ICES REPORT January Saumik Dana and Mary. F. Wheeler

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1 ICES REPORT January 2018 Convergence analysis o ixed stress split iterative scheme or anisotropic poroelasticity with tensor Biot parameter by Saumik Dana and Mary. F. Wheeler The Institute or Computational Engineering and Sciences The University o Texas at Austin Austin, Texas Reerence: Saumik Dana and Mary. F. Wheeler, "Convergence analysis o ixed stress split iterative scheme or anisotropic poroelasticity with tensor Biot parameter," ICES REPORT 18-01, The Institute or Computational Engineering and Sciences, The University o Texas at Austin, January 2018.

2 Convergence analysis o ixed stress split iterative scheme or anisotropic poroelasticity with tensor Biot parameter Saumik Dana a, Mary. F. Wheeler a a Center or Subsurace Modeling, Institute or Computational Engineering and Sciences, UT Austin, Austin, TX Abstract We perorm a convergence analysis o the ixed stress split iterative algorithm or the Biot system modeling coupled low and deormation in anisotropic poroelastic media with ull Biot tensor. The ixed stress split iterative scheme solves the low subproblem with all components o the stress tensor rozen using a multipoint lux mixed inite element method, ollowed by the poromechanics subproblem using a conorming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed or time marching. The convergence analysis is based on studying the equations satisied by the dierence o iterates to show that the ixed stress split iterative scheme or anisotropic poroelasticity with Biot tensor is contractive. Keywords: Anisotropic poroelastic medium, Biot tensor, Fixed stress split iterative coupling, Contractivity 1. Introduction Staggered solution algorithms are those in which operator splitting strategies are used to decompose coupled problems into subproblems which are then solved sequentially in successive iterations until a convergence criterion is met at each time step (Felippa et al. [13], Armero and Simo [3], Turska and Schreler [25], Schreler et al. [21]). The ixed stress split is one such operator splitting strategy or isotropic poroelasticity with scalar Biot parameter in which the low problem is solved irst while reezing the volumetric mean stress o the porous solid. Such an approach has been shown to enjoy numerical and theoretical convergence addresses: saumik@utexas.edu (Saumik Dana), mw@ices.utexas.edu (Mary. F. Wheeler)

3 New time step New coupling iteration Solve low with stress tensor ixed Solve poromechanics yes converged? no Figure 1: Fixed stress split iterative scheme or anisotropic poroelasticity with tensor Biot parameter (Kim et al. [18], Mikelić and Wheeler [19], Castelletto et al. [10], Almani et al. [2]). The subject o this paper is the extension o the decoupling assumption to the case o anisotropic poroelasticity with tensor Biot parameter. The micromechanical analyses or the case o anisotropic poroelasticity (see Carroll [8], Carroll and Katsube [9], Katsube [17]) revealed that the modiication to the stress applied to the porous solid due to the presence o pore luid pressure is not hydrostatic, as it is in the case o isotropic poroelasticity (see Nur and Byerlee [20]). Further, unlike in case o isotropic poroelasticity where the solid-luid coupling parameter is a scalar (see Biot [4], Geertsma [14], Skempton [23], Nur and Byerlee [20]), the coupling parameter in case o anisotropic poroelasticity is a tensor (see Biot [5], Coussy [12]). Intuitively, this implies that the decoupling assumption o reezing the hydrostatic part o the stress tensor during the low solve in every coupling iteration or the case o isotropic poroelasticity requires a generalization or the case o anisotropic poroelasticity. As we shall show in the ensuing convergence analysis, the decoupling assumption o reezing all components o the stress tensor during the low solve in every coupling iteration does enjoy theoretical convergence or the case o anisotropic poroelasticity with tensor Biot parameter. The convergence analysis is motivated by the previous work o Mikelić and Wheeler [19] and Almani et al. [2]. Mikelić and Wheeler [19] proved that the standard ixed stress split scheme or isotropic poroelasticity with scalar Biot parameter is a contraction 2

