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).