ICES REPORT 16-13 May 2016 Multirate Undrained Splitting or Coupled Flow and Geomecanics in Porous Media by Tameem Almani, Kundan Kumar, Mary F. Weeler Te Institute or Computational Engineering and Sciences Te University o Texas at Austin Austin, Texas 78712 Reerence: Tameem Almani, Kundan Kumar, Mary F. Weeler, "Multirate Undrained Splitting or Coupled Flow and Geomecanics in Porous Media," ICES REPORT 16-13, Te Institute or Computational Engineering and Sciences, Te University o Texas at Austin, May 2016.
Multirate Undrained Splitting or Coupled Flow and Geomecanics in Porous Media T. Almani 1, K. Kumar 2, 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 a multirate iterative sceme or te uasi-static Biot euations modelling te coupled low and geomecanics in a porous medium. Te iterative sceme is based on undrained splitting were te low and mecanics euations are decoupled wit te mecanics solve ollowed by te pressure solve. Te multirate sceme proposed ere uses dierent time steps or te two euations, tat is, uses low steps or eac coarse mecanics step and may be interpreted as using a regularization parameter or te mecanics euation. We prove te convergence o te sceme and te proo reveals te appropriate regularization parameter and also te eect o te number o low steps witin coarse mecanics step on te convergence rate. Keywords. poroelasticity; undrained split iterative coupling; multirate sceme; contraction mapping 1 Introduction Coupling o geomecanics and low in poroelastic media as many important applications suc as subsidence events, carbon seuestration, ground water remediation, ydrocarbon production, enanced geotermal systems, solid waste disposal, and biomedical modeling. Starting rom te pioneering work o Terzagi and Biot [1], tere as been active investigation into te coupled geomecanics and low problems [12]. Te Biot model consists o a geomecanics euation coupled to a low model wit te displacement, pressure and low velocity as unknowns. Tere is a uge literature on Biot euations and tey ave been analyzed by a number o autors wo establised existence, uniueness, and regularity, see Sowalter [14], Pillips and Weeler [9] and reerences terein. In contrast to a ully implicit sceme or solving te coupled model o Biot euations, iterative metods are oten employed in practice [2 4]. Te iterative scemes allow te decoupling o te low and mecanics euations and tus oer several attractive eatures suc as use o existing low and mecanics codes, use o appropriate pre-conditioners and solvers or te two models, and ease o implementation. Te design o iterative scemes owever is an important consideration or an eicient, convergent, and robust algoritm. 1
a Single Rate b Multirate Figure 1.1: Flowcart o te undrained split single rate and multirate iterative coupling algoritms. In te single rate sceme, bot low and mecanics sare te exact same time step t. In te multirate sceme, te low iner time step is t, and mecanics coarser time step is t. In addition, oten we can take a coarser time step or te mecanics euation tan or te low. Here, we consider one o te iterative scemes oten used in practice: undrained splitting and propose a multirate iterative sceme, as illustrated by Figure 1.1b. Tis sceme considers a iner time step or te low model and a coarser time step or mecanics low steps or eac mecanics step and ten perorms an iteration between te mecanics and iner low steps. In contrast, in te single rate sceme, as illustrated by Figure 1.1a, te low and