ICES REPORT April Saumik Dana and Mary F. Wheeler

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1 ICES REPORT April 2018 Convergence analysis o ixed stress split iterative scheme or small strain anisotropic poroelastoplasticity: A primer 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 small strain anisotropic poroelastoplasticity: A primer," ICES REPORT 18-05, The Institute or Computational Engineering and Sciences, The University o Texas at Austin, April 2018.

2 1 2 3 CONVERGENCE ANALYSIS OF FIXED STRESS SPLIT ITERATIVE SCHEME FOR SMALL STRAIN ANISOTROPIC POROELASTOPLASTICITY: A PRIMER 4 SAUMIK DANA AND MARY F. WHEELER Abstract. We perorm a convergence analysis o the ixed stress split iterative scheme or the Biot system modeling coupled single phase low and small strain deormation in an anisotropic poroelastoplastic medium. The ixed stress split iterative scheme solves the low subproblem with stress tensor ixed 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 iterative scheme is contractive. 1. Introduction. This report serves as a primer to our eorts in arriving at theoretical convergence estimates or the ixed stress split iterative scheme or small strain anisotropic poroelastoplasticity coupled with single phase low. This work ollows up on our previous work [5], where we arrived at a contraction map or the case o anisotropic poroelasticity with tensor Biot parameter 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 [8]) Ω (i, j = 1, 2, 3) S : T = S ij T ij A ourth order tensor is a linear transormation o a second order tensor to a second order tensor in the ollowing manner (see [8]) (i, j, k, l = 1, 2, 3) PS = T P ijkl S kl = T ij The dyadic product o two second order tensors S and T is given by (see [8]) P = S T P ijkl = S ij T kl Model equations Flow model. Let the boundary Ω = Γ D Γ N where Γ D is the Dirichlet 39 boundary and Γ N is the Neumann boundary. The luid mass conservation equation 40 (2.1) in the presence o deormable and anisotropic porous medium with the Darcy 1 This manuscript is or review purposes only.

3 2 S. DANA, AND M. F. WHEELER New time step New coupling iteration Solve low with stress tensor ixed Solve poromechanics yes converged? no Fig. 1. Fixed stress split iterative scheme or anisotropic poroelastoplasticity with tensor Biot parameter law (2.2) and linear pressure dependence o density (2.3) with boundary conditions (2.4) and initial conditions (2.5) is (2.1) (2.2) ζ t + z = q z = K µ ( p ρ 0g) = κ( p ρ 0 g) 45 (2.3) ρ = ρ 0 (1 + c (p p 0 )) 46 (2.4) p = g on Γ D (0, T ], z n = 0 on Γ N (0, T ] 47 p(x, 0) = p 0 (x), ρ(x, 0) = ρ 0 (x), φ(x, 0) = φ 0 (x) (2.5) ( x Ω) where p : Ω (0, T ] R is the luid pressure, z : Ω (0, T ] R 3 is the luid lux, ζ is 51 the increment in luid content 1, n is the unit outward normal on Γ N, q is the source 52 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 54 measure o the hydraulic conductivity o the pore luid, c is the luid compressibility 55 and T > 0 is the time interval Poromechanics model. Let the boundary Ω = Γ p D Γp N where Γp D is the 57 Dirichlet boundary and Γ p N is the Neumann boundary. Linear momentum balance or 58 the anisotropic porous solid in the quasi-static limit o interest (2.6) with small strain 1 [1] deines the increment in luid content as the measure o the amount o luid which has lowed in and out o a given element attached to the solid rame This manuscript is or review purposes only.

