Permeability Estimation of Engineering-Adapted Clay–Gravel Mixture Based on Binary Granular Fabric

Abstract

Clay–gravel mixture is an increasingly popular material used in geotechnical engineering for its engineering adaptability and easy accessibility. Among various granulometric factors, gravel content plays a critical role in the alteration of mixture microstructure. Its influence on mechanical behavior has been comprehensively investigated, yet the hydraulic models accounting for the paired impact of clay and gravel particles are seldomly discussed. In an effort to enhance the permeability prediction capability of this soil, a generalized binary model derived from a theoretical hydraulic conductivity expression is proposed, with the participation of two fundamental compound seepage models. High accuracy between test and calculation results indicates the reliability of this model, as well as its supremacy over conventional models. The parameter sensitivity analysis demonstrates that the proposed model, being of convincing parametric stability regardless of variant particle size distribution characteristics, has the potential to be applicable to a wide range of engineering-adapted CGMs. The predictive formula for cohesive fraction and the anomaly coefficient, as is integrated into the binary model, are explicitly discussed. Suitable for clay–gravel materials under a transitional soil state for engineering applications, this model provides a quantitative and reasonable evaluation of hydraulic conductivity with high practicality. The above findings might work as a perspective for the credible assessment of structure seepage safety behavior, as well as a quantitative evaluation method regarding the mixing quality of CGMs.

1. Introduction

Soil is a naturally formed, easily accessed material available on earth, and it is normally categorized by grain size from an engineering perspective. The clay–gravel mixture (CGM), which mainly consists of clayey soil and gravelly soil, can be deposited in nature [1], as well as artificialized by deliberate blending [2].

Seepage, strength, and deformation are key engineering properties studied in soil mechanics, and they play a crucial role in structure design and maintenance [3,4,5]. Unlike purely cohesive soils or non-cohesive soils, CGMs represent a typical binary granular fabric, characterized by their complex behavior in both mechanical and permeability aspects [6]. Between them, the mechanical property has attracted continuous academic attention, and it has been found that gravel aggregate content has a comprehensive influence on the compressibility, shear strength, and stress–strain relationship and a variety of indexes [7,8,9]. Shafiee and Yamada [10,11], among other pioneers, conducted dynamic modulus damping tests on kaolin clay mixed with gravel and sandy clay to investigate the effects of gravel content and gravel type. Their findings indicate an increase in the maximum dynamic shear modulus as the gravel content increases. Meidina et al. [12] also probed the influence of gravel content on the damping ratio. A critical gravel content value is found, below which the damping ratio increases with the gravel content. A stable soil skeleton is witnessed upon reaching the critical value, representative of the typical binary granular fabric of CGMs.

The hydraulic characteristics of CGMs, in the meantime, have scarcely been studied in an explicit manner in terms of quantitative modeling. Several empirical formulas, by displaying their applicability in other types of soil mixture, confront the challenges of estimating the permeability coefficient of a CGM [13]. This may be because the particle-size parameters in these equations are often replaced by an equivalent particle size, which is not fully representative of the porous structure characteristics of CGMs [14,15]. Another reason can be attributed to its wide gradation of soil particles, as the empirical equation is typically derived from data pertaining to a specific soil type. Rosas et al. [16] compared calculation results with measured permeability data by adopting 20 empirical equations and found a drastic deviation of up to 500% between the theoretical and experimental values. In other words, the permeability assessment of CGMs utilized in constructions is heavily reliant on traditional seepage experiments, and the research emphasis is commonly placed on the piecewise relationship between the permeability and the gravel content [17]. Regardless of the experimental approach used to determine the hydraulic conductivity of CGMs, it is highly beneficial to foster an understanding of hydraulic connection to intrinsic parameters of clay fraction and gravel fraction. This could be ascribed to the high cost and long duration of permeability testing, as well as the difficulty in soil sampling.

The artificial CGM, due to its rapid engineering popularity and decisive role in project-related safety practices, is focused on in this paper. A typical illustration of such a CGM is shown in Figure 1. Compared to CGMs accumulated in nature, this type of material has a narrow range of gravel content [18,19]. Induced by the inherent variance in the grain diameter, the inhomogeneity with respect to the porous structure also tends to be less dominant because of the admixing process applied for the on-site preparation of artificial CGM. The above features indicate that the hydraulic conductivity investigation of this earth material requires a distinctive approach.

