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Numerical simulation of wormhole propagation with the mixed hybridized discontinuous Galerkin finite element method

Jiansong Zhang, Jiang Zhu, Yiming Wang, Yanyu Liu, Hui Guo

TL;DR

The paper addresses the numerical simulation of wormhole propagation during acidization in carbonate reservoirs. It introduces a combined hybridized mixed DG framework that jointly treats porosity evolution, Darcy flow, and acid transport, featuring a maximum-principle preserving porosity discretization, an upwind HDG scheme for convection-dominated transport, and hybridization to reduce degrees of freedom. The authors establish existence and uniqueness of the coupled scheme and derive optimal error estimates, supported by a convergence theorem that demonstrates robust performance under practical regularity assumptions. Numerical experiments on elliptic, convection-diffusion, and coupled problems confirm the expected convergence rates and showcase the method's ability to capture wormhole development with significant computational efficiency. Overall, the approach provides a scalable, accurate tool for simulating wormhole growth in fractured media, with potential practical impact on reservoir stimulation design.

Abstract

The acid treatment of carbonate reservoirs is a widely employed technique for enhancing the productivity of oil and gas reservoirs. In this paper, we present a novel combined hybridized mixed discontinuous Galerkin (HMDG) finite element method to simulate the dissolution process near the wellbore, commonly referred to as the wormhole phenomenon. The primary contribution of this work lies in the application of hybridization techniques to both the pressure and concentration equations. Additionally, an upwind scheme is utilized to address convection-dominant scenarios, and a ``cut-off" operator is introduced to maintain the boundedness of porosity. Compared to traditional discontinuous Galerkin methods, the proposed approach results in a global system with fewer unknowns and sparser stencils, thereby significantly reducing computational costs. We analyze the existence and uniqueness of the new combined method and derive optimal error estimates using the developed technique. Numerical examples are provided to validate the theoretical analysis.

Numerical simulation of wormhole propagation with the mixed hybridized discontinuous Galerkin finite element method

TL;DR

The paper addresses the numerical simulation of wormhole propagation during acidization in carbonate reservoirs. It introduces a combined hybridized mixed DG framework that jointly treats porosity evolution, Darcy flow, and acid transport, featuring a maximum-principle preserving porosity discretization, an upwind HDG scheme for convection-dominated transport, and hybridization to reduce degrees of freedom. The authors establish existence and uniqueness of the coupled scheme and derive optimal error estimates, supported by a convergence theorem that demonstrates robust performance under practical regularity assumptions. Numerical experiments on elliptic, convection-diffusion, and coupled problems confirm the expected convergence rates and showcase the method's ability to capture wormhole development with significant computational efficiency. Overall, the approach provides a scalable, accurate tool for simulating wormhole growth in fractured media, with potential practical impact on reservoir stimulation design.

Abstract

The acid treatment of carbonate reservoirs is a widely employed technique for enhancing the productivity of oil and gas reservoirs. In this paper, we present a novel combined hybridized mixed discontinuous Galerkin (HMDG) finite element method to simulate the dissolution process near the wellbore, commonly referred to as the wormhole phenomenon. The primary contribution of this work lies in the application of hybridization techniques to both the pressure and concentration equations. Additionally, an upwind scheme is utilized to address convection-dominant scenarios, and a ``cut-off" operator is introduced to maintain the boundedness of porosity. Compared to traditional discontinuous Galerkin methods, the proposed approach results in a global system with fewer unknowns and sparser stencils, thereby significantly reducing computational costs. We analyze the existence and uniqueness of the new combined method and derive optimal error estimates using the developed technique. Numerical examples are provided to validate the theoretical analysis.

Paper Structure

This paper contains 18 sections, 16 theorems, 104 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The formulation of combined method
  3. The discretization of the porosity eqaution
  4. Hybrid mixed finite element method for the pressure and Darcy velocity equations
  5. Mixed hybridized discontinuous Galerkin method for the concentration equation
  6. The combined mixed hybridized discontinuous Galerkin finite element method
  7. Existence and uniqueness
  8. Convergence analysis
  9. The error estimate of the porosity
  10. The error estimates of pressure and velocity
  11. The error estimate of the concentration
  12. Convergence theorem
  13. Numerical examples
  14. Convergence test
  15. Elliptic type pressure equation
  16. ...and 3 more sections

Key Result

Lemma 1

Under the assumpution ass1, we know that $\kappa^{-1}(\phi )$ is bounded and Lipschitz continuous, that is, there exists some constant $C$ such that

Figures (2)

  • Figure 1: Porosity of rock at time $t=10s$ (a), $t=20s$ (b), $t=30s$ (c), $t=40s$ (d) with $d=1e-5$.
  • Figure 2: Concentration of acid at time $t=10s$ (a), $t=20s$ (b), $t=30s$ (c), $t=40s$ (d) with $d=1e-5$.

Theorems & Definitions (23)

  • Lemma 1
  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3: Stability and boundedness of $B_{\mathbf{u}}$
  • Lemma 4: Stability and boundedness of $B_{c}$
  • proof
  • Theorem 2: Existence and Uniqueness
  • proof
  • Lemma 5
  • ...and 13 more