Leaching of Contaminants to Groundwater - Dealing with Contaminated Sites: From Theory Towards Practical Application - page 830

where subscripts refer to spatial discretization and superscript to temporal
discretization (e.g., j and j +1 are for the previous and actual time levels, respec-
tively; see Fig. 18.16 ), z is the spatial step (assumed to

be constant). Notice that this equation contains only one unknown variable (i.e., the
concentration c i j +1 at the new time level), which hence can be evaluated directly
(explicitly) by solving the equation.
By comparison, a fully implicit (backward-in-time) finite difference scheme can
be written as follows

t is the time step, and

J j + 1
i

J j + 1
i

D c j + 1

2 c j + 1
i

c j + 1
i

v c j + 1

c j + 1
i

c j + 1
i

c j
i

2 −

1 −

+

1 −

−

+ 1 / − 1 / 2

i + − 1 i + − 1

=− = −

z ) 2

t z ( 2 z

(18.60)

and an implicit (weighted) finite difference scheme as:

c j + 1
i 1

) c j
i 1

2 c j + 1
i

c j + 1
i

2 c i + c j

i

c j + 1
i

c j
i

D ε

1 −

+ + (1 − ε

1 −

− + − + −

=

z ) 2

t (

c j + 1
i + 1

) c j
i + 1

(18.61)

c j + 1
i −

c j
i −

v ε

1 −

+ (1 − ε

1 −

−

2 z

where is a temporal weighting coefficient. Different finite difference schemes

result depending upon the value of

ε

ε , i.e., an explicit scheme when ε = 0, a Crank-

Nicholson time-centered scheme when ε = 0.5, and a fully implicit scheme when

ε =

1.

18.5.1.2 Finite Elements

Finite element methods can be implemented in very much the same way as finite
differences for one-, two-, and three-dimensional problems. A major advantage of
the finite elements is that they are much easier used to discretize complex two-
and three-dimensional transport domains (Fig. 18.17 ). As an example, Fig. 18.17
shows triangular unstructured finite element grids for a regular rectangular and an
irregular domain as generated with the automated MeshGen2D mesh generator of
HYDRUS-2D (Šimunek et al. 1999b ). Notice that even though the figure on the
right (Fig. 18.17 ) has an irregular soil surface, as well as a tile drain within the
transport domain, MeshGen2D could easily discretize/accommodate this transport
domain using an unstructured triangular finite element mesh.

18.5.2 Existing Models

18.5.2.1 Single-Species Solute Transport Models

A large number of numerical models are now available for evaluating variably-
saturated water flow and contaminant transport processes in the subsurface. Some
of these models are in the public domain, such as MACRO (Jarvis et al. 1994),
SWAP (van Dam et al. 1997 ), UNSATH (Fayer 2000 ), VS2DI (Healy 1990 ), and