Standard and enhanced (BBAR or EAS) low order finite elements applied to the problem of consolidation of two-phase nonlinear continua do not satisfy the LBB condition when the same interpolation functions are used for both displacement and pore pressure fields. Strong spatial pressure oscillations are the main consequence of the LBB condition violation. The class of direct stabilized methods, widely used in the field of fluid mechanics, is a powerful tool to circumvent violation of the aforementioned condition. Three stabilized formulations, designed for the problem of consolidation of fully or partially saturated media are presented: Galerkin/least-squares (GLS) with the least-squares term construction based on the residuum of the local fluid mass conservation equation; the pressure stabilized formulation (FPL) in which the rate of the pore pressure Laplacian (residual free in the incompressibility limit) is added to the fluid mass conservation equation; and, finally, a stabilized formulation in which the stabilization term is constructed based on the rate of the residuum of the local momentum equation. An h-convergence study and test problems for the three stabilized schemes are discussed.