Unformatted text preview: Previous Page ht + Z. = h0 + Z0 otherwise (Case B in Fig. 14.9) where Z0 and h0 = the invert elevation and depth of the flow at the entrance of the downstream sewer taking the outflow from the junction, respectively. For a supercritical flow in a sewer flowing into a point junction, Case C in Fig. 14.9 would not occur. Only a subcase of Case B in Fig. 14.9 with ht < hic exists where Eq. (14.51) applies. For Case D of Fig. 14.9 with submerged exit, Zi + WY) = (PJD + Z0> ZI + D1 (14.52) The flow in the downstream sewer, which takes water out from the junction, may be subcritical, supercritical, or submerged, depending on the geometry and flow conditions. The flow equations are the same as the storage junction outflow Cases IIV given in Table 14.13. 74.5. HYDRAULICS OF A SEWER NETWORK Hydraulically, sewers in a network interact, and the mutual flow interaction must be accounted for to achieve realistic results. In designing the sewers in a network, the constraints and assumptions on sizing sewers as discussed in Sec. 14.3.4 should be noted. The rational method is the most commonly and traditionally used method for the design of sewer sizes. As described in Sec. 14.3.4, each sewer is designed independently without direct, explicit consideration of the flow in other sewers. This can be done because to design a sewer, only the peak discharge, not the entire hydrograph, of the designstorm runoff is required. As previously explained in Sec. 14.3.4, each sewer has its own design storm. The information needed from upstream sewers is only for the alignment and bury depth of the sewer and the flow time to estimate the time of concentration for the determination of the rainfall intensity / for the sewer to be designed. Contrarily, in simulation of flow in an existing or predetermined sewer network for urban stormwater control and management, often the hydrograph, not merely the peak discharge, is needed, and a higher level of hydraulic analysis that considers the interaction of the sewers in the network is required. This network system analysis involves combining the hydraulics of individual sewers as described in Sec. 14.3, together with the hydraulics of junctions described in Sec. 14.4. A sewer network can be considered as a number of nodes joined together by a number of links. The nodes are the manholes, junctions, and network outlets. The links are the sewer pipes. Depending on their locations in the network, the nodes and links can be classified as exterior or interior. The exterior links are the most upstream sewers or the last sewer having the network exit at its downstream end. An exterior sewer has only one end connected to other sewers. Interior links are the sewers inside the network that have both ends connected to other sewers. Exterior nodes are the junctions or manholes connected to the upstream end of the most upstream sewers, or the exit node of the network. An exterior node has only one link connected to it. Interior nodes (junctions) inside the network have more than one link connected to each node. A systematic numeric representation of the nodes or links is important for computer simulation of a network. One approach is to number the links (sewers) according to the branches and the order of the pipes in the branch, similar to Horton's numbering of river systems. Another approach is to identify the links by the node numbers at the two ends of the link. Using the node number identification system, a numbering order technique similar to topographic contour lines called the isonodal line method, can be used. This method was proved effective for computer manipulation of sewer network simulation (Mays, 1978; Yen et al, 1976). There are special hydraulic features of the networks that are often ignored to simulate flow in sewers. Usually, much attention is given to single sewers in the network but little attention to the junctions, and the mutual effects among the sewers are often neglected. If the pipes in a sewer network are all surcharged, it is obvious that the network should be solved as a whole similar to water supply networks. Conversely, if all the pipes carry supercritical flow, one can simply solve the flow starting from the most upstream sewers, complete the solution for the upstream sewers before proceeding to the solution of the downstream sewers in sequence. Likewise, if each one of the sewers in the network has a drop of sufficient height at its exit to ensure a free fall at its downstream end, the downstream boundary condition is specified when the flow is subcritical. Hence, the solution can still progress sewer by sewer starting from the most upstream exterior sewers, having the solution for the upstream sewers completed before proceeding to the next downstream sewer. However, more often than not, flows in sewer networks do not fall into one of these three types just mentioned. For example, for a subcritical sewer flow that can be mathematically represented by the SaintVenant equation or its simplified approximations, the downstream boundary condition at the sewer exit depends on the hydraulic condition of the downstream junction. Except for the network exit, this downstream junction condition by itself is unknown, and its solution is affected by the sewers joining to it. The junction continuity equation is obtained from Eq. (14.47) written in a finite difference form using the average values between time levels n and n + 1: 1(G411 + Q,n + i) + Qj.n + Qj*
+ i  ^ (Hn + i  Hn) = O (14.53) where the summation of i is over the number of the sewers connected to the junction. For a pointtype junction, the last term vanishes. It should be cautioned that if surcharge flow is considered, pointtype junctions should not be used. When the junction is completely filled and ground flooding occurs, A7 is so large that Hn +1 practically equals Hn. The junction dynamic equation depends on the condition between the junction and the connecting pipes, as shown in Table 14.13 and discussed in Sees. 14.4.5 and 14.4.6. Thus, for a simple network consisting of a single fourway junction with three upstream sewers and one downstream sewer and having each sewer divided into two computational reaches, by using an implicit numerical scheme the matrix of the equations to be solved for the 24 unknowns is shown in Fig. 14.22. There is one continuity equation and one momentum equation in its complete or simplified form for each reach. There are two boundary conditions for each sewer, which are described by the junction equations for an interior junction, and by the network outlet or inflow conditions for an exterior junction. Therefore, in general, if there are N sewers in the network and each sewer is divided into M reaches for openchannel flow computation, there are N(2M + 2) algebraic equations to solve for the N(2M + 2) unknowns. If all the sewers of the simple fourway junction network are surcharged and the Preissmann hypothetical slot is not used, there are nine algebraic equations to be solved for the nine unknowns, that is, one surcharge equation and one upstream junction continuity equation for each sewer, plus the network outlet boundary condition. The matrix for implicit solution of the simple network is shown in Fig. 14.23. Likewise, if the simple fourway junction network has one sewer surcharged and the other three sewers under openchannel flow for which each sewer is divided into two computational reaches, there are 20 equations for the 20 unknowns. The corresponding matrix is shown in Fig. 14.24. In general, if a network has N0 sewers under openchannel flow, each divided into M computational reaches, and N5 sewers under surcharge flow, the Equation Unknown FIGURE 14.22 Matrix configuration for 4way junction with all four sewers having openchannel flow. (After Yen, 1986a). Equation Unknown FIGURE 14.23 Matrix configuration for 4way junction with all four sewers having surcharged flow. (After Yen, 1986a). Equation Unknown FIGURE 14.24 Matrix configuration for fourway junction with three sewers having openchannel flow and one sewer surcharged. (After Yen, 1986a). total number of equations is N0(2M + 2) + 2NS if the last (exit) pipe of the network is not surcharged, or N0(IM+ 2) + 2NS + 1 if the exit pipe of the network is surcharged. Thus, it is obvious that the number of unknowns and equations to be handled can easily become large for a sewer network consisting of many sewers. Consequently, the method of solution becomes important to achieve the efficient solution of the entire network. The network solution techniques can be classified into four groups as follows: 1. Cascade method. In this method, the solution is sought sewer by sewer, starting from the most upstream sewers. Each sewer is solved for the entire duration of runoff before moving downstream to solve for the immediately following sewer: that is, the entire inflow hydrograph is routed through the sewer before the immediate downstream sewer is considered. For the noninertia, quasisteady dynamic wave and dynamic wave equations, this cascading routing is possible only for the following two conditions: (1) the flow in all the sewers is supercritical throughout, or (2) the exit flow condition of each sewer is specified, independent of the downstream junction condition—for example, a free fall over a drop of sufficient height at the downstream end of each sewer. However, in some sewer models, the downstream condition of the sewer is arbitrarily assumed—for example, using a forward or backward difference and approaching to normal flow, so that the cascading routing computation can proceed, and the solution, of course, bears no relation to reality. Conversely, in the kinematic wave approximation, only one boundary condition is needed and it is usually the inflow discharge or depth hydrograph of the sewer. At the junction, only the downstream sewer dynamic equation and junction continuity equations are used. The junction dynamic equations for the upstream sewers are ignored. Thus, the computation can proceed downstream sewer by sewer in a cascading manner, completely ignoring the downstream backwater effects. This method of solving each sewer individually for the entire hydrograph before proceeding to solve for the next downstream sewer is relatively simple and inexpensive. But, it is inaccurate if the downstream backwater effect is significant. 