CONSIDERATIONS ABOUT THE WIDTH W IN SWMM MODELING EFFORTS
Prepared by:
1.
Dr. Luis A. Parra, PhD, PE, CPSWQ, ToR, D.WRE.
Director of Water Resources, REC Consultants.
ANTECEDENTS
It is the opinion of the reviewer of the Fanita Ranch project that the method used by this author in the
definition of W for continuous simulation analysis is not valid, and that a new W must be defined in all
areas using only the longest water-path. This author disagrees with the reviewer, specifically in relation
with the previous definition applied to the large magnitude of the sub-catchment areas of this project,
as the designers believe that the parameter selection is sufficiently appropriate considering the existing
knowledge regarding overland flow length L, which is tied to the average width W according to: W·L = A,
with A being the contributing sub-area in the model.
The following Technical Memorandum will explain in detail the logic behind the selection of W in the
Fanita Ranch study, where very large sub-areas were used to determine first hydromodification analysis,
and later, to establish No Net Impact Calculations for Critical Coarse Sediment Yield Analyses.
2.
JUSTIFICATION OF MODELING VARIABLES
2.1
Relationship between Manning’s n, Slope S, Width W and Contributing Area A in the
Governing Equations of Sub-catchment Runoff Modeling
The governing nonlinear differential equation of sheet-flow in a sub-watershed within SWMM is given
by the mass balance equation (3.5) *Reference 1+, parametric in α, defined as α = 1.49·W·S0.5/(A·n) =
1.49·S0.5/(L·n), with L, W and A defined previously, S being the average sheetflow slope and n the
average sheetflow Manning’s coefficient. This means that identical values of α generate identical results.
In other words, and because the contributing area A is the best and most precise defined parameter, for
a given contributing area A, results will be the same as long as W·S0.5/n remains the same (or as long as
S0.5/(L·n) remains constant). It should be noted that higher α generates quicker responses and usually
higher peaks while smaller α generates slower responses and lower peaks (see figure 3-10, [1]).
The previous α explanation is an important fact because the variables W, S, and n are somehow
approximate. For example, even if S is approximately estimated (S is the average sheet-flow slope of the
sub-catchment area) there are large expected variations of S for a large contributing areas in a complex
topography; therefore, it would not be unusual to say that the average slope S can change significantly
within an area. For example, variable slopes from 0.5% up to more than 50% in a 200 acre subwatershed for Fanita Ranch can be observed, which will lead to changes of the average S as large as 9%
to 24% (depending if a geometric average or an arithmetic average is used, and depending on the
relative importance assigned to the slope of the longest water-path). In a similar manner, n can be
justified to be as low as 0.017 or as high as 0.18 (within natural soils, according to [5]), and L (associated
with A/W) can have a justifiable variation of 50 to 500 (according to [1], [3], [6], [8] and [9]).
We believe this preliminary discussion is important because the selection of the parameters in Fanita
Ranch was made to represent the existing conditions in a manner as reliable as possible, according to
the following assumptions:



A conservative value of S (small S) was chosen, initially defined with the typical slope of the
longest water-path because it was defined before the slope analysis was made.
An average value of n was used (0.05 for pervious areas), which corresponds to an average
condition of ¼ gravel and moderate bare soil, 1/3 pasture and average grass, 1/6 dense grasses
and timberland, and ¼ of shrubs and bushes, to represent the high variability of surfaces
according to aerial photography and google satellite images; see [5] for n definition, which is an
approved reference by the Water Board.
A relatively conservative value of L developed by the author of this technical memo was applied,
to (1) standardized the selection of W (W=A/L); (2) provide solid & mathematical bases for the
analysis; (3) to minimize inaccurate and capricious interpretation of W at the convenience of
one side (very conservative) or another (very risky) and (4) to account for changes in length
depending on the percentage of imperviousness and increase the hydrologic response of the
watershed as the imperviousness increases. In this regard, the definition of L is as follows: L =
100·ai + (NRCS-L·UDFCD-L)0.5·(1-ai) = 100·ai + 224·(1-ai) (noticed that NRCS-L = 100 ft; UDFCD-L =
500 ft [1]; geometric mean was used as it is slightly smaller than the average n, so gives more
weight to NRCS-L which shows a more robust researched approach [6], and ai = fraction
impervious. For 100% impervious area, L = 100 ft, and for 100% pervious area L = 224 ft, which
as we will see later in this technical memo is in the conservative side of possible assumptions).
