  This paper is devoted to the stability analysis of a classical three-field
formulation of Biot's consolidation model where the unknown variables are the
displacements, fluid flux (Darcy velocity), and pore pressure. Specific
parameter-dependent norms provide the key in establishing the full
parameter-robust inf-sup stability of the continuous problem. Therefore,
stability results presented here are uniform not only with respect to the
Lam\'e parameter $\lambda$, but also with respect to all the other model
parameters. This allows for the construction of a uniform block diagonal
preconditioner within the framework of operator preconditioning. Stable
discretizations that meet the required conditions for full robustness and
guarantee mass conservation, both locally and globally, are discussed and
corresponding optimal error estimates proven.
