ICES REPORT May Tameem Almani, Kundan Kumar, Gurpreet Singh, Mary F. Wheeler

Transcription

1 ICES REPORT May 2016 Stability o Multirate Explicit Coupling o Geomecanics wit Flow in a Poroelastic Medium by Tameem Almani, Kundan Kumar, Gurpreet Sing, Mary F. Weeler Te Institute or Computational Engineering and Sciences Te University o Texas at Austin Austin, Texas Reerence: Tameem Almani, Kundan Kumar, Gurpreet Sing, Mary F. Weeler, "Stability o Multirate Explicit Coupling o Geomecanics wit Flow in a Poroelastic Medium," ICES REPORT 16-12, Te Institute or Computational Engineering and Sciences, Te University o Texas at Austin, May 2016.

2 Stability o multirate explicit coupling o geomecanics wit low in a poroelastic medium T. Almani 1, K. Kumar 2, G. Sing 1, M. F. Weeler 1 1 Center or Subsurace Modeling, ICES, UT Austin, USA 2 Matematics Institute, University o Bergen, Norway {tameem,gurpreet,mw}@ices.utexas.edu, kundan.kumar@uib.no May 11, 2016 Abstract We consider single rate and multirate explicit scemes or te Biot system modeling coupled low and geomecanics in a poro-elastic medium. Tese scemes are te most widely used in practice tat ollows a sequential procedure in wic te low and mecanics problems are ully decoupled. In suc a sceme, te low problem is solved irst wit time-lagging te displacement term ollowed by te mecanics solve. Te multirate explicit coupling sceme exploits te dierent time scales or te mecanics and low problems by taking multiple iner time steps or low witin one coarse mecanics time step. We provide ully discrete scemes or bot te single and multirate approaces tat use Backward Euler time discretization and mixed spaces or low and conormal Galerkin or mecanics. We perorm a rigorous stability analysis and derive te conditions on reservoir parameters and te number o iner low solves to ensure stability or bot scemes. Furtermore, we investigate te computational time savings or explicit coupling scemes against iterative coupling scemes. Keywords. poroelasticity; Biot; iterative and explicit coupling; multirate sceme; mixed ormulation 1 Introduction Te coupling between subsurace low and reservoir geomecanics plays a critical role in obtaining accurate results or models involving reservoir deormation, surace subsidence, well stability, sand production, waste deposition, ydraulic racturing, CO 2 sequestration, and ydrocarbon recovery [2, 8, 14]. Te quasi-static Biot equations are used to model te subsurace coupled low and mecanics and consists o a system o two coupled linear partial dierential equations, eac o wic is typically associated wit te low and mecanics, respectively. Quite oten in practice, te geomecanics problem as a muc slower evolution tan tat o te low problem. In suc cases, te mecanics problem can cope wit a muc coarser time step compared to te low problem. Te multirate sceme exploits tis dierence in te two equations and allows te low to take several iner time steps beore updating te mecanics and is a natural candidate in tis setting. Figures 1.1a and 1

3 1.1b illustrate te dierences between single rate versus multirate explicit coupling scemes. Figure 1.1a represents a typical single rate sceme, in wic te low and mecanics problems sare te exact same time step. In contrast, Figure 1.1b demonstrates a typical multirate sceme, in wic te low problem takes multiple iner local time steps witin one coarser mecanics time step. Te explicit coupling approac tat we consider ere is te most widely utilized sceme in practice. Te decoupling o te two equations makes it easy to implement and te time marcing witout any iterations leads to a lower computational cost. Te drawback is tat tis sceme is only conditionally stable. For te single rate sceme, te rigorous stability properties ave been investigated in te work o Mikelić and Weeler [15]. However, in te case wen te multiple low time steps are taken or one mecanics time step, it is unclear ow tese stability properties cange. In tis work, we ocus our attention on te explicit coupling approac, establis its stability teoretically or bot ully discrete single rate and multirate scemes, and investigate its computational time savings numerically. Moreover, in contrast to te explicit coupling approac, te iterative coupling approac as been investigated in te past. In tis approac, te two coupled subsystems are solved iteratively by excanging te values o 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 [5, 13, 14]. Multirate iterative coupling scemes, extending te ixed-stress split coupling algoritm, ave been rigorously designed and analyzed in [1, 2]. Unconditional stability o suc scemes ollows immediately by establising teir Banac contraction properties. In addition, multirate iterative coupling scemes, based on te undrained-split coupling algoritm, are sown to be contractive, and tus unconditionally stable [12]. In tis work, we consider explicit coupling scemes and rigorously establis teir stability properties. In addition, we perorm numerical computations on ield scale problems to compare te eiciency and computational perormance o tese two approaces. Te coupled low and geomecanics problem as been intensively investigated in te past pioneered by Terzagi [21] and Biot [3, 4]. Terzagi was te irst to propose an explanation o te soil consolidation process, in wic e assumed tat grains orming te soil are bound togeter by some molecular orces resulting in te ormation o te porous material wit elastic properties. It is te success o Terzag s teory in predicting te settlement o dierent types o soils tat led to te creation o te science o soil mecanics [4]. Biot ten extended Terzag s one dimensional work to te tree-dimensional case, and presented a more rigorous generalized teory o consolidation [4]. A compreensive treatment o poromecanics and te teory o mecanics o porous continua can be ound in [6] by Coussy. Oter nonlinear extensions on te teory o poroelasticity can be ound in [7, 17, 20]. Recently, te work o Mikelić and Weeler [15] establises stability and geometric convergence contraction wit respect to appropriately cosen metrics or dierent low and geomecanics coupling scemes. In addition, Kim et al. [10, 11] ave used von Neumann stability analysis to study te stability and convergence o similar scemes. Te multirate scemes or te non-stationary Stokes-Darcy model ave been investigated in [18, 24]. In tis work o multirate explicit coupling o low wit geomecanics, we establis stability results or bot te single rate and multirate scemes, and investigate teir accuracies and computational time savings numerically. To te best o our knowledge, tis is te irst 2

