Kouli_etal_2008_Groundwater modelling_BOOK.pdf - Pantelis ...

160
Husam Baalousha
The advantages of the finite difference method are that it is easy to implement, well
documented and produces reasonably good results. However, finite difference method has
some disadvantages. The main disadvantage is that it does not fit properly to an irregular
model boundary (Figure 6). In addition, the distribution of grids, their size, and whether they
are of equal size highly affects the accuracy and computation effort. Output accuracy of the
finite difference method is not good in the case of solute transport modelling. Mass balance is
not guaranteed if conductivity or grid spacing varies (Cirpka 1999).
The most widely used finite difference based groundwater model is MODFLOW
(Harbaugh and McDonald 1996).
Box 3: Considerations in selecting the size of the nodal spacing in grid or mesh
design
• Variability of aquifer characteristics (e.g.. conductivity, storativity).
• Variability of hydraulic parameters (e.g. recharge, pumping).
• Curvature of the water table.
• Desired detail around sources and sinks (e.g., rivers).
• Vertical change in head (vertical grid resolution/layers).
5.2.2. Finite Element Method
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 (Figure 7). 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.
The weighted residual method is being widely used in groundwater finite element
problems. The Russian engineer B. G. Galerkin introduced this method in 1915 (Pinder and
Gray 1970). To illustrate the weighted residual approach, consider a groundwater or solute
transport problem. This problem over a domain B can be written as:
L( φ(
x,
y,
z))
− F(
x,
y,
z)
= 0
Equation (5)
Where L is a differential operator, φ(x,y,z) is the dependent variable (i.e. groundwater
head) and F(x,y,z) is a known function.