Hydrological Sciences-Journal-des Sciences Hydrologiques, 45(5) October 2000
709
The relationship between runoff rate and lag time
and the effects of surface treatments at the plot
scale
B. YU, C. W. ROSE, C. C. A. CIESIOLKA & U. CAKURS
Faculty of Environmental Sciences, Griffith University, Nathan, Queensland 4109, Australia
e-mail: [email protected]
Abstract Rainfall and runoff data collected from plots (108 m2) at 1-min intervals
allow accurate determination of the time difference between peak rainfall intensity and
peak runoff rate. Manning's equation for fully turbulent flows implies that a power
function can be used to characterize the relationship between the lag time and peak
runoff rate. An exponent close to -0.4 would validate the Manning's equation for
overland flow and the coefficient can then be used to estimate the roughness
coefficient (Manning's n) during natural storm events. The relationship between the
observed lag time and peak runoff rate was investigated for three different surface
treatments on a loamy sand: (a) bare fallow, (b) farmers' practice—pineapple beds
with bare furrows constructed across the slope, and (c) furrows covered with mulch
from residues of previous pineapple crops. Manning's equation was found to be
applicable for these treatments and Manning's n is consistently between 0.04 and 0.06.
The relationship between lag time and peak runoff rate can be used to model runoff
hydrographs at the plot scale. The effect of mulch cover on flow resistance was found
to be minimal. Velocity measurements using a dye tracing technique at the same site
showed similar results, although the estimated Manning's n was lower. Resistance to
flow was significantly increased only when fresh pineapple leaves from a growing
pineapple crop were in contact with the soil.
Relation entre écoulement et temps de réponse et effets des états de
surface à l'échelle de la parcelle
Résumé Les données de pluie et d'écoulement recueillies sur des parcelles (108 m")
selon un pas de temps d'une minute permettent de déterminer avec précision le temps
écoulé entre le maximum d'intensité de la pluie et celui de l'écoulement. L'équation de
Manning pour les écoulements turbulents implique que la relation entre ce temps et
l'intensité maximale de l'écoulement est une fonction puissance. Un exposant de l'ordre
de 0,4 validerait l'équation de Manning pour le ruissellement de surface et ce coefficient
pourrait alors être utilisé pour estimer le coefficient de rugosité de Manning (Coefficient
n de Manning) au cours d'événements pluvieux naturels. La relation entre le temps
séparant les maximums et le débit de pointe a été étudiée pour trois états de surface d'un
sol argilo-sableux : (a) jachère, (b) sol cultivé—ou les rides sont construits au travers de
la pente, et (c) labour recouvert d'un paillis provenant d'une précédente culture
d'ananas. L'équation de Manning s'avère applicable à ces différents états de surface et le
coefficient n de Manning est toujours compris entre 0,04 et 0,06. La relation entre
l'espacement des maximums et l'écoulement de pointe peut donc être utilisée pour
modéliser les hydrogrammes à l'échelle de la parcelle. Les effets de la couverture de
paillis sur la résistance à l'écoulement se sont révélés minimes. Sur le même site, les
mesures de vitesse utilisant un traceur coloré ont conduit à des résultats similaires, bien
que l'estimation du coefficient n de Manning ait été plus petite. Seules les feuilles
fraîches en contact avec le sol d'une culture d'ananas en pleine croissance ont significativement augmenté la résistance à l'écoulement.
INTRODUCTION
In recent years, a number of attempts were made to describe and model water-driven
soil erosion based on physical principles and our understanding of the processes
Open for discussion until 1 April 2001
710
B. Yu et al.
involved, for example the Water Erosion Prediction Project (WEPP—Laflen et al,
1991, 1997), European Soil Erosion Model (EUROSEM—Morgan et al, 1998) and
Griffith University Erosion System Template (GUEST—Misra & Rose, 1996; Rose et
al, 1997). Common to all of this new generation of physically-based erosion models is
a construct that soil erosion is a rate-dependent process. Sediment concentration, for
example, is related to either shear stress (Foster, 1982; Foster et al, 1995) or stream
power (Hairsine & Rose, 1992a,b). Nearing et al (1997) provided further support for
the role of stream power in rill erosion. In all cases the determination of flow velocity
and depth are required, and the Manning's equation is commonly used to partition unit
discharge into mean flow depth and velocity. Therefore, to predict the rate of soil loss
for assessment of erosion hazard and planning of erosion control activities, the runoff
rate needs to be predicted from rainfall rate using infiltration models and the kinematic
wave approximation for routing the overland flow downslope (Stone et al, 1995; Yu
et al, 1997a).
Lag time can be defined in seven different ways (Hall, 1984), with the time
difference between peak runoff and the mass centre of rainfall excess being the most
widely used. The lag time between rainfall excess and runoff at the plot or catchment
outlet is a critical parameter in runoff routing models (Mein et al, 1974; Boyd &
Bufill, 1989). Following the early work of Laurenson (1964) and Askew (1970), and
much experience with using flood routing models (Pilgrim, 1987), there is a growing
consensus that the lag time is not a constant but varies inversely with the flow rate
(Pilgrim, 1987; Pilgrim & Cordery, 1992). Variable lag time implies basin nonlinearity. However, there are some examples suggesting that basin linearity is
approached during extreme flood events (Pilgrim, 1976; Bates & Pilgrim, 1986).