4 map with respect to appropriately chosen metrics. Almani et al. [2] extended those results to establish convergence o the multirate ixed stress split scheme in which the subproblems are resolved on dierent time scales with the poromechanics subproblem being resolved on a larger time scale. To the best o our knowledge, this is the irst time a contraction map has been rigorously derived or the ixed stress split iterative scheme or anisotropic poroelasticity with tensor Biot parameter. This paper is structured as ollows: Section 2 presents the model equations or low and poromechanics. Section 3 presents the statement o contraction o the ixed stress split iterative scheme or anisotropic poroelasticity with tensor Biot parameter. Section 4 presents the convergence criterion or the algorithm Preliminaries Given a bounded convex domain Ω R 3, we use P k (Ω) to represent the restriction o the space o polynomials o degree less that or equal to k to Ω and Q 1 (Ω) to denote the space o trilinears on Ω. For the sake o convenience, we discard the dierential in the integration o any scalar ield χ over Ω as ollows χ(x) χ(x) dv ( x Ω) Ω Ω Sobolev spaces are based on the space o square integrable unctions on Ω given by L 2 (Ω) { θ : θ 2 Ω := θ 2 < + }, The inner product o two second order tensors S and T is given by (see Gurtin et al. [15]) S : T = S ij T ij (i, j = 1, 2, 3) A ourth order tensor is a linear transormation o a second order tensor to a second order tensor in the ollowing manner (see Gurtin et al. [15]) PS = T P ijkl S kl = T ij (i, j, k, l = 1, 2, 3) Ω 2. Model equations 2.1. Flow model Let the boundary Ω = Γ D Γ N where Γ D is the Dirichlet boundary and Γ N is the Neumann boundary. The luid mass conservation equation (2.1) in the presence o deormable 3

5 and anisotropic porous medium with the Darcy law (2.2) and linear pressure dependence o density (2.3) with boundary conditions (2.4) and initial conditions (2.5) is ζ t + z = q (2.1) z = K µ ( p ρ 0g) = κ( p ρ 0 g) (2.2) ρ = ρ 0 (1 + c (p p 0 )) (2.3) p = g on Γ D (0, T ], z n = 0 on Γ N (0, T ] (2.4) p(x, 0) = p 0 (x), ρ(x, 0) = ρ 0 (x), φ(x, 0) = φ 0 (x) ( x Ω) (2.5) where p : Ω (0, T ] R is the luid pressure, z : Ω (0, T ] R 3 is the luid lux, ζ is the luid content, n is the unit outward normal on Γ N, q is the source or sink term, K is the uniormly symmetric positive deinite absolute permeability tensor, µ is the luid viscosity, ρ 0 is a reerence density, φ is the porosity, κ = K µ is a measure o the hydraulic conductivity o the pore luid, c is the luid compressibility and T > 0 is the time interval Poromechanics model Let the boundary Ω = Γ p D Γp N where Γp D is the Dirichlet boundary and Γp N is the Neumann boundary. Linear momentum balance or the anisotropic porous solid in the quasistatic limit o interest (2.6) with small strain assumption (2.8) with boundary conditions (2.9) and initial condition (2.10) is σ + = 0 (2.6) = ρφg + ρ r (1 φ)g (2.7) ɛ(u) = 1 2 ( u + ( u)t ) (2.8) u n 1 = 0 on Γ p D [0, T ], σt n 2 = t on Γ p N [0, T ] (2.9) u(x, 0) = 0 x Ω (2.10) where u : Ω [0, T ] R 3 is the solid displacement, ρ r is the rock density, is the body orce per unit volume, n 1 is the unit outward normal to Γ p D, n 2 is the unit outward normal to Γ p N, t is the traction speciied on Γp N, ɛ is the strain tensor, σ is the Cauchy stress tensor given by the generalized Hooke s law (see Coussy [12]) σ = σ 0 + Mɛ α(p p 0 ) (2.11) 4

