ues. Normally, a weighted average (e.g., through optimization) of the values is adopted foreach of the coefficients.

It is not infrequent to see a model misused or abused. Sometimes this is due to the lackof understanding about how the model works. Sometimes it is due to the lack of appreci-ation of the operational modes. For example, data used for calibration should not be usedagain for verification. Yet, this situation happens again and again. In such a case of usingthe same data for calibration and verification, the difference between the model output andthe recorded data is simply a reflection of the numerical errors and the deviation of theparticular data set from the weighted average situation.

Not all models require calibration. Presumably, some strictly physically based modelshave their coefficient values assigned based on available information and no calibration isneeded. However, in rainfall-runoff modeling, some degree of spatial and temporal aggre-gation of the physical process is unavoidable. Therefore, calibration is desirable, if notnecessary.

14.9 DETENTIONANDRETENTIONSTORAGE

Detention and retention basins are widely used to control the increased runoff due tourbanization of undeveloped areas. These basins can also offer excellent water quality ben-efits since pollutants are removed from the stormwater runoff through sedimentation,degradation, and other mechanisms, as the runoff is temporarily stored in a basin.Detention basins are sometimes called dry ponds, because they store runoff only duringwet weather. The outlet structures are designed to completely empty the basin after a stormevent. Retention basins are sometimes called wet ponds since they retain a permanent pool.

Flow

Rate

Post development hydrograph (pond inflow)

Required Pond Volume

Pre-developnent peak flow rateRouted post-developnenthydrograph (pond outflow)

FIGURE 14.43 Routing of runoff through detention basin.

Previous Page

Maintenance ShoulderWater Surface

Protection

FlowDutlet Structure

Emergency Spillway

Clay Core

Ahtl-seep CollarPipe bedding

Over-size barrelRiprap Energy Dlssipator

Tailwater

GeotextileTreatment

Endwall

Fill

FIGURE 14.44 Basic elements of detention basin.

14.9.1 Detention Basins

The primary function of a detention basin is to control the quantity of stormwater runoff.Most stormwater management policies require that the postdevelopment peak flow rates bereduced to predevelopment peak flow rates for one or more specified design return periodssuch as 2, 10, and 25 years. Peak flow reduction is achieved by routing thepostdevelopment runoff through a detention basin, that is by detaining the runoff tem-porarily in a basin. Figures 14.43 illustrates the effect of a detention basin on storm waterrunoff.

The schematic diagram given in Fig. 14.44 shows the basic elements of a detentionbasin. In addition, sediment forebays are often used for partial removal of sediments fromthe stormwater runoff before it enters the detention basin. Energy dissipating structuressuch as baffle chutes are used at inlets. Most detention basins also have a trickle flow ditchor gutter in the bottom sloped towards the outlet to provide drainage of the pond bottom.

14.9.1.1 Detention basin design guidelines. Specific design criteria for detention basinsvary in different local ordinances. Some general guidelines are summarized herein.Similar guidelines can be found elsewhere in the literature [ASCE, 1996; FederalHighway Administration (FHWA), 1996; Loganathan et al. 1993; Stahre and Urbonas,1990; Urbonas and Stahre, 1993; Yu and Kaighn, 1992].

The main objective of a detention basin is to control the peak runoff rates. The outfallstructures should be designed to limit the peak outflow rates to allowable rates. A detentionbasin should also provide sufficient volume for temporary storage of runoff. The inlet, outlet,and side slopes should be stabilized where needed to prevent erosion. The side slopes shouldbe 3H/IV or flatter. The channel bottom should be sloped no less than 2 percent toward thetrickle ditch. Detention basin length to width ratio should be no less than 3.0. Outlets shouldhave trash racks. Coarse gravel packing should be provided if a perforated riser outlet is used.An emergency spillway should be built to provide controlled overflow relief for large storms.A 100-year storm event can be used for the emergency spillway design.

14.9.1.2 Outlet structures. Detention basin outlet structures can be of orifice-type, weir-type, or combinations of the two. Schematics of basic outlet structures are shown inFig. 14.45.

Discharge through an orifice outlet is calculated as

Q = k0a0V2^H~o (14.88)

where a0 = the orifice area, k0 = the orifice discharge coefficient, and H0 = the effectivehead. If the orifice is submerged by the tailwater, H0 is the difference between the head-water and tailwater elevations. If the orifice is not submerged by the tailwater, it isassumed that Q = O if the headwater is below the centroid of the orifice. Otherwise, H0is set equal to the difference between the headwater elevation and the orifice centroid.This approximation is acceptable for small orifices. To account for partial flow in largeorifices, the inlet control culvert flow formulation can be used to determine the orificeflow rates. Short outflow pipes smaller than 0.3 m (1.0 ft) in diameter can also be treatedas an orifice provided that H0 is greater than 1.5 times the diameter. Typical values of k0are 0.6 for square-edge uniform entrance conditions, and 0.4 for ragged edge orifices(FHWA, 1996).

Weir-type structures include sharp-crested weirs, broad-crested weirs, spillways, andv-notch weirs. Flow over spillways, broad-crested weirs, and sharp-crested weirs with noend contractions is expressed as

G = W^H/5 (14.89)

RECTANGULAR WElRFRONT VIEW RECTANGULAR WEIRSIDE VIEW

UNSUBMERGEO ORIFICE SUBMERGED ORIFICE

EMBANKMENT

OVER-SlZEDBARRELL

SIDE VIEW OF STAND PIPE OR INLET BOX

FIGURE 14.45 Detention-outlet structures.

where kw = the weir discharge coefficient, Lc = the effective crest length, and H0 = thehead over the weir crest. The weir discharge coefficient depends on the type of the weirand the head. The head over the weir is the difference between the water surface elevationin the detention basin and the weir crest. For sharp-crested weirs with end contractions

Q = kw (Lc - 2H0)V^H0I-* (14.90)

and for V-notch weirs

Q = kv (jj)V2^(tan f) H™ (14.91)

where kv = the V-notch discharge coefficient, $ = the notch angle, and H0 = the head overthe notch bottom.

Riser pipes act like a weir at low heads and like an orifice at higher heads. It is alsopossible that the flow will be controlled by the outflow barrel at even higher heads. Inmany applications, the outflow barrel is oversized to avoid the flow control by the barrel.In that case the outflow through the structure is calculated for a given head both using theweir and orifice flow equations, and the smaller of the two is used. If a trash rack isinstalled, the clear water area should be used in the calculations. It should be noted thatmany engineers design riser pipes so that orifice-type flow will not occur, because it isoften observed that vortices form in the structure under orifice flow conditions.Sometimes antivortex structures are installed to avoid this problem.

Multiple outlets are used if the design criteria require that more than one designstorm be considered. Figure 14.46 displays schematics of several multiple-outlet struc-tures.

14.9.1.3 Stage—storage relationships. The stage-storage relationship is an importantdetention basin characteristic. For regular-shaped basins, this relationship is obtainedfrom the geometry of the basin. For instance, for trapezoidal detention basins that has arectangular base of W X L and a side slope of z, the relationship between the volume, S,and the flow depth d is

S = L Wd + (L + W) zd1 + y z2# (14.92)

FIGURE 14.46 Example multiple outlet structures.

For irregular-shaped detention basins, first the surface area, A5, versus elevation, h,relationship is obtained from the contour maps of the detention basin site. Then

S2 = S1 + (h2 - H1) Asl +

2 As2 (14.93)

where S1 and A51 correspond to elevation hl9 and S2 and As2 correspond to h2. Equation(14.93) is applied to sequent elevations. A more accurate relationship is

S2 = S1 + ̂ A ^A51 + A52 + VA51A52J (14.94)

The stage-storage relationship for most human-made and natural basins can also beapproximated by

S = bhc (14.95)

where b and c = fitting parameters. Figure 14.47 displays approximate relationshipsbetween the parameter b and c, the base area, length to width ratio, and the side slope fortrapezoidal basins.

14.9.1.4 Detention pond design aids. The conventional procedure for the hydraulic designof a detention basin is a trial-and-error procedure, and it consists of the following steps:

1. Calculate the detention basin inflow hydrograph(s) for the design return period(s)being considered. A rainfall-runoff model, such as HECl, TR-20, or SCSHYDRO, canbe used for this purpose. For urbanizing areas, the inflow hydrograph(s) are normallythose calculated for postdevelopment conditions.