Basin lag time is also an important hydrological parameter because it is directly
related to the time of concentration. The latter is needed to use the Rational method for
estimating peak flood discharge. The lag time, like the time of concentration, is often
determined as a power function involving the ratio of a characteristic length and the
square root of the basin slope (Kirpich, 1940; Rowe & Thomas, 1942; Chow, 1962;
NERC, 1975; Watt & Chow, 1985), although the rainfall excess rate has been included
recently in the estimators of the lag time (Papadakis & Kazan, 1987; Aron et al, 1991;
Loukas & Quick, 1996).
Almost all of the early work on lag time or time of concentration was concerned
with small to large watersheds with a lower limit of about 104 m2 in size. At the plot
scale (10~'-102 m ), rainfall simulators were commonly used to produce steady flow
conditions. Among other things, constant runoff rate so generated can be used to
determine Manning's roughness coefficient on a range of land surfaces (Engman,
1986).
At the plot scale, both rainfall and runoff vary greatly in time during natural storm
events. To determine excess rainfall as a function of time can be problematic because
infiltration characteristics have to be assumed and the temporal distribution of rainfall
excess will depend on the particular infiltration equation adopted. Another practical
difficulty with the lag time defined using the mass centre of rainfall excess is the
uncertainty in the runoff rate to which this lag time should be related. Peak rainfall and
runoff rates, in contrast, can be uniquely defined, and it is a reasonable assumption that
peak rainfall excess and peak rainfall occur simultaneously (Yu et al, 1997a). In this
paper, lag time is quantified as the time difference between peak runoff rate recorded
The relationship between runoff rate and lag time and the effects of surface treatments
711
at the plot outlet and peak rainfall intensity, and this lag time is then related to the peak
runoff rate. The lag time determined using field data at the plot scale can be used to
validate the relationship between the hydrological lag time and runoff rate, and
estimate the effective roughness coefficient for Manning's equation for natural storm
events. In addition, the lag time can be used to quantify the storage effect on runoff
rate. The larger the lag time, the greater the attenuation of the runoff rate. Vegetative
cover not only increases the amount of infiltration but also reduces the flow velocity,
lengthens the lag time, and increases the storage effect on runoff rate. Even if the
amount of runoff is the same with different surface treatments, one factor reducing the
rate of soil loss would be the reduced runoff rate.
It is shown herein that simultaneous measurements of rainfall and runoff rates at
the plot scale allow one to define the relationship between lag time and runoff rate, to
quantify the magnitude of the storage effect, and to estimate roughness parameters
such as Manning's n. The approach is illustrated using 1-min rainfall-runoff data
collected from runoff plots at Goomboorian, a pineapple farm in subtropical southeast
Queensland, Australia.
SITE DESCRIPTION, DATA AND METHOD OF ANALYSIS
To examine the relationship between runoff rate and lag time, field data from the
Goomboorian site were used. The site is on a commercial pineapple farm near Gympie
in southeast Queensland, Australia (26°04'S, 152°48'E). This is one of six experimental sites established as part of a multi-country research project on sustainable
agriculture on steeplands in tropical and subtropical environments (Coughlan & Rose,
1997), and this site has produced the best rainfall and runoff data in terms of their
completeness for all treatments. The soil at the site is loamy sand (Typic Eutropept),
and was thought to be so erodible as to preclude agriculture as a suitable land use,
partly due to typical land slopes of 14%. However, the cultivation of pineapples with
modest soil loss has proved possible by using planting beds separated by furrows with
slope <6%, and with attention to safe disposal of runoff. Mean annual rainfall at the
site is 1200 mm with about 70% of rain occurring between November and April. Pan
evaporation at Gympie, some 20 km away, is about 3.8 mm day"1 on average varying
from 5.7 mm day"1 in December to 2.0 mm day"1 in June and July. Runoff plots at the
site all have plot length of 36 m, area 108 m", and slope 5%. Each runoff plot consists
of two pineapple beds and two furrows. The spacing between furrows is 1.5 m. The
study period was from August 1992 to November 1995.
In addition to a bare fallow plot (BP) as a control, two other plots were used to
evaluate the effectiveness of different soil conservation technologies: farmers' practice
(FP), which involves construction of pineapple beds across the slope to reduce furrow
gradient to less than 6%; and an improved conservation methodology which involves
mulching of furrows with residues from previous pineapple crops (mulch cover: MC).
Runoff and sediment leaving these plots passed over a Gerlach trough of low slope (1%)
in which the coarser "bedload" fraction was deposited with the remaining water passing
over a "tipping-bucket" device. The time of bucket tip was recorded by an electronic
data logger, and these data were subsequently converted into runoff rates per unit plot
area in mm h"1 at 1-min intervals. The same data logger was also used to record rainfall
712
B.Yuetal.
rate at the site. A tipping-bucket raingauge with a 200 mm diameter opening was used to
measure the rainfall rate also at 1-min intervals. Methods used for data collection and
management have been described elsewhere (Ciesiolka et al., 1995).
