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Modeling Consistency for Small and Large Watershed Studies James C.Y. Guo 1 and Ken MacKenzie 2

1. Professor and Director, Box 113, Dept. of Civil Engineering, U of Colorado Denver, CO 80217-3364. E-mail James.Guo@UCDenver.edu

2. Senior Manager, Urban Drainage and Flood Control District, Denver, CO 80235 Abstract When conducting a master drainage plan for an urban area, catchments are varied from small to large in size. Although there are many hydrologic methods developed for stormwater predictions, the size of catchment often serves as the basis to select a proper method. As always, the common question is how to quantitatively define the size of small watershed, and how to establish the modeling consistency between the rational method for small catchments and the unit hydrograph method for large watersheds. In this study, the volume-based runoff coefficients used in the rational method are theoretically derived, and then calibrated to achieve the best agreement with the unit hydrograph method. The example of the rational method versus the Colorado Urban Hydrograph Procedure was used to demonstrate how to achieve the modeling consistency for the metro Denver area. The same procedure can be extended into the relationships between the Rational and kinematic wave methods or the Rational and SCS unit hydrograph methods using the local design rainfall depths and soil loss functions. With the established model consistency, the master drainage plan can be implemented using both the rational and unit-graph methods for all sizes of watersheds used for drainage designs. Key Words: Model Consistency, Rational Method, Unit hydrograph Method, Runoff Coefficient, Time of Concentration, CUHP INTRODUCTION Stormwater modeling is sensitive to the size of watersheds (Guo 2012). Often the rational method is recommended for small catchment hydrology studies, while the unit-graph method is more suitable for large watersheds (USWDCM 2001). In the planning stage, a large tributary area is decomposed into small to large sub-areas for stormwater simulations using the unit hydrograph method. Later on, during the design phase, the on-site drainage system associate with a small sub-area is often designed using the rational method (CCRFCD Master Plan Update 1997). Although this is a common practice, without guidance, the question always exists as to how to establish such a model consistency that will guide the use of the rational method to produce good agreement with the unit hydrograph method. The model consistency warrants the integrity of the regional master drainage plan method (Guo and Hsu 2006). This is such a common challenge in every metro urban area where the master drainage plan has been conducted and is in the process to be enforced as a design criterion for storm water drainage and flood mitigation designs. In this study, the volume-based runoff coefficients were theoretically derived with and without a low-impact cascading flow system. A procedure was developed to apply the volume-based runoff coefficient to establish the model consistency between the rational and unit hydrograph methods. In essence, the solution lies in how to calibrate the runoff coefficient, and how to define the threshold size of small watersheds. The relationship between small and large watersheds depends on local rainfall statistics, soil losses, and empirical formulas used in the unit hydrograph method. The application of this procedure was further demonstrated with a case study between the rational method and Colorado Urban Hydrograph Procedure (CUHP 2005). Similarly, the approach presented in this paper can be extended into the relationship between the rational method and other method for large watersheds, such as kinematic wave (SWNN 2005) or SCS unit hydrograph methods (HEC-1 1998). This paper offers an example as to how to implement a regional master drainage planning with a modeling consistency between small and large watershed’s flood flow prediction methods.

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RATIONAL METHOD The rational method is a simplified kinematic wave approach for peak flow estimation. The major variables in the rational method are time of concentration, watershed tributary area, and runoff coefficient. The rational method states as (Kuichling 1889):

QP = C I A (1) in which Qp= peak flow, C= runoff coefficient, A= tributary area, and I= average rainfall intensity. By definition, the flow-based runoff coefficient is determined as:

IA

QC p (2)

To predict a peak flow, the contributing rainfall amount is defined by the Intensity-Duration-Frequency (IDF) curve under the assumption that the period of rainfall amount is equal to the time of concentration as: (3) in which P= index rainfall depth (Hershfield 1961), Tc = time of concentration in minutes, α, β, and γ = IDF rainfall constants. The time of concentration is the travel time for water to flow through the waterway or a ratio of waterway length, L, to average flow velocity, V, as shown in Fig 1. Although the rational method was originally derived for peak flow predictions, it can in fact be expanded for the entire hydrograph prediction (Guo 2001). The flow rate, Q(T), on the hydrograph at time T depends on the contributing rainfall amount from T-Tc to T as:

Tt

TTtc c

tPT

TI )(1

)( where Tc≤ T ≤ Td (4)

)()( TICATQ (5)

in which I(T)= moving average rainfall intensity at time T for a period of Tc prior to T, Q(T) = runoff rate at time T in runoff hydrograph, t= time variable, Td= event duration, and ΔP(t)= incremental rainfall depth at time t.