2. Onesweep explicit solution method. In this method, the flow equations of the sewers and junctions are formulated by using an explicit finite difference scheme such that the flow depth, discharge, or velocity at a given computation station x = i&x and the current time level t = n Af can be solved explicitly from the known information at the previous locations x < i Ax at the same time level, as well as known information at the previous time level t = (n — 1) At. Thus, the solution is sought reach by reach, sewer by sewer, and junction by junction, individually from upstream to downstream, over a given time level for the entire network before progressing to the next time level for another sweep of individual solutions of the sewers and junctions for the entire network. In this method, only one or a few equations are solved for each station, avoiding the large matrix manipulation as in the implicit simultaneous solution of the entire network, and computer programming is relatively direct and simple compared to the simultaneous solution and overlapping segment methods to be discussed later. Nevertheless, this onesweep method bears the drawback of computational stability and accuracy problems of explicit schemes that requires the use of small Af. An example of this approach for sewer flow is the Extended Transport (EXTRAN) Block of Stormwater Management Model (SWMM) (Roesner et al., 1984) in which each sewer is treated as a single computational reach. Variations of this onesweep approach do exist. For example, each reach of a sewer can be solve explicitly, or the sewer is solved implicitly, and then the junction flow condition is solved explicitly. 3. Simultaneous solution method. When the implicit difference formulations of the dynamic wave, quasisteady dynamic wave, or noninertia equations and the corre spending junction equations are applied to the entire sewer network and solved for the unknowns of flow variables together, simultaneous solution is ensured. The simultaneous solution usually involves solving a matrix similar to those shown in Figs. 14.2214.24, but much bigger for most networks. There are numerically stable and efficient solution techniques for sparse matrices of implicit schemes. Nonetheless, since the matrix is not banded, solution of the sparse matrix may still require high computational cost and large computer capacity if the network is large. 4. Overlapping segment method. To avoid the costly implicit simultaneous solution for large networks and still preserve the advantages of stability and numerical efficiency of the implicit schemes, Sevuk (1973) and Yen (1973a) proposed the use of a technique called the overlapping segment method. Similar to the wellknown Hardy Cross method for solving flow in distribution networks, this method is a successive iteration technique. Unlike the Hardy Cross method which applies only to looped networks, the overlapping segment method can be applied to dendritic as well as looptype networks. It decomposes the network into a number of small, overlapped, subnetworks or segments, and solutions are sought for the segments in succession. Thus, it is suitable to be used in programming for models to be applied for simulation of large sewer networks in different locations. A simple, singlestep overlapping segment example of a network consisting of three segments is shown in Fig. 14.25. Each segment is formed by a junction together with all the sewers joining to it. Thus, except for the most upstream and downstream (exterior) sewers, each interior sewer belongs to two segments—as a downstream sewer for one segment and then as an upstream sewer for the sequent segment, that is, "overlapped." Each segment is solved as a unit. The flow equations are first applied to each of the branches of the most upstream segment for which the upstream boundary condition is known and solved numerically and simultaneously with appropriate junction equations for all the sewers in the segment. If the flow is subcritical and the boundary condition at the lower exit of the downstream sewer of the segment is unknown, the forward or backward differences, depending on the numerical scheme, are used as an approximate substitution. Simultaneous numerical solution is obtained for all the sewers and junctions of the segment for each time step, repeating until the entire flow duration is completed. For example, for the network shown in Fig. 14.25a, solutions are first sought for each of the two segments shown in Fig. 14.25Z?. Since the downstream boundary condition of the segment is assumed, the solution for the downstream sewer is doubtful and discarded, whereas the solutions for the upstream sewers are retained. The computation now proceeds to the next immediate downstream segment (e.g., the segment shown in Fig. 14.25c). The upstream sewers of this new segment were the downstream sewers of each of the preceding segments for which solutions have already been obtained. The (a) ( b) (c) FIGURE 14.25 Overlapping segment scheme for network solution. (After Yen, 1986a), inflows into this new segment are given as the outflow from the junctions of the preceding segments. With the inflows known, the solution for this new segment can be obtained. This procedure is repeated successively, segment by segment in the downstream direction, until the entire network is solved. For the last (most downstream) segment of the network, the prescribed boundary condition at the network exit is used. The method of overlapping segments reduces the requirements on computer size and time when solving for large networks. It accounts for downstream backwater effect and simulates flow reversal, if it occurs. Its accuracy and practical usefulness have been demonstrated by Sevuk (1973) and Yen and Akan (1976). Solution by the singlestep overlapping segment method accounts for the downstream backwater effect of subcritical flow only for the adjacent upstream sewers of the junction, but it cannot reflect the backwater effect from the junction to sewers farther upstream if such case occurs. However, by considering the lengthtodepth ratio of actual sewers, in most cases the effect of backwater beyond the immediate upstream branches is small, and hence, this imposes no significant error in routing of sewer flows. For the rare case of two junctions located closely, the overlapping segment method can be modified to perform double iteration by recomputing the upstream segment after the approximate junction condition is computed from the first iteration of the downstream segment. Alternatively and perhaps better, a twostep overlapping procedure can be employed by making a segment containing two neighboring junctions, forming the segment with three levels of sewers instead of two, and retaining only the solutions of the top level sewers for each iteration. Thus, interior sewers are iterated twice. The overlapping segment method can be modified to account for divided sewers or loop networks, in addition to treetype networks. The entrance and exit loss coefficients of the sewers at a junction vary with the submergence YID at the exit or entrance of the sewers. The values of these loss coefficients may be approximated from the information given in Sec. 14.4.2. In order to minimize the chance of computation instability due to discrete values of the loss coefficient K, Pansic (1980) assumed an Scurvetype smooth transition starting from a minimum value K1 at incipient submergence YID = 1. The loss coefficient will attain a maximum value Km at some maximum submergence YJD, for example, at the level of ground flooding. The variation in between is
(Y Y K=K1+a[lJ and
K=K f orl Y Y +D < *£L_ Y +D Y Y "JD^D^T (14.54) [Y i (Y Tl2 +1  [D2{T l
a fr ° (14 55) ' where „,2^^.
