Basics of GroundWater Modelling Part 2
*Types of Models
There are different types of models to simulate groundwater movement and contaminant transport. In general, models can be classified into three categories: physical, analogue and mathematical models. The latter type can be classified further depending on the type of solution.
1-Physical Models
Physical models (e.g. sand tanks) depend on building models in the laboratory to study specific problems of groundwater flow or contaminant transport. These models can demonstrate different hydrogeological phenomena like the cone of depression or artesian flow. In addition to flow, contaminant movement can be investigated through physical models. Though they are useful and easy to set-up, physical models cannot handle complicated real problems.
2-Analogue Models
The equation which describe groundwater flow in isotropic homogenous porous media is called the Laplace’s Equation . This equation is very common in many applications in physical mathematics like heat flow, and electricity. Therefore, comparison between groundwater flow and other fields where the Laplace equation is valid, is possible. The most famous analogue model is the flow of electricity. The electric analogue is based on the similarity between Ohm’s law of electric current flow and Darcy’s law of groundwater movement. As electric current moves from high voltage to lower voltage, so does the groundwater, which moves from high head to lower head. Simple analogue models can easily be setup to study the movement of groundwater flow. More detailed information on analogue models is available (Verruijt, 1970, Anderson and Woessner, 1992, Strack 1989; Fetter 2001).
3-Mathematical Models
Mathematical models are based on the conceptualisation of the groundwater system into a set of equations. These equations are formulated based on boundary conditions, initial conditions, and physical properties of the aquifer. Mathematical models allow an easy and rapid manipulation of complex models. Once the mathematical model is set-up, the resulting equations can be solved either analytically, if the model is simple, or numerically.
*Types of Model Solutions
As discussed in the preceding sections, the mathematical models can be solved either
analytically or numerically. Some approaches use a mixture of analytical and numerical
solutions. The following sections briefly discuss the main types of solutions used in
groundwater modelling.
1-Analytical Solutions
Analytical solutions are available only for simplified groundwater and contaminant transport problems. They were developed before the use of numerical models. The advantages of analytical solutions are that they are easy to apply and produce continuous and accurate results for simple problems. Unlike numerical solutions, analytical solutions give a continuous output at any point in the problem domain. However, analytical solutions make many assumptions like isotropy and homogeneity of an aquifer, which are not valid in general. Analytical solutions; therefore, cannot deal with complex groundwater systems. Examples of analytical solutions are the Toth solution (Toth, 1962) and Theis equation (1941). More details on analytical solutions of groundwater problems can be found in Bear (1979) and Walton (1989).
2-Numerical Solutions
Because analytical solutions of partial differential equations (PDE) implies many assumptions, simplifications and estimations that do not exist in reality, they cannot handle complicated real problems. Numerical methods were developed to cope with the complexity of groundwater systems. Numerical models involve numerical solutions of a set of algebraic equations at discrete head values at selected nodal points . The most widely used numerical methods are finite difference and finite element methods. Other methods have been developed, such as the boundary-element method
3-Finite difference method
Finite difference method (FDM) has been widely used in groundwater studies since the early 1960s. FDM was studied by Newton, Gauss, Bessel and Laplace (Pinder and Gray 1977).
This method was first applied in petroleum engineering and then in other fields. The finite difference method depends on the estimation of a function derivative by a finite difference
The basis of the finite element method is solving integral equations over the model domain. When finite element method is substituted in the partial differential equations, a residual error occurs. The finite element method forces this residual to go to zero. There are different approaches for the finite element method. These are: basis functions, variational principle, Galerkin’s method, and weighted residuals. Detailed description of each method can be found in Pinder and Gray (1970). Finite element method discritises the model domain into elements. These elements can be triangular, rectangular, or prismatic blocks. Mesh design is very important in the finite element method as it significantly affects the convergence and accuracy of the solution. Mesh design in the finite element method is an art more than a science, but there are general rules for better mesh configuration. It is highly recommended to assign nodes at important points like a source or sink, and to refine mesh at areas of interest where variables change rapidly. It is better to keep the mesh configuration as simple as possible. In the case of triangular mesh, a circle intersecting vertices should have its centre in the interior of the triangle.
End of Part 2
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