54
3. Analytic Solutions of Hydrodynamic Dispersion Equations
z
y
o
T P(x,y,z)
I
I
I Y
I
/'
X
I /' 'z
- - - - - - - y
x
FIGURE 3.3. A representation of
the plane sources problem.
The line-souree fundamental solution, Eq. (3.1.15), ean be used for solving the
two-dimensional point-souree problem perpendieular to the line. Similarly,
the plane-souree fundamental solution, Eq. (3.1.17), ean be used for the onedimensional point-souree problem perpendicular to a plane.
It is possible to solve some diffusion problems with simple boundaries
through the super-position of imaginary sources or sinks, as is done in groundwater flow problems. Let us eonsider a two-dimensional diffusion problem
with a straight line boundary through whieh no flow passes. Assurne that a
solute mass m is injeeted instantaneously into the porous medium over a unit
thiekness at the point (0, Yo). To obtain the eoneentration distribution eaused
by the injection, we suppose that there is another source symmetrically 10eated on the other side of the no-flow boundary with the same injection rate.
The aetual eoneentration distribution ean be obtained by superposing the
solutions of the two (line) sourees, that is
C(
) _ m/8 {
[_ x
2
+ (y - YO)2]
[_ x 2 + (y + yo)2]}
X, y, t - 4nDt exp
4Dt
+ exp
4Dt
.
(3.1.18)
3.1.3 Continuous Injection in a Uniform Flow Field
It is assumed that there is a one-dimensional flow in a homogeneous and
isotropie porous medium with a veloeity u along the x-direetion. If this
direetion is taken as the prineipal direetion of dispersion, the adveetiondispersion equation will be
(3,1.19)
Letting
(3.1.20)