2.1 Historical Perspective
Over the years, many attempts have been made to develop models for the analysis of
overland flow. Many engineers, mathematicians and scientists, who have contributed to
the solution of overland flow, have undertaken this challenge. The essential problem to
be solved is to determine the flow off the plane at the downstream end for given physical
conditions and a given pattern of lateral inflow along the plane (Dooge 1973).
Accumulation of excess rainfall allows for water to flow out of the plane; however,
equilibrium conditions can be attained if rainfall excess continues and water begins to
pond on the plane such that inflow is equal to outflow. Once rainfall excess ceases,
equilibrium conditions are lost, water continues to flow out of the plane, and stored
volume begins to deplete back to zero (Ponce 1989).
The St. Venant equations (Ponce 1989) describe the motion of water in overland flow.
These are the equation of continuity (Eq. 1) and the equation of motion (Eq. 2):
𝜕𝑞 𝜕𝑦
+
=𝑖
𝜕𝑥 𝜕𝑡
𝜕𝑢
𝜕𝑢
𝜕𝑦
𝑢
+𝑢
+𝑔
− 𝑔(𝑆0 − 𝑆𝑓 ) + 𝑖 = 0
𝜕𝑡
𝜕𝑥
𝜕𝑥
𝑦
in which q = flow rate per unit width, y = depth of flow, i = lateral inflow (excess
rainfall), u = velocity, g = acceleration due to gravity, So = bottom slope, Sf = friction
slope. These are partial differential equations; however, the nonlinearity of the equation
of motion has defied the search for a solution. The equation of motion is a function of
inertia, pressure gradient, friction, gravity and momentum-source term.
Overland flow modeling has been attempted by Horton (1938), Izzard (1944), Lighthill
and Whitham (1955), Schaake (1965), Wooding (1965), Woolhiser and Liggett (1967),
Dooge (1973), and Ponce (1986). These authors have simplified the equation of motion
to determine an approximation of the behavior of overland flow. The kinematic wave and
diffusion waves have been used to obtain a simplified solution to the overland flow
problem.
Horton (1938) studied surface runoff in reference to soil erosion. In his paper, he explains
that in order for surface runoff to occur, there must be rainfall excess or the difference
between rainfall and infiltration. Moreover, that all existing depressions must be filled to
overflow levels. This suggests that there is a relationship between runoff intensity and
depth of surface detention. This relationship, as expressed by Horton, is simply a power
function (Eq. 3):
𝑞𝑠 = 𝐾𝑠 𝛿 𝑀
In which qs = runoff intensity, Ks = constant dependent on the characteristics of the area,
δ = depth of surface detention along the stream margin, and M = an exponent on the type
of overland flow. Eq. 3 is a storage equation that relates net supply, surface detention and
surface runoff.
Izzard (1944) applied the equation of motion (Eq. 2) to experimental data of paved
surfaces under equilibrium conditions, that is, the data does not include flow from the
rising or falling limbs of the hydrograph. From the studies, Izzard concluded that the
upper end of the surface profile is independent of the overland flow length; thus, it is
appropriate to use detention data for various lengths to develop a relationship between
detention and discharge.
Lighthill and Whitham (1955) introduced the kinematic wave concept, where the solution
arises on the equation of continuity (Eq. 1). The kinematic wave describes the convection
of a quantity. Since the convection equation is of first order, the kinematic wave has only
one wave velocity. This wave velocity is described by Seddon's law (Eq. 4):
𝑐=
1 𝜕𝑄
𝐵 𝜕𝑦
In which B = the local breadth or width of the river.
Schaake (1965) presented a method for synthesizing the hydrograph for paved surfaces in
a watershed. The rainfall-runoff process consists of infiltration, depression storage and
surface runoff. Schaake explains that the synthetic unit hydrograph must be obtained
from knowledge of the physical features of the area. To describe the mechanics of surface
runoff, the equations of gradually varied unsteady flow in open channels (Eq. 1 and 2)
were used. The drainage area was divided into small component parts where the
equations of motion were applied to describe each component. The solution to the
equations was obtained using the finite-difference method. Outflow from one component
becomes inflow for the next, which ultimately represents the overland flow of the entire
area. Schaake's results agreed with those of Izzard (1943).
Wooding (1965) presented an analytical solution for the hydraulic model by the method
of characteristics. The rainfall and infiltration are uniformly distributed over the entire
catchment. The catchment is presented as two rectangular planes joined together to form
a V-shape at which a stream flows and the main characteristics of a catchment are to be
preserved (slope, roughness and flow regime). Wooding's solution is presented as the
kinematic wave approximation to the equations of motion: the continuity equation (Eq. 1)
and the momentum equation is reduced to a depth-discharge relationship similar to that of
Horton's storage equation (Eq. 3).
Woolhiser and Liggett (1967) used a dimensionless approach to developing a solution to
equations (1) and (2). They integrated the governing equations using the finite-difference
method. Comparison with previously performed calculations led to the discovery that
there is no particular rising hydrograph for overland flow, and that the kinematic wave
solution was, for the most part, an accurate solution to the overland flow problem.
Woolhiser and Liggett found that a dimensionless parameter k (Eq. 5) was a suitable
criterion for the choice between the complete equations or the kinematic wave
approximation in which the effects of the length and slope of the plane (channel) can be
related.
