1 Introduction
This work follows up on our previous work (Dana and Wheeler (2018)), where we arrived at convergence estimates for the case of anisotropic poroelasticity with tensor Biot parameter.
1.1 Preliminaries
Given a bounded convex domain , we use to represent the restriction of the space of polynomials of degree less that or equal to to and to denote the space of trilinears on . For the sake of convenience, we discard the differential in the integration of any scalar field over as follows
Sobolev spaces are based on the space of square integrable functions on given by
The inner product of two second order tensors and is given by (see Gurtin et al. (2010))
() 
A fourth order tensor is a linear transformation of a second order tensor to a second order tensor in the following manner (see
Gurtin et al. (2010))() 
The dyadic product of two second order tensors and is given by (see Gurtin et al. (2010))
2 Flow model
Let the boundary where is the Dirichlet boundary and is the Neumann boundary. The fluid mass conservation equation (2.1) in the presence of deformable and anisotropic porous medium with the Darcy law (2.2) and linear pressure dependence of density (2.3) with boundary conditions (2.4) and initial conditions (2.5) is
(2.1)  
(2.2)  
(2.3)  
(2.4)  
(2.5) 
where is the fluid pressure, is the fluid flux, is the increment in fluid content^{1}^{1}1Biot (1962) defines the increment in fluid content as the measure of the amount of fluid which has flowed in and out of a given element attached to the solid frame, is the unit outward normal on , is the source or sink term, is the uniformly symmetric positive definite absolute permeability tensor, is the fluid viscosity, is a reference density, is the porosity, is a measure of the hydraulic conductivity of the pore fluid, is the fluid compressibility and is the time interval.
3 Poromechanics model
One of the chief hypotheses underlying the small strain theory of elastoplasticity is the decomposition of the total strain, , into the sum of an elastic (or reversible) component , and a plastic (or permanent) component, ,
Let the boundary where is the Dirichlet boundary and is the Neumann boundary. Linear momentum balance for the anisotropic porous solid in the quasistatic limit of interest (3.1) with small strain assumption (3.3) with boundary conditions (3.4) and initial condition (3.5) is
(3.1)  
(3.2)  
(3.3)  
(3.4)  
(3.5) 
where is the solid displacement, is the rock density, is the body force per unit volume, is the unit outward normal to , is the unit outward normal to , is the traction specified on , is the strain tensor, is the Cauchy stress tensor given by the generalized Hooke’s law
(3.6) 
where is the fourth order anisotropic elasticity tensor, is the Biot tensor and is the elastoplastic tangent operator (see de Souza Neto et al. (2009)) and is the effective stress. The inverse of the generalized Hooke’s law (3.6) is given by
(3.7) 
where is a generalized Hooke’s law constant and is a generalization of the Skempton pore pressure coefficient (see Skempton (1954)) for anisotropic poroelastoplasticity, and is given by
(3.8) 
3.1 Increment in fluid content
4 Statement of contraction of the fixed stress split scheme for small strain anisotropic poroelastoplasticity with Biot tensor
We use the notations for any quantity evaluated at time level , for any quantity evaluated at the coupling iteration at time level , for the change in the quantity during the flow solve over the coupling iteration at any time level and for the change in the quantity over the coupling iteration at any time level. Let be finite element partition of consisting of distorted hexahedral elements where . The details of the finite element mapping are given in Dana et al. (2018).
4.1 Discrete variational statements for the flow subproblem in terms of coupling iteration differences
Before arriving at the discrete variational statement of the flow model, we impose the fixed stress constraint on the strong form of the mass conservation equation (2.1). In lieu of (3.9), we write (2.1) as
(4.1) 
Using backward Euler in time, the discrete in time form of (4.1) for the coupling iteration in the time step is written as
where is the time step and the source term as well as the terms evaluated at the previous time level do not depend on the coupling iteration count as they are known quantities. The fixed stress constraint implies that gets replaced by i.e. the computation of and is based on the value of stress updated after the poromechanics solve from the previous coupling iteration at the current time level . The modified equation is written as
As a result, the discrete weak form of (2.1) is given by
Replacing by and subtracting the two equations, we get
The weak form of the Darcy law (2.2) for the coupling iteration in the time step is given by
(4.2) 
where is given by
and is given by
We use the divergence theorem to evaluate the first term on RHS of (4.2) as follows
(4.3) 
where we invoke on . In lieu of (4.2) and (4.3), we get
Replacing by and subtracting the two equations, we get
4.2 Discrete variational statement for the poromechanics subproblem in terms of coupling iteration differences
The weak form of the linear momentum balance (3.1) is given by
(4.4) 
where is given by
where is defined, in general, for any integer as
where the derivatives are taken in the sense of distributions and given by
We know from tensor calculus that
(4.5) 
Further, using the divergence theorem and the symmetry of , we arrive at
(4.6) 
We decompose into a symmetric part
and skewsymmetric part
and note that the contraction between a symmetric and skewsymmetric tensor is zero to obtain From (4.4), (4.5), (4.6) and (LABEL:app4), we getwhich, after invoking the traction boundary condition, results in the discrete weak form for the coupling iteration as
Replacing by and subtracting the two equations, we get
4.3 Summary
The discrete variational statements in terms of coupling iteration differences is : find , and such that
(4.7)  
(4.8)  
(4.9) 
where the finite dimensional spaces , and are
where represents the space of constants, represents the space of trilinears and the details of are given in Dana et al. (2018).