An example is given here for a large sub-area of our project: POC2ExArea-C. It is a 129.31 acre sub-area,
soil type C, with n = 0.05, s = 9%; L = 224 ft (and hence W = 25146 ft). Notice that according to the slope
analysis made in later phases of the project, 31.38 acres have a slope less than 10%, 23.71 acres have a
slope between 10-20%, 53.10 acres have a slope of 20-40%, and 21.12 acres have a slope larger than
40%. For sensitivity purposes, a weighted arithmetic mean average on the slope can justify a slope of
24.4% (instead of the 9% used); a different weighted approach of the existing surfaces can justify a
lower n (50% combination of smooth, rough, moderate soil plus gravel and rock soils; 0% dense grasses,
16.7% shrubs and bushes and 33% average grasses can lead to n = 0.0392 instead of n = 0.05 as used)
and in order to obtain the same continuous simulation results an L value can be adjusted to L = 470 ft,
which generates W = 11985 ft. Notice how W reduces to less than 50% than the original value (because
L more than doubles it) and the results remain the same. It is clear from this example than a much less
conservative approach could have been used to further increase α and accelerate the hydrologic
response (for example with n=0.04, S = 20%, L = 100 ft, α = 0.1666 while we use α = 0.04). Similarly, an
unrealistically conservative approach could have been used (n = 0.07; S = 10%; L = 300 ft, α = 0.022). It is
clear that an intermediate value of 0.04 (closer to the conservative minimum α value of 0.0224 than to
the more speculative value of 0.1666) is an adequate selection for the project.
2.2
Context of the Longest Water-path Approach to Define W
The reviewer suggested the designers to inspect EPA SWMM Reference Manual Volume I [1] to
understand that the proper length to be used in the definition of W is that associated with the longest
water-path, which is also the approach used by the BMP Manual. Although this author agrees that this is
often the case, one must understand the context into which such approach was defined, and used it
accordingly.
In [1], typical areas are of the order of 1 to 2 acres, or at most 5 acres in the original W definition. In this
project, the authors are dealing with much larger areas (for example, an area of 438 acres for POC-2B
alone). Clearly, in order to get to the level of detail of the longest water-path discussion as shown in
Figure 3-8 [1] (open-book approach) or Figure 3-9 [1] (different rectangular shapes represented by a
single main course), there is the need to define close to 200 sub-areas for POC-2B alone, a task clearly
out of scope for an analysis of this type. The reader of this Technical Memorandum should notice that all
examples of the EPA SWMM Reference Manual Volume I [1] are associated with small areas (Table 3-3,
A = 0.92 acres; Figure 3.7, A = 1.67 acres (see page 70); Table 3-7, A = 5 acres; Table 6-4, A = 1 acre;
among others). Therefore, the validity of a dimensionless parameter such as L/W must be understood in
this area context. For much larger areas, such as those analyzed in the Fanita Ranch project, L/W only
made sense in a Hortonian context (see Attached Figure 1 in Appendix 1, prepared by the author, and
based on (a) the classic geomorphologic explanation by Horton ([2], [3]) where the entire network of
channels is analyzed, (b) Figure 8.2.4 of the Handbook of Hydrology *10+ and (c) the author’s
interpretation of Figures 3.8 & 3.11 [1] translated into a large sub-area). As a matter of fact, Ree [3]
explains that in the grassed watershed operated by the Stillwater Hydraulic Laboratory the total length
of all drainageways was about 48,900 ft for a contributing area of 206 acres and hence L = 92 ft. In this
light, the approach used in Fanita Ranch is rather conservative, because, for example, a length of about
26,800 ft was used for an area of 138 acres for POC-2A, which is less length per unit area than the
detailed example presented in [3], even when considering that the 206 acres area in question is (a)
vegetated and (b) has a smaller slope than the Fanita project. Those factors typically reduce the initial
setting of the dendritic pattern of drainage (in other words, longer drainageways are expected in poorlyvegetated, high slope areas such as those of Fanita Ranch than in flatter and vegetated areas such as
the area own by the Stillwater Hydraulic Laboratory).
Also, in terms of following the recommendations of the BMP Manual, this author, was the also the
author of the first study made in the County of San Diego to define SWMM parameters [4], from which
the County of San Diego and the City of San Diego BMP Manuals took the parameter recommendations
in Appendix G for hydromodification modeling using SWMM [personal communication from Dr. Tavakol,
formerly at Geosyntec and in charge of SWMM parameter selection for the BMP Manual, and now
Assistant Professor at SDSU]. The context in which Parra & Ponce [4] defined the longest water-path to
determine W was understood in the same context into which reference [1] worked, which is for
relatively simple and small watersheds. Also, the reader should understand that the original work of
Parra & Ponce was the pioneer work on SWMM in the entire County of San Diego, prior to the
experience accumulated since, and was supposed to be a first attempt to define SWMM parameters.
Few parametric improvements have occurred since then (See Walker job in regards to n coefficient [5],
for example), and to this author’s knowledge, not further discussion has occurred into how to improve
W, which according to [1] can also be seen as a calibration parameter. Consequently, it bears no logic
that the original recommendation of this author to provide a proper determination of W was valid
before when the BMP Manual was assembled, and it is still valid for the current versions of the BMP
Manual, but is invalid now in the reviewer’s opinion, in regards to a larger area such as Fanita Ranch.