4 a Single Rate b Multirate Figure 1.1: Flowcart or explicit single rate and multirate time stepping or coupled geomecanics and low problem analysis o te multirate explicit coupling sceme or Biot equations. Te paper is structured as ollows. We present te model and discretization in Section 2. Te single rate and multirate explicit coupling scemes are introduced and analyzed in Section 3. Numerical results are sown in Section 4. Conclusions and outlook are discussed in Section Preliminaries Let Ω be a bounded domain open and connected o IR d, were te dimension d = 2 or 3, wit a Lipscitz continuous boundary Ω, and let Γ be a part o Ω wit positive measure. Wen d = 3, we assume tat te boundary o Γ is also Lipscitz continuous. As usual, we denote by H 1 Ω te classical Sobolev space equipped wit te semi-norm and norm: We also deine: H 1 Ω = {v L 2 Ω ; v L 2 Ω d }, 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. H 1 0 Ω = {v H 1 Ω ; v Ω = 0}, and or 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 + v 2 L 2 Ω 1/2. 3

5 For a vector v in IR d, recall te strain or symmetric gradient tensor εv: εv = 1 2 v + vt. In te sequel we sall use Poincaré s and Korn s inequalities. Poincaré s inequality in H 1 0 Ω reads: Tere exists a constant P Ω depending only on Ω suc tat v H 1 0 Ω, v L 2 Ω P Ω v H 1 Ω. 1.1 Next, recall Korn s irst inequality in H 1 0 Ωd : Tere exists a constant C κ depending only on Ω suc tat v H 1 0 Ω d, v H 1 Ω C κ εv L 2 Ω Model equations, discretization and splitting algoritm We assume a linear, elastic, omogeneous, and isotropic poro-elastic medium Ω R d, d = 2 or 3, in wic te reservoir is saturated wit a sligtly compressible viscous luid. 2.1 Assumptions Te luid is assumed to be sligtly compressible and its density is a linear unction o pressure, wit a constant viscosity µ > 0. Te reerence density o te luid ρ > 0, te Lamé coeicients > 0 and G > 0, te dimensionless Biot coeicient α, and te pore volume ϕ are all positive. Te absolute permeability tensor, K, is assumed to be symmetric, bounded, uniormly positive deinite in space and constant in time. A quasi-static Biot model [4, 8] will be employed in tis work. Te model reads: Find u and p satisying te equations below or all time t ]0, T [: Flow Equation: t 1 M + c ϕ 0 p + α u Mecanics Equations: Boundary Conditions: 1 µ K p ρ,r g η = q in Ω div σ por u, p = in Ω, σ por u, p = σu α p I in Ω, σu = ui + 2 Gεu in Ω u = 0, K p ρ,r g η n = 0 on Ω Initial Condition t = 0 : 1 M + c ϕ 0 p + α u 0 = 1 M + c ϕ 0 p 0 + α u 0. were: g is te gravitational constant, η is te distance in te vertical direction assumed to be constant in time, ρ,r > 0 is a constant reerence density relative to te reerence 4

6 pressure p r, ϕ 0 is te initial porosity, M is te Biot constant, q = were q is a mass source or sink term taking into account injection into or out o te reservoir. We remark tat te above system is linear and coupled troug te Biot coeicient terms. 2.2 Mixed variational ormulation A mixed inite element ormulation or low and a conormal Galerkin ormulation or mecanics will be used. Te mixed ormulation is a locally mass conservative sceme, and allows or explicit lux computation. Te lux is deined as a separate unknown and te low equation is rewritten as a system o irst order equations. Accordingly, or te ully discrete ormulation discrete in time and space, we use a mixed inite element metod or space discretization and a backward-euler time discretization. Let T denote a regular amily o conorming triangular elements o te domain o interest, Ω. Using te lowest order Raviart-Tomas RT spaces, we ave te ollowing discrete spaces V or discrete displacements, Q or discrete pressures, and Z or discrete velocities luxes: q ρ,r 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 o displacements, V, is equipped wit te norm: v V = d v i 2 Ω 1/2. i=1 We also assume tat te iner time step is given by: = t k t k 1. I we denote te total number o timesteps by N, ten te total simulation time is given by T = N, and t i = i, 0 i N denote te discrete time points. For te ully discrete sceme, we ave cosen te Raviart-Tomas spaces or te mixed inite element discretization. However, te proo extends to oter coices or te mixed spaces, and we will state te results or Multipoint Flux Mixed Finite Element MFMFE spaces [23] in Remark 3.6. Remark 2.1 Notation: We adopt te ollowing notations,k denotes te coarser time step iteration index or indexing mecanics coarse time steps, m is te iner local time step iteration index or indexing low ine time steps, stands or te unit iner time step, and q is te ixed number o local low time steps per coarse mecanics time step. 2.3 Fully discrete sceme or single rate As discussed above, using te mixed inite element metod in space and te backward Euler inite dierence metod in time, te weak ormulation o te single rate sceme reads as ollows. 5

