Theoretical analysis and numerical simulation in discrete fracture networks using both the virtual element method and the mortar method

The Discrete Fracture Network (DFN) method is a discrete modeling approach designed to simulate fluid flow and transport phenomena within fractured rock formations. In this method, fractures are represented as 2D planar domains, with fluid flow restricted to the fractures themselves, as the surrounding rock matrix is considered impermeable. A central challenge in these simulations is the geometric handling of the domain, particularly when global or local mesh conformity is necessary. Fluid flow is modeled using Darcy’s law, which governs the hydraulic head equilibrium within each fracture. This equation is coupled through flux balance and continuity conditions at fracture intersections, called in what follows traces. To address the problem, two techniques can be found in literature: the Virtual Element Method (VEM), a highly adaptable mesh-based approach that supports the use of general polygonal and polyhedral meshes, and the Mortar Method, a domain decomposition technique that accommodates non-matching meshes at fracture interfaces. VEM’s key advantages are its strong mathematical foundation, computational efficiency, and accuracy. The Mortar Method is especially useful for multi-physics problems requiring interface flexibility. These methods offer two distinct computational approaches: a fully conforming VEM and a mixed VEM-Mortar method, which differ in their handling of conditions on the traces. The thesis has two primary objectives. First, it undertakes a theoretical analysis of the problem, detailing both methods and their coupling. The innovative contribution here lies in the combined use of these methods to achieve well-posedness of the discrete problem and an a-priori error estimate for the discretization. The thesis also provides a detailed implementation guide, with a particular focus on the matrices involved in the saddle-point problem that emerges. The second objective is the application of the mixed VEM-Mortar method to a benchmark problem. Starting from a C++ code developed for a fully VEM approach, I incorporate in the code the mixed VEM-Mortar method implementation. This work lays the foundation for future research aimed at studying how the error behavior and matrix condition number in the saddle-point problem vary with the number of VEM or Mortar traces used.