204
7. Mathematical Models of Groundwater Quality
tion errors. If the observed data are insufficient in quantity and quality,
or attention is not paid to the effect of the errors of model structure, the
parameters obtained may be very different from their true values.
7.2.2 Field Experiments for Determining Dispersivities
We have mentioned the effect of experimental scale on the identified values of
dispersivities in Section 2.5.1. In what folIows, we will introduce various
laboratory and field experiments for determining the longitudinal dispersivity r:J. L and transverse dispersivity r:J.T' These experiments are classified
according to their scales.
The laboratory scale is usually in the range of 10 0 m. In a classical column
experiment, the water with tracer concentration Co is continuously injected
into a column containing sand or other materials of porous media from one
end to displace the original water in the column without tracer. The concentration and volume of emuent water from other end of the column is measured. Since the experimental conditions may be simulated approximately by
Problem 1 or Problem 2 in Section 3.2.1, the observation results can be
interpreted by the following analytical solution:
C(x't)=~erfc[X-VtJ=_1_ [00
exp(-rJ2)drJ.
Co
2
2jD;t ß J(X-vt)/J2"ihI
(7.2.8)
For a given time t, this equation may be represented as 1 - N[(x - m)/oJ,
where N is anormal distributed function. Its mathematical expectation m
and variance (J are
m = Vt and (J = J2DL t,
(7.2.9)
respectively. From the definition of normal distributed function, we have
N(1) ~ 0.84, N( -1) ~ 0.16.
(7.2.l0)
For a fixed time t, the distance between point X O . 84 and point X O . 16 may be
defined as the width e of the transition zone, where the relative concentration
CjC o = 0.84 at point X O .84 and CjCo = 0.16 at point XO. 16 , respectively.
From Eq. (7.2.10), we have e = 2(J. Based on the observed data, we can obtain
an illustration of the relative concentration versus x at a certain time t, as
shown in Figure 7.6. From Figure 7.6, we can find the locations of points
X O . 84 and XO. 1 6' The distance between them is just the width e of the transition zone. Thus, we have
e 1
(J = 2 = 2(XO.16 - XO.84)·
Using Eq. (7.2.9), we then obtain
1
2
DL = -8 (xo 16 - X o 84) .
t '
.
(7.2.11 )
7. Mathematical Models of Groundwater Quality
tion errors. If the observed data are insufficient in quantity and quality,
or attention is not paid to the effect of the errors of model structure, the
parameters obtained may be very different from their true values.
7.2.2 Field Experiments for Determining Dispersivities
We have mentioned the effect of experimental scale on the identified values of
dispersivities in Section 2.5.1. In what folIows, we will introduce various
laboratory and field experiments for determining the longitudinal dispersivity r:J. L and transverse dispersivity r:J.T' These experiments are classified
according to their scales.
The laboratory scale is usually in the range of 10 0 m. In a classical column
experiment, the water with tracer concentration Co is continuously injected
into a column containing sand or other materials of porous media from one
end to displace the original water in the column without tracer. The concentration and volume of emuent water from other end of the column is measured. Since the experimental conditions may be simulated approximately by
Problem 1 or Problem 2 in Section 3.2.1, the observation results can be
interpreted by the following analytical solution:
C(x't)=~erfc[X-VtJ=_1_ [00
exp(-rJ2)drJ.
Co
2
2jD;t ß J(X-vt)/J2"ihI
(7.2.8)
For a given time t, this equation may be represented as 1 - N[(x - m)/oJ,
where N is anormal distributed function. Its mathematical expectation m
and variance (J are
m = Vt and (J = J2DL t,
(7.2.9)
respectively. From the definition of normal distributed function, we have
N(1) ~ 0.84, N( -1) ~ 0.16.
(7.2.l0)
For a fixed time t, the distance between point X O . 84 and point X O . 16 may be
defined as the width e of the transition zone, where the relative concentration
CjC o = 0.84 at point X O .84 and CjCo = 0.16 at point XO. 16 , respectively.
From Eq. (7.2.10), we have e = 2(J. Based on the observed data, we can obtain
an illustration of the relative concentration versus x at a certain time t, as
shown in Figure 7.6. From Figure 7.6, we can find the locations of points
X O . 84 and XO. 1 6' The distance between them is just the width e of the transition zone. Thus, we have
e 1
(J = 2 = 2(XO.16 - XO.84)·
Using Eq. (7.2.9), we then obtain
1
2
DL = -8 (xo 16 - X o 84) .
t '
.
(7.2.11 )