6 where σ 0 is the in situ stress, M is the ourth order anisotropic elasticity tensor and α is the Biot tensor. The inverse o the generalized Hooke s law (2.11) is given by (see Cheng [11]) ɛ = Cσ + 1 CBp (2.12) 3 where C is the ourth order anisotropic compliance tensor, C(> 0) is a generalized Hooke s law constant and B is a generalization o the Skempton pore pressure coeicient B (see Skempton [22]) or anisotropic poroelasticity Fluid content The luid content ζ is given by (see Cheng [11]) ζ = Cp CB : σ 1 M p + α : ɛ (2.13) where M(> 0) is a generalization o the Biot modulus (see Biot and Willis [6]) or anisotropic poroelasticity. 3. Statement o contraction o the ixed stress split scheme or anisotropic poroelasticity with Biot tensor We use the notations ( ) n+1 or any quantity ( ) evaluated at time level n + 1, ( ) m,n+1 or any quantity ( ) evaluated at the m th coupling iteration at time level n + 1, δ (m) ( ) or the change in the quantity ( ) during the low solve over the (m + 1) th coupling iteration at any time level and δ (m) ( ) or the change in the quantity ( ) over the (m + 1) th coupling iteration at any time level. Let T h be inite element partition o Ω consisting o distorted hexahedral elements E where h = max E T h diam(e). The details o the inite element mapping are given in Appendix A. The discrete variational statements in terms o coupling iteration dierences is : ind δ (m) p h W h, δ (m) z h V h and δ (m) u h U h such that C(δ (m) p h, θ h ) Ω + t( δ (m) z h, θ h ) Ω = C 3 (B : δ(m 1) σ, θ h ) Ω (3.1) (κ 1 δ (m) z h, v h ) Ω = (δ (m) p h, v h ) Ω (3.2) (δ (m) σ : ɛ(q h )) Ω = 0 (3.3) 5

7 where the inite dimensional spaces W h, V h and U h are W h = { } θ h : θ h P 0 (E) E T h V h = { v h : v h E ˆv Ê ˆV(Ê) E T h, v h n = 0 on Γ } N U h = { q h = (u, v, w) E Q 1 (E) E T h, q h = 0 on Γ p } D and the details o ˆV(Ê) are given in Appendix B. The equations (3.1), (3.2) and (3.3) are the discrete variational statements (in terms o coupling iteration dierences) o (2.1), (2.2) and (2.6) respectively. The details o (3.1) and (3.2) are given in Appendix C whereas the details o (3.3) are given in Appendix D. Theorem 3.1. The ixed stress split iterative coupling scheme or anisotropic poroelasticity with Biot tensor in which the low problem is solved irst by reezing all components o the stress tensor is a contraction given by B : δ (m) σ 2 Ω + 2 δ (m) p h 2 Ω + 6 t C κ 1/2 δ (m) z h 2 Ω + 6 C 2 δ(m) ζ δ (m) ζ 2 Ω B : δ (m 1) σ 2 Ω 0 6 C (δ(m) σ : Cδ (m) σ) Ω + 3 C 2 δ(m) ζ 2 Ω where the quantity δ (m) ζ δ (m) ζ 2 Ω is driven to a small value via the convergence criterion or the algorithm. Proo. Step 1: Flow equations Testing (3.1) with θ h δ (m) p h, we get C δ (m) p h 2 Ω + t( δ (m) z h, δ (m) p h ) Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.4) Testing (3.2) with v h δ (m) z h, we get From (3.4) and (3.5), we get κ 1/2 δ (m) z h 2 Ω = (δ (m) p h, δ (m) z h ) Ω (3.5) C δ (m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.6) Step 2: Poromechanics equations Testing (3.3) with q h δ (m) u h, we get (δ (m) σ : δ (m) ɛ) Ω = 0 (3.7) 6

