Culvert and Storm Drain Hydraulics

Culvert and Storm Drain Hydraulics

Inlet and Outlet Control, Overtopping, Gutters, Inlets, HGL

Scope

The tool holds two independent analyses of the FHWA drainage manuals. The culvert part follows Hydraulic Design Series No. 5, Hydraulic Design of Highway Culverts (3rd edition, 2012): for a circular, box, elliptical or pipe-arch barrel with one of 22 inlet configurations it finds the headwater under inlet control and under outlet control, takes the larger as the controlling headwater, reports the outlet velocity, adds the flow over the roadway when the headwater rises above the crest, and draws the performance curve of headwater against discharge. The storm drain part follows Hydraulic Engineering Circular No. 22, Urban Drainage Design Manual (3rd edition, 2009, and the 4th of 2024, which keeps the same equations): the Rational method with an intensity-duration fit, the time of concentration by segments, the spread and interception of a gutter inlet on grade, the sizing of a trunk line of pipes with the cumulative C A and the system time, the inverts from the crown-drop rule, and the energy and hydraulic grade lines through the structures by the FHWA access-hole energy-loss method. Both manuals are U.S. government works; every equation, coefficient table and worked example used here is theirs.

Conventions and units

Lengths, flows and elevations go through the site's unit dictionary (ft, ft³/s in USCS; m, m³/s in SI). Slopes are ratios, Manning's n and the coefficients are plain numbers, times are in minutes, drainage areas in acres (hectares in SI) and rainfall intensities in in/h (mm/h in SI), as the manuals write them. The equations of both manuals carry unit constants (K_u): the tool evaluates them in their English form and converts the inputs and results, so the SI page prints the same physics with the metre values. Elevations are measured from the datum of the entered inverts; the culvert headwater HW is a depth above the inlet invert, the tailwater TW a depth above the outlet invert.

Culvert barrel and critical depth

The barrel section gives the full area A, perimeter p and hydraulic radius R = A/p and, at a depth y, the flow area, wetted perimeter and top width. The circle and the box are exact; the elliptical and pipe-arch barrels are both computed as an ellipse of the entered span and rise (area and top width scale the circle by span/rise, the arc length by quadrature), which HDS-5 Appendix A itself does when it applies the pipe-arch inlet constants to an ellipse; the flat-bottom corner radii of a pipe-arch change the part-full properties by a few percent. The critical depth d_c is the depth at which Q² T / (g A³) = 1, found by bisection; in a closed conduit it cannot exceed the rise, and the tool caps it there as the handbook's critical-depth charts do. The normal depth is the depth at which Manning's equation, Q = (1.486/n) A R^2/3 S^1/2, carries the flow; a circular barrel conveys most at 0.938 D, and a flow above that capacity means the barrel runs full.

Inlet control

Under inlet control the entrance limits the flow and the headwater depends only on the inlet geometry, the discharge and, weakly, the slope. HDS-5 Appendix A gives the equations fitted to the National Bureau of Standards model tests. Unsubmerged (the entrance acts as a weir), up to a flow parameter Q/(A D^0.5) of about 3.5 in ft units: Form (1), HW_i/D = H_c/D + K [Q/(A D^0.5)]^M + K_s S, with H_c = d_c + V_c²/2g the specific head at critical depth; or Form (2), HW_i/D = K [Q/(A D^0.5)]^M. Submerged (the entrance acts as an orifice), above about 4.0: HW_i/D = c [Q/(A D^0.5)]² + Y + K_s S. The constants K, M, c, Y come from Table A.1 (circular pipe, box, bevels and tapered inlets) and Table A.2 (pipe-arches and ellipses) for the inlet configuration chosen; the box constants must not be used for curved shapes nor the reverse, and the tool refuses a mismatch. The slope term K_s S is −0.5 S for every inlet except the mitered ones, +0.7 S; the design nomographs were drawn for a 2 % slope, the tool uses the actual slope as HY-8 does. Between 3.5 and 4.0 the handbook draws a transition tangent to the two curves; the tool joins the two equation values by a straight line, the difference being within the drawing precision. In SI the same equations hold with the flow parameter multiplied by 1.811.

