A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches

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

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

Flow simulations in geology-based Discrete Fracture Networks

International audienceThe underground is a reservoir of natural resources (water, oil and gas, heat,...) and a potential warehouse storage solution. Using these resources and storage facilities in a sustainable way requires a good understanding of the physical, chemical and biological processes happening there. Also, the geometry of the subsurface couples these processes together. Here, numerical models are very useful: they reduce the costs and risks of in situ experiments and allow long-term predictions

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

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

Refinement strategies for polygonal meshes applied to adaptive VEM discretization

In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well and states new issues, here tackled, concerning good quality mesh elements and reliability of the simulations. In this paper we propose several new polygonal refinement strategies and numerically investigate the quality of the meshes generated by an adaptive mesh refinement process, as well as optimal rates of convergence with respect to the number of degrees of freedom. Among the several possible problems in which these strategies can be applied, here we have considered a geometrically complex geophysical problem as test problem that naturally yields to a polygonal mesh and tackled it by the Virtual Element Method. All the adaptive strategies here proposed, but the “Trace Direction strategy”, can be applied to any problem for which a polygonal element method can be useful and any numerical method based on polygonal elements and can generate good quality isotropic mesh elements