# Prediction of Permeability Coefficient k in Sandy Soils Using ANN

^{*}

## Abstract

**:**

_{D}, void ratio e and effective soil diameter d

_{10}. The mean relative error and single maximum value of the relative error for the proposed ANN are following: Mean RE = ±4%, Max RE = 7.59%. The use of the ANN to predict the soil permeability coefficient allows the reduction of the costs and time needed to conduct laboratory or field tests to determine this parameter.

## 1. Introduction

^{−2}÷ 1 × 10

^{−10}m·s

^{−1}.

_{10}), porosity and specific surface area are used [4,5]. However, existing empirical formulas ignore the impact of soil structure, permeability anisotropy and the soil grain shape. Research indicates that the parameter k of the same material may differs significantly if it is estimated based on different empirical formulas [6,7]. Therefore, it is crucial to determine the exact empirical formulas and the correctness of the values of the parameters used, which may affect the calculation results. It is also important that the empirical formulas depend on a greater number of parameters of the analysed soil and the exact determination of the parameters used in empirical formulas.

^{−6}m·s

^{−1}), the BAT probe test is used. This test involves the connection of a piezometer to the probe measuring part with a glass water container in which pressure changes are registered. Parameter k is determined on the basis of pressure changes in a function of time. Depending on the degree of filling the pores with water in the soil, the test may be performed under the conditions of water supply (inflow) or outflow from the probe tip. In the laboratory, parameter k is determined using constant or variable gradient methods. Constant-gradient methods, including the Rowe chamber and Trautwein system and ZW-K2 apparatus, are most often used to evaluate the parameter k in well-permeable soils [11]. Variable-gradient methods, including the flow-pump method, Kamienski tube method and modified oedometer with a burette, are used to evaluate the parameter k in low-permeable soils. Of the above-mentioned variable-gradient methods, the flow-pump method is the most commonly used. In this method, the differences in pressure at the top and bottom of the soil sample are measured after establishing a constant water flow velocity in the test sample. The test is continued until the pressure stabilizes in the soil sample.

## 2. Materials and Methods

_{2}—water level in piezometer 2, z

_{1}—water level in piezometer 1, x

_{2}—distance between well and piezometer 2 and x

_{1}—distance between well and piezometer 1.

_{10}was evaluated. The results of the grain size distribution tests indicate that the analysed soils, according to EN ISO 14688-2: 2006 and EN ISO 14688-2: 2006-Ap2: 2012, are FSa-Fine Sand, MSa-Middle Sand and CSa-Coarse Sand. According to the EN ISO 14688-2: 2006 standard, the tested sands were characterized by a gravel fraction of 0 ÷ 19%, sand fraction of 80 ÷ 98%, silt fraction of 0 ÷ 12% and clay fraction of 0 ÷ 2%.

_{D}of tested soils were determined using cone penetrometer tests (CPT). Void ratios e were determined using electrical resistivity measurement. After connecting the electrodes to the power source and the gauge, several dozen measurements were made with a frequency of 12 s for the same place. The arithmetic mean of the measurement results was taken as the final result of the electrical resistance.

_{D}determined in the field studies using the cone penetrometer tests (CPT) were reached [50,51]. The determination of the permeability coefficient was based on the analysis of the consolidation process in the uniaxial state of strain. For this purpose, consolidometer tests were performed using the flow pump method for soil saturation. Tests were performed with a continuous inflow of water at constant gradients of 0.50. Based on the obtained characteristics, the permeability coefficients k were determined by the Taylor method. The values of the parameter k for particular soils obtained with the same gradients did not differ from each other by more than 5%.

_{D}, void ratio e and effective soil diameter d

_{10}of tested soils can be seen in Table 1.

## 3. Results

_{D}on the permeability coefficients k is significant in the tested soils. Higher values of permeability coefficients k were in soils characterized by a lower relative density I

_{D}, higher void ratio e and lower value of effective soil diameter d

_{10}.

_{D}from two test sites (wells no. 1 and 6), are presented in Figure 3.

_{D}, void ratio e and the effective soil diameter d

_{10.}For instance, the difference in parameter k is two-fold in FSa between wells no. 1 and 6. The impact of the analysed parameters on the parameter k is the highest in fine sands (FSa).

