Simulation Methods


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 / (0.4036 + .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.