Finite element analysis of contaminant transport in groundwater

Abstract

In this paper, we focus on the development of a finite element model for predicting the contaminant concentration governed by the advective–dispersive equation. In this study, we take into account the first-order degradation of the contaminant to realistically model the transport phenomenon in groundwater. To solve the resulting unsteady advection–diffusion equation with production, a finite element model is constructed, which employs a quadratic basis function to approximate the contaminant concentration. The development of a weighted residuals finite element model involves constructing a biased test function to retain the scheme stability for wide ranges of values of the physical coefficients. In the process of constructing the Petrov–Galerkin finite element model for stability reasons, it is desirable to obtain an acceptable degree of accuracy. The method used to retain stability without loss of accuracy is to approximate the differential equation within the semi-discretization framework. After discretizing the time derivative term using the Euler time-stepping scheme, the resulting ordinary differential equation, which involves only the spatial derivative terms, is solved using the nodally exact finite element model. For better control of the user's specified time step and mesh size, full analysis of the discretization scheme is conducted. In this study, both modified equation analysis and Fourier stability analysis are employed to better understand the proposed semi-discretized Petrov–Galerkin finite element model. Validation of the newly proposed model is accomplished through analysis of the results obtained for several test problems. Some of them are amenable to analytic solutions. The rate of convergence of the employed finite element model then can be obtained.