To this end, a theoretical microstructural model based on the binary granular fabric of CGMs is proposed in this study. An integral simplification of this model is thereafter conducted to provide a convenient means of predicting the mixture’s hydraulic conductivity. A series of permeability tests are performed using soils sampled on site, and the prediction accuracy is comparatively analyzed. Variables integrated in the model, including the permeability of cohesive fraction and the anomaly coefficient, are discussed and interpreted. A semi-empirical method for the estimation of the permeability of artificial CGMs is also developed for practicality.

2. Revisiting the Existing Hydraulic Models

Hydraulic conductivity modeling can save time and manpower in engineering applications. So far, these kinds of models are commonly categorized to determine the hydraulic conductivity of cohesionless and cohesive soils [20]. Cohesionless soil is regarded as the ensemble of relatively coarse particles with grain diameters exceeding 0.075 mm. As its hydraulic performance primarily depends on the stacking state yielded under the influence of particle granulometry and stress distribution, an equation derived from retrospective modeling analysis is summarized by Vuković and Soro [21] and can be written as

$$k=f_f\cdot C\cdot f\left(n\right)\cdot d_e^2$$

(1)

where k = the hydraulic conductivity, f_f = the function of fluid properties, C = the sorting coefficient, f(n) = the porosity function, and d_e = the effective grain diameter.

Following this cardinal form, typical expressions of hydraulic conductivity with respect to cohesionless and cohesive soil are listed in Table 1. Despite the similar application of characteristic particle size, as well as the porous state in most cases, the applicable range varies among different models according to the dataset utilized for calibration.

It is, on the other hand, of greater intricacy to describe the permeable properties of cohesive soil. The reasons could be attributed to a more sophisticated micro-structure, with the participation of a double electric layer, together with bound water, within the soil matrix [22,23]. Quantitative modeling in this scenario not only relies on parameters from the grain-size and packing perspectives but also requires a comprehensive granulometric understanding, such as specific surface area (S_s) and liquid limit (w_L) [20,24,25]. Representative formulas that are known to bear a reasonable accuracy for a broad cross-section of cohesive soil are also displayed in Table 1.

Table 1. Existing representative permeability models for cohesionless and cohesive soil.

Existing Representative Permeability Models	Expression	Note
K-C non-cohesive permeability model [26]	$k_G={\displaystyle \frac{1}{C_k}}{\displaystyle \frac{{\rho }_{w}g}{{\mu }_w}}{\displaystyle \frac{e^3}{1+e}}{d_{10}}^2$	Ck = geometrical factor; ρw = water mass density; g = gravitational acceleration; μw = water viscosity; e = void ratio; d10 = diameter for which 10% of all particles are smaller
K-C cohesive permeability model	$k_G={\displaystyle \frac{1}{C_k}}{\displaystyle \frac{{\gamma }_w}{{\mu }_w}}{\displaystyle \frac{1}{s_0^2}}{\displaystyle \frac{e^3}{1+e}}$	γw = unit weight of water; S0 = specific surface area
K-C cohesive revised permeability model [27]	$k_R=C{\displaystyle \frac{e_t^{3m+3}}{{\left(1+e_t\right)}^{5/3m+1}{\left[{\left(1+e_t\right)}^{m+1}-e_t^{m+1}\right]}^{4/3}}}$	C = γw/(CF × ρ2m × μw × S02); CF = dimensionless shape constant; ρm = particle density of soil
Chapuis–Aubertine model [28]	$\mathit{log}k=0.5+\mathit{log}\left({\displaystyle \frac{e^3}{G_s^2{S}_S^2\left(1+e\right)}}\right)$	Gs = dimensionless factor

Table 1. Existing representative permeability models for cohesionless and cohesive soil.

