A Finite Volume Eulerian‐Lagrangian Localized Adjoint Method for Solution of the Contaminant Transport Equations in Two‐Dimensional Multiphase flow Systems

Abstract

An Eulerian‐Lagrangian localized adjoint method (ELLAM) numerical solution is developed for the multiphase contaminant transport equations in two dimensions. The ELLAM uses finite volume test functions in the space‐time domain defined by the characteristics of the hyperbolic part of the governing equation. The use of the characteristics results in an approximation that allows large time steps while still maintaining accurate solutions. This greatly reduces the computational effort required to find solutions to the advection‐dispersion equation. Combination of the finite volume test functions with the conservative form of the governing equation results in a local conservation of mass property. For problems with a constant saturation such as that of contaminant transport in the saturated zone this property is highly advantageous as the resulting numerical approximation conserves mass both globally and locally. For problems with a variable saturation the requirement of local conservation of mass is too stringent, particularly around first‐type boundaries where oscillations can occur because of errors inherent in the numerical determination of fluid velocities and in the backtracking routine. A combined conservative/nonconservative ELLAM is developed with an ELLAM formulation based on the nonconservative form of the governing equation being applied to subdomains intersecting first‐type boundaries and a conservative ELLAM being used for all other subdomains. The combined conservative/nonconservative ELLAM is compared to a Galerkin finite element scheme and is found to have greatly superior performance, requiring far fewer time steps to obtain a solution of equivalent accuracy.

Application of Computational Fluid Dynamics (CFD) for Simulation of Acid Mine Drainage Generation and Subsequent Pollutants Transportation through Groundwater Flow Systems and Rivers

“…This increased flexibility can be used to great advantage in generating grids about arbitrary geometries (Harvard et al, 1999). Although the finite element method is mainly used for the simulation of the contaminant transport problems through groundwater flow systems (see Barovic & Boochs, 1981;Bignoli and Sabbioni, 1981;Fried, 1981;Kerdijk, 1981;Nawalany, 1981;Putti et al, 1990), finite volume techniques have also been used by many researchers for the solution of groundwater flow and solute transport governing equations in saturated and unsaturated flow systems (see Binning and Celia, 1996;Green and Clothier, 1994;Putti et al, 1990;Svensson, 1997). The basic principles and concepts of the finite volume formulation are very simple and easier to understand by engineers than other numerical techniques.…”

“…Balusu (1993) developed a numerical model using a CFD code, FIDAP, to simulate airflow patterns and the respirable dust concentration at a longwall face in underground coal mines. The analysis of contaminant transport in groundwater systems using finite volume techniques have been carried out by Putti et al, (1990) and Binning & Celia (1996). Singh & Doulati Ardejani (2004) developed an one-dimensional numerical finite model using PHOENICS code to simulate long term pyrite oxidation, acid mine drainage generation and transportation of the oxidation products through the backfills of an open cut mine.…”

A summary of numerical methods for time-dependent advection-dominated partial differential equations

“…In this term, the test functions w(x) are standard FEM basis functions on at time t m+1 , but the value of c(x * ; t m ) has to be evaluated by a backtracking method where x * = r(t m ; x; t m+1 ) is the point at the foot corresponding to x at the head [37,43]. For multidimensional problems, the evaluation of this term with a backtracking algorithm requires signiÿcant e ort, due to the need to deÿne the geometry at time t m that requires mapping of points along the boundary of the element and subsequent interpolation and mapping onto the ÿxed spatial grid at the previous time t m [7,74]. This procedure introduces a mass balance error and leads to schemes that fail to conserve mass [15,74,107].…”

Grand Challenge Problems in Environmental Modeling and Remediation: Groundwater Contaminant Transport (Partnerships in Computational Science)

“…All of those previous studies provided important input and guidance, but were restricted to either one spatial dimension or two spatial dimensions. As is evident in Binning and Celia ( 1996a), boundary condition implementation, and evaluation of integrals at the known time level, are the two major difficulties in the ELLAM approach. These difficulties increase as the dimensionality of the problem increases.…”