Section 13: Hydrograph Method
A hydrograph represents runoff as it varies over time at a particular location within the watershed. The area integrated under the hydrograph represents the volume of runoff.
Estimation of a runoff hydrograph, as opposed to the peak rate of runoff, is necessary for watersheds with complex runoff characteristics. The hydrograph method also should be used when storage must be evaluated, as it accounts explicitly for volume and timing of runoff. The hydrograph method has no drainage area size limitation.
Figure 42 shows that in cases for which a statistical distribution cannot be fitted and a regression equation will not predict adequately the design flow, some sort of empirical or conceptual rainfallrunoff model can be used to predict the design flow. Such application is founded on the principle that the AEP of the computed runoff peak or volume is the same as the AEP of the rainfall used as input to (the boundary condition for) the model.
The hydrograph method is applicable for watersheds in which t_{c} is longer than the duration of peak rainfall intensity of the design storm. Precipitation applied to the watershed model is uniform spatially, but varies with time. The hydrograph method accounts for losses (soil infiltration for example) and transforms the remaining (excess) rainfall into a runoff hydrograph at the outlet of the watershed. Figure 49 shows the different components that must be represented to simulate the complete response of a watershed.
Figure 49. Components of the hydrograph method
The method is also applicable to complex watersheds, in which runoff hydrographs for multiple subbasins are computed, then routed to a common point and combined to yield the total runoff hydrograph at that location. Reasons for dividing a large watershed into subbasins include:
 Anchor: #LNGNFNIL
 The subbasins have distinct hydrologic characteristics. Anchor: #JFEKIEIM
 Unique storm depths are appropriate for the different subbasin areas. Anchor: #GGMHKFIN
 Computed hydrographs are needed at more than one location.
Because the resulting runoff hydrograph is a time series of flow values, the method provides a peak flow value as well as volume of runoff. This makes the method suitable for design problems requiring runoff volume as a design parameter.
Successful application of the hydrograph method requires the designer to:
 Anchor: #GGNFFNJE
 Define the temporal and spatial distribution of the desired AEP design storm. Anchor: #LIHLJHKE
 Specify appropriate loss model parameters to compute the amount of precipitation lost to other processes, such as infiltration, and does not run off the watershed. Anchor: #NPGKMGJG
 Specify appropriate parameters to compute runoff hydrograph resulting from excess (not lost) precipitation. Anchor: #GKQIMLJI
 If necessary for the application, specify appropriate parameters to compute the lagged and attenuated hydrograph at downstream locations.
Basic steps to developing and applying a rainfallrunoff model for predicting the required design flow are illustrated in Figure 410. These steps are described in more detail below.
Figure 410. Steps in developing and applying the hydrograph method
Anchor: #i1109089Design Storm Development
A design storm is a precipitation pattern or intensity value defined for design of drainage facilities. Design storms are either based on historical precipitation data or rainfall characteristics in the project area or region. Application of design storms ranges from point precipitation for calculation of peak flows using the rational method to storm hyetographs as input for rainfallrunoff analysis in the hydrograph method. The fundamental assumption using design storms is that precipitation of an AEP yields runoff of the same AEP.
Selection of Storm Duration
Selecting storm duration is the first step in design storm modeling. The appropriate storm duration for stormwater runoff calculations is dependent on the drainage area’s hydrologic response. The selected storm duration should be sufficiently long that the entire drainage area contributes to discharge at the point of interest. Storm duration is defined in terms of time of concentration (t_{c}), which is the time it takes for runoff to travel from the hydraulically most distant point of the watershed to a point of interest within the watershed.
For complete drainage of the area, time for overland flow, channel flow, and storage must be considered. Typically for hydrograph computations the storm duration should be four or five times the time of concentration. Longer duration of storm will not increase the peak discharge substantially, but will contribute greater volume of runoff.
Commonly, a storm duration of 24 hours is used. However the 24hour storm duration should not be used blindly. Runoff from longer and shorter storms should be computed to demonstrate the sensitivity of the design choices.
For TxDOT, the NRCS 24hour storm should be used as a starting point for analysis. However, if the analysis results appear inconsistent with expectations, site performance, or experience, an alternative storm duration should be considered. In that case, the designer should consult the Design Division Hydraulics Branch for advice.
Storm Depth: DepthDurationFrequency (DDF) Relationships
Once the storm duration is selected, the next step is to determine the rainfall depth or intensity of that duration for the selected AEP. DepthDuration Frequency Precipitation for Texas (Asquith 1998) provides procedures to estimate that depth for any location in Texas. The Atlas of DepthDuration Frequency of Precipitation Annual Maxima for Texas (TxDOT 51301011) is an extension of the 1998 study and an update of Technical Paper No. 40: Rainfall Frequency Atlas of the United States (Hershfield 1961), Technical Paper No. 49: 2 to 10Day Precipitation for Return Periods of 2 to 100 Year in the Contiguous United States (Miller 1964), and NOAA NWS Hydro35: 5 to 60 Minute Precipitation Frequency for the Eastern and Central United States (Frederick et al. 1977).
The Atlas of DepthDuration Frequency of Precipitation Annual Maxima for Texas includes 96 maps depicting the spatial variation of the DDF of precipitation annual maxima for Texas. The AEPs represented are 50%, 20%, 10%, 4%, 2%, 1%, 0.4%, and 0.2% (2, 5, 10, 25, 50, 100, 250, and 500years). The storm durations represented are 15 and 30 minutes; 1, 2, 3, 6, and 12 hours; and 1, 2, 3, 5, and 7 days.
IntensityDurationFrequency Relationships
While hydrograph methods require both rainfall depth and temporal distribution, the rational method requires only intensity. The rainfall intensity (I) is the average rainfall rate in inches/hour for a specific rainfall duration and a selected frequency. For drainage areas in Texas, rainfall intensity may be computed by:
 Anchor: #FMNIGGEK
 Using maps in the Atlas of DepthDuration Frequency of Precipitation Annual Maxima for Texas publication to obtain the precipitation depth for a given frequency. Anchor: #KNFNHJNG
 Converting the precipitation depth to a precipitation intensity by dividing the depth by the storm duration. The precipitation is measured in inches/hour.
For example, if the 100year, 6hour depth is 3.2 inches, the precipitation intensity equals 3.2 inches/6 hours = 0.53 inches/hour.
Areal Depth Adjustment
When estimating runoff due to a rainfall event, a uniform areal distribution of rainfall over the watershed is assumed. However, for intense storms, uniform rainfall is unlikely. Rather, rainfall varies across the drainage area. To account for this variation, an areal adjustment is made to convert point depths to an average areal depth. For drainage areas smaller than 10 square miles^{}, the areal adjustment is negligible. For larger areas, point rainfall depths and intensities must be adjusted. Two methods are presented here for use in design of drainage facilities: the first is by the US Weather Bureau and the second is by USGS.
US Weather Bureau Areal Depth Adjustment
The US Weather Bureau (1958) developed Figure 411 from an annual series of rain gauge networks. It shows the percentage of point depths that should be used to yield average areal depths.
Figure 411. Depth area adjustment (US Weather Bureau 1958)
USGS ArealReduction Factors for the Precipitation of the 1Day Design Storm in Texas
Areal reduction factors (ARFs) specific for Texas for a 1day design storm were developed by Asquith (1999). Asquith’s method uses an areal reduction factor that ranges from 0 to 1. The method is a function of watershed characteristics such as size and shape, geographic location, and time of year that the design storm is presumed to occur. The study was based on precipitation monitoring networks in the Austin, Dallas, and Houston areas. If using a 1day design storm, this is the appropriate method of areal reduction to use for design of highway drainage facilities in Texas.
However, the applicability of this method diminishes the farther away from the Austin, Dallas, or Houston areas the study area is and as the duration of the design storm increasingly differs from that of 1 day. For further information and example problems on calculating the ARF, refer to Asquith (1999).
A relationship exists between the point of an annual precipitation maxima and the distance between both the centroid of the watershed and every location radiating out from the centroid. This is assuming the watershed is or nearly so circular. ST(r) is the expected value of the ratio between the depth at some location a distance r from the point of the design storm. T refers to the frequency of the design storm. Equations for ST(r) for the 50% (2year) or smaller AEP vary by proximity to Austin, Dallas, and Houston. For an approximately circular watershed, the ARF is calculated with the following equation:
Equation 424.
Where:
r = variable of integration ranging from 0 to R
 Anchor: #MELKKFNI
 R = radius of the watershed (mi) Anchor: #IEKHKEMN
 S_{2}(r) = estimated 2year or greater depthdistance relation (mi)
The sitespecific equations for S_{2}(r) for differing watershed radii are in Table 412 at the end of this section.
Once the ARF is calculated, the effective depth of the design storm is found by multiplying the ARF by the point precipitation depth found using Atlas of DepthDuration Frequency of Precipitation Annual Maxima for Texas.
For example, an approximately circular watershed in the Dallas area is 50.3 square miles^{} (R = 4 miles). From Table 412:
S_{2} = 1.0000 – 0.06(r) for
S_{2} = 0.9670 – 0.0435(r) for
Substituting the above expressions into Equation 425 gives:
ARF = 0.85
An easier way to determine ARF for circular watersheds is to use the equation from Table 412 in column “ARF for circular watersheds having radius r” for the city and radius of interest. For the previous example (City of Dallas, R = 4 miles), the equation would be:
ARF = 0.96700.290(r) + (0.0440/r^{2})
ARF = 0.85
From the precipitation atlas, the 1% (100year) 1day depth is 9.8 inches. Multiply this depth by 0.85 to obtain the 24hour 1% AEP areally reduced storm depth of 8.3 inches.
If the designer finds that a circular approximation of the watershed is inappropriate for the watershed of interest, the following procedure for noncircular watersheds should be used. The procedure for noncircular watersheds is as follows:
 Anchor: #GLNINIIM
 Represent the watershed as discrete cells; the cells do not have to be the same area. Anchor: #PEHLGEKH
 Locate the cell containing the centroid of the watershed. Anchor: #SFGINKII
 For each cell, calculate the distance to the centroid (r). Anchor: #LGFGLGGK
 Using the distances from Step 3, solve the appropriate equations from Table 412 for S_{2}(r) for each cell. Anchor: #HEEIMEME
 Multiply S_{2}(r) by the corresponding cell area to compute ARF; the area multiplication simply acts as a weight for a weighted mean. Anchor: #EKKFFLKF
 Compute the sum of the cell areas. Anchor: #MRJGIEEG
 Compute the sum of the product of S_{2}(r) and cell area from Step 5. Anchor: #LOIHKGII
 Divide the result of Step 7 by Step 6.
City 
Estimated 2yr or greater depthdistance relation for distance r (mi) 
ARF for circular watersheds having radius r (mi) 
Equation limits 