mecanics problems sare te exact same time step, and te coupling iteration continues until convergence. Bot scemes are iterative in te sense tat or eac coarse mecanics time step, we solve or low iner time steps ollowed by a mecanics step and we urter repeat te process. Details about convergence criteria can be ound in [7]. Te converged solutions solve te coupled time-discrete system consisting o low solves and one mecanics solve. Te low iner solve uses te mecanics at te coarse step and ence, te coupled system is ully implicit. Since te cost o mecanics is oten muc more tan te low, a less mecanics solves leads to computational savings. Our work is motivated by te recent work o Mikelić and Weeler [7,8] were tey ave considered dierent iterative scemes or low and mecanics couplings and establised contractive results in suitable norms, see also [6] or studying te von Neumann stability o iterative algoritms, [13] or multirate scemes or Darcy-Stokes, and [10,11] or relationsip o tese iterative metods to te linearization procedures. 2
2 Model euations, discretization and splitting algoritm 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. 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. Our model assume a linear, elastic, omogeneous, and isotropic poro-elastic medium Ω R d, in wic te reservoir is saturated wit a sligtly compressible viscous luid. Te luid is assumed to be sligtly compressible and its density is a linear unction o pressure, wit a constant viscosity > 0. Te reerence density o te luid ρ > 0, te Lamé coeicients λ > 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 uasi-static Biot model [1] will be employed in tis work. Te model reads: Find u and p satisying te euations below or all time t ]0, T [: Flow Euation: t 1 M + c ϕ 0 p + α u Mecanics Euations: div σ por u, p = in Ω, Boundary Cond.: 1 K p ρ,r g η = in Ω σ por u, p = σu α p I in Ω, σu = λ ui + 2 Gεu in Ω u = 0 on Ω, K p ρ,r g η n = 0 on Γ N, p = 0 on Γ D, Initial Cond. t=0: 1 M + c ϕ 0 p + α u 0 = 1 M + c ϕ 0 p0 + α u 0. were: g is te gravitational constant, η is te distance in te direction o gravity assumed to be constant in time, ρ,r > 0 is a constant reerence density relative to te reerence pressure p r, ϕ 0 is te initial porosity, M is te Biot constant, = ρ,r were 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 involving α. 2.1 Mixed variational ormulation We use a mixed inite element ormulation or low and a conormal Galerkin ormulation or mecanics or te spatial discretization and a backward-euler or te 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 3
velocities luxes: V = {v H 1 Ω d ; T T, v T P 1 d, v Ω = 0}, Q = {p L 2 Ω ; T T, p T P 0 }, Z = { Hdiv; Ω d ; T T, T P 1 d, n = 0 on Ω}. We also assume tat te iner time step is given by: t = t k t k 1. I we denote te total number o 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. Te proo presented ere can be easily extended to oter mixed metod approaces e.g., [15] or Conormal Galerkin discretizations. Notation: 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, t stands or te unit iner time step, and is te ixed number o local low time steps per coarse mecanics time step. We start by analyzing te ully discrete single-rate undrained-split coupling algoritm, in wic bot low and mecanics sare te exact same time step. Tis can be viewed as an extension o te work o Mikelić & Weeler [8], in wic conormal Galerkin is used or low discretization versus mixed orm or te low in tis work, and