BH (1456) 74.6 CAPACITY AND BOTTLENECK DETERMINATION One of the most useful pieces of information to solve urban drainage problems is knowing the flow carrying capacity of the channel or sewer. There are actually various kinds of capacities. There is the capacity for a single sewer. There is the capacity of the sewer net work as a system which usually is different from the capacity of individual sewers. For an open channel, often the maximum steady uniform flow that the channel can carry without spilling over bank is quoted as its capacity. For a sewer, the just about full gravity (openchannel) steady uniform flow is usually quoted. In fact, for a subcritical openchannel flow, the discharge that the channel can carry depends on the downstream water level. For a channel with a range of possible exit water levels, this would require repeated backwater profile computations. For a channel or sewer network that has a number of connected channels, the number of backwater computations can easily become very large making it nearly impossible, if not impractical, for the network capacity determination. Yen and Gonzalez (1994) developed a method to summarize the backwater information of a channel into a hydraulic performance graph from which the network capacity can be determined. Knowing the channel or sewer capacities allows a new approach to solve flood drainage problems by separating them into two parts: The demand part of how much water needs to be drained, which is essentially a hydrology problem. The supply part of how much can the channel or sewer handle, that is, the capacity, which is a hydraulic problem. The flood drainage problem can be analyzed by comparing the two parts and searching for a solution. 14.6.1 Hydraulic Performance Graph A hydraulic performance graph (HPG) is a plot of a set of curves showing the relationship between water depths, yu and yd, at the upstream and downstream ends of the channel reach for different specified discharges, that is, yu = F(yd, Q). Depicted in Figs. 14.26 and 14.27 are the HPGs for a mildslope channel and a steepslope channel, respectively. For a channel with mild slope (that is, Vn > yc where yn is the normal flow depth and yc is the critical depth for the given Q), the HPG has the following main characteristics: 1. The hydraulic performance curves, each for a given discharge, never intercept each other. The curves with higher discharges are located above those with lower discharges. 2. The left bound of the curves indicated as "Ccurve" in Fig. 14.26 represents the locus of critical flow condition at the downstream end of the channel reach. The downstream Ccurve Nline MlRipon Zline FIGURE 14.26 Hydraulic performance graph for mildslope channel. (From Yen and Gonzalez, 1994). NUiu Zline encarve FIGURE 14.27 Hydraulic performance graph for steepslope channel. (From Yen and Gonzalez, 1994). depth yd is computed by using the following equation with yd = yc(Q),:
A3 Q2 f = ^(14.57) C O where Ac and Tc = the flow crosssectional area and water surface width corresponding to the critical depth yc, respectively, and g = the gravitational acceleration. 3. The hydraulic performance curves are bounded at the right by the 45° straight line yu = yd  S0L, designated as Zline in Fig. 14.26, which is the line representing a horizontal water surface and no flow. The performance curves approach asymptotically to the Zline for very large values of yd or yu where the flow has very small convective acceleration. 4. The locus of normal flow depths for the possible discharges is name the Wline. It can be expressed as yu = yd and is located at a distance S0L left to the Zline. 5. The Afline divides the hydraulic performance curves into two regions. The region between the Ccurve and the Wline contains all the possible pairs of upstream and downstream end water depths for which the backwater profiles are M2type (see Table 3.5 or Chow, 1959), whereas the water depth pairs within the region between the Nline and the Zline correspond to all the possible Mltype profiles. The HPGs for horizontalslope and adverseslope channels are similar to those in Fig. 14.26 without the TVline because for these two cases, the normal depth is infinity and imaginary, respectively. The HPG of Sl profiles in a steepslope channel (Fig. 14.27) has the following major characteristics: 1. The hydraulic performance curves of Sl profiles, each for a given discharge, never intercept each other. The curves with higher discharges are located below those with lower discharges. 2. The right bound of the curves indicated as CHcurve in Fig. 14.27 represents the locus of critical flow condition at the upstream end of the channel reach. The upstream depth yu is computed by using Eq. (14.57) with yu = yc(Q). 3. The region of S!profile hydraulic performance curves is bounded on the left by the hydraulic performance curve that corresponds to the discharge Qs which starts at the C14 curve and is asymptotic to the Zline for large yu and yd. The threshold discharge Qs is the discharge for which the channel slope is critical. For smaller discharges, the channel slope becomes mild instead of steep. The value of Qs is determined by setting the critical depth equal to the normal depth, or equivalently the critical discharge equal to the normal flow discharge with yn = yc, which yields #4/3J sn2
(1458) V = fe For a given channel reach, the procedure to establish the hydraulic performance graph HPG is as follows: 1. Determine the ranges of depths or water surface elevations to be considered at the two ends of the channel reach. 2. Determine and plot the Zline for which the water surface elevations are equal at the upstream and downstream ends, or yd = yu + S0L, where S0 is the channel slope and L is the reach length. 3. Determine and plot the Wline which is the 45° line at a distance equal to S0L to the left of the Zline, and on which yd = yu. 4. For a mildslope channel with Ml or M2type backwater profiles, choose a discharge Q, a. Compute the normal flow depth yn = yu = yd and mark this point on the TVline for this Q. b. Compute the critical depth, yc, at the downstream end of the channel reach by using Eq. (14.57). c. Perform the backwater computation for the given Q and yc at the downstream end to determine the corresponding upstream depth yu, the backwater computation can be done by using the standard step method, direct step method, or any other methods described in Sec. 3.5 or in Chow (1959). d. Plot the result of this set of (yu, yd = yc) for the specified Q as one point of the Ccurve on the HPG; it is also the beginning point of the hydraulic performance curve of the chosen discharge. For a steepslope channel with 51type backwater profiles, a. Determine the value of Qs which corresponds to the condition at which the critical depth is equal to the normal depth, that is, yn = yc, or equivalently the critical discharge is equal to the normal flow discharge, that is, Qn = Qc. The value of yn = yc can be obtained by using Eq. (14.58). This depth is the water depth that corresponds to the minimum discharge Q5, for which the channel slope remains steep, and with this depth, Q5 can be computed by using Eq. (14.57). b. For a chosen Q ^ Q5, use Eq. (14.57) to compute the critical depth yc at the upstream end of the channel reach. c. Perform the backwater computation for the specified Q starting with the corresponding yc at the upstream end of the reach to determine the downstream depth yd. d. Plot the result of this set of (yd, yu = yc) for the specified Q as one point of the Cncurve on the HPG, and it serves as the beginning point of the hydraulic performance curve of the chosen discharge. 5. For the Q chosen in Step 4, select a feasible downstream depth yd and perform a backwater computation to determine the upstream depth yu. This pair of (yd, yu) constitutes a point of the hydraulic performance curve for this Q. 6. Repeat Step 5 for a few selected yd's to provide sufficient pairs of (yd, yu) values to plot the hydraulic performance curve for the chosen Q. The curve starts at the Ccurve and approaches asymptotically to the Zline for large yu or yd. For a mildslope channel, the constant Q curve crosses the AMine between the downstream Qcurve and the Zline. For a steepslope channel, the constant Q curve starts at the upstream CMcurve. 7. Select different feasible discharges and repeat Steps 46 to establish the hydraulic performance curves for different Q's. 8. Connect the criticaldepth C points computed for different discharges as the Ccurve on the HPG. Upstream Bank Elevation Upstream Water Surface Elevation [ft] Downstream Bank Elevation Nline Upstream channel bottom elevation Downstream channel bottom elevation Downstream Water Surface Elevation FIGURE 14.28. Example HPG for Reach 1 of Boneyard Creek. A typical Mtype HPG is shown in Fig. 14.28 as an example. 