)( cT

PI

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Fig 1 Hydrographs Prediction Using Rational Method

Aided with Eq’s (4) and (5), the rational method is expanded into a convolution process to integrate a series of individual triangular hydrographs to produce the entire runoff hydrograph (Guo 2001). In this study, the volume-based runoff coefficient is referred to as the volume ratio of runoff hydrograph to rainfall hyetograph as:

d

B

TT

T

TT

T

R

F

TPA

TTQ

V

VC

0

0

)(

)( (6) in which VF= runoff volume under the hydrograph, VR= rainfall volume under the hyetograph, ΔT = incremental time step on runoff hydrograph such as 5 minutes, Td= rainfall duration, and TB= based time of runoff hydrograph. In theory, Eq’s (2) and (6) should yield identical values for runoff coefficients. In practice, the difference between the rainfall hyetograph and IDF curve may result in gaps between Eq’s (2) and (6). In this study, the volume-based runoff coefficient is selected to establish the model consistency between the rational method and Colorado Urban Hydrograph Procedure (CUHP 2005). The CUHP was developed to apply the unit hydrograph protocol to predict storm hydrographs using a convolution process (Sherman 1932). The input parameters for CUHP include catchment area, length of waterway, length to centroid, slope for waterway, and soil losses (CUHP 2005, USWDCM 2001). There are two system identifiers in the rational method: runoff coefficient and time of concentration. To establish the basis for model consistency, it is necessary to develop a procedure to calibrate the rational method for solving these two system identifiers. VOLUME-BASED RUNOFF COEFFICIENT In an urban catchment, the conventional drainage design is a two-flow system that separates the impervious areas from pervious areas. As a result, the storm runoff can be quickly and efficiently collected. In the recent years, under the concept of low-impact development (LID), the cascading flow system is recommended to spread stormwater from the upper impervious area onto the lower pervious area for more infiltrating benefits. In this study, two sets of theoretical runoff coefficients are separately derived for each of these two flow systems.

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Fig 2 Two-flow and Cascading Drainage Systems

Two-Flow System

In a two-flow system, the impervious areas are connected together to deliver stormwater directly into manholes. Pervious areas are linked through swales to pass stormwater to the downstream streets. A two-flow system is essentially composed of two independent flow paths to drain surface runoff. Under a rainfall event, the total rainfall volume on the catchment is:

PAVR                        (7)

where VR= event rainfall volume on catchment, P= event rainfall depth, and A=catchment area. The runoff volumes produced from the pervious and impervious areas are calculated as:

AIDPV avim )( (8)

AIFDPmV avpP )1)(( 0;01 motherwiseVifm p (9)

pmF VVV (10)

Where Vm = runoff volume from impervious area, Vp = runoff volume from pervious area, VF= total runoff volume, Dvi = depression loss on impervious area, Ia = impervious area ratio, Dvp=depression loss on pervious area, F=infiltration amount, and m = 1 if Vp>0 or 0 if Vp≤0. The variable, m, is to warrant that Vp is numerically positive. By definition, the volume-based runoff coefficient is calculated as:

)]1)(1()1[( avp

avi

R

F IP

F

P

DmI

P

Dn

V

VC (11)

0;01 notherwiseCifn

where C= volume-based runoff coefficient, and n = variable to warrant C≥0. Eq (11) is the sum of two flows, and mostly dominated by the runoff volume, Vm, from the impervious areas. The runoff coefficient in Eq (11) is always greater than zero as long as P>Dvi.

