Runoff Routing Methods
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 Created by Unknown User (reynard.juanir), last modified by Abraham Toribio on Feb 26, 2019
SWMM Runoff Nonlinear Reservoir Method
Subcatchments are modeled as idealized rectangular areas with the slope of the catchment perpendicular to the width. Each subcatchment is classified into 3 or 4 subareas as indicated in the following table.
Subarea  Perviousness  Depression Storage  Snow Cover and Extent  

Single Event  Continuous  
A1  Impervious  Yes  Bare  Normally bare, but may have snow cover on 100% of A1 plus A3 
A2  Pervious  Yes  Constant  Snow covered subject to fraction of area areal depletion curve which is snow covered 
A3  Impervious  No  Bare  Same as A1 
A4  Impervious  Yes  100% covered  Snow covered subject to areal depletion curve 
Flow from each subarea moves directly to a node and does not travel over any other subarea. The width of the pervious subarea, A2, is the entire subcatchment width, whereas the width of the impervious subareas A1, A3, A4 are in proportion to the ratio of their area to the total impervious area. Subcatchments are analyzed as spatially lumped nonlinear reservoirs. The routing is performed separately for each of the four subareas (A1A4) of the catchment.
Kinematic Wave (Runoff)
The kinematic wave method for overland flow applies only the kinematic wave component of the St Venant shallow flow equations for momentum and continuity. Similar to the Runoff procedure the Subcatchments are modelled as idealised rectangular areas with the slope of the catchment perpendicular to the width. The width parameter is used as per the Runoff procedure. Both the Runoff and Kinematic wave procedures only apply either the Horton or Green Ampt loss to the pervious percentage of the subarea, as defined by the % impervious data item. The only loss applied to the impervious component is via the depression storage defined for the impervious area in the infiltration dialog. Additional depression loss (defined with infiltration information) can also be applied to the pervious component. This will be an additional loss to the Horton or Green Ampt loss.
Henderson and Wooding (1964) and Wooding (1965) developed a theory for the overland and stream hydrograph. All flows are assumed to obey the equations of continuity and momentum. The continuity and momentum equations for overland kinematic waves reduce to the following two equations, respectively:
Continuity Equation:
and
Momentum Equation:
q = ay^{m} = 1.49/N S_{o}½ y ^{5/3}
Where:
y = y(x,t) = depth of overland flow (ft),
q = q(x,t) = rate of overland flow/unit width,
if = i_{e }= net rainfall rate,
a = conveyance factor = (1/N)S_{o}1/2 when obtained from Manning’s equation,
m = 5/3 when obtained from Manning’s equation,
N = effective roughness coefficient,
S_{o }= average overland slope,
Y_{o }= mean depth of overland flow.
After Bedient and Huber, 1992
The required data for this method is similar to the EPA Runoff method including area (ha or acres), impervious %, subarea width (m or ft) and slope (m/m or ft/ft).
Laurenson
When using Laurenson Hydrology the subcatchment width is by default not used. The catchment roughness utilised when calculating the storage delay parameter “B” is taken from the pervious Manning "n" value for the subcatchment included with the infiltration information. A second method is available by including the Configuration parameter RAFTS under the Configuration menu. This allows % imp to be set to 0 and 100 in the first two separate sub areas. This will allow the infiltration equation to be applied to the total subareas. Width is then used as a factor to control the “B” computation by inputting 0.0 and 100.0 respectively for the fully pervious and impervious sub areas. Appropriate infiltration can then be applied to the respective subareas.
The default procedure (without the RAFTS configuration parameter active) only applies either the Horton or Green Ampt loss to the pervious percentage of the subarea, as defined by the % impervious data item. It is encouraged to split the catchment into two subcatchments, with 0 and 100% impervious as the % imperviousness ratio is only applied to the U parameter of the Laurenson's method.
Routing for a particular subcatchment is carried out using the Muskingum procedure. The storage, however, is a nonlinear function of the discharge, ie:
s = K(q) × q (equation 3)
Where:
s = volume of storage, (hrs × m³/s)
q = instantaneous rate of runoff, (m³/s)
K(q) = storage delay time as a function of q (hours).
The storage function is used in the continuity equation in finite difference form thus:
(i1 + i2) × Δt/2  (q1 + q2) × Δt/2 = s2  s1 (equation 4)
Where:
i1, i2 = inflow at beginning and end of routing period (m³/s)
Δt = routing interval (hrs)
q1,q2 = outflow from the storage at beginning and end of routing period (m³/s)
s1,s2 = storage volume at beginning and end of routing period (hrs × m³/s).
Substituting s2 and s1 in Equation (4) from Equation (3) gives:
q2 = C0i2 + C1i1 + C2q1 (equation 5)
Where:
C0 = C1 = Δt/(2K2 + Δt) (equation 6)
C2 = (2K1  Δt)/(2K2 + Δt) (equation 7)
An iterative solution to Equation (5) is required due to the interrelation between C0, C1, C2, K2 and q2. K1, K2 is defined via Equation (8).
StorageDischarge Relationship
Each subarea is treated as a concentrated conceptual storage. Each storage has a storage delay time described thus:
K(q) = Bq^n (equation 8)
where:
K(q) = subarea storage delay time (hours) as a function of q
q = discharge (m3/s)
B = storage delay time coefficient
n = storage nonlinearity exponent.
Substituting Equation (8) into Equation (3) gives:
s = Bq^n + 1 (equation 9)
Nonlinearity Exponent "n"
The default value for the nonlinearity exponent in RAFTS is .285. RAFTSXP provides the mechanisms to alter this value, usually in respect to rare events involving significant subcatchment overbank flood routing, by:
directly entering an amended value of "n". A value of zero (.001) would indicate linear catchment response and equate with unit hydrograph theory, or
entering an “n” as a function of Q for each rating curve
It would be expected that the default "n" would prevail up to bank full and then, possibly, progressively drop back towards zero during PMF.