𝑘=
𝑆0 𝐿0
𝐻0 𝐹02
In which So = bottom slope, Lo = length of the plane (channel), Ho = normal depth and Fo
= Froude number for normal depth. The objective of the studies presented by Woolhiser
and Liggett was to determine the effects of the dimensionless parameters k and Fo on the
hydrographs. The solutions were compared to that of the kinematic wave solution and the
1946 experimental results of Izzard. When compared to the kinematic wave solutions, the
results showed that for values greater than 10 for parameter k, the kinematic wave
solution was a good approximation (Woolhiser and Liggett 1967).
Dooge (1973) compiled several solutions to overland flow, including that of HortonIzzard, kinematic wave, and Woolhiser and Liggett. The Horton-Izzard approach to the
solution of overland flow modeling is accomplished by replacing the equation of motion
by an outflow-storage relationship. This solution was first applied to natural catchments
by Horton and later applied to paved surfaces by Izzard. The assumption of this method is
that the solution to the whole system is lumped together and treated as a single nonlinear
reservoir.
The kinematic wave solution, on the other hand, is not a lumped solution but rather is
distributed at each point in which a relationship between the flow depth and discharge
can be made. The assumption of the kinematic wave technique to solve the overland flow
problem is that all components of the momentum equation are negligible compared to the
bottom slope and friction slope (Dooge 1973).
Ponce (1989) presented a solution to the equations of motion for modeling of catchment
dynamics that has better convergence properties than the kinematic wave models. Ponce's
solution is the diffusion wave method that matches physical and numerical diffusivities.
This allows for full control of the numerical diffusion and featuring grid independence.
The model is stable and convergent for values of Courant number (Eq. 6) close to 1. The
Courant number is defined as follows:
𝐶=𝑐
Δ𝑡
Δ𝑠
Δ𝑠
In which c = physical celerity, and Δ𝑡 = grid celerity. The diffusion wave method
presented here is independent of grid size; that is, that a coarser grid size resolution can
be used and the solution will still converge.
Orlandini and Rosso (1996) presented a solution to overland flow using the diffusion
wave model, following the model of Ponce (1986). The Muskingum-Cunge method of
variable parameters was used to describe the flow dynamics and a digital elevation model
(DEM) was used to capture the topography and river network structure on storm-flow.
The results obtained through this method provided 98% mass conservation. Moreover, it
was shown that the model was accurate for a wide range of slopes and roughness
characteristics, space-time resolutions, and rainfall excess forcing rates.
2.2 Wave Models
The solution to the equations of motion (Eq. 1 and Eq. 2) can be obtained from the full
dynamic wave method, which is the most accurate representation of free-surface flow.
The full dynamic wave solution to overland flow modeling includes all the terms of the
momentum equation: inertia, pressure gradient, friction, gravity and a momentum-source
term. The full dynamic wave solution does not require linearization; however, it is
complex and failure-prone. An alternative is to resort to various simplifications of the full
equations, in the interest of practicality and/or mathematical tractability (Ponce 1990).
Simplifying the equations, linearizing, and combining into one equation gives rise to
three different methods: (1) kinematic wave, (2) diffusion wave, and (3) diffusion wave
with dynamic component, herein referred to as dynamic wave. Kinematic waves are the
most simplified type of wave; dynamic waves are the most complete. Diffusion waves lie
somewhere in between kinematic and dynamic waves (Ponce 1989).
2.3 Kinematic Wave Modeling
The kinematic wave is obtained from the equation of continuity and the assumption of
uniform flow; thus, the momentum equation reduces to a balance of only gravitational
and frictional forces. The equation of continuity states that the quantity in a small element
of length changes in time at a rate equal to the difference between inflow and outflow
(Lighthill and Whitham 1955).
Uniform flow in open channels is described by the Manning equation:
𝑄=
1 2/3 1/2
𝐴𝑅 𝑆𝑓
𝑛
in which n = Manning's roughness coefficient, A = flow area, R = hydraulic radius,
and Sf = friction slope. The kinematic wave equation is obtained by multiplying the
equation of continuity times the wave celerity, to obtain:
𝜕𝑄
𝜕𝑄
+ (𝛽𝑉)
=0
𝜕𝑡
𝜕𝑥
in which βV = wave celerity. In general, wave celerity varies with discharge, making
Eq. 8 nonlinear (quasilinear). In practice, constant wave celerity may be assumed and the
equation can be solved by analytical and numerical means.
A numerical solution for Eq. 8 can be derived using a first-order-accurate numerical
scheme, in which backward differences is used for both spatial and temporal derivatives.
Using this scheme, the following routing equation is obtained:
𝑛+1
𝑛
𝑄𝑗+1
= 𝐶0 𝑄𝑗𝑛+1 + 𝐶2 𝑄𝑗+1
in which,
𝐶0 =
𝐶
1+𝐶
𝐶2 =
1
1+𝐶
This first-order numerical scheme is stable but extremely diffusive.
