  We propose four-field and five-field Hu--Washizu-type mixed formulations for
nonlinear poroelasticity -- a coupled fluid diffusion and solid deformation
process -- considering that the permeability depends on a linear combination
between fluid pressure and dilation. As the determination of the physical
strains is necessary, the first formulation is written in terms of the primal
unknowns of solid displacement and pore fluid pressure as well as the
poroelastic stress and the infinitesimal strain, and it considers strongly
symmetric Cauchy stresses. The second formulation imposes stress symmetry in a
weak sense and it requires the additional unknown of solid rotation tensor. We
study the unique solvability of the problem using the Banach fixed-point
theory, properties of twofold saddle-point problems, and the
Banach--Ne\v{c}as--Babu\v{s}ka theory. We propose monolithic Galerkin
discretisations based on conforming Arnold--Winther for poroelastic stress and
displacement, and either PEERS or Arnold--Falk--Winther finite element families
for the stress-displacement-rotation field variables. The wellposedness of the
discrete problem is established as well, and we show a priori error estimates
in the natural norms. Some numerical examples are provided to confirm the rates
of convergence predicted by the theory, and we also illustrate the use of the
formulation in some typical tests in Biot poroelasticity.