2.3
Further Discussion and References to Justify L Selection
The selection of a limited L value (which determines a W that can be very large for large sub-catchments
as W = A/L) is supported by the following references and/or discussions:




NRCS modified TR-55 to limit sheet-flow to 100 ft, from an original maximum value of 300 ft,
after an analysis of 12 references in the subject (see attached NRCS study in Appendix 1, at the
end of this Memorandum, [6]). It should be noted that NRCS recommendation is one of the 2
mentioned in the SWMM Manual [1].
Sheetflow distances of up to 100 ft are standard practice in the San Diego region, and 100 ft is
the maximum recommended length in the San Diego County Hydrology Manual [7]
The other reference used in the SWMM Manual *1+ is the Denver’s Urban Drainage and Flood
Control District Reference [8] which is based in equation 6-3 of that manual, calibrated for
Denver’s use. As a matter of fact, *8+ warns in page 6-9 “use of these coefficients and this
procedure outside of the semi-arid climate found in the Denver region may not be valid”.
Therefore, little value should be given to this recommendation that nonetheless was taken into
account to develop a maximum L value.
Another advanced approach for L selection is found in the Hydrology National Engineering
Handbook (USDA, NRCS), chapter 15 [9]. This reference establishes L upper limit according to
equation (15-9) and presents values of L in Table 15-2 as small as 12.5 ft and as long as 172 ft,
with an average of 61 ft. Interestingly, the substitution of this equation on equation 15-8 gives
us Tt (min) = 16.7/P20.5, independent of S and n. Knowing that in Fanita Ranch P2 (2yr, 24 hr
precipitation) is 2 inches, the maximum overland travel time is only 11.8 min, much less than
the reporting time interval (1 hr) for Flow Duration Curve analysis.
3.
CONCLUSIONS


The selection of W cannot be translated into a rigid determination of area divided by length of
longer water-path, especially for large areas in excess of 2 acres where many patters of drainage
appear. W is rather a conceptual width such that when it is multiplied by the average length of
overland flow generates the contributing area. W and L are important because most of the
infiltration process occurs there, before the flow is channelized, and because once overland flow
concludes, travel time is dominated by fast-moving water along the drainage dendritic patter
even in natural terrain.
The values selected in Fanita Ranch are realistic and supported by many sources, most of them
mentioned here. Also, considering the fact that runoff response is actually a function of a
combined parameter α (which also involves Manning’s n, and in minor proportion the overland
slope S), the same α parameter can be justified even if W is reduced by a factor of 2 by more
accurately accounting for S and using a less conservative assumptions to define n. As a
consequence, it is concluded that the parameters selected in the study are representative
because they do not depict neither the large possible value of α nor the smallest possible value,
but a realistic and in the conservative side middle value in between.
4.
REFERENCES
1. EPA: Storm Water Management Model Reference Manual. Volume I, Hydrology (Revised). As of
7/12/2018, available at: https://nepis.epa.gov/Exe/ZyPDF.cgi?Dockey=P100NYRA.txt
2. Horton, R. E. “Erosional Development of Streams and Their Drainage Basins: Hydrophysical
Approach to Quantitative Morphology”, Geol. Soc. Am. Bull. 56, pp. 275-370, 1945.
3. W. O. Ree, A Progress Report on Overland Flow Studies, USDA, Agricultural Research Service,
Southern Plains Branch, 1963.
4. L. Parra, V. Ponce, T. Walker “Review and Analysis of San Diego County Hydromodification
Management Plsn (HMP): Assumptions, Criteria, Methods, & Modeling Tools”. TRWE
Engineering, May 30, 2012. Prepared for San Marcos, Oceanside and Vista.
5. TRWE, 2016: Improving Accuracy in Continuous Simulation Modeling: Guidance for Selection
Pervious Overland Flow Manning’s n Values in the San Diego Region.
6. References on Time of Concentration with Respect to Sheet Flow: W. Merkel, USDA, NRCS,
NW&CC. 2001. https://www.nrcs.usda.gov/Internet/FSE_DOCUMENTS/16/stelprdb1043054.doc
(available as of 7/12/2018 and included here as Appendix 1).
7. San Diego County Hydrology Manual, DPW, FCS. June 2003. As of 7/12/2018, available at:
https://www.sandiegocounty.gov/content/dam/sdc/dpw/FLOOD_CONTROL/floodcontroldocuments/hydro-hydrologymanual.pdf
8. Urban Storm Drainage Criteria Manual. Volume 1. Published 1969; Updated 2016. Urban
Drainage and Flood Control District. Denver, Colorado, 80211. As of 7/12/2018, available at:
https://udfcd.org/wp-content/uploads/uploads/vol1%20criteria%20manual/USDCM%20Volume%201.pdf
9. Part 630 Hydrology National Engineering Handbook. Chapter 15. As of 7/12/2018, available at:
https://www.wcc.nrcs.usda.gov/ftpref/wntsc/H&H/NEHhydrology/ch15.pdf
10. Handbook of Hydrology. David Maidment, Editor. McGraw Hill, Inc. 1992.
5.
