in which cs is the uncontrolled spillway discharge coefficient, hs is the uncontrolled spillway crest, cg is the gated spillway discharge coefficient, hg is the centerline elevation of the gated spillway, cd is the discharge coefficient for flow over the crest of the dam, Ls is the spillway length, and Q, is a constant outflow term that is head independent or it may be a specified discharge time series. The uncontrolled spillway flow or the gated spillway flow can also be represented as a table of head-discharge values. The gate flow may also be specified as a function of time via a known time series for Ag(t). The breach outflow (Qh) is computed as broad-crested weir flow (Fread, 1977, 1985, 1988, 1992; Fread and Lewis, 1998), i.e.,
Qb = cA[3. l&M - hb)L5 + 2.45^,. - hh)2 5] (66)
in which cv is a small correction for velocity of approach, bt is the instantaneous breach bottom width, hi is the elevation of the water surface just upstream of the structure, hb is the elevation of the breach bottom in which hh is assumed to be a function of time (tb) from beginning of the breach formation time (t), z is the side slope of the breach, and ks is the submergence correction factor due to the downstream tailwater elevation (hc), i.e., ks= 1.0 h* < 0.67 (6.7)
Using a parametric description of the breach, the instantaneous breach bottom width (bj) starts at a point at the crest of the dam and enlarges at a linear or nonlinear rate over the failure time (t) until the terminal bottom width (b) is attained and the breach bottom has eroded to the minimum elevation, hbm. The instantaneous bottom elevation of the breach (hb) is described as a function of time (tb) according to the following:
in which hd is the elevation of the top of the dam, hbm is the final elevation of the breach bottom, which is usually, but not necessarily, the bottom of the reservoir or outlet channel bottom, tb is the time since beginning of breach formation, and p is the parameter specifying the degree of nonlinearity, e.g., p = 1 is a linear formation rate, while p = 2 is a nonlinear quadratic rate; the range for p is 1 < p < 4, with the linear rate usually assumed. The interval of time (t) required for the breach to form is given by t = 0.3 Vj153/H(j-9 in which Hd = hb- hbm, Vr is the reservoir volume (acre-ft) from empirical data by Froehlich (1987); the standard error of estimate for z is ±0.9h or ±74% of i (Fread, 1988, 1995). The instantaneous bottom width (bt) of the breach is given by the following:
in which b is the final width of the breach bottom given by b = b — zHd and b = 9.5k0(VrHdf25 from empirical data by Froehlich (1987) in which k0 = 0.7 for piping and kQ = 1.0 for overtopping; the standard error of estimate for b is ±82 ft or ±56% of b (Fread, 1988, 1995).
When simulating a dam failure, the actual breach formation can commence when the reservoir water-surface elevation (h) exceeds a user-specified value, hj . This feature permits the simulation of an overtopping of a dam in which the breach does not form until a sufficient amount of water has passed over the crest of the dam to have eroded away the downstream face of the dam.
If the breach is formed by piping, Eq. (66) is replaced by an orifice equation:
in which hp is the specified centerline elevation of the pipe. Each of the terms in Eq. (65) except Qt may be modified by a submergence correction factor similar to ks that can be computed by Eqs. (67) to (69), but in Eq. (69) hb is replaced by hs, hg, and hd, respectively.
Bridge. If the internal boundary represents highway/railway bridges together with their earthen embankments that cross the floodplain, Eqs. (63) and (64) can still be used although Qs in Eq. (65) is computed by the following contracted bridge flow expression:
in which Cb is a coefficient of bridge flow (Chow, 1959), Cd is the coefficient of flow over the crest of the road embankment, hc is the crest elevation of the embankment, and ks is similar to Eqs. (67) to (69) except hb is replaced by hc. A breach of the embankment is treated the same as with dams.
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