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

An optimization approach for large scale simulations of discrete fracture network flows

In recent papers the authors introduced a new method for simulating subsurface flow in a system of fractures based on a PDE-constrained optimization reformulation, removing all difficulties related to mesh generation and providing an easily parallel approach to the problem. In this paper we further improve the method removing the constraint of having on each fracture a non empty portion of the boundary with Dirichlet boundary conditions. This way, Dirichlet boundary conditions are prescribed only on a possibly small portion of DFN boundary. The proposed generalization of the method in relies on a modified definition of control variables ensuring the non-singularity of the operator on each fracture. A conjugate gradient method is also introduced in order to speed up the minimization proces

Uncertainty quantification in Discrete Fracture Network models: stochastic geometry

We consider the problem of uncertainty quantification analysis of the output of underground flow simulations. We consider in particular fractured media described via the discrete fracture network model; within this framework, we address the relevant case of networks in which the geometry of the fractures is described by stochastic parameters. In this context, due to a possible lack of smoothness in the quantity of interest with respect to the stochastic parameters, well assessed techniques such as stochastic collocation may fail in providing reliable estimates of first-order moments of the quantity of interest. In this paper, we overcome this issue by applying the Multilevel Monte Carlo method, using as underlying solver an extremely robust method

A hybrid mortar virtual element method for discrete fracture network simulations

The most challenging issue in performing underground flow simulations in Discrete Fracture Networks (DFN), is to effectively tackle the geometrical difficulties of the problem. In this work we put forward a new application of the Virtual Element Method combined with the Mortar method for domain decomposition: we exploit the flexibility of the VEM in handling polygonal meshes in order to easily construct meshes conforming to the traces on each fracture, and we resort to the mortar approach in order to ``weakly'' impose continuity of the solution on intersecting fractures. The resulting method replaces the need for matching grids between fractures, so that the meshing process can be performed independently for each fracture. Numerical results show optimal convergence and robustness in handling very complex geometries

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

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