A History-dependent Dynamic Biot Model

TL;DR

This work develops and tests a fully dynamic Biot-Allard poroelastic model that includes memory effects through convolution with a history-dependent permeability tensor $\\mathcal{A}$. It extends the fixed-stress splitting approach to the dynamic, memory-containing setting and uses backward Euler time stepping with Galerkin finite elements in space, including trapezoidal discretization of history terms. Two numerical experiments on the unit square show second-order convergence in pressure and near mesh-independent iteration counts, with convergence influenced by the stabilization parameter $L$. The results provide a robust, scalable solver framework for dynamic poroelasticity with memory, enabling investigation of regimes where permeability memory significantly alters system response.

Abstract

In this work, we consider a fully dynamic Biot model that includes memory effects due to evolving permeability. Time integrals are used to account for the change in structure. We propose an iterative splitting scheme for this model, extending the fixed-stress split for the quasi-static Biot. We use finite elements in space and a backward Euler discretization in time. The performance of the method is demonstrated through a numerical experiment

Figures (5)

• Figure 1: Numerical solutions at $T=1$; (a) Displacement $|u_h|$, (b) Pressure $p$.
• Figure 2: $L^{2}$ error of the pressure with a line (red) representing second-order convergence and the average number of iterations for different mesh sizes.
• Figure 3: Average number of iterations per time step for different stabilization parameters with fixed mesh size $h=1/16$.
• Figure 4: Numerical solutions at $T=0.1$; (a) Displacement $|u_h|$, (b) Pressure $p$.
• Figure 5: $L^{2}$ error of the pressure with a line (red) representing second-order convergence and the average number of iterations for different mesh sizes and time step sizes. Here $h_0=1/16$ and $\tau_{0}=0.1$.

Theorems & Definitions (1)

• remark 1