  We present a finite element discretisation to model the interaction between a
poroelastic structure and an elastic medium. The consolidation problem
considers fully coupled deformations across an interface, ensuring continuity
of displacement and total traction, as well as no-flux for the fluid phase. Our
formulation of the poroelasticity equations incorporates displacement, fluid
pressure, and total pressure, while the elasticity equations adopt a
displacement-pressure formulation. Notably, the transmission conditions at the
interface are enforced without the need for Lagrange multipliers. We
demonstrate the stability and convergence of the divergence-conforming finite
element method across various polynomial degrees. The a priori error bounds
remain robust, even when considering large variations in intricate model
parameters such as Lam\'e constants, permeability, and storativity coefficient.
To enhance computational efficiency and reliability, we develop residual-based
a posteriori error estimators that are independent of the aforementioned
coefficients. Additionally, we devise parameter-robust and optimal block
diagonal preconditioners. Through numerical examples, including adaptive
scenarios, we illustrate the scheme's properties such as convergence and
parameter robustness.
