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).