Coefficients B and n
B is either directly input for each subcatchment or estimated from Equation (10) which was derived by Aitken (1975). The value of B for each subarea is assumed to equal the average value of B for the subcatchment.
where:
B = mean value of coefficient B for subcatchment
A = subcatchment area (km²)
U = fraction of catchment that is urbanised. (Where U = 1.0, the catchment is fully urbanised and when U = 0.0, the catchment is completely rural)
Sc = main drainage slope of subcatchment (%). (The longest path of the subcatchment, starting at subcatchment outlet running up the main channel then if necessary branching off at the furthest tributary, to the top of the subcatchment.)
This equation was initially derived from six urban catchments in Australia with the following ranges applying:
A varied from 0.8 km² to 56 km²
U varied from 0.0 to 1.00
Sc varied from 0.22% to 2.90%.
However over the last fourteen years a wide range of areas, slopes and urbanisation outside these ranges have been tested with a high degree of success. See Sobinoff et al (1983).
For gauged catchments, deduced B values, evaluated as the average value from recorded rainfall/runoff events, should be used in preference to generalised regression estimates.
As 'U' in certain instances can be rather vague, data input in this respect has been amended to include a % impervious parameter for each subcatchment in place of the 'U' term.
The model interprets U in terms of %I based on the following ratios:
I%  U 

0  0 
30  0.7 
50  1.0 
100  2.0* 
* This value is extrapolated from the original data based on limited results from fully impervious areas.
B Modification Factors
Where gauged rainfall/runoff data is available for a range of events it should be used in preference to the above regression equation with modifying factors.
PERN. The original regression equation (Equation (10)) does not differentiate between catchments with the same urbanisation but different roughness. An additional empirical parameter has therefore been added to take pervious subcatchment roughness into account.
The parameter PERN is input as an Mannings 'n' representation of the average subcatchment roughness. B is then modified in accordance with the following table. If PERN is left blank then B is unchanged.
Mannings "n"  Multiplication Factor 

.010  0.4 
.015  0.5 
.025  1.0 
.100  3.0 
BX. During calibration of a gauged catchment an additional parameter BX in the header data is included to modify the calculated or input B by a further multiplication factor. The parameter BX will then uniformly modify all subcatchment B values previously computed, or set (Equation (10)). To invoke BX within this program it is necessary to utilise a Configuration Parameter selectable from the Tools menu.
To set a BX to a value of 1.5 for example add the Configuration Parameter BX=1.5.
SCS Hydrology
As an alternative to the nonlinear runoff routing method employed by Runoff, hydrographs may optionally be generated by the SCS method described in the SCS National Engineering Handbook  Section 4 (NEH4).
Pervious Area Curve Number
The runoff curve number as described in NEH4 Chapter 9. Typical values vary from 20 for regions with high infiltration and interception capacities to 98 for impervious areas.
It is a dimensionless number depending on hydrologic soil group, cover type, treatment, hydrological condition, and antecedent moisture conditions. This number has a valid range from 0 to 100 with typical values from 60 to 90 and 98 for impervious surfaces.
This curve number is only for the pervious area. Internally, an area weighted composite number is generated using this number for the pervious area and 98 for the impervious area. Essentially, the entire area is subject to infiltration using the composite number. Breaking the catchment to the pervious and impervious parts would allow infiltration to be restricted to the pervious area.
Hydrograph Shape Factor
The shape factor or peak rate factor as defined in NEH4 Chapter 16. The typical value is 484 for a hydrograph where the volume under the falling side of the triangular unit hydrograph is equal to 1.67 times the volume under the rising limb of the curvilinear unit hydrograph. Actual values may vary from 100 in very flat swampy country to 800 in steep rocky terrain.
The Soil Conservation Service has been determined the hydrograph shape factor to be 484 for most watersheds. This was the result of analyzing many watersheds of various size and geographic location. It is used in the formulation of peak discharge and the peak of the unit hydrograph:
Qp=484A/tp
in which Qp is the peak discharge in cfs A is the drainage area mi^2 and tp is the time to peak in hr. Several other values have been used for shape factor within Florida and other areas of the country for wetlands, sandy soils and steep terrain. Typical values in Florida include 323 or 256 and values as low as 100. Effectively, the hydrograph shape factor will lengthen the time base of the unit hydrograph and lower the peak. If you are using real rainfall data this parameter can be used for calibration. For example the falling limb time base of the hydrograph will be double when using a shape factor of 300 instead of 484.
A curvilinear or triangular shape may be chosen by selecting the appropriate button. Both shapes are supported for the entire range of hydrograph shape factors.
Time of Concentration
The time of concentration as described in NEH4 Chapter 15 (in mins) is defined in two ways; the time for runoff to travel from the furthermost point of the watershed to the point in question, and the time from the end of excess rainfall to the point of inflection on the trailing limb of the unit hydrograph. Tc can be estimated from several formula such as Kinematicwave. For a constant excess rainfall can be described as:
Tc=C(n^0.6 L^0.6/i^0.4 S^0.3)
in which L is the distance from the upper end of the plane to the point of interest (usually the inlet) n is the Manning resistance coefficient, i is the excess rainfall rate, S is the dimensionless slope of the surface, and C is a constant that depends on units of the other variables. For tc in minutes, i in inches/hr and L in feet C equals 0.938. For tc in minutes i in mm/hr and L in meters C equals 6.99.