8 We now invoke (2.12) to arrive at the expression or change in strain tensor over the (m+1) th coupling iteration as ollows δ (m) ɛ = Cδ (m) σ + C 3 Bδ(m) p h (3.8) Substituting (3.8) in (3.7), we get (δ (m) σ : Cδ (m) σ) Ω + C 3 (B : δ(m) σ, δ (m) p h ) Ω = 0 (3.9) Step 3: Combining low and poromechanics equations Adding (3.6) and (3.9), we get C δ (m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : Cδ (m) σ) Ω + C 3 (B : δ(m) σ, δ (m) p h ) Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.10) Step 4: Variation in luid content In lieu o (2.13), the variation in luid content in the (m + 1) th coupling iteration can be written as As a result, we can write δ (m) ζ = Cδ (m) p h + C 3 B : δ(m) σ (3.11) 1 2C δ(m) ζ 2 Ω C 2 δ(m) p h 2 Ω C 18 B : δ(m) σ 2 Ω = C 3 (B : δ(m) σ, δ (m) p h ) Ω (3.12) From (3.10) and (3.12), we get C δ (m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : Cδ (m) σ) Ω + 1 2C δ(m) ζ 2 Ω C 2 δ(m) p h 2 Ω C 18 B : δ(m) σ 2 Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.13) Adding and subtracting C 6 B : δ(m) σ 2 Ω to the LHS o (3.13) results in C 6 B : δ(m) σ 2 Ω + C 2 δ(m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : Cδ (m) σ) Ω + 1 2C δ(m) ζ 2 Ω C 9 B : δ(m) σ 2 Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.14) In lieu o (2.13) and the ixed stress constraint during the low solve, the variation in luid content during the low solve in the (m + 1) th coupling iteration is given by δ (m) ζ = Cδ (m) p h + C 3 B : δ (m) σ 0 7

9 Further, since the pore pressure is rozen during the poromechanical solve, we have δ (m) p h = δ (m) p h. As a result, we can write Subtracting (3.15) rom (3.11), we can write which implies that In lieu o (3.16), we can write (3.14) as δ (m) ζ = Cδ (m) p h (3.15) δ (m) ζ δ (m) ζ = C 3 B : δ(m) σ 1 C δ(m) ζ δ (m) ζ 2 Ω = C 9 B : δ(m) σ 2 Ω (3.16) C 6 B : δ(m) σ 2 Ω + C 2 δ(m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : Cδ (m) σ) Ω + 1 2C δ(m) ζ 2 Ω 1 C δ(m) ζ δ (m) ζ 2 Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.17) Step 5: Invoking positive-semideiniteness o the compliance tensor The ourth order anisotropic compliance tensor C is positive-semideinite (see Gurtin et al. [15]) in the sense that it obeys S : CS 0 or all symmetric tensors S. In lieu o the above, since the Cauchy stress tensor is symmetric, the ollowing holds true In lieu o (3.18), we can write (3.17) as δ (m) σ : Cδ (m) σ 0 (3.18) 0 C 6 B : δ(m) σ 2 C Ω + 2 δ(m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : Cδ (m) σ) Ω + 1 2C δ(m) ζ 2 Ω 1 C δ(m) ζ δ (m) ζ 2 Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω (3.19) The quantity δ (m) ζ δ (m) ζ 2 Ω on the LHS o (3.19) is driven to a small value via the convergence criterion or the algorithm as explained in Section 4. Step 6: Invoking the Young s inequality Since the sum o the terms on the LHS o (3.19) is nonnegative, the RHS is also nonnegative. We invoke the Young s inequality (see Steele [24]) ab a2 2 + b2 2 ( a, b R) 8