Existing Representative Permeability Models	Expression	Note
K-C non-cohesive permeability model [26]	$k_G={\displaystyle \frac{1}{C_k}}{\displaystyle \frac{{\rho }_{w}g}{{\mu }_w}}{\displaystyle \frac{e^3}{1+e}}{d_{10}}^2$	Ck = geometrical factor; ρw = water mass density; g = gravitational acceleration; μw = water viscosity; e = void ratio; d10 = diameter for which 10% of all particles are smaller
K-C cohesive permeability model	$k_G={\displaystyle \frac{1}{C_k}}{\displaystyle \frac{{\gamma }_w}{{\mu }_w}}{\displaystyle \frac{1}{s_0^2}}{\displaystyle \frac{e^3}{1+e}}$	γw = unit weight of water; S0 = specific surface area
K-C cohesive revised permeability model [27]	$k_R=C{\displaystyle \frac{e_t^{3m+3}}{{\left(1+e_t\right)}^{5/3m+1}{\left[{\left(1+e_t\right)}^{m+1}-e_t^{m+1}\right]}^{4/3}}}$	C = γw/(CF × ρ2m × μw × S02); CF = dimensionless shape constant; ρm = particle density of soil
Chapuis–Aubertine model [28]	$\mathit{log}k=0.5+\mathit{log}\left({\displaystyle \frac{e^3}{G_s^2{S}_S^2\left(1+e\right)}}\right)$	Gs = dimensionless factor

Irrespective of the above thrilling progress, few studies focus on the hydraulic modeling of soil mixtures, especially those comprising an unignorable portion of both cohesionless and cohesive particles. Zhang et al. [29] resorted to using the Kozeny–Carman (K-C) equation to predict the saturated hydraulic conductivity of gravel–sand binary mixtures by adjusting the porosity and effective grain diameter. Schneider et al. [30] proposed a geometric mean model to describe the permeability of the clay–silt mixture with varying clay proportions. Attempts were performed by Gao et al. [31] to develop the conductivity estimation model of well-graded gravelly soil depending on the Kozeny equation integrated with pore structural parameters.

3. Establishment of Hydraulic Model—Binary Granular Fabric

Clay–gravel mixture, which is typically utilized in geotechnical engineering structures, fulfills its qualification standard when the fine content exceeds 15% and the gravel content remains between 30% and 45%. Taking into account the transient structural type in this gravel-content interval [32], the impacts of neither parts can be regarded peripheral. Therefore, a binary granular model has to be conceptualized, so that the hydraulic conductivity of CGMs, which reflects variant soil properties, can be estimated later.

The elemental structure models within the soil mixture are greatly consistent with the ideal layer arrangements of different soils [33]. Stated another way, the layer direction can be either perpendicular or horizontal to the seepage orientation, and no intermediate layer constituted by clay–gravel mixture exists (see Figure 2). These definitions derived from the domain of electricity and previously discussed in past studies are retained in this study. In the parallel arrangement, water can infiltrate separately through both clayey and gravelly soils, as shown in Figure 2a, with the overall hydraulic conductivity being largely determined by the gravel. Conversely, in the series arrangement, water seeps consecutively through two different soils, as illustrated in Figure 2b, with the overall hydraulic conductivity primarily influenced by the clay [34].

In the case of the binary granular fabric formed in series, the hydraulic conductivity of soil mixture has the following expression:

$$K_{mix-s}={\displaystyle \frac{K_{Cl}{K}_{Gr}}{C_{Cl}{K}_{Gr}+C_{Gr}{K}_{Cl}}}$$

(2)

where K_mix-s = the hydraulic conductivity of soil mixture formed in series, K_Cl = the hydraulic conductivity of clayey soil, K_Gr = the hydraulic conductivity of gravelly soil, C_Cl = the proportion of clayey soil by weight, and C_Gr = the proportion of gravelly soil by weight.

In contrast, in the case of binary granular fabric arranged in parallel, the hydraulic conductivity of soil mixture can be expressed as

$$K_{mix-p}=C_{Cl}{K}_{Cl}+C_{Gr}{K}_{Gr}$$

(3)

where K_mix-p = the hydraulic conductivity of the soil mixture formed in parallel.