Lag time was determined by visually examining the hyetograph and hydrograph
and pairing the peak in rainfall intensity with the corresponding peak runoff rate. As an
example, Fig. 1 shows the rainfall intensity and runoff rate during a storm event on
22 February 1993. There were two peaks in the rainfall intensity. For the first peak, the
time between peak rainfall intensity and peak runoff rate was 1 min for both the bare
plot (BP) and farmers' practice (FP). No peak in the runoff rate for the event with
mulch cover (MC) could be clearly defined in relation to the first peak in rainfall
intensity, although the lag time associated with the second peak is easily identifiable
for the same plot with MC.
Over 2000 rainfall-runoff events recorded at 1-min intervals from 26 August 1992
to 23 November 1995 were systematically examined to determine the peak in rainfall
intensity and corresponding peak runoff rate for each treatment. An event was
considered to be a period of rainfall of at least 6 min duration, and separated by a
period of at least 30 min from any other events. The duration of the longest event was
1126 min (approximately 19 h), while the duration of the shortest was 6 min.
A large number of events were initially eliminated from further consideration
because they produced little or no runoff for each of the three treatments. It was found
that peak rainfall intensity for an event of at least 20 mm h"1 was required before any
discernible peaks in runoff could be detected. Consequently, over 90% of the original
events were eliminated.
From experience, judgement needs to be exercised in pairing peak rainfall
intensity and runoff rate. No simple algorithm is available to extract automatically all
the eligible peaks in rainfall and runoff. Therefore, all of the remaining 1-min rainfall
and runoff data were visually examined and only those lags between peak rainfall
120
rainfall
100
E
— - — bare fallow
—x—farmers practice
—•— m ubh cover
CD
JO
"5
c
=s
"cô
|c
CO
DC
50
60
Time (min) since 19:40, 22 Feb 1993
Fig. 1 Hyetograph and corresponding hydrographs at the Goomboorian site during a
storm event beginning at 19:41 h, 22 February 1993.
The relationship between runoff rate and lag time and the effects of surface treatments
713
intensity and resultant peak runoff rate were selected that could be defined with
certainty. Given the method of determining the lag time, the error in the lag is 0.5 min
on average. The standard error in runoff rate was 1.1 mm If1 for the site because of the
discrete nature of the tipping-bucket technology, and the error is independent of the
magnitude of the runoff rate being measured (Yu et al, 1997b). Therefore, the relative
error in runoff rate can be large when the latter is low during small runoff events.
Data on the lag time between peak rainfall and peak runoff, Kp, and peak runoff
rate, Qp, were used to fit a power function of the form:
Kp=aQbp
(1)
The analytical reason for such a functional form is given below. A nonlinear curve
fitting method, known as the Levenberg-Marquardt method (Press et al., 1992), was
used to estimate the coefficient, a, and the exponent, b. The commonly used log
transformation followed by a simple linear regression model was not used because
some of the observed lags for the bare plot were zero. In addition, bias in the lag time
is introduced through re-transformation (Duan, 1983; Ferguson, 1986). Model
performance in this paper is characterized by an efficiency measure, E, which is
defined as (Nash & Sutcliffe, 1970):
E =l-^-
(2)
where N is the number of observed lags for the data set, and Kpi and Kp are the
estimated and the average lag times, respectively. The coefficient of efficiency, E, is
commonly used as a measure of model performance in hydrology (e.g. Loague &
Freeze, 1985) and soil sciences (e.g. Risse et al., 1993). For a linear regression model,
the coefficient of efficiency is identical to the familiar r2. Generally speaking, E is
much less than r2. The value of E can vary from 1, when there is a perfect agreement,
to -oo. It has been shown that the coefficient of efficiency is a much superior measure
of "goodness of fit" for model validation purposes in comparison to the familiar r2 or
the coefficient of determination (Willmott, 1981; Legates & McCabe, 1999).
In addition, a set of velocity measurements using the dye tracing technique
(Abrahams et al, 1986) was used to compute Manning's n for flow in the furrows with
different surface conditions employed in pineapple cultivation. These dye velocity
measurements were taken on the same soil at the Goomboorian site in December 1991,
about eight months prior to the systematic rainfall and runoff measurements described
earlier. A rotameter was used to control the rate of inflow, and runoff downstream was
collected so that any reduction in the flow rate due to infiltration could be determined.
