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

Abstract

Permeability characteristics in coarse-grained soil is pivotal for enhancing the understanding of its seepage behavior and effectively managing it, directly impacting the design, construction, and operational safety of embankment dams. Furthermore, these insights bridge diverse disciplines, including hydrogeology, civil engineering, and environmental science, broadening their application and relevance. In this novel research, we leverage a Convolutional Neural Network (CNN) model to achieve the accurate segmentation of coarse-grained soil CT images, surpassing traditional methods in precision and opening new avenues in soil granulometric analysis. The three-dimensional (3D) models reconstructed from the segmented images attest to the effectiveness of our CNN model, highlighting its potential for automation and precision in soil-particle analysis. Our study uncovers and validates new empirical formulae for the ideal particle size and the discount factor in coarse-grained soils. The robust linear correlation underlying these formulae deepens our understanding of soil granulometric characteristics and predicts their hydraulic behavior under varying gradients. This advancement holds immense value for soil-related engineering and hydraulic applications. Furthermore, our findings underscore the significant influence of granular composition, particularly the concentration of fine particles, on the tortuosity of water-flow paths and the discount factor. The practical implications extend to multiple fields, including water conservancy and geotechnical engineering. Altogether, our research represents a significant step in soil hydrodynamics research, where the CNN model’s application unveils key insights into soil granulometry and hydraulic conductivity, laying a strong foundation for future research and applications.

1. Introduction

Coarse-grained soil, defined by particle sizes from 0.075 to 60 mm, is a prevalent component in natural environments and engineering sites. Its utility as a fundamental constituent in embankment dams, forming key parts like rockfill zones, inverted filters, and cushion layers, is particularly noteworthy. The complex granular composition and variation in particle sizes make coarse-grained soil an interesting yet challenging subject for understanding permeability characteristics. Furthermore, the hydraulic behavior of coarse-grained soil greatly affects the stability and safety of structures built upon it, especially in water conservancy projects. The soil’s permeability substantially dictates its stability, load-bearing capability, and seepage deformation—vital engineering attributes. Consequently, it is of paramount importance in the design, construction, and safety during the operation of embankment dams. Deepening our understanding of the permeability attributes of coarse-grained soil and accurately forecasting its hydraulic conductivity, a metric for seepage capacity, are essential for comprehending and manipulating its seepage behavior. The implications of these findings extend into various disciplines, including hydrogeology, civil engineering, and environmental science.

Hydraulic conductivity is traditionally determined via experimental testing or numerical simulations. Empirical tests, conducted both in laboratories and in field conditions, offer valuable insights, for instance, the mixed-method [1] investigation of Nam et al. into the impact of test conditions on the hydraulic conductivity of natural coarse-grained soils. However, the accuracy of such empirical tests is susceptible to factors like specimen preparation and boundary conditions [2,3,4,5,6,7,8], which demand rigorous operational protocols and equipment. In contrast, numerical simulations leverage finite or discrete elements to model real-world coarse-grained soils [9,10,11,12]. However, these models often rely on parameters that are experimentally elusive, undermining their applicability and limiting experimental validation.

Computed tomography (CT) image analysis offers a promising hybrid approach, combining empirical data with simulation [13,14,15,16,17]. This technique’s efficacy hinges on accurate CT image segmentation, a process where conventional methods such as the adaptive threshold [18,19], hysteresis threshold [20], and watershed segmentation [21,22] methods often fall short when applied to the bulk structures of coarse-grained soils. Coarse-grained soils necessitate CT image analysis that segments each soil particle in adjacent stacks, crucial for extracting parameters like particle size and shape factor, which subsequently inform the study of their impact on hydraulic conductivity. Consequently, a need exists for more refined methods that can accurately and effectively perform this complex analysis.

The advent of deep learning models, particularly convolutional neural networks (CNNs) [23,24], offers a potential solution. By categorizing or labeling each pixel in an image, these models facilitate segmentation into discrete regions or objects. With their inherent ability to learn intricate features directly from data, CNNs can efficiently manage high-resolution, multi-scale, and multi-class images. While these models have gained traction in studying the engineering properties of hydraulic and geotechnical materials, their use has primarily been on solid materials, such as concrete [25,26,27,28,29,30,31,32] and rock cores [33,34,35,36,37]. Comparatively, coarse-grained soils present more complex imaging tasks, necessitating extensive research into the application of deep learning models to soil CT images, accounting for variations in particle size distributions, shapes, and materials. This is paramount in tailoring the model to the unique requirements of coarse-grained soil CT image segmentation.

