## Abstract

Hydraulic conductivity is a parameter dictating groundwater recharge, having dependability on factors related to aquifer properties such as particle size, shape, degree of compaction, grain size distribution and fluid flow properties like viscosity and specific weight. The present study is focused on the effect of the grain size distribution of the particles of the aquifer material on its permeability. In order to investigate variation of permeability with respect to the grain size distribution, experimental investigations are conducted on natural borehole samples and those prepared by mixing borehole samples with known quantities of marble chips within a laminar flow regime. A power function model is developed for the estimation of permeability based on grain size distribution parameters *σ* (standard deviation) and *D*_{50} (median grain size). The results from the developed model show good agreement with experimental data as the values of *R*^{2}, RMSE and MAE for the model are (0.99, 0.007, 0.005) for 5.08 cm dia., (0.99, 0.005, 0.004) for 10.16 cm dia. and (0.97, 0.004, 0.003) for 15.24 cm dia. permeameters respectively. The developed power function model provides an efficient tool to estimate the yield of wells, seepage below earthen structures and design of filters with reasonable accuracy.

## INTRODUCTION

Water is a resource of fundamental importance to mankind and the abstraction of water from the surface and groundwater resources, a prime cause of concern. In order to ensure the constant availability of groundwater, it needs to be recharged periodically by natural or artificial means (Linsley *et al*. 1992). Groundwater recharge and extraction are dependent on the transmitting capacity of substrata or the bed. One of the quantitative measures of the transmitting capacity is permeability or hydraulic conductivity.

Hydraulic conductivity (Cabalar & Akbulut 2016) is the ease with which water flows through a given porous medium, and is found to be dependent invariably both on the physical properties of flowing water and the characteristics of the transmitting medium (Najafzadeh *et al*. 2017a, 2017b). In natural occurrences, the physical properties of the flowing fluid, i.e. viscosity and specific weight, are practically constant and thus the permeability analogous to hydraulic conductivity may be considered to be a function of the properties of the medium alone (Urumović & Urumović 2014). Such medium properties include the particle size, shape, structure, degree of compaction and grain size distribution (Elhakim 2016; Zięba 2017).

In the past, various researchers such as Fancher & Lewis (1933), Bakhmeteff & Feodoroff (1937), Coulson (1949) and Szymkiewicz (2013) applied theoretical and semi-theoretical approaches to modelling the fluid flow through porous media for permeability measurement. Solutions rendered by these studies to flow problems through porous media have certain limitations. One of the difficulties is to control all other factors that influence the permeability while studying the effect of one particular factor, e.g., standard deviation about the mean, geometric standard deviation or dispersion measure such as (*D*_{50} – *D*_{10}) (Harleman *et al.* 1963). Another constraint is the exact simulation of the groundwater flow in strata, keeping its infinite extent intact (Zahiri & Najafzadeh 2017). Because of the small-scale replication of field conditions, a ‘boundary effect’ or ‘wall effect’ is introduced, which alters the values of flow parameters (Kango *et al.* 2017). Graton & Fraser (1935), Rose & Rizk (1949), Franzini (1965), Dudgeon (1967) and others in their studies suggested that the two major influences of the wall on the fluid flow through porous media are (1) the packing disturbance which leads to greater porosity in the immediate vicinity of the wall, subsequently reducing the number of particles blocking the fluid, resulting in greater flow through the walls, dominantly for turbulent flow (Franzini 1965); and (2) the opposing effect of greater shearing stress of the fluid at the wall, which invariably increases resistance to flow, showing greater influence in the laminar flow region (Blake 1922; Bakhmeteff & Feodoroff 1937).

The present study focuses on grain size distribution, which has significant influence on the permeability of the aquifer. Standard deviation (*σ*) and median grain size (*D*_{50}) are two significant parameters representing grain size distribution that are easily obtained for any particular material and substantially influence the behavior of the grain size distribution curves (Hazen 1893). The principal aspect of the present study is to develop and evaluate an empirical relation between permeability and two parameters i.e., median grain size (*D*_{50}) and standard deviation (*σ*), based on experimental investigations (Masch & Denny 1966).

