7.2. Model Calibration and Parameter Estimation
211
test has been given in Problem 3 in Section 3.2.2. Thus, the analytical solution for this problem is:
C(
)
m/n
[(X - Vt)2 y2 ]
X Y t =
exp -
- - -
"
4nV JaLa T
4aL Vt
4aT Vt '
(7.2.24)
where V is the steady one-dimensional flow velocity. Here, we let the directi on of the flow coincide with axis x and injection weIl B be at the origin of
the coordinates.
Let (X, Y) be the coordinates of an observation weIl and C(X, Y, t) be the
concentration of tracer in that weIl. Substituting the following dimensionless
variables:
tR = Vt/aL ,
a = (X 2 /rii + Y 2 /aLaT)1 / 2,
and
into Eq. (7.2.24), we then obtain the concentration of point (X, Y) as:
where
K = tR,max exp[(a 2 + ti,max)/4tR,max],
tR,max = J4 + a 2 - 2.
(7.2.25)
(7.2.26)
(7.2.27)
(7.2.28)
(7.2.29)
Equation (7.2.28) only depends on a dimensionless parameter a, so a group of
typical curves can be plotted against different values of a, see Figure 7.11.
A logarithmic scale is used in the figure for the abscissa t R •
By making a comparison between the observed curve of N s in Figure 7.10
and the typical curves in Figure 7.11, we can identify the best value of a,
marked as aso Let the coordinates of N s be (X s , 0), then from Eq. (7.2.26) we
FIGURE 7.11. The typical curves ofrelative concentration CR(t R ) with parameter a.