# Deep-Learning-Enhanced CT Image Analysis for Predicting Hydraulic Conductivity of Coarse-Grained Soils

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Hydraulic Conductivity Calculation Model for Coarse-Grained Soil Based on Equivalent Simplified Model and Poiseuille’s Law

_{0}) is given by [38]:

_{e}represents the diameter of ideal soil particles; n is the porosity; and β is the particle shape correction factor of coarse-grained soil.

_{0}is the pore pipe’s cross-section, g is the acceleration of gravity, J is the hydraulic gradient, μ is the coefficient of water movement viscosity, and r

_{0}is the radius of the pore pipe.

_{0}pore ducts, the actual overflow area is N

_{0}A

_{0}, which, when expressed in terms of porosity, can be equated as

_{1}is the radius of the coarse-grained soil sample (Figure 1 in Section 2.1).

_{e}ideal soil particles of diameter d

_{e}, both the mass of the coarse-grained soil specimen (M

_{1}; Equation (5)) and the total mass of the ideal soil of the simplified equivalent model (M

_{e}; Equation (6)) are equal.

_{d}and ρ

_{e}are the dry density of coarse-grained soil and the density of the ideal soil of the simplified equivalent model, respectively, and L is the stacking height of coarse-grained soil.

_{e}and α, which are related to equivalent particle size and particle shape and pore structure, respectively. The determination of d

_{e}and α through tests requires a combined CT scan and constant-head permeability test for coarse-grained soils. By analyzing CT images, we can obtain the particle size, volume, and equivalent volume sphere diameter of each particle in the specimen. The hydraulic conductivity of the specimen can be obtained by substituting the hydraulic conductivity of the specimen and d

_{e}into Equation (9). For some coarse-grained soil specimens with a different particle size distribution and porosity, a series of discount factors can be obtained by the above method. Further details about the combined CT scan and permeability test for coarse-grained soils and the method of CT image analysis for coarse-grained soils are outlined in the following section.

#### 2.2. Materials

#### 2.3. Laboratory Test

_{u}) and the curvature coefficient (C

_{c}) of the particle size distribution curve serve as key parameters in evaluating soil grading. A soil grade is considered good when C

_{u}≥ 5 and C

_{c}lies between 1 and 3. Conversely, if the soil grading is poor, it can cause pipe surge phenomena due to the lack of intermediate-sized soil particles. The formulae to calculate C

_{u}and C

_{c}are given by Equations (10) and (11), respectively, with the computation results for each specimen listed in Table 2.

_{10}, d

_{30}and d

_{60}represent the particle sizes corresponding to the 10%, 30%, and 60% mass accumulation percentages in the particle size distribution curve of the coarse-grained soil.

_{u}values less than 5 and C

_{c}values between 0.80 and 1.34, indicating poor grading across all specimens.

- A settlement measurement device was installed at the top of the coarse-grained soil specimens to prevent seepage deformation during the test.
- Aerated water was used to negate the impact of air bubbles on the percolation volume.
- Before the test, the specimen was saturated with bottom-up exhaust under a lower head and soaked for over 8 h to eliminate the influence of non-saturation on the permeability test results.
- The test head from the starting hydraulic slope dropped from 0.05 to 0.30, loaded step by step, with each head level loaded for 20 min before measuring the overflow in the permeameter and recording it.
- The next level of head was loaded only when the overflow in the unit time remained unchanged. This process continued until the test concluded.

#### 2.4. Coarse-Grained Soil CT Image Segmentation Method Based on Convolutional Neural Network

#### 2.4.1. U-Net Structured Convolutional Neural Network

#### 2.4.2. Loss Function

#### 2.4.3. Workflow of Convolutional Neural Network Segmentation Model

#### 2.5. Geometric Characterization of Coarse Soil Particles via CT Image Analysis

_{eq}is the diameter of a sphere with a volume equivalent to the particle.

#### 2.6. Data Analysis

_{e}, α and k. Additionally, the strength and direction of the linear correlation are assessed employing the Pearson correlation coefficient.

_{i}is the independent variable; y

_{i}is the dependent variable; $\overline{x}$ is the mean of x; $\overline{y}$ is the mean of y; and β

_{0}and β

_{1}are defined as the intercept and slope of the coefficients for the least squares line, respectively. Once β₀ and β₁ are calculated, the linear regression model (Equation (21)) is utilized to predict the value of y based on a given x value.

_{X}and σ

_{Y}, and expectations E[X] and E[Y], the Pearson correlation coefficient is computed by the following formula [45]:

_{e}, α, and k are assessed. The integration of the least squares method for linear regression and the Pearson correlation coefficient facilitates a robust evaluation and the characterization of d

_{e}, α, and k.