The infiltration loss was typically about 30% for these experiments. A factor of 2/3
was used to convert measured surface velocity into mean velocity, and the average of
the upstream and downstream flow rates was used to determine the mean unit
discharge. In a previous study, Manning's n was determined for individual flow
measurements (Rose et al, 1997). In this paper, Manning's n was estimated using all
the measurements for a range of flow rates at a given location. Assuming that the mean
water depth can be used for the hydraulic radius, Manning's formula can be written as:
714
B. Yu et al.
2/3
T/5/3
1/2
(3)
= nV
2
1
]
where q is the unit discharge (m s" ), V and 5 are velocity (m s~ ) and slope,
respectively. Thus, Manning's n can be estimated as the slope of the plot of V5h
against qm sm.
s
AN ANALYTICAL EXPRESSION FOR THE LAG TIME
Consider an impervious plane with a uniform slope, s, and length, L. The continuity
equation at a particular point, x, with x = 0 at the top of the slope, is:
yxDx = Qx
(4)
where Vx and Dx are flow velocity and depth, respectively, and Q is runoff rate per unit
area with a dimension of LT"1. If it is assumed that Manning's equation:
V=-Dll3sU2
(5)
is applicable for overland flow, substitution of Dx from equation (4) into equation (5)
yields:
V,
f
r\
x2/5Q2'5
(6)
n
v
j
In the following theory development, a constant rainfall and runoff rate with steadystate conditions is assumed. This assumption is made even when the theory is applied
to peak rainfall and runoff rates. With this assumption the amount of time it takes for
water to travel from a point, x, on the slope to the outlet of the plot is:
x
rc dx
ax
J,V*v
5
Q-2/5(L3,5-X315)
^
(7)
The lag time, K, can be defined as the average time of travel for the plot, or,
K = jJTxax:
nL
o:
(8)
The lag time is related to the mean flow velocity by:
V=LIK
(9)
Also the amount of water yet to run off the plot, or in other words, the amount of water
temporarily stored on the soil surface per unit area, S, is given by:
1
S=-[DXAX
nL
Q-2,5Q = KQ
(10)
Li
The larger the amount of water stored on the soil surface, the greater the lag time, the
greater the storage effect and hence the greater the attenuation of the peak runoff rate.
The relationship between runoff rate and lag time and the effects of surface treatments
715
The expression for the lag time (equation (8)) was derived for steady-state conditions.
Under unsteady conditions when both rainfall and runoff vary in time, the kinematic
wave equation is still appropriate for simulating overland flows (Singh, 1996). The
storage equation:
— = R-Q
at
(11)
where R is the rainfall excess, is commonly combined with a storage-discharge
relationship where storage is only a function of the outflow:
S = kQm
(12)
to route hydrographs through the catchment (Pilgrim & Cordery, 1992). It is important
to note that k in equation (12), as distinct from K in equation (10), is effectively a
constant independent of Q and k is a function of catchment properties only. Numerical
solutions to equation (11) and equation (12) have been widely implemented in flood
routing programmes such as R O R B (Laurenson & Mein, 1988), Watershed Bounded
Network Model (WBNM—Boyd et al., 1987), and Runoff Analysis and Flow Training
Simulation (RAFTS—Goyen, 1988). For a plane, it can be shown that the storage term
in equation (12) can be written as:
S = *
nL
Q315
(13)
using the M a n n i n g ' s equation (Mein et al, 1974). It can b e seen from equations (8)
and (13) that the lag time for the steady-state condition is identical to the lag time for
unsteady flows with the assumption that storage is a function of outflow only.
Historically, while Manning's equation was invoked to justify the use of a power
function for the discharge-storage relationship (Mein et al., 1974), no attempt was
made to estimate an effective Manning's n based on the storage-discharge relationship
because of many compounding factors at the catchment scale. In fact, the exponent in
equation (12), m, may not even be 3/5 for natural catchments, for there is a growing
consensus that the most appropriate value for m is 0.8 for ungauged catchments
(Pilgrim, 1987).
From a quite different angle, Rose et al. (1983), in an attempt to derive an
approximate solution to the kinematic wave equation, assumed that the unit discharge
is linearly proportional to the distance down the plane:
q(x) = ^
x
(14)
With this assumption, an approximate analytical solution follows:
(
R=0 +
/p
R V TV
An"$
P
\L dg'
P + l 1 a dt
(15)
where parameters a and P define a generic water depth vs runoff relationship:
<2 = ^ -
(16)
B. Yu et al.
716
The analytical solution of Rose et al. (1983) is valid for overland flow with rainfall
excess as any known function of time. For the Manning's equation:
cc =
(17a)
n
and
P= |
(17b)
By inserting equation (17) into equation (15), and inserting equation (13) into
equation (11), the resulting equations are identical. While the assumptions are
apparently quite different, the approximate analytical solution of Rose et al. (1983) to
the kinematic wave equation is identical to that based on the assumption that the
storage is a power function of the outflow only. The latter approach has been widely
adopted for catchment flood routing purposes.
Hence, equation (8) can be regarded as an approximate expression for the lag
time both for natural storm events with variable rainfall and runoff rates and under
steady-state conditions. Clearly, the lag time is a function of the runoff rate as
equation (8) indicates. The lag time obtained is defined herein using the peak runoff
rate, Qp, in equation (8), as K(QP). Whilst it is convenient to use K(QP), one needs to
know how this lag time is related to the lag time measured between peak rainfall and
peak runoff, Kp.