APPENDIX LIST


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REFERENCE [6] (PDF of the Document)
References on time of concentration with respect to sheet flow
William Merkel, Hydraulic Engineer
USDA, NRCS, National Water and Climate Center
Beltsville, MD
December 17, 2001
Introduction
Certain references found in the technical literature were reviewed with their statements concerning sheet
flow characteristics. Concern was expressed that the maximum sheet flow length in TR-55 was reduced
from 300 feet to 100 feet in the recently developed Windows TR-55 software system. Technical
information to justify the revision was requested. Though only one of the studies focused on the length of
sheet flow (McCuen and Spiess), several give useful insights on the appropriate length, roughness, and
depth related to sheet flow. For the following references, some direct quotes from the papers are included
along with narrative comments. After a review of the papers, a brief summary of this investigation and
some general conclusions are offered for consideration.
Woodward and Welle, NRCS, Northeast NTC, Hydrology Technical Note N4, 1986.
Quote:
The Manning-kinematic solution is sound, defensible, and easy to use. Therefore, it is recommended that
this equation be used to compute Tt for the overland flow segment. The maximum flow length of 300’
with a most likely length of 100’ should be used in overland flow computations for unpaved areas. Paved
areas may have longer lengths of sheet flow until flow becomes channelized in gutters or low areas of
parking lots. The range of mean depth is 0.002’ for paved areas to 0.02’ for vegetated areas.
Narrative:
One approach to estimating the length of sheet flow may be to define a maximum depth for which sheet
flow applies. With respect to depth, the Manning equation may be used to estimate depth based on
discharge, Manning n and slope (assuming a one foot width). There are several ways to estimate the
discharge at the end of a selected length of sheet flow. A set of 12 references is included with this paper,
which presumes that the statement was based on those references (though the authors did not state which
specific reference).
W. O. Ree, A Progress Report on Overland Flow Studies, USDA, Agricultural Research Service,
Southern Plains Branch, 1963.
Quote:
Overland flow occurs in every watershed to some degree. Whether it is extensive enough or the flow
length great enough to influence the hydrograph needs to be determined. One method for estimating the
average length of overland flow was developed by Robert E. Horton. In his classic work on
geomorphology, he shows that the average length of overland flow, lo, can be estimated by the relation:
lo = 1 / ( 2 Dd )
(1)
Where Dd is the drainage density defined as:
Dd = sum of stream lengths for the watershed / area of the watershed
(2)
Thus, if the length of all channels that are fed directly by overland flow can be measured, it is possible
to estimate the average overland flow length. An attempt was made to do this for the 206-acre grassed
watershed operated by the Stillwater Hydraulic Laboratory. Every waterway visible on an aerial
photograph of the watershed was carefully measured. It was found that the sum of the lengths of all
drainageways was about 48,900 feet. This value and the value for the area were substituted into equations
1 and 2 and the average length of overland flow was calculated to be 92 feet.
The determination of what constituted a drainageway was quite a subjective one. By searching for still
smaller channels it would be possible to produce a different estimate of lo. Nevertheless, it is thought that
the estimate made does indicate the order of magnitude of overland flow and that it is large enough to have
an important role in the determination of the hydrograph of outflow.
Narrative:
The calculation of overland flow (sheet flow) length by equation 1 requires the sum of stream lengths and
area of the watershed to be in consistent units. For example, if Dd is 48,900 feet, the area of the watershed
also should be in square feet or 206 acres times 43,560 square feet per acre. These values result in the
average overland (sheet) flow length of 92 feet.
Even though this relationship was developed using geomorphic data, it also has a physical interpretation. If
all flow concentrations are identified on a map of the watershed, lo represents the average maximum
distance from locations in the watershed to the nearest flow concentration. This applies to distances from
points along the watershed boundary to the nearest flow concentration as well as points within the
watershed which are between flow concentrations. This average value of sheet flow length may vary
according to the definition of flow concentration and how it is measured. It also represents a watershed
average value and not the sheet flow length which falls on the path used to calculate time of concentration.
This procedure is most applicable in small undeveloped watersheds with a relatively homogeneous
drainage network. Using this procedure in developed or partially developed areas has limitations in that the
density of identified flow concentrations may vary significantly within the watershed (for example, dense
in developed and wide-spread in undeveloped areas). The “average” value may not have practical meaning
because of this variability. Dividing the watershed into subwatersheds could be considered in such cases.
The practical use of this procedure is related to getting a general estimate of sheet flow length in a
watershed. As an alternative, after the flow concentrations are identified, sheet flow distances may be
measured at several locations in a watershed to get a similar general idea or estimate. Another use of the
procedure would be to develop estimates of average sheet flow length for various geomorphic regions or
urban development characteristics.
Quote:
Supply rate
Inch/hr
1
2
3
4
Depth at outflow
Inches
0.26
0.39
0.50
0.59
Narrative:
The table above was developed using an overland flow equation developed by Horton which estimates the
depth of overland flow based on Manning n, slope, supply rate (rainfall intensity), and overland flow
length. This gives an order of magnitude of the depth of overland flow of 0.59 inch (0.05 feet) for a 4 inch
per hour rainfall intensity.