7 Deinition 2.2 low equation Find p k+1 Q, and z k+1 Z suc tat, uk uk 1 θ Q, 1 p k+1 M + c ϕ 0 p k, θ + 1 µ z k+1, θ + α q Z, K 1 z k+1, q = p k+1, q + ρ,r gη, q, θ = q, θ Deinition 2.3 mecanics equation Find u k+q V suc tat, v V, 2Gεu k+1, εv + u k+1, v αp k+1, v = k+1, v Fully discrete sceme or multirate Te weak ormulation o te multirate sceme reads as ollows. Deinition 2.4 low equation For 1 m q, ind p m+k Q, and z m+k Z suc tat, 1 θ Q, 1 M + c ϕ 0 p m+k, θ + 1 z m+k µ, θ = α u k q uk q, θ + q, θ, 2.7 q Z, K 1 z m+k, q = p m+k, q + Deinition 2.5 mecanics equation Find u k+q v V, 2Gεu k+q, εv + u k+q ρ,r g η, q, 2.8 V suc tat,, v αp k+q, v = k+q, v. 2.9 wit te initial condition or te irst discrete time step or bot single rate and multirate scemes, p 0 = p Note tat or te multirate sceme, te pressure unknowns p and lux unknowns z are being solved at iner time steps t k+m, m = 0,..., q wereas te mecanics variables u are being solved at t iq, i N. Tereore, or eac mecanics time step o size q, tere are q low solves justiying te nomenclature o multirate. Moreover, te above system o PDEs is linear, decoupled and te inormation excange taking place at te coarse time steps. 3 Analysis o Explicit Coupling Scemes In tis section, we perorm a matematically rigorous analysis o te stability o te single rate and multirate explicit coupling scemes. Recall tat in te single rate case Figure 1.1a, te low and mecanics problems sare te exact same time step. In contrast, in te multirate case Figure 1.1b, te low problem takes q iner local time steps witin one coarser mecanics time step. As as been stated above, te explicit coupling approac is a sequential procedure in wic te low or te mecanics problem is solved irst ollowed by te oter. Tere is no coupling iteration between te two problems. 6

8 3.1 Single Rate Formulation: We start by analyzing te single-rate explicit coupling algoritm, in wic bot low and mecanics sare te same time step. To te best o our knowledge, tis is te irst rigorous matematical analysis o te ully discrete single-rate explicitly coupled Biot system. In addition, te analysis reveals a more general stability condition compared to te one obtained in [16] by elementary means. Te algoritm is given as ollows: Algoritm 1: Single Rate Explicit Coupling Algoritm 1 Given initial conditions u 0 and p0, solve ully implicitly or p1, u1 satisying Biot model 2 or k = 1, 2,... do /* time step index */ 3 First Step: Flow equations 4 Given u k and uk 1 : 5 Solve or p k+1 and z k+1 satisying deinition Second Step: Mecanics equations 7 Given p k+1 and, z k+1 : 8 Solve or u k+1 satisying deinition 2.3 Note tat we begin wit k = 1 and we require bot u 1 and u0 or obtaining p2. In te irst step, we use a ully implicit metod to solve or p 1, u1. Alternatively, to keep te problem decoupled, we can use iterative tecniques suc as ixed stress splitting or undrained splitting [15] Assumptions For notational convenience, we deine β = 1 M + c ϕ 0. For stability to old, we assume te ollowing: Result A 1 β > α2. Our results make explicit te dependence o te stability on te dierence o te above quantities. we ave te ollowing stability result. Teorem 3.1 [Single rate] Under te Assumption A 1 above, te ollowing stability result olds or te single rate explicit coupling sceme or time steps t 0 t k t J : K 1/2 z J+1 µ 2 + K 1/2 z k+1 z k 2 + 2G εu k+1 u k 2 k=1 k=1 + u J+1 u J P 2 C + 2β α2 q + Ω C2 κ 2G k=1 k+1 k=1 k 2 7

9 3.1.3 Stability Analysis: Te proo o te above teorem is carried out in tree steps by considering te low equation, te mecanics equation and ten combining te two togeter. Recall tat β = 1 M + c ϕ 0. Proo. Step 1: Flow equations Testing 2.4 wit θ = p k+1 p k, we obtain β 1 p k+1 p k z k+1 µ, p k+1 p k + α u k uk 1, p k+1 = p k q, p k+1 p k 3.1 Next, we consider te lux equation 2.5. Taking te dierence o two consecutive time steps t = t k+1 and t = t k and testing wit q = z k+1, we obtain: K 1 z k+1 z k, zk+1 = p k+1 p k, zk Substituting 3.2 into 3.1, ater some algebraic manipulations o te resulting term using: aa b = 1 2 a2 b 2 + a b 2, we derive β p k+1 p k K 1/2 z k+1 2µ 2 K 1/2 z k 2 + K 1/2 z k+1 z k 2 + α u k uk 1, p k+1 p k = q, p k+1 p k 3.3 Step 2: Elasticity equation Considering 2.6 or te dierence o two consecutive time steps, t = t k+1 and t = t k, and testing wit v = uk+1 u k, we obtain 2G εu k+1 u k 2 + u k+1 u k 2 α p k+1 = 1 p k, uk+1 u k k+1 k, uk+1 u k 3.4 8

