## Abstract

Weirs are used to control and regulate the flow in open channels. In gabion structures, the flow conditions are more complex due to the complexity of flow through the porous body of a gabion. The present study aims to investigate the water surface profile, the overflow velocity profile, and both the through-flow and overflow ratios. Six physical models of the three downstream slopes (V:H 1:1, 1:2, and 1:3) and two types of rockfill (crushed stone and rounded gravel) were investigated. Results show that for the same discharge, the milder slope model (1:3) shows higher water surface and higher velocity than the steeper slope (1:1) with about 9 and 8% on average respectively. The water surface was 60% higher on the lower steps than on the upper steps at the nappe flow regime. Moreover, the low porosity models show a slightly higher velocity and flow depth than higher porosity models for all sections. Furthermore, increasing the porosity from 0.38 to 0.42 led to about a 27% increment in the through-flow ratio. Finally, four relationships were developed to estimate the through-flow and overflow ratios at the upstream and inner sections of the gabion weir. The suggested relationships can be considered novel relationships.

## HIGHLIGHTS

The water surface profile for different flow regimes was investigated.

The overflow velocity profile for different downstream slopes was investigated.

The through-flow and overflow to total flow ratios were investigated.

A relationship for the ratio of through-flow to total flow and overflow to total flow was developed.

