# Better-Fitted Probability of Hydraulic Conductivity for a Silty Clay Site and Its Effects on Solute Transport

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{−6}m/d to 1.6 × 10

^{−1}m/d, which corresponded to the lithology of silty clay. The magnitude and the range of the hydraulic conductivity increased with the depth. Five probability distribution models were tested with the experimental probability, indicating that a Levy stable distribution was more matched than the log-normal, normal, Weibull or gamma distributions. A simple analytical model and a Monte Carlo technique were used to inspect the effect of the silty clay hydraulic conductivity field on the statistical behavior of the solute transport. The Levy stable distribution likely generates higher peak concentrations and lower peak times compared with the widely-used log-normal distribution. This consequently guides us in describing the transport of contaminations in subsurface mediums.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sampling and Lithology

^{2}at an areal density of 8 boreholes/km

^{2}. The average distance of each pair of two boreholes was 345 m. Each borehole was drilled up to a depth of 10 m, and different samples (3~5 cores per borehole) were collected according to the diversity of the lithology. The cylinder sampler used was 20 cm in length and 110 mm in inner diameter. The undisturbed samples were then moved to a laboratory at Hohai University to test their hydraulic conductivities. In terms of the lithology of sediments in the study area dominated by clay and silty clay, a recommended falling head method [1] applied to the fine material was used to analyze the hydraulic conductivity of the core using a permeameter (TST-55; Φ: 61.8 mm × 40 mm; made in China). The infiltration process for low-K sediments was quite slow, and a successful permeameter test averagely took more than 6 hours in our study. Therefore, eight permeameters were used to test different clay samples, simultaneously. In total, we finished 212 falling head permeameter tests to investigate the hydraulic conductivity of the reservoir sediments.

#### 2.2. Method of Statistics

#### 2.3. Random Modeling of Solute Transport to K Distribution

^{2}), u is the flow velocity (m/d), n

_{e}is the effective porosity, and D

_{L}is the longitudinal dispersion coefficient (m

^{2}/d).

_{p}) when the solute reaches the maximum concentration at a distance from the injection point, and the corresponding concentration is called the peak concentration (C

_{p}). The migration distance x is set when the first derivative equals zero (dC/dt = 0); the peak time (t

_{p}) is written as the following:

_{p}) is acquired via the peak time (t

_{p}):

_{p}and C

_{p}. In terms of unclosed forms of the Levy stable distribution, the CMS (Chambers-Mallows-Stuck) method involved in a MATLAB toolbox developed by Liang and Chen [23] was applied herein to generate the Levy random number.

## 3. Results and Discussion

#### 3.1. Hydraulic Conductivity of the Silty Clay Medium

^{−4}m/d for the first shallow plain fill layer, to 1.6 × 10

^{−1}m/d for the fourth marine silty clay layer. The range of K values for each layer apparently increased from the shallow to the deep. The average hydraulic conductivity had the same pattern, with an increasing range from 6.7 × 10

^{−5}m/d to 3.9 × 10

^{−3}m/d. The standard deviation also increased from 1.68 × 10

^{−4}m/d for the first layer, to 1.92 × 10

^{−2}m/d for the fourth layer.

#### 3.2. Probability Density Function of the Low Hydraulic Conductivity Field

^{−6}to 5.93 × 10

^{−5}, respectively. As $\alpha $ decreases, the frequency of sudden large jumps in the random field increases [32]. The parameter $\gamma $ is known as the scale parameter. It is equal to half the variance when $\alpha $ = 2, and plays a similar role for $\alpha $ < 2 (i.e., it is a measure of the width of the distribution). As $\gamma $ increases, the magnitudes of the sudden large jumps increase [33]. The skewness parameter $\beta $ values for layers 2 and 4 were the same, equal to 1 (the maximum for this parameter), indicating extremely right-skewed distributions for these datasets. $\beta $ for layer 3 equaled 0.39, and the degree of skewness was weaker than for the other two layers. $\delta $, the shift parameter, represents the centering of the distribution, and it is equal to the mean of the distribution only when $\alpha $ > 1 and $\beta $ = 0. The differences of $\delta $ in Table 2 show the different centering of the four layers.

#### 3.3. The Effects of K PDFs on Solute Transport

_{e}) was 0.3, and the longitudinal dispersion coefficient (D

_{L}) was 0.05 m

^{2}/d. Although the porosity and dispersion coefficients were indeed of high variations for subsurface mediums, even for the silt and clay materials, we only focused the influence on the solute transport from the variation of K rather than the porosity and dispersion coefficients. The hydraulic gradient was set to a constant of 1.0, and the velocity (u) become a single-valued function of K; then the explicit expressions of t

_{p}-K and C

_{p}-K could be easily obtained from Equations (5) and (6). The monotonic variations, both of increasing C

_{p}and decreasing t

_{p}, with the increasing K value were observed from Figure 4. High t

_{p}and low C

_{p}values were consistent with the small K, and low t

_{p}and high C

_{p}values corresponded to a strong permeability of high K.