10 Step 3: Combining low and elasticity equations Combining 3.3 wit 3.4 yields β p k+1 p k K 1/2 z k+1 2µ 2 K 1/2 z k 2 + K 1/2 z k+1 z k 2 + 2G εu k+1 u k 2 + u k+1 u k 2 α = u k uk 1, p k+1 p k }{{} + α p k+1 p k, uk+1 u k + q, p k+1 p k }{{}}{{} R 2 R 3 + R 1 1 k+1 k, uk+1 u k }{{} R Denoting by R 1, R 2, R 3, and R 4 te terms on te rigt and side, togeter wit Poincaré s, Korn s, and Young s inequalities, we estimate R 1 α 1 u k 2ɛ u k α ɛ 1 p k p k 2 R 2 α u k+1 2ɛ u k 2 αɛ 2 + p k p k 2 R ɛ 3 q + p k+1 2ɛ 3 2 p k 2 R 4 1 k+1 2ɛ k 2 + ɛ 4 u k u k 2 1 k+1 2ɛ k 2 + ɛ 4PΩ 2 C2 κ εu k u k 2. or ɛ 1, ɛ 2, ɛ 3, and ɛ 4 > 0. Coosing ɛ 1 = ɛ 2 = α, ɛ 3 = 2 α2 β, ɛ 4 = multiplying 3.5 by 2, we derive K 1/2 z k+1 µ 2 K 1/2 z k 2 + K 1/2 z k+1 z k 2 + 2G + u k+1 u k 2 u k u k P + q + Ω 2 C2 κ 2β 2α 2 2G 2G P 2 Ω C2 κ and εu k+1 u k 2 k+1 k Summing up 3.6 or 1 k J, or J time steps, wit telescopic cancellations, we get: K 1/2 z J+1 µ 2 + K 1/2 z k+1 z k 2 + 2G k=1 εu k+1 u k 2 k=1 + u J+1 u J 2 u 1 u 0 2 K + 1/2 z µ + P2 Ω C2 κ 2G 9 k+1 k=1 2 2β α2 2 q k=1 k 2, 3.7

11 Recall tat u 1, z1 ave been computed using te ully implicit time discretization. Using standard a priori estimates or te coupled Biot model, we conclude tat u 1 u0 C 2 and K 1/2 z 1 2 C. Tis completes te derivation. Remark 3.2 Te above proo also provides a way to devise an explicitly coupled algoritm tat is unconditionally stable. For te single rate algoritm, we replace 2.4 by te ollowing equation: low equation Find p k+1 Q and z k+1 Z suc tat, θ Q, 1 M + c ϕ 0 + α2 p k+1 p k, θ + 1 z k+1 µ, θ + α uk uk 1, θ = 3.8 Note tat te stabilisation term α2 pk+1 p k as been added above in contrast to 2.4. Te stability result is ten obtained wit te assumption A 1 relaxed. Te consistence error is expected to be o order O wic is also expected or te sceme. To see te unconditional stability o te new sceme, consider te analog o 3.5 and proceed as in te previous case, β + α2 p k+1 p k K 1/2 z k+1 2µ 2 K 1/2 z k 2 + K 1/2 z k+1 z k 2 + 2G εu k+1 u k 2 + u k+1 u k 2 α = u k uk 1, p k+1 p k }{{} + α p k+1 p k, uk+1 u k + q, p k+1 p k }{{}}{{} R 2 R 3 + R 1 1 k+1 k, uk+1 u k. }{{} R Denoting by R 1, R 2, R 3, and R 4 te terms on te rigt and side, togeter wit Poincaré s, Korn s, and Young s inequalities, we estimate R 1 α 1 u k 2ɛ u k α ɛ 1 p k p k 2 R 2 α u k+1 2ɛ u k 2 αɛ 2 + p k p k 2 R ɛ 3 q + p k+1 2ɛ 3 2 p k 2 R 4 1 k+1 2ɛ k 2 + ɛ 4 u k u k 2 1 k+1 2ɛ k 2 + ɛ 4PΩ 2 C2 κ εu k u k 2. q, θ. 10

