FLOOD05499

• County: Statewide
• Community: State of Colorado
• Basin: Statewide
• Title: Urban Storm Runoff Inlet Hydrograph Study - Volume 5
• Date: 5/1/1975
• Prepared By: Cheng-lung Chen
• Floodplain - Doc Type: Flood Mitigation/Flood Warning/Watershed Restoration

first unsaturated node is controversial because both
Eqs. 3 and 4 do not accurately describe the situation
at the saturation front. Use of Eq. 3 at the saturation
front is only correct when f(t) = r(t), This is the
situation only when t = t , Smith and Woolhiser's
approach in essence is sinillar to the one presented
herein, but is taken without actually solving the
location of the saturation front. Lf. Accuracy of
both the present method and their approach arc
ultimately subjected to the accuracy of the computa-
tion of the f(t) value that can be evaluated by Eq, 10
(or Eq. 10 without Ko in Smith and Woolhiser's
approach), It would be more accurate for the explicit
scheme to compute the exact location of the satura.
tion front such as by using Eq, 17. especially when
the saturation front is close to the soil surface.

Another approach proposed by Fujioka and
Kitamura (1964) assumes a discontinuous propaga.
tion of pore pressure at the saturation front. Horn.
berger and Remson (1970) formulated two internal
moving boundary conditions at the saturation front
basing on that assumption, One of them appears to be
very similar to that obtained by equating Eq, II to
Eq, 16. provided that h(t) in this report can be
regarded as the critical value of pressure head defined
in their study. However, their other internal moving
boundary condition. that IjJ is equal to the critical
value of pressure head. in no way corresponds to the
one defined by Eq, IS. namely IjJ = 0 at the saturation
front. The difference in one of the boundary condi.
tions prescribed at the saturation front was best
explained by Hornberger and Remson as the dif.
ference in the moisture content versus pressure head
relationships that the discontinuous propagation
theory assumes as a first-order discontinuity while the
other theory such as in this study assumes no
existence of such discontinuity. Regardless of
whether or not such discontinuity exists at the
saturation front. the best model will probably be the
one that recognizes an internal boundary (Hornberger
and Remson. 1970).

Regarding the magnitude of the rainfall inten-
sity. r(t). only those which are greater than Ks were
investigated in the present report. However this limit
on r(t) imposed herein can by no means be regarded
as a restriction to the present mathematical model,
Eqs. I through 5, The model can also apply to those
rainfall intensities which are less than Ks, For
example. particularly if r(t)<Ko. the total rate of
change of the moisture content in the soil decreases
and hence the IjJ (e)- and K( e)-relationships in the
drying process should be used instead,

The ponding depth of water. h(t). can be
specified as large (or small) as desired. For example.
Freeze (1969) set a maximum allowable limit on h(t)
while Smith (1972) assumed it always zero, If a

surface runoff model is incorporated with the present
infIltration model for the surface runoff computa.
tion. h(t) has yet to be computed from the surface
runoff model. as illustrated in Smith and Woolhiser's
(1971) study. In the latter case. if the present model
is adopted. h(t) is no longer given. but in fact
becomes part of the solution because of its coupling
with surface water. As a special case, if the soil
surface under consideration is horizontal and of an
infinite areal extent without specifying any upper
limit in h(t). it follows from the mass continuity
principle that

,t
h(t) . J [r(T). f(T)] d' ...' ,(18)
tp

where T is the integration variable for time, L The
differential equation in h(t) corresponding to Eq, 18
can be formulated as

!C~ = r(t) - f(t)
dt

, , , , , , , . ' ,(19)

which is actually the continuity equation for flow of
surface water on the horizontal surface.

Setting an arbitrarily fixed constant value on
h( t) after ponding is physically impossible in reality.
regardless of whether it is zero or not. In the present
study. however. such a hypothetical boundary-value
probiem of rain infiltration was formulated so that
the effect ofh(t) on f(t) could be investigated,

Numerical Model

The finite-difference equation in an explicit
scheme may be formulated by use of specified grid
intervals in the z, t.plane. There are other numerical
schemes (Richtmeyer. 1957) which can be used. In
the present study. the z. t-plane is divided into a mesh
of grid lines with grid or nodal points i = 1. 2. ..'. m
designated along z axis and j = 1.2. ..,. along t axis. as
shown in Figure 2, The interval between the two
distance grid lines is t:;z and that between the two
time grid lines is !:to An association of any given
variable with a given grid point (i. j) in the z. t-plane
will be indicated by subscript. i. and superscript. j.
such as ej, The soil surface will be denoted by i = I.
the wetting front by i = m. and the initial time level
by j = I, A fractional vaJue of subscript or superscript
indicates that the variable under consideration is
evaluated at a point in an indicated fraction of the
way between the two grid points. For instance, i +
1/2 denotes a point halfway between grid points i and
i+ 1.

By following the explicit finite-difference
scheme of Richtmeyer (1957). Eqs, 1.2.3.4. and 6
can be approximated by

18