Austin 











































Dallas 











































Houston 






























Anchor: #i1171585
Rainfall Temporal Distribution
The temporal rainfall distribution is how rainfall intensity varies over time for a single event. The mass rainfall curve, illustrated in Figure 412, is the cumulative precipitation up to a specific time. In drainage design, the storm is divided into time increments, and the average depth during each time increment is estimated, resulting in a rainfall hyetograph as shown in Figure 413.
Figure 412. Example mass rainfall curve from historical storm
Hyetograph Development Procedure
In the rational method the intensity is considered to be uniform over the storm period. Hydrograph techniques, however, account for variability of the intensity throughout a storm. Therefore, when using hydrograph techniques, the designer must determine a rainfall hyetograph: a temporal distribution of the watershed rainfall, as shown in Figure 413.
Figure 413. Rainfall hyetograph
Methods acceptable for developing a rainfall hyetograph for a design storm include the NRCS method, the balanced storm method, and the Texas storm method.
NRCS Hyetograph Development Procedure
The NRCS design storm hyetographs were derived by averaging storm patterns for regions of the U.S. The storms thus represent a pattern distribution of rainfall over a 24hour period to which a design rainfall depth can be applied. The distribution itself is arranged in a critical pattern with the maximum precipitation period occurring just before the midpoint of the storm.
Figure 414 and Table 413 show the NRCS 24hour rainfall distributions for Texas: Type II and Type III. Figure 415 shows the areas in Texas to which these distribution types apply. The distribution represents the fraction of accumulated rainfall (not runoff) accrued with respect to time.
Figure 414. NRCS 24hour rainfall distributions (NRCS 1986)
Time, t (hours) 
Fraction of 24hour rainfall 



Type II 
Type III 
0 
0.000 
0.000 
2 
0.022 
0.020 
4 
0.048 
0.043 
6 
0.080 
0.072 
7 
0.098 
0.089 
8 
0.120 
0.115 
8.5 
0.133 
0.130 
9 
0.147 
0.148 
9.5 
0.163 
0.167 
9.75 
0.172 
0.178 
10 
0.181 
0.189 
10.5 
0.204 
0.216 
11 
0.235 
0.250 
11.5 
0.283 
0.298 
11.75 
0.357 
0.339 
12 
0.663 
0.500 
12.5 
0.735 
0.702 
13 
0.772 
0.751 
13.5 
0.799 
0.785 
14 
0.820 
0.811 
16 
0.880 
0.886 
20 
0.952 
0.957 
24 
1.000 
1.000 
Figure 415. Rainfall distribution types in Texas (TR55 1986)
Use the following steps to develop a rainfall hyetograph:
 Anchor: #FPIHFEKG
 Determine the rainfall depth (P_{d}) for the desired design frequency and location. Anchor: #MGEMMIEN
 Use Figure 415 to determine the distribution type. Anchor: #IKIFKINK
 Select an appropriate time increment for computation of runoff hydrograph ordinates. An increment equal 1/5 or 1/6 of the time of concentration is adequate for most analyses. Anchor: #PJKGIMLI
 Create a table of time and the fraction of rainfall total. Interpolate the rainfall distributions table for the appropriate distribution type. Anchor: #LOOMFFHF
 Multiply the cumulative fractions by the total rainfall depth (from step 1) to get the cumulative depths at various times. Anchor: #JLKIJINK
 Determine the incremental rainfall for each time period by subtracting the cumulative rainfall at the previous time step from the current time step.
Balanced Storm Hyetograph Development Procedure
The triangular temporal distribution, with the peak of the storm located at the center of the hyetograph, is also called balanced storm. It uses DDF values that are based on a statistical analysis of historical data. The procedure for deriving a hyetograph with this method is as follows:
 Anchor: #MFLLKKKI
 For the selected AEP, tabulate rainfall amounts for a storm of a given return period for all durations up to a specified limit (for 24hour, 15minute, 30minute, 1hour, 2hour, 3hour, 6hour, 12hour, 24hour, etc.). Use the maps in Asquith 2004, locating the study area on the appropriate map for the duration and AEP selected for design. Anchor: #GQLLFGGE
 Select an appropriate time interval. An
appropriate time interval is related to the time of concentration
of the watershed. To calculate the time interval, use:
Equation 425.
Where:
 Anchor: #HIGEGMNE
 Δt = time interval Anchor: #NIELGLMI
 t_{c} = time of concentration Anchor: #FLNEJIHJ
 For example, if the time of concentration is 1 hour, Δt = 1/5t_{c} = 1/5 of 1 hour = 12 minutes, or 1/6 of 1 hour = 10 minutes. Choosing 1/5 or 1/6 will not make a significant difference in the distribution of the rainfall; use one fraction or the other to determine a convenient time interval.
Anchor: #MKFHIJHN  For successive times of interval Δt, find
the cumulative rainfall depths from the DDF values. For depths at
time intervals not included in the DDF tables, interpolate depths
for intermediate durations using a loglog interpolation. (Durations
from the table are usually given in hours, but in minutes on the
plot.) For example, given a study area in the northern part of Bexar County,
the loglog plot in Figure 416 shows the 10% depths for the 15,
30, 60, 120,
180, 360, 720, and 1440minute durations included in Asquith
and Roussel 2004. The precipitation depth at 500 minutes is interpolated
as 5.0 inches.
Figure 416. Log time versus log precipitation depth
Anchor: #PKNEKJGJ  Find the incremental depths by subtracting the cumulative depth at a particular time interval from the depth at the previous time interval. Anchor: #ONJFJMII
 Rearrange the incremental depths so that the peak depth is at the center of the storm and the remaining incremental depths alternate (to left and right of peak) in descending order.
For example, in Figure 417, the largest incremental depth for a 24hour storm (1,440 minutes) is placed at the 720minute time interval and the remaining incremental depths are placed about the 720minute interval in alternating decreasing order.
Figure 417. Balanced storm hyetograph
Anchor: #NTEMKLKFTexas Storm Hyetograph Development Procedure
Texas specific dimensionless hyetographs were developed by researchers at USGS, Texas Tech University, University of Houston, and Lamar University (WilliamsSether et al. 2004, Asquith et al. 2005). Two databases were used to estimate the hyetographs: 1) rainfall recorded for more than 1,600 storms over mostly small watersheds as part of historical USGS studies, and 2) hourly rainfall data collection network from the NWS over eastern New Mexico, Oklahoma, and Texas.
Three methods of developing dimensionless hyetographs are presented: 1) triangular dimensionless hyetograph; 2) Lgamma dimensionless hyetograph; and 3) empirical dimensionless hyetograph. Any of these hyetographs can be used for TxDOT design. Brief descriptions of the three methods are presented here. For further information and example problems on the Texas hyetographs, refer to Asquith et al. 2005.
Triangular Dimensionless Hyetograph
A triangular dimensionless hyetograph is presented in Figure 418. The vertical axis represents relative rainfall intensity. The rainfall intensity increases linearly until the time of peak intensity, then decreases linearly until the end of the storm. The triangular hyetograph, in terms of relative cumulative storm depth, is defined by Equations 426 and 427, with values for parameters a and b provided in Table 414.
Equation 426.
Equation 427.
Where:
 Anchor: #QLLGEFJI
 p_{1} = normalized cumulative rainfall depth, (ranging from 0 to 1) for F ranging from 0 to a Anchor: #NKEJGNNJ
 p_{2} = normalized cumulative rainfall depth, (ranging from 0 to 1) for F ranging from a to 1 Anchor: #MLJNEEIN
 F = elapsed time, relative to storm duration, ranging from 0 to 1 Anchor: #LIJNNIII
 a = relative storm duration prior to peak intensity, from Table 414 Anchor: #ENGKLEFK
 b = relative storm duration prior to peak intensity, from Table 414
Figure 418. Triangular dimensionless Texas hyetograph
Triangular hyetograph model parameters 
Storm duration 


512 hours 
1324 hours 
2572 hours 

a 
0.02197 
0.28936 
0.38959 
b 
0.97803 
0.71064 
0.61041 
Based on the storm duration, the designer selects the appropriate parameters a and b for use in Equations 426 and 427. The ordinates of cumulative storm depth, normalized to total storm depth, are thus obtained. Values of rainfall intensity are obtained by computing total storm depth for durations of interest, and dividing by the duration.
Triangular Dimensionless Hyetograph Procedure
The following is an example computation using the triangular dimensionless hyetograph procedure for a 12hour storm with cumulative depth of 8 inches:
 Anchor: #MKFHFFNF
 Express F in Equations 426 and 427 in terms of time t and total storm duration T: F = t / T. Anchor: #FOHIJLMK
 Express p in terms of cumulative rainfall depth d and total storm depth D: p = d / D. Anchor: #JIJKEHME
 Substituting into Equations 426 and 427
gives:
 From Table 414, a = 0.02197 and b = 0.97803.
 Substituting 12 (hours) for T and 8 (inches)
for D gives:
 Simplifying:
These resulting equations provide cumulative depth in inches as a function of elapsed time in hours, as shown in Table 415.
Time, t (hr.) 
Precipitation Depth, d (in.) 
Precipitation Intensity, I (in./hr.) 

0 
0 
0 
0.13 
0.04 
0.33 
0.26 
0.17 
0.99 
0.50 
0.49 
1.32 
0.75 
0.81 
1.29 
1.00 
1.13 
1.26 
2.00 
2.32 
1.19 
3.00 
3.40 
1.08 
4.00 
4.36 
0.97 
5.00 
5.22 
0.85 
6.00 
5.96 
0.74 
7.00 
6.58 
0.62 
8.00 
7.09 
0.51 
9.00 
7.49 
0.40 
10.00 
7.77 
0.28 
11.00 
7.94 
0.17 
12.00 
8.00 
0.06 
Lgamma Dimensionless Hyetograph
Asquith (2003) and Asquith et al. (2005) computed sample Lmoments of 1,659 dimensionless hyetographs for runoffproducing storms. Storms were divided by duration into 3 categories, 0 to 12 hours, 12 to 24 hours, and 24 to 72 hours. Dimensionless hyetographs based on the Lgamma distribution were developed and are defined by:
Equation 428.
Where:
 Anchor: #HOHHJNHF
 e = 2.718282 Anchor: #MMLEGMFF
 p = normalized cumulative rainfall depth, ranging from 0 to 1 Anchor: #GJUEJKNJ
 F = elapsed time, relative to storm duration, ranging from 0 to 1 Anchor: #IKOGMKEH
 b = distribution parameter from Table 416 Anchor: #ENGJMFII
 c = distribution parameter from Table 416
Parameters b and c of the Lgamma distribution for the corresponding storm durations are shown in Table 416. Until specific guidance is developed for selecting parameters for storms of exactly 12 hours and 24 hours, the designer should adopt distribution parameters for the duration range resulting in the more severe runoff condition.
Storm duration 
Lgamma distribution parameters 


b 
c 

0 – 12 hours 
1.262 
1.227 
12 – 24 hours 
0.783 
0.4368 
24  72 hours 
0.3388 
0.8152 
Lgamma Dimensionless Hyetograph Procedure
Use the following steps to develop an Lgamma dimensionless Texas hyetograph for storm duration of 24 hours and a storm depth of 15 inches:
 Anchor: #UFEIMHHI
 Enter the Lgamma
distribution parameters for the selected storm duration into the
following equation:
 Express F in terms of time t and total
storm duration T: F = t / T. Express p in terms of cumulative rainfall
depth d and total storm depth D: p = d / D. Substituting gives:
 Substitute 24 (hours) for T and 15 (inches)
for D:
This equation defines the storm hyetograph. d is the cumulative depth in inches, and t is the elapsed time in hours.
Empirical Dimensionless Hyetograph
Empirical dimensionless hyetographs (WilliamsSether et al. 2004, Asquith et al. 2005) have been developed for application to small drainage areas (less than approximately 160 square miles) in urban and rural areas in Texas. The hyetographs are dimensionless in both duration and depth, and are applicable for storm durations ranging from 0 to 72 hours. The hyetograph shapes are not given by a mathematical expression, but are provided graphically in Figure 419, and are tabulated in Table 417.
Figure 419. Dimensionless hyetographs for 0 to 72 hours storm duration (WilliamsSether et al. 2004)
Storm duration (%) 
1st quartile depth (%) 
2nd quartile depth (%) 
3rd quartile depth (%) 
4th quartile depth (%) 

