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Fractures and thin heterogeneities as Robin-Wentzell interface conditions

Marco Favino

TL;DR

This work develops a reduced modeling approach for fractures and thin heterogeneities in diffusion and Darcy flow by replacing thin inclusions with interface conditions. The key contributions are the derivation of Wentzell-type conditions for the flux jump and Robin-type conditions for the flux average, a general weak formulation suitable for finite element discretization, and extensive numerical validation across conductive and blocking fractures, including networks and varying apertures. The method enables accurate macroscopic behavior without explicit fracture meshes, and naturally extends to fracture intersections and complex networks. It also offers a flexible framework applicable to general diffusion problems with thin heterogeneities and to potential multi-physics extensions, with future work addressing regularity and advanced discretizations such as CutFEM/XFEM.

Abstract

We formally derive interface conditions for modeling fractures in Darcy flow problems and, more generally, thin inclusions in heterogeneous diffusion problems expressed as the divergence of a flux. Through a formal integration of the governing equations within the inclusions, we establish that the resulting interface conditions are of Wentzell type for the flux jump and Robin type for the flux average. Notably, the flux jump condition is unconventional, involving a tangential diffusion operator applied to the average of the solution across the interface. The corresponding weak formulation is introduced, offering a framework that is readily applicable to finite element discretizations. Extensive numerical validation highlights the robustness and versatility of the proposed modeling technique. The results demonstrate its effectiveness in accommodating a wide range of material properties, managing networks of inclusions, and naturally handling fractures with varying apertures -- all without requiring an explicit geometric representation of the fractures.

Fractures and thin heterogeneities as Robin-Wentzell interface conditions

TL;DR

This work develops a reduced modeling approach for fractures and thin heterogeneities in diffusion and Darcy flow by replacing thin inclusions with interface conditions. The key contributions are the derivation of Wentzell-type conditions for the flux jump and Robin-type conditions for the flux average, a general weak formulation suitable for finite element discretization, and extensive numerical validation across conductive and blocking fractures, including networks and varying apertures. The method enables accurate macroscopic behavior without explicit fracture meshes, and naturally extends to fracture intersections and complex networks. It also offers a flexible framework applicable to general diffusion problems with thin heterogeneities and to potential multi-physics extensions, with future work addressing regularity and advanced discretizations such as CutFEM/XFEM.

Abstract

We formally derive interface conditions for modeling fractures in Darcy flow problems and, more generally, thin inclusions in heterogeneous diffusion problems expressed as the divergence of a flux. Through a formal integration of the governing equations within the inclusions, we establish that the resulting interface conditions are of Wentzell type for the flux jump and Robin type for the flux average. Notably, the flux jump condition is unconventional, involving a tangential diffusion operator applied to the average of the solution across the interface. The corresponding weak formulation is introduced, offering a framework that is readily applicable to finite element discretizations. Extensive numerical validation highlights the robustness and versatility of the proposed modeling technique. The results demonstrate its effectiveness in accommodating a wide range of material properties, managing networks of inclusions, and naturally handling fractures with varying apertures -- all without requiring an explicit geometric representation of the fractures.

Paper Structure

This paper contains 26 sections, 60 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A One-Dimensional Example
  3. Geometry of the problem
  4. Heterogeneous Problems with Thin Inclusions
  5. Thin Inclusions as Interface Conditions
  6. Variational and Finite Element Formulation
  7. Mathematical Preliminaries
  8. Decompositions of the domain
  9. Tangential and Normal Differential Operators
  10. Model Problem and Wentzell Boundary Conditions
  11. A General Framework for Interface Conditions in Interface Problems
  12. Primal Weak Formulation of Interface Problems with Wentzell Interface Conditions
  13. Thin inclusions as interface conditions
  14. Statement
  15. Steps in the Derivation
  16. ...and 11 more sections

Figures (10)

  • Figure 1: Illustration of domain decompositions for the heterogeneous and interface problems.
  • Figure 2: Comparison between the solutions obtained with the heterogeneous and interface problems.
  • Figure 3: Illustration of the decompositions of $\Omega$ for the heterogeneous and interface problems.
  • Figure 4: Domain and fracture network for the test case Regular Fracture Network (2D). The left side, in pink, is where flow boundary conditions are imposed, while the right side, in blue, is where pressure boundary conditions are applied. Pressure profiles for the conductive scenario are evaluated along the segments $AA'$ and $BB'$, while a pressure profile for the blocking scenario is evaluated along the segment $CC'$.
  • Figure 5: Pressure distribution for the conductive scenario ($k_f=10^4$) of the test case Regular Fracture Network (2D).
  • ...and 5 more figures