This paper addresses these research gaps through comprehensive CT scanning and infiltration tests on coarse-grained soils of varying grain size distributions and porosity. We study the hydraulic conductivity of blasted coarse-grained soils from the Yalong River Lianghekou Hydropower Station construction site in China. An initial coarse-grained soil hydraulic conductivity calculation model is introduced, derived from a simplified cohesionless soil model and Poiseuille’s law. We propose a CNN-based method for CT image segmentation, focusing on the unique characteristics of coarse-grained soils. The model employs a U-net architecture and is guided by a novel loss function incorporating both grain size and shape perception terms. We verify the accuracy of our segmentation method against a gold standard image and the known grain size distribution of the coarse-grained-soil CT image. The results are utilized to devise a prediction formula for hydraulic conductivity based on CT image analysis. The presented approach offers an efficient, precise, and intelligent method for studying the permeability characteristics of coarse-grained soils, promising to significantly impact the optimization of water conservancy project design and construction.

2. Materials and Methods

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

A simplified equivalent model of coarse-grained soil is depicted in Figure 1, which idealizes the soil as a uniform sphere and abstracts seepage channels as parallel capillaries. Assuming the total volume and total surface area of the pore channels of the simplified model are equal to those of coarse-grained soil, the diameter of the pore channels (d₀) is given by [38]:

$$d_0=\frac{1}{\beta }\frac{2}{3}\frac{n}{1-n}{d}_e$$

(1)

where d_e represents the diameter of ideal soil particles; n is the porosity; and β is the particle shape correction factor of coarse-grained soil.

As depicted in Figure 2, water flow in the single pore pipe of the simplified model adheres to Poiseuille’s law [39], stating that flow resistance in the pore pipe is inversely proportional to the fourth power of the pipe radius and proportional to the length of the pipe and the liquid viscosity coefficient. The average flow velocity of the single pore pipe (V) is then:

$$V=\frac{Q}{A_0}=\frac{gJ}{8\mu }{r}_0^2$$

(2)

where Q is the pore pipe’s flow rate, A₀ 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 r₀ is the radius of the pore pipe.

For a coarse-grained soil overflow section A with N₀ pore ducts, the actual overflow area is N₀A₀, which, when expressed in terms of porosity, can be equated as

$$nA=N{A}_0$$

(3)

$$n\pi r_1^2=N_0\pi r_0^2$$

(4)

where r₁ is the radius of the coarse-grained soil sample (Figure 1 in Section 2.1).

For a coarse-grained soil simplified equivalent model comprising an N_e ideal soil particles of diameter d_e, both the mass of the coarse-grained soil specimen (M₁; Equation (5)) and the total mass of the ideal soil of the simplified equivalent model (M_e; Equation (6)) are equal.

$$M_1={\rho }_d\pi r_1^{2}L$$

(5)

$$M_e=N_e{\rho }_e·\frac{4}{3}\pi {\left(\frac{d_e}{2}\right)}^3=\frac{\pi }{6}{d}_e^3{\rho }_e{N}_e$$

(6)

where ρ_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.

The discount factor α represents the degree of loss in the actual seepage channel of coarse-grained soil caused by pore connectivity and tortuosity, which increases with the complexity of the pore structure of coarse-grained soil and decreases with an increase in pore connectivity. The total surface area of the pore channel of the simplified equivalent model of coarse-grained soil is defined as:

$$S_e=2\pi r_0^{2}L{N}_0=\alpha ·4\pi {\left(\frac{d_e}{2}\right)}^2{N}_1$$

(7)

Combining Equations (2)–(7), we obtain the discount factor as

$$\alpha =\frac{n{d}_e}{3\left(n-1\right)}\sqrt{\frac{ngJ}{8\mu V}}$$

(8)

Inserting Equation (8) into Darcy’s law [40] V = KJ and simplifying it, we find the equation for the hydraulic conductivity (K) of coarse-grained soil:

$$K=\frac{g}{{72\mu \alpha }^2}\frac{n^3}{{(1-n)}^2}{d}_e^2$$

(9)

The difficulty in using Equation (9) to calculate the hydraulic conductivity of coarse-grained soils is establishing the equations for d_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

The coarse-grained soil used for the experiment was sourced from blasted tuff at the construction site of the Lianghekou Rockfill Dam, with a grain size ranging between 1 and 20 mm (Figure 3). Located in Sichuan Province, Southwest China, the 295 m high Lianghekou Rockfill Dam serves as the largest hydropower project in the Tibetan region.

2.3. Laboratory Test

A combination of CT scanning and constant-head permeability tests were carried out on the coarse-grained soil samples. Initially, the specimens were prepared inside a resin-made permeameter, conforming to the test numbers and porosities outlined in

Table 1. Coarse-grained soil sample IDs and porosities.