0.0 
0 
0 
0 
0 
2.5 
8.70 
2.81 
2.51 
3.28 
5.0 
18.81 
5.89 
4.73 
5.16 
7.5 
28.27 
8.89 
6.86 
7.53 
10.0 
36.71 
11.82 
8.96 
9.59 
12.5 
43.93 
14.60 
10.92 
11.30 
15.0 
50.35 
17.32 
12.76 
12.93 
17.5 
55.74 
19.93 
14.41 
14.30 
20.0 
60.57 
22.61 
15.95 
15.98 
22.5 
64.85 
25.34 
17.34 
17.64 
25.0 
68.65 
28.36 
18.66 
19.46 
27.5 
71.70 
31.47 
19.91 
21.27 
30.0 
74.09 
34.77 
21.05 
23.10 
32.5 
75.85 
38.14 
22.08 
24.71 
35.0 
77.23 
41.69 
22.89 
26.30 
37.5 
78.42 
45.34 
23.45 
27.67 
40.0 
79.62 
49.29 
23.77 
28.95 
42.5 
80.86 
53.27 
24.24 
30.19 
45.0 
82.20 
57.39 
25.14 
31.51 
47.5 
83.43 
61.42 
27.11 
32.86 
50.0 
84.59 
65.46 
30.15 
34.27 
52.5 
85.59 
69.27 
34.37 
35.65 
55.0 
86.42 
73.09 
39.28 
36.92 
57.5 
87.12 
76.77 
44.53 
38.02 
60.0 
87.75 
80.35 
49.48 
39.04 
62.5 
88.28 
83.34 
54.35 
39.90 
65.0 
88.85 
85.85 
58.90 
40.72 
67.5 
89.46 
87.80 
63.36 
41.71 
70.0 
90.10 
89.24 
67.56 
43.15 
72.5 
90.81 
90.23 
71.75 
45.03 
75.0 
91.53 
91.15 
75.58 
47.52 
77.5 
92.22 
92.03 
79.15 
50.60 
80.0 
92.87 
92.90 
82.29 
54.45 
82.5 
93.54 
93.81 
85.28 
59.01 
85.0 
94.25 
94.75 
87.93 
64.24 
87.5 
95.01 
95.69 
90.36 
70.27 
90.0 
95.84 
96.66 
92.62 
76.81 
92.5 
96.82 
97.64 
94.86 
83.44 
95.0 
97.90 
98.63 
97.15 
90.01 
97.5 
99.02 
99.65 
98.94 
96.48 
100.0 
100.00 
100.00 
100.00 
100.00 
Before applying the method, the designer determines the appropriate storm depth and duration for the AEP of interest. With the depth and duration defined, four dimensionless hyetographs, corresponding to the 1st, 2nd, 3rd, and 4th quartiles are defined. The quartile defines in which temporal quarter of the storm the majority of precipitation occurs. Until further guidance is provided by research, it is recommended that the designer consider all four quartile hyetographs, and select the one which produces the most severe design condition. Note that the combined 1st through 4th quartile hyetograph shown in Figure 419 is not presently recommended for design.
Confidence limits for the empirical dimensionless hydrographs have been computed for each of the four quartile hyetographs. These are available in the form of hyetographs representing 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, and 90th percentiles confidence limits. The four quartile hydrographs recommended for design are in fact the 50th percentile, or median, percentile hyetographs. Because the hyetographs are dimensionless, all of the percentile hyetographs have the same dimensionless storm depth, but represent variations in the temporal distribution of rainfall during the storm duration. Percentile hyetographs are available and discussed further in WilliamsSether et al. 2004.
Anchor: #i1109862Models for Estimating Losses
Losses refer to the volume of rain falling on a watershed that does not run off. With each model, precipitation loss is found for each computation time interval, and is subtracted from the precipitation depth for that interval. The remaining depth is referred to as precipitation excess. This depth is considered uniformly distributed over a watershed area, so it represents a volume of runoff.
Loss models available to the TxDOT designer include:
 Anchor: #HNLIJFLJ
 Initial and constantrate loss model. Anchor: #KKKNNJGI
 Texas initial and constantrate loss model. Anchor: #GUJHMLHN
 NRCS curve number loss model. Anchor: #EKEMGGHG
 Green and Ampt loss model.
Initial and ConstantRate Loss Model
Basic Concepts and Equations
For the initial loss and constant–rate loss model, no runoff occurs in the watershed until an initial loss capacity has been satisfied, regardless of the rainfall rate. Once the initial loss has been satisfied, a constant potential loss rate occurs for the duration of the storm. This method is a simple approximation of a typical infiltration curve, where the initial loss decays over the storm duration to a final nearconstant loss rate. In the example in Figure 420, the initial loss is satisfied in the first time increment, and the constant rate accounts for losses thereafter.
Figure 420. Initial and constantloss rate model
The initial and constant lossrate model is described mathematically as:
Equation 429.
Equation 430.
Equation 431.
Where:
 Anchor: #LLGLMLNE
 I(t) = rainfall intensity (in./hr.) Anchor: #GPGNFMEF
 f(t) = loss rate (in./hr.) Anchor: #LTHEIMFF
 P(t) = cumulative rainfall depth (in.) at time t Anchor: #FGHNMMNF
 I_{a} = initial loss (in.) Anchor: #FLMEEFNG
 L = constant loss rate (in./hr.)
I_{a} accounts for interception and depression storage, and the initial rate of infiltration at the beginning of the storm event. Interception refers to the capture of rainfall on the leaves and stems of vegetation before it reaches the ground surface. Depression storage is where the ponded rainfall fills small depressions and irregularities in the ground surface. Depression storage eventually infiltrates or evaporates during dryweather periods. Until the accumulated precipitation on the pervious area exceeds the initial loss volume, no runoff occurs.
Estimating Initial Loss and Constant Rate
The initial and constantrate loss model includes one parameter (the constant rate) and one initial condition (the initial loss). Respectively, these represent physical properties of the watershed soils and land use and the antecedent condition.
If the watershed is in a saturated state, I_{a} will approach 0. If the watershed is dry, then I_{a} will increase to represent the maximum precipitation depth that can fall on the watershed with no runoff; this will depend on the watershed terrain, land use, soil types, and soil treatment.
The constant loss rate can be viewed as the ultimate infiltration capacity of the soils. The NRCS classified soils on the basis of this infiltration capacity as presented in Table 418; values in Column 4 represent reasonable estimates of the rates.
Texas Initial and ConstantRate Loss Model
Recent research (TxDOT 041937) developed four computational approaches for estimating initial abstraction (I_{A}) and constant loss (C_{L}) values for watersheds in Texas. The approaches are all based on the analysis of rainfall and runoff data of 92 gauged watersheds in Texas. One of those methods, presented here, allows the designer to compute I_{A} and C_{L} from regression equations:
Equation 432.
Equation 433.
Where:
 Anchor: #ENGIMELG
 I_{A} = initial abstraction (in.) Anchor: #KHGKFEMG
 C_{L} = constant loss rate (in./hr.) Anchor: #FLGLIENH
 L = main channel length (mi.) Anchor: #ITNJMFNG
 D = 0 for undeveloped watersheds, 1 for developed watersheds Anchor: #ENNLFGEH
 R = 0 for nonrocky watersheds, 1 for rocky watersheds Anchor: #HKHNMIGF
 CN = NRCS curve number
In the above equations, L is defined as “the length in streamcourse miles of the longest defined channel shown in a 30meter digital elevation model from the approximate watershed headwaters to the outlet” (TxDOT 041937).
Anchor: #JSGEGKGGNRCS Curve Number Loss Model
NRCS has developed a procedure to divide total depth of rainfall into soil retention, initial abstractions, and effective rainfall. This parameter is referred to as a curve number (CN). The CN is based on soil type, land use, and vegetative cover of the watershed. The maximum possible soil retention is estimated using a parameter that represents the impermeability of the land in a watershed. Theoretically, CN can range from 0 (100% rainfall infiltration) to 100 (impervious). In practice, based on values tabulated in NRCS 1986, the lowest CN the designer will likely encounter is 30, and the maximum CN is 98.
Hydrologic Soil Groups
Soil properties influence the relationship between rainfall and runoff by affecting the rate of infiltration. NRCS divides soils into four hydrologic soil groups based on infiltration rates (Groups AD). Urbanization has an effect on soil groups, as well. See Table 418 for more information.
Soil group 
Description 
Soil type 
Range of loss rates 


(in./hr.) 
(mm/hr.) 

