ICES REPORT May Saumik Dana and Mary F. Wheeler

Transcription

1 ICES REPORT 8-08 May 208 Te correspondence between Voigt and Reuss bounds and te decoupling constraint in a two-grid staggered solution algoritm or coupled low and deormation in eterogeneous poroelastic media by Saumik Dana and Mary F. Weeler Te Institute or Computational Engineering and Sciences Te University o Texas at Austin Austin, Texas 7872 Reerence: Saumik Dana and Mary F. Weeler, "Te correspondence between Voigt and Reuss bounds and te decoupling constraint in a two-grid staggered solution algoritm or coupled low and deormation in eterogeneous poroelastic media," ICES REPORT 8-08, Te Institute or Computational Engineering and Sciences, Te University o Texas at Austin, May 208.

2 2 3 4 THE CORRESPONDENCE BETWEEN VOIGT AND REUSS BOUNDS AND THE DECOUPLING CONSTRAINT IN A TWO-GRID STAGGERED SOLUTION ALGORITHM TO COUPLED FLOW AND DEFORMATION IN HETEROGENEOUS POROELASTIC MEDIA 5 SAUMIK DANA AND MARY F. WHEELER Abstract. We perorm a convergence analysis o a two-grid staggered solution algoritm or te Biot system modeling coupled low and deormation in eterogeneous poroelastic media. Te algoritm irst solves te low subproblem on a ine grid using a mixed inite element metod (by reezing a certain measure o te mean stress ollowed by te poromecanics subproblem on a coarse grid using a conorming Galerkin metod. Restriction operators map te ine scale low solution to te coarse scale poromecanical grid and prolongation operators map te coarse scale poromecanical solution to te ine scale low grid. Te coupling iterations are repeated until convergence and Backward Euler is employed or time marcing. Te analysis is based on studying te equations satisied by te dierence o iterates to sow tat te two-grid sceme is a contraction map under certain conditions. Tose conditions are used to construct te restriction and prolongation operators as well as arrive at coarse scale elastic properties in terms o te ine scale data. We sow tat te adjustable parameter in te measure o te mean stress is linked to te Voigt and Reuss bounds requently encountered in computational omogenization o multipase composites Introduction. Staggered solution algoritms are used to decompose coupled problems into subproblems wic are ten solved sequentially in successive iterations until a convergence criterion is met at eac time step ([0], [3], [24], [23]. Tese algoritms oer avenues or augmentations in wic subproblems associated wit ine scale penomena can be solved on a ine grid and subproblems associated wit coarse scale penomena can be solved on a coarse grid. Consolidation in deep subsurace reservoirs as inerent lengt scale disparities wit ine scale eatures o multipase low restricted to te reservoir and coarse scale eatures o geomecanical deormation associated wit a domain including but not restricted to te reservoir. In lieu o te above, [7] developed a two-grid staggered solution algoritm in wic te low equations are solved on a ine grid and te poromecanics equations are solved on a coarse grid (wit te grids being non-nested in every coupling iteration in every time step and used te classical Mandel s problem ([7], [] to sow tat te sceme is numerically convergent. Tereater, motivated by te previous work o [8] and [2], [8] establised teoretical convergence o te two-grid sceme o [7] or te degenerate case o nested brick grids wit te low and poromecanical domains being identical, as sown in Figure. Te measure o mean stress tat remains ixed during te low solve is ydrostatic part o te total stress, also reered to as te mean stress. Te interesting result o te work o [8] is tat te convergence analysis lends itsel to an expression or coarse scale bulk moduli in terms o ine scale bulk moduli, and urter te coarse scale moduli are a armonic mean o te ine scale moduli. Te armonic mean is exactly te Reuss bound (see [22]. Tis observation leads to a ypotesis tat tere must be a measure o mean stress wic wen ixed during te low solve in te two-grid approac, leads to te aritmetic mean (Voigt bound or coarse scale bulk moduli in terms o ine scale bulk moduli. Te objective o tis work is to examine te link between te decoupling constraint used in te two-grid approac and eective coarse scale property tat te convergence analysis lends itsel to. Wit tat in mind, we deine a measure o mean stress wic equates to te actual mean stress only as a special case. As a result, te staggering in tis work is a generalization o te ixed stress split staggering tat was studied in [8], [2] and [8]. Tis paper is structured as ollows: Section 2 presents te model equations or low