## MATERIALS AND METHODS

The experiments are performed on 11 samples chosen for the study, three of which are natural borehole samples named A, B and C (from a borehole with depth variation from 10 m to 100 m) and the rest are prepared in the laboratory by mixing different quantities of marble chips in the above three samples. Marble chips were used to make the samples representative of the actual borehole material. The details of the samples are given in Table 1.

Sample no. . | Mixing amount . | D_{50} (mm)
. | σ (mm)
. |
---|---|---|---|

1 | Sample B: borehole sand | 0.325 | 0.1162 |

2 | Sample C: borehole sand | 0.320 | 0.1548 |

3 | Sample B (2 kg) + white marble chips of 4 mm size (10 g) | 0.320 | 0.2841 |

4 | Sample B (2 kg) + white marble chips of 4 mm size (40 g) | 0.320 | 0.5287 |

5 | Sample B (2 kg) + white marble chips of 4 mm size (100 g) | 0.320 | 0.8114 |

6 | Sample C (2 kg) + yellow marble chips of 6.3 mm size (50 g) | 0.320 | 0.9483 |

7 | Sample C (2 kg) + yellow marble chips of 6.3 mm size (100 g) | 0.320 | 1.8119 |

8 | Sample C (2 kg) + yellow marble chips of 6.3 mm size (200 g) | 0.320 | 1.8119 |

9 | Sample A (2 kg) + black marble chips of 8 mm size (200 g) | 0.320 | 2.3203 |

10 | Sample A (2 kg) + black marble chips of 8 mm size (500 g) | 0.320 | 3.4394 |

11 | Sample A (2 kg) + black marble chips of 8 mm size (1,000 g) | 0.315 | 4.415 |

Sample no. . | Mixing amount . | D_{50} (mm)
. | σ (mm)
. |
---|---|---|---|

1 | Sample B: borehole sand | 0.325 | 0.1162 |

2 | Sample C: borehole sand | 0.320 | 0.1548 |

3 | Sample B (2 kg) + white marble chips of 4 mm size (10 g) | 0.320 | 0.2841 |

4 | Sample B (2 kg) + white marble chips of 4 mm size (40 g) | 0.320 | 0.5287 |

5 | Sample B (2 kg) + white marble chips of 4 mm size (100 g) | 0.320 | 0.8114 |

6 | Sample C (2 kg) + yellow marble chips of 6.3 mm size (50 g) | 0.320 | 0.9483 |

7 | Sample C (2 kg) + yellow marble chips of 6.3 mm size (100 g) | 0.320 | 1.8119 |

8 | Sample C (2 kg) + yellow marble chips of 6.3 mm size (200 g) | 0.320 | 1.8119 |

9 | Sample A (2 kg) + black marble chips of 8 mm size (200 g) | 0.320 | 2.3203 |

10 | Sample A (2 kg) + black marble chips of 8 mm size (500 g) | 0.320 | 3.4394 |

11 | Sample A (2 kg) + black marble chips of 8 mm size (1,000 g) | 0.315 | 4.415 |

*D*

_{50}, is ascertained using standard guidelines. The standard deviation of each sample is calculated using the relation (Garde & Raju 2000): where

*σ*is standard deviation,

*D*

_{50}is the diameter of grain for which 50% of the material is finer,

*D*

_{i}is the diameter of material retained on a particular sieve and is the fraction of total material, i.e. retained on the sieve, corresponding to

*D*.

