Convergence analysis of fixed stress split iterative scheme for anisotropic poroelasticity with tensor Biot parameter

Abstract

We perform a convergence analysis of the fixed stress split iterative scheme for the Biot system modeling coupled flow and deformation in anisotropic poroelastic media with tensor Biot parameter. The fixed stress split iterative scheme solves the flow subproblem with all components of the stress tensor frozen using a multipoint flux mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the fixed stress split iterative scheme for anisotropic poroelasticity with Biot tensor is contractive. We also demonstrate that the scheme is numerically convergent using the classical Mandel’s problem solution for transverse isotropy.

Appendices

Appendix A: finite element mapping

Let r_i, i = 1,.., 8 be the vertices of E. Now, consider a reference cube $\hat {E}$ with vertices $\hat {\mathbf {r}}_{1}=[0\,\,0\,\,0]^{T}$, $\hat {\mathbf {r}}_{2}=[1\,\,0\,\,0]^{T}$, $\hat {\mathbf {r}}_{3}=[1\,\,1\,\,0]^{T}$, $\hat {\mathbf {r}}_{4}=[0\,\,1\,\,0]^{T}$, $\hat {\mathbf {r}}_{5}=[0\,\,0\,\,1]^{T}$, $\hat {\mathbf {r}}_{6}=[1\,\,0\,\,1]^{T}$, $\hat {\mathbf {r}}_{7}=[1\,\,1\,\,1]^{T}$ and $\hat {\mathbf {r}}_{8}=[0\,\,1\,\,1]^{T}$ as shown in Fig. 3. Let $\hat {\mathbf {x}}=(\hat {x},\hat {y},\hat {z})\in \hat {E}$ and x = (x,y,z) ∈ E. The function $F_{E}(\hat {\mathbf {x}}):\hat {E}\rightarrow E$ is

$$\begin{array}{@{}rcl@{}} F_{E}(\hat{\mathbf{x}})&=&\mathbf{r}_{1}(1\,-\,\hat{x})(1\,-\,\hat{y})(1\,-\,\hat{z})+\mathbf{r}_{2}\hat{x}(1\,-\,\hat{y})(1\,-\,\hat{z})+\mathbf{r}_{3}\hat{x}\hat{y}(1\,-\,\hat{z})\\ &&+\mathbf{r}_{4}(1\,-\,\hat{x})\hat{y}(1\,-\,\hat{z})+\mathbf{r}_{5}(1\,-\,\hat{x})(1\,-\,\hat{y})\hat{z}+\mathbf{r}_{6}\hat{x}(1\,-\,\hat{y})\hat{z}\\ && +\mathbf{r}_{7}\hat{x}\hat{y}\hat{z}+ \mathbf{r}_{8}(1\,-\,\hat{x})\hat{y}\hat{z} \end{array} $$

Fig. 3

Trilinear mapping $F_{E}:\hat {E}\rightarrow E$ for 8 noded distorted hexahedral elements. The faces of E can be non-planar

Denote Jacobian matrix by DF_E and let J_E = det(DF_E). Defining r_ij ≡r_i −r_j, we have

$$\scriptstyle DF_{E}(\hat{\mathbf{x}}) = \left[\begin{array}{ccccccc} \!\mathbf{r}_{21}\,+\,(\mathbf{r}_{34}\,-\,\mathbf{r}_{21})\hat{y} +(\mathbf{r}_{65}\,-\,\mathbf{r}_{21})\hat{z} \,+\,((\mathbf{r}_{21}\,-\,\mathbf{r}_{34}) -(\mathbf{r}_{65}\,-\,\mathbf{r}_{78}))\hat{y}\hat{z};\\ \!\mathbf{r}_{41}\,+\,(\mathbf{r}_{34}\,-\,\mathbf{r}_{21})\hat{x} +(\mathbf{r}_{85}\,-\,\mathbf{r}_{41})\hat{z} \,+\,((\mathbf{r}_{21}\,-\,\mathbf{r}_{34}) -(\mathbf{r}_{65}\,-\,\mathbf{r}_{78}))\hat{x}\hat{z};\\ \!\mathbf{r}_{51}\,+\,(\mathbf{r}_{65}\,-\,\mathbf{r}_{21})\hat{x} +(\mathbf{r}_{85}\,-\,\mathbf{r}_{41})\hat{y} \,+\,((\mathbf{r}_{21}\,-\,\mathbf{r}_{34}) -(\mathbf{r}_{65}\,-\,\mathbf{r}_{78}))\hat{x}\hat{y} \end{array}\!\right] $$