Outlet control

Under outlet control the barrel and the tailwater limit the flow. For a barrel flowing full the losses between the headwater pool and the outlet are the entrance loss k_e V²/2g (Table C.2 gives k_e by entrance type, 0.2 for a groove end or a bevel to 0.9 for a projecting corrugated pipe), the friction loss from Manning's equation, 29 n² L V²/(2g R^1.33) in ft units (19.63 in SI), and the exit loss of one velocity head when the downstream velocity is neglected; H is their sum (eq. 3.5). The headwater is HW_o = h_o + H − L S (eq. 3.6b), where h_o is the hydraulic grade line at the outlet: the tailwater depth when it submerges the outlet, otherwise the FHWA approximation (d_c + D)/2, the point where the extended full-flow hydraulic grade line was found to pierce the outlet plane in many backwater computations; the larger of the two is used. The approximation serves while the barrel flows full over part of its length and is adequate down to a headwater of 0.75 D; below that a backwater profile would be needed and the page says so. The approach velocity head is neglected, as in the handbook's design procedure.

Controlling headwater and outlet velocity

The culvert operates under whichever control gives the higher headwater; the page names it and compares the headwater elevation with the allowable one when entered. The outlet velocity follows Section 3.1.6: under inlet control the profile tends to normal depth down the barrel, so the velocity is Q divided by the flow area at normal depth (the full barrel when the normal depth would exceed the crown), slightly high for a short barrel, as the HY-8 comparison in Design Guideline 1 shows; under outlet control the area is set by the critical depth when the tailwater is below it, by the tailwater depth up to the crown and by the full barrel above it. The tailwater may be entered or taken as the normal depth of a downstream trapezoidal channel at each flow, computed with the Open-Channel Flow tool's routine.

Roadway overtopping

When the headwater rises above the road the overflow is a broad-crested weir, Q_o = C_d L HW_r^1.5 (eq. 3.9, ft units; the coefficient is multiplied by 0.552 in SI) with C_d = k_t C_r from Figure 3.11, which the tool carries as tables digitized from the figure: C_r against the head HW_r for deep overtopping (HW_r/L_r above 0.15, L_r the roadway width in the flow direction) and for shallow overtopping, for paved and gravel surfaces, and the submergence factor k_t against h_t/HW_r, unity until the tailwater rises above about 0.76 of the head. The culvert flow and the road flow must add up to the design discharge: the tool finds the headwater at which they do by bisection, the culvert flow at a given headwater being the inverse of the headwater function. Design Guideline 1.5 is reproduced: overtopping of its 7 ft box begins at about 797 ft³/s (HY-8: 798.5) and at 1,000 ft³/s the road takes 185 ft³/s (HY-8: 183.5) at a headwater of 16.49 ft, the HY-8 value to the digit.

Performance curve

The table and the chart give, for flows from zero to the entered maximum, the inlet control and outlet control headwater elevations, the controlling one, the tailwater, the outlet velocity and, with a road, the split and the headwater that passes the total flow. The inlet control curve rises faster than the outlet control curve at high flows (the orifice law against the full-flow losses), which is why most short culverts on moderate slopes end in inlet control.

Rainfall, runoff and time of concentration

The Rational method, Q = C i A (ft³/s with i in in/h and A in acres; divided by 360 in SI with mm/h and hectares), gives the peak of each inlet area with the runoff coefficient C of HEC-22 Table 3-1 and the intensity at a duration equal to the time of concentration, never shorter than the entered minimum (5 minutes in the manual). The intensity comes from the fit i = a/(t + b)^c to the local IDF curve; the default reproduces the Example 7-3 table of HEC-22 within 0.1 in/h. The time of concentration to the first inlet is the sum of the travel times of the flow segments: sheet flow by the kinematic wave, T_t = 0.933 (n L / S^0.5)^0.6 / i^0.4 minutes (HEC-22 eq. 3-3), whose intensity depends on the time itself and is iterated to a fixed point, or by the TR-55 form with the 2-year 24-hour rainfall; shallow concentrated flow at V = 3.28 k S_p^0.5 with the intercept k of Table 3-3, or on the TR-55 paved and unpaved lines (20.3282 and 16.1345 S^0.5 ft/s, Manning's equation with n = 0.025, R = 0.2 ft and n = 0.05, R = 0.4 ft); and a channel or pipe segment by Manning's equation.

