SECTION VI: FLOOD ROUTING Consider the watershed with 6 sub-basins Q 1 = Q A + Q B (Runoff from A & B) 1
Q 2 = (Q A + Q B ) 2 + Q C + Q D (Routed runoff from Q 1 ) + (Direct runoff from C & D) What causes attenuation? 1) Storage and 2) Friction What causes flow lag? 1) Flood wave travel time (celerity) - f(length, depth, friction, slope) Consider a short reach or reservoir (pond) Continuity Eq. Q in - Q out = S/ t 2
See Appendix Figure 4.2 Bedient Simple Hydrologic Flood Routing Simple methods based on lumped continuity equation f(x)) ("lumped" not Q i - Q o = S/ t Problem: Q i is known, need to solve for Q o What about S? 3
To solve the flood routing problem a relationship between S and Q is needed. Either: (See handout, USGS rating curve) 2) Q = f(s) or S = f(q) directly - Less common than (1) 3) Solve for Q(y) from momentum equation (e.g., Q = ky m, kinematic ), S(A), A = cross-sectional area 4
RESERVOIR ROUTING - simplest hydrologic method - sometimes called "Pond Routing" "Linear" Reservoir Q o = ks k = [1/time] = routing coefficient =f (channel geometry) Q o = outflow linearly related to storage then Q in - Q out = S/ t (Homework # s 4.1, 3, 4, 5, 7) 5
Q in - Q out = d(k -1 Q out )/dt k f(time) Solution: dq out /dt + k Q out = k Q in (t) Can solve analytically (exponential solution) Constant = Q out at t=0 Need functional form for Q i (i.e., Q i (t)) so that it can be integrated over time. For example, could assume Q i = f(sine function) or 6
could use numerical (discrete step) methods Finite Difference Method of Continuity Equation Q in + Q o = ds/dt Subscript 1 implies beginning of a time step Known: Q i1, Q i2, Q o1, S 1 Subscript 2 implies end of a time step Unknown: Q o2, S 2 7
Modified Puls Method (Finite Difference Continuity Eq.) - also called Storage Indication Method (book) - collect knowns and unknowns on opposite sides of the equation To solve: use a relationship between S and Q from rating curve (stagedischarge relation), weir equation, uniform flow assumption, or other information. Can construct a table or graph of Q o = f(s + ( t/2) Q o ) Book has good example (p. 256-260) 8
Consider a river reach shown Consider uniform flow 9
Manning's Equation for uniform flow S eq1 = S bed A = cross-sectional area L = length of channel reach and S = L A = reach storage Then: R = Hydraulic radius = Area/ Wetted perimeter For rectangular channel (y = depth, b = bottom width), Area = y b WP = 2y + b R h = yb/(2y + b) y/2 for b = 2y Q o = Constant y 2/3 S 10
Muskingum Method For Flood Routing S = prism storage + wedge storage = KQ o + Kx (Q i - Q o ) Two Parameters K, x x is not distance Can substitute into continuity 11
Can solve analytically as before if x = 0, linear reservoir In general, 0 x 0.5 note x is higher for more regular ("improved") channels x is lower for more natural channel e.g., natural irregular channels x ~ 0.15 concrete lined, trapezoidal channel 0.3 x 0.4 if x = 0.5, pure translation 12
Finite differences applied to continuity Equation Muskingum Eq. For a linear reservoir x = 0 Must have storage and flow data to evaluate parameters Plot graph shown for trial values of x, keep trying with different x, until loop narrows (approximates a line) Slope = K, S = K [ x Q i + (1 - x ) Q o ] 13
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When storage is a maximum ds/dt = 0 = Q i - Q o, substitute S = K [ Q o + Kx(Q i - Q o )] 15
If we had these at the same time, we could solve Use two slope values and solve for x 16
In the real river reach the parameters are a function of the flow, Q. For this case need parameter estimation for several flow rates (i.e., variable parameter, K(flow), x(flow), Muskingum Method). Probably better to just go to full St. Venant (dynamic) equations. Another method is Muskingum-Cunge Method - A better way for parameter estimation (relate them to physical parameters of the channel) ref Cunge 1969 (Ref 66) let Muskingum parameter, x, now be (x in the following eqs reps. distance) 17
Taylor Series Remember: and Substitute for Q(x + x, t) and dq(x + x, t)/dt in continuity equation (neglect d 3 Q/dt dx 2 term): Use continuity equation of the form: Define: (Muskingum-Cunge) 18
Solve for: LHS Combine d 2 Q/dx 2 terms: 19
Muskingum - Cunge This is a form of the advection - diffusion equation (the bracketed term represents a sort of flow wave diffusivity, D). [RHS] Numerical Diffusivity Likely to be curved: e.g., c = f(flow) Also, approximate c by flood profiles: Hydraulic Flood Routing Large "Dynamic" Rivers "Dynamic" - subjected to rapid fluctuations in flow requiring inclusion of acceleration terms in equations of flow * Must use St. Venant Eqns to adequately describe flow (p. 237-8) 20