216
7. Mathematical Models of Groundwater Quality
dispersion coefficients will not reflect their true values in Eq. (7.2.32). Hutton
and Lightfoot proved that if Eq. (7.2.32) is averaged along the vertical direction and transformed into the form of Eq. (7.2.33), the "apparent velocity" V
and "apparent dispersion coefficient" D will both depend on time, and an
additional "apparent source/sink term" appears. The authors thought that
this analysis could explain why the dispersion coefficients depend on the
propagation distance when using Eq. (7.2.33) to fit the test data. Dominieo
and Robbins (1984) got similar results from numerical analysis. Bear and
Veriijt (1987) obtained the two-dimensional advection-dispersion equation
through averaging the three-dimensional advection-dispersion equation along
the vertical direction. From their formulation we can see that the application
of two-dimensional models may enlarge the dispersivities when the physical
problem is three-dimensional.
Killey and Moltyaner (1988), Moltyaner and Killey (1988a, b) reported the
tracer tests conducted in the Twin Lake aquifer of Canada by Chalk River
Nuclear Laboratories. They used a three-dimensional model to fit the observed concentration distribution. The "scale effect" was not found when
the propagation distance reached 40 m. The longitudinal and transverse
dispersivities obtained are almost identical with the results obtained in
the laboratory scale. Jensen et al. (1993) reported that the "best fitting"
dispersivity parameters for the large-scale tests recentiy conducted in Denmark are: r:t. L = 0.45 m and r:t.T = 0.001 m in the horizontal direction, and
r:t. L = 0.05 m and r:t. T = 0.0005 m in the vertical direction.
It is thus clear that, when making an interpretation of the test data, we
must carefully analyze the conditions of the test, find the structure of the
aquifer, and observe the plumes in detail. Then, we can select an appropriate
model to fit the observations. The model should not be arbitrarily simplified
for convenience. Since the three-dimensional numerical model can take all
the practical conditions into account, it is most suitable for interpreting the
results of field experiments.
In Chapter 2, we have mentioned the review paper of Gelhar et al. (1992),
in which 59 different field sites are summarized. The paper presented the
following conclusions:
• After reanalyzing existing experimental data (for example, eliminating the
effect of using one- or two-dimensional models when the spreading was
three-dimensional in nature), the longitudinal dispersivities associated with
high reliability data only ranged from 0.1 to 10 m.
• On a given scale, the longitudinal dispersivity values were found to range
over 2 to 3 orders of magnitude and the higher reliability data tend to fall
in the lower portion of this range. The variations in dispersivity reflect the
influence of differing degrees of aquifer heterogeneity at different sites.
• The ratio of longitudinal to horizontal trans verse dispersivities ranged
from 10 to 50 for high reliability data. The vertieal trans verse dispersivities
are typically an order of magnitude sm aller than horizontal trans verse
dispersivities.
7. Mathematical Models of Groundwater Quality
dispersion coefficients will not reflect their true values in Eq. (7.2.32). Hutton
and Lightfoot proved that if Eq. (7.2.32) is averaged along the vertical direction and transformed into the form of Eq. (7.2.33), the "apparent velocity" V
and "apparent dispersion coefficient" D will both depend on time, and an
additional "apparent source/sink term" appears. The authors thought that
this analysis could explain why the dispersion coefficients depend on the
propagation distance when using Eq. (7.2.33) to fit the test data. Dominieo
and Robbins (1984) got similar results from numerical analysis. Bear and
Veriijt (1987) obtained the two-dimensional advection-dispersion equation
through averaging the three-dimensional advection-dispersion equation along
the vertical direction. From their formulation we can see that the application
of two-dimensional models may enlarge the dispersivities when the physical
problem is three-dimensional.
Killey and Moltyaner (1988), Moltyaner and Killey (1988a, b) reported the
tracer tests conducted in the Twin Lake aquifer of Canada by Chalk River
Nuclear Laboratories. They used a three-dimensional model to fit the observed concentration distribution. The "scale effect" was not found when
the propagation distance reached 40 m. The longitudinal and transverse
dispersivities obtained are almost identical with the results obtained in
the laboratory scale. Jensen et al. (1993) reported that the "best fitting"
dispersivity parameters for the large-scale tests recentiy conducted in Denmark are: r:t. L = 0.45 m and r:t.T = 0.001 m in the horizontal direction, and
r:t. L = 0.05 m and r:t. T = 0.0005 m in the vertical direction.
It is thus clear that, when making an interpretation of the test data, we
must carefully analyze the conditions of the test, find the structure of the
aquifer, and observe the plumes in detail. Then, we can select an appropriate
model to fit the observations. The model should not be arbitrarily simplified
for convenience. Since the three-dimensional numerical model can take all
the practical conditions into account, it is most suitable for interpreting the
results of field experiments.
In Chapter 2, we have mentioned the review paper of Gelhar et al. (1992),
in which 59 different field sites are summarized. The paper presented the
following conclusions:
• After reanalyzing existing experimental data (for example, eliminating the
effect of using one- or two-dimensional models when the spreading was
three-dimensional in nature), the longitudinal dispersivities associated with
high reliability data only ranged from 0.1 to 10 m.
• On a given scale, the longitudinal dispersivity values were found to range
over 2 to 3 orders of magnitude and the higher reliability data tend to fall
in the lower portion of this range. The variations in dispersivity reflect the
influence of differing degrees of aquifer heterogeneity at different sites.
• The ratio of longitudinal to horizontal trans verse dispersivities ranged
from 10 to 50 for high reliability data. The vertieal trans verse dispersivities
are typically an order of magnitude sm aller than horizontal trans verse
dispersivities.