3
Analytic Solutions of Hydrodynamic
Dispersion Equations
3.1 Superposition of Fundamental Solutions
3.1.1 The Fundamental Solution of a Point Source
Consider an infinite, homogeneous, and isotropie region, where it is assumed
that the flow field is stationary and the solute mass M is injeeted instantaneously at the origin. In this ease, the general hydrodynamie dispersion
equation may be simplified as
(3.1.1)
where o( . ) is the Dirae-o funetion, D is the moleeular diffusion eoeffieient in
porous media. In this ease, the use of a spherie eoordinate system is eonve.
ac
ac
ment. In Eq. (2.6.9), let a(} = 0, at/J = 0, V = 0, and Dd T = D, we than have
ac = D ~ (r 2 ac) + M c:5(r)c:5(t).
at r 2 ar ar
()
The subsidiary eonditions are:
C(r,O) = 0, r > 0,
lim C(r,t) = 0, t > 0.
(3.1.2)
(3.1.3)
(3.1.4)
The souree term in Eq. (3.1.2) ean be removed, when the instantaneous
injeetion is represented by the following eondition:
4n(} IX) Cr 2 dr = M, t ~ 0,
(3.1.5)
where () is the volumetrie fraetion of the fluid, or the effeetive porosity n for a
saturated flow.
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