Case studies
This year's case studies are listed below. Material for the case studies will successively be added to this page.
Case Study 1: Groundwater flow modeling using the finite element method
In environmental engineering, it is of crucial importance to have good computational models for the slow flow of groundwater through the subsurface, in order, for instance, to estimate the transport of contaminants through the groundwater. This case study constitutes an basic introduction to how such modeling can be carried out. We will do the following:
- study the standard model for the flow of groundwater, as derived from the law of mass conservation together with Darcy's law (despite its name, the latter is not really a "law of nature", such as mass or energy conservation, but rather a so-called constitutive model —a mathematical model of material properties—for fluid flow through a porous medium);
- introduce the finite element method for elliptic partial differential equations and discretize the groundwater flow equations using this method;
- learn how to implement finite-element approximations of a given set of partial differential equations in Comsol Multiphysics; and
- use Comsol Multiphysics to solve the problem of calculating the water table (grundvattenytan) and the subsurface flow of water in a given situation.
Recall that the second theme in last year's course Numeriska metoder för civilingenjörer concerned the modeling of the flow of water in a network of water pipes. The first material to go through in preparation for the current case study is a review of that theme:
Network models and linear systems, Part I
The network flow model is a discrete analogue to the groundwater flow equations that we will introduce here. "Discrete" means that pressures and fluxes are only defined at the nodes and edges in the water pipe network. In contrast, in the case of groundwater flow, pressures and flow rates typically vary continuously throughout the domain of interest.
A short introduction to the modeling of fluid flow in porous media is contained in the following document, an excerpt from the Habilitation Thesis of Peter Bastian:
Modeling Immiscible Fluid Flow in Porous Media,
(If you are interested, you may also download the full Habilitationsschrift!)
The Wikipedia articles on Darcy's Law and the Groundwater Flow Equations are also recommended for reference.
We will numerically solve the groundwater flow equations using the finite element method. This method approximates unknown field quantities, such as the pressure in the groundwater flow equations, with piecewise polynomials, defined on a triangulation of the computational domain, that is, a subdivision of the domain in elementary geometrical objects, typically triangles (2D) or tetrahedrons (3D). To obtain equations for the coefficients in this piecewise-polynomial expansion, the finite element method relies on a rewriting of the governing partial differential equations and associated boundary condition in an integral form . All this, and more, is explained in the document
An Introduction to the Finite Element Method for Elliptic Problems,
Lab 1, October 4: Equation-based Modeling in Comsol Multiphysics. This lab constitutes an introduction to the use of equation-based modeling in Comsol. For this lab, you need the lab instruction and an excerpt from the Comsol Multiphysics Modeling Guide.
Lab 2, October 11: Groundwater flow computation: Lab instruction.
Miscellaneous other study material:
- Review: Porous Media Flow . Slides from 2nd lecture.
- Approximations and representations of functions in the Finite Element Method . Slides from 2nd lecture.
- Review and some extensions: FEM . Slides from 3d lecture. This is a corrected version of the documents that were distributed at the lecture.
- Review questions and exercises for Case Study 1. Covers both porous media flow and the finite element method. Handed out at 3d lecture. The same document is also available with answers.
Darcy's law of Case Study 1 says that the apparent velocity field is driven by pressure gradients. Together with mass conservation, we obtained that the pressure head satisfied Laplace's equation. Darcy's law has analogues in many other applications, such as heat conduction in solids (Fourier's law), molecular diffusion processes (Fick's law), and electrostatics (Ohm's law). The physical principle for all of these is one of diffusion.
In this case study we will study a fundamentally very different type of problems, namely models of transport. Large-scale practical examples of this type of problems occurs for instance in coastal ocean modeling. Here, the objective can be to predict, for instance, the effect of potential oil spills or the effect of storm surges. In Part 2 of this case study, we will study the dynamics of a fluid resulting from opening a sluice gate between two water reservoirs at different water levels. As in the previous case study, Part 1 constitutes an introduction to the case study.
Lab 1, November 1: Conservation laws and finite volume methods. Lab instruction and a file plotfvm.m containing a Matlab function to plot piecewise-linear functions.
Lab 2, November 16: A sluice gate opens. Lab instruction, and the Matlab function plotfvm.m (same as used in Lab 1).
Miscellaneous other study material:
- Review: conservation laws and Finite Volume Methods . Slides from lecture 5.
- Review questions and exercises for Case Study 2. The same document with answers.