Another formula for determining tc is the lag equation
Tc = L/0.6
L= l^0.8(S+1)^0.7/1900Y^0.5
where:
Tc = time of concentration in hours
L = lag time in hours
I = hydraulic length of the watershed in feet
Y = Average land slope in percent
S = Potential maximum retention in inches
CN = weighted Curve Number
The time of concentration is required as input although internally the time to peak is used in creating the unit hydrograph. Tp=2/3 Tc for shape factor 484.
Calc: The time of concentration can be calculated using the Time of Concentration Calculator.
Initial Abstraction
The initial abstraction from the precipitation may be represented as an absolute number, that is the total depth of precipitation that is less (in mm or inches) or as a fraction of the amount of precipitation (between 0 to 1).
Initial Abstraction, Ia, contains all the loss terms prior to beginning of runoff. All though it was found to be highly variable for many small agricultural watersheds it was found that
Ia = 0.2 × S
Where
S = (1000/CN)10
S is the potential maximum abstraction.
xp allows the user to use a fraction with .2 being the default or the Initial abstraction can be a depth in inches or mm.
Additional References:
James 1998, Editor "Modeling the Management of Stormwater Impacts" Volume 6 Chapter 23 Author Ashok Pandit and Joanie Regan What is the Impervious Area Curve Number. Publisher CHI Guelph Ontario Canada.
Viessman W. et al 1977, "Introduction to Hydrology", Harper&Row Publishers New York
USDA National Resources Conservation Service, 1986, Urban Hydrology for Small Watersheds, Technical Release 55.
WEF 1992, Manual of Practice FD20 "Design and Construction of Urban Stromwater Management Systems" ASCE and WEF ISBN 0872628558
Unit Hydrographs
The unit hydrograph and quasi unit hydrograph procedures currently available include:
 Nash
 Rational Hydrograph (Triangular Unit Hydrograph)
 Santa Barbara Urban Hydrograph
 Snyder
 Snyder (Alameda modified)
 Time Area
 Clark
When one of the above methods requires a Time to Peak or Time of Concentration this can be calculated using the Time of Concentration Calculator.
Nash Unit Hydrograph Procedure
Nash in 1957 proposed a conceptual catchment model by considering a drainage basin with a series of identical linear reservoirs in series. By routing a unit inflow through the reservoirs, a mathematical equation for instantaneous unit hydrograph can be derived. Two items of data are required to apply this method. These include an Exponent and a Time of concentration. This method requires calibration data to base input data.
Rational Hydrograph
The Rational method as applied by this method uses a unit hydrograph approach applying a deterministic form of the Rational formula Q = CIA. The two items of data needed for this procedure are the subarea runoff coefficient “C” and time of concentration tc. The method utilizes the rainfall concurrent to the time step being computed. The unit hydrograph incorporates a base length equal to two times the time of concentration.
This method is currently not recommended without calibration data or for models with storage nodes and significant attenuation.
This method is NOT recommended as it requires a new Rainfall event for every Tc in the model and Tc’s must be calculated by hand. If the Rational method is needed then the Rational Formula method should be used.
Santa Barbara Urban Hydrograph Procedure
The SBUH method, like the Soil Conservation Service Urban Hydrograph (SCSUH) method, is based on the curve number (CN) approach, and also uses SCS equations for computing soil absorption and precipitation excess. The SCSUH method works by converting the incremental runoff depths (precipitation excess) for a given basin and design storm hydrographs of equal time base according to basin time of concentration and adds them to form the runoff hydrograph. The SBUH method, on the other hand, converts the incremental runoff depths into instantaneous hydrographs which are then routed through an imaginary reservoir with a time delay equal to the basin time of concentration.
The SBUH method was developed by the Santa Barbara County Flood Control and Water Conservation District, California. The SBUH method directly computes a runoff hydrograph without going through an intermediate process (unit hydrograph) as the SCSUH method does.
The SBUH method uses two steps to synthesize the runoff hydrograph:
Step one – computing the instantaneous hydrograph, and step two – computing the runoff hydrograph.
The instantaneous hydrograph, I(t), in cfs, at each step, dt, is computed as follows:
I(t) = 60.5 R(t) A/dt
where:
R(t) = total runoff depth (both impervious and pervious runoffs) at time increment dt, in inches (also known as precipitation excess)
A = area in acres
dt = time interval in minutes*
*A maximum time interval of 10 minutes should be used for all design storms of 24hour duration. A maximum time interval of 60 minutes should be used for the 100year, 7 day design storm.
The runoff hydrograph, Q(t), is then obtained by routing the instantaneous hydrograph I(t), through an imaginary reservoir with a time delay equal to the time of concentration, Tc, of the drainage basin. The following equation estimates the routed flow, Q(t):
Q(t +1) = Q(t) + w[I(t) +I(t +1) – 2Q(t)]
where:
w = dt/(2T + dt)
dt = time interval in minutes
Snyder Unit Hydrograph Procedure
Snyder (1938) was the first to develop a synthetic unit hydrograph based on a study of watersheds in the Appalachian Highlands. In basins ranging from 10 to 10,000 mi^{2}. Snyder’s relationships are
T_{p} = _{Ct}(LL_{c})^{0.3 }(equation 1)
where
t_{p} = basin lag
L = length of the main stream from the outlet to the divide,
L_{c} = Length along the main stream to a point nearest the watershed centroid,
C_{t} = coefficient usually ranging from 1.8 – 2.2 (Ct has been found to vary from 0.4 in mountainous areas to 8.0 along the Gulf of Mexico),
Q_{p} = 640CpA/t_{p}, (equation 2)
Where
Q_{p} = peak discharge of the unit hydrograph
A = drainage area
C_{p} = storage coefficient ranging from 0.4 to 0.8 where larger values of Cp are associated with smaller values of Ct
T_{b} = 3 + t_{p}/8 (equation 3)
Where:
Tb is the time base of the hydrograph. For small waterheds, Eq 3 should be replaced by multiplying tp by a value of from 3 to 5 as a better estimate of Tb. Eqns. 1,2 and 3 define points for a unit hydrograph produced bt an excess rainfall odf duration D = tp/5.5.
