62
3. Analytic Solutions of Hydrodynamic Dispersion Equations
thickness. At t = 0, a certain amount of tracer, m, is injected instantaneously
into the aquifer at the origin. Let us find the time-space distribution of the
concentration of the tracer. The mathematical model of this problem is
oC
02C
02C
oC
at = DL ox2 + DT oy2 - V ox'
C(x, y, 0) = 0, (x, y) # (0,0),
f: f: nCdxdy = m,
C(±oo,y,t) =0, t~O,
C(x, ±oo, t) = 0, t ~ 0.
(3.2.24)
where D L and D T are the longitudinal and transverse dispersion coefficients,
respectively; n is the porosity. If the transformation Y = yJDdDT is
adopted, and the observation is made in the moving coordinate system e =
x - Vt, this problem becomes the transient line source problem mentioned
earlier. The solution, therefore, can be directly obtained from Eq. (3.1.23),
C(x,y,t) =
m/n
ex p [- (x - Vt)2 - LJ.
4ntJD L D T
4DLt
4DTt
(3.2.25)
If the effect of the molecular diffusion is neglected, then D L = (XL V and D T =
(XT V can be substituted into this equation to obtain
m/n
[ (x - Vt)2
y2 ]
C(x,y,t) =
exp -
- - - ,
4nVtJ(XL(XT
4(XLVt
4(XTVt
(3.2.26)
where (XL and (XT are the longitudinal and transverse dispersivities, respectively. Figure 3.6 shows that the concentration contours plotted according to
Eq. (3.2.26) vary with time. The plume of the tracer moves forward along the
x axis and its velocity is equal to the average velocity, V. As time proceeds,
the areal extent of the plume increases. The rates at which the plume expands
y
FIGURE 3.6. Variations of concentration contours with time in the case of transient
injection.