A 
Low runoff potential due to high infiltration rates even when saturated 
Deep sand, deep loess, aggregated silts 
0.300.45 
7.611.4 
B 
Moderately low runoff potential due to moderate infiltration rates when saturated 
Shallow loess, sandy loam 
0.150.30 
3.87.6 
C 
Moderately high runoff potential due to slow infiltration rates Soils in which a layer near the surface impedes the downward movement of water or soils with moderately fine to fine texture 
Clay loams, shallow sandy loam, soils low in organic content, and soils usually high in clay 
0.050.15 
1.33.8 
D 
High runoff potential due to very slow infiltration rates 
Soils that swell significantly when wet, heavy plastic clays, and certain saline soils 
0.000.05 
1.3 
Estimating the CN
Rainfall infiltration losses depend primarily on soil characteristics and land use (surface cover). The NRCS method uses a combination of soil conditions and land use to assign runoff CNs. Suggested runoff curve numbers are provided in Table 419, Table 420, Table 421, and Table 422. Note that CNs are whole numbers.
For a watershed that has variability in land cover and soil type, a composite CN is calculated and weighted by area.
Cover type and hydrologic condition 
Average percent impervious area 
A 
B 
C 
D 

Open space (lawns, parks, golf courses, cemeteries, etc.): 

Poor condition (grass cover < 50%) 

68 
79 
86 
89 
Fair condition (grass cover 50% to 75%) 

49 
69 
79 
84 
Good condition (grass cover > 75%) 

39 
61 
74 
80 
Paved parking lots, roofs, driveways, etc. (excluding rightofway) 

98 
98 
98 
98 
Streets and roads: 

Paved; curbs and storm drains (excluding rightofway) 

98 
98 
98 
98 
Paved; open ditches (including rightofway) 

83 
89 
92 
93 
Gravel (including rightofway) 

76 
85 
89 
91 
Dirt (including rightofway) 

72 
82 
87 
89 
Western desert urban areas: 

Natural desert landscaping (pervious areas only) 

63 
77 
85 
88 
Artificial desert landscaping (impervious weed barrier, desert shrub with 1 to 2in. sand or gravel mulch and basin borders) 

96 
96 
96 
96 
Urban districts: 





Commercial and business 
85 
89 
92 
94 
95 
Industrial 
72 
81 
88 
91 
93 
Residential districts by average lot size: 





1/8 acre or less (townhouses) 
65 
77 
85 
90 
92 
1/4 acre 
38 
61 
75 
83 
87 
1/3 acre 
30 
57 
72 
81 
86 
1/2 acre 
25 
54 
70 
80 
85 
1 acre 
20 
51 
68 
79 
84 
2 acres 
12 
46 
65 
77 
82 
Developing urban areas: Newly graded areas (pervious area only, no vegetation) 

77 
86 
91 
94 
Table 419 notes: Values are for average runoff condition, and I_{a} = 0.2S. The average percent impervious area shown was used to develop the composite CNs. Other assumptions are: impervious areas are directly connected to the drainage system, impervious areas have a CN of 98, and pervious areas are considered equivalent to open space in good hydrologic condition.� 
Cover type 
Treatment 
Hydrologic condition 
A 
B 
C 
D 

Fallow

Bare soil 
 
77 
86 
91 
94 
Crop residue cover (CR) 
Poor Good 
76 74 
85 83 
90 88 
93 90 

Row crops

Straight row (SR)

Poor Good 
72 67 
81 78 
88 85 
91 89 
SR + CR

Poor Good 
71 64 
80 75 
87 82 
90 85 

Contoured (C) 
Poor Good 
70 65 
79 75 
84 82 
88 86 

C + CR

Poor Good 
69 64 
78 74 
83 81 
87 85 

Contoured & terraced (C&T) 
Poor Good 
66 62 
74 71 
80 78 
82 81 

C&T + CR 
Poor Good 
65 61 
73 70 
79 77 
81 80 

Small grain

SR

Poor Good 
65 63 
76 75 
84 83 
88 87 
SR + CR

Poor Good 
64 60 
75 72 
83 80 
86 84 

C

Poor Good 
63 61 
74 73 
82 81 
85 84 

C + CR

Poor Good 
62 60 
73 72 
81 80 
84 83 

C&T

Poor Good 
61 59 
72 70 
79 78 
82 81 

C&T + CR

Poor Good 
60 58 
71 69 
78 77 
81 80 

Closeseeded or broadcast legumes or rotation meadow 
SR 
Poor Good 
66 58 
77 72 
85 81 
89 85 
C 
Poor Good 
64 55 
75 69 
83 78 
85 83 

C&T 
Poor Good 
63 51 
73 67 
80 76 
83 80 

Table 420 notes: Values are for average runoff condition, and I_{a} = 0.2S. Crop residue cover applies only if residue is on at least 5% of the surface throughout the year. Hydrologic condition is based on a combination of factors affecting infiltration and runoff: density and canopy of vegetative areas, amount of yearround cover, amount of grass or closedseeded legumes in rotations, percent of residue cover on land surface (good > 20%), and degree of roughness. Poor = Factors impair infiltration and tend to increase runoff. Good = Factors encourage average and better infiltration and tend to decrease runoff. 
Cover type 
Hydrologic condition 
A 
B 
C 
D 

Pasture, grassland, or rangecontinuous forage for grazing 
Poor Fair Good 
68 49 39 
79 69 61 
86 79 74 
89 84 80 
Meadow – continuous grass, protected from grazing and generally mowed for hay 
 
30 
58 
71 
78 
Brush – brushweedgrass mixture, with brush the major element 
Poor Fair Good 
48 35 30 
67 56 48 
77 70 65 
83 77 73 
Woods – grass combination (orchard or tree farm) 
Poor Fair Good 
57 43 32 
73 65 58 
82 76 72 
86 82 79 
Woods 
Poor Fair Good 
45 36 30 
66 60 55 
77 73 70 
83 79 77 
Farmsteads – buildings, lanes, driveways, and surrounding lots 
 
59 
74 
82 
86 
Table 421 notes: Values are for average runoff condition, and I_{a} = 0.2S. Pasture: Poor is < 50% ground cover or heavily grazed with no mulch, Fair is 50% to 75% ground cover and not heavily grazed, and Good is > 75% ground cover and lightly or only occasionally grazed. Meadow: Poor is < 50% ground cover, Fair is 50% to 75% ground cover, Good is > 75% ground cover. Woods/grass: CNs shown were computed for areas with 50 percent grass (pasture) cover. Other combinations of conditions may be computed from CNs for woods and pasture. Woods: Poor = forest litter, small trees, and brush destroyed by heavy grazing or regular burning. Fair = woods grazed but not burned and with some forest litter covering the soil. Good = woods protected from grazing and with litter and brush adequately covering soil.� 
Cover type 
Hydrologic condition 
A 
B 
C 
D 

Herbaceous—mixture of grass, weeds, and lowgrowing brush, with brush the minor element 
Poor Fair Good 

80 71 62 
87 81 74 
93 89 85 
Oakaspen—mountain brush mixture of oak brush, aspen, mountain mahogany, bitter brush, maple, and other brush 
Poor Fair Good 

66 48 30 
74 57 41 
79 63 48 
Pinyonjuniper—pinyon, juniper, or both; grass understory 
Poor Fair Good 

75 58 41 
85 73 61 
89 80 71 
Sagebrush with grass understory 
Poor Fair Good 

67 51 35 
80 63 47 
85 70 55 
Saltbush, greasewood, creosotebush, blackbrush, bursage, palo verde, mesquite, and cactus 
Poor Fair Good 
63 55 49 
77 72 68 
85 81 79 
88 86 84 
Table 422 notes: Values are for average runoff condition, and I_{a} = 0.2S. Hydrologic Condition: Poor = < 30% ground cover (litter, grass, and brush overstory), Fair = 30% to 70% ground cover, Good = > 70% ground cover. Curve numbers for Group A have been developed only for desert shrub.� 
Soil Retention
The potential maximum retention (S) is calculated as:
Equation 434.
Where:
 Anchor: #NIIEKGGI
 z = 10 for English measurement units, or 254 for metric Anchor: #ENGNLMHG
 CN = runoff curve number
Equation 434 is valid if S is less than the rainfall excess, defined as precipitation (P) minus runoff (R) or S < (PR). This equation was developed mainly for small watersheds from recorded storm data that included total rainfall amount in a calendar day but not its distribution with respect to time. Therefore, this method is appropriate for estimating direct runoff from 24hour or 1day storm rainfall.
Initial Abstraction
The initial abstraction consists of interception by vegetation, infiltration during early parts of the storm, and surface depression storage.
Generally, I_{a} is estimated as:
Equation 435.
Effective Rainfall Runoff Volume
The effective rainfall (or the total rainfall minus the initial abstractions and retention) used for runoff hydrograph computations can be estimated using:
Equation 436.
Where:
 Anchor: #HIMMGEEK
 P_{e} = accumulated excess rainfall (in.) Anchor: #IFPJELEG
 I_{a} = initial abstraction before ponding (in.) Anchor: #PEJHKFMJ
 P = total depth of rainfall (in.) Anchor: #FMTHHJIF
 S = potential maximum depth of water retained in the watershed (in.)
Substituting Equation 435, Equation 436 becomes:
Equation 437.
P_{e} and P have units of depth, P_{e} and P reflect volumes and are often referred to as volumes because it is usually assumed that the same depths occurred over the entire watershed. Therefore P_{e} is considered the volume of direct runoff per unit area, i.e., the rainfall that is neither retained on the surface nor infiltrated into the soil. P_{e} also can be applied sequentially during a storm to compute incremental precipitation for selected time interval Δt.
Climatic Adjustment of CN
NRCS curve numbers, estimated (predicted) using the procedure described in Estimating the CN, may be adjusted to account for the variation of climate within Texas. The adjustment is applied as follows:
Equation 438.
Where:
 Anchor: #MGSNFKFM
 CN_{obs} = CN adjusted for climate Anchor: #MKLKMMLM
 CN_{pred} = Estimated CN from NRCS procedures described in Estimating the CN Anchor: #FOPNIKHG
 CN_{dev} = Deviation of CN_{obs} from CN_{pred} = climatic adjustment factor
In two studies (Halley and McGill 1983, Thompson et al. 2003) CN_{dev} was computed for gauged watersheds in Texas as CN_{obs}  CN_{pred} based on historical rainfall and runoff volumes. These studies show that CN_{dev} varies by location within the state.
The following excerpt (Thompson et al. 2003) guides the designer in selection and application of the appropriate climatic adjustment to the predicted CN.
Given the differences between CN_{obs} and CN_{pred}, it is possible to construct a general adjustment to CN_{pred} such that an approximation of CN_{obs} can be obtained. The large amount of variation in CN_{obs} does not lend to smooth contours or function fits. There is simply an insufficient amount of information for these types of approaches. However, a general adjustment can be implemented using regions with a general adjustment factor. Such an approach was taken and is presented in Figure 421.
The bulk of rainfall and runoff data available for study were measured near the I35 corridor. Therefore, estimates for this region are the most reliable. The greater the distance from the majority of the watershed that were part of this study, then the more uncertainty must be implied about the results. For the south high plains, that area south of the Balcones escarpment, and the coastal plain, there was insufficient data to make any general conclusions.
Application of the tool is straightforward. For areas where adjustment factors are defined (see Figure 421) the analyst should:
 Anchor: #HHKEJMKM
 Determine CN_{pred} using the normal NRCS procedure. Anchor: #OKFFJGGN
 Find the location of the watershed on the design aid (Figure 421). Determine an adjustment factor from the design aid and adjust the curve number. Anchor: #FNKLGKNN
 Examine Figure 422 and find the location of the watershed. Use the location of the watershed to determine nearby study watersheds. Then refer to Figure 422 and Table 423, Table 424, Table 425, Table 426, and Table 427 and determine CN_{pred} and CN_{obs} for study watersheds near the site in question, if any are near the watershed in question. Anchor: #HMMIGILF
 Compare the adjusted curve number with local values of CN_{obs}.
The result should be a range of values that are reasonable for the particular site.
As a comparison, the adjusted curve number from Hailey and McGill (Figure 423) can be used.
A lower bound equivalent to the curve number for AMC I (dry antecedent conditions), or a curve number of 60, which ever is greater, should be considered.
Note that CN values are whole numbers. Rounding of values of CN_{pred} in the tables may be required.
Judgment is required for application of any hydrologic tool. The adjustments presented on Figure 421 are no exception. A lower limit of AMC I may be used to prevent an overadjustment downward. For areas that have few study watersheds, the Hailey and McGill approach should provide some guidance on the amount of reduction to CN_{pred} is appropriate, if any.
Figure 421. Climatic adjustment factor CN_{dev}
Figure 422. Location of CNdev watersheds
Figure 423. Climatic adjustment of CN  comparison of Hailey and McGill adjusted curve numbers, CNH&M, with CNobs. Negative differences indicate that CNH&M is larger than CNobs. Also shown are the lines of equal adjustment to curve number from Hailey and McGill’s (1983) Figure 4.
USGS Gauge ID 
Quad Sheet Name 
CN_{obs} 
CN_{pred} 
CN_{dev} 

