Estimating output flow depth from Rockfill Porous media Open Access

The binomial equation was proved by dimensional analysis (Ward 1964) and by the Navier-Stokes equations (Ahmed & Sunada 1969) and is more accurate and efficient in comparison to the exponential equation (Leps 1973; Stephenson 1979).

SedghiAsl & Rahimi (2011) applied the Darcy-Weisbach friction loss within pipes and combined it with the Manning equation to study flow in the coarse-grained porous media. They proposed a new equation and calibrated the coefficients based on the performed experiments considering the rockfill and flow characteristics. Sadeghian et al. (2013) studied the parallel and convergent nonlinear flows through coarse-grained porous media and concluded that both the parallel and convergent flows are non-Darcy flows for which the binomial and exponential equations are still valid. Salahi et al. (2015) conducted a series of experiments on the pressure columns in the rockfill media to study different hydraulic gradient equations of the non-Darcy flows. They concluded that there is a nonlinear relationship between hydraulic gradient and pore velocity of flow and proposed an empirical equation between the hydraulic gradient and pore velocity. Sedghi-Asl & Ansari (2016) analytically solved free flows by the Dupuit-Forchheimer assumptions and presented dimensionless results. In addition, they compared the results with a series of experimental data and concluded that the analytical solution is capable of calculating the flow profile inside the rockfill media. Many authors investigated flow through rockfill porous media (e.g., Sidiropoulou et al. 2007; Di Nucci 2018). A semi-analytical solution of the nonlinear differential equations constructing a fully saturated porous media was presented by Abbas et al. (2021). The effect of different factors on particle transport in porous media (Zhang et al. 2016; Cui et al. 2017) and solute transport in porous media (Rolston 2007; Agarwal & Sharma 2020) has been investigated. Ostad-Ali-Askari et al. (2019) investigated the effect of climate change on groundwater resources in semiarid and arid areas and have shown adverse effects on groundwater recharge and water level. In this study, climate conditions were predicted for the future period under the emission scenarios of RCP2.6, RCP4.5, and RCP8.5, for an aquifer using MODFLOW. Hu et al. (2019) Kriging-approximation simulated annealing (KASA) optimization algorithm has been used to optimize flow parameters in the porous media. The particle swarm optimization (PSO) algorithm is a population-based evolutionary algorithm and is used in civil engineering and water resources optimization problems such as reservoir performance (Nagesh Kumar & Janga Reddy 2007), water quality management (Lu et al. 2002; Chau 2005; Afshar et al. 2011), optimization of the Muskingum method coefficients (Chu & Chang 2009; Moghaddam et al. 2016; Bazargan & Norouzi 2018; Norouzi & Bazargan 2020; Norouzi & Bazargan 2021) and optimization of the parameters of the porous media equations (Norouzi et al. 2021; Safarian et al. 2021). Therefore, in the present study, the Particle Swarm Optimization (PSO) algorithm was used to optimize the coefficients of the proposed equation.

There are two general methods to analyze steady–non-Darcy flow through rockfill materials:

• a.

  One-dimensional analysis of flow, using gradually-varied flow theory:

Application of this theory in simulation of flow through rockfill materials was investigated by Bari & Hansen (2002) and then by other researchers, such as Bazargan & Shoaei (2006). The results showed that at the end-points of the formed media, particularly at high discharges, there was a significant difference between the observational and computational depth results. Hence, in this method, computations of the longitudinal profile of the water surface are not accurate enough because of the lack of output flow depth as the downstream boundary condition.

• b.

  Two-dimensional analysis of flow, using the Parkin equation:

Parkin, 1969, combined the continuity equation with the exponential function of velocity and hydraulic gradient for the first time and developed an equation as an alternative to Laplace equation. This equation could be solved by knowing the boundary conditions as well as porosity (Arbhabhirama & Dinoy 1973). In other word, without knowing output flow depth as the downstream boundary condition, and the water surface profile as a boundary condition which depends on the output flow depth, accurate solution of the Parkin equation is practically impossible.

Using gradually-varied flow theory in analysis of non-Darcy steady flows is an acceptable engineering method that involves lower computations compared to the Parkin equation and solution of a two-dimensional problem (Bazargan & Shoaei 2006).

In general, the leakage line profile in rockfill materials is important, because:

• 1.

  Output leakage level downstream of the rockfill structures and equivalent drainage is immersed in a particular discharge. If the computed output water level considering the maximum discharge is low, the erosive power of aggregates and, consequently, the possibility of destruction of the downstream slope of the structures would be high.