3 2 S. DANA AND M. F. WHEELER New time step New coupling iteration Solve or pressures on ine grid Project pressures onto coarse grid Solve or displacements on coarse grid Project volumetric strains onto ine grid i converged Ceck or convergence (a i not converged (b Fig.. (a: Two-grid staggered solution algoritm. A measure o te mean stress remains ixed during te low solve. Ater te low solve, te updated pressures are projected onto te coarse scale poromecanics grid. Ater te poromecanics solve, te updated volumetric strains are projected onto te ine scale low grid. (b: One coupling iteration o two-grid sceme. In order to be consistent wit te terminology used in multigrid metods, we reer to projection onto coarse grid as restriction and projection onto ine grid as prolongation and poromecanics, Section 3 presents te statement o contraction o te two-grid ixed stress split iterative sceme, Section 4 presents te details o ow te statement o contraction is used to arrive at restriction and prolongation operators as well as te eective coarse scale moduli, Section 5 presents te two-grid ixed stress split

4 algoritm and Section 6 discusses te link between te decoupling constraint and te Voigt and Reuss bounds... Preliminaries. Given a bounded convex domain Ω R 3, we use Meas(Ω to denote te volume o Ω, P k (Ω to represent te restriction o te space o polynomials o degree less tat or equal to k to Ω and Q (Ω to denote te space o trilinears on Ω. For te sake o convenience, we discard te dierential in te integration o any scalar ield χ over Ω as ollows χ(x χ(x dv ( x Ω Ω Ω Sobolev spaces are based on te space o square integrable unctions on Ω given by L 2 (Ω { θ : θ 2 Ω := θ 2 < + }, 2. Model equations. 2.. Flow model. Te luid mass conservation equation (2. in te presence o deormable porous medium wit te Darcy law (2.2 and linear pressure dependence o density (2.3 wit boundary conditions (2.4 and initial conditions (2.5 is Ω 70 7 (2. (2.2 ζ t + z = q z = K µ ( p ρ 0g = κ( p ρ 0 g 72 (2.3 ρ = ρ 0 ( + c (p p 0 73 (2.4 p = g on Γ D (0, T ], z n = 0 on Γ N (0, T ] (2.5 p(x, 0 = p 0 (x, ρ(x, 0 = ρ 0 (x, φ(x, 0 = φ 0 (x ( x Ω 76 were p : Ω (0, T ] R is te luid pressure, z : Ω (0, T ] R 3 is te luid lux, ɛ 77 is te volumetric strain, Γ D is te Diriclet boundary, n is te unit outward normal 78 on te Neumann boundary Γ N, q is te source or sink term, K is te uniormly 79 symmetric positive deinite absolute permeability tensor, µ is te luid viscosity, ρ 0 80 is a reerence density, κ = K µ is a measure o te ydraulic conductivity o te pore 8 luid, c is te luid compressibility, T > 0 is te time interval, ζ M p + α ɛ is reered to as te luid content (see [5], [2], [9], [6] were α K b 82 K s is te Biot constant 83 (see [4], [], [9] and M is te Biot modulus (see [5] wit K φ 0c+ (α φ 0 ( α b K b 84 being te drained bulk modulus o te pore skeleton and K s being te bulk modulus 85 o te solid grains. For te sake o convenience, we introduce a variable ϕ M + α2 η, 86 were η is an adjustable parameter as we sall in Module Poromecanics model. Te linear momentum balance (2.6 in te quasistatic limit o interest wit te deinition o te total stress (2.7 (see [4] wit te expression or te body orce (2.8 and te small strain assumption (2.9 wit bound-

