A PDE-constrained optimization formulation for discrete fracture network flows

We investigate a new numerical approach for the computation of the 3D flow in a discrete fracture network that does not require a conforming discretization of partial differential equations on complex 3D systems of planar fractures. The discretization within each fracture is performed independently of the discretization of the other fractures and of their intersections. Independent meshing process within each fracture is a very important issue for practical large scale simulations making easier mesh generation. Some numerical simulations are given to show the viability of the method. The resulting approach can be naturally parallelized for dealing with systems with a huge number of fractures

Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations

Simulations of fluid flow in naturally fractured rocks have implications for several subsurface applications, including energy storage and extraction, and waste storage. We are interested in flow in discrete fracture networks, which explicitly represent flow in fracture surfaces, but ignore the impact of the surrounding host rock. Fracture networks, generated from observations or stochastic simulations, will contain intersections of arbitrary length, and intersection lines can further cross, forming a highly complex geometry. As the flow exchange between fractures, thus in the network, takes place in these intersections, an adequate representation of the geometry is critical for simulation accuracy. In practice, the intersection dynamics must be handled by a combination of the simulation grid, which may or may not resolve the intersection lines, and the numerical methods applied on the grid. In this work, we review different classes of numerical approaches proposed in recent years, covering both methods that conform to the grid, and non-matching cases. Specific methods considered herein include finite element, mixed and virtual finite elements and control volume methods. We expose our methods to an extensive set of test cases, ranging from artificial geometries designed to test difficult configurations, to a network extruded from a real fracture outcrop. The main outcome is guidances for choice of simulation models and numerical discretization with a trade off on the computational cost and solution accuracy

Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models

[EN] The simulation of underground flow across intricate fracture networks can be addressed by means of discrete fracture network models. The combination of such models with an optimization formulation allows for the use of nonconforming and independent meshes for each fracture. The arising algebraic problem produces a symmetric saddle-point matrix with a rank-deficient leading block. In our work, we investigate the properties of the system to design a block preconditioning strategy to accelerate the iterative solution of the linearized algebraic problem. The matrix is first permuted and then projected in the symmetric positive-definite Schur-complement space. The proposed strategy is tested in applications of increasing size, in order to investigate its capabilities.Gazzola, L.; Ferronato, M.; Berrone, S.; Pieraccini, S.; Scialò, S. (2022). Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 346-354. https://doi.org/10.4995/YIC2021.2021.12234OCS34635

The Mixed Virtual Element Method on curved edges in two dimensions

In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical feature, such as a portion of domain boundary or an internal interface, may introduce a geometrical error that degrades the expected order of convergence of the scheme. In the present work a suitable VEM approximation space is proposed to consistently handle curvilinear geometrical objects, thus recovering optimal convergence rates. The resulting numerical scheme is presented along with its theoretical analysis and several numerical test cases to validate the proposed approach

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

The virtual element method for discrete fracture network simulations

In this work, an optimization based approach presented in previous work of the authors for Discrete Fracture Network simulations is coupled with the Virtual Element Method (VEM) for the space discretization of the underlying Darcy law. The great flexibility of the VEM in allowing rather general polygonal elements, allow to easily describe irregular solutions starting from a general triangulation which can be built independently of the mesh on other fractures. Only a partial conformity is in fact obtained with this approach. Numerical results performed on several DFN configurations confirm the viability and efficiency of the resulting method

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