12 or ɛ 1, ɛ 2, and ɛ 4 > 0. Coosing ɛ 1 = α, ɛ 2 = α, ɛ 3 = 2β, and ɛ 4 = 2G and multiplying PΩ 2 C2 κ 3.9 by 2, we derive K 1/2 z k+1 µ 2 K 1/2 z k 2 + K 1/2 z k+1 z k 2 + 2G εu k+1 u k 2 + u k+1 u k 2 u k u k P 2 q + Ω C2 κ k+1 2β 2G k and rest o te steps proceeds as ollows. 3.2 Multirate Formulation: Recall tat in te multirate explicit coupling approac, te low problem is solved q times wit a iner time step witin a coarser mecanics time step. Algoritm 2: Multirate Explicit Coupling Algoritm 1 Given initial conditions u 0 and p0, solve implicitly or um, pm, zm, m = 1, 2,..., q satisying ully coupled multirate Biot model 2 or k = q, 2q, 3q,.. do /* mecanics time step iteration index */ 3 First Step: Flow equations 4 Given u k 5 or m = 1, 2,.., q do /* low iner time steps iteration index */ 6 Solve or p m+k and z m+k satisying deinition Second Step: Mecanics equations 8 Given p k+q and, z k+q 9 Solve or u k+q satisying deinition Assumptions Te stability assumption in te multirate case takes te orm: A q β > q + q α2 or q 1, were q is te number o low iner time steps witin one coarse mecanics time step. As in te single rate case, we need to prepare te initial data or starting te time stepping. Accordingly, in te irst step o te multirate algoritm Algoritm 2, or k = 0, and m = 1, 2,..., q, te initial conditions are computed by solving te coupled Biot system wit ully implicit time discretization wit a time step o size or te q coupled solves. Alternatively, decoupled iterative scemes [2, 12] suc as ixed stress iterative single rate sceme can be used to compute u m, pm, zm, m = 1, 2,..., q. Note tat i q = 1, te multirate condition A q is identical to te single rate condition A 1. Our main result is te ollowing stability estimate. 11

13 Teorem 3.3 [Multirate] Under te assumption A q, te ollowing stability result olds or te multirate explicit coupling sceme or mecanics time steps t 0 t k t J, k = q, 2q,..: 2G εu k+q u k 2 K + 1/2 z J+q µ 2 + k=q + u J+q u J 2 q 2 C + 2 β α2 q + q 3.3 Stability Analysis k=q m=1 K 1/2 z m+k 2 P 2 q + Ω C2 κ 2G k=q z m 1+k k+q k=q 2 k Te proo or te stability analysis ollows te same ideas as in te single rate proo, owever te use o multiple time steps requires additional estimates. We ollow te same principle o estimating te low equation ollowed by mecanics equation and ten combining te two togeter to obtain te stability estimates. Proo. Step 1: Flow equations Testing 2.7 wit θ = p m+k β p m+k z m+k µ + α u k q uk q, we get, p m+k, p m+k = q, p m+k 3.12 In te lux equation 2.8, considering te dierence or two consecutive iner time steps t = t m+k and t = t m 1+k, and testing wit q = z m+k, we obtain K 1 z m+k z m 1+k, z m+k = p m+k, z m+k Substituting 3.13 into 3.12, we derive βp m+k 2 + K 1 z m+k µ = α u k q uk q z m 1+k, p m+k, z m+k + q, p m+k Summing across low iner time steps 1 m q, we get use aa b = 1 2 a2 b 2 + a b 2 and te telescopic cancellations β + m=1 m=1 p m+k K 1/2 z m+k 2 + K 1/2 z k+q 2µ 2 K 1/2 z k z m 1+k 2 = α q + q, u k uk q, p m+k m=1 2 p m+k m=

14 Step 2: Elasticity equation Considering 2.9 or te dierence o two consecutive mecanics time steps, t = t k and t = t k+q, and testing wit v = u k+q u k, we obtain 2Gεu k+q u k 2 + u k+q u k 2 αp k+q p k, uk+q u k = Step 3: Combining low and elasticity equations Combining 3.14 wit 3.15 gives β m =1 p m+k 2 + 2Gεu k+q + K 1/2 z k+q 2µ 2 K 1/2 z k 2 + m=1 = α u k q uk q, p m+k m=1 }{{} R 1 + αp k+q p k, uk+q u k }{{ } R 3 + u k 2 + u k+q u k 2 K 1/2 z m+k + q, k+q k, uk+q u k }{{} k+q k, uk+q u k 3.15 z m 1+k p m+k m=1 2 }{{} R 4. R Denoting by R 1 and R 2 te irst two terms on te rigt and side, Youngs s and triangle s inequalities give R 1 α q ɛ1 2 ɛ2 R 2 2 m=1 m=1 p m+k p m+k Using te act tat p k+q p k = m=1 p m+k q 2ɛ 1 u k u k q q 2 q 2ɛ 2 2, togeter wit Young s and triangle s inequalities, te tird term on te rigt and side o 3.16, denoted by R 3, can be written as R 3 αɛ 3 2 m=1 p m+k 2 + qα u k+q 2ɛ u k

15 By Poincaré s, Korn s, and Young s inequalities, te last term on te rigt and side o 3.16, denoted by R 4, can be written as R 4 1 k+q 2ɛ 4 Coosing ɛ 1 = α, ɛ 2 = 2 2 we derive 1 k+q 2ɛ 4 β q K 1/2 z k+q µ 2 K 1/2 z k G k 2 + ɛ 4 u k+q 2 u k 2 k 2 + ɛ 4PΩ 2 C2 κ εu k+q 2 u k 2. α2 + q, ɛ 3 = qα, ɛ 4 = 2G, and multiplying by PΩ 2 C2 κ m=1 K 1/2 z m+k z m 1+k 2 εu k+q u k 2 + u k+q u k 2 u k u k q + q 2 2 β q + q α2 2 P 2 q + Ω C2 κ 2G k+q 2 k We need to impose te ollowing condition: β α2 q + q > 0, wic is noting but te Assumption A q. Summing up equation 3.18 or q k J k is a multiple o q, tat is, k = q, 2q,.., we write 2G εu k+q u k 2 K + 1/2 z J+q µ 2 + K 1/2 z m+k z m 1+k 2 k=q m=1 + u J+q u J 2 K 1/2 z q µ 2 + u q u0 2 k=q q β α2 q + q 2 P 2 q + Ω C2 κ 2G k=q k+q k=q k To estimate te irst two terms on te rigt and side, we need to obtain a priori estimates or te ully implicit sceme or te multirate Biot. Tis a priori estimate is obtained by a sligt variation o te tecnique rom te single rate sceme and yields u q u0 2 Cq 2 2 and K 1/2 z q C. We spare te details o obtaining tese a priori estimates. Putting togeter, we conclude te result. Remark 3.4 As in te single rate case in remark 3.2, te multirate case can also be made unconditionally stable by adding a stabilisation term. In te deinition 2.4, we modiy te low equation 2.7 by adding a stabilisation term γ α2 pm+k p m 1+k, were γ = q +q. Te modiied equation reads: 14