te contraction coeicient obtained ere is optimized. In addition, understanding te strategy o te single-rate proo serves as a good introduction beore tackling te more complicated multirate case. 2.2 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. Deinition 2.1 mecanics euation Find u k+1 V suc tat, v V, 2G εu k+1, εv + λ u k+1, v α p k+1, v =, v. 2.1 and low euation ind p k+1 Q, and z k+1 Z suc tat,, θ θ Q, Z, 1 1 t α t M + c ϕ 0 p k+1 u k+1 K 1 z k+1, = p k, θ + u k p k+1, + wit te initial conditions or te irst discrete time step, + 1 z k+1, θ =, θ, 2.2 ρ,r g η,, 2.3 p 0 = p 0, u 0 = u 0. 2.4 4
2.3 Fully discrete sceme or multirate Te weak ormulation o a multirate sceme reads as ollows. Deinition 2.2 mecanics euation Find u k+ V suc tat, v V, 2G εu k+ and low euation For 1 m, ind p m+k θ Q, Z, 1 1 t, εv + λ u k+, v α p k+, v =, v. 2.5 α t M + c ϕ 0 p m+k u k+ K 1 z m+k, = Q, and z m+k, θ p m 1+k, θ + u k p m+k, + wit te initial conditions or te irst discrete time step, Z suc tat, + 1 z m+k, θ =, θ, 2.6 ρ,r g η,, 2.7 p 0 = p 0, u 0 = u 0. 2.8 3 Single Rate Formulation - Undrained Split Iterative Metod Te sceme starts by solving te mecanics problem ollowed by te low problem, and iterates between te two until convergence is acieved. Te iteration assumes a constant luid mass during te deormation o te structure and can be interpreted as a regularization o mecanics euation. We start by presenting te sceme. In te above, we ave used β = 1 M + c ϕ 0 or te notational convenience. L is a regularization parameter and te corresponding term vanises in te case o convergence. 3.1 Analysis o te single rate undrained sceme Te convergence proo is based on studying te dierence o two successive iterates and deriving te contraction o appropriate uantities in suitable norms. Accordingly, we deine: δξ n,k = ξ n+1,k ξ n,k, were ξ = p, u, or z. Te uantity to be contracted on is a composite one consisting o bot pressure p n,k and volumetric strain terms u n,k. For a particular coupling iteration, n 1, and or time step t k, we deine te uantity to be contracted on as: m n,k = L γ un,k + α γ pn,k, were γ is an adjustable coeicient tat will be selected careully suc tat te sceme acieves contraction on m. Te presence o γ does not alter te contractivity, owever, it simpliies te algebra and provides a systematic tecniue or obtaining similar results or oter problems. 5
Algoritm 1: Undrained Split Single Rate Algoritm 1 or k = 0, 1, 2, 3,.. do /* mecanics time step iteration index */ 2 or n = 1, 2,.. do /* coupling iteration index */ 3 First Step: Mecanics euations 4 Given p n,k+1, solve or u n+1,k+1 satisying assuming an initial value is given or te irst iteration: p 0,k+1 : 2G εu n+1,k+1 λ + L u n+1,k+1 I = α Second Step: Flow euations 5 Given u n+1,k+1, solve or p n+1,k+1 and z n+1,k+1 p n+1,k+1 β t p k pn,k+1 I L u n,k+1 I + 3.1 + 1 z n+1,k+1 = α z n+1,k+1 satisying: u n+1,k+1 t u k + 3.2 = K p n+1,k+1 ρ,r g η 3.3 3.1.1 Weak ormulation o dierence o two successive iterates Considering te dierence between two consecutive iterative coupling iterations, te weak ormulation o euations corresponding to 3.1, 3.2, and 3.3 can be written as ollows. Mecanics step: V suc tat, Given δp n,k+1 rom te previous coupling iteration, ind δu n+1,k+1 v V, 2G εδu n+1,k+1, εv + λ + L δu n+1,k+1, v = α δp n,k+1, v + L δu n,k+1, v, 3.4 Flow step: θ Q, Given δu n+1,k+1, ind δp n+1,k+1 Q, δz n+1,k+1 Z suc tat: β δp n+1,k+1 t, θ + 1 δz n+1,k+1 α, θ = t δu n+1,k+1, θ 3.5 Z, K 1 δz n+1,k+1, = δp n+1,k+1, 3.6 Step 1: Mecanics euation First, we analyze te mecanics euation. Testing 3.4 wit v = δu n+1,k+1, we get: 2G εδu n+1,k+1 2 + λ + L δu n+1,k+1 2 = αδp n,k+1 + L δu n,k+1, δu n+1,k+1 = γδ m n,k+1, δu n+1,k+1 ε δu n+1,k+1 2 1 δ + 2 2ε γ2 m n,k+1 2 6
