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The outflow depth from the radial porous media (inflow to the well) is very useful as the downstream boundary condition and the starting point for water surface profile calculations. Based on the studies, unlike the Stephenson's hypothesis (the outflow depth is equal to the critical depth), the outflow depth from the rockfill media is a coefficient (Γ) of the critical depth. In the present study, using several (large scale and almost real) experimental data in the radial non-Darcy flow condition, dimensional analysis and the particle swarm optimization (PSO) algorithm, an equation was presented to calculate the mentioned coefficient based on upstream water depth (h) and distance between the well center and the upstream (R). Then, using the calculated outflow depth and the 1D flow analysis equations, the water surface profile in the radial non-Darcy condition was calculated for the first time. The results showed that considering an outflow depth equal to the critical depth and using the proposed solution in the present study, the mean relative error (MRE) values of 83.43% and 3.53% were obtained, respectively. In addition, using the proposed solution for different experimental conditions, an average MRE of 2.58% was calculated for the water surface profile.

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  • Using the experimental data with almost real scale.

  • Calculation of the output flow depth from the radial porous media.

  • Providing a relationship based on the upstream water depth (h) and the distance of well center from upstream (R) to calculate the output flow depth.

  • Calculation of water surface profile in radial non-Darcy flow using gradually varied flow theory.

  • Using PSO algorithm in calculations.

dimensional analysis, gradually varied flow theory, one-dimensional analysis, output flow depth, radial porous media, steady flow
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The flow through the groundwater is the type of flow through the porous media (Cui et al. 2021; Wang et al. 2022). Flow in porous media is of two types of laminar flow (in fine-grained materials such as clay and silt) and turbulent flow (in coarse-grained materials such as sand and cobblestone). In fine-grained materials, (linear) Darcy equation is valid between the hydraulic gradient (i) and the flow velocity (V) (McWhorter et al. 1977) (Equation (1)), and in coarse-grained materials, the Darcy equation is not valid anymore (Hansen et al. 1995) and the power and binomial equations (non-Darcy or nonlinear equations, Equations (2) and (3)) are valid (Forchheimer 1901; Leps 1973; Stephenson 1979).
(1)
(2)
(3)
where k is the permeability coefficient of materials (m/s), a (s/m) and b (s2/m2) are values dependent on the properties of the porous media and fluid, and m (s /m) and n (-) are coefficients dependent on the flow as well as the porous media characteristics.

Flow through coarse-grained porous media is classified into parallel flows (such as flow through rockfill dams, gabions, etc.) and radial flows (convergent) (such as flow near drilled wells in coarse-grained alluvial beds, groundwater, etc.). Both types exist in the form of free and pressured flows. An example of radial flows is flow near wells drilled in coarse-grained alluvial beds and aquifers. Groundwater is one of the sources of water storage in the world (Jamie et al. 2019). Groundwater resources can be achieved using drilled deep and semi-deep wells, springs and aqueducts (Akbarpour 2016). Investigation of water flow in the above mentioned cases is very important to evaluate the volume of water that can be exploited from aquifers.

Steady non-Darcy flow is studied using one-dimensional (1D) analysis (gradually varied flow theory) and two-dimensional (2D) analysis (Parkin equation). The governing equations on radial non-Darcy flow (combination of continuity equation in cylindrical coordinates and power equation between hydraulic gradient and flow velocity) are solved using numerical methods of finite differences, finite elements and finite volumes (Sadeghian 2013). Solving the mentioned equations requires boundary conditions and a lot of data and is therefore massive, time consuming and costly. However, gradually varied flow theory requires much less data and is easier and less expensive. (Bari & Hansen 2002) studied non-Darcy flows using the standard step method. Bazargan & Shoaei (2010) studied non-Darcy steady flow using the gradually varied flows theory. Gudarzi et al. (2020) studied the effect of drag force on the accuracy of water surface profile calculations passing through coarse-grained porous media using the gradually varied flows theory. Sadeghian et al. (2013) evaluated the binomial and power equations in calculating the changes in hydraulic gradient based on the flow velocity in the free surface radial flow. Shayannejad & Ebrahimi (2020) proposed an equation between discharge and hydraulic potential in the radial flow of free aquifers using the fractional derivative of the flow velocity relative to the hydraulic gradient under steady state conditions. Venkataraman & Rao (2000) studied the binomial equation in convergent pressured flows and proposed the modified coefficients a and b. Many researchers (Ward 1964; Ahmed & Sunada 1969; Sedghi-Asl & Ansari 2016; Norouzi et al. 2022a) studied parallel non-Darcy flows and few studies have been done on radial non-Darcy flows in the free-surface case. In order to use the binomial equation to calculate the changes in the hydraulic gradient based on the flow velocity in the radial flow case, the coefficients a and b of the parallel flow need to be corrected (Venkataraman & Rao 2000). Whereas the coefficients m and n in the power equation need no corrections and the same values can be used in both parallel and radial flows using optimization, and have better performance in the non-Darcy radial case (Sadeghian et al. 2013). For this reason, the power equation is preferred to the binomial equation in the case of radial non-Darcy flow.

