Mixed finite element formulation for dynamics of porous media

This paper presents a numerical approach for computing solutions to Biot's fully dynamic model of saturated porous media with incompressible solid and fluid phases. Spatial discretization is based on a three-field (u-w-p) formulation, employing a lowest-order Raviart-Thomas mixed element for the fluid Darcy velocity (w) and pressure (p) fields, and a nodal finite element for skeleton displacement field (u). The discretization is constructed based on the natural topology of the variables and satisfies the Ladyženskaja-Babuška-Brezzi stability condition, avoiding locking in the incompressible and undrained limits. Since fluid acceleration is not neglected, the three-field formulation fully captures dynamic behavior even under high-frequency loading phenomena. The importance of consistent initial conditions is addressed for poroelasticity equations with the mass balance constraint, a system of differential algebraic equations. Energy balance is derived for the porous medium and used to assess accuracy of time integration. To demonstrate the performance of the proposed approach, a variety of numerical studies are carried out including verification with analytical and boundary element solutions and analyses of wave propagation, effect of hydraulic conductivity on damping and frequency content, energy balance, mass lumping, mesh pattern and size, and stability. Some discrepancies found in dynamic poroelasticity results in the literature are also explained.