Rectangular Pond LW = 3:1 Rectangular Pond LW = 4:1

bottom aream2orf t2

bottom aream2orf t2

FIGURE 14.47 Detention basin stage-storage parameters. (After Currey and Akan, 1998).

2. Set the hydraulic design criteria. In most applications, the postdevelopment peak(s) arerequired to be reduced to the magnitude(s) of the predevelopment peak(s) for thedesign return period(s). If predevelopment peak(s) are not available, a rainfall-runoffmodel can be used to calculate them. The hydraulic design criteria may also restrict themaximum water surface elevation in the detention basin.

3. Trial design a detention basin. A trial design consists of the stage-storage relationship,and the types, sizes and elevations of the outlet structures.

4. Route the inflow hydrograph(s) through the trial-designed detention basin and checkif the design criteria set are met. If not go back to Step 3. Also, if the criteria are met,but the outflow peak(s) are much smaller than the allowable value(s), then the trialbasin is overdesigned. Again, go back to Step 3. The level-pool routing procedure isadequate for most detention basin design situations. This procedure is based on thesolution of the hydrologic storage routing equation

^jL = I-Q (14.96)

where / = inflow rate and t = time. Unless a computer program is used, Eq. (14.96) issolved by employing a semigraphical method like the storage indication method, whichcan be found in any standard hydrology textbook.

Obviously a good trial design is the key in this procedure. Designing a detention basincan become a tedious and lengthy task if the trial designed basins are not chosen carefully.Various charts and equations are available in the literature that can be used as trial designaids. Most of these aids are based on predetermined solutions to Eq. (14.96) in dimen-sionless form (Akan, 1989a; Akan, 1990; Currey and Akan, 1998; Kessler and Diskin,1991; McEnroe, 1992). Others are based on assumed inflow and outflow shapes (Abt andGrigg, 1978; Aron and Kibler, 1990) or results of numerous routings for many detentionbasins (Soil Conservation Service, 1986; Wycoff and Singh, 1976). Table 14.20 presentsvarious design aid equations, where Ip = the peak inflow rate (peak discharge of postde-velopment hydrograph), Qp = the allowable peak outflow rate, Smax = the required stor-age volume, and SR = the volume of runoff.

The use of these design aids can be illustrated through a simple-example. Suppose therainfall excess resulting from a design rainfall is 3.5 in. over a 2,178,000 ft2 urban water-shed, and the runoff hydrograph has a peak of Ip = 212ft3/s occurring at tp = 30 min =1800 s. A detention basin is to be designed to reduce the peak flow rate to Qp= 120 ft3/s.A weir-type outlet will be used that has kw = 0.40. It is also required that the depth ofwater above the weir crest not to exceed 6.50 ft. A trapezoidal detention basin is suggest-ed width a length-to-width ratio of 4 and sideslopes of 3H/IV. To size the required basin,let the surface area of the detention basin at the weir crest elevation be 40,000 ft2. Thenfrom Fig. 14.47, b = 42,500 and c = 1.055. By definition, SR = (3.5/12)(2,178,000) =635,250 ft3. Using the equations suggested by Currey and Akan (1998) from Table 14.20,we obtain 5max = 302,715 ft3, /zmax = 6.43 ft, and Lc = 2.29 ft. Note that /zmax < 6.50 ft.,so the suggested basin with a base area of 40,000, ft2 should work.

The Soil Conservation Service (SCS), (1986) equations given Table 14.20 are recom-mended if the standard SCS design rainfall hyetographs are to be used. Also, these equa-tions are not restricted to single outlet detention basins. Suppose a detention basin isrequired to control the stormwater runoff for 2-year and 25-year events. Given for the 2-year event are Ip = 91 ft3/s, Qp = 50 ftVs, SR = 408,480 ft3, and for the 25 =year event Ip= 360 ftVs, Qp= 180 ftVs, SR = 928,750 ft3. A two-stage weir outlet is suggested with kw= 0.40, and the maximum water depth above the lower weir crest is not allowed to exceed5 ft. The design is to be based on SCS 24-h Type II rainfall. Suppose a trapezoidal deten-

Table 14.20 Design Aid Equations for Detention and Retention Storage

ReferenceRemarksOutletType(s)

Numberof Outlets

Equation

Wycoff and Singh(1976)

Abt and Grigg(1978)

Baker (1979)

Soil ConservationService (1986)

Soil ConservationService (1986)

Aron and Kibler(1990)

Kessler and Diskin(1991)

Kessler and Diskin(1991)

Based on numerical simulationsTb = time base of inflow hydrograph

Triangular inflow hydrograph,trapezoid outflow hydrograph

Triangular inflow and outflowhydrograph

For SCS 24-h Types I and IArainfall

For SCS 24-h Types II and IIIrainfall

td = storm duration, Tc = time of concentration

Trapezoidal inflow hydrograph, risinglimb of outflow hydrograph is linear,

Constant reservoir surface area, valid for

0.2 < ̂ < 0.9P

Constant reservoir surface area, valid for

0.2 < Qs- < 0.9p

Notspecified

Notspecified

Notspecified

Notspecified

Notspecified

Notspecified

Weirtype

Orificetype

Notspecified

Notspecified

Notspecified

Notspecified

Notspecified

Notspecified

Single

Single

S^ 1.291(1 -Qp/Ipy>™SR (W411

smax _ i _ fcp>SR UJ

Smax Qp

S* /„

%^ - 0.660 - L7^\ + 1.96 ]̂2 - 0.730^)^R (.'P) I lp J I *p J

^p - 0.682 - 1.43(̂ 1 + 1-64Jy^ ~ 0.804 |̂

tj+TSmax = Iptd - G,(JL2-1)

^P = 0.932 - 0.792 ̂^R *p

%^ - 0.872 - 0.861^*R 1P

Table 14.20 (Continued)

ReferenceRemarksOutletTypes(s)

Numberof Outlets

Equation

McEnroe(1992)

McEnroe(1992)

Currey and Akan(1998)

Currey and Akan(1998)

Gamma function inflow hydrograph

Gamma function inflow hydrograph

Gamma function inflow hydrograph,stage-storage relationship: 5 = bhc,h = stage, Lc = weir crest length,kw = weir discharge coefficient,

g = gravitional acceleration

Gamma function inflow hydrograph,stage-storage relationship: S = bhc,h = stage, a0 = orifice area, k0 = orificedischarge coefficient, g = gravitationalacceleration

Weirtype

Orificetype

Weirtype

Orificetype

Single

Single

Single

Single

%*• = 0.98 - 1.17 ^SL + 0.77 [^ F - 0.46 [^ F^R 1P (1P J (1P J

%i = 0.97 - 1.42 Qs. + 0.82 (^) - 0.46 (•%)^R 1P (1P J (1P J

%^ = 0.922 - 0.787 [ l̂^R \ 1P J

( Q V/'0.922 SR - 0.787 =S- SRZ, P"max

V )\l.5/c

L- b Q»O k V2g

0.922^-0.787^^] w *P

( O \0.847 SR - 0.841 ^- SR P

h ~\ J

%^ = 0.847 - 0.841 [ l̂SR { 1P J

( \5/C

b Oa°~ o , +L0.847 SR - 0.841 ̂ SK\ k° V 2g

\ K I Jp

tion basin with a length-to-width ratio of 4 and side slopes of 3H/ IV is suggested. FromTable 14.20, 5max - 928,750 [0.682 - 1.43(180/360) + 1.64(180/36O)2 - 0.804(180/360)3] = 256,800 ft3. Likewise, Smax = 105,400 ft3 for the 2-year event. To deter-mine the base area (or the surface area of the detention basin at the lower crest elevation),use Eq. (14.92) with S = 256,800 ft3, L = 4W, z = 3, and d = 5 ft. Solving the equationfor W, we obtain approximately W = 104 ft, and then L = 416 ft. To size the lower crestfor the 2-year event use Smax = 105,400 ft3. Now substituting S = 105,400 ft3, W = 104ft, L = 416 ft and z = 3 in Eq. (14.92) and solving for d, we obtain the maximum headover the lower crest for the 2-year event as being 2.25 ft. Next, using the weir equation(Eq. 14.89) with Q = 50 ft3/s, kw = 0.40, h = 2.25 ft, and g = 32.2 ft/s2, we obtain L0 =4.61 ft for the lower crest. Let the upper crest be placed 2.30 ft above the lower crest. Tosize the upper crest, the 25-year event is considered. The maximum head over the lowercrest will be 5 ft. At this head the lower crest will discharge 165 ftVs [from Eq. (14.89)].Therefore, the upper crest should be sized to pass (180 — 165) =15 ft3/s under a head of(5.00 - 2.30) = 2.70 ft. From the weir formula [Eq. (14.89)] we obtain Lc = 1.05 ft forthe upper crest.