Quote:
Channel research at Stillwater, Oklahoma, has included a study of low flows. These were conducted in
flat-bottomed channels 3 feet wide and 96 feet long, with a bed slope of 5 percent. The sides of the
channels were low concrete curbs about 0.2 foot high. While these tests were made primarily in connection
with the studies of the hydraulics of vegetation-lined waterways, the low flow data may be useful in the
investigation of overland flow problems.
Narrative:
The report includes results of these tests for different kinds of grass cover and cover density. Results of
one test were plotted which show relationship of discharge to depth at the end of the 96 foot slope. A
discharge of 0.01 cfs / foot of width had a depth of 0.05 feet.
Engman, E. T., Roughness Coefficients for Routing Surface Runoff, ASCE Hydraulics Division
Conference, Frontiers in Hydraulic Engineering, August, 1983.
Quotes:
Data used in this study were collected on plots to evaluate erosion rates and volumes for different soils and
management practices by the Agricultural Research Service-USDA stations in West Lafayette, Indiana,
Oxford, Mississippi, and Tucson, Arizona. The plots were typical of those used in erosion research and
varied in length from about 10 to 20 m and in width from about 1.7 to 4 m. Simulated rainfall was applied
to the plots at a constant intensity that varied from plot to plot and location but were generally was between
5 to 10 cm / hr. Runoff was measured with a flume and continuous stage recorder that provided accurate
timing and the shape of the hydrograph.
In using these roughness values one should be aware of two potential limitations: (1) These values are
valid for so-called sheet flow or overland flow before significant channelization occurs. Thus these data
will be valid for relatively short slopes. Exactly what length of slope these values will be valid for is
unknown at this time. However, as slopes approach 50 to 100 meters in length one would expect
channelization to begin or else very large and unreasonable depths of overland flow would be calculated;
(2) These values include the effect of rain drop impact which tends to increase the effective roughness.
Narrative:
The author includes a table of Manning n values for sheet flow surfaces. Some values from that table are
included in Table 3-1 of TR-55 (1986 printing). The author states these values of Manning n apply to
relatively short slopes. They were developed on plots of 10 to 20 meters in length and so have that limited
applicability. With respect to depth, high Manning n for dense grass and woods would definitely produce
an unreasonable depth if the sheet flow length were extended to 300 feet. This reference reinforces the case
for considering both length and depth of sheet flow simultaneously.
Engman, E. T., Roughness Coefficients for Routing Surface Runoff, ASCE Journal of Irrigation and
Drainage, Vol. 112, No. 1, pages 39-49, February 1986.
Quote:
Information for choosing roughness coefficients is fairly common for streams, channels and canals.
Typically, Manning’s n values can be estimated with guidance from descriptive information and
photographs. However, very little information is available for shallow depth of overland flow over natural
surfaces.
Narrative:
This paper is related to the reference by Engman above. It contains a summary of a literature review
concerning overland flow and additional analyses of plot data. The published table of Manning n values is
somewhat different from the 1983 paper. It contains some different cover, tillage practices, and surface
residue categories as well as some changes in recommended Manning n for some surfaces.
The author reports the mean, standard deviation, and range of Manning n determined for each cover type
and the number of experiments with each cover type. The range of values indicate Manning n can vary by
a factor of about 1.5 to 3 or greater for most surface types. The sheet flow travel time will thus have a
possible wide range based on the estimate of Manning n for the surface. The estimated depth of flow could
also have a wide range of values based on the Manning n selected. TR-55 uses a single value of Manning n
for each cover type. It masks the variability of the underlying data. Using the mean of the various cover
and tillage conditions, the travel time value which is calculated could have an error of plus or minus 25 to
50 percent (assuming the cover type and condition are selected properly).
Quote:
The depth of calculated flow should not become too large. On long flow planes, the routing models may
calculate depths that may be unrealistically large. The users must be aware of this and limit flow plane
lengths. It appears that excessive depths would not be encountered if slope lengths are on the order of 150300 feet (50-100 m).
Narrative:
This quote is similar to that of the other paper by Engman above. However, the depth should be estimated
and length adjusted accordingly. Appropriate depths of overland or sheet flow are addressed in other
reviewed papers.
Parsons, D. A., Depths of Overland Flow, USDA-SCS, SCS-TP-82, July 1949.
Quotes:
Runoff was measured from an area 6 feet wide by 47.5 feet long with sheet metal boundaries. The soil was
Decatur clay loam from north Alabama. It was bare and had been considerably compacted.
One of the 50-foot tilting plots was also used in determination of overland flow depths on an excellent
stand and growth of Korean lespedeza. The vegetation was green and nearly mature.
Narrative:
Several more types of cover and shorter plot lengths (and a few longer plots) were investigated. Results of
this study are again primarily limited to short slopes and extending these results to long slopes is
questionable.