• 2.

  The observed leakage level downstream of the rockfill structures is used as a boundary condition in computation of pore pressure in the rockfill dams and computation of hydraulic conductivity (Hansen 1992).

In the previous studies, values of Γ were calibrated by different researchers using experimental data. Using such data, Sedghi-Asl et al. (2010) obtained the value of Γ coefficient for the angular and rounded materials as 2.3 and 2.4, respectively. There is no doubt that Γ value for open channels that lack rockfill materials is equal to 1. In other words, the difference between values of Γ is due to the fact that in the angular materials, porosity of media is more than for rounded ones and is closer to 1, which is the same as the porosity of open channels (Sedghi-Asl et al. 2010). In another study, Chabokpour & Tokaldani (2018) used their own experimental data and obtained the value of Γ coefficient for a length of 100 cm as 1.83 and 2.05 for aggregate diameters of 16 and 30 mm, respectively; and for a length of 193 cm as 1.58 and 1.84, respectively considering the same aggregate diameters.

In previous studies, coefficient Γ values were presented as numbers that were only accurate enough for the same experimental conditions, and could not be used for different experimental conditions (by changing experimental conditions, the value of the Γ coefficient also changed). In other words, in previous studies, the coefficient Γ value was not presented considering the physical characteristics of the aggregates and the flow characteristics and using dimensional analysis so that it can be used for all experimental conditions. For this reason, in the present study, using dimensional analysis, the particle swarm optimization (PSO) algorithm and experimental data in different conditions (a total of 178 experimental data for rounded, crashed, Glass artificial materials with rhomboid structure, Glass artificial materials with cubic structure, sandy natural materials), an equation was presented to calculate the coefficient Γ considering the physical characteristics of the rockfill as well as the flow that can be used for all experimental conditions with high accuracy. In other words, using the equation presented in the present study, it is possible to calculate the value of the Γ coefficient and consequently, the output flow depth from the rockfill porous media in different conditions with high accuracy.

In the present study, in order to conduct the tests, authors designed and installed a steep flume with a square cross-section (width and height of 0.3 m, length of 5 m) in the Technical Faculty of Zanjan Universtiy (Figure 1). Flow discharge was measured using spillovers (rectangular sharp-edged spillways.) The bottom and walls of the flume were made of Plexiglas, which allows the flow to be observed. To record the longitudinal profile of pressure, 23 piezometers were installed on the bottom with an installation distance of 10 cm. To develop a rockfill porous media, two porous plastic-made sidewalls were installed at both sides of the porous media. Upstream and downstream faces of this container were orthogonal, keeping fixed by sidewalls during the experiments. A schematic view of the channel is shown in Figure 2.

Sieving method was used to make aggregates that are reasonably uniform. Then, in order to obtain a grading curve, ASTM D422-63 and AASHTO T88-70 practices in sampling were adopted. Table 1 presents characteristics of the rockfill materials that were used in this study for 8 samples of rounded and crashed shapes.

In Table 1, uniformity coefficient (C_u) corresponds to and curvature coefficient (C_c) corresponds to ⁠.

Figure 3 shows changes in depth of the input and output flows versus flow discharge for rounded and crashed materials.

In addition, the data registered by Eshkou (2008) in the Laboratory of Amirkabir University, Tehran, Iran, were used in these experiments (Figure 4). Experiments were conducted inside a horizontal flume with a width, height and length of 0.5, 0.75 and 6 meters, respectively. The sidewalls were made of glass; in the middle of the flume, the physical model was developed by putting together single-sized glass balls (marbles) (0.035 diameters) with cubic (porosity of 0.526) and rhomboid structures (porosity of 0.42), or by putting together sandy single-sized natural materials (diameter of 0.011, porosity of 0.42).

Figure 5 shows changes in depth of the input and output flows versus flow discharge for glass artifacts with different structures (porosities) and sandy natural materials.

In general, the data used in the present study (experimental data of steady flow passing through coarse-grained porous media) include the following:

• (1)

  Experimental data of steady flow through Rounded and Crashed materials that were recorded by the authors of the present study in the Laboratory of Zanjan University, Zanjan, Iran.

• (2)

  Experimental data of other researchers that were recorded in the Laboratory of Amirkabir University, Tehran, Iran, considering steady flow condition through Glass artificial materials with cubic structure, Glass artificial materials with rhomboid structure and Sandy natural materials.

It is worth noting that due to the difference in input and output flow depths (Figures 3 and 5), there is developed flow through the coarse-grained porous media. In addition, for the following reasons, the flow depth across the channel width has not changed.