4 3 59 assumption (2.8) with boundary conditions (2.9) and initial condition (2.10) is (2.6) (2.7) (2.8) σ + = 0 = ρφg + ρ r (1 φ)g ɛ(u) = 1 2 ( u + ( u)t ) 63 (2.9) u n 1 = 0 on Γ p D [0, T ], σt n 2 = t on Γ p N [0, T ] (2.10) u(x, 0) = 0 x Ω 66 where u : Ω [0, T ] R 3 is the solid displacement, ρ r is the rock density, is the 67 body orce per unit volume, n 1 is the unit outward normal to Γ p D, n 2 is the unit 68 outward normal to Γ p N, t is the traction speciied on Γp N, ɛ is the strain tensor, σ is 69 the Cauchy stress tensor given by the generalized Hooke s law (2.11) σ = Dɛ e αp D ep ɛ αp where D is the ourth order anisotropic elasticity tensor, α is the Biot tensor and D ep is the elastoplastic tangent operator given in (A.1). The inverse o the generalized Hooke s law (2.11) is given by (2.12) ɛ = D ep 1 (σ + αp) = D ep 1 σ + C 3 Bp where C(> 0) is a generalized Hooke s law constant and B is a generalization o the Skempton pore pressure coeicient B (see [9]) or anisotropic poroelastoplasticity, and is given by (2.13) B 3 C Dep 1 α Increment in luid content. The increment in luid content ζ is given by (see [3]) (2.14) ζ = 1 M p + α : ɛe + φ p Cp + 1 CB : σ + φp where M(> 0) is a generalization o the Biot modulus (see [2]) or anisotropic poroelasticity and φ p is a plastic porosity (see [3]). 3. Statement o contraction o the ixed stress split scheme or small strain anisotropic poroelastoplasticity 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 diam(e). The details o the inite E T h element mapping are given in [4]. 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 This manuscript is or review purposes only.

5 4 S. DANA, AND M. F. WHEELER such that (3.1) C(δ (m) p h, θ h ) Ω + t( δ (m) z h, θ h ) Ω + (δ (m) φ p, θ h ) Ω = C 3 (B : δ(m 1) σ, θ h ) Ω (3.2) (κ 1 δ (m) z h, v h ) Ω = (δ (m) p h, v h ) Ω (3.3) (δ (m) σ : ɛ(q h )) Ω = 0 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 where P 0 represents the space o constants, Q 1 represents the space o trilinears and the details o ˆV( Ê) are given in [4]. 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 B whereas the details o (3.3) are given in Appendix C. 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 } Proo. Step 1: Flow equations Testing (3.1) with θ h δ (m) p h, we get s C δ (m) p h 2 Ω + t( δ (m) z h, δ (m) p h ) Ω + (δ (m) φ p, δ (m) p h ) Ω (3.4) = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω Testing (3.2) with v h δ (m) z h, we get (3.5) κ 1/2 δ (m) z h 2 Ω = (δ (m) p h, δ (m) z h ) Ω From (3.4) and (3.5), we get (3.6) C δ (m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) φ p, δ (m) p h ) Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω Step 2: Poromechanics equations Testing (3.3) with q h δ (m) u h, we get (3.7) (δ (m) σ : δ (m) ɛ) Ω = 0 We now invoke (2.12) to arrive at the expression or change in strain tensor over the (m + 1) th coupling iteration as ollows (3.8) δ (m) ɛ = D ep 1 δ (m) σ + C 3 Bδ(m) p h This manuscript is or review purposes only.

6 Substituting (3.8) in (3.7), we get (3.9) (δ (m) σ : D ep 1 δ (m) σ) Ω + C 3 (B : δ(m) σ, δ (m) p h ) Ω = 0 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) φ p, δ (m) p h ) Ω + (δ (m) σ : D ep 1 δ (m) σ) Ω (3.10) + C 3 (B : δ(m) σ, δ (m) p h ) Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω Step 4: Variation in luid content In lieu o (2.14), the variation in luid content in the (m + 1) th coupling iteration is (3.11) As a result, we can write (3.12) δ (m) ζ = Cδ (m) p h + C 3 B : δ(m) σ + δ (m) φ p 1 2C δ(m) ζ 2 Ω C 2 δ(m) p h 2 Ω C 18 B : δ(m) σ 2 Ω 1 2C δ(m) φ p 2 Ω (δ (m) φ p, δ (m) p h ) Ω 1 3 (B : δ(m) σ, δ (m) φ p ) Ω = C 3 (B : δ(m) σ, δ (m) p h ) Ω From (3.10) and (3.12), we get (3.13) C δ (m) p h 2 Ω + t κ 1/2 δ (m) z h 2 Ω + (δ (m) σ : D ep 1 δ (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 1 2C δ(m) φ p 2 Ω 1 3 (B : δ(m) σ, δ (m) φ p ) Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω 158 Adding and subtracting C 6 B : δ(m) σ 2 Ω to the LHS o (3.13) results in C (3.14) 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 : D ep 1 δ (m) σ) Ω + 1 2C δ(m) ζ 2 Ω C 9 B : δ(m) σ 2 Ω 1 2C δ(m) φ p 2 Ω 1 3 (B : δ(m) σ, δ (m) φ p ) Ω = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω In lieu o (2.14) 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 : 0 δ (m) σ + δ (m) Further, since the pore pressure is rozen during the poromechanical solve, we have p h = δ (m) p h. As a result, we can write δ (m) φ p (3.15) δ (m) ζ = Cδ (m) p h + δ (m) φ p This manuscript is or review purposes only.