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%

, and following the particle-size distribution curves detailed in Figure 4. The particle-size distribution curves of each specimen (Figure 4) aligned with the original grading of the material from the Lianghekou core-wall rockfill dam quarry, with specimen S1 and S7 representing the upper and lower envelopes, respectively. Table 1 lists the minimum porosity of each specimen under the specified particle size distribution. The specimens, each cylindrical with a diameter of 10 cm and a filling height of 8.5 cm, totaled 12 in number.

The non-uniformity coefficient (C_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. Non-uniformity coefficient and curvature coefficient of particle size distribution of coarse-grained soil specimens.

Sample ID	S1	S2	S3	S4	S5	S6
Cu	1.65	3.05	3.58	3.66	3.68	3.05
Cc	1.07	1.20	0.80	1.06	0.89	1.29
Sample ID	S7	S8	S9	S10	S11	S12
Cu	2.38	3.36	3.52	3.11	2.93	3.45
Cc	1.34	1.11	1.14	1.31	1.27	0.98

.

$$C_u=\frac{d_{60}}{d_{10}}$$

(10)

$$C_c=\frac{d_{30}^2}{d_{10}{d}_{60}}$$

(11)

where, d₁₀, d₃₀ and d₆₀ 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.

All specimens, as per Table 2, displayed C_u values less than 5 and C_c values between 0.80 and 1.34, indicating poor grading across all specimens.

Specimens were prepared in the permeameter and scanned using a diondo d2 industrial high-resolution nano-focus CT machine from Germany. The machine has a spatial resolution of 96 μm and scans in a bottom-up direction, creating 1333 scanned slices for each coarse-grained soil specimen. Following this, a constant-head permeability test was conducted on the specimens, following these steps:

• 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

In this research, a convolutional neural network-based CT image segmentation method for coarse-grained soil is proposed, implemented using a Python program that we wrote ourselves. The details are as follows:

2.4.1. U-Net Structured Convolutional Neural Network

Ronneberger et al. [41] proposed a novel Convolutional Neural Network (CNN) named U-Net, a significant advancement in robust and efficient image segmentation. The unique U-shaped structure of U-Net, reminiscent of an autoencoder but exhibiting unique characteristics, divides the network into two separate segments. The schematic diagram of U-Net with an input image resolution of 800 × 800 is shown in Figure 5.

The first segment, known as the Contracting or Downsampling path (forming the left side of the ‘U’), comprises dual 3 × 3 convolutions, each succeeded by a rectified linear unit (ReLU), and a subsequent 2 × 2 max pooling operation with a stride of 2 for downsampling. Notably, each step in this downsampling operation doubles the number of feature channels. Conversely, the Expansive or Upsampling path (making up the right side of the ‘U’) begins with an upsampling of the feature map, followed by a 2 × 2 convolution, known as an “up-convolution”. This up-convolution is concatenated with the corresponding feature map from the downsampling path and is succeeded by dual 3 × 3 convolutions, each followed by a ReLU. This innovative design of U-Net provides a streamlined and efficient process for image segmentation. ‘None’ in Figure 5 indicates that the image at that layer can have any batch size.

2.4.2. Loss Function

The loss function plays a vital role in CNN-based CT image segmentation. This utility function quantifies the discrepancy between predicted segmentation results and the actual or “ground truth” segmentations. Crucially, it allows for model performance evaluation and guides the optimization process, aiming to minimize this loss function.

The choice of an appropriate loss function is essential. In the context of coarse-grained soil CT image segmentation, attention is given to the shape and size of particles. Therefore, the loss function in this code includes shape perception and particle size perception terms.

The shape perception term uses the image similarity index IoU as a judgment metric to compute the shape similarity between the predicted segmentation and the actual segmentation:

$$ShapeLoss\left(A,B\right)=-IoU$$

(12)

where A and B represent the masks of the binary segmentation results of the actual image and the predicted image, respectively, and IoU is the metric for evaluating the similarity between the actual image and the predicted image. The calculation formula for IoU is:

$$IoU=\frac{\left|Predicted\cap GroundTruth\right|}{\left|Predicted\cup GroundTruth\right|}$$

(13)

The particle size perception term’s loss function employs the radius of the equivalent circle of the target particles in the binary segmentation of the true and predicted images as a judgment metric:

$$DiameterLoss\left(A,B\right)=\frac{\left|EqDiameter\left(A\right)-EqDiameter\left(A\right)\right|}{EqDiameter\left(A\right)}$$

(14)

where EqDiameter represents the equivalent circle radius:

$$EqDiameter=\sqrt{\frac{4\left|S\right|}{\pi }}$$

(15)

where S is the area of the target particle.