16 low equation For 1 m q, ind p m+k Q, and z m+k Z suc tat, θ Q, 1 1 M + c ϕ 0 + γα2 α q u k uk q p m+k, θ +, θ + 1 µ z m+k, θ = q, θ Te proo or te unconditional stability ollows te same ideas as in te single rate case and is skipped ere. Remark 3.5 For te numerical simulations we will be using te multipoint lux mixed inite element metod MFMFE [22,23] or te low discretization. All our obtained results remain valid or tis discretization. Indeed, or suc a sceme, te stability results 3.19 translates to, µ + 2G K 1 z J+q εu k+q k=q, z J+q + Q µ k=q m=1 u k 2 + u J+q + u q u0 2 q β α2 q + q K 1 z m+k z m 1+k, z m+k z m 1+k Q u J 2 K 1 z q µ, zq Q 2 P 2 q + Ω C2 κ 2G k=q k+q k=q k 2, 3.21 were K 1.,. Q is te quadrature rule deined in [23] or te MFMFE corresponding spaces. It was sown by Weeler and Yotov in [23], and ten extended to distorted quadrilaterals and exaedra in [22], tat or any z Z, C 1 z 2 K 1 z, z Q C 2 z 2, or a constant C 1, C 2 > 0. Tis immediately leads to a similar stability result. Te same argument olds or single rate case. Remark 3.6 Te well source/sink term q can be assumed to be varying wit discrete ine/coarse time steps, and all obtained results remain valid. 4 Numerical Results 4.1 Iterative vs. Explicit Coupling Scemes In tis section, we compare single rate and multirate explicit coupling scemes versus iterative coupling scemes. Bot scemes are implemented in te Integrated Parallel Accurate Reservoir Simulator IPARS on top o a single-pase low model coupled wit a linear poroelasticity model. Te Multipoint Flux Mixed Finite Element Metod MFMFE is used or low discretization and Conormal Galerkin is used or elasticity discretization. Mikelić and Weeler [15] ave analyzed dierent iterative coupling scemes, and ave sown tat te two oten used tecniques known as te ixed-stress split and te undrained-split coupling algoritms are unconditionally stable. Te numerical computations in [14] sow te relative perormances o te two metods wit ixed stress splitting perorming better. 15

17 Total Simulation time: 64.0 days k xx Range: , md Finer Unit time step: 1.0 days k yy Range: , md Number o grids: grids k zz Range: , md Possion Ratio, ν 0.35 Biot Modulus, M 1.E15 psi Biot s constant, α 0.9 = Eν 1+ν1 2ν 4.32E7 psi Initial porosity, ϕ Flow Boundary Conditions: zero low B.C. Fluid viscosity, µ w 1.0 cp Mecanics B.C.: Initial luid pressure, p psi X+ boundary σ xx = σ n x = 10, 000psi luid compressibility c w : 1.E-6 1/psi X- - boundary u = 0, zero displacement Rock compressibility: 1.E-6 1/psi Y+ - boundary u = 0, zero displacement Rock density: lb m/t 3 Y- - boundary σ yy = σ n y = 2000psi Initial luid density, ρ o lb m/t 3 Z+ - boundary u = 0, zero displacement Young s Modulus E 5.0E7 psi Z- - boundary σ zz = σ n z = 1000psi Table 1: Input Parameters or Brugge Field Model % o Reduction in: q = 1 q = 2 q = 4 q = 8 CPU run time 47.34% 47.62% 44.97% 51.69% Number o low linear iterations 48.50% 52.92% 52.16% 53.55% Number o mecanics linear iterations 52.21% 52.20% 51.50% 52.89% Table 2: Computational savings o explicit coupling scemes versus iterative coupling scemes or dierent values o q te number o low ine time steps witin one coarse mecanics time step. In te multirate case te unconditional stability o tese two scemes ave been studied in [2, 12]. For our numerical tests, we consider te iterative ixed-stress coupling algoritm wen comparing te eiciency o te iterative coupling scemes versus explicit coupling scemes Brugge Fileld Model We consider te Brugge ield model [19] or comparing te accuracy and eiciency o iterative versus explicit coupling scemes. Te model consists o a general exaedral elements capturing te ield geometry, wit 30 bottom-ole pressure speciied wells, 10 o wic are injectors at a pressure o 4600 psi, and 20 are producers at a pressure o 1200 psi. Producers are located at a lower elevation compared to injectors. No low boundary condition is enorced across all external boundaries. For te mecanics model, we apply a mixture o zero displacement and traction boundary conditions. we also include te eects o gravity. Detailed speciications o te input parameters can be ound in Table 1. We note ere tat assumptions A 1 and A q are bot satisied or te single rare and multirate explicit coupling cases q = 1, 2, 4, and 8, respectively. 16

