Agriculture Reference
In-Depth Information
4.12) can be solved using finite-difference methods. The detailed finite-
difference scheme is shown in Appendix 4A. The resulting tri-diagonal
linear equation system was solved using the Thomas algorithm (Press
et al., 1992).
4.2.1.2 CDE with a Linearly Distance-Dependent Dispersivity
By the removal of the bar from the mean travel distance (
x
) in the dispersivity-
mean travel distance relationship, we change the mean travel distance to
a distance from source (
x
) and obtain a distance-dependent dispersivity as
given by
α=
()
xax
(4.13)
2
where now
a
2
is a constant as
a
1
is in Equation 4.5. If we also ignore molecu-
lar diffusion, the dispersion coefficient becomes a function of distance from
source and is thus given by
Dx
()
=α
()
xv
=
axv
(4.14)
2
Therefore, transport for a tracer solute or nonreactive chemical in a one-
dimensional heterogeneous soil system with distance-dependent dispersion
coefficient, under steady-state water flow, is governed by the following equation:
∂
∂
c
t
∂
∂
∂
∂
c
x
∂
∂
c
x
−
()
=
x
Dx
v
(4.15)
Substituting Equation 4.14 into the above governing equation and expand-
ing gives:
2
∂
∂
c
t
∂
∂
−−
∂
c
av
c
x
(4.16)
=
avx
(1
)
2
2
2
x
∂
The corresponding initial and boundary conditions for a finite soil column
can be expressed as:
cxt
(,)0,
=
t
=
0
(4.17)
cxt
(,)
=
c x
,
=
0,
0
<≤
t
T
(4.18)
0
cxt
(,)0,
=
x
=
0,
tT
>
(4.19)
∂
∂
c
x
xL
=
0
(4.20)
=
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