It is worth noting that in the microstructure of authentic CGM material, the distribution of gravel particles adheres to a stochastic pattern [35]. This can be ascribed to the natural process referred to as sedimentation and self-filling, as well as the admixing technologies obliged during the preparation stage. In order to reflect such randomness, a schematized granular model is herein illustrated in Figure 3. In this schematized granular model, it is assumed that the macro-pores within the gravel particles are filled with clay particles. The hydraulic balance analysis regarding the flow rate, as well as the water head loss, in the unit section ΔL is conducted in the seepage direction as follows:

$$Q\left(\Delta L\right)=K_{Cl}{i}_{Cl}\left(\Delta L\right)\Delta L+K_{Gr}{i}_{Gr}\left(\Delta L\right)\Delta L$$

(4)

$$\Delta h=i_{Cl}\left(\Delta L\right)C_{Cl}\left(\Delta L\right)\Delta L+i_{Gr}\left(\Delta L\right)C_{Gr}\left(\Delta L\right)\Delta L$$

(5)

where Q (ΔL) = the flow rate at section ΔL, i_Cl (ΔL) = the hydraulic gradient dispersed within clayey soil, i_Gr (ΔL) = the hydraulic gradient dispersed within gravelly soil, C_Cl (ΔL) = the proportion by weight of clayey soil at section ΔL, and C_Gr (ΔL) = the proportion by weight of gravelly soil at section ΔL.

By applying Darcy’s law to Equations (4) and (5), the hydraulic conductivity of the soil mixture at section ΔL (K_mix (ΔL)) can be written as

$$K_{mix}\left(\Delta L\right)={\displaystyle \frac{K_{Cl}{K}_{Gr}}{C_{Cl}\left(\Delta L\right)K_{Gr}+C_{Gr}\left(\Delta L\right)K_{Cl}}}$$

(6)

Therefore, the global hydraulic conductivity of the soil mixture (K_mix) can be calculated following the formula regarding infinite soil layers arranged in parallel:

$$K_{mix}={\displaystyle \sum K_{mix}\left(\Delta L\right)\frac{\Delta L}{L}}={\displaystyle \frac{K_{Cl}{K}_{Gr}}{L}}{\displaystyle \underset{0}{\overset{L}{\int }}\frac{dl}{C_{Cl}\left(l\right)K_{Gr}+C_{Gr}\left(l\right)K_{Cl}}}$$

(7)

where L = the side length on the upstream facet, C_Cl (l) = the proportion function of clayey soil at different sections, and C_Gr (l) = the proportion function of gravelly soil at different sections.

Equation (7) displays the theoretical calculation method used to determine the global hydraulic conductivity of the soil mixture. Among variant structural constitutions, the arrangements in series and in parallel are two particular conditions. By assuming that C_Cl (l) and C_Gr (l) remain unchanged in the interval of L, Equation (7) can be transformed into Equation (2). Another transformation from Equation (7) to Equation (3) would be committed under the establishment of the piecewise functions C_Cl (l) and C_Gr (l).

However, this equation is difficult to solve, especially considering the intricate granulometric arrangements within the soil mixture. In engineering projects, the qualified treatment of a CGM involves a thorough stirring and mixing process [19]. It is consequently rational to postulate that gravel particles are scattered in a fairly even manner, as well as that C_Cl (l) and C_Gr (l) tend to be two constants that approximate C_Cl and C_Gr, respectively, in most cases. To quantitatively evaluate the probability of abnormal conditions in which a discrepant pattern prevails with respect to C_Cl (l) and C_Gr (l), an anomaly coefficient noted as α_A is herein introduced. The value of α_A ranges from 0 to 1, and this coefficient is capable of differentiating the hydraulic conductivity model of soil mixture from that under series conditions when α_A ≠ 0. Equation (7) can, thus, be expressed as follows:

$$K_{mix}=\left(1-{\alpha }_A\right){\displaystyle \frac{K_{Cl}{K}_{Gr}}{C_{Cl}{K}_{Gr}+C_{Gr}{K}_{Cl}}}+{\displaystyle \frac{K_{Cl}{K}_{Gr}}{L}}{\displaystyle \underset{0}{\overset{{\alpha }_A}{\int }}\frac{dl}{C_{Cl}\left(l\right)K_{Gr}+C_{Gr}\left(l\right)K_{Cl}}}$$