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Cascading Flow System

A lumped system represents a cascading flow process that drains storm water from impervious onto pervious areas. Mathematically, the intercepted runoff volume is directly added to the lower pervious area for more infiltration benefits. Due to the fact that not the entire impervious area can be drained onto the receiving pervious area, a flow interception ratio between zero and one, similar to the routing percentage used in SWMM 2005 (Rossman 2005), is introduced to the runoff volume calculation as:

])1)(()([ AIFDPAIDPrmV avpaviP (12)

where r = flow interception ratio. When r=1, Eq (12) represents a complete flow interception, while r=0, Eq (12) is reduced to Eq (9) for two-flow system. For 0<r <1, the residual runoff volume from the impervious area is directly released to the street as:

AIDPrV avim ))(1( (13)

The resultant runoff coefficient is calculated as:

)]}1)(1()1([)1)(1{( avp

avi

avi

R

F IP

F

P

DI

P

DrmI

P

Drn

V

VC (14)

Setting r=0, Eq(14) is reduced to Eq (11). Numerically, Eq (14) can be reduced to zero if the catchment is under a low development condition. On the contrary, Eq (14) is converged to Eq (11) for a highly urbanized catchment because the lower pervious area is too small to produce any significant infiltration benefits.

TIME OF CONCENTRATION Time of concentration is defined as the flow time required through the watershed, or the duration for the contributing rainfall amount to the peak flow (Kirpitch 1941). The time of concentration is composed of overland flow time and gutter flow time as:

foC TTT 1 (15)

In which TC1 = computed time of concentration in minutes, To = overland flow time in minutes, and Tf = gutter flow time in minutes. There are many empirical formulas developed for Eq (15). As usual, empirical formulas are sensitive to dimensional units. As recommended (FAA 1970, USWDCM 2001), the airport overland flow formula using English units is adopted as:

33.05 )1.1(395.0

o

oo

S

LCT

for overland flow where Lo ≤ L* (16)

Where Lo = overland flow length in feet, C5= runoff coefficient for 5-yr event, So = overland flow slope in feet/feet, and L* = maximum allowable distance in feet such as 300 feet for urban area or 500 feet for rural area. The flow time through street gutters is often estimated using the SCS upland flow method as (NRCS 2013):

6

g

of V

LLT

60

for street gutter flows (17)

ogg SKV (18)

where L= waterway length in feet, Vg= gutter flow velocity, and Kg= conveyance factor of 2.0 feet/sec for gutters ( NRCS 2013). In an urban area, drainage ways are often constructed with drop structures, grade controls, and check dams. As a result, Eq’s 16 and 17 may not reflect the post-development condition. To be conservative, the computed time of concentration based on its overland flow in Eq 16 and gutter flow in Eq 17 needs a further examination with the regional formula for the post-development time of concentration. In the Cities of Denver and Las Vegas (CCRFCD Manual 1999), the regional time of concentration is computed as:

**2 60V

LTTC (19)

oSKV ** (20)

Where TC2= regional time of concentration in minutes, T*= initial overland flow time in minutes, V*= post-development concentrated flow velocity in feet/sec, K*= conveyance factor for concentrated flow in feet/sec. In practice, the design time of concentration is the smaller one between the computed and regional time of concentration as: Tc=min (TC1, TC2) (21) in which Tc= design time of concentration in minutes, In this study, the system variables for the regional time of concentration are T* and K*. Both will be solved by the least square method using the data base built for a case study. CASE STUDY In this case study, the goal is to establish the model consistency between the rational method and CUHP. The data base was selected from the master drainage studies (MacKenzie 2010). It was composed of 293 individual urban small catchments. The ranges of hydrologic parameters are summarized in Table 1.