18 Accumulated CPU Time minutes Accumulated CPU Run Time vs Simulation Period Iterative Coupling - Single Rate Iterative Coupling - Multirate q = 2 Iterative Coupling - Multirate q = 4 Iterative Coupling - Multirate q = 8 Explicit Coupling - Single Rate Explicit Coupling - Multirate q = 2 Explicit Coupling - Multirate q = 4 Explicit Coupling - Multirate q = 8 Accumulated Mecanics Linear Iterations Accumulated # o Mecanics Linear Itrns vs Simulation Period Iterative Coupling - Single Rate Iterative Coupling - Multirate q = 2 Iterative Coupling - Multirate q = 4 Iterative Coupling - Multirate q = 8 Explicit Coupling - Single Rate Explicit Coupling - Multirate q = 2 Explicit Coupling - Multirate q = 4 Explicit Coupling - Multirate q = Simulation Period days a CPU Run Time vs Simulation Days Simulation Time days b Total Number o Mecanics Linear Iterations vs Simulation Days Accumulated Flow Linear Iterations Accumulated # o Flow Linear Itrns vs Simulation Period Iterative Coupling - Single Rate Iterative Coupling - Multirate q = 2 Iterative Coupling - Multirate q = 4 Iterative Coupling - Multirate q = 8 Explicit Coupling - Single Rate Explicit Coupling - Multirate q = 2 Explicit Coupling - Multirate q = 4 Explicit Coupling - Multirate q = Simulation Time days c Total Number o Flow Linear Iterations vs Simulation Days Figure 4.1: Brugge Field Model Numerical Results 17

19 a Pressure ield at 64.0 days Iterative Coupling b Displacement ield at 64.0 days Iterative Coupling c Pressure ield at 64.0 days Explicit Coupling d Displacement ield at 64.0 days Explicit Coupling Figure 4.2: Iterative vs Explicit Coupling Results: Background: Pressure Proile, Arrows: Mecanical Displacements 18

20 4.1.2 Results & Discussion Figure 4.1a sows te accumulated CPU run time or te single rate case q = 1, and or multirate cases: q = 2, 4, and 8, or bot iterative and explicit coupling scemes. In general, or a ixed q, explicit coupling scemes are more eicient, compared to teir counterpart iterative coupling scemes. Tis is expected as explicit scemes eliminate any coupling iteration between te two problems. Tis results in a uge reduction in te total number o low and mecanics linear iterations or explicit coupling scemes, as sown in Figures 4.1c, and 4.1b respectively. Te results obtained sow tat explicit coupling scemes can reduce te accumulative number o low linear iterations or te wole simulation run by almost 50.0% compared to iterative coupling scemes. In addition, te accumulative number o mecanics linear iterations is reduced as well wen comparing an explicit coupling sceme to an iterative sceme or a ixed value o q. As sown in igure 4.1b, te single rate iterative coupling sceme results in te igest number o total mecanics linear iterations. In contrast, te multirate explicit coupling sceme q=8 results in te lowest number o mecanics linear iterations or te wole simulation run. Computational savings o explicit coupling scemes versus iterative coupling scemes are sown in Table 2. Figures 4.2a and 4.2b sow te pressure and displacement ields or te iterative coupling sceme ater 64.0 days o simulation o te Brugge ield case. Figures 4.2c and 4.2d sow te corresponding ields or te explicit coupling sceme. Te solutions or bot te approaces are airly close wit a sligt dierence between te iterative and explicit coupling being more apparent or pressure ields. Te dierences in displacement ields or bot scemes are negligible. 4.2 Validating Teoretical Assumptions In tis section, we try to validate our teoretically induced assumptions or te single rate and multirate explicit coupling scemes against te Frio ield model. Located on te Gul Coast, near Dayton, Texas, at Sout Liberty oil ield, te Frio ield model is a ield-scale problem wit a geometrically callenging geological ormation [9]. Te ield is curved in te dept direction, wit several tin curved aults [9]. In tis work, we only consider te callenging geometry o te ield, and its real permeability distribution. Gravity eects are included in tis model. Oter input speciications are sown in Table Results & Discussion We recall tat or te single rate case, te stability assumption is 1 M + c ϕ 0 > α2 and in te multirate case it reads 1 M + c ϕ 0 > α2 q + q. We consider a particular coice or q = 2 and or te parameters sown in Table 3, our assumption requires 1 M + c ϕ 0 > For te numerical test cases, we consider two dierent compressibility values corresponding to 1 satisying te stability condition and 2 te stability assumption is violated. In te irst case, we coose c = satisying te stability assumption. Te pressure proile ater 4010 simulation days is sown in igure 4.3a. Resulting pressures lie 19