## 4. ANN (Artificial Neural Network) Analysis

#### 4.1. Architecture of ANN

_{1}–X

_{N}, H—number of nodes in the hidden layer, M = Y—number of nodes in the output layer.

_{i}—known values of the tests, y

_{i}—predicted values using ANN.

- -
- Relative error for individual cases:$$R{E}_{i}=\left|\frac{{d}_{i}-{y}_{i}}{{d}_{i}}\right|\xb7100\%$$
- -
- Determination coefficient R
^{2}:$${R}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{P}{\left({d}_{i}-{y}_{i}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{P}{\left({d}_{i}-{\overline{d}}_{i}\right)}^{2}}$$ - -
- Mean absolute error:$$MAE=1-\frac{{{\displaystyle \sum}}_{i=1}^{P}\left|{d}_{i}-{y}_{i}\right|}{P}$$
- -
- Root mean squared error:$$RMS=\sqrt{\frac{1}{P}\xb7{\displaystyle \sum}_{i=1}^{P}{\left({d}_{i}-{y}_{i}\right)}^{2}}$$

_{i}—measured value, y

_{i}—predicted value by ANN, ${\overline{d}}_{i}$—measured mean value in subset.

#### 4.2. Data Sets, Training and Testing the ANN

_{A}= 50 cases. Set A was described by five variables: X

_{1}= soil type Є {FSa; MSa, CSa}; X

_{2}= relative density I

_{D}Є {0.22 ÷ 0.92}; X

_{3}= void ratio e Є {0.405 ÷ 0.728%}; and X

_{4}= effective soil diameter d

_{10}Є {0.04 ÷ 0.63} as well as Y = permeability coefficient k (Table 1). Variable X

_{1}was treated as quality variable “one from N” type and required the use of many input neurons because three soil types were used. Set B consisted of data obtained in consolidometer tests and included n

_{B}= 120 cases. Set B was described by independent variables X

_{1}–X

_{4}, and dependent variable Y = k. The development and training of the ANN was carried out with the use of set B. The selected ANN was used to predict permeability coefficient k on the basis of new data from field tests (set A).

^{2}and the lowest mean values of relative errors RE [52].

^{2}) for the developed ANN 6-8-1, in particular, and the subsets training Tr, testing T and validation V of set B are shown in Table 3. Prediction mean relative error Mean RE of the selected ANN 6-8-1 was a maximum of about ±4% in all subsets of data set B.

#### 4.3. ANN Estimation

_{1}, X

_{2}, X

_{3}and X

_{4}(Table 1) were introduced and Y = k values were estimated using the proposed ANN. Measured values of parameter k determined in pumping tests and values of parameter k predicted using developed ANN 6-8-1 are presented in Table 4. The value of the maximum single relative error Max RE was equal to 7.59%.

## 5. Conclusions

_{D}on the permeability coefficients k is significant in the tested soils. Higher values of permeability coefficients k were found in soils characterized by a lower relative density I

_{D}, higher void ratio e and lower value of effective soil diameter d

_{10}.

_{D}, void ratio e and effective soil diameter d

_{10}. The presented ANN estimates the permeability coefficients k with values of determination coefficient R

^{2}= 0.97, mean relative error RE = ±4% and single maximum relative error Max RE = 7.59%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Chemical composition spectra and sample SEM photos showing the structure for FSa, characterized by similar relative density I

_{D}from two test sites (wells no. 1 and 6): (

**a**) FSa from well no. 1, (

**b**) FSa from well no. 6.

**Table 1.**Grain size distribution, relative densities, void ratios and effective soil diameters d

_{10}of soils.