5 4 S. DANA AND M. F. WHEELER 90 ary conditions (2.0 and initial condition (2. is (2.6 (2.7 (2.8 (2.9 σ + = 0 σ = σ 0 + λ ɛi + 2Gɛ α(p p 0 I = ρφg + ρ r ( φg ɛ(u = 2 ( u + T u 95 (2.0 u n = 0 on Γ p D [0, T ], σt n 2 = t on Γ p N [0, T ] (2. u(x, 0 = 0 ( x Ω 98 were u : Ω [0, T ] R 3 is te solid displacement, ρ r is te rock density, G is te 99 sear modulus, ν is te Poisson s ratio, n is te unit outward normal to te Diriclet 00 boundary Γ p D, n 2 is te unit outward normal to te Neumann boundary Γ p N, α is te 0 Biot parameter, is body orce per unit volume, t is te traction boundary condition, 02 ɛ is te strain tensor, ɛ is te volumetric strain, σ 0 is te in situ stress, λ is te Lame 03 parameter and I te is second order identity tensor Te decoupling assumption. Te basic idea o te two-grid staggered solution strategy is to solve te low system (2.-(2.5 on a ine grid or te pressures at te current coupling iteration based on te value o a certain measure o mean stress rom te previous coupling iteration. We reer to tat measure o mean stress as σ, and is expressed as ollows σ = η ɛ αp were η is an adjustable parameter, wic wen equated to te drained bulk modulus, lends itsel to te total mean stress (reered to as σ v as ollows σ = K b ɛ αp σ v (wen η = K b Tese pressures are ten ed to te poromecanics system (2.6-(2. wic is solved or displacements on a coarse grid tereby updating te stress state. Tis updated stress state is ten ed back to te low system or te next coupling iteration. Since tis strategy condemns te porous solid to ollow a certain stress pat during te low solve, te convergence o te solution algoritm is not automatically guaranteed. It is important to note tat te adjustable η allows or lexibility in te coice o decoupling constraint, and te ixed stress split strategy is only a special case wen te adjustable parameter is identical to te drained bulk modulus i.e. wen η = K b Statement o contraction o te two-grid ixed stress split sceme. 24 Te objective o our analysis is to arrive at a contraction map or te ully discrete 25 two-grid staggered solution algoritm wile taking into account te eterogeneities in 26 te underlying porous medium. Let T represent te ine scale low grid consisting o brick elements and T p H be te coarse scale poromecanical grid consisting o 28 brick elements suc tat r = max diam( max diam(

6 5 ine scale low grid T coarse scale poromecanical grid T p H I Ep { : 8 2 T } Fig. 2. Depiction o nested grids (in a two-dimensional ramework or te sake o convenience. Red dots are vertices o low element(s and green dots are vertices o poromecanical element(s. 3 Since te grids are nested, eac coarse scale poromecanical element T p H can be 32 viewed as a union o low elements belonging to te set I Ep as ollows = were I { : T } I Ep To take into account te underlying eterogeneities in te porous medium, we introduce te notations ( or te value o any material parameter ( at low element and ( or te value o any material parameter ( evaluated at poromecanics element. 3.. Variational statements in terms o coupling iteration dierences. We use te notations ( n+ or any quantity ( evaluated at time level n+, ( m,n+ or any quantity ( evaluated at te m t coupling iteration at time level n+, δ (m ( or te cange in te quantity ( during te low solve in te (m + t coupling iteration at any time level and δ (m ( or te cange in te quantity ( over te (m + t coupling iteration at any time level. Te discrete variational statements in terms o coupling iteration dierences is : ind δ (m p W, δ (m z V and δ (m u H U H suc tat ϕ (δ (m p, θ + t( δ (m z, θ (3. (3.2 = α (δ (m σ, θ (κ δ (m z, v = (δ (m p, v (3.3 2G (e(δ (m u H, e(q H + (δ (m σ, q H = 0 were te inite dimensional spaces W, W H V and U H are given by W = { θ : θ P 0 ( T } W H = { θ H : θ H P 0 ( T p } H V = { v : v ˆv Ê : ˆv Ê ˆV(Ê E T, v n = 0 on Γ } N U H = { q H = (u, v, w : u, v, w Q ( T p H, q H = 0 on Γ p } D

