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7. Mathematical Models of Groundwater Quality
is not equal to the mathematical expectation of the solution. The same
conc1usion has also been reached by Sagar (1978) from his research on a finite
element groundwater flow model. The inaccuracy of Eq. (7.4.15), which is
based on first-order approximation, is obvious now.
The operator approach described above is a useful tool for moment
analysis. Mc1aughlin and Wood (1988) presented a general perturbation
method for coupled groundwater flow and mass transport problems, in
which mathematical expectation E[], covariance matrix Cepep, and crosscovariance matrix Cepp are obtained by solving the corresponding moment
equations.
In recent years, the uncertainties of stochastic models were considered
by many researchers (Vomvoris and Gelhar, 1990; Dagan, 1990; Li and
McLaughlin, 1991; Kapoor and Gelhar, 1994a). In study the uncertainties of
stochastic models, the local dispersion is often neglected. With the assumption of unidirectional flow, we have the following stochastic equation for
concentration C:
oC
oc ,ac
- + V - + V; - = O.
ot
oX 1
oXi
(7.4.31)
The mean equation of Eq. (7.4.31) is
oe
oe
o2e
-+ V - - VA··-~=O
ot
oX 1
IJ oxßx j .
(7.4.32)
Following Kapoor and Gelhar (1994a), we define S = C 2 . From Eq.
(7.4.31), it is easy to find the following equation:
oS
oS
,oS
-+ V-+ V;-=O.
ot
oX 1
oX i
(7.4.33)
Since
(7.4.34)
we have
(7.4.35a)
and
S' = C 2 + 2eC - (Jr
(7.4.35b)
From Eqs. (7.4.9), (7.4.31) and (7.4.33), the following Fickian relationship can
be obtained:
Substituting Eq. (7.4.35) into the above equation, we then have
- -
o(J2
C 2 v;' = - VAij;;-.
uXj
(7.4.36)
(7.4.37)