_{p}) and peak concentration (C

_{p}) could be simulated by Equations (5) and (6).

#### 3.3.1. Effects of K PDFs on Peak Time

^{5}d) was a little greater than that simulated by the log-normal distribution (788.3 d, 9.9 × 10

^{4}d). The minimum values especially between these two distributions were significantly different. This indicated that the Levy stable distribution could generate greater random values of K and then result in a smaller peak time. Additionally, the mean and median values of simulated peak time values from the Levy stable distribution (8.4 × 10

^{4}d and 9.6 × 10

^{4}d) were apparently greater than those of the log-normal distribution (7.7 × 10

^{4}d and 8.8 × 10

^{4}d). The interquartile ranges (IQRs) for the Levy-stable and log-normal distributions were 1.1 × 10

^{4}d and 3.7 ×10

^{4}d, respectively. From Figure 5, the simulation results of peak times from the Levy stable distribution are gathered in the area of greater value, and random numbers of K generated by the Levy stable distribution concentrate in the area of lower value. These results agreed with the high-peak and heavy-tail characteristics of the Levy stable distribution.

^{4}d. Equation (4) explicitly indicates the negative correlation between peak time and flow velocity. In this study, velocity (u) was proportional to K through the assumed 1-D model, and the peak time (t

_{p}) and K had a negative correlation. Section I reflects the simulation result when K was relatively large. As shown in Figure 5, the results calculated by the Levy stable distribution indicated that the probability of small peak time events (red line) was clearly higher than that calculated by the log-normal distribution (black line).

^{4}d and 9.6 × 10

^{4}d, and Section II is shown as the middle part of Figure 5. According to the correlation between the peak time and K mentioned above, this section corresponds to the range for which K was moderate (greater than that in Section I and less than that in Section III). The simulation results indicated that the log-normal distribution achieved a higher probability than the Levy stable distribution. The probability of random events with moderate peak times simulated by the Levy stable distribution was less than that for the log-normal distribution. The peak time of Section III was the highest, representing the smallest K. The simulation results indicated that the probability of high peak time events simulated by the Levy stable distribution was larger than for the log-normal distribution.

_{p}calculated by the simple analytical model had more non-uniform characteristics. The probabilities of a greater peak time occurring (Section I) and smaller values occurring (Section III) were both larger than for under the widely used log-normal distribution. However, the comparison of moderate peak time events (Section II) revealed that the probability calculated by the log-normal distribution was clearly greater than that by the Levy stable distribution. Our results indicate that simulated t

_{p}values from log-normally-distributed K are likely ranged within the medium level, and the Levy stable distribution can produce higher or lower extremes much more easily.

#### 3.3.2. Effects of K PDFs on Peak Concentration

_{p}) simulated by the Levy stable distribution (8.1 g/L, 594.7 g/L) was significantly greater than that simulated by the log-normal distribution (8.1 g/L, 155.9 g/L). The maximum values between these two distributions were significantly different. The average value of the simulated peak concentration from the Levy stable distribution (17.5 g/L) was slightly greater than that from the log-normal distribution (13.5 g/L). However, the median of the Levy stable distribution (9.8 g/L) was slightly less than the counterpart of the log-normal distribution (11.0 g/L). Although the standard deviation of C

_{p}from the Levy stable distribution (43.9 g/L) was much greater than that from the log-normal distribution (7.4 g/L), the Levy stable distribution could generate a concentrated C

_{p}(IQR = 1.9 g/L) compared with the log-normal distribution (IQR = 5.9 g/L), as seen from the shorter box in Figure 7.

_{p}shown in Figure 6, three different sections of C

_{p}are discriminated in Figure 8 (Sections I and II are zoomed in Figure 8B). According to Equation (3), the peak concentration (C

_{p}) had a positive correlation with the flow velocity (u). Since the flow velocity (u) had a positive correlation with the hydraulic conductivity (K), a positive correlation between C

_{p}and K could be inferred. The cumulative probability of random events in Sections I and II contributed approximately 90% of all random samples. In Sections I and II, the difference of cumulative frequency distributions of C

_{p}under two different distributions was small, as shown by Figure 8A. Approximately 90% of simulated peak concentrations were located in the range of 8–25 g/L for both of the two distributions. In Section III, the peak concentration from the log-normal distribution achieved the highest value of 155.9 g/L, and the greater C

_{p}(>155.9 g/L) only appeared under the assumption of Levy-stable-distributed K. The distribution of C

_{p}under the Levy stable distribution was much flatter than that under the log-normal distribution. The Levy stable distribution had better performance to simulate extreme conditions and to reveal the heavy-tailed characteristic of hydraulic conductivity, especially.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 6.**Cumulative frequency distributions of t

_{p}from Levy stable and log-normal distribution K values.