_{i}Specific gravity tests are conducted on the sample material using a pycnometer method to determine the porosity. The hydraulic investigations are conducted to study the resistance to the flow of water offered by a given sample material in 5.08 cm, 10.16 cm and 15.24 cm internal diameter vertical flow constant head permeameters, as shown in Figure S1 (available with the online version of this paper) (ASTM 2011; Kango *et al.* 2017). In the case of 5.08 cm, 10.16 cm and 15.24 cm diameter permeameters, 3.3 kg, 12 kg and 24 kg of sample is used respectively for each test run. Each time the required quantity of material is weighed and divided in ten equal parts. Each part of the sample is given 20 blows in the permeameter before the next layer is placed over it. The height of the sample in the permeameter is noted down. After packing the permeameter and leveling the top of the material, the coupling on the permeameter outer wall is fitted. The material is saturated fully for 12 hours. Outlet valves are slowly closed so that when water enters the manometer tubes, the air is removed. After this, the discharge using the outlet valve and corresponding pressure drop readings for the test length of the sample are recorded from the manometer.

### Sample size and properties

Sieve analysis for different samples is carried out and a plot between percentage finer and sieve size is prepared as shown in Figure 1. The mean size and standard deviation of the particles for sample no. 1 to 11 is detailed in Table 1.

### Test runs

The test runs conducted during investigations involve three observations i.e., discharge to the permeameter, pressure drop across the test length of material and water temperature.

In the test runs on the 11 samples (Table 1), for all the three permeameters, observations are taken in receding order of magnitude of discharge and pressure drop. The test run starts with maximum discharge so that if there is any settlement of the bed due to the impact of the jet of incoming water, it is secured in the beginning. The bed is checked at the end of each run for settlement, if any. The porosity of the bed is determined after making allowance for recorded settlement. The discharge is measured in a measuring flask for a certain period, which is recorded with a stop watch. The flow is then reduced for the next observation. This operation continues until the discharge drops to a certain minimum measurable value. The temperature of the water is recorded at the beginning and at the end of each run.

## RESULTS AND DISCUSSION

Experimental investigations on different samples are conducted to study the interrelation of different aquifer parameters and the effect of parameters on permeability for permeameters of different diameter. Results obtained through investigations are plotted to study and analyze the variation of parameters.

On the basis of experimental investigations, analysis of the variation of friction factor (*Fr*) with Reynolds number (*Re*), and permeability (*K*) with median grain size (*D*_{50}), standard deviation (*σ*), ratio of *σ* to *D*_{50} (*σ*/*D*_{50}) is carried out.

### Variation of Fr vs Re

Variation between *Fr* (friction factor) and *Re* (Reynolds number) for the various samples on permeameter diameters 5.08 cm, 10.16 cm and 15.24 cm have been studied and plotted on a logarithmic scale, however for brevity, the variation for only the 5.08 cm dia. permeameter is shown in Figure 2. A straight line variation indicates that the flow is in the laminar regime with value of *Re* less than 1.

### Variation of K vs D _{50}

The range of *D*_{50} of the samples used in the investigations is 0.315 mm to 0.325 mm, which is small. Variation of *K* with *D*_{50} for different diameter permeameters is shown in Figure 3. The value of *K* varies from 0.42 to 0.071 mm/sec, 0.315 to 0.045 mm/sec and 0.266 to 0.036 mm/sec for pipe diameters 5.08 cm, 10.16 cm and 15.24 cm respectively.

### Variation of K vs σ and σ /D _{50}

The variation of *K* vs standard deviation (*σ*) is plotted for different diameter permeameters i.e., 5.08 cm, 10.16 cm and 15.24 cm in Figure 4. The value of *σ* for the samples used ranges from 0.1162 mm to 4.415 mm, which is a large variation and hence a great variability of *K* with respect to *σ* is observed in the plot between *K* and *σ*.

Figure 4 clearly shows the effect of *σ* on the permeability. From Figure 4 it is clear that as the value of *σ* increases, permeability decreases. As the value of *K* decreases and tends to zero, the curves asymptotically approach the intercept on the *y*-axis.