8154700 
Austin West 
59 
68.9 
9.9 
8155200 
Bee Cave 
65 
70.7 
5.7 
8155300 
Oak Hill 
64 
69.8 
5.8 
8155550 
Austin West 
50 
87.3 
37.3 
8156650 
Austin East 
60 
83.6 
23.6 
8156700 
Austin East 
78 
86.6 
8.6 
8156750 
Austin East 
66 
86.8 
20.8 
8156800 
Austin East 
66 
87 
21 
8157000 
Austin East 
68 
88.3 
20.3 
8157500 
Austin East 
67 
89.1 
22.1 
8158050 
Austin East 
71 
83.9 
12.9 
8158100 
Pflugerville West 
60 
72.6 
12.6 
8158200 
Austin East 
62 
75.6 
13.6 
8158400 
Austin East 
79 
88.9 
9.9 
8158500 
Austin East 
71 
85.6 
14.6 
8158600 
Austin East 
73 
76.7 
3.7 
8158700 
Driftwood 
69 
74.5 
5.5 
8158800 
Buda 
64 
73.3 
9.3 
8158810 
Signal Hill 
64 
69.8 
5.8 
8158820 
Oak Hill 
60 
67.9 
7.9 
8158825 
Oak Hill 
49 
67.2 
18.2 
8158840 
Signal Hill 
74 
69.8 
4.2 
8158860 
Oak Hill 
60 
68 
8 
8158880 
Oak Hill 
67 
79.4 
12.4 
8158920 
Oak Hill 
71 
77.5 
6.5 
8158930 
Oak Hill 
56 
75.2 
19.2 
8158970 
Montopolis 
56 
77.7 
21.7 
8159150 
Pflugerville East 
63 
78.8 
15.8 
USGS Gauge ID 
Quad Sheet Name 
CN_{obs} 
CN_{pred} 
CN_{dev} 

8055580 
Garland 
85 
85.2 
0.2 
8055600 
Dallas 
82 
86.1 
4.1 
8055700 
Dallas 
73 
85.5 
12.5 
8056500 
Dallas 
85 
85.8 
0.8 
8057020 
Dallas 
75 
85.5 
10.5 
8057050 
Oak Cliff 
75 
85.7 
10.7 
8057120 
Addison 
77 
80.2 
3.2 
8057130 
Addison 
89 
82.9 
6.1 
8057140 
Addison 
78 
86.8 
8.8 
8057160 
Addison 
80 
90.3 
10.3 
8057320 
White Rock Lake 
85 
85.7 
0.7 
8057415 
Hutchins 
73 
87.8 
14.8 
8057418 
Oak Cliff 
85 
79.1 
5.9 
8057420 
Oak Cliff 
80 
81 
1 
8057425 
Oak Cliff 
90 
82.9 
7.1 
8057435 
Oak Cliff 
82 
81.1 
0.9 
8057440 
Hutchins 
67 
79.1 
12.1 
8057445 
Hutchins 
60 
86.5 
26.5 
8061620 
Garland 
82 
85 
3 
8061920 
Mesquite 
85 
86 
1 
8061950 
Seagoville 
82 
85.3 
3.3 
Gauge ID 
Quad Sheet Name 
CN_{obs} 
CN_{pred} 
CN_{dev} 

8048520 
Fort Worth 
72 
82.3 
10.3 
8048530 
Fort Worth 
69 
86.7 
17.7 
8048540 
Covington 
73 
88 
15 
8048550 
Haltom City 
74 
91.2 
17.2 
8048600 
Haltom City 
65 
84.3 
19.3 
8048820 
Haltom City 
67 
83.4 
16.4 
8048850 
Haltom City 
72 
83 
11 
USGS Gauge ID 
Quad Sheet Name 
CN_{obs} 
CN_{pred} 
CN_{dev} 

8177600 
Castle Hills 
70 
84.8 
14.8 
8178300 
San Antonio West 
72 
85.7 
13.7 
8178555 
Southton 
75 
84.2 
9.2 
8178600 
Camp Bullis 
60 
79.7 
19.7 
8178640 
Longhorn 
56 
78.4 
22.4 
8178645 
Longhorn 
59 
78.2 
19.2 
8178690 
Longhorn 
78 
84.4 
6.4 
8178736 
San Antonio East 
74 
92.3 
18.3 
8181000 
Helotes 
50 
79.2 
29.2 
8181400 
Helotes 
56 
79.8 
23.8 
8181450 
San Antonio West 
60 
87.3 
27.3 
USGS Gauge ID 
Quadrangle Sheet Name 
CN_{obs} 
CN_{pred} 
CN_{dev} 

8025307 
Fairmount 
53 
55.4 
2.4 
8083420 
Abilene East 
65 
84.7 
19.7 
8088100 
True 
60 
85.9 
25.9 
8093400 
Abbott 
61 
88.1 
27.1 
8116400 
Sugarland 
70 
82.9 
12.9 
8159150 
Pflugerville East 
55 
83.7 
28.7 
8160800 
Freisburg 
56 
67.8 
11.8 
8167600 
Fischer 
51 
74.3 
23.3 
8436520 
Alpine South 
64 
86.4 
22.4 
8435660 
Alpine South 
48 
86.7 
38.7 
8098300 
Rosebud 
88 
80.5 
7.5 
8108200 
Yarrelton 
77 
79.9 
2.9 
8096800 
Bruceville 
62 
80 
18 
8094000 
Bunyan 
60 
78.4 
18.4 
8136900 
Bangs West 
51 
75.8 
24.8 
8137000 
Bangs West 
52 
74.5 
22.5 
8137500 
Trickham 
53 
76.5 
23.5 
8139000 
Placid 
53 
74.6 
21.6 
8140000 
Mercury 
63 
74.4 
11.4 
8182400 
Martinez 
52 
80 
28 
8187000 
Lenz 
53 
83.8 
30.8 
8187900 
Kenedy 
63 
73.3 
10.3 
8050200 
Freemound 
80 
79.6 
0.4 
8057500 
Weston 
80 
78.2 
1.8 
8058000 
Weston 
86 
80.1 
5.9 
8052630 
Marilee 
80 
85.4 
5.4 
8052700 
Aubrey 
74 
84.1 
10.1 
8042650 
Senate 
59 
63.4 
4.4 
8042700 
Lynn Creek 
50 
62.5 
12.5 
8042700 
Senate 
56 
62 
6 
8042700 
Senate 
65 
55.9 
9.1 
8063200 
Coolidge 
70 
79.4 
9.4 
Anchor: #OLLLLEML
Green and Ampt Loss Model
Basic Concepts and Equations
The Green and Ampt loss model is based on a theoretical application of Darcy’s law. The model, first developed in 1911, has the form:
Equation 439.
Where:
 Anchor: #NKJGNMGK
 f = infiltration capacity (in./hr.) Anchor: #MRHFIGLE
 K_{s} = saturated hydraulic conductivity (permeability) (in./hr.) Anchor: #IJKGMFJG
 S_{w} = soil water suction (in.) Anchor: #FNLEEMNJ
 θ_{s} = volumetric water content (water volume per unit soil volume) under saturated conditions Anchor: #KLHHKIFJ
 θ_{i} = volumetric moisture content under initial conditions Anchor: #EKGHHKFG
 F = total accumulated infiltration (in.)
The parameters can be related to soil properties.
Assumptions underlying the Green and Ampt model are the following:
 Anchor: #QMEHLMEH
 As rain continues to fall and water infiltrates, the wetting front advances at the same rate throughout the groundwater system, which produces a welldefined wetting front. Anchor: #NPGHJGNH
 The volumetric water contents, θ_{s} and θ_{i}, remain constant above and below the wetting front as it advances. Anchor: #GLGEJFII
 The soilwater suction immediately below the wetting front remains constant with both time and location as the wetting front advances.
To calculate the infiltration rate at a given time, the cumulative infiltration is calculated using Equation 440 and differences computed in successive cumulative values:
Equation 440.
Where:
 Anchor: #NFNNHKIH
 t = time (hr.)
Equation 440 cannot be solved explicitly. Instead, solution by numerical methods is required. Once F is solved for, the infiltration rate, f, can be solved using Equation 439. These computations are typically performed by hydrologic computer programs equipped with GreenAmpt computational routines. With these programs, the designer is required to specify θ_{s}, S_{w}, and K_{s}.
Estimating GreenAmpt Parameters
To apply the Green and Ampt loss model, the designer must estimate the volumetric moisture content, θ_{s}, the wetting front suction head, S_{w}, and the saturated hydraulic conductivity, K_{s}. Rawls et al. (1993) provide GreenAmpt parameters for several USDA soil textures as shown in Table 428. A range is given for volumetric moisture content in parentheses with typical values for each also listed.
Soil texture class 
Volumetric moisture content under saturated conditions θ_{s} 
Volumetric moisture content under initial conditions θ_{i} 
Wetting front suction head S_{w} 
Saturated hydraulic conductivity K_{s} 

Sand 
0.437 (0.3740.500) 
0.417 (0.3540.480) 
1.95 
9.28 
Loamy sand 
0.437 (0.3630.506) 
0.401 (0.3290.473) 
2.41 
2.35 
Sandy loam 
0.453 (0.3510.555) 
0.412 (0.2830.541) 
4.33 
0.86 
Loam 
0.463 (0.3750.551) 
0.434 (0.3340.534) 
3.50 
0.52 
Silt loam 
0.501 (0.4200.582) 
0.486 (0.3940.578) 
6.57 
0.27 
Sandy clay loam 
0.398 (0.3320.464) 
0.330 (0.2350.425) 
8.60 
0.12 
Clay loam 
0.464 (0.4090.519) 
0.309 (0.2790.501) 
8.22 
0.08 
Silty clay loam 
0.471 (0.4180.524) 
0.432 (0.3470.517) 
10.75 
0.08 
Sandy clay 
0.430 (0.3700.490) 
0.321 (0.2070.435) 
9.41 
0.05 
Silty clay 
0.479 (0.4250.533) 
0.423 (0.3340.512) 
11.50 
0.04 
Clay 
0.475 (0.4270.523) 
0.385 (0.2690.501) 
12.45 
0.02 
Anchor: #FEFELELJ
Capabilities and Limitations of Loss Models
Selecting a loss model and estimating the model parameters are critical steps in estimating runoff. Some pros and cons of the different alternatives are shown in Table 429. These are guidelines and should be used as such. The designer should be familiar with the models and the watershed where applied to determine which loss model is most appropriate.
Model 
Pros 
Cons 

