18 research outputs found
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
Mesh Generation and Flow Simulation in Large Tridimensional Fracture Networks
International audienceFractures in the Earth's subsurface have a strong impact in many physical and chemical phenomena, as their properties are very different from those of the surrounding rocks. They are generally organized as multi-scale structures, which can be modeled by Discrete Fracture Networks (DFNs) that may contain hundreds of thousands of ellipses in the tridi-mensional space. This paper presents our approach to generate meshes of such large DFNs and to simulate single-phase flow problems using these meshes
Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media
In this paper, we present a monolithic multigrid method for the efficient
solution of flow problems in fractured porous media. Specifically, we consider
a mixed-dimensional model which couples Darcy flow in the porous matrix with
Forchheimer flow within the fractures. A suitable finite volume discretization
permits to reduce the coupled problem to a system of nonlinear equations with a
saddle point structure. In order to solve this system, we propose a full
approximation scheme (FAS) multigrid solver that appropriately deals with the
mixed-dimensional nature of the problem by using mixed-dimensional smoothing
and inter-grid transfer operators. Remarkably, the nonlinearity is localized in
the fractures, and no coupling between the porous matrix and the fracture
unknowns is needed in the smoothing procedure. Numerical experiments show that
the proposed multigrid method is robust with respect to the fracture
permeability, the Forchheimer coefficient and the mesh size.Comment: arXiv admin note: text overlap with arXiv:1811.0126
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 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 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