Adaptive Mesh Refinement for a Finite Volume Method for Flow and Transport of Radionuclides in Heterogeneous Porous Media

Abstract

-Adaptive Mesh Refinement for a Finite Volume Method for Flow and Transport of Radionuclides in Heterogeneous Porous Media -In this paper, we consider adaptive numerical simulation of miscible displacement problems in porous media, which are modeled by single phase flow equations. A vertex-centred finite volume method is employed to discretize the coupled system: the Darcy flow equation and the diffusion-convection concentration equation. The convection term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion-dispersion term is discretized by piecewise linear conforming finite elements. We introduce two kinds of indicators, both of them of residual type. The first one is related to time discretization and is local with respect to the time discretization: thus, at each time, it provides an appropriate information for the choice of the next time step. The second is related to space discretization and is local with respect to both the time and space variable and the idea is that at each time it is an efficient tool for mesh adaptivity. An error estimation procedure

A space-time adaptive method for reservoir flows: formulation and one-dimensional application

“…For example, the magnitude of solution gradients can be used to identify important features [7,12,14]. Another approach, based on the magnitude of the residual, has been demonstrated for porous media flows by Klieber using the DG method in [28] and by Amaziane et al using the finite volume method in [1]. Our output-based adaptive method utilizes the dualweighted residual (DWR) approach proposed by Becker and Rannacher [10,11] to obtain both global and local error estimates, which are then used to drive the mesh adaptation.…”

“…This work focuses on h-adaptation, which involves changing the size and shape of elements in the mesh to control the total output error. A widely used strategy is to perform isotropic mesh refinement where selected elements are uniformly refined to decrease the error, as seen in [28,1,14] for flows through heterogeneous porous media. However, for problems involving highly anisotropic features, including the model reservoir flow applications in this work, anisotropic adaptation will be significantly more efficient.…”

Finite element methods for the Darcy-Forchheimer problem coupled with the convection-diffusion-reaction problem

“…In [13], the authors derived an optimal a posteriori error estimate for each of the numerical schemes proposed in [4]. We can also refer to [3] where the authors used a vertex-centred finite volume method to discretize the coupled system. Furthermore, for the time-dependent convection-diffusion-reaction equation coupled with Darcy's equation, we refer to [8,9] where the authors established the corresponding a priori and a posteriori errors.…”

“…Rate of convergence of the velocity in norms L 2 (Ω) 2 and L 3 (Ω)3 . Example (5.1) with algorithm (V ahi ).…”

Space-time mesh adaptation for solute transport in randomly heterogeneous porous media

“…A series of previous works provides examples of implementation of adaptive grids in the context of numerical modeling of flow (Knupp, 1996;Cao and Kitanidis, 1999;Cirpka et al, 1999;Mehl and Hill, 2002;Bresciani et al, 2012;Jayasinghe, 2015) and solute transport scenarios in homogenous (see, e.g., Pepper and Stephenson,1995;Kavetski et al, 2002;Saaltink et al, 2004;Younes and Ackerer, 2010) and heterogeneous (see, e.g., Huang and Zhan, 2005;Klieber and Rivière, 2006;Chueh et al, 2010;Gedeon and Mallants, 2012;Amaziane et al, 2014;Mansell et al, 2002 and references therein) porous media. Amaziane et al (2014) employ both space and time adaptive technique for simulating radionuclide transport in block-wise heterogeneous media. In their approach, these authors did not incorporate the anisotropic features of the solution to guide the spatial adaptation of the grid.…”