Initial and constantloss rate 
Has been successfully applied in many studies throughout the US. Easy to set up and use. Model only requires a few parameters to explain the variation of runoff parameters. 
Difficult to apply to ungauged areas due to lack of direct physical relationship of parameters and watershed properties. Model may be too simple to predict losses within event, even if it does predict total losses well. 
Texas initial and constantloss rate 
Developed specifically from Texas watershed data for application to sites in Texas. Method is product of recent and extensive research. Simple to apply. 
Method is dependent on NRCS CN. Relatively new method, and not yet widely used. 
NRCS CN 
Simple, predictable, and stable. Relies on only one parameter, which varies as a function of soil group, land use, surface condition, and antecedent moisture condition. Widely accepted and applied throughout the U.S. 
Predicted values not in accordance with classical unsaturated flow theory. Infiltration rate will approach zero during a storm of long duration, rather than constant rate as expected. Developed with data from small agricultural watersheds in midwestern US, so applicability elsewhere is uncertain. Default initial abstraction (0.2S) does not depend upon storm characteristics or timing. Thus, if used with design storm, abstraction will be same with 0.5 AEP storm and 0.01 AEP storm. Rainfall intensity not considered. 
Green and Ampt 
Parameters can be estimated for ungauged watersheds from information about soils. 
Not widely used, less experience in professional community. 
Anchor: #KLJGGNKI
Rainfall to Runoff Transform
After the design storm hyetograph is defined, and losses are computed and subtracted from rainfall to compute runoff volume, the time distribution and magnitude of runoff is computed with a rainfall to runoff transform.
Two options are described herein for these direct runoff hydrograph computations:
 Anchor: #MLOIEIEN
 Unit hydrograph (UH) model. This is an empirical model that relies on scaling a pattern of watershed runoff. Anchor: #EFPLEGMK
 Kinematic wave model. This is a conceptual model that computes the overland flow hydrograph method with channel routing methods to convert rainfall to runoff and route it to the point of interest.
Unit Hydrograph Method
A unit hydrograph for a watershed is defined as the discharge hydrograph that results from one unit depth of excess rainfall distributed uniformly, spatially and temporally, over a watershed for a duration of one unit of time. The unit depth of excess precipitation is one inch for English units. The unit of time becomes the time step of the analysis, and is selected as short enough to capture the detail of the storm temporal distribution and rising limb of the unit hydrograph.
The unit hydrograph assumes that the rainfall over a given area does not vary in intensity. If rainfall does vary, the watershed must be divided into smaller subbasins and varying rainfall applied with multiple unit hydrographs. The runoff can then be routed from subbasin to subbasin.
For each time step of the analysis, the unit hydrograph ordinates are multiplied by the excess rainfall depth. The resulting timecoincident ordinates from each resulting hydrograph are summed to produce the total runoff hydrograph for the watershed. This process is shown graphically in Figure 424. Hydrographs a, b, c, and d are 1hour unit hydrographs multiplied by the depth of excess rainfall in the individual 1hour time steps. The total runoff hydrograph resulting from 4 hours of rainfall is the sum of hydrographs a, b, c, and d.
Figure 424. Unit hydrograph superposition (USACE 1994)
Mathematically, the computation of the runoff hydrograph is given by:
Equation 441.
Where:
 Anchor: #NIHNGELL
 n = number of time steps Anchor: #MGOGKFKE
 Q_{n} = the runoff hydrograph ordinate n (at time nΔt) Anchor: #FLPMIKML
 P_{m} = effective rainfall ordinate m (in time interval mΔt) Anchor: #JHHIHGFH
 _{} = computation time interval Anchor: #IRNHGGLI
 Q_{u (nm+1)} = unit hydrograph ordinate (nm+1) (at time (nm+1)Δt) Anchor: #NQHILEFE
 m = number of periods of effective rainfall (of duration Δt) Anchor: #EFMLHFHM
 M = total number of discrete rainfall pulses
Equation 441 simplified becomes Q_{1} = P_{1}U_{1}, Q_{2} = P_{1}U_{2}+P_{2}U_{1}, Q_{3} = P_{1}U_{3}+P_{2}U_{2}+P_{3}U_{1}, etc.
Several different unit hydrograph methods are available to the designer. Each defines a temporal flow distribution. The time to peak flow and general shape of the distribution are defined by parameters specific to each method. The choice of unit hydrograph method will depend on available options within the hydrologic software being used, and also the availability of information from which to estimate the unit hydrograph parameters.
Two unit hydrograph methods commonly used by TxDOT designers are Snyder’s unit hydrograph and the NRCS unit hydrograph. These methods are supported by many rainfallrunoff software programs, which require the designer only to specify the parameters of the method. These two methods are discussed in the following sections.
Snyder’s Unit Hydrograph
Snyder developed a parametric unit hydrograph in 1938, based on research in the Appalachian Highlands using basins 10 to 10,000 square miles^{}. Snyder’s unit hydrograph is described with two parameters: C_{t}, which is a storage or timing coefficient; and C_{p}, which is a peaking coefficient. As C_{t} increases, the peak of the unit hydrograph is delayed. As C_{p} increases, the magnitude of the unit hydrograph peak increases. Both C_{t} and C_{p} must be estimated for the watershed of interest. Values for C_{p} range from 0.4 to 0.8 and generally indicate retention or storage capacity of the watershed.
The peak discharge of the unit hydrograph is given by:
Equation 442.
Where:
 Anchor: #NNHNLFIL
 Q_{p} = peak discharge (cfs/in.) Anchor: #ENHMLNEM
 A = drainage area (mi^{2}) Anchor: #NJPGLILG
 C_{p} = second coefficient of the Snyder method accounting for flood wave and storage conditions Anchor: #VHHNEEIM
 t_{L} = time lag (hr.) from the centroid of rainfall excess to peak of hydrograph
t_{L} is given by:
Equation 443.
Where:
 Anchor: #MMHLIHFJ
 C_{t} = storage coefficient, usually ranging from 1.8 to 2.2 Anchor: #JTHIJEIK
 L = length of main channel (mi) Anchor: #FKLENLEF
 L_{ca} = length along the main channel from watershed outlet to the watershed centroid (mi)
The duration of excess rainfall (t_{d}) can be computed using:
Equation 444.
Equation 444 implies that the relationship between lag time and the duration of excess rainfall is constant. To adjust values of lag time for other values of rainfall excess duration, the following equation should be used:
Equation 445.
Where:
 Anchor: #NKGGKLEI
 t_{La} = adjusted time lag (hr.) Anchor: #FNHHFGLM
 t_{da} = alternative unit hydrograph duration (hr.)
The time base of the unit hydrograph is a function of the lag time:
Equation 446.
Where:
 Anchor: #KPNGGFEF
 t_{b} = time base (days)
The time to peak of the unit hydrograph is calculated by:
Equation 447.
Empirical relations of Snyder’s unit hydrograph were later found to aid the designer in constructing the unit hydrograph (McCuen 1989). The USACE relations, shown in Figure 425, are used to construct the Snyder unit hydrograph using the time to peak (t_{p}), the peak discharge (Q_{p}), the time base (t_{b}), and 2 time parameters, W_{50} and W_{75}. W_{50} and W_{75} are the widths of the unit hydrograph at discharges of 50 percent and 75 percent of the peak discharge. The widths are distributed 1/3 before the peak discharge and 2/3 after.
Figure 425. Snyder’s unit hydrograph
Values for W_{50} and W_{75} are computed using these equations (McCuen 1989):
Equation 448.
Equation 449.
Where:
q_{a} = peak discharge per square mile (i.e., Q_{p}/A, ft^{3}/sec/mi^{2})
Anchor: #PMEMIMJKNRCS Dimensionless Unit Hydrograph
The NRCS unit hydrograph model is based upon an analysis and averaging of a large number of natural unit hydrographs from a broad cross section of geographic locations and hydrologic regions. For convenience, the hydrograph is dimensionless, with discharge ordinates (Q_{u}) divided by the peak discharge (Q_{p}) and the time values (t) divided by the time to peak (t_{p}).
The timebase of the dimensionless unit hydrograph is approximately five times the time to peak, and approximately 3/8 of the total volume occurs before the time to peak. The inflection point on the recession limb occurs at 1.67 times the time to peak, and the hydrograph has a curvilinear shape. The curvilinear hydrograph can be approximated by a triangular hydrograph with similar characteristics.
A curvilinear dimensionless NRCS unit hydrograph is shown in Figure 426.
Figure 426. NRCS dimensionless unit hydrograph
The ordinates of the dimensionless unit hydrograph are provided in Table 430.
t/t_{p} 
Q/Q_{p} 

