New twofold saddle-point formulations for Biot poroelasticity with porosity-dependent permeability

TL;DR

This work develops two Hu–Washizu–type mixed formulations for nonlinear Biot poroelasticity with porosity-dependent permeability, addressing both strong and weak stress symmetry. Existence and uniqueness of weak solutions are established via Banach fixed-point theory and twofold saddle-point analysis under small-data assumptions, with corresponding monolithic Galerkin discretizations using Arnold–Winther, PEERS, and Arnold–Falk–Winther elements. The authors derive Céa-type error estimates and optimal convergence rates, and several numerical tests confirm theoretical predictions and demonstrate practical applicability to swelling, tissue filtration, and Mandel’s effect. The approach provides a robust, high-fidelity framework for highly coupled poroelastic problems with nonlinear permeability and has potential impact in geomechanics and soft-tissue biomechanics where pore-fluid flow and deformation are tightly coupled.

Abstract

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--Nečas--Babuška 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.

Figures (5)

• Figure 6.1: Convergence tests. Approximate solutions computed with the second-order PEERS$_k$-based finite element scheme and rendered on the deformed configuration. Only the non-trivial component of the rotation tensor is shown in the bottom-right panel.
• Figure 6.2: Swelling of a poroelastic block. Approximate infinitesimal strain magnitude, fluid pressure, poroelastic stress magnitude, displacement magnitude and direction arrows, and permeability distribution.
• Figure 6.3: Poroelastic filtration of trabecular meshwork tissue. Approximate infinitesimal strain magnitude, fluid pressure, nonlinear permeability (depending on an initial randomly distributed porosity), two off-diagonal entries of the poroelastic stress, and displacement magnitude with direction arrows.
• Figure 6.4: Mandel's test. Plot over the domain horizontal mid-line of axial and radial strains (first and last components of $\boldsymbol{d}$) normalised through $d^*=0.07$ [--], pore pressure profile normalised by $p^*=60$ [Pa], axial poroelastic stress normalised by $\sigma^* = 4$ [Pa], and patterns of horizontal an vertical velocities (in [m]), for the constant and nonlinear permeability cases. The legend in the top-left panel (indicating the different times) applies to all panels in the figure.
• Figure 6.5: Mandel's test. Variation of pore pressure normalised with $p^*=60$ [Pa], poroelastic stress (axial component normalised with $\sigma^*=-100$ [Pa] and radial component with $\sigma^*=8$ [Pa]) and displacement (vertical component normalised with $u^* = -0.04$ [m] and horizontal component with $u^* = 0.04$ [m]) against time for the constant and nonlinear permeability cases, and recorded at the points $(0,H/2)$ (left) and $(L/2,H)$ (middle). The right panel shows the patterns of pore pressure at the final time on the deformed configuration.

Theorems & Definitions (25)

• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 4.1