14.9.2 Extended Detention Basins

Extended detention basins are effective means of removing particulate pollutants fromurban storm water runoff. As shown in Fig. 14.48, an extended detention pond has twostages. The bottom stage is expected to be inundated frequently. The top stage remains dryexcept during large storms.

Top View

Top Stage (NormallyDry, Maintained asMeadow)

ButlerLandscaped withShrubs for Habitat

25FootBuffer

Side-slopes3:1 Maximum

BottomStage Embankment

Outfall

DETENTION TIME: 24 to 40 HoursDETENTION VOLUME: 0.75 to 1.50 Inches/Impervious Acre

Side View2 Year Water Surface Elevation

Top Stage (Normally Dry)with Hood

Bottom Stage Sized toAccept Runoff Volumeof Mean Storm Encased in Gravel JacketShallow Marsh(6 to 12 inches)

EmergencySpillway

Anti-seepCollars

FIGURE 14.48 Extended detention basin. (After Schueler, 1987).

14.9.2.1 Detention volume and time. An extended detention basin is designed todetain a certain quantity of runoff, sometimes referred to as the water quality volume,for a certain period of time to achieve the targeted level of pollutant removal. The vol-ume to be detained and the duration over which this volume to be released vary in dif-ferent stormwastet management policies. For example, Hampton Roads PlanningDistrict Commission (1992) requires that a quantity of runoff calculated as 0.5 inchtimes the impervious watershed area be released over 30 h in southeastern Virginia.Prince George County Department of Environmental Resources (1984) requires therunoff volume generated from the 1-year, 24-hour storm be released over a minimumof 24 h.

American Society of Civil Engineers (1998) outlines a procedure to size extended basinsserving up to 1.0 km2 (0.6 m3) watersheds. In this procedure, the volume of water to bedetained per unit watershed area, P0, is estimated as

P0 = ar(0.858i3 - 0.78/2 + 0.774/ + 0.04)P6 (14.97)

where ar = a regression coefficient, / = the watershed imperviousness expressed as a frac-tion, and P6 = the mean storm precipitation depth that can be obtained from Fig. 14.49.The value of the regression coefficient ar is 1.109, 1.299, and 1.545 for detention volumerelease times of 12, 24, and 48 h, respectively. Interpolation of these values is allowed fordurations between 24 and 48 h.

The use of this procedure can be illustrated by a simple example. Suppose anextended detention basin is to be designed for a 150-acre watershed in Norfolk,Virginia that is 40 percent impervious. Determine the required size if the detainedrunoff is to be released over 36 hours. From Fig. 14.49, P6 = 0.67 in for Norfolk,Virginia. Because the watershed is 40 percent impervious, / = 0.40. Also, interpolat-ing the ar values between 24 and 48 h, ar = 1.422 for 36 h. Then, from Eq. (14.97), weobtain P0 = 0.27 in. Therefore, the volume of runoff to be detained is (150) (0.27/12)= 3.38 aq-ft = 147,233 ft3. It is advisable to increase this volume by about 20 percentfor sedimentation.

FIGURE 14.49 Mean storm precipitation depth in inches. (After ASCE, 1998).

Pipe Wrapped with Filter FabricFIGURE 14.50 Extended detention pond outlets. (After Schueler, 1987).

14.9.2.2 Extended detention outlet structures. The outlets for extended detention basinsare designed to slowly release the captured runoff from the basin over the specified emp-tying time to allow settling of particulate pollutants. We sometimes refer to these outletsas water quality outlets. Low-flow orifices are often used as outlet structures. Figure 14.50displays various methods for extending detention times. As pointed out by Schueler(1987) and ASCE (1998), however, extended detention outlet structures are generallyprone to clogging. This makes the design of outlet structures difficult since the hydraulicperformance of a clogged outlet will be uncertain and different from what it is designedfor. Regular cleanouts must be performed.

A hydrograph routing approach is probably the best way to size an extended detentionbasin and the water quality outlet. However, this requires an inflow hydrograph. In prac-tice, as discussed in the preceding section, only the volume of captured runoff is consid-ered for pollutant removal. There are no broadly accepted procedures to convert this vol-ume to an inflow hydrograph. Therefore, the water quality outlets are often sized by usingapproximate hydraulics. This can be illustrated by a simple example.

Side View

Trash Rack

Gravel JacketPerforated Riser

Wire Mesh

Barrel

b. Inlet Controlled Perforated Pipe

Extended Detention OrificesReplaceable Cap(or Dean-out

Stone

Gravel

Wire Mesh

ToLOWFlowOrifice

c. Internally Controlled Perforated Pipe

Internal Orifice RegulationGravel

SandTo Low Flow Orifice

Suppose an extended detention basin has a bottom length of 80 ft, a width of 20 ft, andside slopes of 3:1(//:V). The outlet is to be sized so that it will release a water quality vol-ume of 10,200 ft3 over a period of 40 h. To determine the depth of water corresponding tothis volume, Eq. (14.92) is written as

10,200 = (80)(20)J + (80 + 20)3d2 + (4/3)32J3.

By trial and error, d = 3.6 ft. Let the outlet structure be comprised of !/2-in circularragged edge orifice holes cut around a riser pipe. Let the average elevation of the holes be1 ft above the pond bottom. Therefore, the average head over the orifice holes is (3.6 —1.0)/2 = 1.3 ft. To empty 10,200 ft3 over 4 O h = 144,000 s, the average release rate is10,200/144,000 = 0.0708 ftVs. Noting that the orifice area of a 0.5 in. hole is 0.00136 ft2,and k0 = 0.40 for ragged edge orifices, we can write Eq. 14.88 as

0.0708 = AT(0.40)(0.00136)V2(32.2)Vf3

where Af is the number of orifice holes. Solving for N we obtain N = 14.22. Therefore, weuse 14 holes evenly distributed.

14.9.2.3 Extended detention basin design considerations. Additional design considera-tions for extended detention basins can be found in various publications (ASCE, 1998;FHWA, Schueler, 1987; Urbonas and Stahre, 1993). Briefly, the basin should graduallyexpand from the inlet, toward the outlet. A length-to-width ratio of 2 or higher is recom-mended. Side slopes should not be steeper than 3:1 (H:V) and flatter than 20:1 (H:V). Ariprap, concrete, or paved low-flow channel is required to convey trickle flows. Atwostage design is recommended with a 1.5- to 3.0 ft-deep bottom stage and a 2.0- to 6.0-ft-deep upper stage. A wetland marsh created in the bottom stage will help remove solu-ble pollutants that cannot be removed by settling. The detention basin inlet should be pro-tected to prevent erosion. If the outlet is not protected by a gravel pack, some form of trashrack should be used. A sediment forebay is recommended to encourage sediment deposi-tion to occur near the point of inflow

14.9.3 Retention Basins

Retention basins or wet ponds retain a permanent pool during dry weather as shown inFig. 14.51. A high removal rate of sediment, biological oxygen demand (BOD), organicnutrients, and trace metals can be achieved if stormwater is retained in the wet pond longenough. During wet weather, the incoming runoff displaces the old stormwater from thepermanent pool from which significant amounts of pollutants have been removed. Thenew runoff is retained until it is displaced by subsequent storms. The permanent pooltherefore will capture and treat the small and frequently occurring stormwater runoffwhich generally contain high levels of pollutant loading. The storage volume providedabove the permanent pool is used to control the runoff peaks caused by the specifieddesign storm events.

14.9.3.1 Permanent pool volume. Among all the factors influencing the pollutantremoval efficiency of a retention basin, the size of the permanent pool is the most impor-tant. As pointed out by Schueler (1987), in general, "bigger is better." However, after athreshold size is reached, further removal by sedimentation is negligible.

FIGURE 14.51 Retention basin. (After Yu and Kaighn, 1992).