0.0 
0.00 
0.1 
0.03 
0.2 
0.10 
0.3 
0.19 
0.4 
0.31 
0.5 
0.47 
0.6 
0.66 
0.7 
0.82 
0.8 
0.93 
0.9 
0.99 
1.0 
1.00 
1.1 
0.99 
1.2 
0.93 
1.3 
0.86 
1.4 
0.78 
1.5 
0.68 
1.6 
0.56 
1.7 
0.46 
1.8 
0.39 
1.9 
0.33 
2.0 
0.28 
2.2 
0.207 
2.4 
0.147 
2.6 
0.107 
2.8 
0.077 
3.0 
0.055 
3.2 
0.04 
3.4 
0.029 
3.6 
0.021 
3.8 
0.015 
4.0 
0.011 
4.5 
0.005 
5.0 
0.00 
Table 430 notes: Variables are defined as follows: t = time (min.); t_{p} = time to peak of unit hydrograph (min.); Q = discharge (cfs); and Q_{p} = peak discharge of unit hydrograph (cfs). 
The following procedure assumes the area or subarea is reasonably homogeneous. That is, the watershed is subdivided into homogeneous areas. The procedure results in a hydrograph only from the direct uncontrolled area. If the watershed has been subdivided, it might be necessary to perform hydrograph channel routing, storage routing, and hydrograph superposition to determine the hydrograph at the outlet of the watershed.
Application of the NRCS dimensionless unit hydrograph to a watershed produces a sitespecific unit hydrograph model with which storm runoff can be computed. To do this, the basin lag time must be estimated. The time to peak of the unit hydrograph is related to the lag time by:
Equation 450.
Where:
 Anchor: #JMQKIHGM
 t_{p} = time to peak of unit hydrograph (min.) Anchor: #NJHFGKJF
 t_{L} = basin lag time (min.) Anchor: #JJKMFEMH
 Δt = the time interval of the unit hydrograph (min.)
This time interval must be the same as the Δt chosen for the design storm.
The time interval may be calculated by:
Equation 451.
And the lag time is calculated by:
Equation 452.
The peak discharge of the unit hydrograph is calculated by:
Equation 453.
Where:
 Anchor: #JGMEKJMM
 Q_{p} = peak discharge (cfs) Anchor: #ETJLFJFF
 C_{f} = conversion factor (645.33) Anchor: #NGGMKGNK
 K = 0.75 (constant based on geometric shape of dimensionless unit hydrograph) Anchor: #NIFLGHLE
 A = drainage area (mi^{2}); and Anchor: #EVJJGMIH
 t_{p} = time to peak (hr.)
Equation 453 can be simplified to:
Equation 454.
The constant 484, or peak rate constant, defines a unit hydrograph with 3/8 of its area under the rising limb. As the watershed slope becomes very steep (mountainous), the constant in Equation 451 can approach a value of approximately 600. For flat, swampy areas, the constant may decrease to a value of approximately 300. For applications in Texas, the use of the constant 484 is recommended unless specific runoff data indicate a different value is warranted.
After t_{p} and Q_{p} are estimated using Equations 450 and 454, the site specific unit hydrograph may be developed by scaling the dimensionless unit hydrograph.
For example, Table 431 gives values for a basin with Δt = 10 minutes, t_{p} = 40 minutes, and Q_{p} = 400 cfs. Column 1 shows the time interval of 10 minutes. Column 2 is calculated by dividing the time interval by t_{p}, in this case 40 minutes. Values in Column 3 are found by using the t/t_{p} value in Column 2 to find the associated Q_{u}/Q_{p} value from the dimensionless unit hydrograph shown in Figure 426, interpolating if necessary. Column 4 is calculated by multiplying Column 3 by Q_{p}, in this case 400 cfs.
t (min.) 
t/t_{p} 
Q_{u}/Q_{p} 
Q_{u} (cfs) 

