Correcting equations

The corrected backwater curve will be solution of the equation:

\(f(Z^{i}_{j} + \Delta Z^{i}_{j}, Z^{i}_{j+1} + \Delta Z^{i}_{j+1},Q^{i}_{j} + \Delta Q^{i}) = 0\) [2]

After a first order Taylor development of [2], we get:

[2] - [1] => \(a^{i}_{j}.\Delta Z^{i}_{j} + b^{i}_{j}.\Delta Z^{i}_{j+1} + c^{i}_{j}.\Delta Q^{i} = 0\) [3]

with:

\(a_{j}^{i} = \frac{\partial f}{\partial Z^{i}_{j}}\)~ ~ ~ \(b_{j}^{i} = \frac{\partial f}{\partial Z^{i}_{j+1}}\)~ ~ ~ \(c_{j}^{i} = \frac{\partial f}{\partial Q^{i}}\)

We know that aij cannot be nil, since [1] is solved through the Newton method using the ratio\(\frac{f}{a^{i}_{j}}\).

Let:

\(d^{i}_{j} = \frac{b^{i}_{j}}{a^{i}_{j}}\) et \(e^{i}_{j} = \frac{c^{i}_{j}}{a^{i}_{j}}\)

For a reach i with n cross sections, the (n-1) equations [3] give:

\(\Delta Z^{i}_{1} = d^{i}_{1}.\Delta Z^{i}_{2} + e^{i}_{1}.\Delta Q^{i}\)

…

\(\Delta Z^{i}_{j} = d^{i}_{1}.\Delta Z^{i}_{j+1} + e^{i}_{j}.\Delta Q^{i}\)

…

\(\Delta Z^{i}_{n-1} = d^{i}_{n-1}.\Delta Z^{i}_{n} + e^{i}_{n-1}.\Delta Q^{i}\)

After condensation of these equations, we can get one relation representative of the total reach:

\(\Delta Z^{i}_{1} = D^{i}_{n}.\Delta Z^{i}_{n} + E^{i}.\Delta Q^{i}\) [4]

with: - i = index of the reach - 1 = index of the upstream section of the reach - n = index of the downstream section of the reach - \(D^i = \prod_{j=1}^{n} d^i_j\) - \(E^i = D^i \prod_{k=1}^{n} \frac{e^i_k}{\prod_{j=k}^{n} d^i_j}\) ( with: \(d^i_n = 1\) and \(e^i_n = 0\) )

The other available relations are:

For all the nodes of the loop that are not downstream nodes of the loop:

*Discharge continuity

\(\sum_{i=1}^{k} \varepsilon ^i \Delta Q^i =0\) [5]

with: - k = number of reaches of the loop connected to the node - \(\varepsilon ^i = 1\) upstream reach - \(\varepsilon ^i = -1\) downstream reach

*Water elevations equality

\(Z^i_u + \Delta Z^i_u = Z^l_v + \Delta Z^l_v\) [6]

giving k - 1 equations, with: - u, v = 1 if the reach leaves the node and n if the reach joins the node - i, l = reaches connected to the node

For all the downstream nodes of the loop:

\(\Delta Z^i_n = 0\) [7]

if the downstream boundary condition is a given elevation, or

\(\Delta Z^i_n = g(\Delta Q^i)\)

if the downstream boundary condition is a rating curve (in this case only one reach can join the corresponding downstream node).