Mandel's problem as a benchmark for two-dimensional nonlinear poroelasticity

Abstract

In this paper, we consider Mandel's problem in the context of nonlinear single-phase poroelasticity, where it is assumed that the fluid is sightly compressible and porosity and permeability are given functions of the volume strain. In the first part of the paper we prove well-posedness of the time-discrete incremental problem by recasting the equations in an abstract form involving a pseudo-monotone operator. Further, we show existence of a Lyapunov functional yielding a global time discrete solution. In the second part, we investigate numerically the behavior of the poroelastic structure. In particular, we verify the assumptions leading to Mandel's solution. We also demonstrate some consequences of the proposed nonlinearities.

Keywords:

• Nonlinear poroelasticity
• Lyapunov functional
• incompressible fluids and solids
• Mandel's problem
• benchmark

2020 Mathematics Subject Classifications:

• 35M13
• 76Sxx
• 65M22

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

A. M. was partially supported by Darcy Center of Eindhoven University of Technology and Utrecht University, the Netherlands, by the project UPGEO 〈ANR-19-CU05-032 〉 of the French National Research Agency (ANR) and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ‘Investissements d'Avenir’ (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The author deceased in Lyon on 28/11/2020.