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A domain decomposition approach to pore-network modeling of porous media flow

Shuyu Sun, Zhangchengrui Wang, Lei Zhang, Jijing Zhao

TL;DR

The paper tackles accurate pore-scale flow predictions in porous media while addressing the computational cost of fully resolved finite-element models. It introduces the DD-PNM, which solves local Stokes problems inside each pore on body-fitted meshes and couples them through a sparse global interface system defined by Dirichlet-to-Neumann maps, yielding a Schur complement $S$ that is symmetric positive definite. A key contribution is a constructive bridge to classical pore-network models: half-throat conductivities are calibrated from the local DtN maps to recover C-PNM behavior within a unified algebraic framework. Rigorous mathematical results establish solvability and discrete mass conservation, and numerical experiments in 2-D demonstrate high accuracy and robustness, including realistic geometries; the method enables offline-online workflows and parallel computation for large-scale studies.

Abstract

We propose a domain-decomposition pore-network method (DD-PNM) for modeling single-phase Stokes flow in porous media. The method combines the accuracy of finite-element discretizations on body-fitted meshes within pore subdomains with a sparse global coupling enforced through interface unknowns. Local Dirichlet-to-Neumann operators are precomputed from finite-element solutions for each pore subdomain, enabling a global Schur-complement system defined solely on internal interfaces. Rigorous mathematical analysis establishes solvability and discrete mass conservation of the global system. Moreover, we constructively recover classical pore-network models by fitting half-throat conductivities to local Dirichlet-to-Neumann maps, providing a principled bridge between mesh-based and network-based frameworks. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.

A domain decomposition approach to pore-network modeling of porous media flow

TL;DR

The paper tackles accurate pore-scale flow predictions in porous media while addressing the computational cost of fully resolved finite-element models. It introduces the DD-PNM, which solves local Stokes problems inside each pore on body-fitted meshes and couples them through a sparse global interface system defined by Dirichlet-to-Neumann maps, yielding a Schur complement that is symmetric positive definite. A key contribution is a constructive bridge to classical pore-network models: half-throat conductivities are calibrated from the local DtN maps to recover C-PNM behavior within a unified algebraic framework. Rigorous mathematical results establish solvability and discrete mass conservation, and numerical experiments in 2-D demonstrate high accuracy and robustness, including realistic geometries; the method enables offline-online workflows and parallel computation for large-scale studies.

Abstract

We propose a domain-decomposition pore-network method (DD-PNM) for modeling single-phase Stokes flow in porous media. The method combines the accuracy of finite-element discretizations on body-fitted meshes within pore subdomains with a sparse global coupling enforced through interface unknowns. Local Dirichlet-to-Neumann operators are precomputed from finite-element solutions for each pore subdomain, enabling a global Schur-complement system defined solely on internal interfaces. Rigorous mathematical analysis establishes solvability and discrete mass conservation of the global system. Moreover, we constructively recover classical pore-network models by fitting half-throat conductivities to local Dirichlet-to-Neumann maps, providing a principled bridge between mesh-based and network-based frameworks. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.
Paper Structure (16 sections, 7 theorems, 68 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 16 sections, 7 theorems, 68 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Model problem
  4. Classical pore–network modeling (C-PNM)
  5. Reference finite-element formulation
  6. Domain-decomposition PNM (DD-PNM)
  7. Local Stokes solve and Dirichlet-to-Neumann map
  8. Global coupling and Schur complement
  9. Solution algorithm
  10. Computational cost
  11. Mathematical properties of DD-PNM
  12. Numerical results
  13. model problem
  14. Realistic geometry
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 3.1

For every interface-traction vector $\mathbf p=(p_1,\dots,p_m)^{\top}$, the discrete Stokes subproblem in each $\Omega_i$ admits a unique solution.

Figures (12)

  • Figure 1: (a) computational geometry; (b) maximal-ball graph for C-PNM; (c) subdomains $\{\Omega_i\}$ used by DD-PNM; (d) body-fitted FE mesh.
  • Figure 2: C-PNM versus the calibration of C-PNM from DD-PNM of a throat.
  • Figure 3: Model problem; FEM solution.
  • Figure 4: Model problem; DD-PNM versus FE: point-wise error fields.
  • Figure 5: Model problem; DD-PNM with perturbed interfaces: meshes (top) and pressure errors (bottom).
  • ...and 7 more figures

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1: Local well-posedness
  • proof
  • Lemma 3.2: Discrete incompressibility of the unit responses
  • proof
  • Remark 3.3
  • ...and 11 more