Quote:
The formation of rills and gullies, in effect, shortens the distance, L, of overland flow. Instead of
beginning at the upper boundary and increasing progressively until it reaches the lower boundary, the flow
originates at the upper boundary, or divides between rills, and ends, in part, in concentrated flow uphill
from the lower boundary. The concentrated flows in these channels may be much greater in depth than the
average, but the relative area of the land that is covered by these flows is sufficiently small for this factor to
be outweighed by the effect of shortening the distance of overland flow. The mere presence of rills or flow
concentrations rather than their depth seems to be the more important influence.
Narrative:
The authors found some flow concentrations before the end of the 50 foot long plots. Data were analyzed
as if all flow were overland (sheet) flow. Special analytical treatment of flow concentrations was not
reported.
Quote:
When  = 1, the time of concentration, tc, is
tc =
4
43200 D
-- * -----------3
rf
Narrative:
Based on their field experiments, the authors developed an equation for time of concentration (tc), which in
the current context is travel time for sheet flow, based on slope parameters. These include average depth of
flow (D) in feet and runoff rate (rf) in inches per hour. The units for tc in the above equation are seconds.
Interpretations of how D and rf are estimated are contained in the report. The average depth of overland
flow is dependent on the slope, length, and roughness as well as the discharge or runoff rate. The meaning
of the symbol  = 1 is that the runoff rate has reached its maximum after a period of constant rainfall
intensity. The authors used hydraulic theory to develop various relationships for laminar and turbulent
flow and used Reynolds number and kinematic viscosity of water in their derivations.
Ree, Wimberley, and Crow, Manning n and The Overland Flow Equation, Transactions of the
ASAE, Volume 20, Number 1, pages 89-95, 1977.
Quote:
The average length of overland flow was determined by dividing the watershed area by twice the total
length of all waterways.
The delineation of drainageways on a contour map is highly subjective and is the product of the
mapmaker’s ideas and practices. Yet, the calculated length of overland flow depends completely on the
value of the total drainageway length. Thus describing as exactly as possible how drainageways were
determined becomes essential if results are to be meaningful.
Smooth contours were drawn, which fit the survey points chosen, to obtain a good representation of the
topography using a 5-ft contour interval. The drainage pattern was drawn on the finished map extending
the drainageways through the last contour, which indicated a draw or valley.
Watershed
W-1
W-3
W-4
Average Slope (percent)
4.43
5.13
6.66
Length of overland flow (meters)
69.5
61.9
60.0
Narrative:
The same equation was used to estimate overland (sheet) flow length as used in the above Status Report by
Ree. The sheet flow lengths ranged from 200 to 230 feet. Like the authors state, defining the stream
locations is subjective. In the Status Report by Ree, streams were located using aerial photos, and in this
study, they were defined on a hand produced topographic map. It would be interesting to see what sheet
flow lengths would result from an aerial photo determination on the three watersheds. Examining the
contour map of the 92 acre watershed W-3 reproduced in the paper, there are large areas within the
watershed where no stream has been identified. It is entirely possible that some streams were missed which
would cause the overland flow length to be shorter.
Quote:
The Manning n value data for this study were obtained from tests on grass-lined unit channels at the
laboratory. These are flat-bottomed channels (0.91 m wide and 29.26 m long) with a 5 percent bottom
slope.
Narrative:
Again, the study is based on a flow length of about 100 feet. Results are applicable for that length.
Quote:
Fig. 1 (Photograph) An example for a good cover condition in watershed W-3. The average overland flow
equation for this condition is q = 1.48 D ^ 1.22. The equivalent Manning n for a flow depth of 1 cm is
0.31.
Narrative:
The authors make limited reference to the depth of overland flow such as in this caption to the photograph
labeled Figure 1. 1 cm depth is approximately 0.03 foot. In Figure 6, a plot of discharge versus depth
shows depths of 1 cm and less.
Emmett, W., The Hydraulics of Overland Flow on Hillslopes, USGS Professional Paper 662-A, US
Government Printing Office, 1970.
Quotes:
Seven field sites in west central Wyoming were selected for verification of laboratory data. The field
sites were 7 feet wide, about 45 feet long, and approximately represented four slope angles.
The flume used in this investigation was constructed with a plywood bed… The width was 4 feet and
the length was 16 feet. Flume slope was adjustable by hydraulic jacks at the lower end….
Narrative:
Length of overland flow experiments is again limited to under 50 feet. The slopes of the seven field sites
ranged from 2.9 to 33 percent.
Quote:
On nearly flat slopes, microrelief features on the order of only 0.1 foot appeared to dictate the paths of the
flow concentrations. However, on steeper slopes, small microrelief features did not appreciably alter the
down-slope gradient and their influence on concentrations of flow was masked.
As explained earlier, the flow rarely occurred as a uniform sheet of water and the majority of water
travelled downslope in several lateral concentrations of flow; however, these concentrations were not
considered rill flow.
