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A high-order discretization of nonlinear poroelasticity

Michele Botti, Daniele A. Di Pietro, Pierre Sochala

TL;DR

The paper addresses nonlinear poroelasticity in the quasi‑static Biot framework, coupling a nonlinear elasticity model with Darcy flow through $-oldsymbol{ abla}oldsymbol{oldsymbol{ abla}}oldsymbol{ au}+oldsymbol{ abla}p=oldsymbol{f}$ and $C_0 rac{dp}{dt}+oldsymbol{ abla}oldsymbol{ abla}oldsymbol{u}-oldsymbol{ abla}oldsymbol{ abla}oldsymbol{ abla}p=oldsymbol{g}$, with the Biot coefficient set to $oldsymbol{ abla}=1$ for simplicity. It proposes a nonconforming high‑order discretization that uses the Hybrid High‑Order method for nonlinear elasticity and the Symmetric Weighted Interior Penalty discontinuous Galerkin scheme for Darcy flow, applicable in both 2D and 3D on polyhedral meshes. The main contributions are an inf‑sup stable discretization on general meshes, compatibility with arbitrary approximation orders, and the ability to statically condense a large subset of unknowns in linearized steps, along with robustness to vanishing storage $C_0$ and variable permeability. Theoretical results include stability, well‑posedness, and optimal convergence rates, complemented by numerical tests on Cartesian and Voronoi meshes that corroborate the predicted accuracy and demonstrate robustness on nonmatching interfaces. These advances offer a practical and provably reliable tool for simulating nonlinear poroelastic effects in heterogeneous geophysical settings.

Abstract

In this work we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

A high-order discretization of nonlinear poroelasticity

TL;DR

The paper addresses nonlinear poroelasticity in the quasi‑static Biot framework, coupling a nonlinear elasticity model with Darcy flow through and , with the Biot coefficient set to for simplicity. It proposes a nonconforming high‑order discretization that uses the Hybrid High‑Order method for nonlinear elasticity and the Symmetric Weighted Interior Penalty discontinuous Galerkin scheme for Darcy flow, applicable in both 2D and 3D on polyhedral meshes. The main contributions are an inf‑sup stable discretization on general meshes, compatibility with arbitrary approximation orders, and the ability to statically condense a large subset of unknowns in linearized steps, along with robustness to vanishing storage and variable permeability. Theoretical results include stability, well‑posedness, and optimal convergence rates, complemented by numerical tests on Cartesian and Voronoi meshes that corroborate the predicted accuracy and demonstrate robustness on nonmatching interfaces. These advances offer a practical and provably reliable tool for simulating nonlinear poroelastic effects in heterogeneous geophysical settings.

Abstract

In this work we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.

Paper Structure

This paper contains 20 sections, 7 theorems, 124 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Continuous setting
  3. Notation for function spaces
  4. Stress-strain law
  5. Weak formulation
  6. Discrete setting
  7. Space mesh
  8. Time mesh
  9. $L^2$-orthogonal projectors on local and broken polynomial spaces
  10. Discrete spaces
  11. Displacement
  12. Pore pressure
  13. Discretization
  14. Nonlinear elasticity operator
  15. Hydro-mechanical coupling
  16. ...and 5 more sections

Key Result

Proposition 3

There is a real number $C_{\rm K}>0$, only depending on $\Omega$, $d$, and the mesh regularity parameter, such that, for all $\underline{\vec{v}}_{h}\in\underline{\vec{U}}^k_{h,\mathrm{D}}$,

Figures (1)

  • Figure 1: Cartesian and Voronoi meshes for the numerical tests.

Theorems & Definitions (23)

  • Remark 2: Regularity of the fluid content and of the pore pressure
  • Proposition 3: Discrete Korn's first inequality
  • Remark 4: Strain norm
  • proof
  • Proposition 5: Commuting property for the local symmetric gradient reconstruction
  • Remark 6: Approximation properties of the local symmetric gradient reconstruction
  • proof
  • Remark 7: Choice of the stabilization parameter
  • Lemma 8: Fortin operator
  • proof
  • ...and 13 more