## 3. Results

#### 3.1. Accuracy Verification of CT Image-Segmentation Program Based on Convolutional Neural Network

#### 3.1.1. Verification of CT Image-Segmentation Accuracy Based on Convolutional Neural Networks

#### 3.1.2. Comparison of Segmentation Results between CNN Model and Traditional Methods for CT Images of Coarse-Grained Soil

#### 3.2. Equivalent Simplified Model of Ideal Particle Diameter in Coarse-Grained Soil

#### 3.2.1. Three-Dimensional Model Reconstruction of Coarse-Grained Soil Based on CT Image Segmentation Results

#### 3.2.2. Empirical Formula for the Ideal Particle Size of Coarse Soil Particles

_{eq})—with those of real soil particles. This accuracy satisfies the requirements of our study.

_{eq}and Φ (d

_{eq}× Φ) and the diameter of soil particles. Accordingly, we applied Equations (18)–(22) to regress the particle size of soil particles, as shown in Figure 11. A congruent linear relationship appears across all 12 coarse-grained soil specimens. However, due to space limitations, we only present the linear regression analyses of particle sizes for specimens S5 and S6. Figure 11 presents the Pearson correlation coefficients of d

_{eq}× Φ and the particle size for all specimens, exceeding 0.95 for all specimens except S1, which displays a smaller particle size range (5–13 mm) compared to the other specimens. This smaller range results in a slightly lower Pearson correlation coefficient of 0.88; yet, this figure still denotes a substantial linear correlation between these variables.

_{20}, corresponding to a 20% cumulative mass fraction. This view is further corroborated by several other studies; hence, we incorporated this perspective into our research.

_{20}of the specimen, with the equivalent particle diameter d

_{eq}

_{20}corresponding to the 20% cumulative mass fraction, and the average aspect ratio of the specimen. These parameters are presented in Table 4. Our results, depicted in Figure 12, establish a strong linear correlation between d

_{20}and the product of d

_{eq}

_{20}and average aspect ratio, $\overline{\Phi}$.

_{e}is the rational particle size of coarse-grained soil, which is equal to the particle size d

_{eq}

_{20}corresponding to a cumulative mass percentage of 20% in the diameter distribution of equal volume spheres; d

_{20}is the particle size corresponding to a cumulative percentage of the coarse soil mass of 20%; and $\overline{\Phi}$ is the average aspect ratio of coarse-grained soil particles.

#### 3.3. Prediction Formula for Permeability Coefficient of Coarse-Grained Soil Based on CT Image Analysis

#### 3.3.1. Constant-Head Permeability Test Results

#### 3.3.2. Empirical Formula for Discount Factor

#### 3.3.3. Prediction Formula and Accuracy Verification of Hydraulic Conductivity of Coarse-Grained Soil

## 4. Discussion

_{20}, porosity, and aspect ratio of the particles. Moreover, our proposed formula predicting the hydraulic conductivity of coarse-grained soil based on CT image analysis is more precise, demonstrating a higher value for engineering applications. To our knowledge, this is the first study predicting the hydraulic conductivity of coarse-grained soil using CT images segmented based on deep learning models. Taking the practical implications of our findings into account, the accurate prediction of hydraulic conductivity of coarse-grained soils using our model presents transformative potential in the field of water conservancy and geotechnical engineering. Firstly, in embankment dam design, our approach offers a more precise and reliable method to analyze the granulometric characteristics of the soil, thereby aiding in choosing the right type of soil and understanding its behavior under various conditions. This ultimately impacts the safety and longevity of the dam structure. Secondly, in the context of geotechnical engineering, the understanding of granular composition and its impact on the tortuosity of water-flow paths can assist in effective ground water management and the design of structures requiring soil as the foundational support. With a clearer understanding of soil’s hydraulic behavior, engineers can better anticipate and mitigate potential issues related to soil stability, water seepage and deformation under load [64,65,66].

## 5. Conclusions

- The implementation of the CNN model demonstrates unparalleled precision in the segmentation of coarse-grained soil CT images, ascertaining the model’s superiority over traditional segmentation methods. The accuracy of the 3D models reconstructed from these segmented images corroborates the effectiveness of this approach and broadens the prospects of automation and precision in soil particle segmentation.
- We established and validated empirical formulae for the ideal particle size of coarse-grained soil and the discount factor, both predicated on a robust linear correlation found in the study. These novel formulae contribute significantly to understanding the granulometric characteristics of soils and predicting their behavior under various hydraulic gradients, thus providing valuable insights for soil-related engineering and hydraulic applications.
- Our research underlines the strong influence of the granular composition, especially the concentration of fine particles, on the tortuosity of water flow paths and the discount factor. These findings highlight the potential of the CNN model in soil hydrodynamics research and its implications for a variety of fields, including water conservancy and geotechnical engineering.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic diagram of the flow velocity distribution in the pore tube of the simplified equivalent model of coarse-grained soil.