Theorem 4.1.
The fixed stress split iterative coupling scheme for anisotropic poroelasticity with Biot tensor in which the flow problem is solved first by freezing all components of the stress tensor is a contraction given by
Proof.
Step 1: Flow equations
Testing (4.7) with , we get
(4.10) 
Testing (4.8) with , we get
(4.11) 
From (4.10) and (4.11), we get
(4.12) 
Step 2: Poromechanics equations
Testing (4.9) with , we get
(4.13) 
We now invoke (3.7) to arrive at the expression for change in strain tensor over the coupling iteration as follows
(4.14) 
Substituting (4.14) in (4.13), we get
(4.15) 
Step 3: Combining flow and poromechanics equations
Adding (4.12) and (4.15), we get
(4.16) 
Step 4: Variation in fluid content
In lieu of (3.9), the variation in fluid content in the coupling iteration is
(4.17) 
As a result, we can write
(4.18) 
From (4.16) and (4.18), we get
(4.19) 
Adding and subtracting to the LHS of (4.19) results in
(4.20) 
In lieu of (3.9) and the fixed stress constraint during the flow solve, the variation in fluid content during the flow solve in the coupling iteration is given by
Further, since the pore pressure is frozen during the poromechanical solve, we have . As a result, we can write
(4.21) 
Subtracting (4.21) from (4.17), we can write
which implies that
(4.22) 
In lieu of (4.22), we can write (4.20) as
(4.23) 
Step 5: Invoking the Young’s inequality
Since the sum of the terms on the LHS of (4.23) is nonnegative, the RHS is also nonnegative. We invoke the Young’s inequality (see Steele (2004))
for the RHS of (4.23) as follows
(4.24) 
In lieu of (4.24), we write (4.23) as
Step 6: On the agenda

To render a complete statement of contraction, we need to arrive at estimates for the term . In lieu of that, we need to arrive at the inverse of the elastoplastic tangent operator . The exact expression for shall depend on the type of plasticity model (associative or nonassociative) and the yield criterion being employed. As we know the expression for , we intend to invert it by using the ShermanMorrison formula (see Sherman and Morrison (1950)).

In addition, the term needs to be driven to zero to achieve optimal convergence rate. We intend to do that by designing the convergence criterion as that term going to a small positive value. But before we do that, we need to arrive at an expression for the plastic porosity by using certain hypotheses given in Coussy (2004)).
∎
References
 Biot [1962] M. A. Biot. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33(4):1482–1498, 1962.
 Biot and Willis [1957] M. A. Biot and D. G. Willis. The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics, 24:594–601, 1957.
 Coussy [2004] O. Coussy. Poromechanics. Wiley, 2nd edition, 2004.
 Dana and Wheeler [2018] S. Dana and M. F. Wheeler. Convergence analysis of fixed stress split iterative scheme for anisotropic poroelasticity with tensor biot parameter. Computational Geosciences, 22(5):1219–1230, 2018.
 Dana et al. [2018] S. Dana, B. Ganis, and M. F. Wheeler. A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs. Journal of Computational Physics, 352:1–22, 2018.
 de Souza Neto et al. [2009] E. A de Souza Neto, D. Perić, and D. R. J. Owen. Computational Methods for Plasticity Theory and Applications. Wiley, 2009.
 Gurtin et al. [2010] M. E. Gurtin, E. Fried, and L. Anand. The Mechanics and Thermodynamics of Continua. Cambridge University Press, 1 edition, 2010.
 Sherman and Morrison [1950] J. Sherman and W. J. Morrison. Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. The Annals of Mathematical Statistics, 21(1):124–127, 1950.
 Skempton [1954] A. W. Skempton. The porepressure coefficients a and b. Géotechnique, 4(4):143–147, 1954.
 Steele [2004] J. M. Steele. The CauchySchwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Maa Problem Books Series. Cambridge University Press, 2004.
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