Another solution to Eq. 8 can be obtained from applying two first-order schemes, which
are stable for Courant numbers greater than or equal to 1 and less than or equal to 1. The
first scheme (scheme I) is a forward-in-time, backward-in-space solution, stable for
Courant numbers less than or equal to 1. The second scheme (scheme II) is a forward-inspace, backward-in-time solution, stable for Courant numbers greater than or equal to 1.
The routing equation for scheme I is defined by:
𝑛+1
𝑛
𝑄𝑗+1
= 𝐶1 𝑄𝑗𝑛 + 𝐶2 𝑄𝑗+1
in which,
𝐶1 = 𝐶
𝐶2 = 1 − 𝐶
The routing equation for scheme II is:
𝑛+1
𝑄𝑗+1
= 𝐶0 𝑄𝑗𝑛+1 + 𝐶1 𝑄𝑗𝑛
in which,
𝐶0 =
𝐶−1
𝐶
𝐶1 =
1
𝐶
The stability of these methods is contingent upon the Courant number being close to or
equal to 1 while controlling the grid resolution to reduce numerical diffusion. The
kinematic wave’s lack of physical diffusivity is thus applicable to small catchments,
whose calculated hydrograph is translated only and experience minimal diffusion. The
diffusion wave model is a more accurate representation for catchments experiencing
more diffusion.
2.4 Diffusion Wave Modeling
Diffusion wave modeling is an improvement to the kinematic wave modeling technique
due to the addition of the flow depth gradient of the motion equation. The flow depth
gradient explains the naturally occurring diffusion for unsteady free surface flow (Ponce
1989). This method is a function of the friction slope in the planes and the channel slope.
The second-order component of the diffusion wave method is a function of the channel
slope; thus, showing that for very small channel slopes, the diffusion is very large. This
method is thus suitable for more realistic mild channel slopes.
The derivation of the diffusion wave equation is obtained from the assumption of steady
nonuniform flow (friction slope is equal to water surface slope). Unsteady flow using the
diffusion wave method is defined by Eq. 15:
1 2/3
𝑑𝑦 1/2
𝑄 = 𝐴𝑅 (𝑆0 − )
𝑛
𝑑𝑥
in which So - (dy/dx) is the water surface slope and dy/dx represents the natural diffusion
in unsteady open channel flow. From the continuity equation and the assumption of
steady nonuniform flow, the diffusion wave equation is derived:
𝜕𝑄 𝜕𝑄 𝜕𝑄
𝑄0 𝜕 2 𝑄
+
=
𝜕𝑡 𝜕𝐴 𝜕𝑥 2𝐵𝑆0 𝜕𝑥 2
in which the left-hand side of Eq. 16 is the kinematic wave equation, while the partial
derivative term of the right-hand side is the second-order term that describes physical
diffusivity. The coefficient of the second-order term accounts for the channel diffusivity,
also characterized as:
𝑣=
𝑄0
𝑞0
=
2𝐵𝑆0 2𝑆0
The diffusion wave method involves matching numerical to physical diffusivities, and the
solution to Eq. 16 is much like the Muskingum-Cunge method with the addition of the
lateral inflow QL (Eq. 7):
𝑛+1
𝑛
𝑄𝑗+1
= 𝐶0 𝑄𝑗𝑛+1 + 𝐶1 𝑄𝑗𝑛 + 𝐶2 𝑄𝑗+1
+ 𝐶3 𝑄𝐿
in which
𝐶0 =
−1 + 𝐶 + 𝐷
1+𝐶+𝐷
𝐶1 =
1+𝐶−𝐷
1+𝐶+𝐷
𝐶2 =
1−𝐶+𝐷
1+𝐶+𝐷
𝐶3 =
2𝐶
1+𝐶+𝐷
and D is the cell Reynolds number (Eq. 8), a ratio of physical diffusivity to numerical
diffusivity.
𝐷=
𝑞0
𝑆0 𝑐Δ𝑠
The diffusion wave uses physical characteristics to define physical diffusion, while the
kinematic wave counts on numerical diffusion. The diffusion wave method can be
applied to a wider range of problems because most floods waves attenuate slightly due to
physical properties.
2.5 Dynamic Wave Modeling
The kinematic wave equation is derived from the assumption of steady uniform flow, the
diffusion wave from the assumption of steady nonuniform flow and the full dynamic
wave method provides the complete solution to the equations of motion, taking into
account all the terms of the momentum equation. The diffusion wave can be further
improved to include the dynamics of the wave phenomena (Ponce 1986). The diffusion
with dynamic wave (dynamic wave) incorporates the Froude number for neutral stability
1
(𝛽−1)
to the channel diffusivity (Eq. 17). The modified physical diffusivity (hydraulic
dynamic diffusivity) is expressed:
𝑣=
𝑞0
[1 − (𝛽 − 1)2 𝐹 2 ]
2𝑆0
Therefore, the cell Reynolds number can also be modified to account for the Froude
number and β dependence of the physical diffusivity (Ponce 1986):
𝐷=
𝑞0
[1 − (𝛽 − 1)2 𝐹 2 ]
𝑆0 𝑐Δ𝑠
The modified cell Reynolds number (Eq. 25) is added to the diffusion wave routing
equation (Eq. 18), which, under linear conditions, represents a complete description of
the wave dynamics. This formulation is useful only for flows with dynamic wave
properties. Moreover, for a large class of problems of practical interest in hydrology, the
wave scale may well be in the diffusion (or kinematic) range (Ponce 1986).
