Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

Abstract

Summary The parameters in the governing system of partial differential equations of multiple‐network poroelasticity models typically vary over several orders of magnitude, making its stable discretization and efficient solution a challenging task. In this paper, we prove the uniform Ladyzhenskaya–Babuška–Brezzi (LBB) condition and design uniformly stable discretizations and parameter‐robust preconditioners for flux‐based formulations of multiporosity/multipermeability systems. Novel parameter‐matrix‐dependent norms that provide the key for establishing uniform LBB stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ but also to all the other model parameters, such as the permeability coefficients Ki; storage coefficients cpi; network transfer coefficients βi j,i,j = 1,…,n; the scale of the networks n; and the time step size τ. Moreover, strongly mass‐conservative discretizations that meet the required conditions for parameter‐robust LBB stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm‐equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.

Robust fixed stress splitting for Biot’s equations in heterogeneous media

“…Different alternation of iterative cycles in flow and mechanics, i.e., single- [4] and multi-rate schemes [3,5,48,74], multiscale methods [47] and algebraic solvers can be considered. General Schur complement based preconditioners [8,36,37,40,58,58,94,126,127] and preconditioners which are robust with respect to the model parameters [1,9,60,61,77,102,103] are examples of algebraic solvers. For other splitting schemes, see for example the works of Turska et al [118,119].…”

On the optimization of the fixed‐stress splitting for Biot's equations

“…We mention that the fixed‐stress splitting scheme also can be applied to more involved extensions of Biot's equations, for example, including nonlinear water compressibility, unsaturated poroelasticity, the multiple‐network poroelasticity theory, finite‐strain poroplasticity, fractured porous media, and fracture propagation . For nonlinear problems, one combines a linearization technique, eg, the L ‐scheme, with the splitting algorithm; the convergence of the resulting scheme can be proved rigorously .…”

Iterative solvers for Biot model under small and large deformations

“…In terms of modelling, Biot's model has been extended to unsaturated flow [14,37], multiphase flow [27,28,34,36,47], thermo-poroelasticity [20], and reactive transport in porous media [33,48], where nonlinearities arise in the flow model, specifically in the diffusion term, the time derivative term, and/or in Biot's coupling term. The mechanics model can also be extended to the elastoplastic [3,56], the fracture propagation [35], and the hyperelasticity [21,22], where the nonlinearities appear in the constitutive law of the material, in the compatibility condition and/or the conservation of momentum equation.…”