Report content:
Figures 11 through 15 show plots with average flow depth of 0.05 foot or less for the field tests. Table D
includes average flow depths of 0.04 foot or less (with most under 0.02 foot) for laboratory experiments.
Manning n values ranged from less than 0.01 up to 0.10 for the laboratory tests and from 0.10 to greater
than 2.0 with most falling between 0.20 and 1.0 for the natural ground.
Narrative:
Depth of flow was not uniform across the slope because the natural ground surface and laboratory soil
surface was not smooth. The slope of the plots was also not uniform in the longitudinal direction. This
caused differences in depth as flow proceeded downslope. This is typical of natural surfaces in many areas.
Table C shows results of depth measurements at one of the field sites measured at one foot intervals across
the slope and 2 foot intervals downslope. The range of flow depth was 0.0 to 0.09 feet (with most depths
between 0.01 and 0.06 feet) indicating magnitude of microrelief. The surface in the natural ground tests
was fine to moderate grain size (D50 values were from 0.09 to 0.48 mm) and vegetation cover ranged from
8 to 35 percent. Equations were applied with respect to the average depth of flow across the slope. The
depths were shallow enough that what flow concentrations were present, they were not deep enough to
classify them as rills.
McCuen and Spiess, Assessment of Kinematic Wave Time of Concentration, unpublished
manuscript, November, 1993.
Quote:
Current practices use the flow length, L, as the limiting criterion when using a kinematic wave equation
to calculate travel time. According to the 1986 TR-55 documentation, the flow length in Eq. 3 must be less
than 300 feet. Some localities believe 300 feet is too long and limit L to 100 feet. However, there does not
appear to be documented evidence that shows these limits are justified.
The use of flow length alone as a limiting factor for the kinematic wave equation can lead to
circumstances where the kinematic wave assumptions are no longer valid. Overprediction will generally
occur for lengths with high Manning’s n values and/or flat slopes. For instance, lengths of grassed surfaces
of much less than 100 feet may have significant depression storage, as may flat areas. In such a case, the
kinematic wave equation will overpredict the sheet-flow travel time.
Narrative:
This is the prime study focusing on the limits (including sheet flow length) in the use of the Manningkinematic equation for computing sheet flow travel time. Background is given concerning research on the
length of sheet flow and development of the Manning-kinematic travel time equation. Eq. 3 mentioned in
the above quote is the sheet flow travel time equation which is contained in TR-55 (1986 printing). The
approach in this study is to consider the theory and assumptions of kinematic flow and actual field
measurements to develop practical limits on the use of the Manning-kinematic travel time equation similar
(but not the same) as in the NRCS TR-55 computer program.
According to the authors, length alone is not an appropriate as a criterion. The authors mention Manning n
and slope as additional factors to be considered.
Quote:
Times of concentration and watershed characteristics from 59 field and laboratory experiments were
analyzed to determine a suitable limit for the kinematic equation. The data used in this analysis represented
a wide range of watershed sizes, slopes, and ground conditions.
Narrative:
The authors considered data for paved and unpaved surfaces. Range in length was from 12 to 3033 feet.
Range in Manning n was 0.0073 to 0.4. Slopes ranged from 0.001 to 0.162 ft/ft.
Quotes:
The composite parameter nL/ S, where the variables are as described in Eq. 1, will be developed and
assessed in this paper as an accurate and useful criterion when estimating travel times using the kinematic
wave equation. This criterion is a conceptually more rational limit than both the flow length and the
product iL (where i is the rainfall intensity) because it incorporates main properties of sheet flow. Also, the
criterion nL/ S can differentiate a 100-feet flow length with a steep slope and low Manning’s n and the
same length with a flat slope and high Manning’s n, each of which have the same flow length, but quite
different flow conditions.
In summary, the nL/ S criterion provided better goodness of fit statistics than the length. While the
statistics for iL as a criterion were comparable to those of nL/ S, the latter is preferred because it is
composed of variables that are related to the physical processes that underlie kinematic flow. Thus nL/ S
is a more rational criterion than iL for limiting the use of Eq. 2 in estimating sheet-flow travel times. The
analysis for an upper limit for nL/ S of 100 provided the best overall results and suggested an upper limit
of about 100.
Narrative:
Equations 1and 2 mentioned in the above quote are general kinematic time of concentration equations
developed from theory. The term “time of concentration” is used with reference to the uniform slope
length. If this slope length were the sheet flow length for a watershed which also had shallow concentrated
and channel flows the term would be interpreted as “sheet flow travel time”.
Tc = C1 (nL/ S ) ^ C3 / i ^ C2
(1)
Tc = 0.93 (nL/ S) ^ 0.6 / i ^ 0.4
(2)
Tc is the time of concentration in minutes, i is the rainfall intensity (in/hr), L is the length of sheet flow (ft),
n is Manning’s roughness coefficient, S is the slope (ft/ft), and C1, C2, C3 are coefficients.
The authors investigated whether a limit based on sheet flow length (L), the value iL (product of intensity
in inches per hour and sheet flow length), or nL/ S fit the data the best. They decided on the term nL/ S.