3. MODEL DESCRIPTION
3.1 Model Features
Eq. 9 represents a complete description of the wave dynamics under linear conditions;
however, useful only if the flow has dynamic wave properties. For comparison, the
kinematic, diffusion and dynamic wave methods were applied to a typical problem of
catchment dynamics. In this example, like Wooding (1965), Ponce uses two rectangular
planes adjacent to a channel, which he calls an open-book schematization. Different
levels of grid resolution were used, from very fine to very coarse. The results showed that
under the kinematic wave method, the peak flows varied significantly while the results of
the diffusion wave were much more convergent. The dynamic wave results were very
much like the diffusion wave results, proving that the wave was in fact a diffusion wave.
The results for the diffusion wave method clearly show grid independence compared to
the widely used kinematic wave method, which is highly dependent on, fine grid size
resolution.
3.2 Model Components
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3.3 Hydrologic Abstraction
xxx CN
3.4 Model Topology
xxx
3.5 Input Description
xxx
3.6 Output Description
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4. ONLINE DEVELOPMENT
4.1 Rationale
The simulation of catchment dynamics using a physically based overland flow model is
well established in hydraulic and hydrologic engineering practice (Ponce 1986). The
diffusion wave model is an improvement over the kinematic wave model because the
former is grid independent, while the latter is not. In fact, it can be shown that the outflow
hydrograph generated by the diffusion wave model does not vary with the choice of grid
size, while the same statement does not hold true for the kinematic wave model.
This correct numerical behavior is due to the fact that the diffusion wave model matches
the physical diffusivity of Hayami with the numerical diffusivity of Cunge (Ponce 1989).
In addition to matching diffusivities, ONLINE OVERLAND minimizes numerical
dispersion by specifying the grid ratio such that the Courant number is equal to 1. This
leads to a simulation that is as numerically and physically accurate as it is possible under
the open-book schematization. The specification of the dynamic hydraulic diffusivity,
replacing the kinematic hydraulic diffusivity, extends the diffusion wave model to the
realm of dynamic waves (Ponce 1991). This assures the physical accuracy of the overland
flow model through a wide range of Vedernikov numbers.
ONLINE OVERLAND uses the diffusion wave model to calculate overland flow in an
open-book schematization, using one book. The appropriate specification of Courant
number and dynamic hydraulic diffusivity assures a simulation that is as physically and
numerically accurate as it is possible in deterministic catchment modeling.
4.2 Language
HTML User interface. PHP model
4.3 Script Development
xxx
4.4 Grid Independence
Grid independence is described as the ability of a numerical model to provide the same
results, regardless of grid size. A model that is grid independent through a wide range of
grid sizes is considered to be a good model, i.e., one that minimizes numerical diffusion
and dispersion. In this section, the performance of the diffusion wave model is compared
with that of the backward-in-time/backward-in-space kinematic wave model due to Li et
al. (1975).
Figure 401 shows three outflow hydrographs generated with the diffusion wave model
using three different grid configurations. A significant feature of the diffusion wave
model is that it sets the space interval in such a way that the Courant number is always
equal to 1, optimizing numerical diffusion while minimizing numerical dispersion. This
figure shows the effect of grid size on the outflow hydrograph. For this example, the
effective rainfall depth is specified as Pe = 24 cm; the rainfall duration is tr = 12 hr;
therefore, the effective rainfall intensity ie = 2 cm/hr. The drainage area is A = 18 ha. For
these conditions, the peak flow is: Qp = ie A = 1 m3/s. The three calculated outflow
hydrographs correspond to the time intervals Δt = 15, 7.5, and 3.75 min, respectively.
The hydrographs are shown to be essentially the same, confirming the grid independence
of the diffusion wave model.
For purposes of comparison, Figure 402 shows three outflow hydrographs generated with
a kinematic wave model that uses backward-in-space/backward-in-time differences,
while keeping the Courant number equal to 1. Following Li et al. (1975), Ponce (1980)
described the general formulation of this model. Unlike the diffusion wave model, Fig.
402 shows that the Li et al. model is somewhat sensitive to the grid size.
Figure 403 shows four outflow hydrographs generated with a backward-inspace/backward-in-time model that specifies both time interval Δt and space interval Δx
as input, which effectively renders the Courant numbers on planes and channel different
than 1. Unlike the previous two models that were based on Courant numbers being equal
to 1, Fig. 403 shows that this third model is markedly sensitive to the grid size, defying
grid independence. The lack of grid independence is made more apparent when the
rainfall depth is reduced from 24 cm to 12 cm, and the rainfall duration is reduced from
12 hr to 6 hr, while keeping the same effective rainfall intensity of 2 cm/hr, as shown in
Fig. 404.
Cutting in half the rainfall intensity while doubling the overland flow area tested the
effect of watershed size on the grid independence of the diffusion wave and Li et al
models. The effective rainfall depth was specified as 6 cm and the rainfall duration as 6
hr, i.e., an effective rainfall intensity of 1 cm/hr. The overland flow area was increased to
36 hectares, thus maintaining a peak flow of 1 m3/s. Figure 405 shows four outflow
hydrographs generated with the Li et al model, showing substantial grid dependence. On
the other hand, Fig. 406 shows three outflow hydrographs generated with the diffusion
wave model, showing essentially the same results, regardless of grid size.