7 6 S. DANA, AND M. F. WHEELER Subtracting (3.15) rom (3.11), we can write which implies that δ (m) ζ δ (m) ζ = C 3 B : δ(m) σ + δ (m) φ p δ (m) φ p (3.16) 1 C δ(m) ζ δ (m) ζ 2 Ω 1 C δ(m) φ p δ (m) φ p 2 Ω 1 3 (B : δ(m) σ, (δ (m) φ p δ (m) φ p )) Ω = C 9 B : δ(m) σ 2 Ω In lieu o (3.16), we can write (3.14) as 0 {}}{ 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 0? 0 0 {}}{{}}{{}}{ + (δ (m) σ : D ep 1 δ (m) 1 σ) Ω + 2C δ(m) ζ 2 1 Ω + C δ(m) φ p δ (m) φ p 2 Ω driven to zero by convergence criterion [ {}}{ 1 C δ(m) ζ δ (m) ζ 2 Ω + 1 2C δ(m) φ p 2 Ω + 1 ] 3 (B : δ(m) σ, δ (m) φ p ) Ω (3.17) = C 3 (B : δ(m 1) σ, δ (m) p h ) Ω Step 5: Invoking the Young s inequality Since the sum o the terms on the LHS o (3.17) is nonnegative, the RHS is also nonnegative. We invoke the Young s inequality (see [10]) or the RHS o (3.17) as ollows C 3 (B : δ(m 1) σ, δ (m) p h ) Ω C ( B : δ(m 1) σ 2 Ω + 1 ) (3.18) 2 δ(m) p h 2 Ω In lieu o (3.18), we write (3.17) as >0 {}}{ >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 0? >0 0 {}}{{}}{{}}{ + (δ (m) σ : D ep 1 δ (m) 1 σ) Ω + 2C δ(m) ζ 2 1 Ω + C δ(m) φ p δ (m) φ p 2 Ω driven to zero by convergence criterion [ {}}{ 1 C δ(m) ζ δ (m) ζ 2 Ω + 1 2C δ(m) φ p 2 Ω + 1 ] 3 (B : δ(m) σ, δ (m) φ p ) Ω C 6 B : δ(m 1) σ 2 Ω This manuscript is or review purposes only.