No. of Well | Soil | Fraction-EN ISO 14688-1: 2002; EN ISO 14688-2: 2004 (%) | Relative Density I _{D} (-) | Void Ratio e (-) | Effective Soil Diameter d _{10} (mm) | |||
---|---|---|---|---|---|---|---|---|

Gr | Sa | Si | Cl | |||||

1 | FSa | 0 | 91 | 9 | 0 | 0.49 | 0.542 | 0.07 |

2 | FSa | 1 | 90 | 9 | 0 | 0.67 | 0.591 | 0.07 |

3 | FSa | 0 | 92 | 8 | 0 | 0.61 | 0.498 | 0.08 |

4 | FSa | 1 | 92 | 7 | 0 | 0.64 | 0.523 | 0.09 |

5 | FSa | 2 | 90 | 8 | 0 | 0.41 | 0.656 | 0.08 |

6 | FSa | 0 | 94 | 6 | 0 | 0.54 | 0.599 | 0.10 |

7 | FSa | 1 | 93 | 6 | 0 | 0.51 | 0.587 | 0.10 |

8 | FSa | 0 | 97 | 3 | 0 | 0.56 | 0.486 | 0.17 |

9 | FSa | 0 | 95 | 3 | 2 | 0.39 | 0.705 | 0.17 |

10 | FSa | 0 | 95 | 5 | 0 | 0.50 | 0.589 | 0.12 |

11 | FSa | 0 | 86 | 12 | 2 | 0.35 | 0.712 | 0.04 |

12 | FSa | 2 | 86 | 10 | 2 | 0.68 | 0.506 | 0.063 |

13 | FSa | 1 | 89 | 8 | 2 | 0.39 | 0.701 | 0.09 |

14 | FSa | 1 | 88 | 11 | 0 | 0.69 | 0.520 | 0.06 |

15 | FSa | 0 | 94 | 6 | 0 | 0.22 | 0.728 | 0.11 |

16 | FSa | 1 | 85 | 11 | 3 | 0.54 | 0.599 | 0.06 |

17 | FSa | 2 | 90 | 8 | 0 | 0.39 | 0.680 | 0.08 |

18 | FSa | 0 | 89 | 9 | 2 | 0.42 | 0.701 | 0.07 |

19 | FSa | 2 | 90 | 8 | 0 | 0.80 | 0.491 | 0.08 |

20 | FSa | 1 | 87 | 10 | 2 | 0.45 | 0.593 | 0.063 |

21 | MSa | 0 | 99 | 1 | 0 | 0.48 | 0.603 | 0.25 |

22 | MSa | 0 | 97 | 2 | 1 | 0.41 | 0.589 | 0.23 |

23 | MSa | 1 | 96 | 3 | 0 | 0.58 | 0.521 | 0.20 |

24 | MSa | 0 | 97 | 3 | 0 | 0.52 | 0.580 | 0.20 |

25 | MSa | 0 | 98 | 2 | 0 | 0.61 | 0.536 | 0.21 |

26 | MSa | 2 | 94 | 3 | 1 | 0.65 | 0.514 | 0.21 |

27 | MSa | 2 | 92 | 4 | 2 | 0.70 | 0.545 | 0.18 |

28 | MSa | 1 | 95 | 4 | 0 | 0.67 | 0.513 | 0.17 |

29 | MSa | 0 | 98 | 2 | 0 | 0.56 | 0.563 | 0.21 |

30 | MSa | 1 | 98 | 1 | 0 | 0.78 | 0.456 | 0.25 |

31 | MSa | 0 | 97 | 2 | 1 | 0.37 | 0.631 | 0.23 |

32 | MSa | 0 | 95 | 3 | 2 | 0.85 | 0.405 | 0.20 |

33 | MSa | 1 | 97 | 2 | 0 | 0.78 | 0.415 | 0.24 |