Thus, it is concluded that the diffusion wave model features grid independence through a
wide range of grid sizes, unlike the backward-in-space/backward-in-time kinematic wave
model, which does not. This is due to the proper specification of physical and numerical
parameters in the diffusion wave model. Effectively, the diffusion wave model is
physically responsive to the input bottom slopes, while in fact the kinematic wave model
does not include bottom slope as input.
4.5 Script Testing
Testing of the diffusion wave model was accomplished by varying several parameters, to
examine the behavior and sensitivity of the model to these parameters. The parameters
were: (1) rainfall intensity, (2) rainfall duration, (3) overland flow area, (4) fraction of
area on the left plane, and (5) reference discharge fraction.
Running the model for two different rainfall intensities while maintaining the rest of the
parameters constant tested the effect of rainfall intensity. Figure 411 compares two
hydrographs with rainfall intensities of 2 cm/hr and 4 cm/hr, for a 12-hr duration and a
watershed area of 18 ha. As expected, the peak outflow for the rainfall intensity of 4
cm/hr is twice as large as that corresponding to 2 cm/hr. Likewise, the hydrograph
volume for the rainfall intensity of 4 cm/hr is twice as large as that of the 2 cm/hr.
Figure 412 shows the outflow hydrographs for two rainfall durations, having the same
intensity of 2 cm/hr and a watershed area of 18 ha. The rainfall durations are 6 hr and 12
hr. The peak outflow is the same, while the hydrograph volume for the 12-hr duration is
twice as large as that of the 6-hr duration.
Figure 413 shows the outflow hydrographs for three rainfall durations, having the same
intensity of 1 cm/hr and a watershed area of 36 ha. The rainfall durations are 6 hr, 12 hr,
and 24 hr. The peak outflow is the same, while the hydrograph volume for the 12-hr
duration is twice as large as that of the 6-hr duration, and the hydrograph volume for the
24-hr duration is twice as large as that of the 12-hr duration
Running the model for three watershed areas while maintaining a constant rainfall
intensity and rainfall volume tested the effect of overland flow area. The watershed areas
are 18 ha, 36 ha and 72 ha, with a rainfall intensity of 1 cm/hr and a rainfall volume of
86,400 m3. Figure 414 shows three outflow hydrographs for the three watershed areas. It
is observed that the volume remains constant for all three hydrographs. The volume was
preserved by reducing the rainfall depth and duration as the watershed area increased
correspondingly. For the 36-ha area, a constant rainfall intensity (1 cm/hr) and volume
(86,400 m3/s) results in a peak outflow that is twice as large as that of the 18-ha area.
Likewise, the peak outflow for the 72-ha area is twice as large as that of the 36-ha area.
Figure 421 shows the effect of the fraction of area in the left plane on the outflow
hydrograph. Three fractions are considered: 0.5, 0.75 and 1.00. The rainfall depth (24
cm) and effective rainfall duration (12 hr) were kept constant. It is seen that the
hydrographs have the same peak outflow of 1 m3/s and the same volume of 43,200 m3,
regardless of the contribution from the left plane. However, the timing of the hydrograph
response varies with the contribution from the left plane. As expected, the response from
the 0.5 fraction is faster than that of the 0.75 fraction, and the response of the 0.75
fraction is faster than that of the 1.00 fraction.
The model was tested on the fraction of peak flow used to estimate reference discharge.
The fractions used are 0.5, 0.75 and 1.00, while the rainfall intensity (2 cm/hr) remained
constant by keeping the rainfall depth and rainfall duration constant. Figure 422 shows
that the three outflow hydrographs are essentially the same, having the same peak
outflow (1 m3/s) and the same volume (43,200 m3). This test confirms that the results do
not vary with the choice of reference discharge fraction, at least for this example of a
relatively small watershed (18 ha).
The effect of the rating exponent, β, was tested by running the model for three different
values of β while maintaining a constant rainfall intensity and rainfall volume. Figure 431
shows three outflow hydrographs for the following three values of β: β:= 1.50 for
turbulent flow, β = 2.25 for transitional flow and β = 3.00 for laminar flow. The
hydrographs have the same peak outflow (1 m3/s) and volume (43,200 m3) regardless of
the value of β. The timing of the hydrograph response varies with β. As expected, the
response of the turbulent flow case (β = 1.50) is faster than that of the transitional flow (β
= 2.25); likewise, the response of the transitional flow is faster than that of the laminar
flow (β = 3.00).
Figure 432 shows the effect of bottom slope on the outflow hydrograph. For these runs,
the slopes used were 0.005, 0.001, 0.0005 and 0.0001 on planes and channel, while the
rainfall intensity (2 cm/hr) and watershed area (18 ha) remained constant, corresponding
to a maximum peak outflow of 1 m3/s. All four outflow hydrographs conserved volume
(43,200 m3). The outflow hydrographs for the first three slopes (0.005, 0.001 and 0.0005)
reached the maximum peak outflow. However, the fourth outflow hydrograph (0.0001)
did not reach the maximum peak outflow at the end of rainfall (12 hr). As expected, all
four outflow hydrographs show clearly the effect of bottom slope, that is, the hydrograph
response slows down with a decrease in slope.
