# Method of Relating Grain Size Distribution to Hydraulic Conductivity in Dune Sands to Assist in Assessing Managed Aquifer Recharge Projects: Wadi Khulays Dune Field, Western Saudi Arabia

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Methods

#### 2.1. Site Description

#### 2.2. Field Study and Laboratory Methods

^{2}and were analyzed only for grain size distribution (Figure 2B, blue markers).

**Figure 2.**Wadi Khulays dune field sample locations (black arrows show the predominant wind direction). Green markers show the location of dunes selected for full sampling (analysis of grain size, porosity and hydraulic conductivity across the dune). Blue markers show a second sampling for grain size analysis only from dune crests. The blue region corresponds to the analysis shown in Figure 8. Inset: Diagram showing the sample location across individual dunes.

#### 2.3. Mathematical Analysis

_{10}is the effective grain diameter (at which 10% of the sample is finer), g is the gravitational acceleration and $\nu $ is the kinematic viscosity of water.

_{60}/d

_{10}(d

_{60}is the diameter at which 60% of the sample is finer), P

_{1}and P

_{2}are the fitting parameters to be optimized. The inverse problem was formulated in the sense of root mean square error and the objectivity function was accordingly defined as:

**b**= [P

_{1},P

_{2}] is the parameter vector to be estimated, and, n is the number of samples. An inversion the first twenty-five measurements was used to estimate the fitting parameters P

_{1}and P

_{2}. To properly deal with the nonlinearity of the objective function during minimization and, in particular, avoid being trapped in local minima, we adopted a global optimization approach. Hence, we used the shuffled complex evolution (SCE-UA) algorithm for the global optimization. This global optimization algorithm has been widely used in hydrologic modeling [34], and proved to be consistent and efficient for finding the global optimum parameter values of integrated hydrogeophysical and hydrologic models [35,36].

## 3. Results

#### 3.1. Grain Size, Porosity, and Hydraulic Conductivity Analyses

_{10}, mud percentage, measured porosity and measured hydraulic conductivity of the 50 samples collected across the individual windward and leeward dunes are summarized in Figure 3. The grain size statistical properties d

_{10}, mud (%), standard deviation, skewness, and kurtosis, and the measured hydraulic conductivity vary in a regular pattern across the dune profile, while mean grain diameter does not vary in the same pattern (Figure 3). The dunes crests have significantly higher hydraulic conductivities and d

_{10}values, corresponding to the lowest percentage of mud content. Porosity of the dune sands was found to be relatively uniform with a mean of 0.39 and a standard deviation of 0.02. The red-cross marks on the figure show the location of outlier samples.

**Figure 3.**Summary of grain size and hydraulic properties along the cross-section of the dunes. Red crosses are outlier samples.

#### 3.2. Empirical Hydraulic Conductivity Estimates from Grain Size Distribution

**Figure 4.**Estimated hydraulic conductivity using 21 empirical methods for all Wadi Khulays dune sample compared to the measured values (blue line) (

**A**). (

**B**) Comparison of the estimated hydraulic conductivities to the measured values using the modified methods for dune interior dune sands using the modified equations in Rosas et al. [26].

#### 3.3. Statistical Analyses between Sediment Grain Size Properties and Hydraulic Conductivity

^{2}and p are shown in Table 1. The p-values are reported as −log(p) in order to compare the order of magnitude of the correlations, with a significant correlation being considered to have a value of −log(p) of 2 or more (p < 0.01). Even though the R

^{2}values were very low for all the parameters considered (R

^{2}< 0.4), significant p-values were found for the correlation between the hydraulic conductivity and d

_{10}

^{2}, the standard deviation, skewness and kurtosis of the distribution in phi units (−log

_{2}mm; [52]), as well as for the mud percentage of the sample (fraction <0.0625 mm).

**Table 1.**Correlation coefficients between hydraulic conductivity and grain size distribution parameters.

Correlated Parameters | R^{2} | −log(p-Value) |
---|---|---|

K, d_{10}^{2} | 0.34 | 4.9 |

K, mean | 0.04 | 0.8 |

K, standard deviation | 0.30 | 4.4 |

K, skewness | 0.39 | 6.0 |

K, kurtosis | 0.22 | 3.2 |

K, mud | 0.37 | 5.6 |

Mud, mean | 0.16 | 2.4 |

Mud, standard deviation | 0.72 | 14.2 |

Mud, skewness | 0.45 | 7.1 |

Mud, kurtosis | 0.11 | 1.7 |

K, $F={{d}_{10}}^{2}\text{exp}\left(-Mud\right)$ | 0.56 | 4.8 |

^{2}= 0.72) and negatively correlated with the skewness of the distribution (R

^{2}= 0.45) (Figure 5).