7 6 S. DANA AND M. F. WHEELER 58 and te details o ˆV(Ê are given in [7]. Te equations (3., (3.2 and (3.3 are te 59 discrete variational statements (in terms o coupling iteration dierences o (2., 60 (2.2 and (2.6 respectively. Te details o (3. and (3.2 are given in A wereas te 6 details o (3.3 are given in B Restriction and prolongation operators. We introduce te restriction operator R tat maps te ine scale pressure solution onto te coarse scale poromecanics grid and te prolongation operator P tat maps te coarse scale volumetric strain onto te ine scale low grid as ollows R : W W H P : U H W As a result, te measure o te mean stress is deined on te ine and coarse grids as 70 (3.4 σ = η ɛ H α Rp ( T p H σ = η E P ɛ H α E p ( T (3.5 Teorem 3.. In te presence o medium eterogeneities, te two-grid staggered solution algoritm in wic te low subproblem is resolved on a iner grid is a contraction map wit contraction constant γ and given by (3.6 δ (m σ 2 >0 >0 { }}{{ + 4G e(δ (m u H 2 E + }}{ (2K p b η δ (m ɛ H 2 γ< >0 {{ }}{ ( }} { + 2 t κ /2 δ (m z 2 E α 2 E max T + α2 M i te ollowing conditions are satisied. First condition α (Pδ (m ɛ H, δ (m p δ (m σ 2 α (δ (m ɛ H, Rδ (m p = Second condition η δ (m ɛ H 2 Pδ (m ɛ H Tird condition 88 η 2K b ( T p 89 (3.7 H 90 Proo. Step : Flow equations 9 Testing (3. wit θ W suc tat θ = δ (m p T, we get ϕ δ (m p 2 E + 92 t( δ (m z, δ (m p (3.8 = α (δ (m σ, δ (m p

8 7 95 Testing (3.2 wit v V suc tat v δ (m z T, we get κ /2 δ (m z 2 E = 96 (3.9 (δ (m p, δ (m z From (3.8 and (3.9, we get ϕ δ (m p 2 + t κ /2 δ (m z (3.0 = α (δ (m σ, δ (m p Step 2: Invoking te Young s inequality Since te terms on te LHS o (3.0 are strictly positive, te RHS is also strictly positive. We invoke te Young s inequality ab a2 2ε + εb2 2 or te RHS o (3.0 as ollows a, b, ε R, ε > α (δ (m σ, δ (m p 2ε η 2 δ (m σ 2 E + ε 2 α δ(m p 2 E ( E T Since te above inequality is true or any ε > 0, we coose ε = α 2 ϕ α (δ (m σ, δ (m p α2 2 ϕ δ (m σ 2 In lieu o te above, (3.0 is written as ϕ δ (m p 2 E + α 2 2 ϕ wic can also be written as ϕ 2 δ(m p 2 + ( wic, ater noting tat ϕ (3. α 2 + ϕ 2 δ(m p 2 ( E T δ (m σ 2 t κ /2 δ (m z 2 + t κ /2 δ (m z 2 M + α2 δ (m p 2 + α 2 ϕ δ (m σ 2 ϕ 2 δ(m p 2 α 2 2 ϕ > α2 η, can also be written as 2 t κ /2 δ (m z 2 to get δ (m σ 2

9 8 S. DANA AND M. F. WHEELER 226 Step 3: Poromecanics equations 227 Testing (3.3 wit q H Q H suc tat q 228 δ (m u H δ (m ɛ H, we get = 2δ (m u H T p H and noting tat (3.2 4G e(δ (m u H 2 E + 2(δ (m σ p v, δ (m ɛ H = 0 Furter, rom (3.4, we note tat δ (m σ = K b δ (m ɛ H α Rδ (m p T p H. 23 As a result, (3.2 is written as G e(δ (m u H 2 + 2K b δ (m ɛ H (3.3 2α (δ (m ɛ H, Rδ (m p = Step 4: Combining low and poromecanics equations Adding (3. and (3.3, we get α 2 δ (m p 2 + 2K b δ (m ɛ H 2 2 t κ /2 δ (m z 2 + 2α (δ (m ɛ H, Rδ (m p 4G e(δ (m u H (3.4 α 2 ϕ δ (m σ Now, rom (3.5, we note tat δ (m σ 2 = α2 δ(m p 2 + η2 Pδ(m ɛ H 2 2 α (Pδ(m ɛ H, δ (m p ( T 246 wic implies tat α 2 δ (m p 2 (3.5 ( T = δ(m σ 2 Pδ (m ɛ H 2 E + 2α E (Pδ(m ɛ H, δ (m p