### Graphical Abstract

## NOTATIONS

*H*_{porous}porous media thickness at the section of estimation

*h*_{s}the step height

*H*_{w}weir height

*L*weir length

*l*_{s}the step length

*n*porosity of porous media

*Q*_{Over}overflow

*Q*_{Through}through-flow

*Q*_{Total}total flow

*Re*Reynolds number

*X*the distance between the section and the upstream face of the weir

*y*flow depth above the weir

*Y*_{Section}water depth at the section of estimation

## INTRODUCTION

A gabion weir is a type of hydraulic structure that is built by installing gabion baskets together to form the body of the weir. The characteristics of flow over gabion stepped weirs are more complex due to the complexity of the porous body of the weir. Investigating the characteristics of flow through and over the gabion weirs is problematic due to the complexity and variability of the interior structure of the porous media (Fathi-moghaddam *et al.* 2018). Salmasi *et al.* (2012) pointed out that the hydraulics of through-flow have complex flow patterns and have not received adequate attention in the literature. Michioku & Maeno (2005) used a one-dimensional analysis and laboratory experiment to investigate the discharge over a permeable rubble mound weir. Momentum and continuity equations were utilised to analyse the discharge of two layers: overflow above the weir and through-flow in the weir. A formula was obtained as a solution for the water depth, flow velocity, and discharge over the weir. Mohamed (2010) carried out an experimental study to calculate the discharge through and over the rectangular gabion weir. In this study, both the gabion and solid broad-crested weirs were tested to compare the results. Two equations were formulated, using regression analysis, for calculating the total discharge as a function of the depth of water, stone porosity, and weir dimensions. Mohammadpour *et al.* (2013) simulated a three-dimensional flow on gabion weirs. The model solves the RANS equation to estimate the water surface profile over the rectangular weir using the experimental data given by Leu *et al.* (2008) and Mohamed (2010). Results indicated that the standard *k–ɛ* has good accuracy for estimating the water surface profile and upstream streamwise velocity. Tavakol-Sadrabadi & Fathi-moghaddam (2016) studied numerically flow characteristics around triangular gabion weirs by using a three-dimensional CFD code. They studied three triangular models with upstream slopes of 30, 45, and 60°. They also analysed three models with downstream slopes of the same angles, all with the same height of 160 mm. The results of the simulation contained the water surface profile, distribution of velocity, and through the flow to total flow ratio. The main finding was for the same discharge, the weir of the upstream slope of 60° offers the minimum upstream head and minimum downstream velocity, as well as the maximum ratio of through to total flow. Zuhaira *et al.* (2017) used an advanced computational model to simulate the flow over the gabion stepped spillway. The two-dimensional model used the NEWFLUME code to solve the equations of Reynolds-Averaged Navier–Stokes. The model has been validated using the experimental data of Wuthrich & Chanson (2014). They studied the effects of the step configurations and slope on energy dissipation. Results indicate that the downstream slope is an important factor that affects energy dissipation. The low (mild) slope offers a longer contact surface than the high (deep) slope, which increases the rate of energy dissipation. Tavakol-Sadrabadi *et al.* (2018) simulated the flow characteristics of porous weirs with different upstream and downstream slopes numerically. They used the FLOW-3D software to simulate the flow characteristics through and over gabion weirs. The experimental data of Moradi (2015) were used to validate the numerical model. Results showed that increasing the size of the filling materials leads to an increase in the coefficient of discharge, discharge, and the rate of energy dissipation. Furthermore, non-linear multivariable regression was used to predict relationships to estimate the discharge through porous media. Fathi-moghaddam *et al.* (2018) studied the hydraulic performance of gabion weirs numerically. Further, laboratory experiments were conducted to validate the simulation results. Results showed that the size of the rockfill had a significant effect on the amount of through-flow. Furthermore, using a regression technique, two empirical relationships of the through-flow ratio were obtained for trapezoidal and triangular weirs. Reeve *et al.* (2019) studied numerically the hydraulic performance of gabion stepped spillways. The model solves the Reynolds-averaged Navier–Stokes equations. Four gabion models were used (normal, overlap, inclined, and pooled steps) to investigate the energy dissipation and inception point location. The results indicated that the normal steps have a higher rate of energy dissipation than other configurations. This result was inconsistent with the results obtained by Peyras *et al.* (1992). Rajaei *et al.* (2019) studied experimentally the energy dissipation on gabion and impervious spillways. The physical models of one, two, and three steps have been used. Results showed that the through-flow in the gabion spillways has a significant effect on increasing energy dissipations. Moreover, using multivariate regression, a formula to estimate the energy dissipation in terms of the drop number and steps number was obtained. Jalil *et al.* (2019) investigated the flow characteristics of rectangular gabion weirs experimentally. The effects of the weir height and rockfill material sizes on the upstream water depth were studied. Results indicated that the upstream water depth decreased as the rockfill material size increased for through-flow. Moreover, using non-linear regression analysis, three equations were predicted for calculating upstream water depth as a function of discharge, rockfill mean diameter and weir height. Al-Fawzy *et al.* (2020) investigated the energy dissipation on a stepped gabion weir of three steps experimentally. Results indicated that energy dissipation increases with increasing discharge. In addition, the energy dissipation decreases with increasing the ratio of the length of the third step to the total length of the weir. The porosity has a slight inverse proportion to the energy dissipation. Shariq *et al.* (2020) carried out an experimental study on rectangular gabion weirs to verify the accuracy of the relationships between the velocity and hydraulic gradient in porous media. The main finding is that Ergun's equation is more accurate in predicting the hydraulic gradient than other equations. Salmasi *et al.* (2021) studied experimentally the coefficient of discharge for rectangular gabion weirs. Results indicated that the coefficient of discharge on gabion weirs is about 10% more than that on impervious weirs. Multivariable non-linear regression and genetic programming have been used to predict the through-flow equations on gabion weirs for both submerged and free flow conditions. Shariq *et al.* (2022) used an experimental study to investigate the flow characteristics of rectangular gabion and impervious weirs. They proposed two equations to estimate the discharge coefficient for gabion and impervious weirs. Also, results indicated that the water depth above the weir-to-weir height is a significant factor affecting the discharge coefficient on gabion weirs.

In summary, the previous studies focused on the flow characteristics and the through-flow and overflow ratios on rectangular, triangular, and trapezoidal gabion weirs. Gabion stepped weirs have not been studied regarding the flow characteristics and the through-flow and overflow ratios. Therefore, the current study aims to investigate experimentally the characteristics of flow through and over gabion stepped weirs through three aspects. These are the flow surface profile, the velocity profile over gabion stepped weirs, and the through-flow to total flow ratio and overflow to total flow ratio. The studying of through-flow and overflow ratio on gabion stepped weirs can be considered a novel study as it has not been studied before. The experimental study was conducted in the civil laboratory at Deakin University – Australia.

## MATERIALS AND METHODS

### Experimental facilities and instrumentation

### Physical models

*D*

_{50}= 23 mm and rounded gravel of nominal size (26.5–13.2 mm)

*D*

_{50}= 16 mm. The average porosity was 0.42 and 0.38, respectively, which was measured three times by direct method. The gabion baskets were made of 1.5 mm galvanised wire mesh with square openings of 12.7 × 12.7 mm.