The required size of the permanent pool in relation to the contributing watershed areavaries in different stormwater management policies. For example, FHWA (1996) and Yuand Kaign (1992) recommend a permanent pool size three times the water quality volumedefined for extended detention basins. Montgomery County Department ofEnvironmental Protection (1984), Maryland, requires a volume greater than 0.5 in. timesthe total watershed area. ASCE (1998) recommends that Eq. (14.97), be used with a draintime of 12 hours to determine the permanent pool volume. It is also recommended that asurcharge extended detention volume, equal to the permanent pool volume, be providedabove the permanent pool. U.S. Environmental Protection Agency (1986) provides geo-graphically based design curves to determine the permanent pool surface area as percentof the contributing watershed area (see Fig. 14.52). Hartigan (1989) and Walker (1987)treat a retention basin as a small euthrophic lake and employ empirical models to size theretention pond. This procedure is outlined by ASCE (1998).

U.S. Environmental Protection Agency (1986) presented a procedure to evaluate thelong-term pollutant removal efficiency of retention basins depending on the basin size andthe rainfall statistics of the project area. This procedure was developed by DiToro andSmall (1979), and is outlined in various publications (Akan, 1993; Stahre and Urbonas,1990; Urbonas and Stahre, 1993).

14.9.3.2 Retention basin design considerations. Wet ponds can be designed to controlthe peak runoff rates from rare and large storm events if additional storage volume is pro-vided above the permanent pool. The size of the additional volume can be determined byusing the procedures described for detention basins.

The outlet structures for retention basins include a low flow outlet to control the runofffrom frequent storm events and overflow devices to control the runoff from larger storms.Typical outflow structures are shown in Fig. 14.53 (Schueler, 1987).

Minimum 10 foot WideSdfely/Vedelaled Ledge

StabilizedIntel

Pool Should be 3-8 FeetDeep and Wedge-Shaped

Grassed

Riserwith Hood Embankment

Ouifa*

Minimum 25 Foot WideBuffer Around Pool

IO Year Wajef Surface Uteyqllono Vani'wriief'surfiDce EtevallonRiserwllh Hood cme'̂ ay.to

sray

Anil-seepCollarspulflow

S

!!

is

Rocky Mountains, Denver

NortheastSoutheast Basin depth = 3.5 ftRunoff coefficient = 8.20

Basin surface area as % of contributing catchment area

FIGURE 14.52 Design curves for solids settling, for low-density residential land use. (After USEPA, 1986).

•. lnlemaffy Controlled Slotted SlandpipeRemovable Cap

Standing Water

G(VMiJKkM

b. Negatively Stop«d Pipe trom Riser

Embankment

Negatively Sloped Pipe Drews Waterat LMM On* Fool Bdow Pool

tamanmcPDOl

e. HoortetJ Orifice on Riser Concr»U Box Riser

PermanentExtendedOrrRce

Front View Side View

FIGURE 14.53 Retention basin outlet structures. (After Schueler, 1987).

Additional design considerations have been presented by Schueler (1987). In sum-mary, the pond should be wedge-shaped, narrowest at the inlet and widest at the outlet.A minimum length to width ratio of 3:1 should be used. The pond depth should average3-6 ft, with a shallow underwater bench around the pond's perimeter. Side-slopesshould be no steeper than 3:1 (H:V) and not flatter than 20:1 (H:V). If the soils at thepond site are highly permeable, the pond's bottom should be lined by impervious geot-extile or a 6-in clay liner. The inlets and outfalls should be protected by riprap or othermeans to prevent erosion. Wet ponds should be surrounded by a 25-ft buffer strip plant-ed with water-tolerant grasses and shrubs. A sediment forebay should be constructednear the inlet of the pond with extra storage equal to the projected sediment trappingover a 20- to 40-year period.

14.9.4 Computer Models for Detention and Retention Basin Design

As discussed in the preceding sections, a trial-and-error procedure is used for hydraulicdesign of retention and detention basins. A basin is first trial-designed, and then the designhydrographs are routed through the basin to verify if the design criteria are met. Therefore,any reservoir routing computer program can be useful for designing detention and reten-tion basins. The widely known TR-20 (Soil Conservation Service, 1986) and HEC-I(Hydrologic Engineering Center, 1990), for instance, have reservoir routing schemes andcan be used for pond design. These models are in public domain. Commercially availablepond routing software are a lot more user-friendly, and they include Watershed ModelingStandalone (www.eaglepoint.com). The commercially available models allow a variety ofdifferent outlet structures and simulation of multiple storm events. Also available arePONDOPT (www.cahh.com) and BASINOPT (www.cahh.com) which include an analy-sis option for reservoir routing as in the other pond models. These two models also havea unique design option which performs all the iterations internally. The ponds are sizedand the outlet invert elevations and sizes are determined by the program formultiple-return periods.

74. W SEWER HYDRAULIC SIMULATION MODELS

A model is defined here as a method or simulation algorithm that has been coded into a com-puter program for computations and applications. Numerous models have been developedfor sewer networks. These sewer models can be classified in different ways as follows:

1. According to the purpose of the model: (a) design models—hydraulic design, or Optimaldesign, risk-based design; (b) evaluation/predictions models, (c) planning models.

2. According to the objective of the project: (a) flood control or (b) pollution control.3. According to the extent of space consideration: (a) overland surface only, (b) sewer

system only, or (c) sewer system and overland surface.4. According to the nature of wastewater: (a) sanitary sewer models or (b) storm and

combined sewer models.5. According to water-quality considerations: (a) quantity only, (b) quality only (rare),

or (c) quantity and quality.6. According to time considerations of rainfall input: (a) single-event models or

(b) multiple-event continuous models.

7. According to probability considerations: (a) deterministic or (b) probabilistic—purestatistical or stochastic.

8. According to systems concept: (a) lumped system or (b) distributed system.9. According to hydrologic principles considered: (a) hydrologic (principle of mass con-

servation) or (b) hydraulic (principles of conservation of mass and momentum orenergy).

In the first classification, the design models are for the determination of the size ofthe sewers and perhaps also their slope and layout of either a new sewer systems or anextension or modification of an existing system. The evaluation/prediction models arethose used to simulate the flow in an existing or predetermined sewer system for whichthe size, slope, and layout are already specified. Their use is to compute the flow in thesewers to check the adequacy of sewer capacity, system performance, operation, man-agement of pollution abatement, flood mitigation, and so forth. Or, the model may beincorporated as part of a real-time operation system. The planning models are thosemodels used for strategy planning and decision making for urban or regional storm andwaste water management, usually applied to a larger time and spatial frame than thedesign or evaluation models.

The design models design the sewers in a network for a hypothetical future eventwhich is represented by the design storms of specified return period or risk level. Theevaluation/prediction models simulate the runoff produced by a rainstorm of the past, pre-sent, future, or the flow from other sources. The planning models usually consider a rela-tively long continuous period of time covering many rainstorms and dry periods inbetween. The planning models utilize the least hydraulic consideration of flow on over-land areas and in sewers. Often, a simple water budget balance suffices. A typical exam-ple is the STORM model (Hydrologic Engineering Center, 1974). Supposedly, for the pur-pose of reliable flow simulation, the evaluation/prediction models require the highestlevel of hydraulic sophistication and accuracy. However, many lower level models doexist. Due primarily to the discrete sizes of commercial pipes, usually a moderate level ofhydraulics is adequate for the design models (Yen and Sevuk, 1975; Yen et al., 1976).Most of the existing sewer models are evaluation/prediction models. Aside from thedesign models derived from the rational method, there are actually very few true sewerdesign models; among them only two models, ILSD (Yen et al., 1976, 1984) and WASSP(Price, 1982b), have published user's guides and arrangements for release of programs.Some of the evaluation/prediction models have the ability to compute the diameterrequired for gravity flow of a specific discharge. However, they are not true design mod-els because different sewers should be designed for different rainstorms of different dura-tions corresponding to the different time of concentration of the sewers. Hence, manycomputer runs are required to complete the design of a network using these models.

In the last classification, the hydraulic models can further be classified according to thelevel of hydraulics shown in Eq. (14.1) or (14.2) as follows: dynamic wave models, non-inertia models, nonlinear kinematic wave models, and linear kinematic wave models. It isimpossible to summarize and report the hydraulic properties of all the exiting sewer mod-els in this chapter. Therefore, only selected models are made in this presentation. Sincethis article deals with the hydraulics of sewers, in the following section, only thehydraulics of selected models are discussed. For models that allow more than onehydraulic level for flow routing, they are presented according to their respective highesthydraulic level. For information and comparison of the nonhydraulic aspects of the mod-els, the reader should refer to other references such as those by Brandstetter (1976), Chowand Yen (1976), and CoIyer and Pethick (1976) in addition to the original model devel-opers' reports or papers. Models without hydraulic consideration of the sewer flow, such

as the rational method models, are excluded. When water quality transport simulation issought, nearly all the models perform the flow routing first and allow another pollutanttransport model—usually in concentration form—coupled with the routing result for sim-ulation. Only the Storm Water Management Model (SWMM) has the quality portion inte-grated in the model as a modular block.