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. 2022a; Safarian et al. 2021).

For this reason, in the present study, the PSO algorithm was used to optimize the power equation coefficients (m, n) and the coefficient of the presented equation to calculate the output flow depth. Calculation of the water surface profile (1D flow analysis using the gradually varied flows theory) and the depth of the output radial flow from the coarse-grained porous media (the input flow depth to the well) is of great importance as the starting point of the 1D analysis of steady flow calculations. In addition, water surface profile and outflow depth is widely used as the boundary condition in the 2D analysis of steady non-Darcy flow (solving Parkin equation) and to study the flow passing through the gravel media. According to previous studies and literature review, the outflow depth in radial non-Darcy flow condition is not equal to the critical depth. In the present study, for the first time, large-scale (almost real) experimental data, dimensional analysis and PSO algorithm was used to present an equation to calculate the outflow depth from the radial coarse-grained porous media. Then, using the exponential equation of hydraulic gradient changes in terms of the flow velocity (Equation (3), coefficients m and n of which were optimized using the PSO algorithm), the gradually varied flows theory and the calculated outflow depth as the starting point of calculations, the water surface profile was calculated for different experimental data with appropriate accuracy.

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Experimental data

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Detailed study of the flow through the coarse-grained alluvial beds requires experiments in real conditions. It is not possible to install the necessary equipment and perform experiments in natural places for different reasons. In the present study, due to the compatibility of cylindrical coordinates and its adaptation to the physics of the radial flow problems, a device was developed in the laboratory of Bu-Ali-Sina University, Iran, in the form of a semi-cylinder with a diameter of 6 m and a height of 3 m (Figure 1). The dimensions of the device are in large scale so that the experimental process would not be influenced by the scale effects. To measure piezometric pressure, piezometric grids were used. The device has a volume of 14,000 L and a material capacity of approximately 40 tons. Four pumps were installed in parallel at the top of the device to develop the required flow. Coarse-grained river materials with a diameter of 2–10 cm, porosity of 40%, uniformity coefficient of 2.13 and curvature coefficient of 1.016 were used. To perform the experiments, the model was first filled with water to a certain height (53, 60, 70, 85, 95, 110, 120, 140, 150 and 160 cm) by pumping. A flow discharge of 49.94–53.16 L per second was developed in the experiments. Flow characteristics and water surface profile for different heights are given in Table 1. It is worth mentioning that the well radius of the experimental device was 25 cm (the radius of the input depth to the well in Table 1) and the slope of the device was zero.
Table 1

Flow characteristics and water surface profile recorded in the laboratory

Radius (cm)Height of pumped water (cm)
5360708595110120140150160
Discharge (L/s)
50.0349.9450.3850.9551.2453.1652.2252.8052.6852.95
Water surface profiles (cm)
25 415 535 655 813 934 1,085 1,180 1,386 1,488 1,593 
50 476 563 675 831 944 1,094 1,189 1,393 1,493 1,598 
75 491 575 683 839 947 1,097 1,192 1,396 1,495 1,600 
105 494 584 689 845 949 1,099 1,194 1,399 1,497 1,602 
140 494 591 693 849 951 1,101 1,196 1,400 1,498 1,603 
180 507 596 697 853 953 1,103 1,198 1,402 1,499 1,604 
225 516 601 700 856 954 1,104 1,199 1,403 1,499 1,604 
275 522 605 702 859 955 1,105 1,200 1,404 1,500 1,605 
Radius (cm)Height of pumped water (cm)
5360708595110120140150160
Discharge (L/s)
50.0349.9450.3850.9551.2453.1652.2252.8052.6852.95
Water surface profiles (cm)
25 415 535 655 813 934 1,085 1,180 1,386 1,488 1,593 
50 476 563 675 831 944 1,094 1,189 1,393 1,493 1,598 
75 491 575 683 839 947 1,097 1,192 1,396 1,495 1,600 
105 494 584 689 845 949 1,099 1,194 1,399 1,497 1,602 
140 494 591 693 849 951 1,101 1,196 1,400 1,498 1,603 
180 507 596 697 853 953 1,103 1,198 1,402 1,499 1,604 
225 516 601 700 856 954 1,104 1,199 1,403 1,499 1,604 
275 522 605 702 859 955 1,105 1,200 1,404 1,500 1,605 
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Figure 1
Different parts of the experimental device: (a) schematic view of experimental model, (b) front view of experimental model, and (c) placement of parallel pumps on the model.
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Different parts of the experimental device: (a) schematic view of experimental model, (b) front view of experimental model, and (c) placement of parallel pumps on the model.

