Ill-conditioning of finite element poroelasticity equations

The solution to Blot's coupled consolidation theory is usually addressed by the finite element (FE) method thus obtaining a system of first-order diff erential equations which is integrated by the use of an appropriate time ma rching scheme. For small values of the time step the resulting linear syste m may be severely ill-conditioned and hence the solution can prove quite di fficult to achieve. Under such conditions efficient and robust projection s olvers based on Krylov's subspaces which are usually recommended for non-sy mmetric large size problems can exhibit a very slow convergence rate or eve n fail. The present paper investigates the correlation between the ill-cond itioning of FE poroelasticity equations and the time integration step Delta t. An empirical relation is provided for a lower bound Deltat(crit) of Delt at below which ill-conditioning may suddenly occur. The critical time step is larger for soft and low permeable porous media discretized on coarser gr ids. A limiting value for the rock stiffness is found such that for stiffer systems there is no ill-conditioning irrespective of Deltat however small, as is also shown by several numerical examples. Finally, the definition of a different Deltat(crit) as suggested by other authors is reviewed and dis cussed. (C) 2001 Elsevier Science Ltd. All rights reserved.