Table 1 Ranges of Hydrologic Parameters used in Data Base According to the SCS hydrologic guide (NRCS 2013), the infiltration characteristics of soils are classified into Types A, B, and C/D. These 293 catchments were studied under the combinations of Soil A, B, or C/D, catchment’s impervious percentage varied from 5%, 25%, 45%, 65%, 85%, to 99%, and rainfall depth ranging from 2-, 5-, 10-, 50- to 100-yr event. The computer model: CUHP 2005 was used to

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produce runoff hydrographs for 293 catchments. Soil infiltration rates were modeled with Horton’s formula (Horton 1933). Horton’s parameters are listed in Table 2 as recommended in USWDCM 2001:

Table 2 Soil Infiltration Parameters

The task begins with the assumption that all 293 test catchments were covered with Type C/D soils. Under a selected storm event, all catchments were then tested for imperviousness percent of 5, 25, 45, 65, 85, or 99%. For each event, the design rainfall IDF curve is defined as: (22) in which P = one-hr rainfall depth in inches for Denver area as shown in Table 3.

One-hr Recurrence Interval In years Depth 2 5 10 50 100

P (inch) 0.95 1.35 1.6 2.2 2.61 Table 3 One-hr Rainfall Depths for Case Study in Denver Area Repeat the same process for Type A and B soils for 5 storm events. A total of 26379 cases were generated for the calibration process (MacKenzie 2010). Since the one-hr rainfall events are the default index depth to generate the design rainfall distributions in CUHP 2005, the corresponding soil infiltration amount is then determined from the CUHP using t=1 hour. In the numerical procedure, the actual infiltration rate is the smaller one between the soil infiltration potential determined by the Horton’s formula and the design rainfall intensity available (Rossman 2005, Guo 1998). From the aforementioned data base, the average 1-hr infiltration amounts in Table 2 are found to be 1.8 inches for Type A soils, 1.0 inch for Type B soils, and 0.88 inch for Type C soils, according to the computer model of CUHP 2005. As shown in Fig’s 3 and 4, the runoff coefficients are first derived from Eq (11) for Soils A and C/D, and then compared with the flow predictions from CUHP 2005.

SCS Initial Final Decay Impervious Pervious Potential F Actual FSoil infiltration Infiltration Factor Depression Depression t= 1 hr t=1-hr

Type fi fo K Dvi Dvpin/hr in/hr 1/hr in in inch inch

A 5.0 1.0 0.0007 0.1 0.4 5.00 1.80B 4.5 0.6 0.0018 0.1 0.4 4.50 1.00C/D 3.0 0.5 0.0018 0.1 0.4 3.00 0.88

789.0)10(

5.28

cT

PI

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Figure 3 Runoff Coefficients for Two-flow System with Type B Soils

Figure 4 Runoff Coefficients for Two-flow System with Type C/D Soils

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The best fitted values for the two parameters, T* and K*, are determined using the least squared error method as:

2

1

)( jcuhpP

Nj

j

QQMinE

(23)

Where Qcuhp = peak flows generated from CUHP 2005, j= j-th case, N = total number of cases. For a two-flow system, Eq 19 was investigated for Soils A, B, and C/D under the 2-, 10- and 100-yr events with various imperviousness percent from 5 to 85%. Fig 5 is the summary of the best-fitted equation using Eq 24 for the value of K*.

1224.0)(* aIfpsK Ia in percent (24)

Fig 5 Conveyance Parameters for Concentrated Flow Eq 24 reveals that the conveyance factor, K*, varied from 12 to 36 feet/sec according to watershed’s imperviousness percent. The recommended K=20 for paved surface in the SCS upland method (NRCS 2013) is almost the average value for Eq 24. Considering a slope of 1%, Eq 24 sets the limits for the concentrated flow velocity between 1.2 and 3.6 feet/sec. Comparing with Manning’s formula (Chow 1959), the value of K* in Eq 24 includes waterway’s roughness and hydraulic radius. Data scattering in Fig 5 reflects the variations in channel linings and cross sectional geometries in an urban setting. An initial time represents the average overland flow time through the upland area. Repeat the above process to analyze the best fitted values for initial times. Fig. 6 presents the analysis for the value of T* in minutes.

aIT 15.018(min)* Ia in percent (25)

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Fig 6 Initial Times for Concentrated Flow

