  We consider the quasi-static Biot's consolidation model in a three-field
formulation with the three unknown physical quantities of interest being the
displacement $\boldsymbol{u}$ of the solid matrix, the seepage velocity
$\boldsymbol{v}$ of the fluid and the pore pressure $p$. As conservation of
fluid mass is a leading physical principle in poromechanics, we preserve this
property using an $\boldsymbol{H}(\operatorname{div})$-conforming ansatz for
$\boldsymbol{u}$ and $\boldsymbol{v}$ together with an appropriate pressure
space. This results in Stokes and Darcy stability and exact, that is, pointwise
mass conservation of the discrete model.
  The proposed discretization technique combines a hybridized discontinuous
Galerkin method for the elasticity subproblem with a mixed method for the flow
subproblem, also handled by hybridization. The latter allows for a static
condensation step to eliminate the seepage velocity from the system while
preserving mass conservation. The system to be solved finally only contains
degrees of freedom related to $\boldsymbol{u}$ and $p$ resulting from the
hybridization process and thus provides, especially for higher-order
approximations, a very cost-efficient family of physics-oriented space
discretizations for poroelasticity problems.
  We present the construction of the discrete model, theoretical results
related to its uniform well-posedness along with optimal error estimates and
parameter-robust preconditioners as a key tool for developing uniformly
convergent iterative solvers. Finally, the cost-efficiency of the proposed
approach is illustrated in a series of numerical tests for three-dimensional
test cases.
