On simulations of discrete fracture network flows with an optimization-based extended finite element method

Following the approach introduced in [Berrone,Pieraccini,Scialò,2013], we consider the formulation of the problem of fluid flow in a system of fractures as a PDE constrained optimization problem, with discretization performed using suitable extended finite elements; the method allows independent meshes on each fracture, thus completely circumventing meshing problems usually related to the DFN approach. The application of the method to discrete fracture networks of medium complexity is fully analyzed here, accounting for several issues related to viable and reliable implementations of the method in complex problems

Efficient partitioning of conforming virtual element discretizations for large scale discrete fracture network flow parallel solvers

Discrete Fracture Network models are largely used for large scale geological flow simulations in fractured media. For these complex simulations, it is worth investigating suitable numerical methods and tools for efficient parallel solutions on High Performance Computing systems. In this paper we propose and compare different partitioning strategies, that result to be highly efficient and scalable, overperforming the classical mesh partitioning approach used to balance the workload of a conforming mesh among several processes. The proposed DFN-based partitioning strategies rely on the distribution of the fractures among parallel processes. The computational cost of the DFN-based partitionings is very small compared to the cost of classical mesh partitioning and the numerical results prove their effectiveness and good performances in solving linear systems for realistic DFN flow simulations

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

In this paper we propose a modified construction for the polynomial basis on polygons used in the Virtual Element Method (VEM). This construction is alternative to the usual monomial basis used in the classical construction of the VEM and is designed in order to improve numerical stability. For badly shaped elements the construction of the projection matrices required for assembling the local coefficients of the linear system within the VEM discretization of Partial Differential Equations can result very ill conditioned. The proposed approach can be easily implemented within an existing VEM code in order to reduce the possible ill conditioning of the elemental projection matrices. Numerical results applied to an hydro-geological flow simulation that often produces very badly shaped elements show a clear improvement of the quality of the numerical solution, confirming the viability of the approach. The method can be conveniently combined with a classical implementation of the VEM and applied element-wise, thus requiring a rather moderate additional numerical cost

Towards effective flow simulations in realistic Discrete Fracture Networks

We focus on the simulation of underground flow in fractured media, modeled by means of Discrete Fracture Networks. Focusing on a new recent numerical approach proposed by the authors for tackling the problem avoiding mesh generation problems, we further improve the new family of methods making a step further towards effective simulations of large, multi-scale, heterogeneous networks. Namely, we tackle the imposition of Dirichlet boundary conditions in weak form, in such a way that geometrical complexity of the DFN is not an issue; we effectively solve DFN problems with fracture transmissivities spanning many orders of magnitude and approaching zero; furthermore, we address several numerical issues for improving the numerical solution also in quite challenging networks

Orthogonal polynomial bases in the mixed virtual element method

The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly-shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill-conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector-polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly-shaped elements, is proposed to validate our procedures

A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method

A new approach for solving flow in Discrete Fracture Networks (DFN) is developed in this work by means of the Virtual Element Method. Taking advantage of the features of the VEM, we obtain global conformity of all fracture meshes while preserving a fracture-independent meshing process. This new approach is based on a generalization of globally conforming Finite Elements for polygonal meshes that avoids complications arising from the meshing process. The approach is robust enough to treat many DFNs with a large number of fractures with arbitrary positions and orientations, as shown by the simulations. Higher order Virtual Element spaces are also included in the implementation with the corresponding convergence results and accuracy aspects

A three-field based optimization formulation for flow simulations in networks of fractures on non-conforming meshes

A new numerical scheme is proposed for flow computation in complex discrete fracture networks. The method is based on a three-field domain decomposition framework, in which independent variables are introduced at the interfaces generated in the process of decoupling the original problem on the whole network into a set of fracture-local problems. A PDE-constrained formulation is then used to enforce compatibility conditions at the interfaces. The combination of the three-field domain decomposition and of the optimization based coupling strategy results in a novel method which can handle non-conforming meshes, independently built on each geometrical object of the computational domain, and ensures local mass conservation property at fracture intersections, which is of paramount importance for hydro-geological applications. An iterative solver is devised for the method, suitable for parallel implementation on parallel computing architectures

A first-order stabilization-free Virtual Element Method

In this paper, we introduce a new Virtual Element Method (VEM) not requiring any stabilization term based on the usual enhanced first-order VEM space. The new method relies on a modified formulation of the discrete diffusion operator that ensures stability preserving all the properties of the differential operator.(c) 2023 Elsevier Ltd. All rights reserved