We consider a nonlinear elliptic-parabolic system of PDEs that models fluid flow through poro-visco-elastic material. The ability of the fluid to flow within the solid is described by the permeability tensor which varies nonlinearly with the structural dilation. We study the existence of weak solutions in bounded domains with physical, mixed-type boundary conditions, and we account for non-zero volumetric and boundary sources. One principal aim is to investigate the influence of viscoelasticity on the qualitative properties solution. Our analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on the presence of viscoelasticity. Theoretical results are further investigated via numerical simulations; when data are appropriately regular, numerical simulations show that the solutions satisfy the predicted energy estimates. Simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy may become unbounded if the data do not enjoy the time regularity required by the theory. These results are interpreted in the context of pressure
changes in lamina cribrosa (in the human eye), and the connection between these biomechanics and the development of glaucoma.