Figure 441 shows the effect of curve number on the outflow hydrograph. The curve
numbers used for this run were 100 and 80. The rainfall depth (24 cm) and effective
rainfall duration (12 hr) were kept constant. The outflow hydrograph for curve number
100 reached a peak outflow of 1 m3/s corresponding to a volume of 43,200 m3. For curve
number 80 the outflow hydrograph reached a peak outflow of 0.74 m3/s corresponding to
a volume of 31,979.891 m3. The results show that the ratio of the peak outflows (1 m3/s
and 0.74 m3/s) is 0.74. Moreover, this relationship is also true for the total volumes
(43,200 m3 and 31,979.891 m3) corresponding to the two peak outflows.
The model was also tested on the effect of cumulative rainfall distribution on the outflow
hydrograph. The cumulative rainfall was distributed over 2, 3, 4 and 5 points. The
average rainfall intensity (2 cm/hr) remained constant by keeping the rainfall depth (24
cm) and rainfall duration (12 hr) constant. The dimensionless cumulative time (T*) and
dimensionless cumulative depth (D*) distribution are shown in Table 1 for 2, 3, 4, and 5
points. Figure 451 shows the effect of cumulative rainfall distribution on the outflow
hydrograph. The outflow hydrograph for the 2 point distribution reached a maximum
peak outflow of 1 m3/s. For the 3 point distribution, the outflow hydrograph reached a
maximum peak outflow of 1.25 m3/s with an initial rainfall intensity of 1.25 cm/hr and
then reducing to the second peak of 0.75 m3/s with a rainfall intensity of 0.75 cm/hr.
Likewise, the outflow hydrographs for the 4 and 5 point rainfall distribution behave
similarly, having more than one peak representative of the rainfall intensity distribution.
As expected, the four hydrographs conserved volume (43,200 m3) regardless of the local
rainfall intensity.
Table 1. Cumulative rainfall distribution using 2, 3, 4, and 5 points.
Number of
2
3
4
5
points Np
Dimensionle
ss
0
1
0
1
0
0.5
0.7
1
0
0.083
0.5
cumulative
0.916
1
0.5
5
3
7
time T*
Dimensionle
ss
cumulative
0
1
0
0.62
5
1
0
0.62
0.7
5
5
1
0
0.083
0.66
0.916
3
7
7
1
depth D*
Running the model for three different rainfall intensity distributions and three curve
numbers tested the effect of rainfall distribution and curve number. The rainfall
distributions used are shown in Tables 2, 3 and 4 and the curve numbers used for the runs
were 100, 80 and 60. Figure 461 shows the dimensionless cumulative rainfall distribution
for test number 1. The rainfall intensity started at 2.4 cm/hr for the first two hours,
followed by a decrease to 0.6 cm/hr for two hours, then it ceases for the next four hours,
and again the rainfall intensity increases to 0.6 cm/hr and 2.4 cm/hr at two hour intervals.
The total hyetograph is shown in Figure 461a, followed by the outflow hydrographs for
the three curve numbers tested (100, 80 and 60) in Figure 461b. As shown in Fig. 461b,
this rainfall distribution created two peaks in the outflow hydrograph and, as expected,
the volume for curve number 100 is 43,200 m3, for curve number 80 is 31,979.891 m3
and for curve number 60 is 20,370.307 m3.
Table 2. Cumulative rainfall distribution: Test 1.
Cumulati
0.08
0.16
0.2
0.33
0.41
0.
0.58
0.66
0.7
0.83
0.91
1.
ve time
3
7
5
3
7
5
3
7
5
3
7
0
1
2
3
4
5
6
7
8
9
10
11
12
Discrete
time
Cumulati
0.0
ve
0.2
0.2
0.0
0.05
0
0
0
0
5
0.
0.05
0.2
5
2
rainfall
Discrete
4.
4.8
4.8
1.2
1.2
0
0
0
0
1.2
1.2
4.8
rainfall
8
Figure 462 shows the dimensionless cumulative rainfall distribution for test number 2.
Table 3 is the corresponding rainfall distribution for test number 2. The rainfall intensity
for the first four hours was 0.6 cm/hr, followed by an increase to 1.8 cm/hr for four hours,
and again the rainfall intensity decreases to 0.6 cm/hr for the last four hours. The total
hyetograph is shown in Figure 462a, followed by the outflow hydrographs for the three
curve numbers tested (100, 80 and 60) in Fig. 462b. The volume for the outflow
hydrographs remains the same as in test number 1.
Table 3. Cumulative rainfall distribution: Test 2.
Cumulat
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ive time
83
67
5
33
17
83
67
5
33
17
1
2
3
4
5
6
7
8
9
10
11
12
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.0
0.0
0.0
0.0
5
5
5
5
5
5
5
5
5
5
5
5
1.2
1.2
1.2
1.2
3.6
3.6
3.6
3.6
1.2
1.2
1.2
1.2
0.5
1.0
Discrete
time
Cumulat
ive
rainfall
Discrete
rainfall
Figure 463 shows the dimensionless cumulative rainfall distribution for test number 3.