14.6.2 Flow Capacities of a Channel Reach The following representative flow capacities can be defined for an individual channel reach (Yen and Gonzalez, 1994): 1. Absolute maximum carrying capacity of a channel reach (Qamax)—the largest discharge the reach is able to convey when the water depth at its exit cross section is critical while there is no bank overflow. 2. Maximum uniform flow capacity (<2nmax)—the maximum steady, uniform, flow discharge that the reach can convey either as the flow just about to spill overbank, or as the flow reaching a surcharged condition, with the free surface parallel to the channel bottom. 3. Maximum flow capacity for a given exit water level (<2exmax)—for a given tailwater stage, the maximum steady gradually varied openchannel flow discharge that the reach can carry without spilling overbank. Obviously, this capacity varies with the tailwater lever, having gamax as its upper limit. 4. Maximum surchargedflow capacity (<2smax)—for a channel reach with a top cover such that under high flow the openchannel flow changes to pressurized conduit flow, this capacity is the discharge this closedtop reach can convey when the upstream water surface is at the bank=full stage and the downstream water elevation is at the crown level of the opening of the bridge, culvert, or sewer. The water surface profiles corresponding to these four different capacities are shown in Fig. 14.29 for the closedtop Reach 2 in Fig. 14.30 as an example. The value of <2smax is determined from the closedconduit flow rating curve shown in Fig. 14.31, whereas the openchannel flow capacities are determined from Fig. 14.32. The values of Q31113x and <2nmax for the other three openchannel reaches individually are also listed in Table 14.14, which are read from their individual HPG shown in Figs. 14.28, 14.33, and 14.34. The HPGs of channels of similar geometries can be nondimensionalized for more general uses. Shown in Fig. 14.35 is such a graph for openchannel flows in circular sewers with S0LID = 0.05 and Qf/VgCP = 0.224 where S09 L, and D = the slope, length, and diameter of the sewer pipe, respectively; Qf = the justfull steady uniform flow sewer Upper Chord Bridge ElCV 704.30 FIGURE 14.29 Water surface profiles for different hydraulic capacities of closedtop bridge Reach 2 of Boneyard Creek. Reach 4 FOOTBRIDGE Elevation [ft] Reach 3 2 Reach 1 CHANNELBOTTOM Critical Point for OeSfS cfs and Exit Sfaoe * 711 ft Location FIGURE 14.30 Example flow capacities and water surface profiles of Boneyard Creek for two exit water levels. The Bridge UpfJer and Lower Ch rd Elevation Difference is 2.2 ft FIGURE 14.31 Rating curve for Reach 2 of Boneyard Creek. Upper Chord Elevation 712.7 Low Cbord Elevation 710.5 Upstream Water Surface Elevation Iftl Low Chord Elevatio 710.5 Upp r Chord Elev tion 7127 Upstream channel bottom elevation 704.6 Downstream ^hannel bottom levotion 704.3 Downstream Water Surface Elevation [ftl FIGURE 14.32. Example HPG for Reach 2 of Boneyard Creek (7 + 375 to 7 + 435). TABLE 14.14 Maximum Capacities of Individual Reaches of Boneyard Creek Channel Reach Reach 1: Reach 2: Reach 3: Reach 4: Lincoln Ave.—Gregory St. Gregory St. Bridge Gregory St.—Footbridge Footbridge—Loomis Lab (7 + 945) £>amax (ft3/s) 1130 1025 1100 1370 <2nmax (ft3/s) 930 807 800 1070 Gsmax (ft3/s) 1371 capacity; y = the depth of flow, subscripts 1 and 2 = upstream (entrance) and downstream (exit) cross section of the sewer, respectively; and the subscript n = normal (steady uniform) flow. 14.6.3 Bottleneck and Channel System Capacity Determination The bottleneck of network of drainage channels or sewers is the critical location within the network where the water is about to spill overbank or violate specified restriction. Therefore, for a given exit water level, the bottleneck determines the capacity of the channel network as a system without flooding. Different exit water levels may have different bottleneck locations and different system capacities. Because the flows in the channels of Upstream! Bank Elevation 711.8 Upstream Water Surface Elevation [ft] Dqwnstream Bqnk Elevation 7t1.8 Upstream cho nel bottom elevation Downstream c annel bottom elevation Downstream Water Surface Elevation FIGURE 14.33. Example HPG for Reach 3 of Boneyard Creek (7 + 435 to 7 + 575). Upstfeom Bank Elevation Upstream Water Surface Elevation {ft] Downstream Bonk Eleva on Upstream hannel bottom elevation Downstream channel bottom elevation Downstream Water Surface Elevation FIGURE 14.34. Example HPG for Reach 4 of Boneyard Creek (7 + 575 to 7 + 945). FIGURE 14.35. Nondimensional hydraulic performance graph for a sewer pipe. (Yen, 1987). the network mutually interact, this interaction must be accounted for in the system capacity determination. The set of the HPGs and rating curves for the individual reaches of the system can be used together in sequence to determine the bottleneck and capacity of the channels as a system. To apply the HPG method of Yen and Gonzalez (1994) to determine the bottleneck and flow capacity of a drainage channel system, the channel system is first subdivided into reaches such that within each reach the geometry, alignment, and roughness are approximately the same and there is no significant lateral inflow within the reach. For subcritical flow, the system capacity is a function of the tailwater level at the exit of the most downstream reach. The maximum system capacity occurs when depth at the system exit is critical. While the flow capacity of each reach can be determined individually from its HPG, the flow capacity of the channel flowing as a part of the system should be Reachj+l Rea^hjl Retichj FIGURE 14.36. Schematic of lateral runoff contribution along a channel. (After Yen and Gonzalez, 1994). determined by accounting for the backwater effect between the reaches, the losses at the junctions, if any, and the significant lateral flow joining the channel. Concerning the lateral flow entering the channel system at the junctions between reaches, a simple approximate method of Yen and Gonzalez (1994) can be used if no better method is available. As shown schematically in Fig. 14.36; the area drained by channel reach j— 1 is Au and the corresponding discharge is Q11 = CjAn, where C14 is the runoff coefficient for the area drained. The lateral flow joining the channel at the downstream end of reach j— 1 is QL(jiy The sewer delivering the lateral flow peak discharge QL(JD drains an incremental local area AL(M) having a CL(/1) runoff coefficient. Therefore, under the rainstorm with intensity i, QL(j _ 1} = CL(j _ l}iAL(j _ 1}. At the junction between reaches j— 1 a ndy, Q1 = Qu + GLOD Hence, the ratio between the lateral flow and the flow in reach j1 is Qj Cj Ar ^P= L «''/^ C yu A (14.59) For a given tailwater level at the system exit, for each reach the upstream water level is determined from the HPG knowing the downstream water level, progressing reach by reach toward upstream, with junction head loss included if it exists. If the reach is surcharged, the rating curve is used instead of HPG. It may require a trial of several discharges to locate the bottleneck and identify the system capacity iteratively. Details and examples of this procedure can be found in Yen and Gonzalez (1994). By applying this procedure to the example fourreach system shown in Fig. 14.30, with the rating curve of Fig. 14.31 and HPGs of Figs. 14.28, 14.33, and 14.34, the network capacity for exit tailwater level equal to 711.0 ft is determined as 515 fWs. This capacity is controlled by the bottleneck of spilling water overbank near the upstream end of reach 1 as shown in Fig. 14.30. The water surface profile is also shown in this figure. The channel system hydraulic capacity and the location of the bottleneck vary with the water surface elevation at the exit. When the exit water level is low, the bottlenecks tend to locate in the upstream parts of the system. As the exit tailwater level rises, the bottlenecks tend to move downstream as the exit tailwater level rises, and the system capacity decreases as demonstrated by the example shown in Fig. 14.37. Identification of the most likely range of exit tailwater levels and removal of the bottlenecks in this range may be a relatively simple and effective way for system capacity improvement. For the example system shown in Figs. 14.30 and 14.37, removal (raising) of the Gregory Street Bridge (the first two obstacles) improves the system capacity as indicated by the dashed line in Fig. 14.37. If the flood frequency (dischargereturn period) relationship is known, the system capacity curve and the locations of bottlenecks shown in Fig. 14.37 can be converted into a system capacity curve in terms of return period versus exit water level as demonstrated in Fig. 14.38 for the system shown in Fig. 14.30. It can be seen that with the improvement of the Gregory Street Bridge removal, the system absolute maximum capacity is increased from 860 to 970 fWs, or a return period improvement from 25 to 40 years. At a likely exit water level of 