34 | MSa | 2 | 96 | 2 | 0 | 0.63 | 0.520 | 0.24 |

35 | MSa | 1 | 98 | 1 | 0 | 0.56 | 0.547 | 0.26 |

36 | MSa | 0 | 98 | 2 | 0 | 0.82 | 0.436 | 0.21 |

37 | MSa | 1 | 97 | 2 | 0 | 0.39 | 0653 | 0.24 |

38 | MSa | 0 | 97 | 2 | 1 | 0.33 | 0.606 | 0.23 |

39 | MSa | 0 | 96 | 4 | 0 | 0.86 | 0.410 | 0.25 |

40 | MSa | 0 | 98 | 2 | 0 | 0.61 | 0.535 | 0.21 |

41 | MSa | 2 | 96 | 2 | 0 | 0.54 | 0.518 | 0.23 |

42 | MSa | 0 | 98 | 2 | 0 | 0.49 | 0.552 | 0.21 |

43 | CSa | 8 | 92 | 0 | 0 | 0.71 | 0.470 | 0.50 |

44 | CSa | 12 | 87 | 1 | 0 | 0.68 | 0.456 | 0.46 |

45 | CSa | 19 | 81 | 0 | 0 | 0.59 | 0.507 | 0.61 |

46 | CSa | 18 | 82 | 0 | 0 | 0.40 | 0.532 | 0.58 |

47 | CSa | 17 | 80 | 2 | 1 | 0.54 | 0.528 | 0.45 |

48 | CSa | 18 | 82 | 0 | 0 | 0.28 | 0.590 | 0.57 |

49 | CSa | 18 | 82 | 0 | 0 | 0.38 | 0.576 | 0.60 |

50 | CSa | 16 | 82 | 2 | 0 | 0.92 | 0.423 | 0.63 |

No. of Well | Soil | Permeability Coefficient k (m/s) | |
---|---|---|---|

Pumping Test | Consolidometer Test | ||

1 | FSa | 2.32 × 10^{−5} | 2.20 × 10^{−5} |

2 | FSa | 3.69 × 10^{−5} | 3.42 × 10^{−5} |

3 | FSa | 2.10 × 10^{−5} | 2.00 × 10^{−5} |

4 | FSa | 1.24 × 10^{−5} | 1.33 × 10^{−5} |

5 | FSa | 5.77 × 10^{−5} | 5.64 × 10^{−5} |

6 | FSa | 4.68 × 10^{−5} | 4.33 × 10^{−5} |

7 | FSa | 3.79 × 10^{−5} | 3.66 × 10^{−5} |

8 | FSa | 4.40 × 10^{−5} | 3.97 × 10^{−5} |

9 | FSa | 4.78 × 10^{−5} | 4.64 × 10^{−5} |

10 | FSa | 5.59 × 10^{−5} | 5.27 × 10^{−5} |

11 | FSa | 9.32 × 10^{−5} | 9.05 × 10^{−5} |

12 | FSa | 3.85 × 10^{−5} | 3.64 × 10^{−5} |

13 | FSa | 8.48 × 10^{−5} | 8.50 × 10^{−5} |

14 | FSa | 4.54 × 10^{−5} | 4.63 × 10^{−5} |

15 | FSa | 9.86 × 10^{−5} | 9.65 × 10^{−5} |

16 | FSa | 5.08 × 10^{−5} | 4.96 × 10^{−5} |

17 | FSa | 7.20 × 10^{−5} | 7.32 × 10^{−5} |

18 | FSa | 8.64 × 10^{−5} | 8.73 × 10^{−5} |

19 | FSa | 5.30 × 10^{−5} | 4.98 × 10^{−5} |

20 | FSa | 6.75 × 10^{−5} | 6.14 × 10^{−5} |

21 | MSa | 1.69 × 10^{−4} | 1.57 × 10^{−4} |