The loss function combines the standard cross-entropy loss with the shape-aware and diameter-aware terms, enabling the loss-functional supervised model to focus on capturing the shape and size of particles by introducing a weighting factor:

$$L\left(y,\widehat{y},A,B\right)=CE\left(y,\widehat{y}\right)+\alpha ·Shapeloss\left(A,B\right)+\beta ·DiameterLoss(A,B)$$

(16)

where α and β are weight coefficients that ensure a balance between standard cross-entropy loss and additional geometric constraints, which are adaptively adjusted by model learning. CE is the cross-entropy loss function [42]:

$$CE\left(y,\widehat{y}\right)=-sum(y·\mathrm{l}\mathrm{o}\mathrm{g}(\widehat{y}\left)\right)$$

(17)

where y is ground truth image, and $\widehat{y}$ is predicted image.

2.4.3. Workflow of Convolutional Neural Network Segmentation Model

The U-Net CNN workflow designed for the segmentation of coarse-soil CT images is illustrated in Figure 6. This workflow includes six primary steps: preprocessing, generating training and validation sets, training the U-Net model, validation, post-processing, and evaluation.

Stage 1: Preprocessing—initially, anisotropic diffusion filtering is employed to denoise all CT images of the coarse-grained soil samples. After this, images are resampled to achieve a pixel resolution of 800 × 800 per CT image, which aligns with computational memory capabilities. Lastly, all CT images are normalized.

Stage 2: Generating training and validation sets—the VGG Image Annotator (VIA) is used to create a ground truth image set via annotations, aiding the CNN in distinguishing between soil particles and pores. The CT images are then divided into a training set (70% of the images), a validation set (15% of the images), and a testing set (remaining 15%), to facilitate model training and performance evaluation.

Stage 3: U-net model training—the training set is used to train the U-net model, aiming to minimize the loss function Equation (16), thereby enabling it to accurately predict the segmentation of coarse soil CT images.

Stage 4: Validation—the model’s performance is validated during or after the training process using the validation set to avoid the overfitting of the training data.

Stage 5: Post-processing—post-processing involves assigning a ‘0’ value to pixels corresponding to particle parts and a ‘1’ value to pore parts in the CT image. This process results in a binary image output for the segmentation results.

Stage 6: Evaluation—the model’s performance is evaluated using metrics such as Intersection over Union (IoU), Precision, Recall, Accuracy, and Specificity, as detailed in

Table 3. Metric to evaluate the performance of model.

Metric	Expression [43]	Range
IoU	$IoU=\frac{\left|Predicted \cap GroundTruth\right|}{\left|Predicted \cup GroundTruth\right|}$	Metric is between 0 and 1, and the closer it is to 1, the better the model performs.
Precision	$Precision=\frac{TruePositives}{TruePositives + FalsePositives}$
Recall	$Recall=\frac{TruePositives}{TruePositives + FalseNegativies}$
Accuracy	$Accuracy=\frac{TruePositives + TrueNegativies}{TotalPredictions}$
Specificity	$Specificity=\frac{TrueNegativies}{TrueNegatives + FalsePositives}$

. If all these parameters exceed 0.95, the training is concluded, and the model, along with its weight coefficients, is saved. If not, the model returns to Stage 3 for retraining.

To evaluate our model’s effectiveness, we used a set of metrics as detailed in Table 2. Intersection over Union (IoU) is a key parameter, quantifying the overlap between predicted and ground truth areas, thus providing insights into its localization accuracy. Precision measures the model’s ability to avoid false positives, highlighting the reliability of the model’s positive predictions. Conversely, Recall evaluates the model’s ability to identify all positive instances, offering insights into its skill in avoiding false negatives. Accuracy encapsulates overall model performance, defined as the proportion of correct predictions over the entire dataset, although interpretability may be impaired in situations with imbalanced datasets. Lastly, we use Specificity to indicate the model’s skill in correctly identifying negative instances, contributing to minimizing false alarms. Taken together, these metrics offer a comprehensive evaluation of the model’s performance, illuminating various aspects of its classification competency.

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

This section delineates the procedure to build three-dimensional (3D) models of soil particles following the segmentation of coarse-grained soil CT images using a U-net structured CNN. Furthermore, it describes the methodology employed to calculate the major axis, minor axis, volume, and isovolumetric sphere diameter of each particle within the 3D model. The computational framework comprises two principal components: 3D modeling and morphological calculations.

Three-dimensional modelling: After segmentation, each binary image slice, where ‘1′ represents soil particles and ‘0′ represents pores, is sequentially stacked to create a three-dimensional voxel model. A voxel, or volume pixel, represents a point within the 3D grid of the object, encompassing spatial relationships and physical quantities. Within this constructed model, each soil particle manifests as a 3D object, discernible by neighboring voxels designated a value of ‘1′.