Figure 5 shows the variation of *K* vs *σ*/*D*_{50} for an almost constant mean diameter (*D*_{50}). It is observed from Figure 5 that as the value of *σ*/*D*_{50} increases, the permeability decreases. This trend is in agreement with the findings of Krumbein & Monk (1943), which establish that for a constant mean diameter the permeability decreases as the standard deviation increases. In the *σ*/*D*_{50} range which is used in the present work the curve is concave upwards and asymptotically touches a lower limiting value. Investigations on the variation between *K* and *σ*/*D*_{50} for an almost constant mean diameter (*D*_{50}) provide the basis for the development of a power function model for estimation of permeability.

#### Development of power function model based on relationship between permeability and statistical parameters

For the estimation of permeability, an empirical relationship i.e., a power function model is developed involving the parametrical ratio *σ*/*D*_{50} for almost constant *D*_{50}.

The developed statistical power function model contains coefficients of 0, 1, 2, 3 and 4 degree of the parameter *σ*/*D*_{50}. The relationship is developed by using the principle of least squares.

where

*ω*= factor that takes into account the effect of the extent of the medium, packing of material near the solid wall, roughness of confinement and particle roughness= 1.25 for pipe dia. 5.08 cm

= 0.95 for pipe dia. 10.16 cm

= 0.81 for pipe dia. 15.24 cm.

Factor *ω* of the power function model is introduced, which gives the cumulative effect of the extent of the medium, packing of material near the solid wall, roughness of confinement and particle roughness. It can be observed that the numerical value of *ω* is maximum for the minimum dia. permeameter and minimum for the maximum dia. permeameter. This observation indicates that as the extent of the material increases, the magnitude of *ω* is insignificant.

*K*

_{exp}) and the model-generated (

*K*

_{emp}) permeability values, as shown in Figure S4 (available with the online version of this paper). The values of the percentage error between the two are found to be less than 6.67% in any case, which establishes the efficacy of the power function model developed in the present study.

Sample no. . | σ/D_{50}
. | Permeameter diameter 5.08 cm . | Permeameter diameter 10.16 cm . | Permeameter diameter 15.24 cm . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

K_{exp}
. | K_{emp}
. | % error . | K_{exp}
. | K_{emp}
. | % error . | K_{exp}
. | K_{emp}
. | % error . | ||

1 | 0.358 | 0.42 | 0.405 | 3.54 | 0.315 | 0.308 | 2.25 | 0.266 | 0.263 | 1.31 |

2 | 0.484 | 0.387 | 0.391 | −0.97 | 0.295 | 0.297 | −0.67 | 0.254 | 0.253 | 0.31 |

3 | 0.888 | 0.335 | 0.348 | −4.01 | 0.252 | 0.265 | −5.08 | 0.216 | 0.226 | −4.53 |

4 | 1.652 | 0.282 | 0.283 | −0.25 | 0.207 | 0.215 | −3.80 | 0.175 | 0.183 | −4.68 |

5 | 2.536 | 0.236 | 0.226 | 4.24 | 0.174 | 0.172 | 1.29 | 0.147 | 0.146 | 0.37 |

6 | 2.963 | 0.208 | 0.205 | 1.63 | 0.157 | 0.156 | 0.95 | 0.134 | 0.133 | 1.05 |

7 | 4.109 | 0.167 | 0.162 | 3.07 | 0.126 | 0.123 | 2.36 | 0.108 | 0.105 | 2.88 |

8 | 5.662 | 0.132 | 0.127 | 4.02 | 0.096 | 0.096 | −0.30 | 0.081 | 0.082 | −1.36 |

9 | 7.215 | 0.108 | 0.104 | 3.32 | 0.075 | 0.079 | −5.81 | 0.062 | 0.064 | −3.23 |

10 | 10.748 | 0.078 | 0.076 | 2.19 | 0.05 | 0.053 | −6.00 | 0.039 | 0.041 | −5.13 |

11 | 14.016 | 0.071 | 0.073 | −2.82 | 0.045 | 0.048 | −6.67 | 0.036 | 0.038 | −5.56 |

