Ill-conditioning of finite element poroelasticity equations

The solution to Biot's coupled consolidation theory is usually addressed by the Finite Element (FE) method thus obtaining a system of first-order differential equations which is integrated by the use of an appropriate time marching scheme. For small values of the time step the resulting linear system may be severely ill-conditioned and hence the solution can prove quite difficult to achieve. Under such conditions efficient and robust projection solvers based on Krylov's subspaces which are usually recommended for non-symmetric large size problems can exhibit a very slow convergence rate or even fail. The present paper investigates the correlation between the ill-conditioning of FE poroelasticity equations and the time integration step Dt. An empirical relation is provided for a lower bound Dt_crit of Dt below which ill-conditioning may suddenly occur. The critical time step is larger for soft and low permeable porous media discretized on coarser grids. A limiting value for the rock stiffness is found such that for stiffer systems there is no ill-conditioning irrespective of Dt however small, as is also shown by several numerical examples. Finally, the definition of a different $\Delta t_{crit}$ as suggested by other authors is reviewed and discussed