Table 4 is the corresponding rainfall distribution for test number 3. The rainfall intensity
for the first six hours was 0.4 cm/hr, followed by an increase to 1.2 cm/hr for four hours,
and an increase to 2.4 cm/hr for the last two hours. The total hyetograph is shown in
Figure 463a, followed by the outflow hydrographs for the three curve numbers tested
(100, 80 and 60) in Fig. 463b. The volume for the outflow hydrographs remained the
same for all three tests regardless of how the rainfall intensity was distributed.
Table 4. Cumulative rainfall distribution: Test 3.
Cumulat
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.
83
67
5
33
17
0
0.5
ive time
83
67
5
33
17
Discrete
1
1
2
3
4
5
6
7
8
9
10
11
time
2
Cumulat
0.0
0.0
0.0
0.0
0.0
0.0
ive
0.
0.1
33
33
33
33
33
0.1
0.1
0.1
0.2
33
2
rainfall
Discrete
4.
0.8
0.8
0.8
0.8
0.8
0.8
2.4
2.4
2.4
2.4
4.8
rainfall
8
4.6 Conclusions
The preceding model results show that the diffusion wave overland flow model is well
behaved, and properly responsive to variations in rainfall distribution and hydrologic
abstractions, as represented by the NRCS runoff curve number. The calculated
hydrographs depict the hyetograph variability and the hydrologic abstraction in a correct
and predictable manner.
5. MODEL APPLICATION
5.1 Effect of Bottom Slope
The catchment response to bed slope is examined by varying the bed slope on the planes
and the channel simultaneously. The bed slope on the planes and the channel was varied
by running the model with nine (9) different slopes from mild to steep: (1) 0.0001 (2)
0.0002 (3) 0.0005 (4) 0.001 (5) 0.002 (6) 0.005 (7) 0.01 (8) 0.02 (9) 0.05. Figures 471a,
471b and 471c show the outflow hydrographs for the nine (9) slopes tested. Fig. 471a
shows the detailed rising of the outflow hydrograph for the first 12-hr period. Fig. 471b
shows a 24-hr plot that shows the peaks reached by all the hydrographs. Fig. 471c shows
the complete outflow hydrographs, including the receding limbs. The outflow hydrograph
rises in an asymptotic behavior as the slopes increase. For the very mild slopes (0.0005
and 0.002), the outflow hydrographs did not attain equilibrium outflow (1 m3/s), because
of the slow rise of the hydrograph and the longer time of concentration. On the other side
of the spectrum, the very steep slopes (0.01, 0.02, and 0.05) attained equilibrium outflow
with a fast rise of the hydrograph and a shorter time of concentration, representing the
asymptotic behavior of the kinematic wave.
5.2 Effect of Manning's n
The outflow hydrograph response to the effect of Manning's n on the planes and the
channel is examined. Five Manning's n for the planes and the channel were used to
observe the response of the outflow hydrograph: (1) npl = 0.1, nch = 0.015, (2) npl = 0.2,
nch = 0.030, (3) npl = 0.3, nch = 0.050, (4) npl = 0.4, nch = 0.070, and (5) npl = 0.8, nch =
0.10. Figure 472 shows five outflow hydrographs corresponding to the Manning's n
values selected. For low Manning's n values, the outflow hydrographs have a faster rising
limb, attaining equilibrium outflow (1 m3/s) with a shorter time of concentration. On the
other hand, for high Manning's n values, the outflow hydrographs have a longer time of
concentration and attain equilibrium outflow at a later time. Moreover, for very high
Manning coefficients in the plane and the channel, the outflow hydrograph did not attain
equilibrium outflow. As expected, all five outflow hydrographs show the effect of
Manning's n, such that the hydrograph response is delayed with an increase in Manning's
n.
5.3 Effect of overland flow area and channel length
Figure 473 shows three outflow hydrographs for three different overland flow areas and a
constant rainfall intensity (2 cm/hr). The overland flow areas used were 18 ha, 36 ha, and
72 ha. The plane width was kept constant while the length of the channel was increased
relative to the area (400 m, 800 m, and 1600 m). From Fig. 473 it can be seen that the
outflow hydrographs attained equilibrium outflow after 5 hours for the two larger areas
(36 ha and 72 ha), while the overland flow through the 18 ha area reached equilibrium
conditions 30 minutes before, at 4.5 hours. The 18 ha area attained an outflow of 1 m3/s;
the 36 ha area attained an outflow twice as that of the 18 ha area (2 m3/s); and the 72 ha
area attained an outflow twice as that of the 32 ha area (4 m3/s). Moreover, the outflow
volume increased relative to the areas, which can also be observed through the outflow
hydrographs. As expected, the outflow hydrographs describe well the response to an
increase of overland flow area.
