Simulations are performed using the algorithms summarized below. Additional detail is provided in the Version 1 model development
report and version 2 update report (see
References ).
Watershed
Runoff
Runoff is driven by rainfall & snowmelt. Runoff from pervious areas is simulated using version of SCS curve number method (USDA,1964), as invoked in GWLF model (Haith et al., 1992). The antecedent moisture condition (AMC) is adjusted based upon 5-day antecedent rainfall + snowmelt.
Percolation from pervious areas is estimated by difference (rainfall -runoff). Percolation is not considered unless explicitly routed to an aquifer. Evapotranspiration is computed from air temp, day length, & month using method described by Haith & Shoemaker (1987).
All runoff is routed directly to downstream devices (without lag). This assumes that the time of concentration is small in relation to the precipitation time step (1 hr). For large watersheds, predicted watershed response will be overestimated. To retard watershed response, direct runoff to a "pipe" device with a positive time-of-concentration.
SnowFall / SnowMelt Simulation
The snow simulation is essentially a water balance with melting governed by SCS degree-day equation.
Tair = mean daily air temperature (deg-F)
S(t) =
snowpack at end of hour t (inches, water equivalent)
M(t) = snowmelt occurring
in hour t (inches)
P(t) = total precipitation in hour t (inches)
R(t) =
rainfall occurring in hour t (inches)
X(t) = snowfall occurring in hour t
(inches)
Tsnow = air temperature generating snowfall (deg-F)
Tmelt =
minimum air temperature for snowmelt (deg-F)
SMCoef = snowmelt coefficient (inches/degrees F-day)
If Tair <= Tsnow then
X(t) = P(t)
R(t)= 0
else
X(t) = 0
R(t) = P(t)
endif
S(t) = S(t-1) + X(t) - M(t)
M(t) = MIN [ MAX [ 0 , SMCoef ( Tair - Tmelt )/24 ] , S(t-1) + X(t) ]
The sum of M(t) + R(t) drives runoff simulation from pervious &
impervious areas.
Runoff from Frozen Soils
The Frozen Soil ( Tfreeze, ' Edit ET/Snowmelt ' screen) can be adjusted to control the rate of runoff from pervious areas when the soil is likely to be frozen.
At the start of each event, P8 computes the 5-day-average antecedent air temperature (TAir). If TAir < TFreeze, the following adjustments are made to the runoff simulation for the duration of the event:
-> Antecedent Moisture Condition = 3
-> Maximum Abstraction computed from Curve Number is multiplied by the Scale Factor for maximum abstraction specified on the ' Edit ET/Snowmelt ' screen. The scale factor would range from 0-1. If = 0, the soil will be treated as completely impervious. If = 1, the effect of soil freezing on max abstraction would be ignored.
This capability has been included to permit simulation of conditions in northern climates (e.g., long cold spell followed by rainfall). To turn this option off, set Tfreeze to a very low number (e.g.,-50).
Impervious Area Runoff Simulation
Runoff from impervious areas is governed by the following equations:
Cum rain+melt:
Y(t) =
Y(t-1) + dY(t)
Excess rain+melt: Et = MAX [ ( Y(t)-Si ) , 0 ]
Runoff: ri(t) = Fi ( E(t) - E(t-1))
Infiltration: qi(t) = (1 - Fi ) (E(t) - E(t-1) )
where,
Y(t) = cumulative rainfall + snowmelt at end of hour t
in current event (in)
dY(t) = incremental rainfall + snowmelt occuring in hour t
(in)
Si = impervious depression storage for watershed i (inches)
Fi =
runoff coefficient for impervious areas in watershed i (dimensionless)
E(t) =
cumulative excess rainfall + snowmelt at end of hour t (inches)
ri(t) =
impervious runoff rate in hour t (inches/hr)
qi(t) = infiltration rate from impervious area in hour t
(inches/hr)
Particle washoff is governed by sum of ri(t) & qi(t).
Runoff from the impervious watershed starts after the cumulative event rainfall + snowmelt exceeds the specified depression storage. Runoff coefficients specified on watershed input screen apply when the cumulative rainfall+snowmelt is less than the "breakpoint" specified on the general input screen (default value = 0.8 inches). Above that value, a runoff coefficient of 1.0 is used. This feature can be disabled by setting the breakpoint to a high value, e.g. 999. See also SLAMM calibrations.