14.10.1 Hydraulic Properties of Selected DynamicWave Sewer Models

Well=known models, in which the highest hydraulic level, the dynamic wave simula-tion is employed, are listed in Table 14.21. In the table, the subscript o denotes the sewerreceiving outflow from the junction. All these models were developed for flow simula-tion rather than for design of sewers in a network. CAREDAS, UNSTDY,HYDROWORKS, and MOUSE are proprietary models. Among the nonproprietarymodels, only two [ISS and Stormwater Management Models Extended Transportation(SWMM-EXTRAN)] have user's manuals published and available to the public. Fordynamic wave and noninertia models, the junction conditions and surcharge transitionconditions—if surcharge is allowed—are important for reliable and realistic simulationof the flow. However, for most of the models listed in Table 14.21, information aboutthe details and assumption on the surcharge transition and on junctions is inadequatelygiven. Also, except ISS which cannot handle flow having a Froude number greater than1.6, it is not known whether the other models can handle supercritical flow with rollwaves, and if so, what assumptions are involved. In the following, dynamic wave mod-els listed in Table 14.21 are briefly discussed in three groups, namely, the explicitscheme model (SWMM-EXTRAN), the models that handle only open-channel sewerflows, and the models that handle both open-channel and surcharge sewer flows. Theallowed network size given in Table 14.21 is that indicated in the quoted literature. Formost models, this number has been increased with later developments.

14.10.1.1 Explicit scheme model: SWMM-EXTRAN. The Storm Water ManagementModel (SWMM) developed under continuous support of the U.S. EnvironmentalProtection Agency is one of the best known among all the sewer models. The ExtendedTransport block (EXTRAN) (Roesner and Shubinski, 1982; Roesner et al., 1984) wasadded to the SWMM Version III to provide the model with dynamic wave simulationcapability. The entire sewer length is considered as a single computational reach, and thedynamic wave equation is written in backward time difference between the time levels n+ 1 and n for the sewer, and expressed explicitly as

[ on2 Af Vl _ _ A — A1 + 2W^ IV«' Q.+ 2VnM+ Vn "'" *-AfZ'Z1An ) ^

- h -h, (14.98)- 8\ "'" AfLJ

where all the symbols have been defined previously, the subscript u = the upstream endof the sewer (that is, entrance) and d = the downstream end of the sewer (that is, exit),the bar indicates the average of values at the entrance and exit, and presumably AA = An+1— An is also the average of the values at the sewer ends. The junction condition used isthe continuity equation, Eq. (14.47), expressed explicitly in terms of the depth and dis-charge values at the time nAf as

TABLE 14.21 Summary of Hydraulic Properties of Selected Dynamic Wave Sewer Network Models

work ReferencesNetSurcharge FlowModel Open-Channel Flow Interior JunctionSizeSurcharge Numerical Solution'tention Equations TransitionNumerical Para Sf Sewer Solution De

Scheme meters Downstream Scheme Storage Condition Hydraulics Scheme Scheme Reported or•ammedProgrCondition

200 Roesner andShubinski

Explicit One sweep,pipe by pipe

Employ open-channel

Z<2 = dsldt All pipesand Hf = h0 entering a

Explicit h, Q Manning Junction water One sweep, YesEXTRAN surface or pipe by pipe

SWMM-

(1982);Roesner et al.

nations;h junction

junction are eqifull or the wit

assumedcondition

(1984)

54 Sevuk et al(1973); Sevukand Yen

(1982)

28 (for Chevereaudynamic et al. (1978);

wave) Cunge andMazaudou(1984)

26 Sjoberg(1976, 1982)

head computedusing assumedadjustmentfactors, excesswater lost

NA

highestenteringpipe issubmerged

Yes ZG = <fa/# and NAH0 = H- (W2g)

or Z<2 = O andh, = h0

ISS Characte- h, V Darcy- Junction water Simultaneousristic Weisbach surface or on overlapping

critical depth segments;pipe by pipe

Preissmann slotYes ZQ = dsldt h/d>Q.9land /ij = h0

CAREDAS Four-point h, Q Chezy or Junction water Simultaneousimplicit Manning surface (double sweep)

Preissmann slotYes Zg - dsldt Q >Qfand

H0 = H- (KV1IIg)o r Z Q ^ O

and H1 = h0

DAGVL-A Implicit, h, Q Manning Junction water Simultaneouscontinuity surface or (double sweep)six-point, critical depthmomentumfour-point,w = 0.55

TABLE 14.21 (Continued) Summary of Hydraulic Properties of Selected Dynamic Wave Sewer Network Models

Surcharge Flow Network ReferencesInterior JunctionOpen-Channel FlowModelNumerical Solution ^e

Scheme Scheme Reported orProgrammed

SurchargeHydraulics

TransitionCondition

Detention EquationsStorage

SolutionScheme

SewerDownstream

Condition

Numerical Para SfScheme meters

Preissmann slot 300 Book et al(1982); Chen

and Chai(1991);Labadie et al.(1978)

Preissmann slot 40 JoMe(1984a, b)

Quasi-steady Implicit Simultaneous 10 Pansic (1980)dynamic,

junction headlosses considered

Preissmann slot (87) Abbott et al(1982); DHI

(1994); Hoff-Clausen et al

(1982)Preissmann slot 5000 Wallingford

Software(1991, 1997)

£0 - dsldt Not givenand /i. = h0

^LQ = O Not givenand hi = h0

Yes Zg = dsldt and Q > QforH0 = H-(KV1IZg) submerged

or ZQ = O exitand hi = h0

Yes 1,Q = dsldt Not givenand

H0 = H- (KV2Kg)or "LQ = Oand h{ = H0

Yes "LQ = dsldtand

H0 = H- (KWFIg)

H, Q Manning Junction water Simultaneous Yesimplicit surface or (double sweep)

sluice gate

h, Q Manning Junction water Simultaneous Noimplicit, surfacew = 0.55,0.6, 0.75,or 1.0Four-point h, Q Manning Junction water Simultaneousimplicit, surface orw = 0.55 critical depth

Six-point h, Q Manning Junction water Simultaneousimplicit, surface (double sweep)w -0.5

Four-point h, Q Colebrook- Junction water Simultaneousimplicit White or surface or (double sweep)

Manning critical depth

UNSTDY Four-point

Joliffe Four-point

SURDYN

MOUSE

SPIDA/HYDRO-WORKS

Hn + 1 = Hn + ̂ - QX + Qj>n) (14.99)Aj

where the subscript j indicates junction. The junction dynamic relation is simplified as acommon water surface [Eq. (14.51)]. Equations (14.98) and (14.99) are solved explicitlyby using a modified Euler method and half-step and full-step calculations. Courant's sta-bility criterion is adopted to select the computational Af.

In EXTRAN, when a junction is surcharged, instead of properly applying the conti-nuity equation (Eq. 14.53), it assumes the point-type junction continuity relationship(Eq. 14.50) applies. On the basis of this point-type junction continuity equation, anexpression of the junction water head is derived through an improper application of thechain rule of differentiation, for which a Taylor expansion would have been more appro-priate. The unsatisfactory result was apparently recognized, and remedies were attempt-ed through the introduction of an adjustment factor and the assumption on the numeri-cal iterations to either reach a maximum number set by the user or the algebraic sum ofthe inflows and outflows of a junction being less than a tolerance. In an earlier versionof EXTRAN that was applied to a project in San Francisco, California, an attempt wasmade to artificially modify the geometry of the junction so that numerical solutioncould be obtained.

The SWMM-EXTRAN, with its explicit difference formulation, solves the flow sewerby sewer by using the one-sweep explicit solution method with no need for simultaneoussolution of the sewers of the network. Therefore, it is relatively easy to program.Nonetheless, because of the assumptions on the surcharge condition, and also the stabili-ty and convergence (accuracy) problems of the explicit scheme for the open-channel con-dition, on a theoretical basis EXTRAN is inferior to other dynamic wave models listed inTable 14.21. The other models, of course, have their share of problems concerning theassumptions on the transitions between open-channel and surcharge flows, betweensupercritical and subcritical flows, and on roll waves.