710.75 ft, the system capacity is increased from 655 to 730 ft3/s, or a return period improvement from 11 to 15 years. Thus, it is obvious that when the connecting reaches are considered interacting as a system, the overall channel capacity is different from any of the capacity values of the individual reaches. For an openchannel system, the backwater effects of connecting reaches usually prevent the exit depth of interior reaches to become critical. Therefore, the absolute maximum capacity, Qamax, of a reach serves as the upper bound provided open Critic I Bank Elevation at Exit Reach = 711.20 ft Water Surface Elevation at the Exit [ft] C iticol Sta ions S ation 7+ 75, Downstream of Gregory St. Bridge S ation 7+ 35, Upstream of Gregory St. Bridge S ation 7+ 75, Footbridge Improveme t by Removal of Gregory S . Bridge Discharge at the Exit FIGURE 14.37 Example hydraulic capacity curve of Boneyard Creek. Water Surface Elevation at Exit Station 7+105 [ft] Critical Bank Elevation at Exit Reach = 711.20 ft CriticalnS7+375, Downstream of Gregory St. Bride Statio tations pstream Station 7+435, Uootbridge of Gregory St. Bridge Station 7+575,y FRemoval of Gregory St. Bridge Improvement b Return Period [Years] FIGURE 14.38 Example flood stage frequency of Boneyard Creek. channel flow prevails in the reach and also in adjacent reaches upstream and downstream. For a closedtop reach, the upper bound is the larger of Q8n^x and Qsmax. For an openchannel reach connected to a closedtop reach at either its upstream or downstream, or both, the upper bound is the larger between Q31n^ and the largest discharge allowed under submerged exit or entrance condition of the openchannel reach. The capacity of the system of reaches as a whole cannot exceed the smallest of the upper bound of the individual reaches just mentioned, adjusted for lateral flow entering the interior reaches in the channel system. However, the location of the bottleneck, which determines the capacity of the channel as a system, may not and often is not in this reach of smallest Q90110, upper bound, and in such a case, the system capacity is smaller than this smallest upper bound. 74.7 HYDRAULICS OF OVERLAND FLOW 14.7.1 Overland Flow and Resistance Equations Runoff on overland surface is usually nonuniform, unsteady, openchannel flow. For the depth and velocity ranges encountered in urban surface runoff, the flow can be laminar or turbulent, subcritical or supercritical, stable or unstable, as depicted in Fig. 14.39 for Velocity, m/t Subcriticol Turbulent Unstoble Supercritical Turbultnt Subcritical Laminar Supercritical Laminar •Unstable Supercritical Laminar FIGURE 14.39 Depthvelocity relation for steady openchannel plane flow. (From Yen, 1986b). Weisbach Resistance Coefficient f Wide open channel Full pipe Reynolds Number R FIGURE 14.40 Weisbach resistance coefficient for steady uniform flow in open channels. (FromYen, 1991). steady flow as an approximation. Mathematically, the flow can be described by the onedimensional continuity relationship [Eq. (14.4) or Eq. (14.5)] and momentum relationship [Eq. (14.1) or Eq. (14.2)] or their approximations, if applicable] or the corresponding twodimensional flow equations (see Yen, 1996, Tables 25.6 and 25.7). These equations are equally applicable to impervious and pervious surfaces, gutters, and pavement. For a highly pervious surface where the infiltrating subsurface flow interacts with the free surface flow, a conjunctive surfacesubsurface flow simulation such as those developed by Akan and Yen (198Ia) and Morita et al. (1996) should be used. Conversely, for drainage design of streets, roads, roadside gutters and inlets, the flow time of concentration is so short that often the flow can be regarded as steady and the design can proceed as such. Traditionally used design procedures with this assumption have been described in Chap. 13. The resistance coefficient in Weisbach form can be determined from Fig. 14.40 (Yen, 1991) which is a modified form of the Moody diagram for twodimensional steady uniform flow over rigid impervious boundary. The curves can be expressed in equation form for the Reynolds number R = VRIv < 500 as f=24/R for 500 < R < 30,000 as /= 0.224/fl^5 and for R > 30,000 with kJR < 0.05 as /= IL1Og fA. + i rf 2 7 4 [ g(uR R™)\ Correspondingly, expressed in terms of Manning's n, for R < 500
n=Rm (14 62) (14.60) (14.61)  ^rg Jit (14 63)  for 500 < R <30,000 as K . n = Rl/6 —,=VO028 R~m Vg
and for R > 30,000 with kJR < 0.05 as R116 Kn \ (k 1.951]1 TvT vF H W + **l (14.64) (14 65)  For the third region of fully developed turbulent flow (Eq. 14.65), it is well known that n can be regarded approximately as a constant (Yen, 1991) and its value can be estimated from standard tables such as in Chow (1959) or Table 3.3. For shallow overland flow under rainfall, raindrops bring in mass, momentum, and energy input into the flow, and hence, the resistance coefficient is modified. Based on the result of a regression analysis by Shen and Li (1973), the values of n a nd/for R < 900 can be estimated from the following nondimensional relationship f
=8 (Ve n Y f ( i >4 24 + 66 Hr — H I ° ^(V^J R T= (Kn R^) J <1466) For a higher Reynolds number, Eq. (14.62) or constant n applies. One of the frequent purposes for overland flow simulation is to determine the peak discharge and its time of occurrence. For a continuous rainfall, the time to reach the peak discharge when all the areas within the watershed contribute is one definition of the time of concentration. A popular method of solving such overland flow problems is the kinematic wave approximation of the SaintVenant equations because it is relatively simple, easy to solve, and requires only one boundary condition for solution, whereas the other two higher level approximations require two boundary conditions (Sec. 14.3.4). Its biggest drawback is its incapability to account for the backwater effect from downstream. Such backwater effect does exist in urban subcritical overland flow, for example, when the street surface flow joins the gutter flow, and the backwater effect from inlet catch basin. Despite the heterogeneous nature of urban catchments, a fundamental understanding of the overland surface runoff process can be gained through the consideration of the runoff of rainfall excess on a sloped, homogeneous, relatively smooth, plane surface. After the initial losses are satisfied, rain water starts to accumulate on the surface. Initially when the amount of water is small and the surface tension effect is predominant, the water may be held as isolated pots without occurrence of flow, as one would observe on a glass surface with a small amount of water. As rain water supply continues, the surface tension can no longer overcome the gravity force and the momentum input of the raindrops along the slope of the surface. The individual water pools merge and flow starts downslope. One should be wise and extremely careful to select the appropriate simplified equations to solve overland flow problems. For instance, if the geometry of a short street gutter is well defined, the hydraulic characteristics of the inlet catch basin downstream of the gutter are known, and a relatively reasonable accurate result is desired, the kinematic wave approximation will not be acceptable and at least the noninertia approximation should be used. Conversely, when simulating a whole block conceptually as a flow plane, there is no reason to use the SaintVenant or noninertia equations because the grossly aggregated information of the block is incompatible with the sophisticated equations. Likewise, when representing a long flow surface as an impervious or pervious surface which is described merely by its length, width, slope, and overall average surface roughness type, the kinematic wave approximation usually suffices, whereas the noninertia or SaintVenant equations overkill because the downstream backwater effect is insignificant except for a small stretch at the very downstream. 14.7.2 Kinematic Wave Modeling of Overland Flow Despite its heterogeneous nature, urban overland surface is often hypothetically conceived as a collection of wide planes in modeling. For most overland flows, the depth is relatively small compared to flow length and the downstream backwater effect is insignificant; hence, the kinematic water approximation is applicable. For a wide open channel where the hydraulic radius R is equal to the flow depth Y, the kinematic wave momentum equation, S0 = S^ can be simplified as V = aYm~l or in terms of discharge per unit width of the channel ^1 as q1 = aY™ (14.68) where m = 5/3 and a = ^KnS0 In for the Manning formula, m = 3/2 and a = (SgS0//)1'2 for the DarcyWeisbach formula, and m = 3/2 and a = VCS0 for the Chezy formula. If either Eq. (14.67) or Eq. (14.68) is solved together with the continuity relationship [Eqs. (14.4) or (14.5)] in a nonlinear form, the simplification is a nonlinear kinematic wave approximation, often simply referred to as kinematic wave. If Eq. (14.67) or Eq. (14.68) is solved together with a simplified, linear form of Eqs. (14.4) or (14.5), the simplification is a linear kinematic wave approximation. Combining Eq. (14.67) with Eq. (14.5) and assuming a to be independent of x, we obtain ~\y ~\ \y \Y + (aym) = — + maYm  i = i (14.69) dt dx fa dx where the rainfall excess ie is ie = i~f