by Young s ineuality. For ε = λ + L, we obtain, 4G λ + L εδu n+1,k+1 Step 2: Flow euation 2 + δu n+1,k+1 2 γ 2 δ m n,k+1 λ + L 2 2. 3.7 Testing 3.5, wit θ = δp n+1,k+1, and multiplying by t, we get: recall β = 1 M + c ϕ 0 βδp n+1,k+1 2 + t δz n+1,k+1, δp n+1,k+1 = α δu n+1,k+1, δp n+1,k+1 3.8 Testing 3.6 wit = δz n+1,k+1, we get and taking te dierence between tem, we get K 1/2 δz n+1,k+1 Substituting 3.9 into 3.8, we ave β δp n+1,k+1 2 = δp n+1,k+1 2 + t K 1/2 δz n+1,k+1 2 + α Step 3: Combining Mecanics and Flow, δz n+1,k+1. 3.9 δu n+1,k+1, δp n+1,k+1 = 0. 3.10 Multiplying 3.10 by anoter ree parameter c 2 and adding 3.7, we obtain 4G εδu n+1,k+1 λ + L { 2 + c 2 βδp n+1,k+1 2 + c 2 α δu n+1,k+1, δp n+1,k+1 + } δu n+1,k+1 2 + c2 t K 1/2 δz n+1,k+1 2 γ 2 δ m n,k+1 λ + L 2 2. 3.11 Step 4: Identiying te parameters Below we provide te procedure or determining te tree adjustable parameters c 2, γ, and L yielding a contraction. Tese parameters sould be cosen suc tat te terms on te let and side o 3.11 remain positive, and te sceme acieves contraction on m. Clearly, δ m n+1,k+1 2 L 2 = δu n+1,k+1 γ 2 2 + α2 δp n+1,k+1 γ 2 2 + 2αL γ 2 δp n+1,k+1, δu n+1,k+1. Matcing coeicients by comparing wit te terms in te curly brackets in 3.11 provides us wit te ollowing conditions: L 2 γ 2 = 1, α 2 γ 2 = c2 β, 2αL γ 2 = c 2 α. Tis gives, L = γ = α2 2β, c2 = 2 L. Substituting in 3.11 leads to a contraction actor o te orm L 2, λ+l wic is strictly less tan one. 7
3.2 Main result: Optimized Contraction or Single rate Our main result summarises te above contraction result. Teorem 3.1 Wit L = α2 2β and c2 = 4β, te undrained single rate iterative sceme deined α 2 by 3.1-3.3 is a contraction given by 4G εδu n+1,k+1 λ + L 2 + c2 t K 1/2 δz n+1,k+1 2 + δ m n+1,k+1 2 L 2 δ m n,k+1 2. λ + L 3.12 Furtermore, te seuences deined by tis sceme converge to te uniue solution o te weak ormulation 2.1 2.3. Remark 3.2 Te above contraction result implies tat te composite uantity m n+1,k+1, symmetric strain εu n+1,k+1, and lux z n+1,k+1 converge at a geometric rate. Relatively straigtorward arguments tat include standard mixed metod or controlling pressure by lux, Korn s ineuality to control te H 1 norm by te L 2 norm o te symmetric strain tensor, imply te uniue convergence o p n+1,k+1, u n+1,k+1 in L 2 and H 1 norms respectively. Remark 3.3 Te above remark 3.2 can also be used to strengten te contraction coeicient in teorem 3.1. Using triangle s ineuality, δ m n,k L γ δun,k + α γ δpn,k and using standard mixed metod to estimate te pressure by te lux and te Korn ineuality, we obtain δ m n,k C n,k δz εδu + n,k. 3.13 Now consider 3.12 in te teorem 3.1 and denote te irst two terms on te let by I n+1,k+1, tat is, I n+1,k+1 = 4G εδu n+1,k+1 λ + L 2 + c2 t K 1/2 δz n+1,k+1 2. Te above ineuality 3.13 can be rewritten as using a generic C, δ m n+1,k+1 2 C 2 I n+1,k+1. 3.14 Te ineuality 3.12 takes te orm C 2 + 1 δ m n+1,k+1 2 L 2 δ m n,k+1 2 λ + L yielding an improved contraction constant, δ m n+1,k+1 2 C L 2 δ m n,k+1 2. 3.15 C + 1 λ + L In practice, it is diicult to estimate C, owever, te above computations sow te relative contributions o te extra positive terms in 3.12 aect te contraction result observed in practice. 8