A NUMERICAL RELATIONSHIP BETWEEN Kp AND K(QP)
In order to derive a relationship between the lag time determined using equation (8)
with Q = Qp, and the lag time between peak rainfall and runoff, the storage equation
(equation (11)) in combination with the storage-discharge relationship (equation (12))
was solved numerically to determine the timing and the magnitude of peak runoff for a
range of input conditions. The following input variables were used for this numerical
simulation: peak rainfall excess rate = 1, 2, 5, 10, 20, 50, 100 mm If1; time to peak
rainfall = 30, 60, 120 min; a symmetric triangular pattern for rainfall excess;
Manning's n = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, L = 36 m, s = 0.05. The magnitude of
these input variables was chosen to reflect the prevailing hydrological conditions at the
Goomboorian site. The time interval was set to 100th of the time-to-peak rainfall for
the numerical simulation. The storage equation was solved using the finite-difference
method to determine the peak runoff rate, equation (8) and the calculated peak runoff
rate were used to determine K(QP). It follows from equation (15) that the peak runoff
rate intersects the falling limb of the excess rainfall graph. The timing of the peak
runoff can therefore be determined more precisely by the time at which excess rainfall
rate equals the peak runoff rate. Figure 2 shows a good linear relationship between
K(QP) a °d Kp. The empirical relationship between the two variables is estimated to be:
KP=0.41K(QP)
£ = 0.99
(18)
Hence, the lag time between peak rainfall and runoff is about half of that determined
using equation (8) and peak runoff rate. The relationship is based on the data for Kp up
to 15 min (Fig. 2) to be consistent with the range in the observed lag time between
The relationship between runoff rate and lag time and the effects of surface treatments
5
10
15
20
25
111
30
CO
"-1
Analytical lag time based on peak runoff rate (min)
Fig. 2 The relationship between the computed lag time between peak rainfall and
runoff for idealized storm events (see text) and the lag time determined analytically
using equation (8) and peak runoff rate.
peak rainfall and peak runoff for the site (see the section on results below).
It is supposed that power functions of the form given in equation (1) and
encouraged by equation (8), i.e.:
Kp=aQbp
can fit the field data on the lag time and peak runoff rate. A comparison of equations
(1), (8) and (18) shows that the exponent b in equation (1) close to -0.4 should lend
support for the application of the Manning's equation to overland flow. From
equations (1), (8) and (18), an estimate of the Manning's n is given by:
L
8a
5X
(19)
with the exponent held constant at -0.4 and X = 0.47 from equation (18). On the other
hand, a significant departure of the exponent b from -0.4 would suggest that the
Manning's equation may not be appropriate for overland flow. Instead, for example,
laminar flow may be the dominant flow regime at the site.
In summary, an analytical expression for lag time as a function of the runoff rate,
slope length, slope steepness and Manning's n is derived. The lag time is applicable for
steady flows as well as unsteady flows assuming that the storage is a function of the
outflow only, a common practice for flood routing through the catchment. The lag
time, when evaluated using the peak runoff rate, can be related to the lag time between
peak rainfall excess and peak runoff. If one assumes that peak rainfall excess and peak
rainfall occur simultaneously, measured lag time between peak rainfall and peak
runoff, and measured peak runoff rate can then be used to indicate the magnitude of
the storage effect and the extent to which Manning's equation is of a suitable form to
describe shallow overland flow. If Manning's equation is found to be applicable,
Manning's n can be estimated for runoff plots.
718
B. Yu et al.
RESULTS
Results on the lag time in relation to peak runoff rate are presented in three main
sections. Firstly, each treatment was considered separately using all the available data
on lag time and peak runoff rate. The relationship between these two quantities was
developed for each treatment independently of other treatments. In the second section,
only those events for which peak runoff rate can be clearly and unambiguously defined
for all of the three treatments were considered, so that the effects of different
treatments could be compared. Finally, Manning's n estimated using the data on peak
runoff rate and lag time was compared to that obtained using velocity measurements at
the same site.
Considering all available data
A total of 123 pairs of peak rainfall intensity and corresponding peak runoff rate could
be clearly defined for BP (bare plot). The number of pairs of peak rainfall intensity and
runoff rate for FP (farmers' practice) and MC (with mulch cover) was lower because
the peak runoff rate was lower and more difficult to define for these treatments. In
Fig. 3, power functions are used to describe the relationship between lag time and peak
runoff rate for individual treatments. The lag time varied from 0 up to 6 min for the
bare plot. The power function relationship between peak runoff rate and lag time has a
coefficient of efficiency of 0.59 for the bare plot. Table 1 summarizes the estimated
parameter values and model performance for all three treatments. The power function
fits the data from FP best and MC data worst. For the mulch-covered plot, most of the
observed peak runoff rates were very low, and it appears that the lag time can assume
any value up to 25 min at low runoff rate. The relationship between the peak runoff
rate and the lag time is poorly defined, with the coefficient of efficiency only 0.14. The
problem of defining the lag time when the runoff rate is low is further confounded
because the error in the observed runoff rate at 1-min intervals is of the same order of
magnitude as runoff rate itself when the latter approaches 1.1 mm h"1. The estimated
exponent for the power functions varies around -0.4. When the exponent was held
constant at -0.4, the estimated coefficient, a, turned out to be consistently around 6.67.1 and the corresponding Manning's n using equation (19) assuming a planar slope
would be between 0.043 and 0.049 (Table 1).