14.10.1.2 Dynamic wave model handling only open-channel flow: ISS. The IllinoisStorm Sewer System Simulation (ISS) model (Sevuk et al. 1973) solves the dynamicwave equation using the first-order scheme of the method characteristics. The Saint-Venant equations [Eqs. (14.2) and (14.5)] or similar type partial differential equations aretransformed mathematically into two sets of characteristic equations, each set consistingof a pair of ordinary differential equations which are solved numerically using a semi-implicit scheme. The formulation can be found in Sevuk and Yen (1982).

The junction conditions used for a storage junction are Eq. (14.47), together with theequations in Table 14.15, for sewer exits, and for sewer entrances Eq. (14.48) with K1 =O, that is,

H=(W2g) + h + Z (14.100)

For a point junction, the equations are Eq. (14.50), together with Eqs. (14.51) or (14.52).The ISS model program considers direct backwater effects for up to three sewers in ajunction. For junction with more than three joining sewers, the excess sewers (preferablythose with small backwater effects from the junction) are treated as direct inflow, that is,Qj in Eqs. (14.47) or (14.50). The flow in the network is solved by using the overlappingsegment method. The outlet of the network can be any one of the following: (1) a free fall,(2) flow continuing to approach normal flow, (3) a stage hydrograph h = f(i), (4) a ratingcurve Q = f(h), (5) a velocity-depth relationship v = f(h\ and (5) a discharge-time rela-tionship Q = f(t).

When used to compute the required pipe diameter of a sewer, ISS is the only modelamong those listed in Table 14.21 that uses a maximum depth criterion to ensure gravityflow in the sewer for the design situation. Other models compute the required pipe diam-eter on the basis of the peak discharge that does not guarantee gravity flow because, forunsteady flow, maximum depth usually does not occur at the same time as the maximumdischarge in a sewer. The ISS model can easily be modified to account also for surchargeflow by adding the Preissmann hypothetical slot.

14.10.1.3 Dynamic wave models handling both open-channel and surcharge flows.Among the seven models belonging to this group listed in Table 14.21, four of them—CAREDAS, UNSTDY, Joliffe, and HYDROWORKS—are numerically similar, using afour-point implicit scheme and adopting the Preissmann fictitious open slot to simulatesurcharge flow. Details of the four-point implicit scheme can be found in Liggett andCunge (1975) and Lai (1986). In fact, the same four-point numerical scheme is also usedin SURDYN (Pansic, 1980). SURDYN is the only model in this group of seven that sim-ulates the surcharge flow by using the standard pressurized conduit approach and solvingit simultaneously with the open-channel flow. The surcharge equation used in this modelis a quasi-steady dynamic equation obtained by dropping the local acceleration (9V/3f)term in Eq. (14.2). For the rising transition from open-channel flow, surcharge is assumedto occur when the discharge exceeds Qf or when the pipe exit is submerged. Falling tran-sition from surcharge to open-channel flow is assumed to occur when the pipe entranceis not submerged, when the discharge falls less than Qf or when the pipe exit is not sub-merged. Pansic (1980) reported that the model simulates the unsteady flow reasonablywell. But oscillations often occur at transitions between open-channel and surcharge con-ditions. This oscillation problem is partly numerical, partly hydraulic, and partly due toassumptions.

Among these models, HYDROWORKS, MOUSE, UNSTDY, and CAREDAS are pro-prietary. They are briefly introduced in the following:

1. HYDROWORKS. The dynamic wave sewer flow routing option of HYDROWORKSis based on an earlier model SPIDA from the same company, Hydraulics Research, inEngland. HYDROWORKS also contains noninertial (WALLRUS) and nonlinear kine-matic wave (WASSP-SIM) sewer routing options. The model can handle a looped-typenetwork as well as a dendritic type. For dynamic wave routing, the inertia terms arelinearly phased out from a Froude number equal to 0.8-1.1. Essentially, for supercriti-cal flow, the noninertia approximation is used. For pressurized flow, the hypotheticalslot width is assumed one-twentieth of the maximum pipe diameter.

2. MOUSE. This model was upgraded from Danish Hydraulic Institute's (DHI) System11-sewer (SIl-S) model. It was first released in 1985 and subsequently updated withpersonal computer PC technology advancements. It uses the Abbott-Ionescu six-pointimplicit scheme (Abbott and Basco, 1990) which is relatively stable and consistent butcostly in computation. The model allows loop network. In addition to dynamic waverouting, it also has noninertia (identified in the model as diffusion wave) and kinematicwave routing options for sewers.

3. UNSTDY. The UNSTDY model uses four-point noncentral implicit schemes to solvethe Saint-Venant equations for subcritical flow. Supercritical flow is simulated byusing the kinematic wave approximation. The model can solve a looped network in thesystem.

TABLE 14.22 Summary of Hydraulic Properties of Noninertia Sewer Network Models

ReferencesSurchargeInterior JunctionsOpen-Channel FlowModelFlowJunction

EquationDetentionStorage

SolutionScheme

Sewer DownstreamCondition

Parameters SfNumericalScheme

Geiger andDorsch (1980);Klym et al.(1972); Vogeland Klym(1973)Sjoberg (1982)

DHI (1994)

Pagliara andYen (1997)

Standardpressurizedpipe flow

Preissmann slot

Preissmann slot

Preissmann slot

ZQ = O andht = h0

ZQ = dsldt andh0 = H -(KV2Vg) orZQ = O and h{ = h0

ZQ = Q andhf = H0.ZQ = O and ht = h0 orZQ = dsldt andH0 = H- (KW2g)

No

Yes

Yes

Yes

Pipe by pipe

Simultaneous(double sweep)

Simultaneous(double sweep)Simultaneous,over-lappingsegment

Unspecified orrating curve

Junction watersurface orcritical depth

Junction watersurface

Junction watersurface orcritical depth

Colebrook-White

Manning

Manning

Manning

h,Q

Implicit, six-point h, Qcontinuityfour-point momentumw = 0.55Six-point implicit h, Qw = 0.5Four-point implicit h, Q

HVM

DAGVL-DIFF

MOUSE

NISN

4. CAREDAS. This is one of the earliest full dynamic wave sewer flow routing modelsdeveloped by SOGREAH at Grenoble, France. This is the first model to incorporate thePreissmann slot to simulate surcharge flow. In applying CAREDAS, a sewer network isfirst checked for the sewers with sufficiently steep slope for which the kinematic waveequation can be applied as an adequate approximation. The dynamic wave model isapplied to each group of the connected, gently sloped sewers.

14.10.2 Hydraulic Properties of Noninertia Sewer Models

The noninertia approximation of the unsteady flow momentum equation [Eq. (14.1)] isprobably the most efficient option among the dynamic-wave momentum equation optionsto solve unsteady open-channel sewer flow problems. It accounts for downstream back-water effect, and it allows reversal flow. Computationally, it is much simpler than the fulldynamic wave option. It is only for rare highly unsteady cases that the noninertia optionis inadequate and the full dynamic wave or the exact momentum options are required.However, only a few noninertia sewer models have been developed; only four are report-ed in the literature and they are summarized in Table 14.22.

The proprietary HVM-QQS model was developed by Dorsch Consult (Klym et al.,1972; Vogel and Klym, 1973) at Munich, Germany. It has been misquoted as a dynam-ic wave model (Brandstetter, 1976). Examination of the equations [Eqs. (3) and (4) inVogel and Klym, 1973] reveals that, in fact, it is a noninertia model. It was stated thatto avoid simultaneous solution of all the sewers in the network, further assumptionswere made. One assumption is to let Sf = S0(QfQ0)

2, where Q0 is defined as a normalflow discharge corresponding to S0, but it is not clear what depth is used in computingQ0. Another issue that the sewer downstream boundary condition at the exit is eitherunspecified or a rating curve h = h(Q), or the exit depth hydrograph h(f) is known. Infact, with unspecified downstream boundary condition, this model does not reallyaccount for the backwater effect, and thus, it omits one of the important advantageousproperties of the noninertia model. No information is given on whether the flow equa-tions are solved implicitly or explicitly.

The DAGVL-DIFF model was developed at the Chalmers University of Technology(Sjoberg, 1982) at Goteborg, Sweden. The equations in the model are solved in a mannersimilar to the dynamic wave model DAGVL-A and were found generally satisfactory. Nofurther development or support of the DAGVL models has been provided since the devel-opment of S11S/MOUSE.

