Theory

This section presents the theoretical foundations on which the program is based. Explain the concepts, assumptions, and formulations used in the different analysis methods implemented.

The content is intended to provide a clear understanding of how the calculation is performed within the software, facilitating both the interpretation of the results and the technical validation of the models.

Manning’s equation

The Manning equation is an empirical formula used in hydraulics to analyze open channel flow. It was developed from experimental observations and allows the calculation of hydraulic parameters such as flow velocity and flow depth.

V = \frac{1}{n} R_h^{2/3} S^{1/2}

Where:

$Q$ : Flow rate ( $\text{m}^3\text{/s}$ ).
$V$ : Average flow velocity ( $\text{m/s}$ ).
$n$ : Manning roughness coefficient, which depends on the characteristics of the channel material (dimensionless).
$R_h$ : Hydraulic radius ( $\text{m}$ ), defined as $R_h = \frac{A}{P}$ , where $A$ is the flow area and $P$ is the wetted perimeter.
$S$ : Channel slope ( $\text{m/m}$ ).

To calculate the flow rate ( $Q$ ), the following relationship is used:

Q = VA

Where:

$V$ : Average flow velocity ( $\text{m/s}$ ).
$A$ : Flow area ( $\text{m}^2$ ).

Froude number

The Froude number ( $Fr$ ) determines the flow regime (subcritical, critical or supercritical):

Fr = \frac{V}{\sqrt{g D}}

Where:

$V$ : Average flow velocity ( $\text{m/s}$ ).
$g$ : Acceleration due to gravity ( $\text{m/s}^2$ ).
$D$ : Flow depth ( $\text{m}$ ).

Based on its value, the classification is:

Subcritical flow ( $Fr < 1$ )
Critical flow ( $Fr = 1$ )
Critical upper flow ( $Fr > 1$ )

Specific energy

The specific energy ( $E$ ) in an open channel is calculated as follows:

E = \frac{V^2}{2 g} + h

Where:

$E$ : Specific energy ( $\text{m}$ ).
$V$ : Average flow velocity ( $\text{m/s}$ ).
$g$ : Acceleration due to gravity ( $\text{m/s}^2$ ).
$h$ : Elevation of the water surface relative to a reference ( $\text{m}$ ).

Specific energy has two components:

Kinetic energy: Represented by the term $\\frac{V^2}{2g}$ , it reflects the energy associated with the movement of the water.
Potential energy: Represented by the term $h$ , it reflects the energy due to the rise of flow.

Appendix: Manning’s Roughness Coefficients

The following is a reference table of Manning roughness coefficients ( $n$ ), compiled from the work of Ven Te Chow:

