Mathematical Theory of Nonlinear Single-Phase Poroelasticity

In this paper, we study the equations of nonlinear poroelasticity derived from mixture theory. They describe the quasi-static mechanical behavior of a fluid saturated porous medium. The nonlinearity arises from the compressibility of the fluid and from the dependence of porosity and permeability on the divergence of the displacement. We point some limitations of the model. In our approach, we discretize the quasi-static formulation in time and first consider the corresponding incremental problem. For this, we prove existence of a solution using Brézis’ theory of pseudo-monotone operators. Generalizing Biot’s free energy to the nonlinear setting, we construct a Lyapunov functional, yielding global stability. This allows us to construct bounds that are uniform with respect to the time step. In the case when dissipative interface effects between the fluid and the solid are taken into account, we consider the continuous time case in the limit when the time step tends to zero. This yields existence of a weak free energy solution.