Effect of Scale - Transport and Fate of Chemicals in Soils: Principles and Applications - page 111

4.2.1 Generalized Equations and Simulations

4.2.1.1 CDE with a Linearly Time-Dependent Dispersivity

Fo r a linearly time-dependent dispersivity in terms of mean travel distance
x , a representative model can be expressed as (Pickens and Grisak, 1981b):

α= () xax

(4.5)

1

where a 1 is a dimensionless constant. Physically, a 1 is the slope of dispersivity-
mean travel distance line. If we ignore molecular diffusion, the dispersion
coefficient can be written as:

2

Dt () =α () xv = axvavt =

(4.6)

1 1

where v is mean pore water velocity. Accordingly, the governing equation
in a heterogeneous system with a time-dependent dispersion coefficient is
given by:

∂
∂

c
t

∂
∂

∂
∂

c
x

∂
∂

c
x

−

=

x Dt

() v

(4.7)

Clearly, the dispersion coefficient as determined by Equation (4.6) is con-
stant for the entire domain at any fixed time. Therefore, the governing equa-
tion can be rewritten as:

2

∂
∂

c
t

∂
∂

c ∂

∂

c
x

2

= avt − v

(4.8) 1

2

x

In addition, the appropriate initial and boundary conditions for a finitely
long soil column can be expressed as:

cxt (,)0, = t = 0

(4.9)

=− ∂
∂

c
x

2

vc xt (,) vc avt , x = 0, 0 < ≤ t T

(4.10)

0 1

∂
∂

c
x

2

vc xt (,) =− avt , x = 0, t > T

(4.11)

1

∂
∂

c
x xL

= 0

(4.12)

=

where L is the length of the soil column, T is the input pulse duration, and
c 0 is the solute concentration in the input pulse. The governing equation
(4.8) subject to initial and boundary conditions (Equations 4.9 through