Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fixed stress splitting approach for contact problems in a porous medium

Tameem Almani, Kundan Kumar

TL;DR

This work addresses a quasi-static poroelastic contact problem in a porous medium governed by the Biot equations with frictionless Signorini boundary conditions. It introduces a fully discrete fixed-stress splitting scheme that decouples flow and mechanics and uses backward Euler time stepping with Raviart-Thomas finite elements for the flow and a conforming discretization for the mechanics. A contraction analysis proves the scheme is contractive, yielding a bound that depends on beta = 1/(M alpha^2) + (c_f varphi_0)/alpha^2 + 1/lambda and a regularization term alpha^2/lambda; the contraction constant is (1/(lambda beta))^2. The results provide a stable, convergent method for simulating flow–geomechanics interactions in fractured porous media, with planned extensions to frictional contact and numerical validation.

Abstract

We consider a poromechanics model including frictionless contact mechanics. The resulting model consists of the Biot equations with contact boundary conditions leading to a variational inequality modelling mechanical deformations coupled to a linear parabolic flow equation. We propose a fully discrete iterative scheme for solving this model. This scheme decoupled the flow and mechanics equations and extends the fixed-stress splitting scheme for the Biot equations. We use finite elements in space and a backward Euler discretization in time. We show that the fixed stress split scheme is a contraction.

Fixed stress splitting approach for contact problems in a porous medium

TL;DR

This work addresses a quasi-static poroelastic contact problem in a porous medium governed by the Biot equations with frictionless Signorini boundary conditions. It introduces a fully discrete fixed-stress splitting scheme that decouples flow and mechanics and uses backward Euler time stepping with Raviart-Thomas finite elements for the flow and a conforming discretization for the mechanics. A contraction analysis proves the scheme is contractive, yielding a bound that depends on beta = 1/(M alpha^2) + (c_f varphi_0)/alpha^2 + 1/lambda and a regularization term alpha^2/lambda; the contraction constant is (1/(lambda beta))^2. The results provide a stable, convergent method for simulating flow–geomechanics interactions in fractured porous media, with planned extensions to frictional contact and numerical validation.

Abstract

We consider a poromechanics model including frictionless contact mechanics. The resulting model consists of the Biot equations with contact boundary conditions leading to a variational inequality modelling mechanical deformations coupled to a linear parabolic flow equation. We propose a fully discrete iterative scheme for solving this model. This scheme decoupled the flow and mechanics equations and extends the fixed-stress splitting scheme for the Biot equations. We use finite elements in space and a backward Euler discretization in time. We show that the fixed stress split scheme is a contraction.
Paper Structure (6 sections, 1 theorem, 18 equations)

This paper contains 6 sections, 1 theorem, 18 equations.

Table of Contents

  1. Introduction
  2. Model equations
  3. Mixed variational formulation
  4. Fixed stress split for frictionless contact problem
  5. Proof of Contraction
  6. Conclusion and future work

Key Result

theorem 1

Let $\beta = \frac{1}{M \alpha^2 } + \frac{c_f}{\alpha^2} \varphi_0 + \frac{1}{\lambda}$, the iterative scheme defined by eq:nf_eq4_fs_sr - eq:fixedstress2 is a contraction given by

Theorems & Definitions (4)

  • theorem 1
  • remark 1
  • remark 2
  • remark 3