Remark 3.4 Te single rate undrained split iterative coupling sceme as been rigorously analyzed by Mikelić and Weeler [8]. In teir analysis, a contraction on te luid mass per bulk volume, deined by: m = m 0 + ρ 0 α u + ρ 0 M p p 0, as been obtained or a continuous in time and space ormulation. Te value o te introduced ree parameter L as been cosen a priori to be L = Mα 2. Following teir approac, and adapting to our ully discrete ormulation mixed orm or low, and conormal Galerkin or mecanics, we ave te ollowing result: Teorem 3.5 [Mikelić & Weeler [8]] For L = Mα 2, ϕ n,k := ϕ 0 +α u n,k + 1 M pn,k p 0, and δ denoting te dierence o two successive iterates, te undrained split sceme as given in algoritm 1 is a contraction given by 4Gα 2 εδu n+1,k λ+mα 2 2 + 2 t K 1/2 M δz n+1,k 2 + 2c ϕ 0 M δp n+1,k 2 + δϕ n+1,k+1 2 Mα 2 2 δϕ n,k+1 λ+mα 2 2. Furtermore, te seuences deined by tis sceme converge to te uniue solution o te weak ormulation 2.1 2.3. We note tat our proo is optimized in te sense tat it reveals te optimal value o te parameter L = α2 2 2β, wic leads to a sarper contraction estimates: L λ+l = α 2 M 2. 2λ+2λMc ϕ 0 +α 2 M 9
4 Multirate Formulation - Undrained Split Iterative Metod Te sceme starts by solving te mecanics problem ollowed by a seuence o low problems, and iterates between te two until convergence is acieved. Te iteration assumes a constant luid mass during te deormation o te structure and can be interpreted as a regularization o mecanics euation. We start by presenting te sceme. Algoritm 2: Undrained Split Multirate Algoritm 1 or k = 0,, 2, 3,.. do /* mecanics time step iteration index */ 2 or n = 1, 2,.. do /* coupling iteration index */ 3 First Step: Mecanics euations 4 Given p n,k+, solve or u n+1,k+ satisying assuming an initial value is given or te irst iteration: p 0,k+ : 2G εu n+1,k+ λ + L u n+1,k+ I = α pn,k+ I L u n,k+ I + 4.1 Second Step: Flow euations 5 Given u n+1,k+ 6 or m = 1, 2,.., do /* low iner time steps iteration index */ 7 Solve or p n+1,m+k and z n+1,m+k satisying: p n+1,m+k β p n+1,m 1+k + 1 t z n+1,m+k z n+1,m+k α = u n+1,k+ u k t + 4.2 = K p n+1,m+k ρ,r g η 4.3 In te above, we ave used β = 1 M + c ϕ 0 or te notational convenience. L is a regularization parameter and te corresponding term vanises in te case o convergence. 4.1 Analysis o te multirate undrained sceme Following a similar approac to tat o te single rate case, te convergence proo is based on studying te dierence o two successive iterates and deriving te contraction o appropriate uantities in suitable norms. We recall tat δξ n,k = ξ n+1,k ξ n,k, were ξ = p, u, or z. It is interesting tat te contracting uantity is a composite one consisting o bot pressure p n+1,k+m and volumetric strain terms u n+1,k+. For a particular coupling iteration, n 1, and between two coarse mecanics time steps t k and t k+, we deine te uantity to be contracted on as: m n+1,k+m = L γ un+1,k+ + α γ pn+1,k+m p n+1,k+m 1, or 1 m, 10