7 (14.67) (14.70) where / = the rain intensity on the water surface and/ = infiltration rate at channel bottom, that is, land surface. Since from Eq. (14.68) with a independent of x, 3g/3f = maYm~l(dY/^t). Substitution of this relation into the continuity equation yields another popular form of the kinematic wave approximation used in modeling: ^i+ 1^1 = ^ dt c dx c where c = maYm~l Assuming that a and m are both constants, Eqs. (14.69) and (14.71) yield _£ = c = mapni = my at (14.73) (14.72) (14.71) I, = ^ = iemV dt — = ie
A and
dt (14.75) (14.76) §=i For an initially dry surface ( 7 = 0 for O ^ x ^ L at t = O) under constant rainfall excess, ie and zero depth at the upstream end, integration of Eq. (14.76) yields Y=ijt Substituting this equation into Eq. (14.73) and integrating, one has x = X0 + aiem~l f* (14.79) (14.78) Let X0 = O, the equilibrium peak discharge time, te, can be obtained with x = L where L is the total length of the overland surface: t, = \^\m \ai*1) At this time, the discharge per unit width from the overland surface is 9,t = & and the discharge for O < t < tf is q,L = a(ietY> (14.82) (14.81) (14.80) FIGURE 14.41 Sketch of kinematicwave water surface profile during buildup times. The water surface profile during the buildup period O < t < te based on the above kinematic wave analysis is shown in Fig. 14.41. Generalized charts are available in the literature to determine the peak kinematic overland flow rate from infiltrating surfaces for which ie is not constant (Akan, 1985a, 1985b, 1988). 14.7.3 Time of Concentration The equilibrium time given in Eq. (14.80) for Manning's formula is often referred to as the kinematic wave time of concentration: H^f''0'4
(1483) For infiltrating overland flow planes, Akan (1989) obtained numerical solution to the kinematic overland flow and the Green and Ampt infiltration equations and fitted the following equation to the numerical results by regression: tc = (^TT (i  K)°A + S.lOA^P/Kl  St)i ~233 \Kn Vb0/ (14.84) where K is the soil hydraulic conductivity, () is porosity, Pf is characteristic suction head, and S1 is the initial degree of saturation of the soil. Note that this equation reduced to Eq. (14.83) for impervious surfaces with K=O. Morgali and Linsley (1965) numerically solved the SaintVenant equations instead of the kinematic wave approximation for a number of runoffs from idealized catchment surface, and the results were regressed to give the following equation for the time of concentration in minutes, n0.605 T0.593 ^ = ^5^03« (1485> where K = 0.99 for English units with L in feet, and i in in./h. Izzard (1946) provided the following equation based on his experiments of artificial rain on sloped surfaces: tc = 4lfo.0007/1/3 + 41 fc ?y * ^ « A ^V <14'86) for English units with iL < 500, L in ft and / in in./h, and C is the rational formula runoff coefficient. In practical applications, often the rain intensity / is unknown a priori. Hence, tc of Eqs. (14.83)(14.86) is computed iteratively with the aid of a rainfall intensity relationship. For overland surfaces of regular geometry beyond the twodimensional surface just discussed, formulas for peak discharge and time of equilibrium estimation can be found in Akan (1985c) and Singh (1996). In addition to the hydraulicbased equations for time of concentration, a number of hydrologicbased empirical time of concentration formulas also exist (Kibler, 1982). Equations (14.83) and (14.85), when applied to actual catchments, usually yield a tc value smaller than found empirically. There are a number of possible reasons to cause this discrepancy (Yen, 1987), including the following: 1. The catchment surface is usually not homogeneous as is assumed in the derivation of Eqs. (14.83) and (14.85), the surface undulation is far more than the sandequivalent roughness assumed in the derivation. 2. For shallow depth Manning's n is not a constant (Yen, 1991) and raindrop impact increases n. 3. The sensitivity to rain input /°4 is far more than reality. In the derivation, the input i is assumed as evenly distributed over the surface and without momentum, a pattern different from real rainfall. 4. Equations (14.83) and (14.85) are based on the time reaching the peak flow considering the influence of the flood wave propagation, different from the water particle travel time along the longest (or largest LA/SJ flow path. 5. The hydraulic time of peak flow measured from the commencement of rainfall excess, whereas the hydrologic time of concentration measured from the commencement of rainfall. Considering the aforementioned factors and to eliminate the rain intensity iteration process, Yen and Chow (1983) proposed the following formula for the overland flow time of concentration: ( NJ >6
HwJ
(i4 87)  where K is a constant and W is an overland texture factor, similar to Manning's n but modified for heterogeneous nature of overland surfaces. The values of K and N, modified slightly from their originally proposed values, are given in Tables 14.15 and 14.16, respectively. 14.8 MODELING OF CATCHMENT RUNOFF 14.8.1 Scientific Fineness versus Practical Simplicity Urban catchment runoff comes mostly from rainfall excess, that is, rainfall minus abstractions. The contribution of prompt subsurface return flow is usually negligible. (This is not necessarily the case for sewers, where leakage through joints and cracked pipes could be TABLE 14.15 Values of K for Yen and Chow Formula Light rain Rain intensity For L0 in feet with For L0 in metres with (in./h) (mm/h) t0 in hours ^inmin t0 in hours f 0 inmin < 0.8 <20 0.025 1.5 0.050 3.0 Moderate rain Heavy Rain 0.81.2 2030 0.018 1.1 0.036 2.2 >1.2 >30 0.012 0.7 0.024 1.4 Source: From Yen and Chow (1983). TABLE 14.16 Overland Texture Factor N for Eq. (14.86) Overland Surface Smooth asphalt pavement Smooth impervious surface Tar and sand pavement Concrete pavement Tar and gravel pavement Rough impervious surface Smooth bare packed soil Moderate bare packed soil Rough bare packed soil Gravel soil Mowed poor grass Average grass, closely clipped sod Pasture Timberland Dense grass Shrubs and bushes Land use Business Semibusiness Industrial Dense residential Suburban residential Parks and lawns Source: From Yen and Chow (1983). Low 0.010 0.011 0.012 0.012 0.014 0.015 0.017 0.025 0.032 0.025 0.030 0.040 0.040 0.060 0.060 0.080 Medium 0.012 0.013 0.014 0.015 0.017 0.019 0.021 0.030 0.038 0.032 0.038 0.050 0.055 0.090 0.090 0.120 High 0.015 0.015 0.016 0.017 0.020 0.023 0.025 0.035 0.045 0.045 0.045 0.060 0.070 0.120 0.120 0.180 0.014 0.022 0.020 0.025 0.030 0.040 0.022 0.035 0.035 0.040 0.055 0.075 0.035 0.050 0.050 0.060 0.080 0.120 considerable.) Among the abstractions, infiltration is by far the most significant. On a rainstormevent basis, evapotranspiration is relatively negligible. Interception varies with land use and seasons. Depression storage is a matter of definition and subsequent method of estimation. Quantitative information on the initial losses—interception and depression storage—can be found in, for example, Chow (1964) or Maidment (1993). At any rate, for a heavy rainstorm, the amount of initial losses is relatively small. However, it should be noted that in terms of pollution, or runoff on an annual basis, the contributions of light rainstorms are also significant. The hydrologic characteristics of urban catchments vary with land uses and seasons. The surface may range from the relatively impervious surfaces such as streets, sidewalks, driveways, parking lots, and roofs to pervious surfaces such as lawns, gardens, bare soil, Rain Roofs Lawns & Soils Sidewalks & Road Driveways Surface Parking Lots Street gutters Ditches Surface detention Surface retention Inlet catch basins Sewer pipes & Drainage Channels FIGURE 14.42. Elements of urban catchment. (After Yen, 1987). and parks. Rainfall excess on these surfaces are drained directly or indirectly through adjacent different types of surfaces and gutters into inlet catch basins, and then into sewers or channels (Fig. 14.42)]. The geometric composition of these different types of surfaces in forming a city block or catchment is usually random. This random heterogeneous nature of urban catchment surface imposes great difficulty in precise simulation of rainstorm runoff. Essentially, each surface requires a special, individual, "custom made" treatment which is costly and impractical in terms of both data and computation requirements. From the scientific viewpoint, existing knowledge appears to allow detailed scientific and quantitative simulation of each of the rainfall abstraction and surface flow processes than the current practice in urban drainage. Such simulation has not been incorporated in engineering practice mainly due to the conflicts between the detailed truthfulness in the scientific approach and the need for efficiency, simplicity, and tolerable accuracy in the practical procedures. In practice, various assumptions are explicitly or implicitly made so that some degree of simplification can be achieved for the sake of practical application and analysis. In the design of the size of most drainage facilities, usually knowing the design peak discharge, Qp, suffices. Conversely, for operation, planning, stormwater quality control and design involving runoff volume (such as detention storage), the discharge or stage hydrograph of the design rainstorm is needed. For the former, Qp, traditionally simple hydrologic methods such as the rational method can be used. For the latter, the runoff hydrograph can be determined using a hydrologic or hydraulic simulation model. Hydraulicbased simulation models employ a momentum or energy equation [either Eqs. (14.1) or (14.2), or any of the simplified approximations], together with the continuity equation; whereas hydrologic models do not consider momentum or energy equations. 