Figure 1
Different parts of the experimental device: (a) schematic view of experimental model, (b) front view of experimental model, and (c) placement of parallel pumps on the model.
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Different parts of the experimental device: (a) schematic view of experimental model, (b) front view of experimental model, and (c) placement of parallel pumps on the model.

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Depth of output flow from the rockfill media

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Stephenson (1979) considered the output flow depth through rockfill drainage equal to the critical depth in case of steady flow (Equation (4)).
(4)
where yc = critical depth, q = discharge in unit width of the rockfill, n = porosity of materials, and g = gravity acceleration.
However, experimental tests conducted by other researchers proved that Equation (4) is inaccurate. In this regard, Sedghi-Asl et al. (2010) corrected the equation based on the experimental data by applying Γ coefficient (Equation (5)).
(5)

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 which 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 the 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. Norouzi et al. (2022b) used experimental data in different conditions as well as PSO algorithm and dimensional analysis to present an equation to calculate the coefficient based on the physical properties of the aggregates and parallel flow characteristics in rockfill media.

In the present study, using the mentioned experimental data, dimensional analysis and PSO algorithm, an equation was presented to calculate the coefficient in the case of radial non-Darcy flow.

Gradually varied flows theory

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The gradually varied flows theory in the standard step method is as Equation (6).
(6)
(7)
where E is the specific energy of the flow (Equation (7)), y is flow depth, V is flow velocity, g is the acceleration of gravity, S0 is bed slope, Δx is the spatial step, and Sf is the slope of the energy line.
The slope of the energy line is one of the main parameters in the analysis of steady flow using the gradually varied flows theory, which is expressed as Equation (8) in the rockfill material (Stephenson 1979; Bari & Hansen 2002; Bazargan & Shoaei 2006).
(8)
In rockfill materials, flow velocity (V) and, consequently, the velocity head can be neglected relative to the other parameters of Equation (8), and the slope of the energy line (Sf) is equal to the hydraulic gradient (i) (Equation (9)) (Stephenson 1979; Bari & Hansen 2002; Bazargan & Shoaei 2006).
(9)
In radial non-Darcy flow in free surface case, Equation (10) was used for the steady flow analysis.
(10)
where Q is discharge (m3/s), R is flow radius (m) (distance between the desired point and the well center), m (s/m) and n (−) are coefficients of the power equation (according to Table 1), θ is the convergence angle of the flow (rad) and Rθ is the arc length. It is worth noting that θ = 180 degrees because of the semi-cylindrical shape of the device.

Given the above equation, flow depth at a point must be available to start the calculations. According to the solution presented in the present study to calculate the depth of the output flow from the porous media, the calculations started from downstream (output flow from the radial media or input flow to the well).

Particle Swarm Optimization (PSO) algorithm

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This algorithm was first developed and introduced by (Eberhart & Kennedy 1995). PSO is a population-based searching algorithm that is inspired by the nature like Genetic Algorithm (GA), Ant Colony and Artificial Bee Colony and is developed based on the collective intelligence and social behavior of the 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 power equation (m, n), and 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).

To evaluate the optimum values of the mentioned coefficients, minimization of the mean relative error (MRE), which is defined using Equation (11), was used as the objective function in the PSO algorithm.
(11)
where and are the observed (experimental) and calculated values. The flowchart used in the present study is presented in Figure 2.
Figure 2
Flowchart of the present study stages.
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Flowchart of the present study stages.

Figure 2
Flowchart of the present study stages.
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Flowchart of the present study stages.

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The present study consisted of the following stages:

  • 1.

    Assuming an output flow depth from the radial porous media equal to the critical depth (Equation (4)):

    According to the hypothesis by Stephenson (1979), the depth of the output flow from the rockfill media is equal to the critical depth. A review of studies conducted on parallel non-Darcy flow indicated that the output flow depth in this case is significantly different from the critical depth. The studies conducted in the present study in the case of radial non-Darcy flow also indicated this inaccuracy of the mentioned equation.

