27 research outputs found
Improving the robustness of the control volume finite element method with application to multiphase porous media flow
Control volume finite element methods (CVFEMs) have been proposed to simulate flow in heterogeneous porous media because they are better able to capture complex geometries using unstructured meshes. However, producing good quality meshes in such models is nontrivial and may sometimes be impossible, especially when all or parts of the domains have very large aspect ratio. A novel CVFEM is proposed here that uses a control volume representation for pressure and yields significant improvements in the quality of the pressure matrix. The method is initially evaluated and then applied to a series of test cases using unstructured (triangular/tetrahedral) meshes, and numerical results are in good agreement with semianalytically obtained solutions. The convergence of the pressure matrix is then studied using complex, heterogeneous example problems. The results demonstrate that the new formulation yields a pressure matrix than can be solved efficiently even on highly distorted, tetrahedral meshes in models of heterogeneous porous media with large permeability contrasts. The new approach allows effective application of CVFEM in such models
A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension
Finite volume methods for problems involving second order operators with full
diffusion matrix can be used thanks to the definition of a discrete gradient
for piecewise constant functions on unstructured meshes satisfying an
orthogonality condition. This discrete gradient is shown to satisfy a strong
convergence property on the interpolation of regular functions, and a weak one
on functions bounded for a discrete norm. To highlight the importance of
both properties, the convergence of the finite volume scheme on a homogeneous
Dirichlet problem with full diffusion matrix is proven, and an error estimate
is provided. Numerical tests show the actual accuracy of the method
The gradient discretisation method for linear advection problems
We adapt the Gradient Discretisation Method (GDM), originally designed for
elliptic and parabolic partial differential equations, to the case of a linear
scalar hyperbolic equations. This enables the simultaneous design and
convergence analysis of various numerical schemes, corresponding to the methods
known to be GDMs, such as finite elements (conforming or non-conforming,
standard or mass-lumped), finite volumes on rectangular or simplicial grids,
and other recent methods developed for general polytopal meshes. The scheme is
of centred type, with added linear or non-linear numerical diffusion. We
complement the convergence analysis with numerical tests based on the
mass-lumped P1 conforming and non conforming finite element and on the hybrid
finite volume method
A Discontinuous Control Volume Finite Element Method for Multi-Phase Flow in Heterogeneous Porous Media
We present a new, high-order, control-volume-finite-element (CVFE) method for multiphase porous media flow with discontinuous 1st-order representation for pressure and discontinuous 2nd-order representation for velocity. The method has been implemented using unstructured tetrahedral meshes to discretize space. The method locally and globally conserves mass. However, unlike conventional CVFE formulations, the method presented here does not require the use of control volumes (CVs) that span the boundaries between domains with differing material properties. We demonstrate that the approach accurately preserves discontinuous saturation changes caused by permeability variations across such boundaries, allowing efficient simulation of flow in highly heterogeneous models. Moreover, accurate solutions are obtained at significantly lower computational cost than using conventional CVFE methods. We resolve a long-standing problem associated with the use of classical CVFE methods to model flow in highly heterogeneous porous media
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A multilevel multiscale mimetic method for an anisotropic infiltration problem
Modeling of multiphase flow and transport in highly heterogeneous porous media must capture a broad range of coupled spatial and temporal scales. Recently, a hierarchical approach dubbed the Multilevel Multiscale Mimetic (M3) method, was developed to simulate two-phase flow in porous media. The M{sup 3} method is locally mass conserving at all levels in its hierarchy, it supports unstructured polygonal grids and full tensor permeabilities, and it can achieve large coarsening factors. In this work we consider infiltration of water into a two-dimensional layered medium. The grid is aligned with the layers but not the coordinate axes. We demonstrate that with an efficient temporal updating strategy for the coarsening parameters, fine-scale accuracy of prominent features in the flow is maintained by the M{sup 3} method
Convergence analysis of a vertex-centered finite volume scheme for a copper heap leaching model
In this paper a two-dimensional solute transport model is considered to simulate the leaching of copper ore tailing
using sulfuric acid as the leaching agent. The mathematical model consists in a system of differential equations: two
diffusion–convection-reaction equations with Neumann boundary conditions, and one ordinary differential equation.
The numerical scheme consists in a combination of finite volume and finite element methods. A Godunov scheme is
used for the convection term and an P1-FEM for the diffusion term. The convergence analysis is based on standard
compactness results in L2. Some numerical examples illustrate the effectiveness of the scheme
The gradient discretisation method for linear advection problems
International audienceWe adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped P1 conforming and non conforming finite element and on the hybrid finite volume method