Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 78

These conditions are for a solute or a tracer solution that is applied at
a specified rate from a perfectly mixed inlet reservoir to the soil surface.
Continuity of the solute flux across the inlet boundary leads directly to the
above third-type boundary conditions. Third-type boundary conditions
were used by Brenner (1962) and Lindstrom et al., 1967), among others.
Proper formulation of the exit boundary condition for displacement
through finite laboratory columns and soils at the field scale are needed. The
boundary condition at some depth L in the soil profile is often expressed as
(Danckwerts, 1953):

∂
∂

C
z

= 0, zL t = , ≥ 0

(3.46)

This second-type boundary condition is used to deal with solute effluent
from soils having finite depths, such as laboratory miscible displacement col-
umns, and describes a zero concentration gradient at z = L .
It is often convenient to solve the CDE where a semi-infinite rather than a
finite length ( L ) of the soil is assumed. Under such circumstances, the appro-
priate condition for a semi-infinite medium is needed. Specifically, for semi-
infinite systems in the field, we need a boundary condition that specifies
solute behavior at large depth ( z → ∞ ). Such a boundary condition may be
expressed as C = constant (commonly zero) as z → ∞ . An appropriate formu-
lation of this boundary is

∂
∂

C
z

= ∞≥ 0, z , t 0

(3.47)

which is identical to that for a finite soil length ( L ). Kreft and Zuber (1978) and
van Genuchten and Wierenga (1986) presented a discussion of the various
types of boundary conditions for solute transport problems.

3.4 Exact Solutions
Analytical or exact solutions to the CDE (Equation 3.34) subject to the appro-
priate boundary and initial conditions are available for a limited number
of situations, whereas the majority of the solute transport problems must
be solved using numerical approximation methods. In general, whenever
the form of the retention reaction is linear, an exact or closed-form solu-
tion is obtainable. A number of closed-form solutions are available in the
literature and are compiled by van Genuchten and Alves (1982). Since sev-
eral boundary conditions are commonly used, we limit our discussion to