236
7. Mathematical Models of Groundwater Quality
a long distance or a long time. In recent years, some field tests are designed
to verify the validity of the new theory. We have mentioned these tests in
Section 7.2.3. In what folIows, we will give a short introduction to the statistic
theory. For a systematical discussion, the reader may refer to Dagan (1989).
New developments in this area can be found in Neuman and Zhang (1990),
Rubin (1990), Rubin (1991), Dagan and Neuman (1991), Rehfeldt and Gelhar
(1992), Russo (1993), Naff (1994), and Kapoor and Gelhar (1994a,b).
7.4.2 The Heterogeneity of Natural Formations
Natural geological formations in local and regional scales are always heterogeneous. It is unrealistic to describe the detail of this kind of porous medium
by any deterministic parameters. In the statistical framework, complex
spatial structures are represented by random functions. For example, the
hydraulic conductivity K(x) of an aquifer can be viewed as a spatial random
field. According to an analysis offield data, Freeze (1975) discovered that the
probability distribution of Y(x) = In K(x) is close to a normal distribution.
This argument was supported by the work of Hoeksema and Kitanidis
(1985), in which data from the literature for about twenty aquifers in the
United States were analyzed.
Anormal random function Y(x) may be completely characterized by its
first two moments: the mathematical expectation (mean), E[Y(x)], and the
covariance function, Cyy(xi, Xj) between two locations Xi and Xj' When Y =
E[Y(x)] is a constant and Cyy(X i , Xj) only depends the separation vector
r = Xi - Xj' the random function Y(x) is said to be stationary or statistie
homogeneous. When the mean Y(x) is not a constant, we can express Y(x) into
two terms:
Y(X) = Y(x) + f(x),
(7.4.2)
where f(x) is called aperturbation. It represents the fluctuation of Y(x)
around its mean Y(x). Since E[f(x)] == 0, f(x) may be considered as a stationary random field.
In the literature, the following stationary three-dimensional covariance
function is often adopted to represent the spatial variability of hydraulic
conductivity:
(7.4.3)
where (1; is the variance of Y = InK, (r 1 ,r 2 ,r 3 ) are the coordinates of r,
1 1 ,1 2 ,1 3 are called eorrelation seales in the three coordinate directions. When
1 1 , 1 2 , 1 3 are not equal, the spatial variability is anisotropie. For the isotropie
case, we have 1 1 = 12 = 13 = Iy , and Eq. (7.4.3) reduces to
Cyy(r) = (1; exp( - r/ly ),
(7.4.4)
where r is the distance of two locations. Correlation length Iy is an important
measure of the heterogeneous scale. When the distance of two locations is
7. Mathematical Models of Groundwater Quality
a long distance or a long time. In recent years, some field tests are designed
to verify the validity of the new theory. We have mentioned these tests in
Section 7.2.3. In what folIows, we will give a short introduction to the statistic
theory. For a systematical discussion, the reader may refer to Dagan (1989).
New developments in this area can be found in Neuman and Zhang (1990),
Rubin (1990), Rubin (1991), Dagan and Neuman (1991), Rehfeldt and Gelhar
(1992), Russo (1993), Naff (1994), and Kapoor and Gelhar (1994a,b).
7.4.2 The Heterogeneity of Natural Formations
Natural geological formations in local and regional scales are always heterogeneous. It is unrealistic to describe the detail of this kind of porous medium
by any deterministic parameters. In the statistical framework, complex
spatial structures are represented by random functions. For example, the
hydraulic conductivity K(x) of an aquifer can be viewed as a spatial random
field. According to an analysis offield data, Freeze (1975) discovered that the
probability distribution of Y(x) = In K(x) is close to a normal distribution.
This argument was supported by the work of Hoeksema and Kitanidis
(1985), in which data from the literature for about twenty aquifers in the
United States were analyzed.
Anormal random function Y(x) may be completely characterized by its
first two moments: the mathematical expectation (mean), E[Y(x)], and the
covariance function, Cyy(xi, Xj) between two locations Xi and Xj' When Y =
E[Y(x)] is a constant and Cyy(X i , Xj) only depends the separation vector
r = Xi - Xj' the random function Y(x) is said to be stationary or statistie
homogeneous. When the mean Y(x) is not a constant, we can express Y(x) into
two terms:
Y(X) = Y(x) + f(x),
(7.4.2)
where f(x) is called aperturbation. It represents the fluctuation of Y(x)
around its mean Y(x). Since E[f(x)] == 0, f(x) may be considered as a stationary random field.
In the literature, the following stationary three-dimensional covariance
function is often adopted to represent the spatial variability of hydraulic
conductivity:
(7.4.3)
where (1; is the variance of Y = InK, (r 1 ,r 2 ,r 3 ) are the coordinates of r,
1 1 ,1 2 ,1 3 are called eorrelation seales in the three coordinate directions. When
1 1 , 1 2 , 1 3 are not equal, the spatial variability is anisotropie. For the isotropie
case, we have 1 1 = 12 = 13 = Iy , and Eq. (7.4.3) reduces to
Cyy(r) = (1; exp( - r/ly ),
(7.4.4)
where r is the distance of two locations. Correlation length Iy is an important
measure of the heterogeneous scale. When the distance of two locations is