Advances in Numerical Analysis

Volume 2016 (2016), Article ID 1945958, 12 pages

http://dx.doi.org/10.1155/2016/1945958

## A Finite Volume-Complete Flux Scheme for a Polluted Groundwater Site

^{1}Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands^{2}Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India

Received 5 April 2016; Revised 10 July 2016; Accepted 17 July 2016

Academic Editor: Hassan Safouhi

Copyright © 2016 M. F. P. ten Eikelder et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a model of a polluted groundwater site. The model consists of a coupled system of advection-diffusion-reaction equations for the groundwater level and the concentration of the pollutant. We use the complete flux scheme for the space discretization in combination with the -method for time integration and we prove a new stability result for the scheme. Numerical results are computed for the Guarani Aquifer in South America and they show good agreement with results in literature.

#### 1. Introduction

There is a great interest in developing groundwater models in the last decades. Groundwater is an important component of water resource systems. For example, more than half of the United States population depend on groundwater for drinking water [1]. Unfortunately, since groundwater is susceptible to pollutants, the quality of groundwater can be poor and is sometimes even deteriorating. Groundwater pollution occurs because of, among other things, disposal of industrial wastes such as gasoline, oil, and chemicals and agricultural activities such as the use of fertilizers. Also, uncontrolled hazardous waste, chemicals, road salts, and contaminants from dumps can make their way down into the groundwater [2]. Moreover, the groundwater is currently withdrawing which makes the quality even worse. This all indicates that groundwater might be unsafe for human use.

To solve these problems, clean water can be injected into aquifers and polluted groundwater can be pumped out. For this, the location of injecting and pumping as well as their rates is of great importance. Decisions have to be made for the proper locations and the total volume of groundwater that should be injected and/or pumped. Numerical simulations of groundwater flow enable us to make predictions about groundwater level and the amount of pollution and are therefore very useful. Therefore, mathematical models are needed for groundwater flow.

In groundwater modeling many different types of models are considered, for example, flow models, which describe the hydraulic head, or solute transport models, which describe the concentration level [3]. Our model is a combination of both a flow model and a solute transport model. Thus, the model includes both the concentration of the pollutant and the groundwater level. New in our model is that we include the effect of pumping and injecting water.

Our model consists of two equations, a pure diffusion equation for the groundwater level and an equation of advection-diffusion-reaction type that describes the concentration of the pollutants. For space discretization of the advection-diffusion-reaction equation we use the finite volume-complete flux scheme developed in [4, 5]. This scheme has proven to be a robust method to discretize advection-diffusion-reaction equations, which is uniformly second-order accurate, even for dominant advection, yet it does not produce any spurious oscillations, which makes the scheme very suitable for groundwater simulations. For the diffusion equation, the complete flux scheme reduces to the central difference scheme, which we use to discretize the groundwater level equation. We have applied the complete flux scheme to various problems from continuum physics, such as the flow of multicomponent mixtures and plasmas [6, 7], incompressible fluid flow [8], or the numerical simulation of plankton populations [9].

For the temporal integration the -method is used. New stability conditions for the combination of the -method with the finite volume-complete flux scheme for spatial discretization are derived in this paper.

We have organized our paper as follows. First, in Section 2, we describe our groundwater model. Next, in Section 3, we describe two versions of the complete flux scheme for our model. In Section 4 we derive stability conditions for the fully discrete scheme. The numerical results are discussed in Section 5. Finally, we end with conclusions in Section 6.

#### 2. Mathematical Model

The Guarani Aquifer in South America is one of the largest aquifers in the world. It is a very important source of drinking water in Argentina, Brazil, Paraguay, and Uruguay and should obviously not be polluted. However, the aquifer is very vulnerable to pollution due to land-surface activities [10]. Therefore it is interesting to define a model problem for the transport of polluted water based on the parameters of this aquifer [11–14].

We consider a groundwater site next to an aquifer. The groundwater in this basin is polluted and flowing towards the aquifer. Since the polluted water should not enter the basin we assume a well at the boundary between basin and aquifer, where all polluted water is removed. We restrict ourselves to a one-dimensional domain , with the width of the basin, adjacent to the aquifer at . The soil in the basin consists mainly of clay and clay-like material, so we choose a uniform one-layer model consisting of clay only. We measure the groundwater level in the layer relative to a flat impermeable bottom. Furthermore, we are not particularly interested in the detailed composition of the polluted water, and therefore we only consider one typical polluting species, namely, lodide. The unknowns of our model are therefore the groundwater level in the layer and the concentration of lodide. A schematic picture of the groundwater site is shown in Figure 1.