Denote inverse mapping by $F_{E}^{-1}$, its Jacobian matrix by $DF_{E}^{-1}$ and let $J_{F_{E}^{-1}}=det(DF_{E}^{-1})$ such that

$$DF_{E}^{-1}(\mathbf{x})=(DF_{E})^{-1}(\hat{\mathbf{x}});\qquad J_{F_{E}^{-1}}(\mathbf{x})=(J_{E})^{-1}(\hat{\mathbf{x}}) $$

Let ϕ(x) be any function defined on E and $\hat {\phi }(\hat {\mathbf {x}})$ be its corresponding definition on $\hat {E}$. Then we have

$$ \nabla \phi = (DF_{E}^{-1})^{T}(\mathbf{x})\,\hat{\nabla} \hat{\phi} = (DF_{E})^{-T}(\hat{\mathbf{x}})\,\hat{\nabla} \hat{\phi} $$

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Appendix B: enhanced BDDF₁ spaces

For the sake of clarity, we provide a brief description of the mixed finite element spaces used in the flow model. Let $\mathbf {V}^{*}_{h}\times W_{h}$ be the lowest order BDDF₁ MFE spaces on hexahedra (see Brezzi et al. [10]). On the reference unit cube, these spaces are defined as

$$\begin{array}{@{}rcl@{}} \hat{\mathbf{V}}^{*}(\hat{E})\!&=&\!P_{1}(\hat{E}) \,+\,r_{0}\,curl(0,0,\hat{x}\hat{y}\hat{z})^{T} \,+\,r_{1}\,curl(0,0,\hat{x}\hat{y}^{2})^{T}\\ &&+s_{0}\,curl(\hat{x}\hat{y}\hat{z},0,0)^{T}+s_{1}\,curl(\hat{y}\hat{z}^{2},0,0)^{T}\\&& +t_{0}\,curl(0,\hat{x}\hat{y}\hat{z},0)^{T}+t_{1}\,curl(0,\hat{x}^{2}\hat{z},0)^{T}\\ \hat{W}(\hat{E})\!&=&\!\mathbb{P}_{0}(\hat{E}) \end{array} $$

with the following properties

$$\hat{\nabla} \cdot \hat{\mathbf{V}}^{*}(\hat{E})=\hat{W}(\hat{E}), \quad \text{and} \quad \forall \hat{\mathbf{v}}\in \hat{\mathbf{V}}^{*}(\hat{E}),\,\forall \hat{e}\subset \partial \hat{E},\,\,\hat{\mathbf{v}}\cdot \hat{\mathbf{n}}_{\hat{e}}\in P_{1}(\hat{e}) $$

The multipoint flux approximation procedure requires on each face one velocity degree of freedom to be associated with each vertex. Since the BDDF₁ space $\mathbf {V}^{*}_{h}$ has only three degrees of freedom per face, it is augmented with six more degrees of freedom (one extra degree of freedom per face). Since the constant divergence, the linear independence of the shape functions and the continuity of the normal component across the element faces are to be preserved, six curl terms are added [19]. Let V_h × W_h be the enhanced BDDF₁ spaces on hexahedra. On the reference unit cube, these spaces are

$$\begin{array}{@{}rcl@{}} \hat{\mathbf{V}}(\hat{E})\!&=&\!\hat{\mathbf{V}}^{*}(\hat{E}) \,+\,r_{2}\,curl({\kern-.3pt}0{\kern-.3pt},0{\kern-.3pt},\hat{x}^{2}\hat{z})^{T} \,+\,r_{3}\,curl(0,0,\hat{x}^{2}\hat{y}\hat{z})^{T} \\ &&+s_{2}\,curl(\hat{x}\hat{y}^{2},0,0)^{T}+s_{3}\,curl(\hat{x}\hat{y}^{2}\hat{z}^{2},0,0)^{T}\\&& +t_{2}\,curl(0,\hat{y}\hat{z}^{2},0)^{T} +t_{3}\,curl(0,\hat{x}\hat{y}\hat{z}^{2},0)^{T}\\ \hat{W}(\hat{E})\!&=&\!\mathbb{P}_{0}(\hat{E}) \end{array} $$

