24
S. C. Lessoff and P. Indelman
soil and the immobile residual water, is a well-known phenomenon (Bresler et al.
1982), that is quantified here in terms of flow and transport parameters. Because
W, K" and Or are all random variables, solute will penetrate to a great depth in
those flow tubes that are transmissive with very low residual moisture content.
Thus when data are available, the model may be able to relate the tails of the
distributions of Or and Ks to the presence of small quantities of tracer at great
depth. Measurements at Bet Dagan showed only trace quantities of solute at depth
(Dror et al. 1999b). These small quantities should have only a minor effect on the
mean concentration.
3.2 Statistical Averaging
We have shown that the vertical and temporal distribution of the solute
concentration C(z, t) depends on the four sets of parameters: soil hydraulic
properties (Ks, Os, 0" f3,), solute reactive properties (A., Kd), conditions of irrigation
(r, Wk ) and injected solute mass(Mo). Generally, these parameters may vary in
space and may be cross-correlated or have a spatial correlation structure. The
mean values of the flow and transport variables are computed by integrating over
the joint probability distribution function (pdf.) of the model parameters (Dagan
and Bresler 1979).
The limited number of sampling depths and times precluded the
characterization of the present transport experiment in terms of spatial and/or
temporal moments of the solute concentrations. Consequently, the present analysis
is restricted to characterizing the measured concentration profiles C(t,z) at a given
elapsed time t in terms of horizontal averages, i.e., the mean Lj Cy(t)INp , where the
subscript i (i= 1 to N 1) is the index of the depth intervals and j (j= 1 to N p ) is the
index of core locations in the horizontal plane. In our experiment N=7 for the
three sampling dates, and N p=28.
In view of the limitations of the input data and the approximations inherent in
the simplified model, we consider variability only in the parameters K" Or, Wk and
Mo. For reasons given below, each of these parameters is treated as an
independent random variable characterized by its pdf. In line with other studies,
we chose Ks and Or for stochastic treatment because they are highly variable and
because steady transport is highly sensitive to their values (Dagan and Bresler
1979; Mallants et al. 1996). The parameters Wk and Mo are treated stochastically
because they are expected to be an important source of concentration variability.
The properties A. and Os are treated as deterministic because we expect that in our
test transport is slightly sensitive to their variability. The parameter fJ is treated as
deterministic because the best available estimates of fJ are its deterministic
effective field value. The parameter Kd was treated as deterministic because it
varies only slightly in the horizontal direction and the effect of variability of Kd
will be strongly dependent on its correlation to flow variables (Cvetkovic et al.
1998), which is not well known.
S. C. Lessoff and P. Indelman
soil and the immobile residual water, is a well-known phenomenon (Bresler et al.
1982), that is quantified here in terms of flow and transport parameters. Because
W, K" and Or are all random variables, solute will penetrate to a great depth in
those flow tubes that are transmissive with very low residual moisture content.
Thus when data are available, the model may be able to relate the tails of the
distributions of Or and Ks to the presence of small quantities of tracer at great
depth. Measurements at Bet Dagan showed only trace quantities of solute at depth
(Dror et al. 1999b). These small quantities should have only a minor effect on the
mean concentration.
3.2 Statistical Averaging
We have shown that the vertical and temporal distribution of the solute
concentration C(z, t) depends on the four sets of parameters: soil hydraulic
properties (Ks, Os, 0" f3,), solute reactive properties (A., Kd), conditions of irrigation
(r, Wk ) and injected solute mass(Mo). Generally, these parameters may vary in
space and may be cross-correlated or have a spatial correlation structure. The
mean values of the flow and transport variables are computed by integrating over
the joint probability distribution function (pdf.) of the model parameters (Dagan
and Bresler 1979).
The limited number of sampling depths and times precluded the
characterization of the present transport experiment in terms of spatial and/or
temporal moments of the solute concentrations. Consequently, the present analysis
is restricted to characterizing the measured concentration profiles C(t,z) at a given
elapsed time t in terms of horizontal averages, i.e., the mean Lj Cy(t)INp , where the
subscript i (i= 1 to N 1) is the index of the depth intervals and j (j= 1 to N p ) is the
index of core locations in the horizontal plane. In our experiment N=7 for the
three sampling dates, and N p=28.
In view of the limitations of the input data and the approximations inherent in
the simplified model, we consider variability only in the parameters K" Or, Wk and
Mo. For reasons given below, each of these parameters is treated as an
independent random variable characterized by its pdf. In line with other studies,
we chose Ks and Or for stochastic treatment because they are highly variable and
because steady transport is highly sensitive to their values (Dagan and Bresler
1979; Mallants et al. 1996). The parameters Wk and Mo are treated stochastically
because they are expected to be an important source of concentration variability.
The properties A. and Os are treated as deterministic because we expect that in our
test transport is slightly sensitive to their variability. The parameter fJ is treated as
deterministic because the best available estimates of fJ are its deterministic
effective field value. The parameter Kd was treated as deterministic because it
varies only slightly in the horizontal direction and the effect of variability of Kd
will be strongly dependent on its correlation to flow variables (Cvetkovic et al.
1998), which is not well known.