3.2. Some Canonical Problems Having Analytic Solutions
63
in the longitudinal and transverse directions depend on the dispersion coefficients D L and D T .
Problem 4. Two-dimensional Dispersion of the Tracer in a Unidirectional
Flow Field with Continuous Injection
The basic assumptions in Problem 4 are the same as that in Problem 3,
except that starting at t = 0 the fluid containing the tracer with concentration Co is continuously injected into the aquifer at the origin with rate q.
Since continuous injection may be considered as aseries of transient injections, we can integrate the solution of Problem 3, Eq. (3.2.25), over time t to
obtain the solution of Problem 4 as folIows:
(3.2.27)
Letting
x2
y2
V2
a = D L + D T ; b = 4D L '
(3.2.28)
and using the variable transformation u = a/4y/, Eq. (3.2.27) is transformed
into
C(x, y, t) =
~ exp (2 VX ) foo exp (- u _ ab) du . (3.2.29)
4n DLD T
DL aJ41
U
U
The above equation can be rewritten as
C(x,y,t) =
~exp(;x)[W(O,Jcib) - W(bt,Jcib)]' (3.2.30)
4n DLD T
DL
where
f.
oo
(
r
2
)de
W(u,r) = u exp -e - 4e T
(3.2.31)
is the weIl-known Hantush leaky weIl function. Letting t -.. 00, the asymptotic
concentration distribution of this case is
Coq
(vx) (X 2 V 2 y2V2)
=
exp -
K o - - + - -
2nJD L D T
2DL
4Df
4DLDT '
(3.2.32)
where K o is the zero-order modified Bessel function of the second kind.