Morphological calculations: After the 3D model construction, morphological parameters—major axis, minor axis, volume, and isovolumetric sphere diameter—are calculated for each distinct soil particle. The major and minor axes are determined by evaluating the eigenvalues of the covariance matrix corresponding to the voxel coordinates of the particle—the maximum eigenvalue aligns with the longest axis (major axis), while the minimum aligns with the shortest (minor axis). The volume of the particle is ascertained by counting the number of constituent voxels, and given the isotropic nature of the voxels, this count is multiplied by the voxel volume. The isovolumetric sphere diameter—corresponding to the diameter of a sphere with volume equivalent to the particle—is calculated using Equation (18). These computations enable the extraction of essential morphological parameters from the 3D model, providing critical insights into the physical characteristics of soil particles. The computed parameters for each particle can be exported and stored for subsequent analysis.

$$d_{eq}=\sqrt[3]{\frac{6V_s}{\pi }}$$

(18)

where vs. represents the volume of the particle; and d_eq is the diameter of a sphere with a volume equivalent to the particle.

In summary, this proposed methodology offers a comprehensive approach, facilitating the construction of a 3D model of soil particles post-image-segmentation, while also enabling the calculation of vital particle characteristics. This paves the way for a more nuanced analysis and interpretation of soil properties.

2.6. Data Analysis

The least squares method is utilized for the linear regression analysis of variables d_e, α and k. Additionally, the strength and direction of the linear correlation are assessed employing the Pearson correlation coefficient.

The least squares method, which is commonly applied for the approximation of solutions in overdetermined systems—where there are more equations than unknowns—operates by minimizing the sum of the squares of the residuals. These residuals represent discrepancies between observed and estimated values, and the method facilitates the optimal linear fit for a given dataset. The method is implemented through the following formula [44]:

$${\beta }_1=\frac{\sum \left(x_i-\overline{x}\right)(y_i-\overline{y})}{\sum {\left(x_i-\overline{x}\right)}^2}$$

(19)

$${\beta }_0=\overline{y}-{\beta }_1\overline{x}$$

(20)

where n is the number of observations and i is between 1 and n; x_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 β₀ and β₁ 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.

$$y={\beta }_0+{\beta }_{1}x$$

(21)

In parallel, the Pearson correlation coefficient (R) is employed to quantify the degree of the linear relationship between pairs of variables. For any pair of random variables, denoted as (X,Y), with standard deviations σ_X and σ_Y, and expectations E[X] and E[Y], the Pearson correlation coefficient is computed by the following formula [45]:

$$R=\frac{E\left[\right(X-E\left[X\right]\left)\right(y-E\left[Y\right])}{{\sigma }_X{\sigma }_Y}$$

(22)

The value of the Pearson correlation coefficient ranges from −1 to 1. A value of 1 implies a strong positive correlation, −1 signifies a strong negative correlation, and 0 indicates no correlation. With this statistical tool, the strength and direction of the linear relationship between d_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

We present a CNN-based approach designed specifically for CT image segmentation of coarse-grained soil. The efficacy of this method is evaluated via five crucial metrics: Intersection over Union (IoU), Recall, Accuracy, Precision, and Specificity. As these metrics approach a value of 1, it indicates optimal prediction and segmentation accuracy. Higher values correspond to more precise outcomes of image segmentation, as detailed in Table 3.

Figure 7 visualizes the segmentation results of our validation set alongside a five-metric evaluation of the ground-truth image. Remarkably, all metrics exceed a value of 0.95 within 300 iterations of the CNN model, demonstrating the robustness of our methodology. It shows its capability to accurately segment CT images of coarse-grained soil, thereby meeting the objectives of this study.

When examining the clarity of segmented images, a pattern emerges: there seems to be a linear correlation between clarity and the number of CNN model iterations, resulting in a large oscillation amplitude. The maximum oscillation amplitude can be attributed to the balance between precision and recall that the CNN model strives to optimize during segmentation.

As the CNN model iteratively learns and adapts, it improves its ability to distinguish between true positive and false positive regions in the image, thereby enhancing image clarity. The increase in clarity over the iterations suggests that the CNN model is not only identifying regions of interest but is also refining the precision of their boundaries. The observed oscillation may serve as an insightful indicator of the model’s learning progress, signifying an improvement in both accuracy and refinement in identifying and defining segmented region boundaries.

Our results underscore the effectiveness and potential applicability of our novel CNN-based method. Given its proficiency in the CT image segmentation of coarse-grained soil, it opens up exciting opportunities for future research in this field.