were γ is an adjustable coeicient tat will be selected careully suc tat Banac contraction olds on m. As in te single-rate case, te contractivity o te sceme is not altered by te presence o γ, as it only simpliies te algebra, and scales te te uantity o contraction is a way suc tat te contraction coeicient is most optimized. 4.1.1 Weak ormulation o dierence o two successive iterates Considering te dierence between one local low iteration and its corresponding local low iteration in te previous coupling iteration, and te dierence between two consecutive mecanics coupling iterations, te weak ormulation o euations corresponding to 4.1, 4.2, and 4.3 can be written as ollows. Mecanics step: V suc tat, Flow step: tat: Given δp n,k+ rom te previous coupling iteration, ind δu n+1,k+ v V, 2G εδu n+1,k+, εv + λ + L δu n+1,k+, v = α δp n,k+, v + L δu n,k+, v, 4.4 Given δu n+1,k+, or 1 m, ind δp n+1,m+k Q, δz n+1,m+k Z suc δp n+1,m+k θ Q, β, θ t + 1 δz n+1,m+k, θ = α t δu n+1,k+, θ 4.5 Z, K 1 δz n+1,m+k, = δp n+1,m+k, 4.6 Step 1: Mecanics euation First, we analyze te mecanics euation. Testing 4.4 wit v = δu n+1,k+ by noting tat 2G εδu n+1,k+ = αδp n,k+ = ε 2 2 + λ + L δu n+1,k+ 2 + L δu n,k+, δu n+1,k+ α δp n,m+k δp n,m 1+k δu n+1,k+ 2 + 1 2ε γ2 δp n,m+k δp n,m 1+k L + δun,k+ δ m n,k+m ε = λ + L, we obtain ater some simpliications, 4G λ + L εδu n+1,k+ = δp n,k+ 2 + δu n+1,k+ 2 2, δu n+1,k+, we get: and using Young s ineuality. For γ 2 λ + L 2 δ m n,k+m 2. 4.7 11
Step 2: Flow euation Testing 4.5, wit θ = δp n+1,m+k β = 1 M + c ϕ 0 β δp n+1,m+k α δu n+1,k+, δp n+1,m+k, and multiplying by t, we get: recall 2 + t δz n+1,m+k, δp n+1,m+k = 4.8 Now, consider 4.6 or two consecutive local low iner time steps, t = t m+k, and t = t m 1+k, and test wit = δz n+1,m+k and taking te dierence between tem, we get 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. 4.9 Substituting 4.9 into 4.8, we ave β δp n+1,m+k 2 + t K 1 δz n+1,m+k α δu n+1,k+ By Young s ineuality, wit urter simpliications, δz n+1,m 1+k, δp n+1,m+k βδp n+1,m+k 2 + α δu n+1,k+, δp n+1,m+k t K 1/2 δz n+1,m+k 2 2 K 1/2 δz n+1,m 1+k 2 + K 1/2 δz n+1,m+k, δz n+1,m+k =. δz n+1,m 1+k 2 = 0. Summing or local low time steps and ater some simpliications telescopic cancellations togeter wit te act tat δz n+1,k = 0, we get β δp n+1,m+k t K 1/2 δz n+1,k+ 2 2 + t 2 2 + α δu n+1,k+ K 1/2 δz n+1,m+k, δp n+1,m+k δz n+1,m 1+k 2 = 0. 4.10 12
Step 3: Combining Mecanics and Flow Multiplying 4.10 by anoter ree parameter c 2 and adding 4.10, we obtain 4G λ + L + c2 α εδu n+1,k+ 2 + δu n+1,k+ { c 2 β, δp n+1,m+k δp n+1,m+k 2 + δu n+1,k+ 2} + c2 t K 1/2 δz n+1,k+ 2 2 + c2 t K 1/2 δz n+1,m+k 2 δz n+1,m 1+k 2 γ 2 λ + L 2 δ m n,k+m 2. 4.11 Step 4: Identiying te parameters Note tat we ave tree ree parameters: c 2, γ, and L. Below we provide te procedure or determining tese parameters yielding a contraction. Tese parameters sould be cosen suc tat te terms on te let and side o 4.11 remain positive, and te sceme acieves contraction on m. Clearly, δ m n+1,k+m 2 = L2 δu n+1,k+ 2 γ 2 2 + α2 p n+1,k+m γ 2 p n+1,k+m 1 + 2αL γ 2 p n+1,k+m p n+1,k+m 1, δu n+1,k+. Matcing coeicients by comparing wit te terms in te curly brackets in 4.11 provides L us 2 α = 1, 2 c 2 β, and 2αL = c2 α α2 2 γ 2 γ 2 γ 2. Tis gives, L = γ, L 2β and since te contraction actor is monotone wit respect to L, its minimum is acieved wen we coose, L = α2 2β implying γ = α2 2β and c2 = 42 β α 2. 2 L Using above in 4.11 we note tat te contraction actor is 2 and is smaller wen 2 λ+l 2 is larger. Also, wen = 1, te above contraction rate reduces to tat o te single rate case [8] wen te time steps or te mecanics and low are te same. 4.2 Main result: Contraction or Multirate Our main result summarises te above contraction result. Teorem 4.1 Wit L = α2 2β and c2 = 42 β, te undrained multirate iterative sceme α 2 13
