The equation of the water surface profile in a reach can be written as follows:

= - Sf + (k-1) [10]

with:
Sf =
g = 9.81 m s^-2
n : Manning coefficient
R : hydraulic radius (m)
A : cross section area (m²)
H : total head (m)
q : lateral inflow (q > 0, k = 0) or outflow (q < 0, k = 1), (m² s^-1)
S_f : linear head losses (m^2/3 s^-2)
Q : discharge (m³ s^-1)

For solving this equation, an upstream boundary condition in terms of discharge and a downstream boundary condition in terms of water surface elevation are required.

In addition, the lateral inflow, and the hydraulic roughness coefficient along the canal should be known. As the equation does not have an analytical solution, in the general case, it is discretized in order to obtain a numerical solution. Knowing the upstream discharge and the downstream water elevation, the water surface profile is integrated, step by step starting from the downstream end.

Figure 16

Integrating equation [10] between sections i and j:

+ + = 0

i.e.:

Hj - Hi - (k-1) q ( + ) + Dxij = 0 [11]

Equation [11] can be written as follows:

Hi(Zi)= Hj + DH(Zi)

A subcritical solution exists if the curves Hi(Zi) and Hj + DH(Zi) intersect.

For this, it is necessary that: d = Hj + DH(ZCi) - Hi(ZCi) > 0

ZCi is the critical elevation defined at i by = 1

d > 0 : Subcritical solution F,
d < 0 : Supercritical solution. The critical depth is assumed systematically. The water surface profile is therefore overestimated.

Figure 17

If a solution does exist, one has to numerically solve an equation of the form f(Zi)=0. The numerical methods used are presented in section II.5.