  We propose, analyze and implement a virtual element discretization for an
interfacial poroelasticity-elasticity consolidation problem. The formulation of
the time-dependent poroelasticity equations uses displacement, fluid pressure,
and total pressure, and the elasticity equations are written in the
displacement-pressure formulation. The construction of the virtual element
scheme does not require Lagrange multipliers to impose the transmission
conditions (continuity of displacement and total traction, and no-flux for the
fluid) on the interface. We show the stability and convergence of the virtual
element method for different polynomial degrees, and the error bounds are
robust with respect to delicate model parameters (such as Lame constants,
permeability, and storativity coefficient). Finally, we provide numerical
examples that illustrate the properties of the scheme.
