Abstract
In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate subproblems. The coupled discretization has the following key properties, the combination of which is novel: (1) The variables for the pressure and displacement are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure per cell center). (2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces and mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and the convergence of the method.
Highlights
Deformable porous media are becoming increasingly important in applications
The discretization has the following properties: (i) The discretizations of the fluid and mechanical subproblems are identical to the decoupled finite volume (FV) discretizations. (ii) When local variables are eliminated by static condensation, the resulting system of equations is in terms of cell-centered displacement and pressure only. (iii) We show that the co-located discretization is naturally stable in the sense of Definition 1 without the addition of any artificial stabilization term or stabilization parameter. (iv) The discretization is consistent with the variational FV formulation of the Biot system
We will define by capital letters the bilinear forms with the FV interpolations suppressed, such that, e.g., AD(uT, v) = aD(ΠuFV,uuT, v). The exceptions to this rule are the coupling term, which for notational consistency is denoted by BDT,2 = aD(ΠuFV,p pT, v), and the local consistency operator, which is denoted by ΔD(pT, r) = bD(ΠuFV,p pT, r)
Summary
Deformable porous media are becoming increasingly important in applications. In particular, the emergence of strongly engineered geological systems such as CO2 storage [31, 33], geothermal energy [35], and shale-gas extraction all require analysis of the coupling of fluid flow and deformation. The exceptions to this rule are the coupling term, which for notational consistency is denoted by BDT ,2 = aD(ΠuFV,p pT , v), and the local consistency operator, which is denoted by ΔD(pT , r) = bD(ΠuFV,p pT , r) This allows us to define our cell-centered discretizations compactly. Terms with similar scaling have previously been introduced artificially in order to obtain a stable discretization of equations (1) (see, e.g., [11, 16]) We note that this term is essential for the stability method of the current scheme, numerical experiments indicate that it is not sufficient to guarantee monotonicity of the resulting pressure solution. For the solid deformation problem, a similar local coercivity condition is needed
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