Sample no. . | σ/D_{50}
. | Permeameter diameter 5.08 cm . | Permeameter diameter 10.16 cm . | Permeameter diameter 15.24 cm . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

K_{exp}
. | K_{emp}
. | % error . | K_{exp}
. | K_{emp}
. | % error . | K_{exp}
. | K_{emp}
. | % error . | ||

1 | 0.358 | 0.42 | 0.405 | 3.54 | 0.315 | 0.308 | 2.25 | 0.266 | 0.263 | 1.31 |

2 | 0.484 | 0.387 | 0.391 | −0.97 | 0.295 | 0.297 | −0.67 | 0.254 | 0.253 | 0.31 |

3 | 0.888 | 0.335 | 0.348 | −4.01 | 0.252 | 0.265 | −5.08 | 0.216 | 0.226 | −4.53 |

4 | 1.652 | 0.282 | 0.283 | −0.25 | 0.207 | 0.215 | −3.80 | 0.175 | 0.183 | −4.68 |

5 | 2.536 | 0.236 | 0.226 | 4.24 | 0.174 | 0.172 | 1.29 | 0.147 | 0.146 | 0.37 |

6 | 2.963 | 0.208 | 0.205 | 1.63 | 0.157 | 0.156 | 0.95 | 0.134 | 0.133 | 1.05 |

7 | 4.109 | 0.167 | 0.162 | 3.07 | 0.126 | 0.123 | 2.36 | 0.108 | 0.105 | 2.88 |

8 | 5.662 | 0.132 | 0.127 | 4.02 | 0.096 | 0.096 | −0.30 | 0.081 | 0.082 | −1.36 |

9 | 7.215 | 0.108 | 0.104 | 3.32 | 0.075 | 0.079 | −5.81 | 0.062 | 0.064 | −3.23 |

10 | 10.748 | 0.078 | 0.076 | 2.19 | 0.05 | 0.053 | −6.00 | 0.039 | 0.041 | −5.13 |

11 | 14.016 | 0.071 | 0.073 | −2.82 | 0.045 | 0.048 | −6.67 | 0.036 | 0.038 | −5.56 |

Different statistical parameters, i.e., *R*^{2}, *R* (correlation coefficient), RMSE (root mean square error), BIAS, MAE (mean absolute error) and SI (scatter index) (Najafzadeh *et al*. 2017a, 2017b) are determined to compare experimentally obtained (*K*_{exp}) and model-obtained (*K*_{emp}) permeability values. The *R*^{2}, *R*, RMSE, BIAS, MAE, SI values for the model-obtained and experimental values are 0.9964, 0.999, 0.007, −0.002, 0.005, 0.104 for 5.08 cm, 0.9967, 0.999, 0.005, 0.001, 0.004, 0.116 for 10.16 cm, and 0.9707, 0.998, 0.004, 0.001, 0.003, 0.095 for 15.24 cm diameter permeameters respectively. The range of obtained statistical parameters indicates a high degree of correlation between experimental and model-obtained permeability values and further strengthens the usefulness of the developed power function model.

## CONCLUSIONS

The present study is focused on developing a power function model for the estimation of permeability (*K*) involving grain size distribution parameters *D*_{50} and *σ*. The model incorporates a factor ‘*ω*’, which takes into account the effect of the extent of the medium, packing of material near the solid wall, roughness of confinement and particle roughness. The minimum value of *ω* for the maximum dia. permeameter indicates that as the extent of material increases, the magnitude of *ω* is insignificant, as in the case of a natural aquifer. A high degree of correlation (*R*^{2}, *R*), low error statistics (RMSE, MAE and BIAS), and satisfactory scatter index with maximum variability of 6.67% between experimental and model-obtained values of permeability establish the efficacy of the developed power function model. This study recommends further research on the effect of particle roughness and roughness of confinement on hydraulic conductivity.

ASTM Annual CDs of Standards, vol. 4 (8), West Conshohocken, PA, USA