CWR 6537 Subsurface Contaminant Hydrology
1
ADVECTIVEDISPERSIVE SOLUTE TRANSPORT EQUATION
:
ANALYTICAL SOLUTIONS
A.
Lecture Goal:
To understand the concepts that surround the solution of the advectivedispersive solute
transport equation for “ideal” solutes by examining several boundary and initial conditions;
dimensionless solution approaches; and time vs. space perspectives.
B.
AdvectiveDispersive Transport of “Ideal” Solutes:
An "Ideal" solute is nonreactive
(no adsorptiondesorption, or ionexchange with the solid matrix)
and conservative
(no degradation or precipitation; no gains owing to dissolution etc.)
The partial differential equation for describing onedimensional transport of ideal solutes
during steady water flow in saturated (or uniformly unsaturated) homogeneous porous media is:
x
c
v

x
C
D
=
t
C
2
2
∂
∂
∂
∂
∂
∂
(1)
where
C = liquidphase solute concentration (M·L
3
·T
1
);
D = (longitudinal) hydrodynamic dispersion coefficient along the x direction, (L
2
·T
1
); and
v
= average porewater velocity along the x direction, (L).
Note: the concentration term, C, is referenced here to the aqueousphase concentration, but for
convenience, we have dropped the subscript
w.
We have also dropped the subscript
m
from
dimension x; however, we are still using the bulk media frame of reference.
Finally, we have
adopted the use of the average porewater velocity rather than the actual velocity distribution, f(v).
Thus, D here represents the hydrodynamic dispersion term (i.e, the combined effects of molecular
diffusion and mechanical dispersion).
What changes in Eq (1) would be needed for solute transport during steady water flow in a
homogeneous porous medium that was
variably saturated
?
C.
Initial and Boundary Conditions:
A variety of initial and boundary conditions may be specified now for solving Eq.[1].
For
example, the porous medium may be:
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CWR 6537 Subsurface Contaminant Hydrology
2
(i)
Infinite medium(
∞
< x < +
∞
); recall the diffusion case
(ii)
SemiInfinite medium (0
≤
x < +
∞
); as in a very long
column
(iii)
Finite medium (0
≤
x
≤
L); as is a short
laboratory column
How do we decide what is a short column or a very long column??
Similarly, the initial solute concentration, C (x,0), may be:
(i)
C (x,0) = f(x); solute concentration varying with distance
(ii)
C (x,0) = C
i
; constant solute concentration with distance
(iii)
C (x,0) = 0; no solute initially in the medium
Finally, various boundary conditions can also be specified.
The concentration at x = 0 (i.e., the
surface), C (0,t), may be specified as:
(i)
C (0,t) = f(t); timevarying solute concentration at x=0
(ii)
C (0,t) = C
o
; constant solute concentration at x=0
These same conditions can also be stated at x= L, the bottom boundary, (i.e., C(L,t) or C(+
∞
,t)).
The surface or bottom boundary condition can also be specified as flux boundary condition.
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 Spring '08
 Jawitz
 Boundary value problem, Brenner, Subsurface Contaminant Hydrology

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