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3. Analytic Solutions of Hydrodynamic Dispersion Equations
Problem 5. Two- and Three-dimensional Dispersions of a Tracer in a
Semi-infinite Region
Leij and Dane (1990) considered a two-dimensional dispersion problem of a
tracer in a semi-plane, which is as follows:
C(x, y, 0) = 0,
° < x < 00, -00 < y < 00,
OCI =0
ox x=oo '
-00 < y < 00, t > 0,
(3.2.33)
C(O, y, t) = g(y), -00 < y < 00, t > 0,
~~ Iy= ±oo = 0, ° < x < 00, t > 0,
where
g(y) =
1
2(CL + CR)' y = 0,
(3.2.34)
CR'
Y > 0,
and CL and C R are given constants. Using the Laplace and Fourier transforms successively, Eq. (3.2.33) may be translated into a boundary value
problem of an ordinary differential equation with parameters 0( and p:
2 ~
~
d C
dC
2
~
DL dx2 - V dx - (0( DT + p)C = 0,
dCI =0
dx x=oo '
(3.2.35)
C(O, 0(, p) = O/p,
where Cis the Fourier transform of C*, i.e.,
~
1 Joo
C(x,O(,p) = F[C*] = M:
exp(iO(y)C*(x,y,p)dy;
v 2n -00
(3.2.36)
and C* is the Laplace transform of solution C of the original problem, i.e.,
C* = L[C] = Ioo e-PtC(x,y,t)dt,
(3.2.37)
and 0 is the Fourier transform of g. After the solution of Eq. (3.2.35) is
obtained, the Laplace and Fourier inverse transforms are used successively,