deined by 4.1-4.3 is a contraction given by c 2 t K 1/2 δz n+1,k+ 2 2 + c2 t 2 + δ m n+1,k+m 2 + 4G λ + L εδu n+1,k+ K 1/2 δz n+1,m+k 2 L 2 2 λ + L 2 δz n+1,m 1+k 2 δ m n,k+m 2. 4.12 Furtermore, te seuences deined by tis sceme converge to te uniue solution o te weak ormulation 2.5 2.7. 4.3 Convergence to discrete multirate ormulation From te result obtained above, we establis convergence o te seuences generated by te multirate undrained split algoritm and sow tat te converged uantities satisy te weak ormulation 2.5 2.8. Te proo uses te matematical induction or te iner low euations combined wit te contraction estimates obtained above. Lemma 4.2 For every coarser mecanics time step, t = t k, tere exist a limit unction u k suc tat u n,k u k strongly in H 1 Ω d. Proo. Te contraction result in teorem 4.1 implies tat or a coarser time step t = t k, εδu n+1,k converges geometrically to zero. Using Korn s ineuality, tis implies tat xi u n+1,k, i = 1, 2, 3 is a Caucy seuence converging geometrically to a uniue limit in L 2 Ω d. It ollows immediately tat u n+1,k is a Caucy seuence converging geometrically to a uniue limit in H 1 Ω d, being a Hilbert space. Lemma 4.3 For every two consecutive coarser mecanics time steps, t = t k, and t = t k+, and or every 1 m, tere exist limit unctions 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 o te above spaces. Proo. Te contraction result in teorem 4.1 implies tat te uantities K 1/2 δz n+1,m+k δz n+1,m 1+k 2, and δ m n+1,m+k 2 converge geometrically to zero. It ollows tat or 1 m, K 1/2 δz n+1,m+k δz n+1,m 1+k 2, and δ m n+1,m+k 2 converge ge- ometrically to zero. Moreover, by 4.3, and Poincaré ineuality, 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 or every 1 m, te iner time step dierences p n,m+k 14
p n,m 1+k, z n,m+k z n,m 1+k, and te uantity deined by m n,m+k are Caucy seuences in L 2 Ω. We will sow strong convergence o te pressure seuence by induction. Te proo o strong convergence o te lux seuence ollows in te same way. Given an initial pressure value or t = t 0 : p n,0 = p 0, rom te above discussion, p n,1 p 0 is a Caucy seuence in L 2 Ω, and, in turn, p n,1 is a Caucy seuence in te complete space L 2 Ω, and as a uniue limit. Tis completes te base case or induction. For te inductive ypotesis, we assume tat or any coarser mecanics time step t = t k, and or any 1 m, p n,m+k is a Caucy seuence converging to a uniue 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 seuence converging to a uniue limit in L 2 Ω. Consider te two Caucy seuences 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 or a, b L 2 Ω. It ollows tat Tus, p n,m+k+2 p n,m+k+2 p n,m+k+1 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. Tereore, we obtain tat or all coarser mecanics time steps t = t k, and or 1 m, p n,m+k, z n,m+k are Caucy seuences converging geometrically to uniue limits in L 2 Ω. For te divergence o te lux, we note tat 4.5 amounts to te ollowing euality a.e. in L 2 Ω: δz n+1,m+k Te convergence o z n+1,m+k p n+1,m+k p n+1,m+k p n+1,m+k p n+1,m+k = β t δpn+1,m+k α t δun+1,k+. and m n+1,m+k and m n+1,m+k in L 2 Ω ollows rom te convergence o te dierence in L 2 Ω, establised above te convergence o implies te convergence o u n+1,k+ by deini- and z n,m+k converging geometrically to uniue limits tion. Tus, we ave bot z n,m+k in L 2 Ω, and ence z n+1,k converges to a uniue limit in Hdiv, Ω d. It remains to pass to te limit in 2.5 2.7. Tis is straigtorward since te euations are linear and all operators involved are continuous in te spaces invoked in te statements o Lemmas 4.2 and 4.3. Moreover te convergences are strong. Tereore, we easily retrieve te ully discrete multirate ormulation. Remark 4.4 As in te single rate case discussed in remark 3.3, in te multirate case too te contraction in teorem 4.1 can be improved. Te above proo already provides te arguments 15
