we consider a finite element discretizations of the biot's model in poroelasticity with lowest order (mini and stabilized p1-p1) elements. we show convergence of discrete schemes which are implicit in time and use these types of elements in space. we also address the issue related to the presence of non-physical oscillations in the pressure approximations for low permeabilities and/or small time steps and present numerical results confirming the monotone behavior of two stabilized schemes. this is a joint work with carmen rodrigo, francisco gaspar (univ. zaragoza) and xiaozhe hu (tufts).