  • 2.

    Using Equation (5) to calculate the depth of the output flow from the radial porous media:

    According to previous studies, the output flow depth in the case of non-Darcy flow is a coefficient (⁠⁠) of critical depth. In the present study, using large-scale (almost real) experimental data, dimensional analysis and PSO algorithm, an equation was presented based on upstream flow depth (h) and the distance between the center of the well and upstream (R) to calculate and, consequently, the output flow depth from the radial non-Darcy media.

  • 3.

    Calculation of coefficients m and n of the power equation:

    The calculation of hydraulic gradient changes (i) based on the flow velocity (V), which indicates the nonlinearity of the flow passing through the coarse-grained porous media, is of great importance in the 1D and 2D analyzes of the steady flow. In the case of radial non-Darcy flow, the power equation (Equation (3)) is more efficient than the binomial equation (Equation (2)) (Sadeghian et al. 2013). In the present study, using the PSO algorithm, the coefficients m and n of the power equation were optimized for 10 different upstream pumped depths.

  • 4.

    One-dimensional analysis of radial non-Darcy flow in steady flow case:

    Calculation of water surface profile is widely used to study flow through the radial non-Darcy media and also as the main boundary condition in 2D analysis of the mentioned flow. To use the theory of gradually varied flow, the flow depth at a point is needed to start the calculations. The depth of the output flow from the porous media (input depth to the well), which is calculated using the solution presented in step (2), is the starting point of the calculations. In other words, in the present study, using computational output flow depth, 1D flow analysis was started from downstream of the coarse-grained porous media.

Depth of output flow from the porous media in the case of radial flow

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If the output flow depth in the case of radial non-Darcy flow is considered equal to the critical depth according to the hypothesis by Stephenson (1979), the obtained results are as shown in Figure 3 (flow depth changes versus discharge). In other words, in this case, the mean relative error is equal to 83.43%, indicating the significant difference between the critical depth and the output flow depth.
Figure 3
Changes in experimental output flow depth and critical depth versus flow discharge.
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Changes in experimental output flow depth and critical depth versus flow discharge.

Figure 3
Changes in experimental output flow depth and critical depth versus flow discharge.
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Changes in experimental output flow depth and critical depth versus flow discharge.

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According to Equation (5), the output flow depth from the radial rockfill media is a coefficient of critical depth. The studies performed in the present study indicated that the mentioned coefficient is a function of upstream flow depth (h) and the distance between the center of the well (downstream) and upstream (the length of the rockfill media) (R). In other words, using experimental data for different upstream water heights, Equation (12) was presented to calculate the coefficient ⁠.
(12)

Using the PSO algorithm, the value of the J coefficient was optimized as 20.75. In other words, the value of the J coefficient was optimized so that the computational and observational (recorded in the laboratory) output flow depths would be as consistent as possible.

Figure 4 shows changes in the computational and experimental output flow depths versus discharge in the case of using Equation (12) to calculate the coefficient and, consequently, the output flow depth. In other words, by using the solution presented in the present study to calculate the output flow depth in the case of radial non-Darcy flow, a mean relative error of 3.53% was calculated.
Figure 4
Changes in the observational and computational output flow depths versus flow discharge (using the solution presented in the present study).
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Changes in the observational and computational output flow depths versus flow discharge (using the solution presented in the present study).

Figure 4
Changes in the observational and computational output flow depths versus flow discharge (using the solution presented in the present study).
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Changes in the observational and computational output flow depths versus flow discharge (using the solution presented in the present study).

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The values of discharge, observational and computational output flow depths and critical depths are listed in Table 2.

Table 2

Observational and computational values of the output flow depths

Height of pumped water (cm)Q (L/s)Output water depth (cm)
ExperimentalCriticalComputed
53 50.03 41.5 13.72 50.3 
60 49.94 53.5 13.71 56.9 
70 50.38 65.5 13.79 66.8 
85 50.95 81.3 13.89 81.7 
95 51.24 93.4 13.95 91.6 
110 53.16 108.5 14.29 108.7 
120 52.22 118 14.12 117.2 
140 52.80 138.6 14.23 137.8 
150 52.68 148.8 14.21 147.4 
160 52.95 159.3 14.25 157.7 
MRE %   83.43 3.83 
Height of pumped water (cm)Q (L/s)Output water depth (cm)
ExperimentalCriticalComputed
53 50.03 41.5 13.72 50.3 
60 49.94 53.5 13.71 56.9 
70 50.38 65.5 13.79 66.8 
85 50.95 81.3 13.89 81.7 
95 51.24 93.4 13.95 91.6 
110 53.16 108.5 14.29 108.7 
120 52.22 118 14.12 117.2 
140 52.80 138.6 14.23 137.8 
150 52.68 148.8 14.21 147.4 
160 52.95 159.3 14.25 157.7 
MRE %   83.43 3.83 
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One-dimensional analysis of steady flow in the case of radial non-Darcy flow