If this limit were set at 100, an example of the application follows. For a surface of dense grass, n = 0.24,
and slope of 0.02 ft/ft, the maximum length of sheet flow would be 59 ft (corresponding to a value of
nL/S of 100). Inserting these values into the TR-55 sheet flow travel time equation and assuming the 2year 24 hour rainfall is 3 inches, the travel time would be 0.16 hour.
Quote:
The SCS Kinematic Wave Equation. Equation 3 has the same structure as the generalized model of
Equation 1. However, it has different coefficients and uses the 2-year 24-hour rainfall depth rather than the
intensity for the time of concentration (Welle and Woodward, 1986). The use of a depth that is not
dependent on the time of concentration is desirable because it simplifies the solution procedure by
eliminating the need to iterate. However, the exponents of 0.8 amd 0.5 in Eq. 3, rather than 0.6 and 0.4 in
Eq. 2, produce an equation that must be dimensionally balanced through the coefficients. The SCS
equation could not be tested with our data because the 2-year 24-hour depth was not obtainable for many of
the data samples, obviously for the laboratory data. Thus the limit for nL/ S of 100 cannot be applied to
the SCS equation, and use of this limit may result in inaccurate estimates of Tc with Equation 3. The larger
exponent of 0.8 for the nL/ S term suggests that the limiting value for Eq. 3 would be less than the limit of
100 for Eq. 2. Equating the two nL/ S terms of Eqs. 2 and 3 with their respective exponents of 0.6 and 0.8
yields a limit of 31.6 for the SCS kinematic wave equation. However this was not tested and it ignores the
differences in the two rainfall terms. Further study of this is needed.
Narrative:
The authors have some reservations on application of a limit of 100 on the value of nL/ S for use in the
SCS (NRCS) kinematic wave travel time equation. Equations 2 and 3 give reasonably close results for
travel time especially in the NRCS Type 2 rainfall distribution region. If Manning n is 0.24, slope is 0.02
ft/ft, L = 59 feet, and 2-year 24 hour rainfall is 3 inches, the travel time using Eq. 3 (NRCS equation) is
0.16 hour. Since Eq. 2 uses intensity instead of 2-year 24-hour rainfall, intensity needs to be estimated for
a duration equal to the travel time. At the location of Cincinnati, Ohio, where the 2-year 24-hour rainfall is
3 inches (TP-40 atlas), the 2-year 5-minute rainfall is 0.44 inches (intensity of 5.3 inches per hour) and the
2-year 15-minute rainfall is 0.85 inches (intensity of 3.4 inches per hour). The short duration rainfalls were
read from maps in the NOAA NWS Hydro – 35 publication. Since 0.16 hour is approximately 10 minutes,
interpolating the intensity gives approximately 4.35 inches per hour. Substituting this intensity of 4.35
inches per hour along with n = 0.24, S = 0.02, and L = 59 feet into Eq. 2 produces a travel time of 0.14
hour. Further study of the author’s reservations could be made but this example indicates the limit is
applicable to the NRCS kinematic wave travel time equation.
Summary of Investigations
Although this investigation did not include an exhaustive search of the literature, enough references were
studied in order to get a general overview of status of knowledge and practice concerning sheet flow
characteristics. A number of additional references were studied but not included in this review because
they did not add significant technical information or insights. Most of the studies considered the major
aspects of sheet flow; length, slope, roughness, and a number of related factors; soil, vegetation, rainfall
intensity, geomorphology, etc. Most also considered theory of hydraulics and kinematic or flow routing.
That no definitive results are stated, does not reflect on the quality of the research but on the complexity of
the problem. The studies focusing on depth of sheet flow give general guidelines on what is realistic in the
field. Defining the point where small flow concentrations become what may be called shallow
concentrated flow is a key to analyzing this problem. Complications in defining this point include
variations in soil type, vegetation (or lack of it), slope, and rainfall intensity.
Defining conditions used in the various experiments is important because, especially when gathering data
and developing various models, one needs to be careful not to extrapolate the results beyond the conditions
where they were measured and developed.
The experiments on unpaved areas clearly focused on relatively short sheet flow lengths. Experiments on
paved areas focused on a wider range of lengths. Whether these support a limit on sheet flow length of 100
feet is not definitive but, especially for unpaved areas, it appears reasonable until further research can be
completed. One commonality of the studies was the shallow depth being considered, generally less than
0.1 foot. Studies of Manning n indicated roughness values were significantly greater for sheet flow than for
channel flow.
The concept of linking hydraulic and hydrologic theory to measured data will lead to the best formulation
of models to analyze sheet flow and also the best guidelines for using them in engineering practice.
Further investigation of sheet flow characteristics is needed. A method to consider length, slope, depth,
and roughness is practical and feasible. With all the variability across the country with respect to soil, land
use, climate, geomorphology, etc, there is no substitute for investigating sheet flow characteristics of the
watershed in the field.