Environmental Engineering Reference
In-Depth Information
18.4 Analytical Models
18.4.1 Analytical Approaches
Many
analytical solutions
have been derived in the past of the contaminant transport
equations, and they are now widely used for analyzing contaminant transport during
steady-state flow (Šimunek
2005
). Although a large number of analytical solutions
also exist for the unsaturated flow equation, they generally can be applied only to
relatively simple flow problems. The majority of applications for water flow in the
vadose zone require a numerical solution of the Richards equation.
Analytical methods are representative of the classical mathematical approach for
solving differential equations to produce an exact solution for a particular prob-
lem. Analytical models usually lead to an explicit equation for the concentration (or
the pressure head, water content, or temperature) at a particular time and location.
One hence can evaluate the concentration directly without time stepping typical
of numerical methods. While exceptions exist (e.g., Liu et al.
2000
), analytical
solutions usually can be derived only for simplified transport systems involving
linearized governing equations, homogeneous soils, simplified geometries of the
transport domain, and constant or highly simplified initial and boundary conditions.
Unfortunately, analytical solutions for more complex situations, such as for tran-
sient water flow or nonequilibrium contaminant transport with nonlinear reactions,
are generally not available and/or cannot be derived, in which case numerical models
must be adopted (Šimunek
2005
).
Analytical solutions are usually obtained by applying various transformations
(e.g., Laplace, Fourier or other transforms) to the governing equations, invoking a
separation of variables, and/or using the Green's function approach (e.g., Leij et al.
2000
).
18.4.2 Existing Models
18.4.2.1 One-Dimensional Models
Some of the more popular one-dimensional analytical transport models have been
CFITM (Van Genuchten
1980b
), CFITIM (Van Genuchten
1981
), CXTFIT (Parker
and Van Genuchten 1984), and CXTFIT2 (Toride et al
.
1995
). While CFITM con-
siders only one-dimensional equilibrium transport in both finite and semi-infinite
domains, CFITIM additionally considers physical and chemical nonequilibrium
transport (i.e., the two-region mobile-immobile model for physical nonequilibrium
and the two-site sorption model for chemical nonequilibrium). CXTFIT expanded
the capabilities of CFITIM by considering more general initial and boundary con-
ditions, as well as degradation processes. CXTFIT2 (Toride et al.
1995
), an updated
version of CXTFIT, solves both direct and inverse problems for three different
one-dimensional transport models:
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