22 | MSa | 2.97 × 10^{−4} | 2.93 × 10^{−4} |

23 | MSa | 2.28 × 10^{−4} | 2.12 × 10^{−4} |

24 | MSa | 1.49 × 10^{−4} | 1.45 × 10^{−4} |

25 | MSa | 1.32 × 10^{−4} | 1.32 × 10^{−4} |

26 | MSa | 1.35 × 10^{−4} | 1.25 × 10^{−4} |

27 | MSa | 1.48 × 10^{−4} | 1.50 × 10^{−4} |

28 | MSa | 1.36 × 10^{−4} | 1.22 × 10^{−4} |

29 | MSa | 2.20 × 10^{−4} | 2.03 × 10^{−4} |

30 | MSa | 1.17 × 10^{−4} | 1.16 × 10^{−4} |

31 | MSa | 2.78 × 10^{−4} | 2.65 × 10^{−4} |

32 | MSa | 1.45 × 10^{−4} | 1.26 × 10^{−4} |

33 | MSa | 1.63 × 10^{−4} | 1.55 × 10^{−4} |

34 | MSa | 2.08 × 10^{−4} | 1.92 × 10^{−4} |

35 | MSa | 2.23 × 10^{−4} | 2.05 × 10^{−4} |

36 | MSa | 1.85 × 10^{−4} | 1.64 × 10^{−4} |

37 | MSa | 2.89 × 10^{−4} | 2.57 × 10^{−4} |

38 | MSa | 2.54 × 10^{−4} | 2.36 × 10^{−4} |

39 | MSa | 1.29 × 10^{−4} | 1.16 × 10^{−4} |

40 | MSa | 1.98 × 10^{−4} | 1.97 × 10^{−4} |

41 | MSa | 1.75 × 10^{−4} | 1.63 × 10^{−4} |

42 | MSa | 1.70 × 10^{−4} | 1.61 × 10^{−4} |

43 | CSa | 3.73 × 10^{−4} | 3.68 × 10^{−4} |

44 | CSa | 4.14 × 10^{−4} | 3.84 × 10^{−4} |

45 | CSa | 4.85 × 10^{−4} | 4.78 × 10^{−4} |

46 | CSa | 6.28 × 10^{−4} | 5.89 × 10^{−4} |

47 | CSa | 5.84 × 10^{−4} | 5.80 × 10^{−4} |

48 | CSa | 7.05 × 10^{−4} | 6.98 × 10^{−4} |

49 | CSa | 6.97 × 10^{−4} | 7.02 × 10^{−4} |

50 | CSa | 3.24 × 10^{−4} | 3.29 × 10^{−4} |

**Table 3.**Errors measures for ANN 6-8-1 in the subsets training Tr, testing T, validation V of set B.

Errors | Subset Tr | Subset T | Subset V |
---|---|---|---|

RMS | 0.0098 | 0.0096 | 0.0084 |

MAE | 0.0215 | 0.0204 | 0.0119 |

R^{2} | 0.976 | 0.976 | 0.976 |

**Table 4.**Measured values of permeability coefficient k from field tests and predicted values of k using developed ANN 6-8-1.

No. of Wells | Soil | Measured Values of k in Pumping Tests d _{i} (m/s) | Predicted Values of k Based on ANN 6-8-1 y _{i} (m/s) | Relative Errors of Individual Case RE _{i} (%) |
---|---|---|---|---|