14.8.2 Modeling Procedure Physically based simulation of the catchment rainfallrunoff process considers the water transport processes in the elements or components (Fig. 14.42) and their relative distribution within the catchment. Because of the number of elements in a catchment and the amount of computations involved, such simulations are usually done using a computerbased model. In formulating a physically based model, the following process phases are considered after the rainfall input has been determined: 1. Decomposition of the catchment. In this phase, one determines the level of subdivision of the catchment and components used to represent a subcatchment. Should the catchment be divided merely into subcatchments? Should the pervious and impervious surfaces be considered separately? Should the street gutters and inlet catch basins be considered specifically? How should the roof contribution in the model be treated? How are the different types of surfaces in a subcatchment related to one another? Should the detention ponds be treated separately and individually? The more the different surfaces and components are aggregated as a unit, the simpler the model, but the greater the loss of physical reality. 2. Methods for abstractions. In this phase, the methods to calculate the losses due to interception, depression storage, and infiltration are selected. One should consider if the abstraction values are assigned catchment wide or if they should be allowed to vary for different subcatchments and different types of surfaces. The latter is more physically satisfactory, but it also requires more input information. It should be decided if the water detained on the overland surface contributes to infiltration when rain supply is insufficient. If multipleevent continuous modeling is being considered, methods to calculate evapotranspiration and infiltrability recovery should also be included. 3. Runoff from subcatchments. In this phase, the method of transforming rain excess water to runoff and the routing of runoff on the surfaces and subcatchments to the inlet catch basins is selected in accordance with the level of subdivision of the catchment. In hydraulically based models, for routing runoff in a catchment, in addition to the continuity equation, a flow momentum equation of some form is used. The continuity equation can be based on the crosssectional averaged form [Eqs. (14.4) or (14.5)]. The momentum equation can be the full dynamic wave equation or any of its simplifications shown in Eqs. (14.1) or (14.2). These equations are applied to the elements of an urban catchment, step by step in sequence as shown in Fig. 14.42 for each time step to yield the runoff hydrograph of the catchment. It is not necessary to use the same routing method for the different elements and types of surfaces in a catchment. For example for an aggregated pervious surface, the timearea method, at most a kinematic wave routing usually would suffice because of the gross representation of its hydraulic characteristics. Conversely, for a street pavement, gutter, and inlet catch basin system, a kinematic wave routing may not be sufficient to provide realistic results because of its relatively well=defined geometric properties and the mutual backwater effects, and hence a noninertia routing may be desirable. It is obviously impractical to apply the highly sophisticated routing methods to each of the overland surfaces and gutters. The construction cost per unit length of such surfaces and gutters is small. And the cost of collection of data needed for such sophisticated computation methods is high. Moreover, the hydraulic characteristics of the overland surface change with time, depending on cleaning and season. On the other hand, the total length of streets and gutters in a catchment and in a city may be considerable. It is also desirable to have reasonably accurate inlet hydrographs as the input to the sewer system. Besides, for pollution control, the estimate of pollutant transport depends on the runoff estimation. Therefore, selection of the most appropriate overland routing method for a model and a catchment is a difficult task requiring delicate balance. Hydrologic simulation models for catchment runoff are not discussed here. They can be found in Chow (1964), Maidment (1993), and many other hydrology books. They range from distributed system model similar to the hydraulicbased model described above in Steps 13, but the flow velocity in Step (3) is estimated by using some empirical techniques; or in the lumped hydrologic system models such as the conceptual reservoirchannel models of the Nash/Dooge type or the unit hydrograph methods the runoff is estimated from an assumed relationship without considering the physical process (Yen, 1986b). In most cases, synchronized runoff and rainfall data are needed for derivation of the catchment unit hydrograph or calibration of the lumped system model; single=event data sets are difficult to obtain and if available often are not sufficiently reliable. For a city or a region, the urban surfaces tend to have some degree of similarity, especially in the United States where many cities have standardized square or rectangular blocks. It is, therefore, possible to group the urban surfaces into typical blocks. Reliable simulations can be made to establish unit hydrographs for the typical blocks or subcatchments. A few of such typical unit hydrographs should be sufficient for a catchment. In later applications, the use of the unit hydrographs provides relatively accurate results avoiding the repetitive costly sophisticated routing computation for individual rainstorms and blocks. Yen et al. (1977) first proposed this approach for a catchment in San Francisco using a nonlinear kinematic wave routing for the surface and gutter flow. Akan and Yen (1980) developed nondimensional unit hydrographs for streetgutterinlet systems using dynamic wave routing and considering specifically the inlet capacity allowing by pass flow. Harms (1982) took a similar concept using a semiempirical approach to establish 1min unit hydrographs. However, the unit hydrograph theory suffers from the linearity assumption between rainfall excess and surface runoff, making it inaccurate when the depth of the simulated rainstorm is significantly different from that of the rainstorm the unit hydrograph is derived. Lee and Yen (1997) introduced a hydraulic element of kinematicwave based flow time determination on geomorphologically represented catchment subdivision for derivation of the catchment instantaneous unit hydrograph, making it a hydraulic distributed model and allowing derivation of unit hydrographs for ungaged catchments. 14.8.3 Selected Catchment Hydraulic Simulation Models There exist many urban rainfallrunoff models. A summary of the important features of selected hydraulicbased urban catchment models, mostly nonproprietary, is given in Table 14.17. Some models cover both catchment surface and sewer network parts; only the catchment part is summarized in this table. The sewer part is given in Tables 14.21 and 14.22. One should refer to the original references for details and objectives of these models. Similar hydrologybased watershed models that have also been applied to urban catchments, such as SCSTR55 and TR20, Hydrologic Engineering Center (HECI) (or its replacement HECHMS), and RORB are not presented here. 14.8.4 Verification and Calibration of Models Models should never be used without being tested and verified. It has happened again and again that in the enthusiasm in model development, models are used without verification. All models have their own assumptions and simplifications. Moreover, most urban runoff models contain coefficients, exponents, or adjustment factors that require calibration with data to determine their values. Besides verification and application for predictions, there are other operational modes of models such as those shown in Table 14.18. In calibration, we try to determine the most TABLE 14.17 Summary of Selected Urban Catchment Surface Runoff Models Model Input Rainfall Abstractions Allow Initial Pervious Impervious Pervious Area Areal Losses Area Area Distribution Infiltration Contribution Contribution Horton's formula