Fig 6 implies that the imbedded initial time in the CUHP/Rational methods is 18 minutes under Ia=0%, and it is gradually reduced to 3 minute under Ia=100%. The initial time represents the length of sheet flows. Eq 25 reveals the nature of urban catchment as to why the higher the watershed imperviousness, the shorter the overland flow length. VERIFICATION OF CONSISTENCY FOR PEAK FLOWS

In this study, Eq (11) is recommended for calculating the volume-based runoff coefficients. Eq’s (18) through (25) are recommended for determining the time of concentration at the project site. Fig 7 presents the comparison for sample data analyses. The difference in peak flow predictions between the rational and CUHP methods is negligible for catchments <90 acres. The difference increases as the catchment’s area increases. Beyond 150 acres, the difference becomes greater than 10% or more. Therefore, for the applications of CUHP in the Metro Denver area, the demarcation between small and large watersheds is recommended to be 90 acres. As a result of this case study, the recommendation is that for any catchment<90 acres, the rational method using the volume-based runoff coefficients can reproduce the peak flows published on the master drainage plan based on the computer model of CUHP 2005, while for any catchment>150 acres, the rational method will underestimate the peak flows. Therefore, the definition of small watershed for the CUHP is identified to be 90 acres, and the selection of hydrologic method shall observe this recommendation in order to warrant the integrity of the regional master drainage plan.

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Fig. 7 Model Consistency between Rational Method and CUHP for Peak Flow Predictions CONCLUSION Small watershed is defined based on its quick response of runoff generation to rainfall. In practice, the selection of modeling methods for stormwater simulations requires a quantitative basis to define the size of small watershed. The procedure presented in this study provides a guide as to how to establish the model consistency through the selection of hydrologic parameters. Based on the case study, the area of 90 acres serves as the demarcation between small and large watersheds in applications of CUHP and Rational methods. In this study, the runoff coefficient in the rational method is defined as the volume ratio between runoff hydrograph and rainfall hyetograph. A new set of runoff coefficients was theoretically derived and then evaluated with the computer model of CUHP 2005. The time of concentration was composed of the concentrated flow time and the overland flow time. The conveyance factor for concentrated flow is increased from 12 to 36 feet/sec, and the initial overland flow time is reduced from 18 to 3 minutes, according to watershed’s imperviousness. The volume-based runoff coefficients are determined using the local hydrologic parameters that are directly related to the ratios of hydrologic losses to design rainfall depth. As demonstrated in the case study, the one-hr precipitation depth and infiltration amount are recommended for determining the runoff coefficients. In practice, the soil infiltration loss function used in the unit hydrograph method is critically important when establishing the model consistency. All figures and examples in this paper were produced for a two-flow system. A preliminary investigation on the cascading flow system was also conducted in this study for flow interception ratios varied from 0.0 (no interception) to 1.0 (100% interception). As shown in Fig 8 using Type C/D soils as an example, the

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impact of additional infiltration benefits through the cascading flows are limited to the 2-yr event with an imperviousness <45%. There is no further investigations were conducted because the impact of cascading flow on runoff coefficients is limited and insignificant.

Fig 8 Impact of Cascading Flows with Various Flow Interceptions on C/D Soils

AppendixI:REFERENCES

CCRFCD Manual (1999). “Hydrologic Criteria and Design Manual,” published by Clark County Regional Flood Control District, Las Vegas, Nevada. CCRFCD Master Plan Update for Las Vegas Valley (1997). Clark County Regional Flood Control District, Las Vegas, Nevada. Chow, V. T. (1959). “Open Channel hydraulics”, McGrow Hill, New York.

CUHP (2005). “User Manual for Colorado Urban Hydrograph Procedure”, Urban Drainage and Flood Control District, Denver, Colorado

SWMM (2005). “Storm Water Management Model (SWMM),” EPA, Cincinnati, OH.