10 9 250 Substituting (3.5 in (3.4, we get 25 δ (m σ 2 >0 >0 { }}{{ + 4G e(δ (m u H 2 E + }}{ (2K p b η δ (m ɛ H Set = 0 to obtain expressions or η and Pδ(m ɛ H T [ { }}{ + 2α (Pδ (m ɛ H, δ (m p ] 2α (δ (m ɛ H, Rδ (m p [ Turns out to be 0 in lieu o Caucy Scwartz inequality { }}{ + η δ (m ɛ H 2 E ] η p Pδ (m ɛ H (3.6 >0 { }}{ + 2 t κ /2 δ (m z 2 E γ δ (m σ Te statement (3.6 is a contraction map in a sense tat δ (0 σ 2 > δ ( σ 2 > δ (2 σ 2 > wit contraction constant γ given by γ ( α 2 ( E max = max T η ϕ M α 2 + α2 < provided te ollowing are true (3.7 α (Pδ (m ɛ H, δ (m p α (δ (m ɛ H, Rδ (m p = (3.8 η δ (m ɛ H 2 Pδ (m ɛ H (3.9 η 2K b ( T p H Te objective now is to satisy te conditions (3.7 and (3.8 or te convergence o te two-grid staggered solution algoritm. 4. Satisaction o conditions or te convergence o te ully discrete two-grid staggered solution algoritm. Corollary 4.. Satisaction o te decoupling constraint during te low solve at bot scales leads to te ollowing expressions or te upscaled pore pressures Rδ (m p = η α I Ep α E δ (m Meas( p Meas( ( T p H

11 0 S. DANA AND M. F. WHEELER Proo. Step : Using te act tat pore pressure is rozen during te poromecanical solve Since te pore pressure is rozen during te poromecanical solve, te total pore pressure cange in a coupling iteration is te same as te pore pressure cange calculated during te low solve in te coupling iteration as ollows (4. Rδ (m 280 δ (m (4.2 p = Rδ (m p ( T p H p = δ (m p ( T Step 2: Applying te decoupling constraint on bot scales Now, te decoupling constraint implies tat tere is no cange in te measure o te mean stress o te system during te low solve. Tis naturally implies tat (4.3 (m δ σ = 0 ( T p H In lieu o (3.4, we write te above as (4.4 (m (η Epδ ɛ H α Rδ (m p = 0 ( T p H wic, in lieu o (4., can be written as (4.5 (m δ ɛ H = α Rδ (m p Meas( ( T p H η Denoting δ (m P ɛ H is te cange in volume o eac element o I Ep, we now impose te decoupling constraint on eac element o I Ep as ollows (m (m δ σ (η E Pδ ɛ H α δ (m p = 0 ( I Ep wic, in lieu o (4.2, can be written as (4.6 (m Pδ ɛ H = α δ (m p Meas( ( T Step 3: Using te act tat te cange in volume measured on bot scales sould be identical Te term δ (m ɛ H is te cange in volume o during te low solve in te (m + t coupling iteration. Tis naturally equates te sum o corresponding canges in volumes o te elements o I Ep as ollows (m δ ɛ H (4.7 (m Pδ ɛ H ( T p H From (4.6 and (4.7, we get (4.8 (m δ ɛ H = I Ep I Ep α E δ (m p Meas( ( T p H η

12 From (4.5 and (4.8, we get α E δ (m p Meas( = α Rδ (m p Meas( ( T p H η I Ep wic results in (4.9 Rδ (m p = η α I Ep α E δ (m Meas( p Meas( ( T p H Corollary 4.2. Satisaction o te condition (3.7 leads to te ollowing expressions or te eective bulk moduli or te coarse scale poromecanical solve η = Meas( η I Ep Meas( ( T p H and te ollowing expressions or te downscaled volumetric strains Pδ (m ɛ H = η Meas( δ (m ɛ H ( I T p H Proo. Step : Recasting te irst term on LHS o (3.7 We start by modiying te irst term on LHS o (3.7 as ollows (4.0 α (Pδ (m ɛ H, δ (m p = α δ (m p Pδ (m ɛ H Meas( were we note tat δ (m p W. Since a low element T 329 ated wit a poromecanical element via I Ep, we can write 330 α δ (m p Pδ (m ɛ H Meas( in uniquely associ = α δ (m p Pδ (m ɛ H Meas( I Ep In lieu o te above, we write (4.0 as (4. α (Pδ (m ɛ H, δ (m p = T T p H I Ep α δ (m p Pδ (m ɛ H Meas( Step 2: Recasting te second term on LHS o (3.7 Next, we modiy te second term on LHS o (3.7 as ollows α (δ (m ɛ H, Rδ (m p (4.2 2α Rδ (m p δ (m ɛ H were we note tat Rδ (m p W H. In lieu o (4. and (4.2, te irst condition given by (3.7 is rewritten as (4.3 I Ep α δ (m p Pδ (m ɛ H Meas( = α Rδ (m p δ (m ɛ H