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Water surface profile calculation is used to study water flow behavior. In addition, with a known water surface profile as the main boundary condition in the 2D analysis of steady flow (solving the Parkin equation), upstream and downstream boundary conditions will also be available. For this reason, in the present study, using large-scale (almost real) experimental data and the theory of gradually varied flow, the water surface profile in the case of radial non-Darcy flow was calculated. It is worth mentioning that in previous studies, the theory of gradually varied flow was used in calculating the water surface profile in the case of parallel non-Darcy flow. In order to calculate the water surface profile, using the experimental data for different upstream pumping heights and the PSO algorithm, the coefficients m and n of the power equation were first optimized. Then, using optimized coefficients and the equations of the gradually varied flow theory and the calculated values of the output flow depth (calculated values in Table 2 in the case of using the solution presented in the present study), the water surface profile calculations in the case of radial non-Darcy flow were done from downstream (entering the well). The optimized values of the coefficients m and n are given in Table 3. The observational and computational values of water surface profile and the MRE are also shown in Table 4.

Table 3

The values of the coefficients m and n of the power equation for the different upstream pumped heights

Height of pumped water (cm)m (s/m)n (–)
53 2.481 1.274 
60 2.039 1.220 
70 3.453 1.444 
85 2.633 1.279 
95 0.996 1.248 
110 1.157 1.251 
120 1.324 1.253 
140 2.861 1,399 
150 3.202 1.503 
160 3.547 1.504 
Height of pumped water (cm)m (s/m)n (–)
53 2.481 1.274 
60 2.039 1.220 
70 3.453 1.444 
85 2.633 1.279 
95 0.996 1.248 
110 1.157 1.251 
120 1.324 1.253 
140 2.861 1,399 
150 3.202 1.503 
160 3.547 1.504 
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Table 4

