Mathematical Modeling of Groundwater Pollution

The chemical and biological constituents contained in groundwater depend on two factors: the natural environment of groundwater storage and move­ment, and human activities. Precipitation infiltration and surface water per­colation are the natural sources of groundwater. The total dissolved solids (TDS) of precipitation is generally very low, but its chemical components will be changed when infiltrated through soil beds by a series of actions, such as solution, oxidation, reduction, ion exchange, and so on. The infiltration and percolation water will be involved in groundwater movement in both the vertical and the lateral directions in the aquifer. During this process, the TDS of groundwater will continually increase as rocks and minerals are dissolved into the water. Human activities may change the natural process and cause groundwater to contain organisms, hydrocarbons, heavy metals and other harmful matter. Groundwater, therefore, should be looked upon as a multicomponent fluid. The content of each component in groundwater can be expressed by its concentration, i.e., a mass of certain component contained in unit volume of water (M/L ³). If the concentration of component a is written as C _a , then the standard of water quality for a certain use can be written in the following common form: \( \begin{gathered} {C_{\alpha ,\min }}\, < \,{C_\alpha }\, < {C_{\alpha ,\max }}, \hfill \\ \,\,\,\,\,\,\,\left( {\alpha \, = \,1,\,2,\,...,\,n} \right) \hfill \\ \end{gathered} \) where C_α,min and C_{α, max} are the given lower and upper limits, respectively, of the concentration of component α, and n is the total number of components considered. According to the actual situation, the components may refer to either single ions or multi-ion compositions.

When fluid flows through the interconnected voids and passages of a porous medium, the walls of these voids and passages form many small tunnels, and the fluid flows inside them. The study of physical phenomena in a porous medium on such a scale (pore scale) is called the microscopic method. Due to the complexity of the micro-geometry of porous media, it is unrealistic to study the details of the flow on this scale. Therefore, one has to describe the flow phenomena in porous media on a macroscopic rather than microscopic basis. The spatial average method is a way to transfer properties of porous media from the microscopic level to the macroscopic level.

Consider the following two-dimensional advection-dispersion equation which is subject to the initial condition boundary conditions and where (R) is the flow domain, (Γ₁) and (Γ₂) are boundary sections of (R), f is a given function in (R), g ₁ and g ₂ are given functions along (Γ₁) and (Γ₂), respectively, and n _x and n _y are components of the unit outer normal vector to the boundary (Γ₂). Equation (5.1.3) expresses the boundary condition of given concentration, i.e., the first-type boundary condition, while Eq. (5.1.4) expresses the boundary condition of given dispersion flux, i.e., the second-type boundary condition.

When the advection term dominates in the advection-dispersion equation, most traditional numerical methods will encounter difficulties. In Chapter 4, we have stated that when the FDM is used for solving advection-dominated problems, two kinds of errors, numerical dispersion and overshoot, will occur. As a result, oscillations will appear around concentration fronts and steep concentration fronts cannot be accurately calculated. In Chapter 5, we pointed out that the Galerkin FEM cannot avoid the above two kinds of errors, either. Consequently, improvement of the accuracy and stability of numerical solutions has become an important subject in current research.

We have introduced the mechanism of hydrodynamic dispersion, derived the hydrodynamic dispersion equations, and presented various methods for solving this kind of equation in previous chapters. Let us now consider how to construct and solve the general hydrodynamic dispersion models, or the advection-dispersion models.

In the application of mathematical models to practical groundwater pollution problems, the following steps should be included:

This book gives a systematic discussion on the theories and methods of contaminant transport in porous media. Emphases are placed on how to construct mathematical models and use numerical solutions. Model calibration and model applications are also discussed in certain depth.