For other rainfall excess durations D’, an adjusted formula for tp becomes:
T_{p}’ = t_{p} + 0.25 (D’ – D) (equation 4)
Where:
Tp’ is the adjusted lag time for duration D’. Once the three quantities tp, Qp, and tb are known, unit hydrograph can be sketched so that the area under the curve represents 1,0 in of direct runoff from the watershed.
In this application the two items of data are:
C_{p} = Storage coefficient plus
T_{p} = catchment lag
The percentage of imperviousness is not an input value for the Synder Runoff method. By design the percentage of imperviousness should be set to 0% for this reason.
Once the percentage is set to 0% it means that the whole catchment will be subject to infiltration. If using the Curve Number method for infiltration, the impervious parts of the catchment can be accounted for by modifying the curve number (CN) that is used for the infiltration accordingly. We recommend a CN value of 98 for the impervious part of the catchment. Users can also check the total infiltration volume in Table R5 of the output file.
Snyder (Alameda) Unit Hydrograph Procedure  old
The Snyder (Alameda) method is the Snyder procedure as applied by Alameda County in California where the individual parameters more suited to Alameda region are computed from catchment characteristics.
This method requires four items of data:
 Stream Length (L) mile
 Centroid Length (Lc) mile
 Stream Slope (S) ft/mile
 Basin “N” (roughness)
where
Lag = K.N ( (L . Lc) / S ^{1/2}) ^{0.38}
This data allows the other Synder parameters of t_{p}, Q_{p}, and T_{b} to be computed directly.
Time Area Method
Timearea methods utilise a convolution of the rainfall excess hyetograph with a timearea diagram representing the progressive area contributions within a catchment in set time increments. Separate hydrographs are generated for the impervious and pervious surfaces within the catchment. These are combined to estimate the total flow inputs to individual subcatchment entries to the underground urban drain network.
The timearea method dates from the research of Ross in 1922. Networked urban drainage adoptions of the procedure however only date back to 1963. This computerised program known as the TRRL Method was developed by the U.K. Transport and Road Research Laboratory, described by Watkins (1963). In the U.S. Terstriep and Stall (1974) further developed the method to include pervious runoff. In South Africa Watson (1981) made a number of additional changes particularly to the way infiltration was estimated. Between 1982 and 1986 O’Loughlin (1988) using Watson’s model as a basis carried out extensive changes once again to formulate a computerised package known as ILSAX. Over the intervening years to the present the ILSAX model has had wide usage throughout Australia particularly for routine urban drainage design. The subcatchment runoff estimating procedure still utilises the basic timearea method to estimate both pervious and impervious portion runoff.
In this application the time/area procedure assumes a linear time area relationship for the subarea and is based on an input time of concentration. It is recommended in urban catchments to define two subarea elements one each to represent the impervious and pervious areas respectively. Values for Impervious% should both be set to 0 (zero) so the input infiltration using either Horton or Green Ampt applies to the full defined area of each. Depression storage loss as defined in the infiltration dialog is currently unused.
The only input necessary for this procedure is the time of concentration for the subarea.
Clark Unit Hydrograph
Enter:
Tc, time of concentration, min for the subcatchment
R, storage coefficient, hr, must be => 0.5 × Runoff time step.
The Clark method (1945) requires three parameters to calculate a unit hydrograph: Tc, the time of concentration for the basin, R, a storage coefficient, and a timearea curve. A timearea curve defines the cumulative area of the watershed contributing runoff to the subcatchment outlet as a function of time (expressed as a proportion of Tc).
xswmm utilizes a dimensionless time area curve:
AI =  1.414 T ^1.5  for 0 <= T < 0.5 
1 – AI =  1.414 (1 – T) ^1.5  for 0.5 < T <1 
where:
AI = cumulative area as a fraction of total subcatchment area and
T = fraction of time of concentration.
The ordinates of the timearea curve are converted to volume of runoff per second for unit excess and interpolated to the given time interval. The resulting translation hydrograph is then routed through a linear reservoir to simulate the storage effects of the basin; and the resulting unit hydrograph for instantaneous excess is averaged to produce the hydrograph for unit excess occurring in the given time interval.
The linear reservoir routing is accomplished using the general equation:
Q(2) = CA × I × CB × Q(1)
The routing coefficients are calculated from:
CA = Delta T / (R + 0.5 × Delta T)
CB = 1  CA
Qungr = 0.5 [Q(1) + Q(2)]
where:
Q(2) = instantaneous flow at end of period,
Q(1) = instantaneous flow at the beginning of period,
I = ordinate of the translation hydrograph,
Delta T = computation time interval in hours (also duration of unit excess),
R = basin storage factor in hours, and
Qungr = the unit hydrograph ordinate at end of computation interval.
The computation of unit hydrograph ordinates is terminated when its volume exceeds 0.995 inch (mm) or 150 ordinates, whichever occurs first.
Rational Formula Hydrology
Runoff Coefficient
Pervious Runoff C. The Runoff Coefficient will be required if the Direct or Alameda method for calculating C is selected in the Rational Formula Global Database.