10 or the RHS o (3.19) as ollows C 3 (B : δ(m 1) σ, δ (m) p h ) Ω C 3 ( 1 2 B : δ(m 1) σ 2 Ω + 1 ) 2 δ(m) p h 2 Ω (3.20) In lieu o (3.20), we write (3.19) as 0 C 6 B : δ(m) σ 2 C Ω + 2 δ(m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : Cδ (m) σ) Ω + 1 C δ(m) ζ δ (m) ζ 2 Ω C 6 B : δ(m 1) σ 2 Ω + C 6 δ(m) p h 2 Ω which can also be written as B : δ (m) σ 2 Ω + 2 δ (m) p h 2 Ω C δ(m) ζ 2 Ω 6 t C κ 1/2 δ (m) z h 2 6 Ω + C (δ(m) σ : Cδ (m) 3 σ) Ω + C 2 δ(m) ζ 2 Ω 6 C 2 δ(m) ζ δ (m) ζ 2 Ω B : δ (m 1) σ 2 Ω (3.21) 4. Convergence criterion or the ixed stress split algorithm or anisotropic poroelasticity with Biot tensor In lieu o (2.13), the variation in luid content δ (m) ζ measured during the low solve in the (m + 1) th coupling iteration at any time step is δ (m) ζ = 1 M δ(m) p h + α : δ (m) ɛ = ζp m+1 ζ m (4.1) where ζ m is the luid content at the end o the previous or m th coupling iteration and ζ m+1 p serves as the predictor to the luid content at the end o the current or (m + 1) th coupling iteration. Similarly, the variation in luid content δ (m) ζ over the (m+1) th coupling iteration (including the low solve and poromechanics solve) at any time step is δ (m) ζ = 1 M δ(m) p h + α : δ (m) ɛ = ζ m+1 c ζ m (4.2) where ζc m+1 serves as the corrector to ζp m+1. Subtracting (4.1) rom (4.2), we get δ (m) ζ δ (m) ζ ζc m+1 ζp m+1 = α : (δ (m) ɛ δ (m) ɛ) = α : (ɛ m+1 c ɛ m+1 p ) 9

11 which implies that the dierence between the predicted value and the corrected value o the luid content at the end o the (m + 1) th coupling iteration is equal to the dierence between the predicted value (ɛ m+1 p ) and the corrected value (ɛ m+1 c ) o the strain tensor at the end o the (m + 1) th coupling iteration scaled by the Biot tensor α. In lieu o the above, the stopping criterion or coupling iterations at any time step is δ (m) ζ δ (m) ζ m+1 c where T OL > 0 is a pre-speciied tolerance. ζ ζ c m+1 ζ m+1 p T OL L L ζ m+1 c Appendix A. Finite element mapping ˆr 8 ˆr 7 r 8 r 7 ˆr 5 ˆr 6 ˆr 4 ˆr 3 F E r r 6 5 r 4 r 3 ˆr 1 ˆr 2 r 1 r 2 Figure A.2: Trilinear mapping F E : Ê E or 8 noded distorted hexahedral elements. The aces o E can be non-planar. Let r i, i = 1,.., 8 be the vertices o E. Now consider a reerence cube Ê with vertices ˆr 1 = [0 0 0] T, ˆr 2 = [1 0 0] T, ˆr 3 = [1 1 0] T, ˆr 4 = [0 1 0] T, ˆr 5 = [0 0 1] T, ˆr 6 = [1 0 1] T, ˆr 7 = [1 1 1] T and ˆr 8 = [0 1 1] T as shown in Figure A.2. Let ˆx = (ˆx, ŷ, ẑ) Ê and x = (x, y, z) E. The unction F E (ˆx) : Ê E is F E (ˆx) = r 1 (1 ˆx)(1 ŷ)(1 ẑ) + r 2ˆx(1 ŷ)(1 ẑ) + r 3ˆxŷ(1 ẑ) + r 4 (1 ˆx)ŷ(1 ẑ) + r 5 (1 ˆx)(1 ŷ)ẑ + r 6ˆx(1 ŷ)ẑ + r 7ˆxŷẑ + r 8 (1 ˆx)ŷẑ Denote Jacobian matrix by DF E and let J E = det(df E ). Deining r ij r i r j, we have DF E (ˆx)= r 21 + (r 34 r 21 )ŷ + (r 65 r 21 )ẑ + ((r 21 r 34 ) (r 65 r 78 ))ŷẑ; r 41 + (r 34 r 21 )ˆx + (r 85 r 41 )ẑ + ((r 21 r 34 ) (r 65 r 78 ))ˆxẑ; r 51 + (r 65 r 21 )ˆx + (r 85 r 41 )ŷ + ((r 21 r 34 ) (r 65 r 78 ))ˆxŷ 10