**Figure 3.**Materials. (

**a**) Materials used for the tests. (

**b**) The construction site of the Lianghekou core-wall rockfill dam.

**Figure 9.**Three-dimensional particle model of coarse-grained soil reconstructed from CT images segmented using the CNN model.

**Figure 11.**Linear regression results of coarse soil particle size. (

**a**) Linear regression results of particle size for S5 sample. (

**b**) Linear regression results of particle size for S6 sample. (

**c**) Pearson correlation coefficient for linear regression of particle size of all samples.

**Figure 13.**Permeability test results of coarse-grained soil samples (test water temperature at 20 °C).

Sample ID | S1 | S2 | S3 | S4 | S5 | S6 |
---|---|---|---|---|---|---|

Porosity | 40% | 30% | 40% | 40% | 38% | 35% |

Sample ID | S7 | S8 | S9 | S10 | S11 | S12 |

Porosity | 38% | 35% | 35% | 32% | 30% | 35% |

**Table 2.**Non-uniformity coefficient and curvature coefficient of particle size distribution of coarse-grained soil specimens.

Sample ID | S1 | S2 | S3 | S4 | S5 | S6 |
---|---|---|---|---|---|---|

C_{u} | 1.65 | 3.05 | 3.58 | 3.66 | 3.68 | 3.05 |

C_{c} | 1.07 | 1.20 | 0.80 | 1.06 | 0.89 | 1.29 |

Sample ID | S7 | S8 | S9 | S10 | S11 | S12 |

C_{u} | 2.38 | 3.36 | 3.52 | 3.11 | 2.93 | 3.45 |

C_{c} | 1.34 | 1.11 | 1.14 | 1.31 | 1.27 | 0.98 |

Metric | Expression [43] | Range |
---|---|---|

IoU | $IoU=\frac{\left|Predicted\cap Ground\text{}Truth\right|}{\left|Predicted\cup Ground\text{}Truth\right|}$ | Metric is between 0 and 1, and the closer it is to 1, the better the model performs. |

Precision | $Precision=\frac{True\text{}Positives}{True\text{}Positives+False\text{}Positives}$ | |

Recall | $Recall=\frac{True\text{}Positives}{True\text{}Positives+False\text{}Negativies}$ | |

Accuracy | $Accuracy=\frac{True\text{}Positives+True\text{}Negativies}{Total\text{}Predictions}$ | |

Specificity | $Specificity=\frac{True\text{}Negativies}{True\text{}Negatives+False\text{}Positives}$ |

Sample ID | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 | S9 | S10 | S11 | S12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

d_{eq}_{20} (mm) | 6.26 | 2.71 | 1.72 | 2.21 | 2.01 | 2.46 | 3.29 | 1.19 | 1.24 | 2.30 | 2.90 | 2.30 |

$\overline{\Phi}$ | 2.31 | 1.93 | 2.03 | 1.95 | 2.04 | 1.86 | 1.99 | 1.98 | 2.03 | 1.93 | 1.92 | 2.00 |

d_{20} (mm) | 10.29 | 5.20 | 3.94 | 4.49 | 4.16 | 4.82 | 6.34 | 3.66 | 3.74 | 4.69 | 5.48 | 4.55 |

Sample ID | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 | S9 | S10 | S11 | S12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

K (cm/s) | 3.45 | 2.30 | 4.58 | 4.30 | 5.81 | 1.37 | 2.70 | 2.72 | 2.62 | 0.86 | 0.99 | 3.04 |

α | 162.29 | 48.79 | 39.42 | 52.27 | 36.65 | 77.88 | 87.99 | 26.74 | 28.39 | 76.80 | 79.58 | 48.88 |

**Table 6.**Comparative analysis of the accuracy of various hydraulic conductivity calculation formulae.

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## Share and Cite

**MDPI and ACS Style**

Peng, J.; Shen, Z.; Zhang, W.; Song, W.
Deep-Learning-Enhanced CT Image Analysis for Predicting Hydraulic Conductivity of Coarse-Grained Soils. *Water* **2023**, *15*, 2623.
https://doi.org/10.3390/w15142623

**AMA Style**

Peng J, Shen Z, Zhang W, Song W.
Deep-Learning-Enhanced CT Image Analysis for Predicting Hydraulic Conductivity of Coarse-Grained Soils. *Water*. 2023; 15(14):2623.
https://doi.org/10.3390/w15142623

**Chicago/Turabian Style**

Peng, Jiayi, Zhenzhong Shen, Wenbing Zhang, and Wen Song.
2023. "Deep-Learning-Enhanced CT Image Analysis for Predicting Hydraulic Conductivity of Coarse-Grained Soils" *Water* 15, no. 14: 2623.
https://doi.org/10.3390/w15142623