**Figure 8.**Cumulative frequency distribution of C

_{p}from Levy stable and log-normal distribution K values (8A shows the full extent and 8B shows the part of C

_{p}less than 25 g/L).

Geologic Material | First Layer | Second Layer | Third Layer | Fourth Layer | |
---|---|---|---|---|---|

K (m/d) | Min | 2 × 10^{−6} | 4 × 10^{−6} | 9 × 10^{−6} | 6 × 10^{−6} |

Max | 6.8 × 10^{−4} | 1.6 × 10^{−3} | 1.3 × 10^{−2} | 1.6 × 10^{−1} | |

Mean | 6.7 × 10^{−5} | 9.5 × 10^{−5} | 5.6 × 10^{−4} | 3.9 × 10^{−3} | |

SD | 1.68 × 10^{−4} | 2.79 × 10^{−4} | 2.12 × 10^{−3} | 1.92 × 10^{−2} | |

Sample size | 18 | 46 | 36 | 112 | |

Lithology | Plain fill | Continental silty clay | Mud–silt clay | Marine silty clay |

Type of PDF | Second Layer | Third Layer | Fourth Layer | |
---|---|---|---|---|

Normal | K–S | No | No | No |

A–D | No | No | No | |

Fitted Parameters | $\mu $ = 9.55 × 10^{−5} | $\mu $ = 5.63 × 10^{−4} | $\mu $ = 0.0039 | |

$\sigma $ = 2.70 × 10^{−4c} | $\sigma $ = 2.1 × 10^{−3} | $\sigma $ = 0.0192 | ||

Log-normal | K–S | Yes | Yes | No |

A–D | Yes | Yes | No | |

Fitted Parameters | $\mu $ = −10.59 | $\mu $ = −9.08 | $\mu $ = −8.25 | |

$\sigma $ = 1.33 | $\sigma $ = 1.44 | $\sigma $ = 1.91 | ||

Levy | K–S | Yes | Yes | Yes |

A–D | Yes | Yes | Yes | |

Fitted Parameters | $\alpha $ = 0.46 | $\alpha $ = 0.52 | $\alpha $ = 0.52 | |

$\beta $ = 1 | $\beta $ = 0.39 | $\beta $ = 1 | ||

$\gamma $ = 4.48 × 10^{−6} | $\gamma $ = 2.10 × 10^{−5} | $\gamma $ = 5.93 × 10^{−5} | ||

$\delta $ = 7.47 × 10^{−6} | $\delta $ = 5.30 × 10^{−5} | $\delta $ = 2.10 × 10^{−6} | ||

Gamma | K–S | No | No | No |

A–D | No | No | No | |

Fitted Parameters | a = 0.48 | a = 0.41 | a = 0.26 | |

b = 0.0002 | b = 0.0014 | b = 0.015 | ||

Weibull | K–S | No | Yes | No |

A–D | No | No | No | |

Fitted Parameters | a = 0.0001 | a = 0.0002 | a = 0.0007 | |

b = 0.6 | b = 0.56 | b = 0.43 |

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**MDPI and ACS Style**

Lu, C.; Qin, W.; Zhao, G.; Zhang, Y.; Wang, W.
Better-Fitted Probability of Hydraulic Conductivity for a Silty Clay Site and Its Effects on Solute Transport. *Water* **2017**, *9*, 466.
https://doi.org/10.3390/w9070466

**AMA Style**

Lu C, Qin W, Zhao G, Zhang Y, Wang W.
Better-Fitted Probability of Hydraulic Conductivity for a Silty Clay Site and Its Effects on Solute Transport. *Water*. 2017; 9(7):466.
https://doi.org/10.3390/w9070466

**Chicago/Turabian Style**

Lu, Chengpeng, Wei Qin, Gang Zhao, Ying Zhang, and Wenpeng Wang.
2017. "Better-Fitted Probability of Hydraulic Conductivity for a Silty Clay Site and Its Effects on Solute Transport" *Water* 9, no. 7: 466.
https://doi.org/10.3390/w9070466