The proprietary DHI (1994) model MOUSE contains noninertia and kinematic wavesewer routing options in addition to dynamic wave routing. The noninertia option simu-lates the flow the same way as the dynamic wave option; thus, it does not take full advan-tage of the simplicity and computational efficiency of the noninertia modeling.

The NISN model (Pagliara and Yen, 1997) utilizes the overlapping segment method tosolve for the flow in a network. For each segment, the flow equations are solved simulta-neously using the Preissmann four-point implicit scheme. Junction storage and headlossare allowed. There is no network size limit for this model.

14.10.3 Nonlinear Kinematic Wave Models

Unlike the dynamic wave and noninertia sewer models, there exist many kinematic wavemodels. Only a few nonlinear kinematic wave models are listed in Table 14.23 for dis-

TABLE 14.23 Sewer Hydraulic Properties of Selected Nonlinear Kinematic Wave Models

ReferencesSurcharge FlowTransitionInteriorOpen-Channel FlowModelSolutionScheme

InteriorJuntion

SurchargeHydraulics

ConditionJunctionSolutionScheme

NumericalScheme

Sewer Parameters SfHydraulics

Dawdy et al.(1978)

Yen andSevuk (1975),Yen et al.(1976)Yen et al(1976)Price(1982a, b)

Huber andHeaney(1982), Huberet al (1984);Metcalf &Eddy Inc. etal (1971)

Pipe by pipe

NA

NA

Implicitsimultaneousrelaxation

Pipe by pipe

Store excesswater, releaselaterNA

NA

H calculated,headlossesconsidered

Store excesswater, releaselater

Q = Qf

NA

NA

Unsteadydynamicequation

Q = Qf

Q>Qf

NA

NA

Q>Qfor assumedsubmergenceconditions

Q>Qf

I1Q = dsldt

ZQ = O

2(2 = 0

IQ = dsldt

ZQ = Ounless storageblock is used

Cascade

Cascade

Cascade

One sweep,pipe by pipe

One sweep,pipe bypipe

Explicit

Four-pointimplicit

Four-pointimplicitQuasiexplicit

Four-pointimplicit,w = 0.55

Manning

Manning

Manning

Darcy-WeisbachandColebrook-WhiteManning

Nonlinear A, QkinematicwaveNonlinear h, Qkinematicwave

Muskingum- h, QCungeMuskingum- h, QCunge

Improved A, Qnonlinearkinematicwave

USGS

ILSD-B2

ILSD-B3

WASSP-SIM

SWMM-TRANS-PORT

cussion. All of the models listed in these tables, except the USGS model, have provisionto compute the required diameter for a specified discharge using the Manning or Darcy-Weisbach formula. All the nonlinear kinematic wave models listed in Table 14.23 consid-er the backwater effect from upstream (entrance) of the sewer within the realm of a sin-gle sewer and not beyond, and not the backwater effect from downstream (sewer exit).

The kinematic wave models, unable to compute reliably the sewer flow cross-sectionarea A, depth /*, and velocity V, are of questionable usefulness in coupling with a water-quality equation for water-quality evaluation. Unless the downstream backwater effect isalways insignificant, otherwise a water-quantity model having a hydraulic level of nonin-ertia approximation or higher should be used.

Nonlinear kinematic wave models may be classified further according to the man-ner the flow equations are formulated for solution. The first group includes the mod-els solving directly the nonlinear kinematic wave equations. The first two models inTable 14.23, [U.S. Geological Survey's Distributed Routing Rainfall-Runoff Model(USGS) (Dawdy et al., 1978)] and [Illinois Least-Cost Sewer System Design Model,option B2 (ILSD-B2) (Yen et al., 1976)] belong to this group. The second groupincludes the models that solve an explicit linear algebraic equation of the Muskingumequation form. The Illinois Least-Cost Sewer System Design Model, option B3 (ILSD-B3) (Yen et al., 1976) and the British Hydraulics Research's Wallingford Storm SewerDesign and Analysis Package Simulation Method (WASSP-SIM) (Price, 1982a,b)belong to this group. The third group consists of the models using other modified non-linear kinematic wave equations for solution such as the TRANSPORT Block inSWMM (Metcalf and Eddy, et al., 1971).

1. ILSD-B2 and USGS models. In the first group, the continuity equation is written as afinite difference algebraic equation of one variable (usually h or Q) or two variables(e.g., /z, Q or A, Q) and solved iteratively with the aid of the simplified momentumequation, S0 = S^ where Sf is approximated by Manning's or similar formulas to relatethe depth or area to discharge. A formulation used in Yen and Sevuk (1975) and adopt-ed in ILSD-B3 is given in the following as an example. Noting that B(h) = 3A/d/z andG(h) = dQ/dh, Eq. (14.4) can be rewritten as

B(h)^ + G(h)^ = Q (14.101)

For partially filled circular pipes (Fig. 14.3),(<$>}B(h) = Dsm\j\ (14.102)

and by using Manning's formula

•w-^fhswFr-1]

-^^('-¥rHf)+=Wi?-1)]where the central angle <|) in radians is (Fig. 14.3):

ty = 2cos -1 [1 - (2h/D)] (14.104)Consider the four computational grid points boxed by the time levels n and n + 1 andspace levels i and i + 1, Eq. (14.101) can be transformed into the following implicitfour-point forward-difference equation:

2^ [(fi,> + 1 + », + 1, H- lXfct. + 1 + A, + U + , - fcu - fc, + !,„)]

+ ̂ [(Gu + i + C, + u + ,)(*, + ,„ + ,- A4. + ,)] = O (14.105)

This equation is nonlinear only with respect to the unknown flow depth hi + l t t l + l sinceBi + i,H + i and Gi + ltH + 1 are both expressed in terms of the depth [Eqs. (14.102) and(14.103)], and hence it can readily be solved by using Newton's iteration method. Thesolution proceeds sewer by sewer from upstream toward downstream. Within eachsewer, the flows for all the reaches are determined for a given time before proceedingto the next time step.In ILSD, there are actually several sewer flow routing schemes of different hydrauliclevels, including options B2 and B3 listed in Table 14.23 and the option of hydrographtime lag adopted in ILSD-I and 2. The objective of ILSD is to develop an efficient andpractical optimization model for the least cost system design of sewer networks.Therefore, the sensitivity and significance of the sophistication of hydraulics on opti-mal design of sewer systems were investigated. It was found that for the purpose ofdesigning sewers, because of the discrete sizes of commercially available pipes, unso-phisticated hydraulic schemes often suffice, and hence the hydrograph time lagmethod, instead of options B2 and B3, is adopted in ILSD-I and 2. Yen and Sevuk(1975) also arrived at a similar conclusion that for design, a low hydraulic level rout-ing method is often acceptable, whereas for evaluation and simulation of flow in sew-ers, a high hydraulic level routing is usually required.In the USGS model, the finite difference equation is formulated from Eq. (14.47) sim-ilar to ILSD-B2. However, the nonlinear relation between QlQf and A/Af is approxi-mated by a straight line, and the flow area A is expressed explicitly as

4+ U + I = M 1 1 + „ 4 + , . J (14.106)

Hence, solution for all the reaches within a sewer must be obtained at each time for thetime increments. However, for the sewers in a network, the solution technique can beeither the cascade method or the one-sweep method. No information on which one isused in the model is given in the literature.

