Geoscience Reference
In-Depth Information
25
20
n
regression
15
3+2/b
2+2/b
10
5
0
0
1
2
3
4
5
6
1/b
Fig. 8.31 Dependence of
n
in Equation (8.36) on the exponent
b
in the power form of the soil-water
retention relationship (8.14). The points are the experimental values from Mualem's (1978)
data collection. The regression line is
n
=
2.18
+
2.51/
b
. Also shown are the lines obtained with
Equations (8.44) and (8.49). (The parameter
b
tends to be smaller for finer-textured soils.)
Example 8.7. Calculation with the power function
Again, this result can be readily integrated with (8.14) to yield
2Ge[(2
σ/γ
)
θ
0
(1
−
S
r
)
b
]
2
(2
b
+
c
+
2) (
b
+
c
+
2)
H
2
+
c
b
k
=
S
2
+
(2
+
c
)
/
b
e
(8.52)
which is in the form of (8.36) with
n
=
2
+
(2
+
c
)
/
b
. A comparison with available exper-
imental data for the relative permeability collected by Mualem (1978), revealed (Brutsaert,
2000) that a value of
c
=
0.5 produces good agreement, or
2
.
5
b
n
=
2
+
(8.53)
As illustrated in Figure 8.31, Equation (8.53) yields practically the same fit with the data as
the regression relationship
n
=
2.18
+
2.51
/
b
, with a correlation coefficient of
r
=
0.75.
8.4
FIELD EQUATIONS OF MASS AND MOMENTUM
CONSERVATION
8.4.1
Constant-density fluid in a rigid porous material
Equation of continuity
In a porous medium, the infinitesimally small control volume, for which the continuity
equation (1.8) is derived, consists of both pore space and solid matter. Therefore, the
amount of fluid mass per unit volume is given by (
) in the case of water. Similarly,
the mass flux per unit area of bulk porous material, comprising pores and solid matter, is
givenby(
ρ
w
θ
ρ
w
q
), in which
q
(or
q
i
) is the specific flux as used in Darcy's law (8.19). Thus
the equation of continuity (1.8), for a fluid with constant density but variable saturation,
