A Global Existence Result for Weakly Coupled Two-Phase Poromechanics
Jakub Wiktor Both, Clément Cancès · SIAM Journal on Mathematical Analysis · 2025
Abstract. Multiphase poromechanics describes the evolution of multiphase flow in deformable porous media. Mathematical models for such multiphysics systems are inherently 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 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 a twofold regularization, first, relaxing the porosity constraint to provide strong monotonicity of the underlying energy, and second, adding coercivity by mitigating the degeneracy in the relative permeability through uniform positivity. This finally enables a Faedo–Galerkin approach and compactness arguments including vanishing regularization. This comprises the first global existence result for multiphase poromechanics accounting for degeneracies that are consistent with the multiphase nature of the flow.