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Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

Imam Wijaya, Hirofumi Notsu

TL;DR

This work analyzes a Navier–Stokes type model for non-stationary flow in non-homogeneous porous media derived via volume averaging, incorporating a nonlinear Forchheimer drag term $B(u,\phi)$ to capture inertial effects. It develops a second-order in time Lagrange–Galerkin scheme using Adams–Bashforth time stepping to solve the model, and proves $L^{2}$-stability with energy estimates that rely on coercivity induced by the drag term. Numerical validation includes a manufactured-solution test showing near second-order convergence in time and space, and two non-homogeneous porosity scenarios illustrating realistic flow patterns: higher porosity regions exhibit faster velocities and irregular porosity yields qualitative behavior consistent with physical intuition. The results provide a stable, higher-order computational framework for simulating non-stationary flow in porous media with spatially varying porosity, with potential impact on geothermal reservoir modeling and related applications.

Abstract

The purposes of this work are to study the $L^{2}$-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically.The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.

Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

TL;DR

This work analyzes a Navier–Stokes type model for non-stationary flow in non-homogeneous porous media derived via volume averaging, incorporating a nonlinear Forchheimer drag term to capture inertial effects. It develops a second-order in time Lagrange–Galerkin scheme using Adams–Bashforth time stepping to solve the model, and proves -stability with energy estimates that rely on coercivity induced by the drag term. Numerical validation includes a manufactured-solution test showing near second-order convergence in time and space, and two non-homogeneous porosity scenarios illustrating realistic flow patterns: higher porosity regions exhibit faster velocities and irregular porosity yields qualitative behavior consistent with physical intuition. The results provide a stable, higher-order computational framework for simulating non-stationary flow in porous media with spatially varying porosity, with potential impact on geothermal reservoir modeling and related applications.

Abstract

The purposes of this work are to study the -stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically.The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.

Paper Structure

This paper contains 9 sections, 4 theorems, 48 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Statement of the problem
  4. Stability estimates
  5. A Lagrange--Galerkin scheme
  6. Numerical results
  7. Order of Convergence
  8. Simulation with non-homogeneous porosity
  9. Conclusion

Key Result

Theorem 4.1

Suppose that Hypotheses hyp1 and hyp2 hold true. Assume $g=0$. Suppose that $(u,p) \in (C^1([0,T];L^2(\Omega)^d) \cap L^2(0,T;V))\times L^2(0,T;L^2(\Omega))$ satisfies prob_weak. Then, it holds that where $\beta_0 > 0$ is a positive constant to be defined in ieq:Korn below.

Figures (6)

  • Figure 1: Representative elementary volume (REV)
  • Figure 2: The order of convergence for scheme \ref{['scheme']}.
  • Figure 3: The boundary conditions and the finite element mesh.
  • Figure 4: Time evolution of velocity magnitude.
  • Figure 5: Computation domain and porosity value distribution
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • Corollary 4.2: Stability estimates
  • Lemma 4.3: Korn's inequality, 2021
  • Lemma 4.4
  • proof
  • proof : Proof of Theorem \ref{['thm']}
  • proof : Proof of Corollary \ref{['cor']}