1 | FSa | 2.32 × 10^{−5} | 2.25 × 10^{−5} | 3.02 |

2 | FSa | 3.69 × 10^{−5} | 3.58 × 10^{−5} | 2.98 |

3 | FSa | 2.10 × 10^{−5} | 2.23 × 10^{−5} | 6.19 |

4 | FSa | 1.24 × 10^{−5} | 1.31 × 10^{−5} | 5.65 |

5 | FSa | 5.77 × 10^{−5} | 5.70 × 10^{−5} | 1.21 |

6 | FSa | 4.68 × 10^{−5} | 4.65 × 10^{−5} | 0.64 |

7 | FSa | 3.79 × 10^{−5} | 3.80 × 10^{−5} | 0.26 |

8 | FSa | 4.40 × 10^{−5} | 4.28 × 10^{−5} | 2.73 |

9 | FSa | 4.78 × 10^{−5} | 4.69 × 10^{−5} | 1.88 |

10 | FSa | 5.59 × 10^{−5} | 5.45 × 10^{−5} | 2.50 |

11 | FSa | 9.32 × 10^{−5} | 9.28 × 10^{−5} | 0.43 |

12 | FSa | 3.85 × 10^{−5} | 3.84 × 10^{−5} | 0.26 |

13 | FSa | 8.48 × 10^{−5} | 8.48 × 10^{−5} | 0 |

14 | FSa | 4.54 × 10^{−5} | 4.53 × 10^{−5} | 0.22 |

15 | FSa | 9.86 × 10^{−5} | 9.82 × 10^{−5} | 0.41 |

16 | FSa | 5.08 × 10^{−5} | 5.03 × 10^{−5} | 0.98 |

17 | FSa | 7.20 × 10^{−5} | 6.98 × 10^{−5} | 3.06 |

18 | FSa | 8.64 × 10^{−5} | 8.63 × 10^{−5} | 0.12 |

19 | FSa | 5.30 × 10^{−5} | 5.46 × 10^{−5} | 3.02 |

20 | FSa | 6.75 × 10^{−5} | 6.84 × 10^{−5} | 1.33 |

21 | MSa | 1.69 × 10^{−4} | 1.65 × 10^{−4} | 2.37 |

22 | MSa | 2.97 × 10^{−4} | 3.05 × 10^{−4} | 2.69 |

23 | MSa | 2.28 × 10^{−4} | 2.22 × 10^{−4} | 2.63 |

24 | MSa | 1.49 × 10^{−4} | 1.49 × 10^{−4} | 0 |

25 | MSa | 1.32 × 10^{−4} | 1.31 × 10^{−4} | 0.76 |

26 | MSa | 1.35 × 10^{−4} | 1.33 × 10^{−4} | 1.48 |

27 | MSa | 1.48 × 10^{−4} | 1.47 × 10^{−4} | 0.68 |

28 | MSa | 1.36 × 10^{−4} | 1.30 × 10^{−4} | 4.41 |

29 | MSa | 2.20 × 10^{−4} | 2.18 × 10^{−4} | 0.91 |

30 | MSa | 1.17 × 10^{−4} | 1.16 × 10^{−4} | 0.85 |

31 | MSa | 2.78 × 10^{−4} | 2.73 × 10^{−4} | 1.80 |

32 | MSa | 1.45 × 10^{−4} | 1.34 × 10^{−4} | 7.59 |

33 | MSa | 1.63 × 10^{−4} | 1.60 × 10^{−4} | 1.84 |

34 | MSa | 2.08 × 10^{−4} | 2.05 × 10^{−4} | 1.44 |

35 | MSa | 2.23 × 10^{−4} | 2.28 × 10^{−4} | 2.24 |

36 | MSa | 1.85 × 10^{−4} | 1.84 × 10^{−4} | 0.54 |

37 | MSa | 2.89 × 10^{−4} | 2.89 × 10^{−4} | 0 |

38 | MSa | 2.54 × 10^{−4} | 2.53 × 10^{−4} | 0.39 |

39 | MSa | 1.29 × 10^{−4} | 1.21 × 10^{−4} | 6.20 |

40 | MSa | 1.98 × 10^{−4} | 1.99 × 10^{−4} | 0.51 |

41 | MSa | 1.75 × 10^{−4} | 1.75 × 10^{−4} | 0 |

42 | MSa | 1.70 × 10^{−4} | 1.68 × 10^{−4} | 1.18 |

43 | CSa | 3.73 × 10^{−4} | 3.71 × 10^{−4} | 0.54 |

44 | CSa | 4.14 × 10^{−4} | 4.25 × 10^{−4} | 2.66 |

45 | CSa | 4.85 × 10^{−4} | 4. 73 × 10^{−4} | 2.47 |

46 | CSa | 6.28 × 10^{−4} | 6.62 × 10^{−4} | 5.41 |

47 | CSa | 5.84 × 10^{−4} | 5.84 × 10^{−4} | 0 |

48 | CSa | 7.05 × 10^{−4} | 7.03 × 10^{−4} | 0.28 |

49 | CSa | 6.97 × 10^{−4} | 6.75 × 10^{−4} | 3.16 |

50 | CSa | 3.24 × 10^{−4} | 3.20 × 10^{−4} | 1.23 |

Max RE _{32} = 7.59% |

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**MDPI and ACS Style**

Wrzesiński, G.; Markiewicz, A.
Prediction of Permeability Coefficient *k* in Sandy Soils Using ANN. *Sustainability* **2022**, *14*, 6736.
https://doi.org/10.3390/su14116736

**AMA Style**

Wrzesiński G, Markiewicz A.
Prediction of Permeability Coefficient *k* in Sandy Soils Using ANN. *Sustainability*. 2022; 14(11):6736.
https://doi.org/10.3390/su14116736

**Chicago/Turabian Style**

Wrzesiński, Grzegorz, and Anna Markiewicz.
2022. "Prediction of Permeability Coefficient *k* in Sandy Soils Using ANN" *Sustainability* 14, no. 11: 6736.
https://doi.org/10.3390/su14116736