13 2 S. DANA AND M. F. WHEELER Step 3: Substituting te expression or upscaled pore pressures Substituting te expression (4.9 or te upscaled pore pressure in (4.3, we get = I Ep α δ (m p Pδ (m ɛ H Meas( α Rδ (m p {}}{ η α E δ (m Meas( p α Meas( I Ep wic implies tat ( Pδ (m ɛ H η Meas( I Ep δ (m ɛ H δ (m ɛ H α δ (m p Meas( = wic, in lieu o te linear independence o te basis p ( T o te pressure 353 space on te ine scale low grid, implies tat Pδ (m ɛ H η 354 Meas( δ (m ɛ H = 0 ( I T p H implying tat (4.4 Pδ (m ɛ H = η Meas( δ (m ɛ H ( I E p T p H Step 4: Using te act tat te cange in volume measured on bot scales sould be identical Te cange in volume o over te (m + t coupling iteration equates te sum o corresponding canges in volumes o te elements o I Ep as ollows (4.5 δ (m ɛ H = I Ep Pδ (m ɛ H = In lieu o (4.4 and (4.5, we get (4.6 δ (m ɛ H = wic inally leads to (4.7 η δ (m ɛ H I Ep Pδ (m ɛ H Meas( ( T p H I Ep = Meas( η E I Ep Meas( η Ep Meas( Meas( ( T p H ( T p H Corollary 4.3. Te Caucy-Scwartz inequality, along wit te obtained expressions or eective coarse scale bulk moduli (4.7 and downscaled volumetric strains (4.4, guarantees te satisaction o te condition (3.8.

14 Proo. Step : Recasting (3.8 in lieu o (4.4 and (4.7 Te condition (3.8 given by η δ (m ɛ H 2 E η p Pδ (m ɛ H 2 E can be written as η δ (m ɛ H 2 Pδ (m ɛ H 2 Meas( T H wic can also be written as [ η δ (m ɛ H 2 E η p Pδ (m ɛ ] H 2 Meas( 0 I Ep wic, in lieu o (4.4, can also be written as [ η δ (m ɛ H 2 I Ep wic can also be written as [ η δ (m ɛ H 2 ( η Meas( δ (m ɛ H Pδ (m ɛ H 2 {}}{ ( η Meas( δ (m ɛ 2 H Meas(E ] 0 2 η wic, in lieu o (4.7, can be written as wic can be inally written as (4.8 I Ep Meas( ] Meas( 0 [ η δ (m ɛ ( H 2 E η p Meas( δ (m ɛ 2 H η η [ η δ (m ɛ ( H 2 E η p Meas( δ (m ɛ 2 ] H 0 ] 0 Step 2: Applying te Caucy-Scwartz inequality Te Caucy-Scwartz inequality (see [20] states tat i S is a measurable subset o R 3 and and g are measurable real-valued or complex-valued unctions on S, ten te ollowing is true ( S 2 g 2 S g 2 S

15 4 S. DANA AND M. F. WHEELER Replacing S by, by δ (m ɛ H and g by, we get δ (m ɛ ( H 2 E p Meas( δ (m ɛ 2 H ( T p wic can be written as η δ (m ɛ H 2 wic implies tat H ( η Meas( δ (m ɛ 2 H 0 ( T p H [ η δ (m ɛ ( H 2 E η p Meas( δ (m ɛ 2 ] H 0 wic is identical to (4.8. Tus, provided te downscaled volumetric strains are computed in accordance wit (4.4 and eective coarse scale bulk moduli are computed in accordance wit (4.7, te Caucy-Scwartz inequality guarantees te satisaction o te condition ( Te Voigt bound, te Reuss bound and te contraction constant. Te contraction constant is given by ( α 2 E γ = max T + < α2 M It is clear to see tat te minimum value o contraction constant is obtained wen te adjustable parameter takes te maximum possible value. To interrogate te maximum value tat te adjustable parameter can acieve, we look at te tird condition or te satisaction o te contractivity given by η 2K b ( T p H It is clear wen η = 2K b, we obtain te minimum contraction constant tus implying astest convergence o te staggered solution algoritm. Te expression (4.8 or te coarse scale moduli in terms on ine scale data is given by (5. η = Meas( η E I Ep Meas( ( T p H Te ollowing cases arise Te adjustable parameter is equal to twice te drained bulk modulus i.e. η 2K b 2K b = = K b 2K I Ep b = K I Ep b Meas( Meas( Meas( Meas( ( T p H ( T p H In tis case, te coarse scale bulk moduli are armonic mean o te ine scale data, tus representing te Reuss bound