Gutter flow and spread

Izzard integrated Manning's equation across a triangular gutter section, where the hydraulic radius misdescribes a section whose top width may be forty times the depth, and obtained Q = (0.56/n) S_x^1.67 S_L^0.5 T^2.67 (HEC-22 eq. 4-2, ft units; 0.376 in SI), which the tool solves for the spread T. A composite section with a depressed gutter of width W and cross slope S_w = S_x + a/W carries Q_s beyond the gutter, from the same equation on T − W, and the whole flow Q = Q_s/(1 − E_o), where the frontal flow ratio E_o = Q_w/Q of eq. 4-4 depends on S_w/S_x and T/W; the spread for an entered flow is found by bisection, replacing the trial-and-error of Example 4-2. The depth at the curb is T S_x plus the depression, the flow area ½ T² S_x + ½ a W and the mean velocity Q/A, as Example 4-7 computes them.

Inlet interception

A grate on grade intercepts all the frontal flow until the gutter velocity reaches the splash-over velocity V_o of its bar pattern and length, above which R_f = 1 − 0.09 (V − V_o) (eq. 4-18), and a part of the side flow, R_s = 1/[1 + 0.15 V^1.8/(S_x L^2.3)] (eq. 4-19); the efficiency is E = R_f E_o + R_s (1 − E_o) and the interception Q_i = E Q. The splash-over velocities are read from Figure 7.8 of the 4th edition (Chart 5 of the 3rd), traced from the figure for the seven grates tested by the Bureau of Reclamation for FHWA: the parallel bar P-1-7/8 and P-1-1/8, the P-1-7/8-4 with lateral rods, the curved vane, the 30° and 45° tilt bars and the reticuline grate; the parallel bar grate passes 8.2 ft/s before splashing over at 2 ft of length, the reticuline only 4.2. A curb opening intercepts everything when it is at least L_T = 0.6 Q^0.42 S_L^0.3 [1/(n S_e)]^0.6 long (eq. 4-22, ft units; 0.817 in SI), with the equivalent cross slope S_e = S_x + (a/W) E_o of a depressed gutter (eq. 4-24), and a shorter one E = 1 − (1 − L/L_T)^1.8 (eq. 4-23). A sweeper combination inlet is computed as the manual's Example 7.4 does: the opening upstream of the grate takes its share of the whole gutter flow, and the grate then works on the remainder at the spread that remainder produces.

Storm drain sizing and inverts

Along the trunk line from the upstream end each pipe carries the flow of the cumulative C A at the intensity of the system time, the larger of the inlet time of its own area and the time of concentration of the upstream pipes plus their travel times (HEC-22 Section 7.2.2); the design flow of a lower pipe is therefore less than the sum of the inlet peaks above it. The diameter for full flow follows Manning's equation for a circular pipe, D = [Q n/(0.46 S^0.5)]^0.375 (eq. 7-1, ft units), rounded up to the next standard size, at least the entered minimum and never smaller than the pipe above it (a size reduction inside a system is a clogging hazard). The full-flow capacity and velocity use the manual's constants 0.46 and 0.59; the part-full velocity, normal depth and critical depth come from the exact solutions of the circular section rather than from the hydraulic-elements chart the manual reads, which differ from them by a few percent. The first invert is set by the ground, the minimum cover and the diameter; at each following structure the invert drops by K_ah V²/2g, the approximate structure loss of eq. 7-11 with the coefficients of Table 7-5, as Example 7-3 does to set its preliminary profile; the last pipe is set from the entered outfall invert.

Energy and hydraulic grade lines

The grade lines are computed from the outfall upstream by the procedure of HEC-22 Section 7.5. At the downstream end of a pipe the controlling level is the tailwater (at the outfall) or the energy level in the structure below (EGL_a). If it submerges the pipe crown the pipe flows full: the energy line is that level plus the exit loss, one full velocity head at an outfall (eq. 7-4 with no downstream velocity) or 0.4 V²/2g into a structure (eq. 7-32). If it lies below the critical depth the pipe is not affected and its energy line is the normal depth plus the velocity head; between critical and normal depth the higher of the two rules governs; between normal depth and the crown the face depth is set by the level and the velocity head is that of the face area. Up the pipe, a full or subcritical pipe adds the friction loss S_f L, with S_f = [Q n/(0.46 D^2.67)]² (eq. 7-3) for full flow and the pipe slope part full, and the inlet end is checked again against the crown, the normal and the critical depth; a supercritical pipe carries no losses upstream and its inlet end sits at normal depth plus velocity head. The energy should never decrease up the line, and the manual notes that a supercritical pipe whose outlet is drowned holds a hydraulic jump somewhere inside.