Runoff Curve Number. The Runoff Curve Number will only be required if AR&R 1977 (the Ordon Curve method) is selected in the Rational Formula Global Database.
TC Calculation
Additional Travel Time. Travel time from outlet of subcatchment to the connecting node (minutes). This option allows for defining odd shaped catchments, or catchments whose flow concentrates at points remote to the node itself. This time does NOT represent the gutter time component of a time of concentration, but the additional time beyond the time of concentration. The typical use of this value is to represent a remote catchment connected by a pipe or drain without explicitly modeling the connecting conduit itself. To allow for gutter travel time, an equivalent length of overland path should be added to the overland flow length.
Time of concentration. Time of Concentration, Tc can be input as a constant value (minutes) or calculated using one of the equations listed in the TC Calculator. A minimum Time of Concentration value of 5 minutes is customary and required.
Time of Constant Flow. The program creates a hydrograph from the rational formula. The duration of the rising and recession limbs of the hydrograph are each equal to the time of concentration. The duration of the period of constant flow is entered in this field (in minutes). To simulate constant flow enter a duration equal to or longer than the Job Control simulation time in this field.
When using the Rational Formula, Q=kCIA, where k=1.008 for English units and is often omitted from the equation. In SI units, however, k=0.00278 and should not be omitted from the equation. It is also critical to enter proper values for C and I.
Time of Concentration Methods
Friend's Equation
where;
B = Const (72 US Units, 107 metric)
S = Average catchment slope (Percent, %)
L = Length of overland flow (ft, m)
n = Horton's roughness (similar but not identical to Manning’s n)
 Paved 0.015
 Bare Soil .0275
 Poorly Grassed .035
 Average Grass .045
 Dense grass 0.06
Kinematic Wave
where:
L = Length of overland flow (ft, m)
S = Average catchment slope (ft/ft, m/m)
i = Rainfall Intensity (in/hr, mm/hr)
n = Surface roughness (similar but not identical to Manning’s n)
 Concrete or Asphalt 0.010  0.013
 Bare Sand 0.010  0.016
 Graveled Surface 0.012  0.030
 Bare ClayLoam Soil (eroded) 0.012  0.033
 Sparse Vegetation 0.053  0.130
 Short Grass Prairie 0.100  0.200
 Lawns 0.170  0.480
Izzard's Formula
Where:
L = Length of overland flow (ft, m)
S = Average catchment slope (ft/ft, m/m)
i = Rainfall Intensity (in/hr, mm/hr)
C_{r} = Retardance Coefficient
 Very smooth asphalt 0.007
 Tar and sand pavement 0.0075
 Concrete 0.012
 Tar and gravel pavement 0.017
 Short grass 0.046
 Dense grass 0.06
Alameda Method
where;
L = Length of overland flow (ft, m)
V = Overland flow velocity (m/s, ft/s)
Kerby's Equation
where;
L = Length of overland flow (ft, m)
S = Average catchment slope (ft/ft, m/m)
n = Retardance Roughness
 Smooth pavement 0.02
 Poor grass, bare soil 0.30
 Average grass 0.40
 Dense grass 0.80
Kirpich’s Equation
where:
L = Length of overland flow (ft, m)
S = Average catchment slope (ft/ft, m/m)
Bransby Williams Method
where:
L = Length of overland flow (ft)
A = Catchment area (acres)
S = Average catchment slope (ft/ft)
When working in metric units (of meters, hectares, and dimensionless slope m/m), the input is converted automatically to US units for use in this equation. Use meters, hectares and dimensionless slope for a metric unit model.
US Federal Aviation Authority
where;
L = Length of overland flow (ft, m)
S = Average catchment slope (ft/ft, m/m)
C = Runoff coefficient
 Business district 0.70  0.95
 Single family residential 0.30  0.50
 Asphalt/Concrete 0.70  0.95
 Sandy soil lawn 0.05  0.20
 Heavy soil lawn 0.13  0.35
 Brick 0.70  0.85
TR55 Methods
From the Technical Release 55 (TR55) – Urban Hydrology for Small Watersheds manual – Chapter 3, we have included Sheet Flow, Shallow Concentrated Flow and Channel Flow Time of Concentration calculation options. Additional details and recommendations on these methods and parameters can be reviewed by browsing to the USDA NRCS TR55 Manual.
Sheet Flow
Shallow
Channel Flow
UK Hydrology
There are five major hydrology methods available within UK Hydrology. The following methods were derived in the UK though may be applicable elsewhere.
The following references are applicable for each method:
General – The SUDS Manual, CIRIA C697, London, 2007.
Variable PR – A New Runoff Volume Model. WaPUG User Note No.28, Osborne, M. Version 3, March 2009.
Wallingford and Variable PR  Design and Analysis of Urban Storm Drainage: The Wallingford Procedure, Volume 1 – Principles, methods and practice. Department of Environment, Wallingford. 1981.
ReFH – Flood Estimation Handbook Supplementary Report No. 1: The revitalised FSR/FEH rainfallrunoff method, Kjeldsen, Centre for Ecology & Hydrology, 2007.
FEH – Flood Estimation Handbook, Volume 4, Reed, Faulkner, HoughtonCarr, Robson & Bayliss, Institute for Hydrology, 1999.
FSR – Flood Studies Report Volumes 15, Institute of Hydrology, 1975 and Design of Flood Storage Reservoirs, CIRIA Book 14, 1993.
It is important to note that some data fields may be shared between hydrology methods. These shared data field values may not be the same for all hydrology methods and should be tailored for the specific UK Hydrology method selected.