with the following properties

$$\hat{\nabla} \cdot \hat{\mathbf{V}}(\hat{E})=\hat{W}(\hat{E}), \quad \text{and} \quad \forall \hat{\mathbf{v}}\in \hat{\mathbf{V}}(\hat{E}),\,\,\forall \hat{e}\subset \partial \hat{E},\,\,\hat{\mathbf{v}}\cdot \hat{\mathbf{n}}_{\hat{e}}\in \mathbb{Q}_{1}(\hat{e}) $$

where $\mathbb {Q}_{1}$ is the space of bilinear functions. Since $dim\,\mathbb {Q}_{1}(\hat {e})= 4$, the dimension of $\hat {\mathbf {V}}(\hat {E})$ is 24 as shown in Fig. 4.

Fig. 4

Degrees of freedom and basis functions for the enhanced BDDF₁ velocity space on hexahedra

Appendix C: 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 (1). In lieu of Eq. 13, we write (1) as

$$\begin{array}{@{}rcl@{}} &&\frac{\partial}{\partial t}(Cp+\frac{C}{3}\mathbf{B}:\boldsymbol{\sigma})+\nabla \cdot \mathbf{z}=q\\ &&C\frac{\partial p}{\partial t} +\nabla \cdot \mathbf{z}=q-\frac{C}{3}\mathbf{B}:\frac{\partial \boldsymbol{\sigma}}{\partial t} \end{array} $$

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Using backward Euler in time, the discrete in time form of Eq. 40 for the m^th coupling iteration in the (n + 1)^th time step is written as

$$C\frac{1}{{\Delta} t}(p^{m,n + 1}-p^{n}) +\nabla \cdot \mathbf{z}^{m,n + 1}=q^{n + 1}-\frac{1}{{\Delta} t}\frac{C}{3}\mathbf{B}:(\boldsymbol{\sigma}^{m,n + 1}-\boldsymbol{\sigma}^{n}) $$

where Δt 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 fixed stress constraint implies that σ^m,n+ 1 gets replaced by σ^{m− 1,n+ 1} i.e. the computation of p^m,n+ 1 and z^m,n+ 1 is based on the value of stress updated after the poromechanics solve from the previous coupling iteration m − 1 at the current time level n + 1. The modified equation is written as

$$C(p^{m,n + 1}-p^{n})+{\Delta} t\nabla \cdot \mathbf{z}^{m,n + 1}={\Delta} t q^{n + 1}-\frac{C}{3}\mathbf{B}:(\boldsymbol{\sigma}^{m,n + 1}-\boldsymbol{\sigma}^{n}) $$

As a result, the discrete weak form of Eq. 1 is given by

$$\begin{array}{@{}rcl@{}} &&C(p_{h}^{m,n + 1}-{p_{h}^{n}},\theta_{h})_{{\Omega}}+{\Delta} t(\nabla \cdot \mathbf{z}_{h}^{m,n + 1},\theta_{h})_{{\Omega}}\\ &=&{\Delta} t(q^{n + 1},\theta_{h})_{{\Omega}}-\frac{C}{3}(\mathbf{B}:(\boldsymbol{\sigma}^{m-1,n + 1}-\boldsymbol{\sigma}^{n}),\theta_{h})_{{\Omega}} \end{array} $$

Replacing m by m + 1 and subtracting the two equations, we get

$$C(\delta^{(m)}p_{h},\theta_{h})_{{\Omega}}+{\Delta} t(\nabla \cdot \delta^{(m)}\mathbf{z}_{h},\theta_{h})_{{\Omega}}=-\frac{C}{3}(\mathbf{B}:\delta^{(m-1)}\boldsymbol{\sigma},\theta_{h})_{{\Omega}} $$