Weir slope V:H . | Model number . | |
---|---|---|

Crushed stone (n = 0.42)
. | Rounded gravel (n = 0.38)
. | |

1:1 | N-CS-1 | N-RG-1 |

1:2 | N-CS-2 | N-RG-2 |

1:3 | N-CS-3 | N-RG-3 |

Weir slope V:H . | Model number . | |
---|---|---|

Crushed stone (n = 0.42)
. | Rounded gravel (n = 0.38)
. | |

1:1 | N-CS-1 | N-RG-1 |

1:2 | N-CS-2 | N-RG-2 |

1:3 | N-CS-3 | N-RG-3 |

### Measurements

*Q*is the through-flow,

_{Through}*Q*is the total flow, and

_{Total}*Q*is the overflow.

_{Over}### Error analysis and uncertainty

^{3}/s. All measurements were repeated three times to obtain the true value and avoid errors. However, according to the accuracy of each instrument used in the measurement, there was an error in each measured value. The common technique to show the closest range of values to the true value is: [measurement = best estimate ± uncertainty]. The relative or fractional uncertainty has been used to report the precision quantitively:

For the flow depth measurements, the measured depth ranged between 4–558 mm, the average depth is 281 mm, and the uncertainty is ±0.1 mm. Then the relative uncertainty equals 0.04%. The measured velocity ranged between 0.25–2.5 m/s, the average velocity is 1.5 m/s, and the uncertainty is ±0.03 m/s. Therefore, the relative uncertainty is equal to 2%. Finally, the measured discharge ranged between 0.003–0.0525 m^{3}/s, the average discharge is 0.0278 m^{3}/s, and the uncertainty is ±0.001 m^{3}/s. The relative uncertainty is equal to 3.6%.

## RESULTS AND DISCUSSION

### Water surface profile

### Velocity profile

### Flow through and over gabion stepped weir

### Through-flow ratio and overflow ratio

*Re*), the porosity of porous media (

*n*), and the ratio of weir length to weir height . The experimental data obtained in the present study for the models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3 at section 1 have been used in the regression analysis to develop the following empirical equation for estimating the through-flow ratio at the upstream section of the weir:

The correlation coefficient is *R**=* 0.98, the root mean square error RMSE = 0.026, and the mean absolute percentage error (*MAPE*) = 4.20. The performance of the estimated through-flow ratio obtained using Equation (2) against the experimentally calculated ratio is presented in Figure 10(a). As shown in the figure, the estimated values of the through-flow ratio using Equation (2) describe the through-flow ratio. Accordingly, the suggested equation can be recommended to estimate the through-flow ratio with high accuracy.

*Re*), the porosity of porous media (

*n*), the ratio of porous media thickness at the section of estimation to the distance between the section and upstream face of the weir , and the ratio of water depth at the section of estimation to the distance between the section and upstream face of the weir , as shown in Figure 9. The obtained general empirical equation:

The correlation coefficient is *R**=* 0.93, the root means square error (*RMSE*) = 0.063, and the MAPE = 11.20. Figure 10(b) presents the performance of the estimated through-flow ratio obtained using Equation (3) against the experimentally calculated ratio. It can be stated that Equation (3) estimates the values of the through-flow ratio at the weir crest edge and each step of the edge with high reliability. The negative sign of the Reynolds number in Equations (2) and (3 indicated that the through-flow ratio reduces when the flow rate increases.

*Re*), the porosity of porous media (

*n*), and the ratio of flow depth above the weir to weir length . The experimental data of the models N-RG-1, N-CS-1, N-RG-2, N-CS-2, N-RG-3, and N-CS-3 at section 1 have been used in the regression analysis to develop the following empirical equation for estimating the overflow ratio at the upstream section of the weir:

The correlation coefficient is *R**=* 0.98, the *RMSE* = 0.027, and the MAPE = 4.81. Figure 10(c) shows the performance of the overflow ratio calculated by using Equation (4) against the experimentally calculated ratio. As shown in the figure, the estimated value of the overflow ratio is calculated by using Equation (4), which well-represents the overflow ratio. Thus, the suggested equation can be used to estimate the overflow ratio with high accuracy.

*Re*), the porosity of porous media (

*n*), the ratio of porous media thickness , the ratio of water depth to distance , and the weir downstream slope . Where

*h*is the step height and

_{s}*l*is the step length. The developed equation is as follows:

_{s}The correlation coefficient is *R**=* 0.95, the *RMSE* = 0.056, and the MAPE = 10.47. Figure 10(d) presents the performance of the estimated overflow ratio obtained by using Equation (5) against the experimentally calculated ratio. According to the figure, Equation (5) can estimate the values of the overflow ratio at the weir crest edge and each step edge with high accuracy. The positive sign of the Reynolds number in Equations (4) and (5 indicated that the overflow ratio increases as the flow rate increases.

