  We consider finite element discretizations of the Biot's consolidation model
in poroelasticity with MINI and stabilized P1-P1 elements. We analyze the
convergence of the fully discrete model based on spatial discretization with
these types of finite elements and implicit Euler method in time. We also
address the issue related to the presence of non-physical oscillations in the
pressure approximation for low permeabilities and/or small time steps. We show
that even in 1D a Stokes-stable finite element pair fails to provide a monotone
discretization for the pressure in such regimes. We then introduce a
stabilization term which removes the oscillations. We present numerical results
confirming the monotone behavior of the stabilized schemes.