2. SWMM-TRANSPORT. Only one model in the third group of modified nonlinear kine-matic wave models is listed in Table 14.23. The SWMM is a comprehensive urbanstorm water runoff quality and quantity simulation model for evaluation and manage-ment. A good summary of the model is given in Huber and Dickinson (1988), Huberand Heaney (1982), and Metcalf and Eddy et al. (1971). It has two sewer flow routingoptions, TRANSPORT and EXTRAN, not counting the crude gutter-type routing inthe RUNOFF block. EXTRAN was discussed above. TRANSPORT is the originalsewer-routing submodel built in the progam. In TRANSPORT, the continuity equationis first normalized using the just-full steady uniform flow discharge Qf and area Ap thenthe equation is written in finite differences and expressed as a linear function of thenormalized unknowns AlAf and Q/Qf at the grid point x = (i + I)Ax andr = (n + I)Ar:

(&Qf)t + i, + i + C1(AIAf), + u + ! + C2 = O (14.107)

where C1 and C2 = functions of known quantities. From the simplified dynamic equa-tion Sf = S0 and Manning's formula, we have

(QIQ) = ARVlAfif" = /(AM7) (14.108)

Accordingly, curves of normalized discharge-area relationship Q/Qf versus A/Af forsteady uniform flow in pipes of different cross-sectional geometries are establishedand solved together with Eq. (14.107) for Q/Qfand AIA^ In the kinematic wave methodof solving Eqs. (14.107) and (14.108), in addition to the initial condition, only oneboundary condition is needed, which is usually the inflow hydrograph at the sewerentrance. No downstream boundary condition is required, and hence, no backwatereffect from the downstream can be accounted for if the flow is subcritical. However,in TRANSPORT through a formulation of friction slope calculation using the previoustime values at the spatially forward point, the downstream backwater effect is partial-ly accounted for at one time step behind. In routing the unsteady nonuniform flow byusing Eqs. (14.107) and (14.108), the value of Qf is not calculated as the steady uni-form full-pipe discharge. Instead, it is adjusted by assuming that

s -s _ a* _ v*v-s _ *, + j,-*u _ v<2 + l,~

vf, a4109)5'~5° & Ia* ° —S— —2iAJ— (14J09)

To improve computational stability, it is further assumed in TRANSPORT that at anyiteration k, Q^ is taken as the average of previous and current values: that is,

Q = Q + —^— AR™~/" 9 >TA* - i) O M A / A V J f2 2nVA* (14.110)

[ V2 V2 1i/2C Ar 4- h - h + '' '<* " 1} - * + M* ~ D^0AX + ft.(jt _ 1} ft. + 1)(jt _ 1} + — —

where all the values of h and V are those at the previous time wAr that are known if theone-sweep or implicit solution method is used to solve for the flow in individual sew-ers at incremental times. Incorporating Eq. (14.109) for S7 in Manning's formula yieldsa quasi-steady dynamic wave approximation instead of the kinematic wave. Thus, useof Eq. (14.110) to compute Qf indirectly gives a partial consideration of the down-stream backwater effect with a time lag. This improvement of the kinematic waveapproximation makes SWMM TRANSPORT hydraulically more attractive than thestandard nonlinear kinematic wave models. Presumably, the partial accounting of thedownstream backwater effect is effective as long as the flow does not change rapidlywith time, and no hydraulic jump or hydraulic drop is allowed. A hydraulic compari-son of EXTRAN to improvement and advantages over TRANSPORT has not beenreported and would be interesting.Nonetheless, since the downstream boundary condition is not truly accounted for, it isrecommended in SWMM TRANSPORT that for a sewer with a large downstream stor-age element from which the backwater effect is severe, the water surface is assumed ashorizontal from the storage element going backward until it intercepts the sewer invert.Moreover, when the sewer slope is steep, presumably implying high-velocity super-critical flow, the flood may simply be translated through the sewer without routing, thatis, shifting of the hydrograph without time lag. Also, if the backwater effect is expect-ed to be small and the sewer is circular in cross section, the gutter flow routing methodin the RUNOFF Block may be applied to the sewer as an approximation.In SWMM, large junctions with significant storage capacity and storage facilities arecalled storage elements, equivalent to the case of storage junction (that is, dsldt J= O),which was discussed above. Only the continuity equation, Eq. (14.35), is used in stor-age element routing. No dynamic equation is considered except for the cases with weiror orifice outlets. Small junctions are treated as point-type junctions with dsldt = O.

3. ILSD-B3 and WASSP-SIM. In the second group of nonlinear kinematic wave models,both ILSD-B3 and WASSP-SIM adopt the Muskingum-Cunge method.The Muskingum routing formula can be written for discharge at x = (i + I)Ar andt = (n + I)Ar as

Qt + m +i = C1Q11n + C2G411 + ! + C3Q1- + u (14.111)

in which

c = KX + 0.5Ar (U m}Cl K(I -X) + 0.5Ar (I4.li2a)

yi 0.5Ar A.X /1/1 11 ou\c' W-x> + osti <l4-"2b)

fi-g:3;S%where A^ is known as the storage constant having a dimension of time and X a factorexpressing the relative importance of inflow. Cunge (1969) showed that by taking K andAr as constants, Eq. (14.111) is an approximate solution of the nonlinear kinematic waveequation [Eqs. (14.4) and (14.102) or Eq. (14.104)]. He further demonstrated that Eq.(14.111) can be considered as an approximate solution of Eq. (14.104) if

K = &x/c (14.113)and 1

X = ^-(e/cA*) (14.114)

where £ is the "diffusion" coefficient and c is the celerity of the flood peak that can beapproximated as the length of the reach divided by the flood peak travel time through thereach. Assuming K = Ar and denoting a = 1 — 2X, Eq. (14.111) can be rewritten as

a^ + . = f^ct. + I^ct. + , + 2^a+l. (14.H5)

In the traditional Muskingum method, X and, consequently, a are regarded as constant. Inthe Muskingum method as modified by Cunge, a is allowed to vary according to the chan-nel geometry and is computed as

a = KQfS0(Ax)2B (14.116)

in which B is the surface width of the flow and S0 the sewer slope. The values of a arerestricted to being between O and 1 so that C1, C2, and C3 in Eq. (14.112) will not be neg-ative. It is the variation of a, and hence C1, C2, and C3, that classifies the Muskingum-Cunge method as a nonlinear kinematic wave approximation.

The Muskingum-Cunge method offers two advantages over the standard nonlinearkinematic wave methods. First, the solution is obtained through a linear algebraic equa-tion [Eq. (14.111) or Eqs. (14.115) and 14.116)] instead of a partial differential equation,permitting the entire hydrograph to be obtained at successive cross sections instead ofsolving for the flow over the entire length of the sewer pipe for each time step as for thestandard nonlinear kinematic wave method. Second, because of the use of Eq. (14.116), alimited degree of wave attenuation is included, permitting a more flexible choice of thetime and space increments for the computations as compared to the standard nonlinearkinematic wave method.

In ILSD-B3, the coefficient a in Eq. (14.115) is computed at each grid point by usingEq. (14.116), while B and K both change with respect to time and space. The values of Kare computed by using Eq. (14.113) with the celerity c evaluated by

c = ag/aA (i4.ii7)

or for a partially filled pipe using Manning's formula

c = *™*L S0- D» f 1 - ™4f [5 + sin WiV™l* - l)l (14.118)n \ 4> / L \2 / \ * /J

The initial flow condition is the specified base flow as in ILSD-B2. The upstreamboundary condition of the sewer is the given inflow hydrograph. The flow depth and othergeometric parameters at the sewer entrance can be computed from the geometric equa-tions given in Fig. 14.3. The junction condition used is the continuity relationship, Eq.(14.53). The solution is obtained over the entire time period at a flow cross section beforeproceeding to the next cross section. The solution then proceeds downstream section bysection and then sewer by sewer in a cascading sequence. More details on the computa-tional procedure of ILSD-B3 can be found in Yen et al. (1976).

The British model WASSP is a sewer design and analysis package consisting of foursubmodels (Price, 1982b): A modified rational method for design of sewers, a hydrographmethod for design of sewers using the Muskingum-Cunge routing, an optimal designmethod, and a simulation method using the fixed parameter Muskingum-Cunge techniquefor open-channel routing in sewers and the unsteady dynamic equation for surcharge flowcomputations. Open-channel flow is routed using Eq. (14.111) with the coefficients C1,C2, and C3 expressed as functions of c and JLI = QIIBS0. In computation, c is taken as thefull-pipe velocity and (I is evaluated at hlD = 0.6. Sewers under open-channel flow aresolved pipe by pipe, using a directionally explicit algorithm to calculate the discharge atthe sewer exit. The space increment Ax along the sewer is selected automatically in termsof Af to enhance computational accuracy. Connected surcharged sewers are solved simul-taneously. For surcharge flow, a time increment as small as a few seconds may be neces-sary if surges occur. The transition between open-channel flow and surcharge flow isassumed to occur when the discharge exceeds Q^ when the sewer entrance and exit aresubmerged, or when the water depth in the junction is higher than the sewer flow depthplus the entrance or exit headloss (Bettess et al num., 1978). At a junction, only the con-tinuity equation is considered for open-channel flow. For surcharge flow, in addition to thecontinuity equation, junction headloss is considered and incorporated into the surchargeunsteady dynamic wave equation. The headloss coefficient is assumed to be 0.15 for ajunction with straight pipes, 0.50 for 30° bend pipes, and 0.90 for 60° bend pipes. Somedetails of WASSP-SIM are reported in Price (1982b).

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