Table 2 shows the results of the estimated regression line including the estimated coefficients, the standard error of the coefficients, the calculated t-statistic, the corresponding *p*-value, and the bounds of the 95% confidence intervals.

Variable . | Coefficients . | Standard Error . | t Stat . | P-value
. | Lower 95% . | Upper 95% . |
---|---|---|---|---|---|---|

Equation (2) for estimating the through–flow ratio in section 1 | ||||||

Intercept | 2.203265 | 0.268499831 | 8.20583343 | 1.01761 × 10^{–6} | 1.627390028 | 2.779139755 |

–0.71844 | 0.051171953 | –14.03965409 | 1.21738 × 10^{–9} | –0.828189438 | –0.608683592 | |

n | 3.63647 | 0.343960924 | 10.57233606 | 4.66833 × 10^{–8} | 2.898747667 | 4.374193288 |

–0.0091 | 0.00938583 | –0.96978954 | 0.348608548 | –0.029232882 | 0.011028323 | |

Equation (3) for estimating the through–flow ratio at the edge of the weir crest and each step | ||||||

Intercept | 1.978494766 | 0.359399821 | 5.504996526 | 6.28928 × 10^{–7} | 1.261129721 | 2.695859811 |

–0.547659485 | 0.081396574 | –6.72828672 | 4.62728 × 10^{–9} | –0.710127726 | –0.385191244 | |

n | 2.011168407 | 0.38753714 | 5.189614615 | 2.12966 × 10^{–6} | 1.237641037 | 2.784695776 |

1.093646271 | 0.223567104 | 4.891803183 | 6.57458 × 10^{–6} | 0.647404464 | 1.539888079 | |

–0.906794736 | 0.210234452 | –4.313254681 | 5.41151 × 10^{–5} | –1.326424459 | –0.487165012 | |

Equation (4) for estimating the overflow ratio in section 1 | ||||||

Intercept | –1.350067951 | 0.30493922 | –4.427334577 | 0.000573841 | –2.00409753 | –0.696038372 |

0.7775316 | 0.07075 | 10.98984594 | 2.86706 × 10^{–8} | 0.625787942 | 0.929275259 | |

n | –3.811979807 | 0.340880674 | –11.18273959 | 2.30036 × 10^{–8} | –4.543096139 | –3.080863475 |

–0.229292729 | 0.184662603 | –1.241684703 | 0.234753651 | –0.625354621 | 0.166769163 | |

Equation (5) for estimating the overflow ratio at the edge of the weir crest and each step | ||||||

Intercept | –0.681457476 | 0.307359032 | –2.217138279 | 0.030063029 | –1.295119584 | –0.067795368 |

0.484827239 | 0.069465634 | 6.979382573 | 1.76248 × 10^{–9} | 0.346134625 | 0.623519853 | |

n | –1.905798548 | 0.326576253 | –5.835692367 | 1.7781 × 10^{–7} | –2.557829075 | –1.253768021 |

–1.329685449 | 0.193151081 | –6.884172975 | 2.60149 × 10^{–9} | –1.715324024 | –0.944046874 | |

1.133872385 | 0.181852553 | 6.235119415 | 3.61663 × 10^{–8} | 0.770792048 | 1.496952722 | |

–0.122931895 | 0.022950286 | –5.356442806 | 1.1557 × 10^{–6} | –0.168753619 | –0.07711017 |

Variable . | Coefficients . | Standard Error . | t Stat . | P-value
. | Lower 95% . | Upper 95% . |
---|---|---|---|---|---|---|

Equation (2) for estimating the through–flow ratio in section 1 | ||||||

Intercept | 2.203265 | 0.268499831 | 8.20583343 | 1.01761 × 10^{–6} | 1.627390028 | 2.779139755 |

–0.71844 | 0.051171953 | –14.03965409 | 1.21738 × 10^{–9} | –0.828189438 | –0.608683592 | |

n | 3.63647 | 0.343960924 | 10.57233606 | 4.66833 × 10^{–8} | 2.898747667 | 4.374193288 |

–0.0091 | 0.00938583 | –0.96978954 | 0.348608548 | –0.029232882 | 0.011028323 | |

Equation (3) for estimating the through–flow ratio at the edge of the weir crest and each step | ||||||

