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A five field formulation for flow simulations in porous media with fractures and barriers via an optimization based domain decomposition method

Stefano Scialò

TL;DR

This work addresses 3D-2D Darcy flow in a Discrete Fracture and Matrix (DFM) model, where fractures can act as barriers or conductive paths. It introduces a PDE-constrained optimization with a five-field domain decomposition, using interface variables $(\Psi^+,\Psi^-,\Psi^F)$ to decouple the 3D matrix problem from the 2D fracture problem and to accommodate pressure discontinuities. A novel extension to non permeable fractures is achieved via filtration-like coupling and XFEM enrichment, with a discrete problem proven well posed and solved efficiently in a matrix-free fashion. Four numerical validations demonstrate accuracy against analytic solutions and reference 2D/3D solutions, confirming the method’s viability on non-conforming meshes and in the presence of barriers. Overall, the framework enables accurate, scalable simulation of fractured porous media with complex fracture–matrix interactions.

Abstract

The present work deals with the numerical resolution of coupled 3D-2D problems arising from the simulation of fluid flow in fractured porous media modeled via the Discrete Fracture and Matrix (DFM) model. According to the DFM model, fractures are represented as planar interfaces immersed in a 3D porous matrix and can behave as preferential flow paths, in the case of conductive fractures, or can actually be a barrier for the flow, when, instead, the permeability in the normal-to-fracture direction is small compared to the permeability of the matrix. Consequently, the pressure solution in a DFM can be discontinuous across a barrier, as a result of the geometrical dimensional reduction operated on the fracture. The present work is aimed at developing a numerical scheme suitable for the simulation of the flow in a DFM with fractures and barriers, using a mesh for the 3D matrix non conforming to the fractures and that is ready for domain decomposition. This is achieved starting from a PDE-constrained optimization method, currently available in literature only for conductive fractures in a DFM. First, a novel formulation of the optimization problem is defined to account for non permeable fractures. These are described by a filtration-like coupling at the interface with the surrounding porous matrix. Also the extended finite element method with discontinuous enrichment functions is used to reproduce the pressure solution in the matrix around a barrier. The method is presented here in its simplest form, for clarity of exposition, i.e. considering the case of a single fracture in a 3D domain, also providing a proof of the well posedness of the resulting discrete problem. Four validation examples are proposed to show the viability and the effectiveness of the method.

A five field formulation for flow simulations in porous media with fractures and barriers via an optimization based domain decomposition method

TL;DR

This work addresses 3D-2D Darcy flow in a Discrete Fracture and Matrix (DFM) model, where fractures can act as barriers or conductive paths. It introduces a PDE-constrained optimization with a five-field domain decomposition, using interface variables to decouple the 3D matrix problem from the 2D fracture problem and to accommodate pressure discontinuities. A novel extension to non permeable fractures is achieved via filtration-like coupling and XFEM enrichment, with a discrete problem proven well posed and solved efficiently in a matrix-free fashion. Four numerical validations demonstrate accuracy against analytic solutions and reference 2D/3D solutions, confirming the method’s viability on non-conforming meshes and in the presence of barriers. Overall, the framework enables accurate, scalable simulation of fractured porous media with complex fracture–matrix interactions.

Abstract

The present work deals with the numerical resolution of coupled 3D-2D problems arising from the simulation of fluid flow in fractured porous media modeled via the Discrete Fracture and Matrix (DFM) model. According to the DFM model, fractures are represented as planar interfaces immersed in a 3D porous matrix and can behave as preferential flow paths, in the case of conductive fractures, or can actually be a barrier for the flow, when, instead, the permeability in the normal-to-fracture direction is small compared to the permeability of the matrix. Consequently, the pressure solution in a DFM can be discontinuous across a barrier, as a result of the geometrical dimensional reduction operated on the fracture. The present work is aimed at developing a numerical scheme suitable for the simulation of the flow in a DFM with fractures and barriers, using a mesh for the 3D matrix non conforming to the fractures and that is ready for domain decomposition. This is achieved starting from a PDE-constrained optimization method, currently available in literature only for conductive fractures in a DFM. First, a novel formulation of the optimization problem is defined to account for non permeable fractures. These are described by a filtration-like coupling at the interface with the surrounding porous matrix. Also the extended finite element method with discontinuous enrichment functions is used to reproduce the pressure solution in the matrix around a barrier. The method is presented here in its simplest form, for clarity of exposition, i.e. considering the case of a single fracture in a 3D domain, also providing a proof of the well posedness of the resulting discrete problem. Four validation examples are proposed to show the viability and the effectiveness of the method.
Paper Structure (8 sections, 1 theorem, 36 equations, 14 figures, 1 table)

This paper contains 8 sections, 1 theorem, 36 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem description
  3. Optimization based domain decomposition
  4. Discrete problem
  5. Well posedness of the discrete problem
  6. Discretization choices
  7. Numerical results
  8. Conclusions

Key Result

Lemma 1

Let matrix ${\bf{C}}\in\mathbb{R}^{\mathcal{N} \times \mathcal{N}^{\text{tot}}}$ be as in Cmatrix, and matrix ${\bf{G}} \in\mathbb{\mathcal{N}^{\text{tot}}\times\mathcal{N}^{\text{tot}}}$ be as in Gmatrix, then ${\bf{C}}$ is full row rank and $\ker{({\bf{G}} )}\cap\ker{({\bf{C}})}=\bf{0}$.

Figures (14)

  • Figure 1: Simple DFM example with nomenclature
  • Figure 2: Selection of elements to be enriched for an example embedded fracture: elements in $\mathcal{T}^\mathcal{D}_\mathcal{E}$ in red on the left and elements in $\mathcal{T}^\mathcal{D}_\mathcal{H}$ in blue on the right.
  • Figure 3: test0: solution on the 3D domain (left) and slice plot of the solution on a plane orthogonal to $F$ (right).
  • Figure 4: test1: solution on the 3D domain (left) and slice plot of the solution on a plane orthogonal to $F$ (right)
  • Figure 5: test1: Error convergence curves against mesh refinement
  • ...and 9 more figures

Theorems & Definitions (2)

  • Lemma 1
  • proof