[Paper Review] A topology-motivated mixed finite element method for dynamic response of porous media

Zahrasadat Lotfian, Mettupalayam V. Sivaselvan|arXiv (Cornell University)|Jun 22, 2015

Numerical methods in engineering52 references3 citations

TL;DR

This paper proposes a topology-motivated mixed finite element method for modeling dynamic poroelasticity in saturated porous media, combining Raviart-Thomas elements for fluid flux and pressure (w-p field) with standard Galerkin finite elements for solid displacement (u field). The method ensures stability and accurate stress-pore pressure coupling via a three-field weak formulation, achieving optimal convergence for both displacement and fluid variables under Biot-Willis coefficient α=1.

ABSTRACT

In this paper, we propose a numerical method for computing solutions to Biot's fully dynamic model of incompressible saturated porous media [Biot;1956]. Our spatial discretization scheme is based on the three-field formulation (u-w-p) and the coupling of a lowest order Raviart-Thomas mixed element [Raviart,Thomas;1977] for fluid variable fields (w, p ) and a nodal Galerkin finite element for skeleton variable field (u). These mixed spaces are constructed based on the natural topology of the variables; hence, are physically compatible and able to exactly model the kind of continuity which is expected. The method automatically satisfies the well known LBB (inf-sup) stability condition and avoids locking that usually occurs in the numerical computations in the incompressible limit and very low hydraulic conductivity. In contrast to the majority of approaches, our three-field formulation can fully capture dynamic behavior of porous media even in high frequency loading phenomena with considerable fluid acceleration such as liquefaction and biomechanics of porous tissues under rapid external loading. Moreover, we address the importance of consistent initial conditions for poroelasticity equations with the incompressibility constraint, which represent a system of differential algebraic equations. The energy balance equation is derived for the full porous medium and used to assess the stability and accuracy of our time integration. To highlight the capabilities of our method, a variety of numerical studies are provided including verification with analytical and boundary element solutions, wave propagation analyses, hydraulic conductivity effects on damping and frequency content, energy balance analyses, mass lumping considerations, effects of mesh pattern and size, and stability analyses. We also explain some discrepancies commonly found in dynamic poroelasticity results in the literature.

Motivation & Objective

• To develop a stable, mixed finite element formulation for dynamic poroelasticity in porous media with coupled solid and fluid responses.
• To address the numerical instability issues common in mixed finite element methods for Biot's equations by leveraging topological constraints in element selection.
• To ensure accurate representation of effective stress and fluid flux through a three-field (u-w-p) weak formulation.
• To achieve optimal convergence rates for both displacement and fluid variables using mixed finite element spaces.
• To provide a robust numerical framework for simulating saturated porous media under dynamic loading, particularly relevant for geomechanics and soil dynamics.

Proposed method

• The method employs a three-field weak formulation based on displacement (u), fluid flux (w), and pore pressure (p), derived from Biot's equations of dynamic poroelasticity.
• Raviart-Thomas (RT) elements are used for the w-p field to ensure local mass conservation and correct flux approximation, especially important for Darcy's law.
• Standard nodal Galerkin finite elements are used for the displacement field u to ensure continuity and optimal convergence for the solid skeleton.
• The effective stress concept σᵗ = σ - αpI is applied, with α = 1 for soil, linking total stress to skeleton stress and pore pressure.
• Boundary conditions are split into essential (Dirichlet) and natural (Neumann) types for both solid (u, σᵗ·n) and fluid (w·n, p) fields, ensuring well-posedness.
• The resulting system is a differential-algebraic equation (DAE) that is spatially discretized and solved via time integration schemes.

Experimental results

Research questions

• RQ1 How can a stable mixed finite element method be constructed for dynamic poroelasticity with coupled solid and fluid responses?
• RQ2 What finite element spaces are optimal for approximating displacement, fluid flux, and pore pressure while preserving physical consistency?
• RQ3 How does the choice of Raviart-Thomas elements for w and p improve accuracy and stability compared to standard Galerkin elements?
• RQ4 What is the convergence behavior of the mixed formulation under standard benchmark conditions?
• RQ5 How does the topology of the finite element spaces influence the stability and accuracy of the solution?

Key findings

• The mixed finite element formulation achieves optimal convergence rates for both displacement and fluid variables, as expected from the mathematical theory of mixed methods.
• The use of Raviart-Thomas elements for the w-p field ensures local mass conservation and accurate flux approximation, critical for modeling fluid flow in porous media.
• The three-field formulation with u, w, and p as independent variables provides a stable and consistent framework for dynamic poroelasticity, avoiding locking and oscillations.
• The effective stress model σᵗ = σ - αpI is successfully implemented, with α = 1, ensuring correct coupling between mechanical and hydraulic responses.
• The boundary conditions are properly enforced via disjoint Dirichlet and Neumann subsets on the domain boundary, ensuring well-posedness of the weak formulation.
• The method is robust for dynamic problems, as it avoids the numerical instabilities common in standard Galerkin formulations of Biot’s equations.