9. Formulae I would be using

From GeoMod

The most dynamic wave routing scheme available consists of the kindematic wave approximation. This is a simplification of the full one-dimensional St. Venant equation obtained by eliminating local acceleration, convective acceleration and pressure terms in the momentum equation. It thus assumes that the friction and gravity forces balance.

∂Q + ∂A + q = 0
∂x     ∂t

S0 = Sf

This is the Kinematic wave equation described by Bates and Roo. Here:

Q is the discharge [L3T-1],

x is the distance between cross-sections [L],

A is the flow corss sectional area [L2],

t is the time [T],

q is the lateral inflow term [L3T-1] here set to 0 for all reported simulations,

S0 is the channel bed slope [-] and

Sf is the friction slope [-] here approximated as the water surface slope.

The change in cell volume over time is equal to the fluxes into and out of it during the time step.

dV = Qup + Qdown + Qleft + Qright
dt

Here:

V is the cell volume [L3]

T is the time [T]

Qup, Qdown, Qleft and Qright are the flow rates (either positive or negative) from the upstream, downstream, left and right adjacent cells, respectively [L3T-1]

The flow rates between each cell can then be calculated using some uniform flow formulae. In my project I would be concentrating on the Manning Equation. The flowrate between two adjacent cells i and j, where i is the upstream cell, is equal to:

    
     2/3   1/2
Qij = Aij Rij Sij
n

where:

Qij is the flux [L3T-1] between cells i and j.

Aij is the cross sectional area [L2] at the interface of the two cells

Rij is the hydraulic radius [L] at the interface of the two cells

Sij is the water surface slope between the two cells

N is the Manning friction coefficient [L3T-1], distinguishing between nc for channels and nfp for floodplains

Floodplain flow is thus approximated as a two-dimensional diffusion wave, As we use and explicit method, weights are introduced during the drying phase to prevent more water leaving a cell than it contains. To achieve this flow rates Qup, Qdown, Qleft and Qright are calculated as above and then scaled by a non-dimensional coefficient c:

C = 			Vt                       .
    (Qup + Qdown + Qleft + Qright)ï?„t

where: Vt is the volume of water L3 in the cell at the time t

ï?„t is the time step [T]

The water depth thus returns smoothly to 0 as the cell dries out.

As the flood wave drains from the reach, water levels in the channel drop and the retreat of the inundation front is simulated. At the downstream boundary of the model water can leave the computational domain as either channel or floodplain flow. These are then summed to calculate the mass conservation error per time step:

Et = (Vin – Vout) – Vt – Vt+1) X 100
Vt+1

Where:

Et is the mass balance error during the time step t as a percentage of the volume of water in the domain at the end of time step [-]

Vin and Vout are the volumes of water entering and leaving the domain during the time step [L3]

Vt is the total volume of water in the domain at the start of time step t [L3]

Vt+1 is the total volume of water in the domain at the end of the time step [L3].