16 Te adjustable parameter is equal to inverse o te drained bulk modulus i.e. η K b K b = I Ep K be Meas( Meas( ( T p H In tis case, te coarse scale bulk moduli are aritmetic mean o te ine scale data, tus representing te Voigt bound We already know tat te Reuss and Voigt bounds on eective moduli yield te lower and upper bounds or te elastic strain energy or multipase composites respectively (see [22]. In lieu o tat, we state tat te adjustable parameter is bounded above by te drained bulk modulus and below by te inverse o bulk modulus as ollows K b η 2K b 6. Conclusions and outlook. Te link we establised between te measure o te mean stress used in te decoupling constraint and te Voigt and Reuss bounds as interesting connotations or te imposed omogeneous boundary conditions used to arrive at eective properties in te computational omogenization o multipase composites. We know tat stress uniorm boundary conditions on te mesoscale lead to te Reuss bound on te eective property at te macroscale wile te kinematic uniorm boundary conditions on te mesoscale lead to te Voigt bound on te eective property at te macroscale ([3], [4], [5], [6], [2], [25]. We also know tat periodic boundary conditions on te mesoscale lead to te most accurate eective properties at te macroscale. In case o te two-grid approac, te ine scale low grid is te mesoscale wile te coarse scale poromecanical grid is te macroscale. Wen te adjustable parameter takes upon te value o twice te drained bulk modulus, we obtain te Reuss bound corresponding to stress uniorm boundary conditions on te mesoscale. Similarly, wen te adjustable parameter takes upon te value o te inverse o te drained bulk modulus, we obtain te Voigt bound corresponding to kinematic uniorm boundary conditions on te mesoscale. By an extension o tat logic, we expect a certain value o te adjustable parameter tat corresponds to te periodic boundary conditions imposed on te mesoscale tereby lending itsel to te most accurate estimate o te macroscale eective property, and tereby lending itsel to te astest convergence o te two-grid staggered solution algoritm. Appendix A. Discrete variational statements or te low subproblem in terms o coupling iteration dierences. Beore arriving at te discrete variational statement o te low model, we impose te decoupling constraint on te strong orm o te mass conservation equation (2.. Invoking te relation σ = η ɛ αp, we get (A. ( t M p + α ϕ ({}}{ M + α2 η ( σ + αp η + z = q p t + z = q α η σ t

17 6 S. DANA AND M. F. WHEELER Using backward Euler in time, te discrete in time orm o (A. or te m t coupling iteration in te (n + t time step is written as ϕ t (pm,n+ p n + z n+ = q n+ α η t ( σm,n+ σ n were t is te time step and te source term as well as te terms evaluated at te previous time level n do not depend on te coupling iteration count as tey are known quantities. Te decoupling constraint implies tat σ m,n+ gets replaced by σ m,n+ i.e. te computation o p m,n+ and z m,n+ is based on te value o σ updated ater te poromecanics solve rom te previous coupling iteration m at te current time level n +. Te modiied equation is written as ϕ(p m,n+ p n + t z m,n+ = tq n+ α η ( σm,n+ σ n As a result, te discrete variational statement o (2. in te presence o medium eterogeneities is ϕ (p m,n+ p n, θ + t( z m,n+, θ (A.2 = t(q n+, θ α ( σ m,n+ σ n, θ Replacing m by m + in (A.2 and subtracting te two equations, we get ϕ (δ (m p, θ + t( δ (m z, θ = α (δ (m σ, θ Te weak orm o te Darcy law (2.2 or te m t coupling iteration in te (n + t time step is (A.3 were V(Ω is given by and H(div, Ω is given by (κ z m,n+, v Ω = ( p m,n+, v Ω + (ρ 0 g, v Ω v V(Ω V(Ω H(div, Ω { v : v n = 0 on Γ N H(div, Ω { v : v (L 2 (Ω 3, v L 2 (Ω } We use te divergence teorem to evaluate te irst term on RHS o (A.3 as ollows } (A.4 ( p m,n+, v Ω = (, p m,n+ v Ω (p m,n+, v Ω = (p m,n+, v n Ω (p m,n+, v Ω = (g, v n Γ (p m,n+, v Ω D were we invoke v n = 0 on Γ N. In lieu o (A.3 and (A.4, we get (κ z m,n+, v Ω = (g, v n Γ + (p m,n+, v Ω + (ρ 0 g, v Ω D