12 Denote inverse mapping by F 1 1, its Jacobian matrix by DF such that E E and let J F 1 E = det(df 1 E ) DF 1 E (x) = (DF E) 1 (ˆx); J F 1(x) = (J E ) 1 (ˆx) Let φ(x) be any unction deined on E and ˆφ(ˆx) be its corresponding deinition on Ê. Then we have E φ = (DF 1 E )T (x) ˆ ˆφ = (DF E ) T (ˆx) ˆ ˆφ (A.1) Appendix B. Enhanced BDDF 1 spaces ˆv 51 ˆv 52 ˆv 11 ˆv 53 ˆv 83 ˆv 81 ˆv 82 ˆv 41 ˆv 42 ˆv 43 ˆv 62 ˆv 63 ˆv 73 ˆv61 ˆv 21 ˆv ˆv ˆv 22 ˆv 23 ˆv 72 ˆv 32 ˆv 31 ˆv 33 ˆv 71 F E v 51 v 52 v 41 v 43 v 83 v 81 v 82 v 53 v 61 v 62 v 42 v 73 v 63 v 71 v 11 v 12 v 13 v22 v 23 v 21 v 72 v 32 v 33 v 31 Figure B.3: Degrees o reedom and basis unctions or the enhanced BDDF 1 velocity space on hexahedra. For the sake o clarity, we provide a brie description o the mixed inite element spaces used in the low model. Let V h W h be the lowest order BDDF 1 MFE spaces on hexahedra (see Brezzi et al. [7]). On the reerence unit cube these spaces are deined as ˆV (Ê) = P 1(Ê) + r 0 curl(0, 0, ˆxŷẑ) T + r 1 curl(0, 0, ˆxŷ 2 ) T + s 0 curl(ˆxŷẑ, 0, 0) T + s 1 curl(ŷẑ 2, 0, 0) T + t 0 curl(0, ˆxŷẑ, 0) T + t 1 curl(0, ˆx 2 ẑ, 0) T Ŵ (Ê) = P0(Ê) with the ollowing properties ˆ ˆV (Ê) = Ŵ (Ê), and ˆv ˆV (Ê), ê Ê, ˆv ˆn ê P 1 (ê) The multipoint lux approximation procedure requires on each ace one velocity degree o reedom to be associated with each vertex. Since the BDDF 1 space V h has only three 11

13 degrees o reedom per ace, it is augmented with six more degrees o reedom (one extra degree o reedom per ace). Since the constant divergence, the linear independence o the shape unctions and the continuity o the normal component across the element aces are to be preserved, six curl terms are added (Ingram et al. [16]). Let V h W h be the enhanced BDDF 1 spaces on hexahedra. On the reerence unit cube these spaces are ˆV(Ê) = ˆV (Ê) + r 2 curl(0, 0, ˆx 2 ẑ) T + r 3 curl(0, 0, ˆx 2 ŷẑ) T + s 2 curl(ˆxŷ 2, 0, 0) T + s 3 curl(ˆxŷ 2 ẑ 2, 0, 0) T + t 2 curl(0, ŷẑ 2, 0) T + t 3 curl(0, ˆxŷẑ 2, 0) T Ŵ (Ê) = P0(Ê) with the ollowing properties ˆ ˆV(Ê) = Ŵ (Ê), and ˆv ˆV(Ê), ê Ê, ˆv ˆn ê Q 1 (ê) where Q 1 is the space o bilinear unctions. Since dim Q 1 (ê) = 4, the dimension o ˆV(Ê) is 24 as shown in Figure B.3. Appendix C. Discrete variational statements or the low subproblem in terms o coupling iteration dierences Beore arriving at the discrete variational statement o the low model, we impose the ixed stress constraint on the strong orm o the mass conservation equation (2.1). In lieu o (2.13), we write (2.1) as t (Cp + C 3 B : σ) + z = q C p t + z = q C 3 B : σ t (C.1) Using backward Euler in time, the discrete in time orm o (C.1) or the m th coupling iteration in the (n + 1) th time step is written as C 1 t (pm,n+1 p n ) + z m,n+1 = q n+1 1 C t 3 B : (σm,n+1 σ n ) where t is the time step and the source term as well as the terms evaluated at the previous time level n do not depend on the coupling iteration count as they are known quantities. The ixed stress constraint implies that σ m,n+1 gets replaced by σ m 1,n+1 i.e. the computation 12