Table 1 The relationship between peak runoff rate and lag time for various treatments using all the
available data at the Goomboorian site.
No. of peaks
Average peak rainfall intensity (mm If1)
Average peak runoff rate (mm h"1)
Average lag time (min)
a
b
E
a(b = -0.4)
E(b = -0.4)
Manning's n
Bare fallow
123
59.4
35.4
2.04
10.8 ±1.3
-0.567 ± 0.050
0.59
7.09 ± 0.26
0.54
0.049
Farmers' practice
96
63.7
20.4
3.19
6.29 ± 0.26
-0.340 ± 0.025
0.70
6.59 + 0.23
0.68
0.043
Mulch cover
35
75.8
11.2
5.96
7.01 ± 0.95
-0.247 ±0.125
0.14
6.84 ± 0.93
0.12
0.046
The relationship between runoff rate and lag time and the effects of surface treatments
719
0
{a}: Bare fallow
a-
10. 8±1 .33
b = ~0.567±0.050
£ = 0.59
°\
0 0
0 O^S^O
0
o
o
(b) Farmers practice
a=
6.29±0.26
b = -0.340±0.25
•
E = 0.70
-
—
(c) M u l c h c o v e r
A
a = 7.01±0.95
b = -0.247±0.125
£ = 0.14
i.
A
A^A
A
0
A
A
A\
A
A
A
10
20
30
A
A
40
50
60
70
Peak runoff rate (mm/h)
Fig. 3 The relationship between peak runoff rate and lag time for three different
surface treatments: (a) bare fallow; (b) farmers' practice and (c) mulch cover.
B. Yu et al.
720
Using clearly defined peak values for all treatments
Considering only those events with the peak runoff rate clearly defined for all three
treatments, a more rigorous comparison of the effects of each treatment can be made.
Out of the 123 events with peak runoff rate clearly defined for the bare plot, only 31
events can be used to compare with all treatments (Table 2). As expected, those events
with peak runoff rate clearly defined for all three treatments tend to be larger events
with higher peak rainfall rates. Comparing Tables 1 and 2, the average peak rainfall
intensity of the 31 events is some 34% higher than the original 123 events. The
average peak runoff rate for the 31 events for the bare plot was 51.3 mm h"1, 45%
higher than the average peak runoff rate of the original 123 runoff events. The average
peak runoff rate for the FP plot was 53% of that for the bare plot for these 31 events.
The average peak runoff rate was even lower for the plot with mulch cover, only 24%
of that for the bare plot. With the decrease in the runoff rate, as expected there is a
concomitant increase in the lag time. The increase in the lag time relative to the bare
plot is by a factor of 1.68 for FP, and 3.42 for MC.
Table 2 summarizes estimated parameters and model performance for individual
treatments as well as various combinations of data for different treatments using data
on lag time and peak runoff rate for these 31 events. Figure 4 shows the relationship
between the peak runoff rate and lag time for all three treatments for the 31 events. The
curve shows the best fit of the combined data set with the exponent b held at -0.4. The
power function fits data well for both the BP and FP plots with similar parameter
values. While the coefficient a is not significantly different among different treatments,
Fig. 4 shows that in the flow range of 20-60 mm h"1, the lag time for the mulch-covered
plot is markedly higher than that for the other two treatments, suggesting a higher
roughness coefficient for a given flow rate in this range. As noted in the previous
section, the power function does not fit well the data for the MC plot. However, the
estimated exponent for this treatment, just as for the other two treatments, is not significantly different from -0.4 (Fig. 5). The exponent being close to -0.4 again suggests that
the Manning's equation may still be applicable to the MC treatment, although
Table 2 The relationship between peak runoff rate and lag time for 31 events with concurrent
observations of the lag time for all three treatments. The average peak rainfall intensity was 79.7 mm h"1
for all treatments.
Treatments
BF
EP
Average peak
runoff rate
(mm h'1)
51.3
27.1
Average lag
1.77
time (min)
2.97
a
11.0±2.7
7.52 + 0.52
b
-0.538 ±0.088 -0.400 ±0.041
0.59
0.82
E
a(b = -0.4)
7.45 10.61
7.52 ±0.34
E(b = -0.4)
0.55
0.82
Manning's n
0.053
0.054
BF: bare fallow; FP: farmers' practice; MC: mulch
MC
Combined data
set
(BF + FP)
Combined data
set
(BF + FP + MC)
12.5
39.2
30.3
2.37
7.75 ±0.50
-0.42010.031
0.76
7.50 ±0.30
0.76
0.054
3.60
7.4410.56
-0.378 1 0.054
0.41
7.5010.54
0.41
0.054
6.06
7.58 ±1.10
-0.289 ±0.137
0.17
7.50 ±1.08
0.16
0.054
cover.