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

Besides quantitative image similarity indicators, we emphasize visual manual inspection to ascertain the accuracy of CT image segmentation. Manual validation was carried out on a randomly selected subset comprising 30% of the test set predicted images after the CNN model training iterations. This was executed alongside the calculation of metrics for all test-set predicted CT images and their corresponding ground truth images. The results affirmed the CNN model’s competence in distinguishing coarse-grained soil particles into separate entities, thereby meeting the analytical criteria for this study.

Due to space limitations, we present two randomly selected CT image segmentation results predicted by the CNN model. Initially displayed as a binary black and white image, the CT segmentation result undergoes color representation to enhance the distinction between adjacent particles. Effective segmentation by the CNN model is, thus, manifested if adjacent particles exhibit distinct colors. Notably, the assignment of particle colors is arbitrary and independent of particle size. We also applied adaptive threshold segmentation [46] and marker-based watershed segmentation [47] methods for comparative analysis on identical CT images to validate the superiority of our CNN model-based CT image segmentation approach. Figure 8 showcases the results of CT image segmentation of coarse-grained soil via the CNN model, adaptive threshold segmentation method, and marker-based watershed segmentation method.

Figure 8 shows that the CNN model provides the most precise segmentation, with particles distinctly defined. In contrast, the adaptive threshold segmentation method falls short, while the marker-based watershed method yields intermediate results. These differences primarily stem from the varying capabilities of the three methods in handling edge pixels of smaller particles in the images.

The marker-based watershed segmentation algorithm retains a degree of accuracy due to its adherence to the dilation principle, but it becomes less efficient with an increase in fine particles and complex pore structures in the CT image. The simultaneous rise in phase-change areas and the smoothing of pixel intensity distribution leads to the failure of both the adaptive threshold segmentation method and eigenvalue segmentation method, due to the diminished edge contrast of fine particles in the image. Although the marker-based watershed segmentation method outperforms the threshold segmentation method, the accuracy of its segmentation results falls short of the study requirements.

In contrast, our CNN model leverages deep learning, freeing it from reliance on image pixel intensity distribution characteristics for segmentation. Instead, it mimics the process of human truth image segmentation through extensive data training, enabling the accurate prediction of segmentation outcomes for new images. Therefore, the CNN model’s segmentation results demonstrate its remarkable efficacy.

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

Upon segmenting the coarse-grained soil CT images from our twelve samples using the CNN model, we reconstructed the segmentation results through voxelization to create corresponding three-dimensional particle models (Figure 9). To confirm the accuracy of these models and further validate the segmentation accuracy of the CNN model, we calculated the non-uniformity and curvature coefficients of the particle size distribution within each model. We then carried out a comparative error analysis against the actual non-uniformity and curvature coefficients of the respective coarse-grained soil samples, as listed in Table 2. An error margin of less than 2% proves that the accuracy of the three-dimensional models meets the research requirements.

Figure 10 illustrates the relative errors in particle size distribution for all the three-dimensional particle models corresponding to the twelve soil samples. It is apparent that the relative error between the calculated non-uniformity and curvature coefficients and their true values for each sample is under the acceptable 2% threshold. This verifies the high accuracy of the models and not only corroborates the precision of the three-dimensional models but also reinforces the segmentation accuracy of the CNN model. This is a significant outcome because it suggests that the CNN model has successfully learned to recognize and segment different sizes of particles in the soil samples, despite their complex and variable nature. Therefore, these results reinforce the viability and reliability of using deep learning models like CNN for soil-particle segmentation and subsequent 3D modeling in soil studies, thereby broadening the prospects of automating and enhancing precision in such processes.

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

An error margin of less than 2% between the 3D models of coarse-grained soil particles, obtained from CT images segmented via the CNN model, and actual specimens, attests to the close alignment of geometric parameters from the 3D models—such as the long axis, short axis, aspect ratio (Φ), and diameter of the equal volume sphere (d_eq)—with those of real soil particles. This accuracy satisfies the requirements of our study.

Our data processing reveals a compelling linear relationship between the product of d_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.

Kozeny and Terzaghi et al. [48,49,50,51,52], from their research on the permeability tests of cohesionless soils along with theoretical analyses, concluded that the hydraulic conductivity of cohesionless soils should be represented by the particle diameter d₂₀, corresponding to a 20% cumulative mass fraction. This view is further corroborated by several other studies; hence, we incorporated this perspective into our research.

In this regard, we performed a linear regression analysis using d₂₀ of the specimen, with the equivalent particle diameter d_eq20 corresponding to the 20% cumulative mass fraction, and the average aspect ratio of the specimen. These parameters are presented in

Table 4. Characteristic particle size and shape factor of coarse-grained soil samples.