At each structure the FHWA access-hole method (Kilgore 2006, eqs. 7-12 to 7-32) replaces the single-coefficient losses of earlier practice. From the outflow pipe's energy head E_i = EGL_i − invert and the discharge intensity DI = Q/[A (g D_o)^0.5], the initial energy level in the structure is the largest of three analogies: outlet control, E_i + 0.2 V²/2g (zero when the outflow pipe is supercritical); a submerged opening, D_o DI²; an unsubmerged one, 1.6 D_o DI^0.67. Three additive adjustments scale the difference E_ai − E_i: benching, with C_B from Table 7-6 (negative, since a bench guides the flow; interpolated between the submerged and unsubmerged values for E_ai/D_o between 1 and 2.5); angled inflow, C_θ = 4.5 (ΣQ_j/Q_o) cos(θ_w/2) with the flow-weighted angle of the non-plunging inflow pipes (180° is straight through); and plunging inflow, C_P = Σ Q_k h_k/Q_o with the relative plunge height h_k = (z_k − E_ai)/D_o of each inflow whose invert z_k lies above the initial level (capped at 10 D_o); the surface inflow of an inlet plunges from the ground. The structure level E_a = E_ai + H_a is never taken below E_i, and EGL_a = E_a + invert is compared with the ground or grate elevation; the manual takes it as the water level in the structure for that check. The next pipe upstream then starts from EGL_a.

Reading the results and their precision

The default culvert is HDS-5 Design Guideline 1.3: 200 ft³/s through 200 ft of 54 in concrete pipe with a groove end projecting on a 1 % slope, inlet invert at 100 ft, tailwater 3.5 ft, allowable headwater 108 ft. The handbook reads its nomographs as HW_i/D = 1.77 (headwater elevation 108.0 ft), an outlet control loss of 4.7 ft (elevation 107.0 ft) and an outlet velocity of 15.3 ft/s; HY-8 solves the same equations to 7.9 ft of headwater and 14.9 ft/s. The equations here give 8.10 ft (108.10 ft, 0.1 ft above the allowable, which the nomograph reading hides), 4.70 ft and 15.2 ft/s; the nomographs carry the ±10 % stated in Appendix A.4, and the design-form values of DG 1.3, 1.4 and 1.5 are reproduced within that band, the HY-8 values within 1 %. The default storm drain is HEC-22 Example 7-3 (the 4th edition's Example 9.2): three inlets and an access hole on four pipes to a drowned outfall; the flows, diameters and inverts agree with the manual's sheet (which rounds each C A to two decimals), and the energy levels 333.68, 345.81, 355.85 and 366.85 ft at structures 43 to 40 are reproduced within the manual's two-decimal rounding and chart readings. The test script records every comparison with its tolerance; the Example 4-7, 4-8 and 4-9a inlets and the Example 3-2 time of concentration are reproduced to the printed digits.

Limits

Culvert: a single straight barrel without inlet depression, tapered inlet, bend, junction or grate losses (HDS-5 Chapters 4 and 5); the outlet control approximation rather than a backwater profile; no approach or downstream velocity heads; the ellipse geometry for pipe-arches; the 22 inlet configurations of Tables A.1 and A.2 (the South Dakota boxes, open-bottom arches and embedded shapes of Tables A.3 to A.6 are not included); the transition zone as a straight line. Storm drain: one trunk line with inlets at its structures (no branching laterals; an angled trunk is handled by the inflow angle); the Rational method, which the manual limits to small areas; the approximate crown drop for the preliminary inverts; the grade lines by the manual's hand procedure (no hydraulic jump location, no surcharge routing above the ground); inlets on grade only (sag inlets, slotted drains and bypass routing between inlets are not computed); the splash-over velocities traced from the figure within about 0.1 ft/s.