5.4 Effect of rainfall intensity
The outflow hydrograph response to rainfall intensity is tested by varying the rainfall
intensity while keeping all variables constant. The rainfall intensities used were 2 cm/hr,
4 cm/hr and 8 cm/hr. Figure 474 shows the response outflow hydrographs for the three
different rainfall intensities tested on a constant overland flow area (18 ha). All three
outflow hydrographs attained equilibrium conditions: 1 m3/s for the 2 cm/hr rainfall
intensity, 2 m3/s for the 4 cm/hr rainfall intensity and 4 m3/s for the 8 cm/hr rainfall
intensity. The outflow hydrograph response for all three intensities is very similar to the
outflow hydrographs generated by varying the overland flow area (Fig. 473). Equilibrium
conditions were attained at about the same time: 4.5 hours for the two lower intensities (2
cm/hr and 4 cm/hr), while the 8 cm/hr intensity produced an outflow hydrograph which
reached equilibrium conditions 30 minutes after, at 5 hours. The outflow volumes also
increase as the rainfall intensity increases over the 18 ha overland flow area. The outflow
hydrograph response to varying the rainfall intensity accurately shows what is expected.
5.5 Effect of rainfall intensity and overland flow area
The rainfall intensity and overland flow areas are varied to examine the response of the
outflow hydrographs. The rainfall intensities and overland flow areas used were 2 cm/hr
and 18-ha, 4 cm/hr and 9-ha, 8 cm/hr and 4.5-ha. Figure 475a shows a plot of the rising
outflow hydrographs (12 hr) for the series of three rainfall intensities and overland flow
areas tested. Figure 475b shows the complete outflow hydrographs, including the
receding limb. It can be seen that the outflow hydrographs for all three rainfall intensities
and overland flow areas attained equilibrium outflow of 1 m3/s, while conserving the
outflow volume of 43,200 m3. The peak of the outflow hydrograph was kept constant by
reducing the overland flow area while increasing the rainfall intensity, to keep their
product the same. Moreover, the outflow hydrograph for the 8 cm/hr rainfall intensity had
a faster response than that of the 4 cm/hr. Likewise, the 4 cm/hr rainfall intensity outflow
hydrograph had a faster response than that of the 2 cm/hr rainfall intensity outflow
hydrograph.
5.6 Effect of rating exponent β
The outflow hydrograph response to the rating exponent, β, is shown in detail in Figure
476a for the first 12 hours. Four different values for β were used to obtain the outflow
hydrograph response shown in Fig. 476a: β = 1.5 for turbulent flow, β = 2.0 and β = 2.5
for mixed laminar-turbulent flow and β = 3.0 for laminar flow. All four outflow
hydrographs attained maximum peak outflow of 1 m3/s while conserving the outflow
volume of 43,200 m3. It can be observed that for laminar conditions (β = 3) the outflow
hydrograph response was somewhat slower, attaining maximum peak outflow at 11
hours, compared to that of turbulent flow (β = 1.5), which achieved the maximum peak
outflow much faster at 2.5 hours. Likewise, for the mixed laminar-turbulent flows (β =
2.0 and β = 2.5), the outflow hydrograph response was in between those of laminar flow
(β = 3.0) and turbulent flow (β = 1.5).
5.7 Conclusions
The model applications examined here confirm the overall soundness of the diffusion
wave overland flow model. The calculated outflow hydrographs clearly depict the
expected results, obtained by running the model with typical variations in the following
parameters: (1) bottom slope, (2) Manning's coefficient n, (3) overland flow area, (4)
rainfall intensity, (5) rainfall intensity and overland flow area, keeping their product
constant, and (6) rating exponent β. Its is shown that the model is responsive to the
parametric variation and predictable in the results obtained.
6. ANALYSIS
[ Summary and Conclusions ] [ References ] [ Appendices ] • [ Top ] [
Acknowledgements ] [ Introduction ] [ Theoretical Background ] [ Model
Description ] [ Online Development ] [ Model Application ]
6.1 Time of Concentration
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6.2 Model Sensitivity to Physical Parameters
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7. SUMMARY AND CONCLUSIONS
[ References ] [ Appendices ] • [ Top ] [ Acknowledgements ] [ Introduction ] [
Theoretical Background ] [ Model Description ] [ Online Development ] [ Model
Application ] [ Analysis ]
7.1 Summary
xxx
7.2 Summary and Conclusions
xxx
7.3 Recommendations
xxx
REFERENCES
[ Appendices ] • [ Top ] [ Acknowledgements ] [ Introduction ] [ Theoretical
Background ] [ Model Description ] [ Online Development ] [ Model Application
] [ Analysis ] [ Summary and Conclusions ]
Wooding, R.A., 1965. A Hydraulic Model for the Catchment-Stream Problem. Journal of
Hydrology, 3(3). pp. 254-267.
Ponce, V.M., 1986. Diffusion Wave Modeling of Catchment Dynamics Journal of
Hydraulic Engineering, 112(8). pp. 716-727.
Orlandini, S., and R. Rosso, 1996. Diffusion Wave Modeling of Distributed Catchment
Dynamics. Journal of Hydrologic Engineering, 1(3). pp. 103-113.
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xxx
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Dooge, J.C.,1973. Linear Theory of Hydrologic Systems. Lecture 9: Mathematical
Simulation of Surface Flow. USDA Technical Bulletin No. 1468.
Woolhiser, D.A. and J.A. Liggett, 1967. Unsteady, One-Dimensional Flow over a Plane the Rising Hydrograph. Water Resources Research, 23(3). pp. 753-771.
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