reuired. Note tat using triangle s ineuality, δ m n+1,k+m L γ δun+1,k+ + α γ δpn+1,k+m δp n+1,k+m 1, and using standard mixed metod to estimate te pressure by te lux and te Korn ineuality, we obtain δ m n+1,k+m 2 C εδu n+1,k+ 2 + C εδu n+1,k+ 2 + K 1/2 δz n+1,m+k K 1/2 δz n+1,m+k Now consider 3.12 in te teorem 4.1 and denote I n+1,k+ I n+1,k+ = 4G λ + L εδu n+1,k+ 2 + c2 t 2 δz n+1,m 1+k 2 δz n+1,m 1+k 2 + K 1/2 δz n+1,k+ 2. 4.13 K 1/2 δz n+1,m+k + c2 t K 1/2 δz n+1,k+ 2 2. Te above ineuality 4.13 can be rewritten as using a generic C, Te ineuality 4.12 takes te orm C 2 + 1 δz n+1,m 1+k 2 δ m n+1,k+m 2 C 2 I n+1,k+. 4.14 δ m n+1,k+m 2 L 2 δ m n,k+m λ + L yielding an improved contraction constant, δ m n+1,k+1 2 C L 2 δ m n,k+1 2. 4.15 C + 1 λ + L In practice, it is diicult to estimate C, owever, te above computations sow te relative contributions o te extra positive terms in 3.12 aect te contraction result observed in practice. 5 Conclusions and outlook We ave considered single rate and multirate iterative coupling scemes or te seuential coupling o low wit mecanics based on te undrained split iterative coupling algoritm. 2 16
For bot scemes, we ave proved Banac ixed-point contraction, and convergence to te weak solution o te corresponding ully discrete sceme ollows immediately. Te multirate is a natural extension o te single rate sceme, and contracts on a composite uantity consisting o pressure and volumetric strain terms. Contraction proos are optimal in te sense tat contraction uantities are scaled suc tat more terms on te let and side are absorbed. Compared to previously obtained results [8], our derived contraction coeicients are saper. To te best o our knowledge, tis is te irst time a contraction result as been rigorously obtained or te multirate undrained split iterative coupling sceme. However, It sould be noted tat our analysis limits to one coarser time step and we ave not considered and investigated te propagation o error due to spatial and temporal discretizations. Tese error estimates providing te convergence rate can be studied and analyzed, or example, in te spirit o [5]. Furter, te nonlinear extensions o tese scemes, teir matematical analyses and computational perormance are interesting uestions tat will be addressed in uture work. Moreover, te perormance o tese algoritms sould be investigated numerically, and appropriate convergence stopping criteria or sould be devised accordingly. 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 UTA10-000444. Reerences [1] M. A. Biot. General teory o tree-dimensional consolidation. J. Appl. Pys., 122:155 164, 1941. [2] 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, 2003. [3] X. Gai, S. Sun, M. F. Weeler, and H. Klie. A timestepping sceme or coupled reservoir low and geomecanics on nonmatcing grids. In SPE Annual Tecnical Conerence and Exibition, 2005. SPE97054. [4] V. Girault, K. Kumar, and M. F. Weeler. Convergence o iterative coupling o geomecanics wit low in a ractured poroelastic medium. Ices report 15-05, Institute or Computational Engineering and Sciences, Te University o Texas at Austin, Austin, Texas, 2015. [5] 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, 2013. 17
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