The relationship between runoff rate and lag time and the effects of surface treatments
25-
o
n
A
721
Bare fallow
Farmers practice
Mulchcover
20,K = 7 . 5 Q
CD
CO
IS o
5-
K = 0.923
- 1 —
20
40
— I —
100
80
60
140
120
Peak runoff rate (mm/h)
Fig. 4 The relationship between peak runoff rate and lag time for 31 runoff events. For
these events the lag time can be clearly defined for each of the three treatments.
o.o.
-0.1 .
-0.2.
Mulch cover
• C o m b i n e d data set
-0.3-0.4.
o
Q.
X
-0.5.
Farmers practice
111
-0.6-|
Bare f a l b w
-0.7
— .
0
1
10
.
1
20
1
1
30
.
1
1
40
1
50
1—
60
Mean peak runoff rate (mm/h)
Fig. 5 The exponent b in Kp = aQpb as a function of the mean peak runoff rate. The
error bars represent one standard deviation.
the roughness coefficient estimated using equation (19) would contain a fair degree of
uncertainty. Figure 5 also shows that the exponent tends to decrease as the average peak
runoff rate increases. The bare plot has the lowest value of the exponent and MC has the
highest. This decrease in the exponent may be related to the lag time approaching to a
constant at high runoff rate, thus forcing the power functions fitted as in Fig. 3 to have a
sharp turn, so resulting in a lower magnitude of exponent.
722
B. Yu et al.
For BP and FP, Fig. 4 suggests that the lag time appears to approach a constant
value at high runoff rate. A simple linear model between the peak runoff rate and lag
time was used to determine whether or not there is a threshold runoff rate above which
the lag time can be regarded as practically constant. The linear slope of this
relationship is negative and significantly different from zero when the threshold runoff
rate is below 40 mm h"1. Using data for only those events with a peak runoff rate in
excess of 40 mm h"1 (N - 26), the linear slope between the lag time and peak runoff
rate is not significantly different from zero at 0.02 level using the standard f-test. For a
threshold runoff rate of 50 mm h"1 (JV = 21), the slope is not even significantly different
from zero at the 0.1 level. It can therefore be said that a constant lag can be assumed
for practical purposes for these two treatments when the runoff rate exceeds
40-50 mm h"1. The average lag time when the peak runoff rate exceeds 40 mm h'1 is
0.923 min and this average lag time is also shown in Fig. 4 for comparison with the
power function fit. It can be seen that the horizontal line fits these data points with
higher peak runoff rate at a higher peak runoff rate just as well as the power function.
A constant lag time would suggest a linear relationship between storage and runoff rate
(equations (8) and (10)), a considerable simplification for runoff routing purposes. For
runoff and soil erosion modelling, high runoff rate is of particular interest because high
soil loss mostly occurs during storm events with high rates of runoff. Thus, in order to
derive physically-based erosion models a simple linear storage-runoff rate relationship
may be appropriate to estimate runoff rate.
Alternative estimation of Manning's n
An alternative way of estimating Manning's n is to take the ratio of the average lag
time to the average of ÇT ' and to compute Manning's n using equation (19) with this
ratio as the coefficient a. This method is simpler in comparison with the nonlinear
regression method, and the results may even be more robust because only the averages
are used in calculating Manning's n. The obvious disadvantage of this method is that
validity of Manning's formula is assumed a priori, and whether the exponent is equal
to -0.4 cannot be independently tested. Estimated Manning's n using this method is
summarized in Table 3 for different treatments and for different data sets. Comparing
Tables 2 and 3 shows that the average value of n for various treatments is very similar
to that using the nonlinear regression method. The difference in n is about 4% using all
the available data. For the sub-data set with concurrent observations for all three
treatments, the difference in average n is less than 1%. As would be expected, the
Manning's n estimated using this simple method tends to increase with surface cover.
The treatment with furrows covered with mulch has the largest n value in comparison
Table 3 Estimated Manning's n using average peak runoff rate and average lag time.
Treatments
BF
FP
MC
Combined
Full data set:
No. peaks
n
nln (bare)
123
96
35
254
0.045
0.045
0.053
0.047
1
1.00
1.16
-
Subset with concurrent observations for all three
treatments:
n
No. peaks
nln (bare)
31
31
31
93
0.048
0.053
0.061
0.056
1
1.09
1.26
-
The relationship between runoff rate and lag time and the effects of surface treatments
723
with other two treatments with n being smallest for the bare plot. Although this pattern
in Manning's n seems to make physical sense, support for the pattern is equivocal
because the trend did not show up clearly using the nonlinear regression method.
Manning's n using velocity measurements
SI~\
7/'-?