18 As a result, te discrete variational statement o (2.2 in te presence o medium eterogeneities is (κ z m,n+, v (p m,n+, v 520 (A.5 = (ρ 0 g, v (g, v n E Γ D Replacing m by m + in (A.5 and subtracting te two equations, we get (κ δ (m z, v = (δ (m p, v Appendix B. Discrete variational statement or te poromecanics subproblem in terms o coupling iteration dierences. Te weak orm o te linear momentum balance (2.6 is given by (B. were U(Ω is given by ( σ, q Ω + ( q Ω = 0 ( q U(Ω U(Ω { q = (u, v, w : u, v, w H (Ω, q = 0 on Γ p D were H m (Ω is deined, in general, or any integer m 0 as H m (Ω { w : D α w L 2 (Ω α m }, were te derivatives are taken in te sense o distributions and given by D α w = α w, α = α + + α n,.. xαn n x α We know rom tensor calculus tat (B.2 ( σ, q (, σq (σ, q Furter, using te divergence teorem and te symmetry o σ, we arrive at } (B.3 (, σq Ω (q, σn Ω We decompose q into a symmetric part ( q s 2( q + ( q T ɛ(q and skew-symmetric part ( q ss and note tat te contraction between a symmetric and skew-symmetric tensor is zero to obtain (B.4 σ : q σ : ( q s + 0 σ : ( q ss = σ : ɛ(q From (B., (B.2, (B.3 and (B.4, we get (σn, q Ω (σ, ɛ(q Ω + (, q Ω = 0 wic, ater invoking te boundary condition σn = t on Γ p N results in (B.5 (t, q Γ p N (σ, ɛ(q Ω + (, q Ω = 0

19 8 S. DANA AND M. F. WHEELER 557 Te stress tensor σ and strain tensor ɛ(q are written as σ = s + 3 tr(σi = s + σ vi; ɛ(q = e(q + 3 tr(ɛ(qi = e(q + 3 ɛ(qi were s is te deviatoric stress tensor, e(q is te deviatoric strain tensor and σ v is te mean stress. Using te above relations, we can write (B.6 σ : ɛ(q = ( s + σ v I : ( e(q + 3 ɛ(qi = s : e(q + s : 3 ɛ(qi + σ vi : e(q + σ v I : 3 ɛ(qi = s : e(q + 3 ɛ(q 0 tr(s + σ v 0 tr(e(q + 3σ v 3 ɛ(q = s : e(q + σ v ɛ(q were we note tat te contraction o any second order tensor wit te identity tensor I is equal to te trace o te tensor and urter, te trace o a deviatoric tensor is zero resulting in tr(s = 0 and tr(e(q = 0. Substituting (B.6 in (B.5, we get (B.7 (t, q Γ p N (s, e(q Ω (σ v, ɛ(q Ω + (, q Ω = 0 Te deviatoric strain tensor is obtained as s = σ 3 tr(σi = σ 0 + λ ɛi + 2Gɛ α(p p 0 I 3 tr( σ 0 + λ ɛi + 2Gɛ α(p p 0 I I (B.8 = s 0 + 2G ( ɛ 3 tr(ɛi = s 0 + 2Ge Substituting (B.8 in (B.7, we get (t, q Γ p N (s 0, e(q Ω (2Ge, e(q Ω (σ v, ɛ(q Ω + (, q Ω = 0 As a result, te discrete variational statement o te linear momentum balance (2.6 or te m t coupling iteration in te (n + t time step in te presence o medium eterogeneities is written as 2G (e(u m,n+ H, e(q H + (σv m,n+, q H (B.9 = (, q H + (t, q H Γ p N (s 0, e(q H Replacing m by m + in (B.9 and subtracting te two equations, we get 2G (e(δ (m u H, e(q H + (δ (m σ v, q H = REFERENCES [] Y. Abousleiman, A. H. D. Ceng, L. Cui, E. Detournay, and J. C. Roegiers, Mandel s problem revisited, Géotecnique, 46 (996, pp

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