Variable PR Method (Variable PR)
The Variable PR runoff method, sometimes known as New PR or New UK method, uses the nonlinear reservoir routing hydrology method as per the Wallingford Procedure runoff method. The volume of percentage runoff varies during the rainfall event to take account of changes to the catchment wetness during the storm. The Variable PR method routing equation is shown below:
PR=IF * PIMP + (100IF * PIMP) * NAPI/PF
Routing Coefficient (KR). The Variable PR runoff method routing coefficient (KR) is typically input as a value 0.5 or less when used in rural catchments.
Effective Impervious Factor (IF). The Effective Impervious Factor is used to give an effective contributing impervious area based on the impervious area runoff. Recommended Values for the Effective Impervious Factor (IF) are shown below:
Surface condition  Effective impervious area factor, IF 

Poor  0.45 
Fair  0.60 
Good  0.75 
Maximum Moisture Depth (PF). The Maximum Moisture Depth is typically set at 200mm.
Net Antecedent Precipitation Index (NAPI). The Net Antecedent Precipitation Index (NAPI) is the 30 day antecedent precipitation index, which is directly dependent on the soil type. Starting NAPI and Maximum Soil Moisture are required. The Net Antecedent Precipitation Index is defined by the following equation:
Where:
k  Soil Type 
0.10  1 
0.50  2 
0.70  3 
0.90  4 
0.99  5 
t  model timestep (hours)
NP  Rain less unsatisfied DEPSTOG in previous timestep (DEPSTOG – depression storage).
After the depression storage is filled, evaporation will be subtracted from the rainfall and the remaining amount will be routed through the same reservoir than in the Wallingford Method. Qvol is multiplied with AREA to get the outflow volume in cubic meter.
WRAP Soil (%). The WRAP SOIL percentages define the index of the water holding capacity of the soil (0.15 tot 0.5), based on the FSR WRAP parameter obtained from FSR. The formula for calculating the WRAP Soil index is shown below:
WRAP SOIL Index = (Soil 1 % * 0.15) + (Soil 2 % * 0.3) + (Soil 3 % * 0.4) + (Soil 4 % * 0.45) + (Soil 5 % * 0.5).
There are five possible WRAP SOIL percentages which can be input to define the subcatchment soil conditions. The WRAP SOIL percentages are input as fractions and the sum of which must equal 100% (i.e.: Soil 1 % + Soil 2 %...+ Soil 5 % = 1.0). The W.R.A.P. soil classifications are defined as follows:
Classification  Description 

1 

2 

3 

4  Clayey, or loamy over clayey soils with an impermeable layer at shallow depth 
5  Soils of the wet uplands with:

Wallingford Procedure (Wallingford)
The Wallingford Procedure runoff method (Wallingford) is a nonlinear reservoir routing method, This method can be simulated with any runoff time step though using the appropriate time step is essential for simulating the correct reservoir routing. The percentage of runoff is fixed throughout the storm. The Wallingford Procedure runoff method routing equation is shown below:
PR = 0.829PIMP+25SOIL+0.078UCWI20.7
Routing Coefficient (KR). The Variable PR runoff method routing coefficient (KR) is typically input as a value 0.5 or less when used in rural catchments.
Antecedent Wetness Condition (UCWI). The Antecedent Wetness Condition (Urban Catchment Wetness Index  UCWI), which is a composite of two antecedent wetness parameters and is given by:
UCWI = 125 +8API_{5} – SMD
Where
API_{5" }= 5day antecedent precipitation index (mm)
SMD = soil moisture deficit (mm)
Alternatively, design values are provided by referring to average annual rainfall for a given location.
WRAP SOIL (%). The WRAP SOIL percentages define the index of the water holding capacity of the soil (0.15 tot 0.5), based on the FSR WRAP parameter obtained from FSR. The formula for calculating the WRAP Soil index is shown below:
WRAP SOIL Index = (Soil 1 % * 0.15) + (Soil 2 % * 0.3) + (Soil 3 % * 0.4) + (Soil 4 % * 0.45) + (Soil 5 % * 0.5).
There are five possible WRAP SOIL percentages which can be input to define the subcatchment soil conditions. The WRAP SOIL percentages are input as fractions and the sum of which must equal 100% (i.e.: Soil 1 % + Soil 2 %...+ Soil 5 % = 1.0). The W.R.A.P. soil classifications are defined as follows:
Classification  Description 

1 

2 

3 

4  Clayey, or loamy over clayey soils with an impermeable layer at shallow depth 
5  Soils of the wet uplands with:

It is noted that while using the Wallingford Procedure runoff method the following subcatchment slope values are recommended:
1.25%  mild slope
2.75%  medium slope
4.0%  steep slope
Revitalised FSR/FEH Method (ReFH)
The Revitalised FSR/FEH Method is a unit hydrograph method which refines the original FSR/FEH Method. The base flow component is dynamic in this method, it is computed by a routing model of the runoff. Tthe runoff time step has to fulfill the condition 0.02*TP <= timestep <= 0.2*TP. In contrast to the original FSR method this method updates the percentage runoff in each time step. The difference between net and effective rain is infiltration.
Season and Return Period. This input field allows the user to specify in which season the simulation will occur. Recommended seasons are; Winter  November to April, and Summer  May to October. The Return Period represents the number of months in the season. The initial soil moisture and initial baseflow values will vary depending on the season selected.
Mean Drainage Path Slope (DPSBAR). The Mean Drainage Path Slope represents the average subcatchment drainage path slope.
Extent of Urban and Suburban Land Cover (URBEXT). For the Extent of Urban and Suburban Land Cover either select the 1990 or 2000 values if available or simply enter a userdefined value. Please note that the year 2000 values are only available if using the FEH CDROM Version 2. If values for 1990 or 2000 are zero, please check the values as they may not be available from the FEH data.