8 On the agenda. Use o the Sherman-Morrison ormula to arrive at the estimates o the positive deiniteness o the elastoplastic compliance tensor Design o the convergence criterion Appendix A. Mathematical theory o small strain elastoplasticity. The important phenomenological aspects o small strain elastoplasticity are The existence o an elastic domain, i.e. a range o stresses within which the behaviour o the material can be considered as purely elastic, without evolution o permanent (plastic) strains. The elastic domain is delimited by the so-called yield stress. I the material is urther loaded at the yield stress, then plastic yielding (or plastic low), i.e. evolution o plastic strains, takes place. Accompanying the evolution o the plastic strain, an evolution o the yield stress itsel is also observed. This phenomenon is known as hardening. The basic components o a general elastoplastic constitutive model are the elastoplastic strain decomposition; a yield criterion, stated with the use o a yield unction; a plastic low rule deining the evolution o the plastic strain; a hardening law, characterising the evolution o the yield limit; and the elastoplastic tangent operator A.1. Additive decomposition o the strain tensor. One o the chie hypotheses underlying the small strain theory o plasticity is the decomposition o the total strain, ɛ, into the sum o an elastic (or reversible) component ɛ e, and a plastic (or permanent) component, ɛ p, ɛ = ɛ e + ɛ p A.2. The yield criterion and the yield surace. A scalar yield unction Φ(σ, A) is introduced where σ is the stress tensor and A is a set o thermodynamical orces. The yield unction deines the elastic domain as the set E = {σ Φ(σ, A) < 0} o stresses or which plastic yielding is not possible. Any stress lying in the elastic domain or on its boundary is said to be plastically admissible. We then deine the set o plastically admissible stresses (or plastically admissible domain) as E = {σ Φ(σ, A) 0} The yield locus, i.e. the set o stresses or which plastic yielding may occur, is the boundary o the elastic domain, where Φ(σ, A) = 0. The yield locus in this case is represented by a hypersurace in the space o stresses. This hypersurace is termed the yield surace and is deined as Y = {σ Φ(σ, A) = 0} For stress levels within the elastic domain, only elastic straining may occur, whereas on its boundary (at the yield stress), either elastic unloading or plastic yielding (or plastic loading) takes place. This yield criterion can be expressed by I Φ < 0 = ɛ p = 0 { ɛ I Φ = 0 = p = 0, elastic unloading ɛ p 0, plastic loading This manuscript is or review purposes only.

9 8 S. DANA, AND M. F. WHEELER σ 3 σ 1 = σ 2 = σ 3 σ 3 σ 1 = σ 2 = σ 3 Φ = 0 Φ = 0 σ 2 σ 2 σ 1 σ 1 Fig. 2. M-C yield surace on the let and D-P yield surace to the right; in principal stress space A.2.1. The Drucker-Prager yield criterion. The Drucker-Prager yield criterion has been proposed by [7] as a smooth approximation to the Mohr-Coulomb law, which states that plastic yielding is the result o rictional sliding between material particles. The M-C criterion is given as Φ = (cosθ 1 3 sinθ sinφ) J 2 + σ hyd sinφ c cosφ where φ is the angle o internal riction, c is the cohesion, σ hyd is the hydrostatic stress given by and θ is the Lode angle given by σ hyd = 1 3 tr(σ) θ = 1 3 sin 1 ( 3 3J3 2J 3/2 2 where J 2 and J 3 are stress deviator invariants given by ) J 2 = 1 2 s : s 259 J3 = det s and s is the stress deviator given by s = σ 1 3 (trσ)i = σ σhyd I The D-P criterion is given as Φ = J 2 + ησ hyd ξc where the parameters η and ξ are chosen according to the required approximation to the M-C criterion. This manuscript is or review purposes only.

10 9 _ǫ σ Φ σ 1 = σ 2 = σ 3 associative D-P σ 3 Φ = 0 σ 2 σ 1 Fig. 3. Associative D-P low vector at the cone surace and at the apex A.3. Plastic low rule. The plastic low rule is given as ɛ p = γn where γ is the plastic multiplier and N is the low vector. A.3.1. Flow vector. It is oten convenient to deine the low rule in terms o a low (or plastic) potential. The starting point o such an approach is to postulate the existence o a low potential with general orm Ψ(σ, A), rom which the low vector is obtained as N Ψ σ A.3.2. The plastic multiplier. The plastic multiplier is non-negative γ 0 and satisies the complementarity condition, Φ γ = 0 A.3.3. Associative and non-associative plasticity. A plasticity model is classed as associative i the yield unction Φ is taken as the low potential, i.e The low vector is then given by Ψ Φ N Φ σ 293 Any other choice o low potential characterises a non-associative (or non-associated) 294 plasticity model. At points where Φ is non-dierentiable in σ, the isosuraces o Φ in 295 the space o stresses contain a singularity (corner) where the normal direction is not 296 uniquely deined. A typical situation is schematically illustrated in Figure 4 where two 297 distinct normals, N 1 and N 2, are assumed to exist. In this case, the subdierential 298 o Φ with respect to σ, denoted σ Φ Φ σ, is the set o vectors contained in the cone 299 deined by all linear combinations (with positive coeicients) o N 1 and N 2. This manuscript is or review purposes only.