Curve Number Adjustment based on Antecedent Moisture Condition
Reference: GWLF Model (Haith et al, 1992)
P5 = 5-day antecedent rainfall +
snowmelt (prior to start of event)
T5 = 5-day antecedent
average air temperature at start of event (deg-F)
RAMC2,
RAMC3 = P5 value corresponding to AMC 2 & 3 (inches)
CN1,CN2,CN3 = curve numbers for amc 1, 2, & 3 for
current event
TFREEZE = T5 value forcing AMC 3 (deg-F)
RAMC2 & RAMC3 defined
separately for growing & non-growing seasons.
CN1 =
CN2 / (2.334 - .01334 CN2 )
CN3 = CN2 / (0.04036
+ .0059 CN2 )
IF (T5 < TFREEZE) or (Snowmelt Event) or (P5 >= RAMC3), then
CN = CN3
Else If P5 <= RAMC2 then
CN = CN1 + (CN2 - CN1)*P5/RAMC2
Else
CN =
CN2 + (CN3 - CN2)*(P5 - RAMC2)/(RAMC3 - RAMC2)
Endif
Watershed Loadings
Loadings from pervious areas are computed by applying a fixed concentration to the computed runoff volume for each particle class. If percolation is routed to an aquifer, the concentration in percolating flow is reduced by the filtration efficiency defined for each particle class.
Loadings from impervious areas are computed using two techniques:
Loads resulting from these mechanisms are totaled.
For each watershed, computed loadings are multiplied
by a constant factor 'Pollutant Load Factor'. This factor (normally = 1) can be
used to adjust for differences in loading intensity due to land use, for
example, if sufficient data are available.
Particle Buildup & Washoff
The differential equation describing buildup & washoff is:
d B
----- = L - k B - f s B - a B r
c
d t
where, in consistent
units:
B = buildup or
accumulation on impervious surface
L = rate of deposition
k = rate of
decay due to non-runoff processes
s = rate of street sweeping
f =
efficiency of street sweeping (fraction removed per pass)
a = washoff coefficient =
SWMM "RCOEFX" (Huber & Dikinson, 1988)
c = washoff exponent
= SWMM "WASHPO" (Huber & Dikinson, 1988)
r = runoff
intensity from impervious surfaces
Values are updated using the analytical solution of
this equation for each time step. At the start of the simulation, B values are
set equal to one day's worth of deposition. It is recommended that the
street sweeping feature be used only for vacuum devices. See further
information on the street sweeping routine and
calibration.
Device Outflows
Flow routing is performed in
downstream order using a modified second-order Runge- Kutta technique (Bedient
& Huber, 1986). For each device, outlet, & timestep the relationship
between volume & outflow is represented by:
Q = d0 + d1 V
where, in consistent units,
Q = outflow
V = current device volume
d0, d1 = intercept & slope of outflow vs. storage
relationship
d0 & d1
values are updated at each time step, based upon the
elevation/area/volume/outflow table for the device.
Linearization of the storage/outflow relationship permits analytical solution of the device flow balance at each time step:
d V
--- = Qin - SUM [ Qout ], Qout =~ d0 + d1
V
d t
Analytical solution for volume increase, not shown
here for lack of space:
V2 - V1 = F(V,t)
where, in consistent units:
V = device volume, V1 at start, V2 at end of time
step
Qout = outflow for given device & outlet
Qin = total inflows
(from watersheds & upstream devices)
SUM = sum over device outlets
(exfiltration, normal, spillway)
d0,d1 = intercept & slope of Qout vs. V relationship
Because d0 & d1 may vary with V, a two-stage
procedure is used to estimate volume derivative:
Vm = V1 + .5 F(V1,t)
V2 = V1 + F(Vm,t)
In order to
maintain continuities of the water and mass balances, the minimum water
depth in each device is constrained to 0.01 feet. If the water balance
calculation attempts to drive the water depth below that value, the device
outflows (normal, spillway, infiltration) are reduced by the same percentage in
order to maintain the minimum water depth. In other words, the
stage/discharge table is over-ridden. Particle removal rates are set to
zero under these conditions.