Computational and observational (experimental) values of water surface profile

Radius (m)Height of pumped water (cm)
53
60
70
85
95
Discharge (L/s)
50.03
49.94
50.38
50.95
51.24
ObservedComputedObservedComputedObservedComputedObservedComputedObservedComputed
0.25 0.415 0.503 0.535 0.569 0.655 0.668 0.813 0.817 0.934 0.916 
0.5 0.476 0.534 0.563 0.594 0.675 0.688 0.831 0.835 0.944 0.923 
0.75 0.491 0.548 0.575 0.606 0.683 0.696 0.839 0.843 0.947 0.926 
105 0.494 0.558 0.584 0.614 0.689 0.702 0.845 0.849 0.949 0.928 
1.4 0.494 0.565 0.591 0.621 0.693 0.706 0.849 0.854 0.951 0.929 
1.8 0.507 0.571 0.596 0.626 0.697 0.709 0.853 0.858 0.953 0.931 
2.25 0.516 0.576 0.601 0.631 0.700 0.711 0.856 0.861 0.954 0.933 
2.75 0.522 0.579 0.605 0.634 0.702 0.713 0.859 0.863 0.955 0.934 
MRE% 12.56 5.29 1.80 0.54 2.19 
Radius (m)
Height of pumped water (cm)
110
120
140
150
160
Discharge (L/s)
53.16
52.22
52.80
52.68
52.95
Observed
Computed
Observed
Computed
Observed
Computed
Observed
Computed
Observed
Computed
0.25 1.085 1.087 1.18 1.172 1.386 1.378 1.488 1.474 1.593 1.577 
0.5 1.094 1.093 1.189 1.178 1.393 1.385 1.493 1.479 1.598 1.582 
0.75 1.097 1.096 1.192 1.181 1.396 1.388 1.495 1.481 1.600 1.584 
105 1.099 1.099 1.194 1.184 1.399 1.390 1.497 1.483 1.602 1.586 
1.4 1.101 1.100 1.196 1.186 1.400 1.392 1.498 1.484 1.603 1.587 
1.8 1.103 1.102 1.198 1.187 1.402 1.393 1.499 1.485 1.604 1.588 
2.25 1.104 1.103 1.199 1.188 1.403 1.394 1.499 1.485 1.604 1.588 
2.75 1.105 1.104 1.200 1.189 1.404 1.395 1.500 1.486 1.605 1.589 
MRE% 0.07 0.85 0.58 0.94 1.00 
Radius (m)Height of pumped water (cm)
53
60
70
85
95
Discharge (L/s)
50.03
49.94
50.38
50.95
51.24
ObservedComputedObservedComputedObservedComputedObservedComputedObservedComputed
0.25 0.415 0.503 0.535 0.569 0.655 0.668 0.813 0.817 0.934 0.916 
0.5 0.476 0.534 0.563 0.594 0.675 0.688 0.831 0.835 0.944 0.923 
0.75 0.491 0.548 0.575 0.606 0.683 0.696 0.839 0.843 0.947 0.926 
105 0.494 0.558 0.584 0.614 0.689 0.702 0.845 0.849 0.949 0.928 
1.4 0.494 0.565 0.591 0.621 0.693 0.706 0.849 0.854 0.951 0.929 
1.8 0.507 0.571 0.596 0.626 0.697 0.709 0.853 0.858 0.953 0.931 
2.25 0.516 0.576 0.601 0.631 0.700 0.711 0.856 0.861 0.954 0.933 
2.75 0.522 0.579 0.605 0.634 0.702 0.713 0.859 0.863 0.955 0.934 
MRE% 12.56 5.29 1.80 0.54 2.19 
Radius (m)
Height of pumped water (cm)
110
120
140
150
160
Discharge (L/s)
53.16
52.22
52.80
52.68
52.95
Observed
Computed
Observed
Computed
Observed
Computed
Observed
Computed
Observed
Computed
0.25 1.085 1.087 1.18 1.172 1.386 1.378 1.488 1.474 1.593 1.577 
0.5 1.094 1.093 1.189 1.178 1.393 1.385 1.493 1.479 1.598 1.582 
0.75 1.097 1.096 1.192 1.181 1.396 1.388 1.495 1.481 1.600 1.584 
105 1.099 1.099 1.194 1.184 1.399 1.390 1.497 1.483 1.602 1.586 
1.4 1.101 1.100 1.196 1.186 1.400 1.392 1.498 1.484 1.603 1.587 
1.8 1.103 1.102 1.198 1.187 1.402 1.393 1.499 1.485 1.604 1.588 
2.25 1.104 1.103 1.199 1.188 1.403 1.394 1.499 1.485 1.604 1.588 
2.75 1.105 1.104 1.200 1.189 1.404 1.395 1.500 1.486 1.605 1.589 
MRE% 0.07 0.85 0.58 0.94 1.00 
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According to Table 4, for the heights of the pumped water upstream equal to 53, 60, 70, 85, 95, 110, 120, 140, 150 and 160 cm, MRE values for the calculated and observed water surface profiles were calculated as 12.56, 5.29, 1.80, 0.54, 2.19, 0.07, 0.85, 0.58, 0.94 and 1%, respectively. In other words, the theory of gradually varied flow in the 1D analysis of steady flow in the case of radial non-Darcy flow is accurate and efficient enough.

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Calculation of the water surface profile and the outflow depth from the porous media in the steady-radial condition is widely used in 1D and 2D flow analysis. In the present study, using large-scale (almost real) experimental data (large-scale semi-cylindrical test device with a diameter of 6 m and a height of 3 m considering 10 different pumped water heights in upstream of the rockfill media), first by using dimensional analysis and the PSO algorithm, the coefficient (the ratio between the outflow depth from the rockfill media and the critical depth) was optimized as a function of the upstream water depth (h) and the distance between the well center and the upstream (R). Then, using the optimized m and n coefficients using the PSO algorithm, the calculated depth of the outflow (the starting point of the calculations) and the gradually varied flows theory, the water surface profile of the steady flow in the radial non-Darcy condition was calculated. The results showed that the MRE of the calculated outflow depth from the radial porous media using the proposed solution improved by 95.8% compared to the case of assuming an outflow depth equal to the critical depth. In addition, the average MRE for 10 different depths of pumped water in the upstream of the rockfill media was calculated as 2.58%. In other words, the solution presented in the present study is suitably accurate in the analysis of steady flow in radial non-Darcy condition.

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

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The authors declare there is no conflict.

