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Convergence analysis of a nonconforming virtual element method for compressible miscible displacement problems in porous media

Sarvesh Kumar, Devika Shylaja

TL;DR

The paper addresses the convergence analysis of a nonconforming virtual element method for a nonlinear, coupled compressible miscible displacement problem in porous media. It couples an $H(div)$-conforming VEM for velocity with a nonconforming VEM for concentration and uses discontinuous polynomial pressure discretization, all paired with a backward Euler time-stepping scheme. The authors prove an optimal a priori error bound of order $h^{k+1}$ in space and first order in time for the velocity, pressure, and concentration, under regularity assumptions and with $\tau = \mathcal{O}(h^{k+1})$, and validate the theory with extensive numerical experiments on polygonal meshes. This work demonstrates the effectiveness and reliability of VEM on general polygonal grids for simulating realistic miscible displacement processes in reservoir-scale porous media.

Abstract

This article presents a priori error estimates of the miscible displacement of one compressible fluid by another in a porous medium. The study utilizes the $H(\rm div)$ conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. Error estimates are established for velocity, pressure and concentration. The article also includes numerical results that validate the theoretical estimates.

Convergence analysis of a nonconforming virtual element method for compressible miscible displacement problems in porous media

TL;DR

The paper addresses the convergence analysis of a nonconforming virtual element method for a nonlinear, coupled compressible miscible displacement problem in porous media. It couples an -conforming VEM for velocity with a nonconforming VEM for concentration and uses discontinuous polynomial pressure discretization, all paired with a backward Euler time-stepping scheme. The authors prove an optimal a priori error bound of order in space and first order in time for the velocity, pressure, and concentration, under regularity assumptions and with , and validate the theory with extensive numerical experiments on polygonal meshes. This work demonstrates the effectiveness and reliability of VEM on general polygonal grids for simulating realistic miscible displacement processes in reservoir-scale porous media.

Abstract

This article presents a priori error estimates of the miscible displacement of one compressible fluid by another in a porous medium. The study utilizes the conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. Error estimates are established for velocity, pressure and concentration. The article also includes numerical results that validate the theoretical estimates.
Paper Structure (15 sections, 14 theorems, 151 equations, 9 figures, 5 tables)

This paper contains 15 sections, 14 theorems, 151 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Weak formulation
  3. Main result
  4. The virtual element method
  5. Discrete spaces
  6. Semi-discrete formulation
  7. Energy projection for velocity and pressure
  8. Energy projection for concentration
  9. Fully discrete formulation
  10. Error estimates
  11. Error estimates for velocity and pressure
  12. Error estimates for concentration
  13. Numerical Results
  14. Example 1
  15. Example 2

Key Result

Lemma 2.1

The following properties hold for the bilinear forms in defn.bilinearVeiga_miscibledisplacement_2021: where $\eta_\mathcal{D}$ is a positive constant depending only on $d_m$, $d_\ell$, and $d_t$.

Figures (9)

  • Figure 1: Square Mesh
  • Figure 2: Triangular Mesh
  • Figure 3: Concave Mesh
  • Figure 4: Structured Voronoi Mesh
  • Figure 5: Random Voronoi Mesh
  • ...and 4 more figures

Theorems & Definitions (18)

  • Lemma 2.1: Properties of the bilinear forms
  • Theorem 2.2
  • Lemma 3.1: Approximation properties
  • Lemma 3.2: Properties of the discrete bilinear forms
  • Lemma 3.3: Auxiliary result
  • Lemma 3.4
  • Corollary 3.5
  • Lemma 3.6
  • Lemma 3.7
  • proof
  • ...and 8 more