Proportion of Time when Soil Moisture Deficit was below 6mm (PROPWET). This parameter represents the proportion of time when the soil moisture deficit was below 6mm during the years between 19611990, as defined by Met Office Rainfall and Evaporation Calculation System (MORECS).
Mean Drainage Path Length (DPLBAR). The Mean Drainage Path Length represents the average subcatchment drainage distance.
Average Annual Rainfall (SAAR). This input parameter is the average annual rainfall value from between 1961 to 1990.
Base Flow Index (BFIHOST). The Base Flow Index represents the baseflow index derived from Hydrology of Soil Types (HOST) soils data.
Overrule Option. The Time to Peak and Baseflow calculations can be specified using the Overrule option. The user input of these parameters is optional. If they are not specified by the user, the engine will generate its own values based on other input parameters. Note that if a user defined baseflow is used it is a constant baseflow and does not vary with the rainfall.
Loss Model (Rainfall). The Loss Model used to derive Net Rainfall
Net rainfall, p_{t}, measured in mm, is needed for the convolution of the unit hydrograph. It is derived from the equation below where p_{t}/P_{t} represents percentage runoff using P, the design rainfall derived above. Calculate the following parameters:
Calculate Maximum Soil Moisture Capacity, C_{max}, (mm):
Eq.6
Calculate Initial Soil Moisture Capacity, C_{ini}, (mm):
If season is Winter use:
Eq.7
If season is Summer use:
Eq.8
If C_{ini}/0.5C_{max} < 1, then C_{ini} = 0.
The soil moisture storage C_{t} (mm) is calculated from:
Eq.9
With C_{t} at time zero equal to Alpha_{T} x C_{ini}.
Correction factor, Alpha_{T}:
Alpha_{t} = 1 for return periods less than 5 years
If season is Winter use:
Eq.10
If season is Summer use:
Eq.11
Calculate the net rainfall, p_{t}:
If C_{t} < C_{max, }use the following equations:
For t = 1:
Eq.12
For t = 2, 3,…
Eq.13
If soil storage filled during the event, where C_{t} = C_{max}, use:
Eq.14
Clarification for calculation of the first two timesteps:
At time zero, t = 0:
Eq.15
First timestep: t =1
Eq.16
using C_{0} calculated at t = 0, giving:
Eq.17
Second timestep: t=2
Eq.18
using C_{1} calculated at t = 1, giving:
Eq.19
and so on.
Convert net rainfall values to correct units
Net rainfall values should be divided by 10 to convert them from mm into cm for convolution with the unit hydrograph.
Hydrograph
Instantaneous Unit Hydrograph Calculations:
Calculate Time to Peak, Tp from Eq. 1 above:
Eq.1
Calculate other parameters necessary to draw the ReFH Kinked Triangle unit hydrograph.
Eq. 20
Eq. 21
Eq. 22
Eq. 23
Eq. 24
Eq. 25
Plot the unit hydrograph using points (Tp, U_{p}), (2Tp, U_{2TP}), (TB, 0)
Finding the unit hydrograph ordinates
If t ≤ Tp use:
Eq. 26
If Tp ≤ t ≤ 2Tp use:
Eq. 27
If 2Tp ≤ t ≤ TB use:
Eq. 28
Converting the IUH to a TUH
To convert the IUH to a TUH (which is the UH for a finite rainfall duration) calculate the area under the graph between timesteps effectively applying an SCurve to the hydrograph. If the peak and kink of the unit hydrograph do not fall on an exact timestep the area will need to be corrected. These areas should be used for the convolution, for example the area under the graph between timestep 0 and timestep 1 should be used as the unit hydrograph ordinate for timestep 1.
The area under the entire curve should be equal to Tp.
Before convoluting the unit hydrograph, the areas must be multiplied by a factor of 2.78*AREA/Tp. This converts the units into m^{3}/s/cm ready for convolution.
Baseflow Model
Calculate Baseflow Equation.
Calculate Baseflow Recharge Parameter, BR
Eq. 30
Calculate Baseflow Lag, BL (hours):
Eq. 31
BL is calculated in hours but should be divided through by 24 to give it in days to be used in the baseflow equation above and in the baseflow constant equations below.
Calculate Initial Baseflow, BF_{0}, (m^{3}/s):
The following equation can be used for either season:
Eq. 32
If season is Winter use:
Eq. 33
If season is Summer use:
Eq. 34
Baseflow Constant Equations:
Eq. 35
Eq. 36
Eq. 37
Where Delta is the timestep in days.
Use these parameters to calculate the baseflow profile:
Eq. 38
where q_{t} is direct runoff from the routing model and K_{1}, K_{2}, and K_{3} are constants and functions of the baseflow parameters BR and BL.
Flood Estimation Handbook (FEH)
The Flood Estimate Handbook (FEH) hydrology method is a unit hydrograph method where a constant baseflow is added to the runoff, similar to the Flood Studies Report hydrology method (FSR). The catchment descriptors can be obtained from the FEH CDROM which is available from http://www.hydrosolutions.co.uk. The runoff time step has to fulfill the condition 0.02*TP <= time step <= 0.2*TP.
Mean Drainage Path Slope (DPSBAR). The Mean Drainage Path Slope represents the average subcatchment drainage path slope.
Extent of Urban and Suburban Land Cover (URBEXT). For the Extent of Urban and Suburban Land Cover either select the 1990 or 2000 values if available or simply enter a userdefined value. Please note that the year 2000 values are only available if using the FEH CDROM Version 2. If values for 1990 or 2000 are zero, please check the values as they may not be available from the FEH data.