Sample ID	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12
deq20 (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
d20 (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

. Our results, depicted in Figure 12, establish a strong linear correlation between d₂₀ and the product of d_eq20 and average aspect ratio, $\overline{\Phi }$.

The regression line’s intercept and slope in Figure 12 suggest that the ideal equation for the particle size of the coarse-grained soil aligns with Equation (23). This finding contributes to a more nuanced understanding of the granulometric characteristics of coarse-grained soils and provides an empirically derived equation for predicting particle size, helping to better inform soil-related engineering and environmental applications.

$$d_e=\frac{d_{20}-2.17}{0.57\overline{\Phi }}$$

(23)

where d_e is the rational particle size of coarse-grained soil, which is equal to the particle size d_eq20 corresponding to a cumulative mass percentage of 20% in the diameter distribution of equal volume spheres; d₂₀ 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

The results from the constant-head permeability tests conducted on coarse-grained soil specimens are displayed in Figure 13. According to Darcy’s Law (Equation (24)) [40], a linear relationship exists between the water velocity (V) and the hydraulic gradient (J). This relationship enables us to calculate the hydraulic conductivity of the specimens, which are presented in

Table 5. Hydraulic gradient and discount factor of coarse-grained soil samples.

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

.

$$V=KJ$$

(24)

where V is the water flow velocity, J is the hydraulic gradient, and K is hydraulic conductivity.

Upon referring to the characteristic particle size of the coarse-grained soil specimens from Table 2 and using our empirically derived Equation (9) for estimating the hydraulic conductivity of coarse-grained soil, we can compute the discount factor (α) for the specimens. These computed values are also enumerated in Table 5. This calculated discount factor and hydraulic conductivity are essential parameters in understanding and predicting the behavior of coarse-grained soil under varying hydraulic gradients.

3.3.2. Empirical Formula for Discount Factor

Table 4 and Table 5 indicate a distinct trend: as the ideal particle size of the coarse-grained soil samples increases, there is a concurrent rise in the discount factor. To better define this relationship, we conducted a linear regression analysis using the least squares method. As illustrated in Figure 14, the resulting Pearson correlation coefficient is a robust 0.95, signifying a strong linear correlation between the discount factor and the ideal soil particle size. Therefore, the empirical formula for the discount factor of coarse-grained soil fitted in this article is shown in Equation (25).

$$\alpha =-4.31+26.85d_e$$

(25)

This relationship is deeply rooted in the soil’s granular composition, with the particle size playing a significant role. A higher concentration of fine particles within a unit volume of coarse-grained soil invariably augments the tortuosity of water-flow paths. This labyrinth-like network intensifies the interactions between the water and particles, causing greater water loss as it traverses the soil, thereby increasing the discount factor—a measure of this water loss. Moreover, particle size and distribution have a substantial impact on the soil’s void ratio, which in turn affects hydraulic conductivity. Typically, larger particles lead to a higher void ratio and greater pore spaces. Conversely, the presence of fine particles tends to fill these voids, reducing total pore space and causing an increased tortuosity in water-flow paths. This elevated tortuosity contributes to a higher reduction coefficient, necessitating more energy for water to pass through the soil matrix.

While the primary focus of this study lies in examining the influence of particle size on the discount factor, it is important to acknowledge that the soil’s pore structure has a fundamental role in this relationship. Future research will explore in greater depth the specific effects of pore structure parameters on the discount factor, shedding light on the complex interactions between particle size, pore structure, and water movement in coarse-grained soils. Such insights will advance our understanding of soil hydrodynamics and its implications for a variety of fields, including water conservancy and geotechnical engineering.

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

Following the segmentation of coarse-grained soil specimens’ CT images via the CNN model, we successfully constructed their respective 3D particle models. An examination of these particle models’ parameters allowed us to derive the empirical Equation (23), which represents the ideal particle size of coarse-grained soil. Furthermore, through an analysis of the infiltration tests conducted on coarse-grained soil specimens, we deduced the empirical Equation (25), illustrating the reduction factor. Integrating Equations (23) and (25) into our established model for calculating the hydraulic conductivity of coarse-grained soil (Equation (9)), we derived a predictive formula for coarse-grained soil hydraulic conductivity, based on CT image analysis, as

$$K=\frac{g}{72\mu {\alpha }^2}·\frac{n^3}{{(1-n)}^2}·\frac{{(d_{20}-2.17)}^2}{0.32{\overline{\Phi }}^2}$$

(26)

among which, the discount factor α is

$$\alpha =-4.31+\frac{83.91{(d_{20}-2.17)}^2}{{\overline{\Phi }}^2}$$

(27)