The weak form of the Darcy law (2) for the m^th coupling iteration in the (n + 1)^th time step is given by

$$ (\boldsymbol{\kappa}^{-1}\mathbf{z}^{m,n + 1},\mathbf{v})_{{\Omega}}\,=\,-(\nabla p^{m,n + 1},\mathbf{v})_{{\Omega}}+(\rho_{0} \mathbf{g},\mathbf{v})_{{\Omega}}\,\, \!\forall\,\,\mathbf{v}\!\in\! \mathbf{V}({\Omega}) $$

(41)

where V(Ω) is given by

$$\mathbf{V}({\Omega})\equiv \mathbf{H}(div,{\Omega})\cap \big\{\mathbf{v}:\mathbf{v}\cdot \mathbf{n}= 0\,\,\text{on}\,\,{{\Gamma}_{N}^{f}}\big\} $$

and H(div,Ω) is given by

$$\mathbf{H}(div,{\Omega})\equiv\big\{\mathbf{v}:\mathbf{v}\in (L^{2}({\Omega}))^{3},\nabla \cdot \mathbf{v}\in L^{2}({\Omega}) \big\} $$

We use the divergence theorem to evaluate the first term on RHS of Eq. 41 as follows

$$\begin{array}{@{}rcl@{}} (\nabla p^{m,n + 1},\mathbf{v})_{{\Omega}}&=&(\nabla,p^{m,n + 1}\mathbf{v})_{{\Omega}}-(p^{m,n + 1},\nabla \cdot \mathbf{v})_{{\Omega}}\\ &=&(p^{m,n + 1},\mathbf{v}\cdot \mathbf{n})_{\partial {\Omega}} -(p^{m,n + 1},\nabla \cdot \mathbf{v})_{{\Omega}}\\ &=&(g,\mathbf{v}\cdot \mathbf{n})_{{{\Gamma}_{D}^{f}}} -(p^{m,n + 1},\nabla \cdot \mathbf{v})_{{\Omega}} \end{array} $$

(42)

where we invoke v ⋅n = 0 on ${{\Gamma }_{N}^{f}}$. In lieu of Eqs. 41 and 42, we get

$$\begin{array}{@{}rcl@{}} (\boldsymbol{\kappa}^{-1}\mathbf{z}^{m,n + 1},\mathbf{v})_{{\Omega}}=-(g,\mathbf{v}\cdot \mathbf{n})_{{{\Gamma}_{D}^{f}}}+(p^{m,n + 1},\nabla \cdot \mathbf{v})_{{\Omega}}+(\rho_{0} \mathbf{g},\mathbf{v})_{{\Omega}} \end{array} $$

Replacing m by m + 1 and subtracting the two equations, we get

$$ (\boldsymbol{\kappa}^{-1}\delta^{(m)} \mathbf{z}_{h}, \mathbf{v}_{h})_{{\Omega}}=(\delta^{(m)} p_{h},\nabla \cdot \mathbf{v}_{h})_{{\Omega}} $$

Appendix D: discrete variational statement for the poromechanics subproblem in terms of coupling iteration differences

The weak form of the linear momentum balance (6) is given by

$$ (\nabla \cdot \boldsymbol{\sigma},\mathbf{q})_{{\Omega}}+(\mathbf{f}\cdot \mathbf{q})_{{\Omega}}= 0\qquad (\forall\,\,\mathbf{q}\in \mathbf{U}({\Omega})) $$

(43)

where U(Ω) is given by

$$\mathbf{U}({\Omega})\equiv \big\{\mathbf{q}=(u,v,w):u,v,w\in H^{1}({\Omega}),\mathbf{q}=\mathbf{0}\,\,\text{on}\,\,{{\Gamma}_{D}^{p}}\big\} $$

where H^m(Ω) is defined, in general, for any integer m ≥ 0 as

$$H^{m}({\Omega})\equiv\big\{w:D^{\alpha}w\in L^{2}({\Omega})\,\,\forall |\alpha| \leq m \big\}, $$

where the derivatives are taken in the sense of distributions and given by

$$D^{\alpha}w=\frac{\partial^{|\alpha|}w}{\partial x_{1}^{\alpha_{1}}..\partial x_{n}^{\alpha_{n}}},\,\,|\alpha|=\alpha_{1}+\cdots+\alpha_{n}, $$

