Geoscience Reference
In-Depth Information
rainfall intensity,
Q
p
/
A
=
CP
(9.102)
where
A
is the area of the catchment;
C
is a constant, also called the runoff coefficient,
which ranges between 0 and 1 (see Table 12.2), depending on the nature of the surface.
Although the basic approach was proposed some 150 years ago (Mulvany, 1850; Dooge,
1957), various versions of the Rational Method are still in common use in the design
of road culverts and other structures draining small areas of a few square kilometers at
most. Equation (9.102) suggests that the rainfall loss rate is simply proportional to the
rainfall intensity, and equal to [(1
C
)
P
]. Physically, the assumption of proportional
losses appears to be more compatible with the early stages of rainfall infiltration (see the
second condition of Equations (9.5)) combined with interception losses (see Equations
(3.14) and (3.19)) for short precipitation events. In contrast, the constant loss rate methods
appear to reflect conditions during longer lasting events (see the third of Equations (9.5))
with eventually a near-constant infiltration capacity. It probably also explains the dif-
ferences in the sizes of the catchments for which both methods have been applied in
engineering. The Rational Method is treated in greater detail in Section 12.2.2.
−
9.6
Capillary rise and evaporation at the soil surface
The water that evaporates at the soil surface is transported to the surface through the under-
lying layers of the soil profile. This transport takes place both in the liquid and in the vapor
phase; moreover, as evaporation is driven by radiation and other energy inputs, the transport
involves not only water pressure gradients, but often also temperature gradients with a soil
heat flux. However, as already discussed in Section 8.3.3, in many situations of hydrologic
interest, some important features of the evaporation at the soil surface can be obtained on
the basis of the isothermal flow equation, viz. Darcy's law (8.19). In particular, two flow
problems have been the subject of previous research, that have practical relevance to soil
surface evaporation; these are steady capillary rise from a water table to the surface, and
unsteady desorption from a deep soil profile without a water table.
9.6.1
Steady capillary rise from a water table
This situation occurs when the water table is maintained at a constant level, from which
water flows upward through the soil profile to the soil surface, where it is taken away
by evaporation under constant atmospheric conditions. Under steady conditions in the soil
profile (
∂θ/∂
t
)
=
0, and the vertical flux is everywhere given by
q
z
=
E
. Thus, for a vertical
coordinate system pointing upward with
z
=
0 at the water table where
p
w
=
0, one obtains
from Equation (8.19)
x
=
p
w
1
γ
w
dx
[1
+
E
/
k
(
x
)]
z
=−
(9.103)
0
in which
x
is a dummy variable representing the water pressure. This can be readily inte-
grated for a uniform soil profile, provided the capillary conductivity
k
=
k
(
H
) is known as
a function of the soil water suction
H
(
=−
p
w
/γ
w
). Gardner (1958) presented solutions of
(9.103) with (8.37) for values of the parameter
c
=
1, 3
/
2, 2, 3 and 4.