FAA (1970), U. S. Department of Transportation, Federal Aviation Administration. "Airport Drainage," Report 150/5320-5B, U.S. Government Printing Office, Washington D.C., Guo, James C.Y. (2012). “Storm Centering Approach for Flood Predictions from Large Watersheds”, Journal of Hydrologic Engineering, Vol. 17, No. 9, September 1, Guo, James C.Y. and Hsu, E.S.C. (2006). “Hydrologic Modeling Consistency and Sensitivity to Watershed Size”, J. of PB Network for Water Engineering and Management, Vol 21, No 3, Issue No. 64, December. Guo, James C.Y. (2001) "Rational Hydrograph Method", ASCE of Hydrologic Engineering, Vol 6, No. 4, July/August, pp 352-357

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Guo, James C.Y. (1998). “Overland Flow on a Pervious Surface,” IWRA International J. of Water, Vol 23, No 2, June.pp 91-96.

HEC-1 (1998). “HEC-1 Flood hydrograph Package – User’s Manual, ” Corps of Engineers, the Hydrologic Engineering Center, Davis, California. Hershfield, D. M. (1961). “Technical Paper No. 40 –Rainfall Frequency Atlas of the United States for Durations from 30 Minutes to 24 Hours and Return Periods from 1 to 100 Years” United States Department of Commerce, Weather Bureau. Washington, D.C. Horton, R.E. (1933). "The Role of Infiltration in the Hydrologic Cycle," Trans. Am., Geophys, Union, Vol 14, pp 446-460 Kirpitch, Z.P., (1941). "Time of Concentration for Small Agricultural Watersheds," Civil Engineering, ASCE, Vol 10, No 6, June, pp 362. Kuichling, E. (1889). "The Relation between Rainfall and the Discharge of Sewers in Populous Districts," Trans. ASCE, Vol 20, pp 1-56. MacKenzie, K. A. (2010). “Full-Spectrum Detention for Stormwater Quality Improvement and Mitigation of the Hydrologic Impact of Development”, Master Thesis, Department of Civil Engineering, U of Colorado Denver. NRCS (2013). “Soil Survey”, http://soils.usda.gov/survey/, Washington D.C.

Rossman, L. A. (2005). “Storm Water Management Model User’s Manual Version 5”, Water Supply and Water Resources Division, National Risk Management Research Laboratory, Cincinnati, OH.

Sherman, L.K. (1932). "Stream Flow from Rainfall by Unit-graph Method," Engineering News-Record, Vol 108, April 7, pp 501-505 USWDCM (2001), “Urban Storm Water Drainage Criteria Manuals,” Volume 1, published by Urban Drainage and Flood Control District, Denver, Colorado.

APPENDIX II.

A= tributary area C= runoff coefficient Dvi = depression loss on impervious area, Dvp=depression loss on pervious area F=infiltration amount, Ia = impervious area ratio I= average rainfall intensity I(T)= moving average rainfall intensity at time T for a period of Tc prior to T, K*= conveyance factor for concentrated flow in feet/sec. Kg= conveyance factor of 2.0 feet/sec for gutter Lo = overland flow length in feet, L* = maximum allowable distance L= waterway length in feet. m = 1 if Vp>0 or 0 if Vp≤0. n = variable to warrant C≥0 r = flow interception ratio of Vm P= index rainfall depth or one hr rainfall depth in inches for Denver area ΔP(t)= incremental rainfall depth at time t. Q(T) = runoff rate at time T in runoff hydrograph, Qp= peak flow from rational method So = waterway slope in feet/feet, TC2= regional time of concentration in minutes,

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Tc= design time of concentration in minutes, T*= initial overland flow time in minutes, TC1 = computed time of concentration in minutes, To = overland flow time in minutes, Tf = gutter flow time in minutes. Tc = time of concentration in minutes, TB= based time of runoff hydrograph, t= time variable, Td= event duration, ΔT = incremental time step on runoff hydrograph such as 5 minutes. V = average flow velocity through the waterway V*= post-development concentrated flow velocity in feet/second, VF= total runoff volume or runoff volume under the hydrograph, VR= rainfall volume under the hyetograph, Vm = runoff volume from impervious area, Vp = runoff volume from pervious area, Vg= gutter flow velocity α, β, and γ = IDF rainfall constants.