11 N 1 N 2 10 S. DANA, AND M. F. σ Φ _ǫ p Fig. 4. Plastic strain increment at corners A.4. Hardening law. Hardening is characterised by a dependence o yield stress level upon the history o plastic straining to which the body has been subjected. Hardening is represented by changes in the hardening thermodynamical orce, A, during plastic yielding. These changes may, in general, aect the size, shape and orientation o the yield surace, deined by Φ(σ, A) = 0. A.4.1. Isotropic hardening. A plasticity model is said to be isotropic hardening i the evolution o the yield surace is such that, at any state o hardening, it corresponds to a uniorm (isotropic) expansion o the initial yield surace, without translation. A.5. The elastoplastic tangent operator. The elastoplastic tangent operator in (2.11) or the associated plasticity model is given by (A.1) D ep = D ( 1 H p + Φ σ : D Φ D Φ ) σ D Φ σ σ where H p is the hardening modulus. The reader is reered to [6] or a derivation o the expression or the elastoplastic tangent operator. Appendix B. 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 (2.14), we write (2.1) as (B.1) t (Cp + C 3 B : σ + φp ) + z = q C p t + z = q C 3 B : σ t φp t This manuscript is or review purposes only.

12 Using backward Euler in time, the discrete in time orm o (B.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 ) 1 m,n+1 t (φp φ pn ) 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 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 ) (φ pm,n+1 φ pn ) 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 + (φ pm,n+1 φ pn, 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 + (δ (m) φ p, 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 (B.2) where V(Ω) 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) and H(div, Ω) is given by 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 the divergence theorem to evaluate the irst term on RHS o (B.2) as ollows } (B.3) ( 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 where we invoke v n = 0 on Γ N. In lieu o (B.2) and (B.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 This manuscript is or review purposes only.

13 12 S. DANA, AND M. F. WHEELER Appendix C. 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 (C.1) ( σ, q) Ω + ( q) Ω = 0 ( q U(Ω)) where U(Ω) is given by 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 D α w = α w, α = α α n, 1.. xαn n x α1 We know rom tensor calculus that (C.2) ( σ, q) Ω (, σq) Ω (σ : q) Ω 382 Further, using the divergence theorem and the symmetry o σ, we arrive at (C.3) (, σq) Ω (q, σn) Ω We decompose q into a symmetric part ( q) s 2( ) 1 q + ( q) T ɛ(q) and skew-symmetric part ( q) ss and note that the contraction between a symmetric and skew-symmetric tensor is zero to obtain From (C.1), (C.2), (C.3) and (??), we get (σn, q) Ω (σ : ɛ(q)) Ω + (, q) Ω = 0 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 = REFERENCES [1] M. A. Biot, Mechanics o deormation and acoustic propagation in porous media, Journal o Applied Physics, 33 (1962), pp [2] M. A. Biot and D. G. Willis, The elastic coeicients o the theory o consolidation, Journal o Applied Mechanics, 24 (1957), pp [3] O. Coussy, Poromechanics, Wiley, 2nd ed., [4] S. Dana, B. Ganis, and M. F. Wheeler, A multiscale ixed stress split iterative scheme or coupled low and poromechanics in deep subsurace reservoirs, Journal o Computational Physics, 352 (2018), pp This manuscript is or review purposes only.

14 [5] S. Dana and M. F. Wheeler, Convergence analysis o ixed stress split iterative scheme or anisotropic poroelasticity with tensor biot parameter, ICES REPORT 18-01, The University o Texas at Austin, (2018). [6] E. A. de Souza Neto, D. Perić, and D. R. J. Owen, Computational Methods or Plasticity Theory and Applications, Wiley, [7] D. C. Drucker and W. Prager, Soil mechanics and plastic analysis or limit design, Quarterly o Applied Mathematics, 10 (1952), pp [8] M. E. Gurtin, E. Fried, and L. Anand, The Mechanics and Thermodynamics o Continua, Cambridge University Press, 1 ed., [9] A. W. Skempton, The pore-pressure coeicients a and b, Géotechnique, 4 (1954), pp [10] J. M. Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art o Mathematical Inequalities, Maa Problem Books Series., Cambridge University Press, This manuscript is or review purposes only.