Ahmed
N.
 &
Sunada
D. K.
1969
Nonlinear flow in porous media
.
Journal of the Hydraulics Division
95
(
6
),
1847
–
1858
.
Google Scholar
Crossref
Search ADS
 
Akbarpour
A.
2016
Simulation-Optimization quantitative and qualitative model operation of aquifer in order to adjust pollutant concentrations using Cuckoo algorithm
.
Journal of Water and Soil Conservation
23
(
5
),
87
–
103
.
(In Persian)
.
Google Scholar
 
Bari
R.
 &
Hansen
D.
2002
Application of gradually-varied flow algorithms to simulate buried streams
.
Journal of Hydraulic Research
40
(
6
),
673
–
683
.
Google Scholar
Crossref
Search ADS
 
Bazargan
J.
 &
Shoaei
S. M.
2006
Application of gradually varied flow algorithms to simulate buried streams
.
IAHR Journal of Hydraulic Research
44
(
1
),
138
–
141
.
Google Scholar
Crossref
Search ADS
 
Bazargan
J.
 &
Shoaei
S.
2010
Analysis of non-darcy flow in rock fill materials using gradually varied flow method
.
Journal of Civil and Surveying Engineering
44
(
2
), 131–139.
Google Scholar
 
Chabokpour
J.
 &
Tokaldani
E.
2018
Experimental-numerical simulation of longitudinal water surface profile through large porous media
.
Iranian Water Researches Journal
11
(
3
),
81
–
90
.
(In Persian)
.
Google Scholar
 
Chau
K.
2005
A split-step PSO algorithm in prediction of water quality pollution
. In: Wang, J., Liao, X. F. & Yi, Z. (eds)
International Symposium on Neural Networks
.
Springer
,
Berlin, Heidelberg, Germany
, pp.
1034
–
1039
.
Google Scholar
Crossref
Search ADS
 
Chu
H. J.
 &
Chang
L. C.
2009
Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model
.
Journal of Hydrologic Engineering
14
(
9
),
1024
–
1027
.
Google Scholar
Crossref
Search ADS
 
Clerc
M.
 &
Kennedy
J.
2002
The particle swarm-explosion, stability, and convergence in a multidimensional complex space
.
IEEE Transactions on Evolutionary Computation
6
(
1
),
58
–
73
.
Google Scholar
Crossref
Search ADS
 
Cui
X.
,
Zhu
C.
,
Hu
M.
,
Wang
R.
 &
Liu
H.
2021
Permeability of porous media in coral reefs
.
Bulletin of Engineering Geology and the Environment
80
(
6
),
5111
–
5126
. .
Google Scholar
Crossref
Search ADS
 
Di Cesare
N.
,
Chamoret
D.
 &
Domaszewski
M.
2015
A new hybrid PSO algorithm based on a stochastic Markov chain model
.
Advances in Engineering Software
90
,
127
–
137
.
Google Scholar
Crossref
Search ADS
 
Eberhart
R.
 &
Kennedy
J.
1995
A new optimizer using particle swarm theory
. In:
MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science
.
IEEE
, pp.
39
–
43
.
Google Scholar
Crossref
Search ADS
 
Forchheimer
P.
1901
Wasserbewagung Drunch Boden, Z.Ver, Deutsh. Ing., 45, 1782–1788
.
Gudarzi
M.
,
Bazargan
J.
 &
Shoaei
S.
2020
Longitude profile analysis of water table in rockfill materials using gradually varied flow theory with consideration of drag force
.
Iranian Journal of Soil and Water Research
51
(
2
),
403
–
415
.
doi:10.22059/ijswr.2019.287292.668295. (In Persian)
.
Google Scholar
 
Gurarslan
G.
 &
Karahan
H.
2011
Parameter estimation technique for the nonlinear muskingum flood routing model
. In
6th EWRA International Symposium-Water Engineering and Management in A Changing Environment
,
Catania, Italy
.
Google Scholar
 
Jamie
M.
,
ahmadianfar
I.
 &
raeisi isa abadi
A.
2019
A numerical IMPES discontinuous Galerkin method for immiscible groundwater contaminations flow using Lax-Wendroff scheme
.
Journal of Water and Soil Conservation
26
(
2
),
1
–
27
.
doi:10.22069/jwsc.2019.15769.3096. (In Persian)
.
Google Scholar
 