We know from tensor calculus that

$$ (\nabla \cdot \boldsymbol{\sigma},\mathbf{q})_{{\Omega}}\equiv (\nabla ,\boldsymbol{\sigma}\mathbf{q})_{{\Omega}}-(\boldsymbol{\sigma}:\nabla \mathbf{q})_{{\Omega}} $$

(44)

Further, using the divergence theorem and the symmetry of σ, we arrive at

$$ (\nabla ,\boldsymbol{\sigma}\mathbf{q})_{{\Omega}}\equiv (\mathbf{q},\boldsymbol{\sigma}\mathbf{n})_{\partial {\Omega}} $$

(45)

We decompose ∇q into a symmetric part $(\nabla \mathbf {q})_{s}\equiv \frac {1}{2}\big (\nabla \mathbf {q}+(\nabla \mathbf {q})^{T}\big )\equiv \boldsymbol {\epsilon }(\mathbf {q})$ and skew-symmetric part (∇q)_ss and note that the contraction between a symmetric and skew-symmetric tensor is zero to obtain

From Eqs. 43, 44, 45 and 46, we get

$$(\boldsymbol{\sigma}\mathbf{n},\mathbf{q})_{\partial {\Omega}} - (\boldsymbol{\sigma}:\boldsymbol{\epsilon}(\mathbf{q}))_{{\Omega}} + (\mathbf{f},\mathbf{q})_{{\Omega}}= 0 $$

which, after invoking the traction boundary condition, results in the discrete weak form for the m^th coupling iteration as

$$ (\mathbf{t}^{n + 1},\mathbf{q}_{h})_{{{\Gamma}_{N}^{p}}} - (\boldsymbol{\sigma}^{m,n + 1}:\boldsymbol{\epsilon}(\mathbf{q}_{h}))_{{\Omega}} + (\mathbf{f}^{n + 1},\mathbf{q}_{h})_{{\Omega}}= 0 $$

Replacing m by m + 1 and subtracting the two equations, we get

$$ (\delta^{(m)} \boldsymbol{\sigma}:\boldsymbol{\epsilon}(\mathbf{q}_{h}))_{{\Omega}}= 0 $$

Appendix E: contracted notation

The generalized Hooke’s law (11) written in indicial notation

$$\sigma_{ij}=\sigma_{0_{ij}}+\mathbb{M}_{ijkl}\epsilon_{kl}-\alpha_{ij}(p-p_{0}) $$

is rewritten in contracted notation as

$$\sigma_{\beta}=\sigma_{0_{\beta}}+\mathbb{M}_{\beta\gamma}\epsilon_{\gamma}-\alpha_{\beta}(p-p_{0}) $$

where the transformation is accomplished by replacing the subscripts ij (or kl) by β (or γ) using the following rules

$$\begin{array}{ccc} ij\,\,(\text{or}\,\,kl) & \longleftrightarrow & \beta\,\,(\text{or}\,\,\gamma)\\ 11 & \longleftrightarrow & 1\\ 22 & \longleftrightarrow & 2\\ 33 & \longleftrightarrow & 3\\ 23\,\,(\text{or}\,\,32) & \longleftrightarrow & 4\\ 31\,\,(\text{or}\,\,13) & \longleftrightarrow & 5\\ 12\,\,(\text{or}\,\,21) & \longleftrightarrow & 6 \end{array} $$

In other words, the stress and strain tensors are represented as

$$\begin{array}{@{}rcl@{}} &&\boldsymbol{\sigma}= \left\{\begin{array}{cccccc} \sigma_{xx} & \sigma_{yy} & \sigma_{zz} & \sigma_{yz} & \sigma_{xz} & \sigma_{xy} \end{array}\right\}^{T},\\ &&\boldsymbol{\epsilon}= \left\{\begin{array}{cccccc} \epsilon_{xx} & \epsilon_{yy} & \epsilon_{zz} & \epsilon_{yz} & \epsilon_{xz} & \epsilon_{xy} \end{array}\right\}^{T} \end{array} $$