This section aims to probe the precision and applicability of various models for predicting the hydraulic conductivity of coarse-grained soils. Existing models mainly fall into two categories. The first type comprises empirical formulae based on pore structure and particle characteristics, exemplified by the Kozeny–Carman equation [53]. The second category includes empirical models derived from experimental data and statistical analysis, such as the Terzaghi [52] and Hazen [54] equations. In this study, we compared our derived model for predicting the hydraulic conductivity of coarse-grained soils (based on CT image analysis) against the aforementioned eight formulae. This comparison entailed calculating the hydraulic conductivity for the coarse-grained soil specimens using each formula, and subsequently comparing these values against the actual hydraulic conductivities derived from permeability tests. To assess the predictive performance of these formulae, we utilized the Pearson correlation coefficient as an evaluation index, with possible values ranging from −1 to 1. A coefficient closer to 1 indicates a strong positive correlation, while a value closer to −1 denotes a strong negative correlation. A value near 0 suggests a lack of any significant correlation.

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

Formula	Equation (25)	KC [53]	Terzaghi [52]	Hazen [54]	Safari [55]	Chapuis [56]	Cote [57]
K Pearson’s r	0.91	0.70	0.06	-0.01	0.75	0.72	0.28

lists the Pearson correlation coefficients between the hydraulic conductivity predictions from the eight formulae and the actual hydraulic conductivities of the specimens. As shown in Table 6, our proposed formula for predicting the hydraulic conductivity of coarse-grained soil provides the most accurate predictions. It attains a Pearson correlation coefficient of 0.91 with the actual hydraulic conductivity values.

4. Discussion

Existing methods for determining the hydraulic conductivity of coarse-grained soils fall into two main categories: experimental methods and numerical simulation methods. Experimental methods typically utilize a constant-head test to directly measure hydraulic conductivity [2,3,4,5,6,7,8]. However, these methods are often time-consuming and labor-intensive, with the accuracy of the results heavily reliant on sample preparation and boundary conditions. Numerical simulation methods, on the other hand, employ finite elements or discrete elements to mimic actual coarse-grained soils [9,10,11,12]. By resolving the fluid flow equation, the hydraulic conductivity can be calculated. Yet, conclusions drawn from these methods often lack experimental validation. A more effective approach to predicting the hydraulic conductivity of coarse-grained soils is to leverage CT image analysis. This method marries experimental results with simulations, thereby offering a more comprehensive and accurate evaluation. The accuracy of such predictions hinges on the precision of the CT image segmentation. However, as coarse-grained soils exhibit bulk structures and their CT image pixel distributions are complex, the accuracy of traditional image-segmentation methods often falls short of the research requirements for coarse-grained soils.

This paper presents a novel method for segmenting CT images of coarse-grained soil using deep learning models. Based on the segmentation results, we created a three-dimensional particle model of coarse-grained soil and extracted the geometric feature parameters of the particles. Subsequently, we fitted a prediction formula for the hydraulic conductivity of coarse-grained soil using CT image analysis. We discovered that the U-Net convolutional neural network, under the supervision of a specially tailored loss function with particle size and shape perception terms, can effectively capture the profiles of coarse soil particles. After adequate training, the network can precisely segment coarse soil. The formula predicting the hydraulic conductivity of coarse-grained soil, based on CT image analysis and fitting, reveals a particular pattern in coarse-grained soil’s hydraulic conductivity when the particle size range remains constant. If the particle size distribution is uneven, smaller particles can fill in the pores formed within the large particle skeleton, leading to lower porosity and permeability. For a constant particle size distribution, a larger aspect ratio of coarse-grained soil (indicating more needle-shaped particles) results in a smaller discount factor, representing the loss degree of actual seepage channel due to pore connectivity, and hence lower permeability.

Our results further the research conducted by other scholars through seepage tests [58,59] and discrete element simulations [60,61,62,63], microscopically confirming, based on CT images, that the hydraulic conductivity of coarse-grained soil primarily depends on the characteristic particle size d₂₀, 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].

However, it is important to note some limitations. Although we have accurately segmented the CT images of coarse-grained soils using U-Net convolutional neural networks, the study of network structure optimization and enhancing learning efficiency remains unexplored. Future research should aim at optimizing the deep learning model for studying coarse-grained soil CT images, with a focus on improving learning efficiency.

5. Conclusions

In this study, we innovatively utilized a Convolutional Neural Network (CNN) model for the accurate segmentation of coarse-grained soil CT images, making significant strides in soil granulometric analysis. Our approach outperforms traditional methods, revealing new empirical formulae to comprehend the granulometric characteristics and hydraulic conductivity of coarse-grained soils, thereby offering fresh insights into soil hydrodynamics. Some valuable conclusion are as follows:

• 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.