Standard Runoff (%) (SPRHOST). The digital FEH data has 29 soil categories while the FSR had only 5.
SPR = SPRHOST = SPR_{1}HOST_{1}+ …..+SPR_{29}HOST_{29}
SPRHOST is calculated by the FEH CD program and written to the .CSV file.
Where measured flow records are available the SPR may be calculated from the Base Flow Index (BFI) from research published in FSSR 16 and it is not covered here – see also FEH, Volume 4, Chapter 1.
At the moment XPSWMM only offers direct input for SPR
Proportion of Time when Soil Moisture Deficit was below 6mm (PROPWET). This parameter represents the proportion of time when the soil moisture deficit was below 6mm during the years between 19611990, as defined by Met Office Rainfall and Evaporation Calculation System (MORECS).
Mean Drainage Path Length (DPLBAR). The Mean Drainage Path Length represents the average subcatchment drainage distance.
Average Annual Rainfall (SAAR). This input parameter is the average annual rainfall value from between 1961 to 1990.
Overrule Option. The Time to Peak, Catchment Wetness Index (CWI) and Baseflow calculations can be specified using the Overrule option. The user input of these parameters is optional. If they are not specified by the user, the engine will generate its own values based on other input parameters. Note that if a user defined baseflow is used it is a constant baseflow and does not vary with the rainfall.
Computed Variables and Hydrograph
TP0 (hrs)
Time to peak is computed based on the following formula:
TP0 = 4.27 * DPSBAR^{0.35} * PROPWET^{0.5} * DPLBAR0.54 * (1+URBEXT)^{5.77}
TPt (hrs)
Time to rise of the unit hydrograph as defined by:
TPt = TP0 + T/2, T time step
QP (m³/s 10mm)
The peak flow for the unit hydrograph as calculated by:
TB (hrs)
Time base of the unit hydrograph as calculated by:
TB = 2.52 * TPt
Base Flow (m³/s)
Base flow for the unit hydrograph as calculated by:
BASEFLOW=AREA [* (33 * (CWI  125) + 3 * SAAR + 5.5] * 10^{5 }
PR (%)
Total percentage run off of the catchment is defined as follows:
where
, and
, if P > 40mm and
otherwise, P is the rainfall depth.
.
Flood Studies Report (FSR)
The Flood Studies Report runoff method (FSR) is a unit hydrograph method where a constant baseflow is added to the runoff. Catchment descriptors should be obtained from measured data. The runoff time step has to fulfill the condition 0.02*TP <= time step <= 0.2*TP.
Height at 85% from Catchment Outlet (H85). This input parameter, in conjunction with the Height at 10% from Catchment Outlet (H10), allows the main channel slope (S1085) to be calculated. The values are the heights at 10% and 85% of the main channel length upstream from the catchment outlet.
Height at 10% from Catchment Outlet (H10). This input parameter, in conjunction with the Height at 85% from Catchment Outlet (H85), allows the main channel slope (S1085) to be calculated. The values are the heights at 10% and 85% of the main channel length upstream from the catchment outlet.
Extent of Urban and Suburban Land Cover
Standard Runoff (%) (SPR%). This value can be calculated from Figure I.4.18 (first revision) (Institute of Hydrology, 1978) which shows the various soil types. The soil classifications are also available from the Walingford Procedure, Volume 3, Maps, “Winter rain acceptance potential”. The equation was first published in FSSR 16, 1985.
SPR = 10S1 + 30S2 + 37S3 + 47S4 + 50S5
where
S1 – S5 are the proportions of the catchment covered by soil classes 15 respectively.
Where measured flow records are available, the SPR may be calculated from the Base Flow Index (BFI) from research published in FSSR 16 and it is not covered here – see also FEH, Volume 4, Chapter 1.
Length of Main Channel (MSL). The Length of the Main Channel or Stream (MSL) measured in metres on a 1:25000 Ordnance Survey Map. It is normally measured in kilometres (and represented in the formulae in km) but it is input in metres to be consistent with other data input.
Average Annual Rainfall (SAAR). This input parameter is the average annual rainfall value from 1941 to 1970 from FSR Figure II.3.1 (Natural Environment Research Council, 1975). The data is also available from the Wallingford Procedure, Volume 3 Maps and the FEH CD.
Overrule Option. The Time to Peak, Catchment Wetness Index (CWI) and Baseflow calculations can be specified using the Overrule option. The user input of these parameters is optional. If they are not specified by the user, the engine will generate its own values based on other input parameters. Note that if a user defined baseflow is used it is a constant baseflow and does not vary with the rainfall.
Computed Variables and Hydrograph
TP0 (hrs) This is the instantaneous time to peak of the unit hydrograph. Ciria14 uses the equation from FSSR 16 published in 1985. However IH 124 published a more recent equation that outperforms FSSR 16 on small, partly urbanised catchments. The IH equation also performed well on the FSSR 16 data set and therefore it is recommended for general use i.e. on small and large catchments, rural and partly urban. For these reasons the IH 124 formula has been used in the software.
Time to peak when calculated from catchment characteristics is:
TPt (hrs). Time to rise of the unit hydrograph as defined by:
TPt = TP0 + T/2, T time step
QP (m³/s 10mm). The peak flow for the unit hydrograph as calculated by:
TB (hrs). Time base of the unit hydrograph as calculated by:
TB = 2.52 * TPt
Base Flow (m³/s). Base flow for the unit hydrograph as calculated by:
PR (%). Total percentage run off of the catchment is defined as follows:
where
, and , if P > 40mm and
otherwise P is the rainfall depth.
.