Device Mass Balances
Each device is assumed to be completely mixed for the purposes of
computing particle masses & concentrations, using the following equations:
B = Q/V +
K1 + K2 Cm + U A/V
d M
--- = W - B M
d t
Analytical Solution:
If B>0 Then
M2 = W/B + (M1 - W/B) exp(-Bt )
else
M2 = M1 + W t
endif
where, in consistent units:
B = sum of
first-order loss terms Cm = average concentration during step
V = avg. device
volume in step M = particle mass in device t = time
W = total inflow load to
device (from watersheds & upstream devices)
Q = average outflow from
device (from flow balance)
U = particle settling velocity
A = average device
surface area
K1,K2 = first & second order decay
coefficients
Average values of V, W, Q, & A are used in each time step. Technique is similar to that used in the SWMM Transport Block (Huber & Dikinson, 1988), except based upon mass rather than concentration.
Concentrations are solved as
follows:
C2 = M2/V2
Cm = [ W + (M1 - M2)/t ] V / B (from mass balance)
where,
C2, V2 = concentration & volume at end of time step
Cm = average concentration during time step (used for downstream routing)
If a nonzero 2nd-order decay rate is specified, 3
iterations are performed, updating the first-order loss term (B) each time based
upon the average concentration (CM) computed in the previous iteration.
Detention Pond Outlet Hydraulics
The normal outlet of a detention pond (releases from
flood storage pool) can be of four types:
Standard hydraulic equations are used to compute orifice (Qo) & weir (Qw) flows (cfs) at a given head (h, feet) (Bedient & Huber, 1988). English units (feet, cfs) are used.
Orifice:
Qo = Co Ao ( 2 g h )1/2
Ao= orifice area (ft2)
Co = orifice
coefficient
Typical ~ .6
Submerged
Culverts with Sharp-Edged Entrance ~ .65
Submerged
Culverts with Well-Rounded Entrance ~ 1.0
Weir:
Qw = Cw Lw h 3/2
Lw = weir length (ft)
Cw = weir
coefficient
Typical ~ 3.0-3.3
Road/Highway Embankment ~ 2.5-2.8
A perforated riser consists of a number of holes of a given diameter spaced evenly over a given height. The specified orifice discharge coefficient (Co) is also used in computing the riser drawdown curve.
See Hulsing, H., "Measurement of Peak Discharge at Dams by Indirect Method", USGS, Techniques of Water Resources Investigations, Book 3, Chapter A5, 1967.
Swale/Buffer Hydraulics
Flow velocities in
swales & buffers are computed using Manning's equation for overland flow
(Bedient & Huber, 1988):
Manning's equation:
v = 1.49 r 2/3 s1/2 / n
where,
v = flow velocity (ft/sec)
s = slope (ft/ft)
r = hydraulic radius (ft)
n = Manning's roughness coef.
Alternative formulations that incorporate a dependence of n on depth or flow/width can also be selected on the device input screen .
Trapezoidal geometry is assumed in computing the hydraulic radius for a given flow depth, bottom width, & side slope. The device routing table ('List Inputs Stage/Discharge) lists the computed velocity (v) as a function of elevation. Peak flow velocities during a given simulation can also be listed ('List Hydraulics')
Experience with the model indicates that particle
removal efficiencies are relatively insensitive to s & n, except at high
infiltration rates.
One limitation of the model is
that it does not simulate re-suspension of previously settled particles in
devices which are subject to high flow velocities. This may be a problem in dry
ponds, swales, & buffers under certain conditions, particularly if
vegetation is sparse & high flow velocities are encountered.
The 'List Inputs
Stage/Discharge' table contains flow velocities as a function of elevation based
upon the specified hydraulic parameters. The 'List Hydraulics' table
contains the maximum velocity that occurs during the simulation.
Velocities less than 4-5 ft/sec
(RIDEM, 1988) or 2-3 ft/sec (MDWRA,1984) are recommended for avoiding erosion in
grassed swales. Swales/buffers should be sized or otherwise designed to
avoid velocities in this range.
As for all devices, the minimum water depth is constrained to 0.01 ft to maintain continuity of the water & mass balances (see Device Outflows above).
See typical parameter values for Swale/Buffer devices.