Channel type and description	Minimum	Normal	Maximum
Natural rivers - minor streams (upper width at flood stage < 30 meters (100 feet))
1. Main channels
a. Clean, straight, full stage, no rapids or deep pools	0.025	0.030	0.033
b. Same as the previous one, but with more rocks and weeds	0.030	0.035	0.040
c. Clean, winding, with some pools and shallows	0.033	0.040	0.045
d. Same as the previous one, but with some weeds and stones	0.035	0.045	0.050
e. Similar to the previous one, lower stages, steeper slopes, and inefficient sections	0.040	0.048	0.055
f. Same as “d”, with more stones	0.045	0.050	0.060
g. Slow sections, with weeds and deep pools	0.050	0.070	0.080
h. Sections very full of weeds, deep pools or flood zones with abundant vegetation and bushes	0.075	0.100	0.150
2. Mountain rivers, without vegetation in the channel, with generally steep banks, trees and shrubs submerged in upper stages
a. Bottom: gravel, pebbles and a few large rocks	0.030	0.040	0.050
b. Background: pebbles with large rocks	0.040	0.050	0.070
3. Floodplains
a. Grasslands, without shrubs
1. Short grass	0.025	0.030	0.035
2. High grass	0.030	0.035	0.050
b. Cultivated areas
1. No Crops	0.020	0.030	0.040
2. Mature row crops	0.025	0.035	0.045
3. Mature field crops	0.030	0.040	0.050
c. Bushland
1. Scattered bushes, dense undergrowth	0.035	0.050	0.070
2. Shrubs and light trees, in winter	0.035	0.050	0.060
3. Shrubs and light trees, in summer	0.040	0.060	0.080
4. Medium to dense scrubland, in winter	0.045	0.070	0.110
5. Medium to dense scrubland, in summer	0.070	0.100	0.160
d. Trees
1. Dense willows, summer, straight	0.110	0.150	0.200
2. Clear land with stumps, no shoots	0.030	0.040	0.050
3. Same as the previous one, but with abundant shoot growth.	0.050	0.060	0.080
4. Dense forest, few fallen trees, little undergrowth, flooding under branches	0.080	0.100	0.120
5. Same as 4. with flooding reaching the branches	0.100	0.120	0.160
4. Excavated or dredged canals
a. Ground, straight and even
1. Clean, freshly completed	0.016	0.018	0.020
2. Clean, after weathering	0.018	0.022	0.025
3. Gravel, uniform section, clean	0.022	0.025	0.030
4. With short grass, few weeds	0.022	0.027	0.033
b. Winding and slow land
1. Without vegetation	0.023	0.025	0.030
2. With grass, some weeds	0.025	0.030	0.033
3. Dense weeds or aquatic plants in deep channels	0.030	0.035	0.040
4. Dirt bottom and rubble sides	0.028	0.030	0.035
5. Stony bottom and banks with weeds	0.025	0.035	0.040
6. Pebble bottom and clean sides	0.030	0.040	0.050
c. Excavated with draglines or dredged
1. Without vegetation	0.025	0.028	0.033
2. With light scrub on the banks	0.035	0.050	0.060
d. Rock cuts
1. Smooth and even	0.025	0.035	0.040
2. Irregular and bumpy	0.035	0.040	0.050
e. Unmaintained canals, uncut weeds and brush
1. Dense weeds, height equal to the depth of the flow	0.050	0.080	0.120
2. Clean background, bushes on the sides	0.040	0.050	0.080
3. Same as above, highest level of flow	0.045	0.070	0.110
4. Dense scrubland, high flow level	0.080	0.100	0.140
5. Lined or constructed channels
a. Cement
1. Smooth surface	0.010	0.011	0.013
2. With mortar	0.011	0.013	0.015
b. Wood
1. Sawn, untreated	0.010	0.012	0.014
2. Sawn, creosoted	0.011	0.012	0.015
3. Without brushing	0.011	0.013	0.015
4. Boards with slats	0.012	0.015	0.018
5. Lined with roofing paper	0.010	0.014	0.017
c. Concrete
1. Trowel finish	0.011	0.013	0.015
2. Floating finish	0.013	0.015	0.016
3. With gravel at the bottom	0.015	0.017	0.020
4. Unfinished	0.014	0.017	0.020
5. With gunite, good section	0.016	0.019	0.023
6. With gunite, wavy section	0.018	0.022	0.025
7. On uniform excavated rock	0.017	0.020
8. On irregularly excavated rock	0.022	0.027
d. Concrete bottom with floating finish and sides of:
1. Hewn stone in mortar	0.015	0.017	0.020
2. Random stone in mortar	0.017	0.020	0.024
3. Masonry with rubble, plastered	0.016	0.020	0.024
4. Masonry with rubble	0.020	0.025	0.030
5. Dry or rocky rubble	0.020	0.030	0.035
e. Gravel bottom with sides of:
1. Molded concrete	0.017	0.020	0.025
2. Random stone in mortar	0.020	0.023	0.026
3. Dry or rocky rubble	0.023	0.033	0.036
f. Brick
1. Glaze	0.011	0.013	0.015
2. In cement mortar	0.012	0.015	0.018
g. Masonry
1. Cemented rubble	0.017	0.025	0.030
2. Dry rubble	0.023	0.032	0.035
h. Hewn stone/stone paving	0.013	0.015	0.017
i. Asphalt
1. Smooth	0.013	0.013
2. Rough	0.016	0.016
j. Plant covering	0.030		0.500

References

Chow, V. W. (1959). Open channel hydraulics. McGraw-Hill.