• (1)

  The side walls of the channel were made of Plexiglas and using photography, changes in the flow depth on both side walls of the channel were recorded and, upon examination, a similar water surface profile was observed.

• (2)

  To study the changes in the piezometric head across the channel width, two piezometers were used. The values recorded by both piezometers were almost the same.

This algorithm was first developed and introduced by Ebrahart & Kennedy (1995). PSO is a population-based searching algorithm that is inspired by nature, like the Genetic Algorithm (GA), Ant Colony and Artificial Bee Colony, and is developed based on the collective intelligence and social behavior of bird flocking or fish schooling. The advantages of this algorithm include simplicity of the structure and implementation, small number of controllable parameters, high convergence speed and high computational efficiency. The basic idea of PSO is based on the assumption that potential solutions are flown through hyperspace with acceleration towards more optimum solutions. Each particle adjusts its flying according to the experiences of both itself and its companions. During the process, the overall best value attained by all the particles within the group and the coordinates of each element in hyperspace associated with its previous best fitness solution are recorded in the memory (Chau 2007; Nagesh Kumar & Janga Reddy 2007).

Efficiency, high convergence speed and proper accuracy of the PSO algorithm have been examined and approved in previous studies. Therefore, the PSO algorithm was selected to optimize the coefficients of the proposed equation. The details of PSO can be obtained elsewhere (Clerc & Kennedy 2002; Gurarslan & Karahan 2011; Karahan 2012; Di Cesare et al. 2015).

A description of the Particle Swarm Optimization (PSO) algorithm is provided in the appendix.

For one-dimensional and two-dimensional analysis of steady flow in a coarse-grained porous media, calculation of the output flow depth as the downstream boundary condition is of utmost importance. In the present study, using dimensional analysis, particle swarm optimization (PSO) algorithm and experimental data in different conditions, an equation was presented to calculate the Γ coefficient in Equation (5), which is highly accurate for all experimental conditions.

In general, the present study consists of the following stages:

• (1)

  According to Stephenson (1979) (the output flow depth from the coarse-grained porous media is equal to the critical depth), the recorded output flow depth from the porous media under different experimental conditions (a total of 178 data recorded under different experimental conditions) was compared with the critical depth (Equation (4)).

• (2)

  Since the output flow depth from the coarse-grained porous media is very different from the critical depth, and according to Equation (5) (the output flow depth from the coarse-grained porous media is as a coefficient of the critical depth and in previous studies, the coefficient was presented as numbers that were only accurate enough for that experimental condition) and studies performed, the Γ coefficient is a function of the average particle size (d₅₀), the length of the rockfill perimeter (L), flow discharge (q), porosity (n), upstream depth (Y_up) and critical depth (Y_c). For this reason, in the present study, using dimensional analysis, PSO algorithm and experimental data in different conditions, an equation was presented to calculate the Γ coefficient with high accuracy and efficiency in all experimental conditions.

It is worth noting that in both Sections 1 and 2, the calculated output flow depth was compared with the recorded one in the experimental condition for rounded, crashed, Glass artificial materials with rhomboid structure, Glass artificial materials with cubic structure, and sandy natural materials first separately and, then, for all materials together.

According to the equation developed by Stephenson (1979), output flow depth from rockfill porous media in steady flow condition is equal to the critical depth (Equation (4)). Figure 7 shows changes in depth of the observational output flow and critical depth versus flow discharge for all experimental data.

As can be seen from Figure 7, there is a significant difference between the observational output flow depth and critical depth. Values of MRE for rounded, crashed, glass artificial materials with cubic and rhomboid structure, as well as sandy natural materials are presented in Table 2.

According to Table 2, if the output flow depth from the coarse-grained porous media is considered to be equal to the critical depth, the mean relative error (MRE) for rounded, crashed, Glass artificial materials with rhomboid structure, Glass artificial materials with cubic structure, and sandy natural materials was calculated as 84.40, 83.81, 60.62, 67.68 and 74.82%, respectively, and 69.96% for all experimental data together. As such, output flow depth through rockfill porous media is not equal to the critical depth.

To optimize the coefficients of Equation (8) (J, B, H, N), the Particle Swarm Optimization (PSO) algorithm and Mean Relative Error (MRE) were adopted as objective functions (Equation (6)). As such, the above-mentioned coefficients were optimized in such a way that the computational and observational output flow depths result in minimum possible MREs.