1/7
Figure 6 shows a plot of V against q s using the measured velocity on different
surfaces at the site. The slope of the scatter plot would be the estimated Manning's n
(Table 4). It can be seen that the relationship is similar for essentially bare soil and for
the farmers' practice with varying degree of surface cover and Manning's n of between
0.02 and 0.03. However, the velocity is considerable less at the same unit discharge
when the overhanging pineapple leaves are in contact with the soil surface (Fig. 6).
0.002 - -,'
Fig. 6 The relationship between V3/3 and qm s11 for different surface conditions.
Straight lines represent regression results from Table 4.
Table 4 Estimated Manning's n using measured velocity for given unit discharge.
Location
Measurement date
Surface condition
Slope (%)
No. of measurements
Manning's n
E (= r2)
Width/depth ratio
Within 30 m of the
experimental site
19 December 1991
Farmers' practice; age
of pineapple was 16
months; the surface was
essentially bare with
minimum leaf litter in
the furrows
3.2-3.8
13
0.028 ±0.001
0.89
23-91
At the same site
At the same site
19 December 1991
Farmers' practice; age of
pineapple was 33
months; about 30% of
the surface of the
furrows was covered by
dry pineapple mulched
leaves (leaves broken up)
5
6
0.024 ± 0.001
0.98
19 December 1991
Farmers' practice; age
of pineapple was 33
months; leaves were not
pruned back, resulting in
about 40-45% cover
caused by leaves
contacting the soil
4.7
3
0.176 ±0.012
0.99
15-16
25-tt
724
B. Yu et al.
The estimated Manning's n is an order of magnitude larger than that estimated for the
bare and mulch-covered soils (Table 4). It seems that, as pineapple leaves are broken
up and mulched and the vegetative matter is in close contact with the soil, the
resistance of the mulch is not greatly different from bare soil particles and aggregates.
While such surface contact cover with mulch can be a very effective means of soil
erosion control, even on very steep slopes, as shown by Paningbatan et al. (1995),
overland velocity for a given unit discharge does not vary much from that occurring on
a bare soil surface.
DISCUSSION
Previous investigation at the plot scale involves mostly constant-rate rainfall
simulators. While rainfall simulators allow a greater control over the rainfall intensity
and duration, measurements of rainfall and runoff rates at small time intervals during
natural storm events allow a closer examination of the dynamics of the overland flow.
It was demonstrated that the relationship between lag time and runoff rate can be
particularly useful because such a relationship tests the validity of the Manning's
equation for overland flow during natural events and the same relationship can then be
used to estimate Manning's n. The lag time between peak rainfall and peak runoff has
the advantage over other measures of the lag time involving the mass centre of the
rainfall excess because this lag time can be accurately determined as long as both
rainfall and runoff are measured using the same data logger. The estimated Manning's
n for the soil is broadly consistent with that determined using velocity measurements,
and probably more realistic and useful given that the data on the lag time and peak
runoff rate were collected during natural storm events. Furthermore, as shown
elsewhere (Yu et al, 1997a), the lag time needs to be taken into account when
modelling runoff rate at small temporal and spatial scales. Although a constant lag
time may be adequate when one is most interested in peak runoff rate for soil loss
predictions, a variable lag time as a function of runoff rate would be more desirable
when the runoff rate is particularly low.
There are also certain limitations to this investigation. For example, a plane
geometry was assumed; the actual furrow geometry and its effects on flow rate were
not taken into consideration. The issue in relation to furrow geometry was complicated
because it was not known how the water surface width would vary downslope. In
addition, although the estimated Manning's n of 0.04-0.06 is comparable to 0.02-0.03
using velocity measurements, the extent to which rainfall impacts and furrow geometry
may have affected the estimated Manning's n for natural storm events is not clear.
Finally, the lack of a statistically significant difference in Manning's n between
different treatments (especially as calculated in Table 2) was surprising because it is
commonly believed that mulch cover can greatly retard the flow, hence increase the
lag time. Although there is some support from velocity measurements that the increase
in the roughness due to the mulch at the site is limited unless there are substantial
amounts of live vegetative matter in the furrows, the lack of a significant difference in
the estimated n is also likely to be a result of the poor relationship between peak runoff
rate and the lag time, and the great uncertainty in the estimated n for this particular
treatment.
The relationship between runoff rate and lag time and the effects of surface treatments
725
CONCLUSIONS
Forty months rainfall and runoff data were examined at 1-min intervals and all of the
identifiable peak rainfall intensity and corresponding peak runoff rate values were
extracted. The relationships between peak runoff rate and the lag time between peak
rainfall and runoff suggest that Manning equation is valid for overland flow during
natural rainfall events, and Manning's n was estimated to be around 0,04-0.06 for the
site and for all of the three treatments. The effect on flow resistance of somewhat
decomposed mulch cover associated with farmers' practice is minimal. This
observation is corroborated by velocity measurements made at the same site. Flow
resistance was significantly increased only when fresh pineapple leaves were in contact
with the soil surface.
Acknowledgements Financial support from Australian Centre for International
Agricultural Research (ACIAR) (PN9201 & PN\LWR2\96\216) is gratefully
acknowledged. The authors thank two reviewers for their constructive comments.
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