Leps
T. M.
1973
Flow through rockfill, Embankment-dam engineering casagrande volume edited by Hirschfeld, RC and Poulos, SJ
.
Lu
W. Z.
,
Fan
H. Y.
,
Leung
A. Y. T.
 &
Wong
J. C. K.
2002
Analysis of pollutant levels in central Hong Kong applying neural network method with particle swarm optimization
.
Environmental Monitoring and Assessment
79
(
3
),
217
–
230
. .
Google Scholar
Crossref
Search ADS
PubMed
 
McWhorter
D. B.
,
Sunada
D. K.
 &
Sunada
D. K.
1977
Ground-Water Hydrology and Hydraulics
.
Water Resources Publication LLC. U.S. Library, Fort Collins, CO, USA
.
Google Scholar
 
Moghaddam
A.
,
Behmanesh
J.
 &
Farsijani
A.
2016
Parameters estimation for the new four-parameter nonlinear Muskingum model using the particle swarm optimization
.
Water Resources Management
30
(
7
),
2143
–
2160
. .
Google Scholar
Crossref
Search ADS
 
Nagesh Kumar
D.
 &
Janga Reddy
M.
2007
Multipurpose reservoir operation using particle swarm optimization
.
Journal of Water Resources Planning and Management
133
(
3
),
192
–
201
.
Google Scholar
Crossref
Search ADS
 
Norouzi
H.
,
Bazargan
J.
,
Azhang
F.
 &
Nasiri
R.
2022a
Experimental study of drag coefficient in non-darcy steady and unsteady flow conditions in rockfill
.
Stochastic Environmental Research and Risk Assessment
36
(
2
),
543
–
562
.
Google Scholar
Crossref
Search ADS
 
Norouzi
H.
,
Hasani
M. H.
,
Bazargan
J.
 &
Shoaei
S. M.
2022b
Estimating output flow depth from Rockfill Porous media
.
Water Supply
22
(
2
),
1796
–
1809
.
Google Scholar
Crossref
Search ADS
 
Sadeghian
J.
2013
Analysis of Radial Flows in Coarse Alluvial Beds
.
PhD Thesis
,
College of Agriculture and Natural Resources, University of Tehran, Tehran, Iran
.
(In Persian)
.
Google Scholar
 
Sadeghian
J.
,
Khayat Kholghi
M.
,
Horfar
A.
 &
Bazargan
J.
2013
Comparison of binomial and power equations in radial non-Darcy flows in coarse porous media
.
Journal of Water Sciences Research
5
(
1
),
65
–
75
.
Google Scholar
 
Safarian
M.
,
Norouzi
H.
 &
Bazargan
J.
2021
Study of hydraulic gradient and velocity changes of unsteady flow through coarse porous media
.
International Journal of River Basin Management
1
–
14
. https://doi.org/10.1080/15715124.2021.1879096.
Google Scholar
 
Sedghi-Asl
M.
 &
Ansari
I.
2016
Adoption of extended dupuit–Forchheimer assumptions to non-darcy flow problems
.
Transport in Porous Media
113
(
3
),
457
–
469
.
Google Scholar
Crossref
Search ADS
 
Sedghi-Asl
M.
,
Rahimi
H.
,
Farhoudi
J.
 &
Samani
J. M. V.
2010
Analysis of the water surface profiles through coarse porous medium
.
Iranian Water Researches Journal
4
(
7
),
81
–
88
.
(In Persian)
.
Google Scholar
 
Shayannejad
M.
 &
Ebrahimi
A.
2020
Hydraulic investigation of non Darcy radial flow in unconfined aquifers in steady state
.
Iranian Journal of Irrigation & Drainage
13
(
6
),
1580
–
1588
.
(In Persian)
.
Google Scholar
 
Stephenson
D. J.
1979
Rockfill in Hydraulic Engineering. Elsevier, Philadelphia, PA, USA
.
Venkataraman
P.
 &
Rao
P. R. M.
2000
Validation of Forchheimer's law for flow through porous media with converging boundaries
.
Journal of Hydraulic Engineering
126
(
1
),
63
–
71
.
Google Scholar
Crossref
Search ADS
 
Wang
S. J.
,
Nguyen
Q. C.
,
Lu
Y. C.
,
Doyoro
Y. G.
 &
Tran
D. H.
2022
Evaluation of geological model uncertainty caused by data sufficiency using groundwater flow and land subsidence modeling as example
.
Bulletin of Engineering Geology and the Environment
81
(
8
),
1
–
21
.
Google Scholar
 
Ward
J. C.
1964
Turbulent flow in porous media
.
Journal of the Hydraulics Division
90
(
5
),
1
–
12
.
Google Scholar
Crossref
Search ADS
 
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