Tracking Groundwater Pollution
This is a teaching module presenting an introduction to modeling ground water pollution, directed to undergraduate students in applied mathematics. It begins with a brief discussion of Darcy's law concerning the flow of a fluid through a porous medium. A mathematical model that uses field data to track ground-water contamination is presented. Students working with the module will write a MATLAB script to obtain a numerical solution of the model and apply it to investigate a real event of groundwater pollution.
The primary goals of this activity are twofold.
- To increase student's motivation to learn mathematics by making it relevant to their lives; connecting mathematics learning to the goals and interests that students bring to college; and showing how mathematics relates to other disciplines, important civic questions, and technological challenges.
- To develop basic skills in programming and scientific computing by writing their own MATLAB code with graphical representation of the solutions.
Context for Use
This module is directed to undergraduate students in applied mathematics, as in a course of mathematical modeling or in a course of calculus with differential equations. Some knowledge of Multivariable Calculus is expected. I have used versions of this material as an end of course group project, but it could also be used as individual project.
Description and Teaching Materials
The Tracking Groundwater Pollution Handout begins with description of Darcy's law concerning the flow of a fluid through a porous medium, and the formula for the interstitial velocity or, simply, velocity of groundwater. It follows by the presentation of a study case of groundwater pollution in a landfill that is currently under the responsibility of the Minnesota Pollution Control Agency (MPCA). Readings of a collection of monitor wells are presented on a table and in the data spreadsheet WDE Landfill Data.
Levels of pollutants such as arsenic, benzene, manganese, and vinyl chloride, exceeding health risk limits, were found in several of the monitor wells. The growing population surrounding the landfill and the presence of a creek near by is a matter of special concern to the surrounding communities and the MPCA. Geographical markers used to model the trajectory of the creek are provided in the spreadsheet Coon Creek Data.
We use the surface modeling tool gridfit, developed by John D'Errico, to fit a surface of the form z=f(x,y) to the data of the monitor wells. This is include in the Surface Fitting MATLAB script below, together with the Head Contour Map MATLAB script that uses MATLAB's function contour to plot the corresponding head contour maps.
The last section includes the description of two assignments. The first one is to develop a MATLAB script to obtain a numerical solution to the model. In the second assignment students apply the model to track the trajectory of groundwater from one of the monitor wells to determine if the flow reaches the creek, and in this case to estimate the time it would take for pollutants to travel from the monitor well to the creek.
The Groundwater Flow Path (private instructor-only file) is a MATLAB script that uses the ODE solver ode45 to obtain numerical solutions of the system of differential equations of the model, with corresponding graphical outputs. It calls the MATLAB function Groundwater Vector Field (private instructor-only file) to compute the vector field of the ODE system.
Teaching Notes and Tips
Although the module includes a self-contained description of the basics of groundwater modeling, I usually dedicate a few lectures during the semester to discuss the material. The references provided below are a good supplement to the material discussed in the module.
Basic skills in programming with MATLAB are expected to complete the project. These skills could be developed during the semester by integrating MATLAB as the main simulation tool within the course. Depending on the level of expertise required, instructors may or may not choose to provide the included private instructors-only MATLAB files to their students.
I use SENCER SALG pre- and post-assessment to assess the changes in student's perception of mathematics, its interaction with other disciplines, and its role in addressing relevant social issues. The SENCER SALG, http://salgsite.org/, is an online free course-evaluation tool that allows college-level instructors to gather learning-focused feedback from students. Instructors are guided through a wizard, using an adaptable template, to design their own instrument.
References and Resources
- C. R. Hadlock, Mathematical Modeling in the Environment, The Mathematical Association of America (1998)
- C.R. Hadlock, Underground Mathematics
, The College Mathematics Journal, Vol. 44, No. 5 (November 2013), pp. 364-375 .