Yes Yes Surface Runoff Routing Street Gutter Method User's Selected Inlet Manual References Catch Basin Yes Keifer et al. (1978); Tholin and Keifer (1960) Yes Papadakis and Preul (1972); Univ. of Cincinnati (1970) Chicago Single hydrograph hyetograph No Depression storage by exponential function Cincinnati Single No Depression hyetograph storage by exponential function Modified Izzard's Linear kinematic No wave storage routing with Manning's formula No Horton's Strips Strips Storage routing Continuity eq. formula, froni with constant of steady rain only depth detention spatially storage function varied flow and Manning 's formula No ILLUDAS Single No Different Horton's Area and entry Area and Timearea with No hyetograph constants for formula,fromline of direct entry line Izzard's time impervious rain only contributing of direct formula or surface, area of contributing kinematic wave and previous surfaces supplemental surface surface IUSR Hyetographs Yes Depression Horton's Divided into strips with Nonlinear Nonlinear Yes storage formula,frominput length, width, slope, kinematic kinematic wave rain only and roughness wave routing routing with Manning's formula Yes Terstreip and Stall (1974) Yes Chow and Yen (1976) TABLE 14.17 Model (Continued) Input Rainfall Abstractions Pervious Impervious Pervious Allow Initial Area Area Area Areal Losses Infiltration Contribution Contribution Distribution Surface Runoff Routing Street Gutter Method User's Selected Inlet Manual References Catch Basin SWMM Hyetographs Yes Depression storage Single Ehime Stormwater hyetograph Runoff
CTH Single hyetograph Belgrade Single hyetograph MOUSE Hyetographs WALLRUS/ Hyetographs HYDROWORKS Linear kinematic Linear kinematic No Yes Huber and Dickinson Horton's Divided into strips with wave, storage (1988); Huber and or Green input length, width, slope, wave, storage equation with and roughnessn routing with Heaney (1982); and uniform depth Manning's formula Metcalf&Eddy,etal. Ampt's continuity and continuity (1971) formulas equations equation and Manning's formula No No Toyokuni and Kinematic wave No No Depression By graph Divided into strips Watanabe (1984) storage by based on graph Horton's formula No No Arnell(1980) Kinematic wave Continuity No Depression Horton's Divided into unitwidth strips with input length, storage for formula slope, and roughness impervious surface by exponential function Nonlinear kinematic No No Radojkovic and Nonlinear Horton's No Strips Strips Maksimovic (1984) wave routing kinematic formula or wave routing coupled with subsurface flow Nonlinear No Yes DHI (1994) No Yes Strips Strips kinematic wave routing No No Yes Wallingford Conceptual twolinear reservoir simulation Yes Software (1997) TABLE 14.18 Modes of Operation of Models Mode Prediction Calibration Verification Validation Detection Parameter identification Sensitivity Reliability Input Known Known Known Known ? Known Known Known Transformation Parameters Coefficient Values Known Known Known Known Known ? Known Known Known ? Known Known Known Known (?) Known Known Output ? Known ?(/Known?) Applicable ? Known Known ?(Per unit change of parameter or coefficient) ?(Over likely ranges of parameters) Source: From Yen (1986b). suitable values of the coefficients of the parameters (variables) knowing the input and output from observed data. In verification, we have the parameters and their coefficient values all determined for the model, and we have the data on both input and output. The input is run through the model to produce output, which is compared to the known output in the data set to verify the agreement between the computed and observed outputs. On the other hand, verification is different from validation. Validation is to ascertain if the correct equation or model is used to solve the problem. Verification is to find out if the equation or model is solved correctly. No model can do everything. For example, a good flow simulation model may not produce a good design of the drainage system. Conversely, a good design model may not—and often need not—be an accurate flow simulation model. Therefore, models should be verified and validated according to their objectives and their applications. In verifying a model, the verification criteria should be setup to confirm with the model objectives. For example, the verification can be made according to the peak discharge, time to peak, or to the fitting of the hydrograph as desired by the objective. Various verification fitting error measures have been suggested in the literature (ASCE Task Committee, 1993; Yen, 1982). Some measures are listed in Table 14.19. In the table, the magnitude parameter Q can be discharge, depth, velocity, or concentration as appropriate to the problem investigated. The subscript p denotes the peak magnitude of the time graph. The subscript m represents the measured or true values used as the gauge for the curve fitting and verification of the model simulation. The selection of the error measures to evaluate the merit of simulation models depends on the objective of the simulation. For example, if the accuracy of the peak rate and peaking time are of paramount importance, ee and e, would be the most appropriate error measures. If the overall fitting of the curves is the main objective, eRMS would be the most important measure, while eT1, £Va, e,p, and ee could be used as auxiliary measures. In calibration, since the reliability of a single set of data is uncertain, the more sets of data used, the better. Different sets of data would produce different sets of coefficient val TABLE 14.19 Simulation or Measurement Error Measures Error measure Magnitude errors Peak rate error Mean rate error Cumulative volume error Definition
^Qp = (G1,  G^) /Q^, c^CGGJ/G™ c v = ( V V J/ Vm Remarks* V = \ Qdt  ^GA* *o Vn=J QJt^ZQnA* O ' Q, = ^Q2dt^ ^ Ze 1 2 Ar 2™ = f JV^1HXA' wi Q mi Absolute volume error Rate moment error €. = ^m^U e  Q L l ^ T ^2Ia c ^ P m/ eCr = ( c,Gj/e™ Rootmeansquare ^RMS = f [ff (C CJ2J'" /M L O J f[Z(c,H' H Time errors Peakrate time error Peak time firstmoment error Graph dispersion error e.p = (^  ^J/u ^ = (T!  ^l J/T1M 6^ = (G  Gw)/Gm ^ f ^ H ? 4 + T>' Second moments with respect to £pm: G = l /Jfi(/^A
G = 4m JVc.  w*  ^) Source: From Yen (1982). * V or Vm becomes total volume if t =flowduration considered, T. Subscript m = measured or reference base values. Subscript i = summation index. Subscript p = peak magnitude of the time graph. ues. Normally, a weighted average (e.g., through optimization) of the values is adopted for each of the coefficients. It is not infrequent to see a model misused or abused. Sometimes this is due to the lack of understanding about how the model works. Sometimes it is due to the lack of appreciation of the operational modes. For example, data used for calibration should not be used again for verification. Yet, this situation happens again and again. In such a case of using the same data for calibration and verification, the difference between the model output and the recorded data is simply a reflection of the numerical errors and the deviation of the particular data set from the weighted average situation. Not all models require calibration. Presumably, some strictly physically based models have their coefficient values assigned based on available information and no calibration is needed. However, in rainfallrunoff modeling, some degree of spatial and temporal aggregation of the physical process is unavoidable. Therefore, calibration is desirable, if not necessary. 14.9 DETENTIONANDRETENTIONSTORAGE Detention and retention basins are widely used to control the increased runoff due to urbanization of undeveloped areas. These basins can also offer excellent water quality benefits 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 during wet weather. The outlet structures are designed to completely empty the basin after a storm event. Retention basins are sometimes called wet ponds since they retain a permanent pool. Post development hydrograph (pond inflow) Required Pond Volume Flow Rate Predevelopnent peak flow rate Routed postdevelopnent hydrograph (pond outflow) FIGURE 14.43 Routing of runoff through detention basin.
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 Lee,Liang
 Fluid Dynamics, yen, HPG, boneyard creek

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