Intercept | 1.978494766 | 0.359399821 | 5.504996526 | 6.28928 × 10^{–7} | 1.261129721 | 2.695859811 |

–0.547659485 | 0.081396574 | –6.72828672 | 4.62728 × 10^{–9} | –0.710127726 | –0.385191244 | |

n | 2.011168407 | 0.38753714 | 5.189614615 | 2.12966 × 10^{–6} | 1.237641037 | 2.784695776 |

1.093646271 | 0.223567104 | 4.891803183 | 6.57458 × 10^{–6} | 0.647404464 | 1.539888079 | |

–0.906794736 | 0.210234452 | –4.313254681 | 5.41151 × 10^{–5} | –1.326424459 | –0.487165012 | |

Equation (4) for estimating the overflow ratio in section 1 | ||||||

Intercept | –1.350067951 | 0.30493922 | –4.427334577 | 0.000573841 | –2.00409753 | –0.696038372 |

0.7775316 | 0.07075 | 10.98984594 | 2.86706 × 10^{–8} | 0.625787942 | 0.929275259 | |

n | –3.811979807 | 0.340880674 | –11.18273959 | 2.30036 × 10^{–8} | –4.543096139 | –3.080863475 |

–0.229292729 | 0.184662603 | –1.241684703 | 0.234753651 | –0.625354621 | 0.166769163 | |

Equation (5) for estimating the overflow ratio at the edge of the weir crest and each step | ||||||

Intercept | –0.681457476 | 0.307359032 | –2.217138279 | 0.030063029 | –1.295119584 | –0.067795368 |

0.484827239 | 0.069465634 | 6.979382573 | 1.76248 × 10^{–9} | 0.346134625 | 0.623519853 | |

n | –1.905798548 | 0.326576253 | –5.835692367 | 1.7781 × 10^{–7} | –2.557829075 | –1.253768021 |

–1.329685449 | 0.193151081 | –6.884172975 | 2.60149 × 10^{–9} | –1.715324024 | –0.944046874 | |

1.133872385 | 0.181852553 | 6.235119415 | 3.61663 × 10^{–8} | 0.770792048 | 1.496952722 | |

–0.122931895 | 0.022950286 | –5.356442806 | 1.1557 × 10^{–6} | –0.168753619 | –0.07711017 |

Equations (2)–(5) have been obtained from the normal steps gabion weirs of normal downstream slopes, which ranged between 1:1 and 1:3. Further, the discharge per unit width (*q*) ranged from 0.05 to 0.105 , and the porosity ranged from 0.38 to 0.42. Therefore, it is recommended to estimate the overflow ratio within the above limitations.

## CONCLUSIONS

The current study investigated the flow characteristics on gabion stepped weir for different downstream slopes and rockfill materials experimentally. The water surface profile, the velocity profile over the gabion stepped weir, and the through-flow and overflow ratio have been investigated. Results indicated that the milder slope model (slope 1:3) shows higher water surface and overflow velocity than the steeper slope (slope 1:1) for all flow regimes. This is because the milder slope models have a longer flow path inside and over the weir than the stepper slope models, which creates more internal and surface flow resistance. Moreover, the flow depth and overflow velocity were slightly higher on models of lower porosity (0.38) than on models of higher porosity (0.42). At the nappe flow regime, the water depth was higher on the lower steps than on the upper steps by about 60%. This was due to flow seeping through the vertical face of the steps. Consequently, an accumulative increment of overflow on the lower steps. For all flow regimes, increasing the porosity of porous media resulted in an increase in the through-flow ratio. The through-flow ratio was higher at low discharge (nappe flow regime) and decreased as the discharge increased (skimming flow regime). Furthermore, on the weir crest, the through-flow ratio was higher at the downstream edge than the upstream one due to water entering inside the porous body near the brink. While on the step edges, the through-flow ratio decreased towards the weir toe due to water seeping from the vertical face of each step. The multilinear regression analysis has been used to develop two empirical equations to estimate the through-flow ratio and the overflow ratio at the upstream section of the weir. Furthermore, two equations have been developed to estimate the through-flow ratio and the overflow ratio at the inner sections of the weir. The suggested equations can estimate the through-flow and overflow ratios with high accuracy. The significance of Equations (3) and (5) is the ability to estimate the flow ratio at any section of the weir.

## ACKNOWLEDGEMENTS

The authors acknowledge the support provided by the technical staff in the School of Engineering at Deakin University, Australia.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.