupling scemes as well. Loose tolerance values reduce te number of iterative coupling iterations per coarse mecanics time step, wic in turn reduces te overall running time. It is a tradeoff between te desired level of accuracy versus computational efficiency and is problem dependent. Altoug te teory provided in tis work and te numerical example are for single pase flow, we anticipate tat multirate iterative coupling scemes will be of more importance for nonlinear flow problems coupled wit geomecanics, as nonlinearities in te flow problem impose restrictions on te flow time step size. Te multirate iterative coupling sceme would be a natural candidate for suc nonlinear flow problems coupled wit geomecanics. 6 Conclusions and outlook We ave considered two multirate iterative coupling scemes based on te fixed stress split iterative coupling algoritm. For bot scemes, we ave proved Banac fixed-point contraction and convergence to te weak solution of te corresponding multirate fully discrete sceme. Te first sceme is a natural extension of te single rate sceme, and contracts on a composite quantity consisting of pressure and volumetric strain terms. Its proof is optimal in te sense tat te contraction quantity is scaled suc tat more terms on te left and side are absorbed. Te second sceme exibits te feature tat it contracts on te same volumetric mean total stress defined in te single rate sceme. Tis is accomplis by successively correcting te fluxes during even coupling iterations. In addition, tese flux corrections vanis as te coupling iteration approaces convergence. In contrast to te first multirate sceme, te modified multirate sceme as te same rate of convergence as in te single rate sceme. Our analysis limits to one coarser time step and we ave not studied te propagation of error 24