Appendix F: material symmetry

A material is said to possess a symmetry with respect to an orthogonal transformation χ if the elasticity tensor $\mathbb {M}$ is invariant under the orthogonal transformation χ as follows (see Ting [31])

$$\mathbb{M}^{\prime}=\mathbf{K}\mathbb{M}\mathbf{K}^{T}\equiv \mathbb{M} $$

where $\mathbb {M}^{\prime }$ is the transformed elasticity tensor and K is given by

$$\begin{array}{@{}rcl@{}} \mathbf{K}&=& \left[\begin{array}{ccccccc} \chi_{11}^{2} & \chi_{12}^{2} & \chi_{13}^{2} & 2\chi_{12}\chi_{13} & 2\chi_{13}\chi_{11} & 2\chi_{11}\chi_{12}\\ \chi_{21}^{2} & \chi_{22}^{2} & \chi_{23}^{2} & 2\chi_{22}\chi_{23} & 2\chi_{23}\chi_{21} & 2\chi_{21}\chi_{22}\\ \chi_{31}^{2} & \chi_{32}^{2} & \chi_{33}^{2} & 2\chi_{32}\chi_{33} & 2\chi_{33}\chi_{31} & 2\chi_{31}\chi_{32}\\ \chi_{21}\chi_{31} & \chi_{22}\chi_{32} & \chi_{23}\chi_{33}\\ \chi_{31}\chi_{11} & \chi_{32}\chi_{12} & \chi_{33}\chi_{13} & & \mathbf{K}^{\prime} & \\ \chi_{11}\chi_{21} & \chi_{12}\chi_{22} & \chi_{13}\chi_{23} \end{array}\right]\\ \mathbf{K}^{\prime}&=& \left[\begin{array}{ccccccc} \chi_{22}\chi_{33}+\chi_{23}\chi_{32} & \chi_{23}\chi_{31}+\chi_{21}\chi_{33} & \chi_{21}\chi_{32}+\chi_{22}\chi_{31}\\ \chi_{32}\chi_{13}+\chi_{33}\chi_{12} & \chi_{33}\chi_{11}+\chi_{31}\chi_{13} & \chi_{31}\chi_{12}+\chi_{32}\chi_{11}\\ \chi_{12}\chi_{23}+\chi_{13}\chi_{22} & \chi_{13}\chi_{21}+\chi_{11}\chi_{23} & \chi_{11}\chi_{22}+\chi_{12}\chi_{21} \end{array}\right] \end{array} $$

where χ given by

$$\boldsymbol{\chi}=\mathbf{I}-2\mathbf{n}\mathbf{n}^{T} $$

is a reflection with respect to a plane whose normal is n. The plane is also referred to as the plane of material symmetry.

1.1 F.1 Transverse isotropy

The symmetry planes can be one of the following three possibilities

• 1. x = 0 plane and any plane that contains the x-axis

• 2. y = 0 plane and any plane that contains the y-axis

• 3. z = 0 plane and any plane that contains the z-axis

For the Mandel’s problem (see Fig. 2), the material symmetry planes are the second possibility (y = 0 plane and any plane that contains the y-axis). In lieu of that, the elasticity tensor $\mathbb {M}$ reduces to

$$ \left[\begin{array}{ccccccc} \mathbb{M}_{11} & \mathbb{M}_{12} & \mathbb{M}_{13} & 0 & 0 & 0\\ \mathbb{M}_{12} & \mathbb{M}_{22} & \mathbb{M}_{12} & 0 & 0 & 0\\ \mathbb{M}_{13} & \mathbb{M}_{12} & \mathbb{M}_{11} & 0 & 0 & 0\\ 0 & 0 & 0 & 2\mathbb{M}_{44} & 0 & 0\\ 0 & 0 & 0 & 0 & (\mathbb{M}_{11}-\mathbb{M}_{13}) & 0\\ 0 & 0 & 0 & 0 & 0 & 2\mathbb{M}_{44} \end{array}\right] $$

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with five independent components $\mathbb {M}_{11}$, $\mathbb {M}_{12}$, $\mathbb {M}_{13}$, $\mathbb {M}_{22}$ and $\mathbb {M}_{44}$.