0 
0.00 
0.000 
0 
10 
0.25 
0.145 
58 
20 
0.50 
0.470 
188 
30 
0.75 
0.875 
350 
40 
1.00 
1.000 
400 
50 
1.25 
0.895 
358 
60 
1.50 
0.680 
272 
70 
1.75 
0.425 
170 
80 
2.00 
0.280 
112 
90 
2.25 
0.192 
77 
100 
2.50 
0.127 
51 
110 
2.75 
0.085 
34 
120 
3.00 
0.055 
22 
130 
3.25 
0.037 
15 
140 
3.50 
0.025 
10 
150 
3.75 
0.017 
7 
160 
4.00 
0.011 
4 
170 
4.25 
0.008 
3 
180 
4.50 
0.005 
2 
190 
4.75 
0.003 
1 
200 
5.00 
0.000 
0 
The capabilities and limitations of the NRCS unit hydrograph model include the following:
 Anchor: #JFNHLEKH
 This method should be used only for a 24hour storm. Anchor: #LOIEIFIK
 This method does not account for variation in rainfall intensity or duration over the watershed. Anchor: #EMMLFGHE
 Baseflow is accounted for separately.
The example sitespecific unit hydrograph is shown in Figure 427.
Figure 427. Example sitespecific unit hydrograph
Anchor: #GKMHHIEKKinematic Wave Overland Flow Model
A kinematic wave model is a conceptual model of watershed response that uses laws of conservation of mass and momentum to simulate overland and channelized flows. The model represents the watershed as a wide open channel, with inflow equal to the excess precipitation. Then it simulates unsteady channel flow over the surface to compute the watershed runoff hydrograph. The watershed is represented as a set of overland flow planes and collector channels.
In kinematic wave modeling, the watershed shown in Figure 428(a) is represented in Figure 428(b) as series of overland flow planes (gray areas) and a collector channel (dashed line). The collector channel conveys flow to the watershed outlet.
Figure 428. Kinematic wave model representation of a watershed (USACE 2000)
The equations used to define conservation of mass and momentum are the Saint Venant equations. The conservation of mass equation is:
Equation 455.
Where:
 Anchor: #NKNFEHGL
 A = cross sectional area of flow (ft^{2}, m^{2}) Anchor: #IOMLFJEG
 T = time (sec.) Anchor: #MOKGGNEG
 Q = flow rate (cfs, m^{3}/sec.) Anchor: #PEFFFJLI
 x = distance along the flow path (ft, m) Anchor: #MKEEJMLM
 q_{o} = lateral discharge added to the flow path per unit length of the flow path (cfs/ft, m^{3}/sec./m)
The momentum equation energy gradient is approximated by:
Equation 456.
Where:
 Anchor: #JFIKKKMJ
 α and β = coefficients related to the physical properties of the watershed.
Substituting Equation 456 into Equation 455 yields a single partial differential equation in Q:
Equation 457.
Where:
 Anchor: #SNGFEFGI
 q_{L} = lateral inflow (cfs/ft, m^{3}/s/m)
Equation 454 can be expressed in terms of Manning’s n, wetted perimeter, and bed slope by substituting the following expression for into Equation 456:
Equation 458.
Where:
 Anchor: #GHLHFMGN
 n = Manning’s roughness coefficient Anchor: #INJIMFLF
 P = wetted perimeter (ft, m) Anchor: #JHKIJFHL
 S_{o} = flow plane slope (ft/ft, m/m)
The solution to the resulting equation, its terms, and basic concepts are detailed in Chow (1959) and other texts.
Anchor: #HUFNIFLKHydrograph Routing
In some cases, the watershed of interest will be divided into subbasins. This is necessary when ground conditions vary significantly between subbasin areas, or when the total watershed area is sufficiently large that variations in precipitation depth within the watershed must be modeled. A rainfallrunoff method (unit hydrograph or kinematic wave) will produce a flow hydrograph at the outlet of each subbasin. Before these hydrographs can be summed to represent flow at the watershed outlet, the effects of travel time and channel/floodplain storage between the subbasin outlets and watershed outlet must be accounted for. The process of starting with a hydrograph at a location and recomputing the hydrograph at a downstream location is called hydrograph routing.
Figure 429 shows an example of a hydrograph at upstream location A, and the routed hydrograph at downstream location B. The resulting delay in flood peak is the travel time of the flood hydrograph. The resulting decrease in magnitude of the flood peak is the attenuation of the flood hydrograph.
Figure 429. Hydrograph routing (USACE 1994)
There are two general methods for routing hydrographs: hydrologic and hydraulic. The methods are distinguished by which equations are solved to compute the routed hydrograph.
Hydrologic methods solve the equation of continuity (conservation of mass), and typically rely on a second relationship (such as relation of storage to outflow) to complete the solution. The equation of continuity can be written as:
Equation 459.
Where:
 Anchor: #NKGLFIMF
 I = average inflow to reach or storage area during Δt Anchor: #ERNNLMHM
 O = average outflow to reach or storage area during Δt Anchor: #IILKGGMG
 S = storage in reach or storage area Anchor: #JLLEFMKG
 Δt = time step
Hydrologic methods are generally most appropriate for steep slope conditions with no significant backwater effects. Hydrologic routing methods include (USACE 1994):
 Anchor: #MKKJLMLE
 Modified Puls—for a single reservoir or channel modeled as series of levelpool reservoirs. Anchor: #KPHGMLEH
 Muskingum—channel modeled as a series of slopedpool reservoirs. Anchor: #ITNNNJHK
 MuskingumCunge—enhanced version of Muskingum method incorporating channel geometry and roughness information.
Most hydrologic software applications capable of multibasin analysis offer a selection of hydrologic routing methods.
Hydraulic routing methods solve the Saint Venant equations. These are the onedimensional equations of continuity (Equation 460) and conservation of momentum (Equation 461) written for openchannel flow. The equations are valid for gradually varied unsteady flow.
The onedimensional equation of continuity is:
Equation 460.
Where:
 Anchor: #IIGLMJLK
 A = crosssectional flow area Anchor: #MGFHEHGI
 V = average velocity of water Anchor: #KMEMIKME
 x = distance along channel Anchor: #MJFHHMMK
 B = water surface width Anchor: #MNOJNLGF
 y = depth of water Anchor: #QKGFHIMI
 t = time Anchor: #LHHFENLJ
 q = lateral inflow per unit length of channel
The onedimensional equation of conservation of momentum is:
Equation 461.
Where:
 Anchor: #HRJJLMHN
 S_{f} = friction slope Anchor: #HGIGFHEJ
 S_{o} = channel bed slope Anchor: #JHEHIGLG
 g = acceleration due to gravity
Hydraulic routing methods are computationally more intensive than hydrologic methods and are distinguished by which terms in the momentum equation (Equation 461) are included (not neglected) in the solution algorithm. Hydraulic routing methods include (USACE 1994):
 Anchor: #ITJGELLK
 Dynamic wave (all terms of St. Venant equations) Anchor: #KOMINNIH
 Diffusion wave Anchor: #LUFLNFFE
 Kinematic wave
Onedimensional unsteady openchannel flow software applications implicitly route hydrographs from one location to another by solving for depth and velocity at all locations (cross sections) in a stream reach (or network of reaches) for every time step. The hydraulic routing method employed is defined by the solution algorithm of the software application. Some applications allow the user to select which hydraulic routing method is used, while other applications support only one method.
The most robust routing method (in terms of steep/mild stream slope and with/without backwater effects) is dynamic wave routing.
Selection of Routing Method
Selection of an appropriate routing method depends on several factors. The application of any method will be improved if observed data are available for calibration/verification of routing parameters. Generally, hydrologic methods are most suitable for steeper reaches having little or no backwater effects resulting from high stages downstream of the routing reach. Hydraulic methods are generally more appropriate for a wider range of channel slopes, including gradual slopes, and can accommodate backwater effects. The exception to this is the MuskingumCunge method, which does not perform well with steeprising hydrographs in gradual slopes, or backwater conditions. Of all methods, only the dynamic wave routing method is appropriate for steep and gradual slopes, as well as with or without backwater conditions.
As a baseline approach, the designer may consider using the MuskingumCunge method in cases having steep slope (greater than 10 feet per mile) and no backwater effects. This method, which is described in Chow (1988) and Fread (1993), has the advantage that it will incorporate the shape of the cross section into computations. In some cases, cross section data may be obtained from existing hydraulic models of the reach. If channel geometry data are unavailable, then the Muskingum or modified Puls methods, which are described below, may be used. However, these two methods should be avoided for channel routing applications unless observed data area available for calibration/verification of routing parameters.
In cases having backwater that significantly affect the storageoutflow relationship of the routing reach, and thereby significantly affect the routed hydrograph, the dynamic wave, diffusion wave, and modified Puls methods are appropriate.
All methods, except for kinematic wave, are appropriate in cases having a channel slope between 2 to 10 feet per mile, no backwater effects, and satisfying the condition given by Equation 462 (USACE 1994):
Equation 462.
Where:
 Anchor: #KNEGJGLJ
 T = hydrograph duration (s) Anchor: #ILHMEGJJ
 S_{o} = average friction or slope (ft/ft) Anchor: #MJFKKHEH
 u_{o} = mean velocity (ft/s) Anchor: #FITNIJMG
 d_{o} = average flow depth (ft)
Only the dynamic wave, diffusion wave, and MuskingumCunge methods are appropriate in cases having a channel slope less than 2 feet per mile, no backwater effects, and satisfying the condition given by Equation 463 (USACE 1994):
Equation 463.
Where:
 Anchor: #LULENJMH
 g = 32.2 ft/s
In cases having a channel slope less than 2 feet per mile, no backwater effects, and not satisfying the condition given by Equation 463, then only the dynamic wave method is appropriate.
It may be tempting for the designer to select the dynamic wave routing method as a general approach for all conditions. However, the designer will find that the substantial amount of information (detailed and closelyspaced cross section geometry data) required to construct a onedimensional unsteady flow model, and the significant time required to ensure that the model is running properly without numerical instabilities, will provide motivation to identify a suitable hydrologic routing method when appropriate. If hydrologic methods are not appropriate for the case under consideration, then an unsteady flow model may be required to properly route flows.
Anchor: #ITLFHKGIReservoir Versus Channel Routing
Inflow hydrographs can be routed through reservoirs using a simple (single reservoir) hydrologic routing method, such as the modified Puls storage method. This is because the relationship between storage and discharge is unique (singlevalued). In other words, the storage in the reservoir is fully described by the stage in the reservoir because the surface of the reservoir is the same shape and slope during the rising and falling limbs of the hydrograph.
Hydrologic routing methods may also be used for channel routing. A channel does not have a singlevalued storageoutflow curve. Instead, the storageoutflow relation is looped, as shown in Figure 430. As a result, a hydrologic routing method employing a single reservoir representation cannot be used.
Figure 430. Looped storage outflow relation (USACE 1994)
The levelpool limitation of hydrologic routing methods is overcome by representing the channel as a series of reservoirs. These are termed subreaches, or steps, within the routing reach. Another enhancement to the levelpool approach, employed by the Muskingum method, is to represent the storage in each reservoir as a combination of prism storage (similar to levelpool reservoir) and wedge storage (additional sloped water on top of prism).
An estimate of the number of routing steps required for a hydrologic channel routing method is given by (USACE 1994):
Equation 464.
Where:
 Anchor: #NIJGLFFE
 n = number of routing steps Anchor: #NMIFEGNK
 K = floodwave travel time through the reach (min.) Anchor: #KNLLEKJF
 Δt = time step (min.)
K in the above equation is given by:
Equation 465.
Where:
 Anchor: #FLPFNEGF
 L = length of routing reach (ft) Anchor: #GKMKGFKI
 V_{W} = flood wave velocity (ft/s)
V_{W} may be approximated as equal to the average channel velocity during the flood hydrograph. A better estimate of V_{W} is given by Seddon’s law applied to a cross section representative of the routing reach (USACE 1994):
Equation 466.
Where:
 Anchor: #KKNKLNLE
 B = top with of the channel water surface (ft) Anchor: #KIIKGEGM
 Q = channel discharge (cfs) as function of elevation y Anchor: #NLMHJIIJ
 = slope of the discharge rating curve (ft^{2}/s)
Two hydrologic routing methods and their application are discussed further in the following sections: the modified Puls method for reservoir routing, and the Muskingum method for channel routing.
Modified Puls Method Reservoir Routing
Basic Concepts and Equations
The basic storage routing equation states that mass is conserved and can be expressed as follows: Average inflow  average outflow = Rate of change in storage
In numerical form, this statement of flow continuity can be written as:
Equation 467.
Where:
 Anchor: #MFEMIKGI
 I_{t} = inflow at time step number t Anchor: #LLEFGMFL
 I_{t+1} = inflow at time step number t + 1 Anchor: #FLGLEEMK
 O_{t} = outflow at time step number t Anchor: #MIFJHHHF
 O_{t+1} = outflow at time step number t + 1 Anchor: #OIHNIFNL
 S_{t} = storage in the reservoir at time step number t Anchor: #JIKEELNI
 S_{t+1} = storage in the reservoir at time step number t + 1 Anchor: #GJHNJMII
 = the time increment Anchor: #GPEMLFFI
 t = time step number
In Equation 464 there are two unknowns: O_{t+1} and S_{t+1}. In order to solve Equation 464, either a second equation with O_{t+1} and S_{t+1 }is required, or a relationship between O_{t+1} and S_{t+1 }is needed. The storageindication approach is the latter and is presented here. First, it is convenient to rewrite the routing equation as:
Equation 468.
In this form, all terms known at time t are on the right hand side of the equation and unknowns are on the left. If a singlevalued storageoutflow curve can be determined for the routing reach, then for any value of O_{t+1}, the corresponding value of S_{t+1} will be known. This reduces the number of unknown parameters in Equation 465 from two (O_{t+1} and S_{t+1}) to one (O_{t+1}).
Use of the storage routing method requires the designer to determine the relationship between storage and outflow. This is simply the volume of water held by the reservoir, storage facility, or pond as a function of the water surface elevation or depth. For a reservoir or storage facility, this information is often available from the reservoir sponsor or owner.
For a pond or lake or where the stagestorage relation is not available, a relationship between storage and outflow can be derived from considerations of physical properties of channel or pond and simple hydraulic models of outlet works or relationship of flow and water surface elevation. These physical properties include:
 Anchor: #MQFMIHKJ
 Ratings of the primary and/or emergency spillway of a reservoir. Anchor: #OLJFHJHG
 Pump flow characteristics in a pump station. Anchor: #FQMIGLEE
 Hydraulic performance curve of a culvert or bridge on a highway. Anchor: #HMFLFMKL
 Hydraulic performance curve of a weir and orifice outlet of a detention pond.
With the stagestorage relation established, a storage indication curve corresponding to the left side of Equation 468 is developed. The relationship is described in the form of O versus (2S/ΔT) + O. An example of a storage indication curve is provided in Figure 431.
Figure 431. Sample storageindication relation
The form of Equation 468 shown above is useful because the terms on the left side of the equation are known. With the relation between the outflow and storage determined (Figure 431), the ordinates on the outflow hydrograph can be determined directly.
Storage Routing Procedure
Use the following steps to route an inflow flood runoff hydrograph through a storage system such as a reservoir or detention pond:
 Anchor: #UFHHMJNK
 Acquire or develop a design flood runoff hydrograph for the project site watershed. Anchor: #PIKHEGKM
 Acquire or develop a stagestorage relation. Anchor: #KLNGFNIE
 Acquire or develop a stageoutflow relationship. Anchor: #QFGFHKFF
 Develop a storageoutflow relationship. Anchor: #QFGFKFLF
 Assume an initial value for O_{t} as equal to I_{t}. At time step one (t = 1), assume an initial value for O_{t} as equal to I_{t}. Usually, at time step one, inflow equals zero, so outflow will be zero and 2S_{1}/ΔT  O_{1} equals zero. Note that to start, t + 1 in the next step is 2. Anchor: #LIJELMIH
 Compute 2S_{t+1}/ΔT + O_{t+1} using Equation 468. Anchor: #QLLNNJFI
 Interpolate to find the value of outflow. From the storageoutflow relation, interpolate to find the value of outflow (O_{t+1}) at (2S_{t+1})/(ΔT)+O_{t+1} from step 6. Anchor: #MFENFFMK
 Determine the value of (2S_{t+1})/(ΔT)O_{t+1}. Use the relation (2S_{t+1})/(ΔT)O_{t+1} = (2S_{t+1})/(ΔT)+O_{t+1}  2O_{t+1}. Anchor: #HIHMGMJM
 Assign the next time step to the value of t, e.g., for the first run through set t = 2. Anchor: #MLKLIJGN
 Repeat steps 6 through 9 until the outflow value (O_{t+1}) approaches zero. Anchor: #IENKKGEF
 Plot the inflow and outflow hydrographs. The peak outflow value should always coincide with a point on the receding limb of the inflow hydrograph. Anchor: #NIGJLMLG
 Check conservation of mass to help verify
success of the process. Use Equation 469 to compare the inflow
volume to the sum of retained and outflow volumes:
Equation 469.
Where:
 Anchor: #JHFFGIHI
 S_{r} = difference in starting and ending storage (ft^{3} or m^{3}) Anchor: #ITNLIEJF
 ΣI_{t} = sum of inflow hydrograph ordinates (cfs or m^{3}/s) Anchor: #RMMNFHJK
 ΣO_{t} = sum of outflow hydrograph ordinates (cfs or m^{3}/s)
Muskingum Method Channel Routing
Routing of flood hydrographs by means of channel routing procedures is useful in instances where computed hydrographs are at points other than the points of interest. This is also true in those instances where the channel profile or plan is changed in such a way as to alter the natural velocity or channel storage characteristics. Routing estimates the effect of a channel reach on an inflow hydrograph. This section describes the Muskingum method equations, a lumped flow routing technique that approximates storage effects in the form of a prism and wedge component (Chow 1988).
The Muskingum method also solves the equation of continuity. With the Muskingum method, the storage in the channel is considered the sum of two components: prism storage and wedge storage (Figure 432).
Figure 432. Muskingum prism and wedge storage
The constants K and X are used to relate the prism component, KO, and wedge component, KX(IO), to the inflow and outflow of the reach:
Equation 470.
Where:
 Anchor: #KFJKGFIE
 S = total storage (ft^{3} or m^{3}) Anchor: #MHLMKIHH
 K = a proportionality constant representing the time of travel of a flood wave to traverse the reach (s). Often, this is set to the average travel time through the reach. Anchor: #EJOFELGL
 X = a weighting factor describing the backwater storage effects approximated as a wedge Anchor: #GLQHJEGE
 I = inflow (cfs or m^{3}/s) Anchor: #IOIJNHFH
 O = outflow (cfs or m^{3}/s)
The value of X depends on the amount of wedge storage; when X = 0, there is no backwater (reservoir type storage), and when X = 0.5, the storage is described as a full wedge. The weighting factor, X, ranges from 0 to 0.3 in natural streams. A value of 0.2 is typical.
Equation 468 represents the time rate of change of storage as the following:
Equation 471.
Where:
 Anchor: #GKGNJJLE
 ΔT = time interval usually ranging from 0.3K to K Anchor: #LLLGMNFL
 t = time step number
Combining Equation 470 with Equation 471 yields the Muskingum flow routing equation:
Equation 472.
Where:
Equation 473.
Equation 474.
Equation 475.
By definition, the sum of C_{1}, C_{2}, and C_{3} is 1. If measured inflow and outflow hydrographs are available, K and X can be estimated using Equation 471. Calculate X by plotting the numerator on the vertical axis and the denominator on the horizontal axis, and adjusting X until the loop collapses into a single line. The slope of the line equals K:
Equation 476.
The designer may also estimate K and X using the MuskingumCunge method described in Chow 1988 or Fread 1993.