Iterative Methods for Coupled Flow and Geomechanics in Unsaturated Porous Media

Abstract

Multiphase poromechanics describes the evolution of multiphase flow in deformable porous media. Mathematical models for such multiphysics system are inheritely nonlinear, potentially degenerate and fully coupled systems of partial differential equations. In this work, we present a thermodynamically consistent multiphase poromechanics model falling into the category of Biot equations and obeying to a generalized gradient flow structure. It involves capillarity effects, degenerate relative permeabilities, and gravity effects. In addition to established models it introduces a Lagrange multiplier associated to a bound constraint on the effective porosity in particular ensuring its positivity. We establish existence of global weak solutions under the assumption of a weak coupling strength, implicitly utilizing the gradient flow structure, as well as regularization, a Faedo-Galerkin approach and compactness arguments. This comprises the first global existence result for multiphase poromechanics accounting for degeneracies that are consistent with the multiphase nature of the flow.

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

“…The convergence analysis of the iterative solvers proposed cannot be addressed with standard techniques [11,14,15,37,39]. This is due to the non-linearities being non-monotone.…”

“…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.…”

“…A theoretical investigation on the optimal choice for this parameter is performed in [53]. The linearization is based on either Newton's method, or the L-scheme [37,43,47] or a combination of them [14,37]. For monolithic and splitting schemes based solely on L-scheme, we refer to [11].…”

A Finite-Volume-Based Module for Unsaturated Poroelasticity

“…Given the complexity of the resulting model, it is imperative to use robust discretization techniques in a flexible computational setting. The fully coupled system is not commonly treated by numerical software, and the few available codes are limited to the use of finite-element methods [26] or mixed finite-element methods [7]. In this module, we propose the use of finite-volume methods (FVM), which are inherently conservative while keeping the advantages of robust discretization schemes; i.e., flexibility in representing complex domains.…”