  In this work, we consider a hybrid mixed finite element method for Biot's
model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's
velocity system of Biot's model presented in \cite{C. Niu} is not uniformly
stable with respect to the physical parameters, resulting in some issues in
numerical simulations. To alleviate such problems, following \cite{V. Girault},
we stabilize the hybrid scheme with face bubble functions and show the
well-posedness with respect to physical and discretization parameters, which
provide optimal error estimates of the stabilized method. We introduce a
perturbation of the bilinear form of the displacement which allows for the
elimination of the bubble functions. Together with eliminating Darcy's velocity
by hybridization, we obtain an eliminated system whose size is the same as the
classical P1-RT0-P0 discretization. Based on the well-posedness of the
eliminated system, we design block preconditioners that are parameter-robust.
Numerical experiments are presented to confirm the theoretical results of the
stabilized scheme as well as the block preconditioners.
