Contaminant Fate and Reactive Transport in Groundwater - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 866

transport equations. Therefore, some sort of spatial and temporal discretization of
the problem governing equations has to be performed. The result of the discretiza-
tion is the approximation of the governing partial differential equations with a set of
algebraic equations typically solved by a computer code.
The often cited numerical codes such as MODFLOW (McDonald and Harbaugh
1988 ) and FEMWATER (Yeh and Ward 1980 ) are used to solve the governing
three-dimensional groundwater flow equation (Eq. ( 19.15 )) and to calculate the dis-
tribution of hydraulic heads which represent a prerequisite for the computation of
the velocity field necessary for the solution of the contaminant transport equation.
The numerical solution of contaminant transport equations is an area of active
research. An exhaustive discussion of numerical methods, however, is beyond the
scope of this topic chapter. In the following a brief overview of principal numerical
techniques and strategies is presented. For further details the interested reader can
refer to the literature (e.g. Barry et al. 2002 ; Bear 1979 ; Steefel and MacQuarrie
1996 ; Zheng and Bennet 2002 ; Zheng and Wang 1999 , among others).
Most numerical methods for solving the advection-dispersion equation can be
classified as Eulerian, Lagrangian or mixed Eulerian-Lagrangian. In Eulerian meth-
ods, the transport equation is solved in a fixed spatial grid. Finite difference,
finite element and finite volume methods are primary examples for this class
of solution methods. Eulerian methods are generally mass conservative and han-
dle dispersion-dominated problems both accurately and efficiently. However for
advection-dominated problems, encountered in many field situations, an Eulerian
method may be susceptible to excessive numerical errors (Zheng and Wang 1999 ).
A typical error is “numerical dispersion”, which indicates the artificial, non-physical
dispersion resulting from the numerical approximation associated with the dis-
cretization of the model domain. In order to overcome these problems and minimize
numerical errors, restrictively small grid spacing and time-steps may be required.
Alternatively, higher order finite difference (or finite element) methods such as the
total-variation-diminishing (TVD) method can be used.
In the Lagrangian approach, the transport equation is solved in either a deform-
ing grid or a deforming coordinate in a fixed grid through particle tracking. This
approach provides a highly efficient solution to advection-dominated problems vir-
tually free of numerical dispersion. However, problems such as local mass balance
errors and numerical instabilities may arise.
Mixed Eulerian-Lagrangian methods (e.g. the widely used methods of charac-
teristics) attempt to combine the advantages of both approaches by solving the
advection term with a Lagrangian method (particle tracking) and the dispersion
term and the other terms with an Eulerian approach (finite difference or finite
elements). Concerning the solution of coupled reactive transport problems, two
different strategies can be distinguished:

• one-step or global implicit approach, which solves the governing transport equa-

tions, including transport and reaction terms, simultaneously. An example of
contaminant reactive transport model using this method is MIN3P (Mayer et al.
2002 );