14 o p m,n+1 and z m,n+1 is based on the value o stress updated ater the poromechanics solve rom the previous coupling iteration m 1 at the current time level n + 1. The modiied equation is written as C(p m,n+1 p n ) + t z m,n+1 = tq n+1 C 3 B : (σm,n+1 σ n ) As a result, the discrete weak orm o (2.1) is given by C(p m,n+1 h p n h, θ h) Ω + t( z m,n+1 h, θ h ) Ω = t(q n+1, θ h ) Ω C 3 (B : (σm 1,n+1 σ n ), θ h ) Ω Replacing m by m + 1 and subtracting the two equations, we get C(δ (m) p h, θ h ) Ω + t( δ (m) z h, θ h ) Ω = C 3 (B : δ(m 1) σ, θ h ) Ω The weak orm o the Darcy law (2.2) or the m th coupling iteration in the (n + 1) th time step is given by (κ 1 z m,n+1, v) Ω = ( p m,n+1, v) Ω + (ρ 0 g, v) Ω v V(Ω) (C.2) where V(Ω) is given by V(Ω) H(div, Ω) { v : v n = 0 on Γ } N and H(div, Ω) is given by H(div, Ω) { v : v (L 2 (Ω)) 3, v L 2 (Ω) } We use the divergence theorem to evaluate the irst term on RHS o (C.2) as ollows ( p m,n+1, v) Ω = (, p m,n+1 v) Ω (p m,n+1, v) Ω = (p m,n+1, v n) Ω (p m,n+1, v) Ω = (g, v n) Γ (p m,n+1, v) Ω D (C.3) where we invoke v n = 0 on Γ N. In lieu o (C.2) and (C.3), we get (κ 1 z m,n+1, v) Ω = (g, v n) Γ + (p m,n+1, v) Ω + (ρ 0 g, v) Ω D Replacing m by m + 1 and subtracting the two equations, we get (κ 1 δ (m) z h, v h ) Ω = (δ (m) p h, v h ) Ω 13

15 Appendix D. Discrete variational statement or the poromechanics subproblem in terms o coupling iteration dierences The weak orm o the linear momentum balance (2.6) is given by where U(Ω) is given by ( σ, q) Ω + ( q) Ω = 0 ( q U(Ω)) (D.1) U(Ω) { q = (u, v, w) : u, v, w H 1 (Ω), q = 0 on Γ p } D where 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 }, where the derivatives are taken in the sense o distributions and given by We know rom tensor calculus that D α α w w = x α, α = α α n, 1.. xαn n ( σ, q) Ω (, σq) Ω (σ : q) Ω (D.2) Further, using the divergence theorem and the symmetry o σ, we arrive at (, σq) Ω (q, σn) Ω (D.3) We decompose q into a symmetric part ( q) s 1 2( q + ( q) T ) ɛ(q) and skewsymmetric part ( q) ss and note that the contraction between a symmetric and skewsymmetric tensor is zero to obtain From (D.1), (D.2), (D.3) and (D.4), we get σ : q σ : ( q) s + 0 σ : ( q) ss = σ : ɛ(q) (σn, q) Ω (σ : ɛ(q)) Ω + (, q) Ω = 0 (D.4) which, ater invoking the traction boundary condition, results in the discrete weak orm or the m th coupling iteration as (t n+1, q h ) Γ p N (σm,n+1 : ɛ(q h )) Ω + ( n+1, q h ) Ω = 0 Replacing m by m + 1 and subtracting the two equations, we get (δ (m) σ : ɛ(q h )) Ω = 0 14

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