21 Wells: 3 production wells, 6 injection well Injection well 1: Pressure speciied, psi Injection well 2: Pressure speciied, psi Injection well 3: Pressure speciied, psi Injection well 4: Pressure speciied, psi Injection well 5: Pressure speciied, psi Injection well 6: Pressure speciied, psi Production well 1: Pressure speciied, psi Production well 2: Pressure speciied, psi Production well 3: Pressure speciied, psi Total Simulation time: days Finer low Unit time step: 1.0 days Coarse mecanics time step: 2.0 days q = 2 Number o grids: 891 grids Permeabilities: k xx, k yy, k zz igly varying, range: 5.27E-10, 3.10E+3 md Initial porosity, ϕ 0: 0.2 Fluid viscosity, µ : 1.0 cp Initial pressure, p 0: psi Fluid compressibilities: Case 1, condition is satisied, c : 1.E-4 1/psi Case 2, condition is not satisied, c : 1.E-13 1/psi Case 3, condition is not satisied, c : 1.E-8 1/psi Rock compressibility: 1.E-6 1/psi Rock density: lb m/t 3 Initial luid density, ρ : lb m/t 3 Young s Modulus E: 1.E5 psi Possion Ratio, ν: 0.3 Biot s constant, α: 0.7 Biot Modulus, M: 1.0E16 psi Eν = 1+ν1 2ν psi Flow Boundary Conditions: no low 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 = 2000psi Z+ - boundary EBCZZ1: u = 0, zero displacement Z- - boundary EBCZZN1: σ zz = σ n z = 1000psi Table 3: Input Parameters or Frio Field Model 20

22 a Pressure proile wen te compressibility o te luid satisies te derived stability condition c = Results are pysically correct, and lie between te expected range o values. b Pressure proile wen te compressibility o te luid does not satisy te derived stability condition c = Results are not pysically correct, as pressure values drop below zero. Figure 4.3: Pressure proiles o te multirate explicit coupling sceme q=2 or te Frio ield model. in te expected range o values, based on wells injection and production rates speciied in table 3. Next, we consider te case wen we coose c = , tat strongly violates te stability condition. In tis case, te coupling iteration did not converge, as a result o producing extremely ig pressure values in magnitudes, and tat, in turn, triggered te pore-volume values o grid blocks to exceed teir corresponding bulk-volume values, wic is pysically meaningless. To urter test te eect o compressibility, we increase te compressibility and coose c = still violating te stability condition. In tis case, te pressure proiles ater 4010 simulation days are sown or two compressibilities in igure 4.3b. It is clear rom te igure tat te pressure proiles are unpysical since te pressure values drop below zero. Given te values o te initial pressure, and wells injection and production rates speciied in table 3, tis is a non-pysical solution. 5 Conclusions and outlook In tis work, we considered single rate and multirate explicit coupling scemes or coupling low wit geomecanics in poro-elastic media. We derived stability criteria or bot multirate and single rate scemes and derived te assumptions on reservoir parameters or te stability to old. In addition, we perorm te numerical experiments were we compare te time savings in te explicit coupling scemes compared to te iterative ixed stress scemes. Te multirate iterative scemes ave been proven to be geometrically convergent. Our computational results sow tat, i te parameters satisy te stability condition, explicit coupling scemes reduce CPU run time eiciently as compared to iterative scemes. 21

23 Acknowledgements TA is unded by Saudi Aramco. We tank Paulo Zunino and Ivan Yotov or elpul discussions. KK would like to acknowledge te support o Statoil Akademia Grant Bergen. Te autors would like to acknowledge te CSM Industrial Ailiates program, DOE grant ER25617, and ConocoPillips grant UTA Reerences [1] T. Almani, A. H. Dogru, K. Kumar, G. Sing, and M. F. Weeler. Convergence o multirate iterative coupling o geomecanics wit low in a poroelastic medium. Saudi Aramco Journal o Tecnology, Spring [2] T. Almani, K. Kumar, A. H. Dogru, G. Sing, and M. F. Weeler. Convergence analysis o multirate ixed-stress split iterative scemes or coupling low wit geomecanics. Ices report 16-07, Institute or Computational Engineering and Sciences, Te University o Texas at Austin, Austin, Texas, [3] M. A. Biot. Consolidation settlement under a rectangular load distribution. J. Appl. Pys., 125: , [4] M. A. Biot. General teory o tree-dimensional consolidation. J. Appl. Pys., 122: , [5] L. Y. Cin, L. K. Tomas, J. E. Sylte, and R. G. Pierson. Iterative coupled analysis o geomecanics and luid low or rock compaction in reservoir simulation. Oil and Gas Science and Tecnology, 575: , [6] O. Coussy. Mecanics o Porous Continua. Wiley, West Sussex PO19 1UD, England, [7] X. Gai, R. H. Dean, M. F. Weeler, and R. Liu. Coupled geomecanical and reservoir modeling on parallel computers. In Te SPE Reservoir Simulation Symposium, Houston, Texas, Feb. 3-5, [8] V. Girault, M. F. Weeler, B. Ganis, and M. Mear. A lubrication racture model in a poro-elastic medium. Tecnical report, Te Institute or Computational Engineering and Sciences, Te University o Texas at Austin, [9] M. Juntunen and M. F. Weeler. Two-pase low in complicated geometries - modeling te rio data using improved computational meses. Computational Geosciences, 17: , [10] J. Kim, H. A. Tcelepi, and R. Juanes. Stability and convergence o sequential metods or coupled low and geomecanics: ixed-stress and ixed-strain splits. Comput. Metods Appl. Mec. Engrg., : ,

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