The optimized values of J, B, H, N and values of MRE, which were obtained by using the proposed equation in the present study (Equation (5) in which Γ is computed using Equation (8)), are presented in Table 3 both separately for each material category and all materials as a whole (or all experimental data).

Unlike open channels, flow leaves coarse-grained porous media with a specific energy greater than the critical energy and, consequently, with a greater depth than the critical depth. In order to accurately calculate the output flow depth from the coarse-grained porous media, the Γ coefficient must be multiplied by Equation (4), according to Equation (5). In previous studies, the value of the Γ coefficient has been calculated as numbers that were accurate only for that special experimental conditions. While in the present study, using dimensional analysis, the PSO algorithm, and experimental data in different conditions, an equation was presented to calculate the Γ coefficient with high accuracy and efficiency for all experimental conditions. According to Table 3, if the presented equation is used to calculate the Γ coefficient and consequently the output flow depth, the mean relative error (MRE) for rounded, crashed, Glass artificial materials with rhomboid structure, Glass artificial materials with cubic structure, and sandy natural materials was calculated as 5.49, 4.72, 6.24, 4.41 and 6.42%, respectively and as 8.99% for all experimental data together.

Figure 8 shows changes in the observational output flow depth (observed in experiments) and computational output flow depth (using the proposed equation in the present study) versus flow discharge for different materials.

Mean values of each term of the proposed equation (Equation (8)) relative to the computational output flow depth (Y_e) are presented in Table 4.

As can be seen from Table 4, these terms have the least effect on computation of the output flow depth: the second term in rounded and crashed materials, the third terms in glass artificial materials with cubic and rhomboid structures, and the second term in sandy natural materials. When just one of the J, B, H, N coefficients was optimized for all materials, all terms of the proposed equation in the present study (Equation (8)) affected computation of the output flow depth, and this is why Equation (8) is presented for computation of the Γ coefficient.

Comparison of flow behavior in open channels and through rockfill materials revealed that since in open channels there is no rockfill to withstand the flow, it is directly affected by the fluid weight. For this reason, flow with minimum specific energy (critical energy) leaves the cross-section and therefore, output flow depth is equal to the critical depth. In other words, fluid weight makes the output flow depth equal to the critical depth. In flow through rockfill media, however, rather than total weight of the fluid, a fraction of it is exerted to the aggregates and therefore, the output flow depth is increased more than that of open channels, and it is steadily higher than the critical depth.

Coarse-grained (rockfill) porous media is of great importance in various fields such as civil engineering and hydraulic structures. Calculation of the output flow depth from the rockfill media is of great importance in one-dimensional and two-dimensional analysis of steady flow. The presence of rockfill causes flow to leave the coarse-grained media with a higher specific energy than the critical energy and, consequently, a greater depth than the critical depth. For this reason, the Stephenson hypothesis (equality of the output flow depth from the rockfill (according to Equation (4)) to the critical depth) is not highly accurate in calculation of the output flow depth from the rockfill. To increase the accuracy of Equation (4), the coefficient Γ was multiplied by the critical depth (Equation (5)). In previous studies, the coefficient Γ has been presented as different numbers that were suitably accurate and efficient only for that special experimental condition. While in the present study, using dimensional analysis, the PSO algorithm and experimental data in different conditions (a total of 178 experimental data for rounded, crashed, Glass artificial materials with rhomboid structure, Glass artificial materials with cubic structure, and sandy natural materials), an equation was presented to calculate the mentioned coefficient in all experimental conditions with suitable accuracy and efficiency. In other words, the relationship presented in the present study includes the physical characteristics of the coarse-grained porous media and flow characteristics and is highly accurate in calculation of the output flow depth from the coarse-grained porous media in different conditions.

The results of the present study include the following:

• (1)

  If, according to Stephenson, the output flow depth from the coarse-grained porous media was considered to be equal to the critical depth, the mean relative error (MRE) for rounded, crashed, Glass artificial materials with rhomboid structure, Glass artificial materials with cubic structure, and sandy natural materials was calculated as 84.40, 83.81, 60.62, 67.68, and 74.82%, respectively and as 69.96% for all experimental data together.

• (2)

  If the equation presented in the present study (Equation (8)) was used to calculate the coefficient Γ and, consequently, the output flow depth from the coarse-grained porous media, the mean relative error (MRE) of 5.49, 4.72, 6.24, 4.41 and 6.42% was calculated for the mentioned materials, respectively, and 8.99% for all experimental data together. In other words, the equation presented in the present study improved the MRE by 93, 94, 90, 93, 91, and 87%, respectively, compared to the case of using the equation presented by Stephenson.

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