**Figure 5.**Relationship between mud percentage and other grain size distribution statistical parameters.

#### 3.4. Optimization of the Target Equations by Inclusion of a Second Parameter

_{1}and P

_{2}) for Equation (2). Figure 6A,B shows the convergence of the P

_{1}and P

_{2}parameters, respectively, to the global optimum values. For the inversion, the optimization parameter space was set to −1 × 10

^{−2}≤ P

_{1}≤ 1 × 10

^{−2}and −10 ≤ P

_{2}≤ 10. In the first 180 iterations the optimization algorithm searched different parameters set in the provided parameter space and later converged to the optimum values. The stopping criterion for the inversion was specified with respect to the convergence criterion in which an optimum was assumed to be reached when the objective function did not improve by more than 0.01% in 10 successive evolution loops. The estimated parameters P

_{1}and P

_{2}were 4.0 × 10

^{−4}and −0.595.

_{1}and P

_{2}parameter plane. The range of each parameter has been divided into 500 discrete values resulting in 250,000 objective function calculations for the plot. The white star marker corresponds to the solution found by the SCE optimization algorithm. Response surface analysis of the objective function is important as it provides valuable insights into the uniqueness of the inverse solution, the sensitivity of the model to the different parameters, and parameter correlations. The topography of the objective function shows that the parameters P

_{1}and P

_{2}are not correlated and a U-valley can be observed in the direction of the P

_{2}parameter. Furthermore, in the direction of parameter P

_{1}, the V-shape valley of decreasing and increasing trend of the objective function shows that the objective function is more sensitive to the parameter P

_{1}as compared to the P

_{2}.

^{2}and RMSE indicate that the new equation estimates modeled hydraulic conductivity close to the measured hydraulic conductivity. The RMSE improvement is calculated with respect to Equation (2).

Equation | R^{2} | RMSE (m/day) | RMSE Improvement (%) |
---|---|---|---|

Beyer | 0.60 | 3.04 | 25% |

Harleman | 0.59 | 3.01 | 23% |

Hazen | 0.59 | 3.03 | 24% |

Kozeny-Carmen | 0.36 | 3.38 | 38% |

Equation (2) | 0.61 | 2.44 |

## 4. Discussion

_{10}value, the mud percentage and the hydraulic conductivities; dune crests showed higher hydraulic conductivities and d

_{10}values, as well the lowest mud content, which were progressively lower (hydraulic conductivity and d

_{10}) and higher (mud percentage) towards the extremities of the dune.

_{10}value and a function of the porosity [26]. The porosity of the dune sands from the Wadi Khulays dune field was determined to be relatively uniform and therefore the effect of porosity on the hydraulic conductivity could not be evaluated.

**Figure 8.**Spatial distribution of the hydraulic conductivity along the dune crests in a selected site within the dune field.

## 5. Conclusions

_{10}value, which is commonly used in empirical equations for determining the hydraulic conductivity. While the relationship of the hydraulic conductivity and the skewness is significant, its influence can be accounted for by the use of a proxy parameter, the mud content, which showed high correlations with both the skewness and hydraulic conductivity of the samples. Finally, the porosity of the dune sands is very uniform at a high porosity of 0.39, with a standard deviation of 0.02. Therefore, an accurate estimation of the hydraulic conductivity can be obtained by employing an empirical equation using the d

_{10}value, the mud percentage of the sample and two optimized coefficients. This approach can be used to study the spatial distribution of dune sands in larger areas. Application of the new empirical equation developed from this research will improve estimates of hydraulic conductivity for interior dune fields, but may not be applied to coastal dune fields that tend to contain coarser sand, have a higher d

_{10}value, and contain little or no mud.

## Acknowledgments

## Authors Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Lopez, O.M.; Jadoon, K.Z.; Missimer, T.M.
Method of Relating Grain Size Distribution to Hydraulic Conductivity in Dune Sands to Assist in Assessing Managed Aquifer Recharge Projects: Wadi Khulays Dune Field, Western Saudi Arabia. *Water* **2015**, *7*, 6411-6426.
https://doi.org/10.3390/w7116411

**AMA Style**

Lopez OM, Jadoon KZ, Missimer TM.
Method of Relating Grain Size Distribution to Hydraulic Conductivity in Dune Sands to Assist in Assessing Managed Aquifer Recharge Projects: Wadi Khulays Dune Field, Western Saudi Arabia. *Water*. 2015; 7(11):6411-6426.
https://doi.org/10.3390/w7116411

**Chicago/Turabian Style**

Lopez, Oliver M., Khan Z. Jadoon, and Thomas M. Missimer.
2015. "Method of Relating Grain Size Distribution to Hydraulic Conductivity in Dune Sands to Assist in Assessing Managed Aquifer Recharge Projects: Wadi Khulays Dune Field, Western Saudi Arabia" *Water* 7, no. 11: 6411-6426.
https://doi.org/10.3390/w7116411