(8)

This formula, as before, fails to provide a feasible method for the determination of the hydraulic conductivity of the soil mixture, so a further simplification with regard to the integral term is necessitated. Past studies show that when the proportion of gravel content exceeds a threshold value (varies from 25% to 30%), the gravel particles transition from the floating state to the contacting state [36]. The extreme condition concerning the seepage channel emerges when the conduit is formed by gravel throughout. Given the strict control of gravel content in the utilization of CGM, the most undesirable circumstance is considered to simplify Equation (8). In other words, it is idealized that flow channels under abnormal conditions can be delineated as gravel-formed seepage channels and clay-formed seepage channels, and the percentage of these two channels equals C_Gr and C_Cl, respectively. Then, a relatively intuitive relationship can be established:

$$K_{mix}=\left(1-{\alpha }_A\right){\displaystyle \frac{K_{Cl}{K}_{Gr}}{C_{Cl}{K}_{Gr}+C_{Gr}{K}_{Cl}}}+{\alpha }_A\left(C_{Cl}{K}_{Cl}+C_{Gr}{K}_{Gr}\right)$$

(9)

It is worth remarking that when α_A equals 1, a mathematical coincidence lies between Equation (9) and Equation (3). However, such consistency in terms of mathematical expression does not indicate that real soil mixture is arranged in parallel, because the prerequisite of equation transformation involves the idealization of seepage channels. This simplification method merely facilitates the quantitative evaluation of hydraulic conductivity, with no intention of fully reflecting the sophisticated microstructure within a porous medium.

As is revisited in the previous section, different permeability models for non-cohesive soil have distinctive ranges of application. Chapuis [20] conducted a comprehensive review on predictions of saturated hydraulic conductivity and ascertained the supremacy of the K-C formula in a broad gradation range. As later examined and verified by numerous experimental studies [26], this semi-empirical approach delineates the hydraulic conductivity by the function-of-void ratio and effective grain size. Given the facile determination of both parameters, the K-C model is, therefore, relied on to obtain K_Gr.

In Equation (9), another important coefficient, K_Cl, is of equal significance, if not more important. In fact, the engineering participation of a CGM is primarily because of its low-permeability characteristics compared to cohesionless material. The cohesive content within the mixture contributes to this hydraulic behavior, so that the preference of prediction models for cohesive soil is explicitly investigated. As they are widely recognized to be of high prediction accuracy, two formulas referenced as the K-C cohesive model and the Chapuis–Aubertin model are used [20]. In light of the non-uniform force chain constituted among coarse particles and fine particles, a modified K-C model incorporating the concept of the effective-void ratio is adopted for comparison [27,37]. To further enhance the practicality of the method proposed in this work, a correlated equation is applied to estimate S_s with the following expression [28]:

$${\displaystyle \frac{1}{S_s}}={\displaystyle \frac{1.3513}{{\omega }_L}}-0.0089$$

(10)

where w_L = the liquid limit of clayey soil.

4. Variant-Head Seepage Tests

4.1. Soil Intrinsic Properties

The experimental test material was collected from a high embankment dam in Markam County, Tibet Region. Utilized as the filling material of core wall, this target soil, in its peculiar form, is a blend of clayey soil and gravelly soil. In-field sampling was conducted at different locations of an elongated-shaped core wall at varied depths. The massive use of mechanical vibration, which accelerates the admixing, as well as the tamping, process could inevitably result in an inhomogeneous distribution of gravel particles. For this reason, target samples drilled by boreholes consisted of variant proportions of gravel content.

To investigate the hydraulic properties of CGMs reconstituted in dam projects, a conventional variant-head permeameter was applied according to GB/T 50123-2019 [38]. As a result of the specimen dimension, gravel particles with grain sizes larger than 20 mm were sieved and removed. The physical properties and grain size distribution (GSD) curves of the resultant mixtures are shown in

Table 2. Physical properties and permeability results of CGMs.

No.	Specific Gravity Gs	Dry Density Ρd (G/Cm3)	Optimum Moisture Content Ωop (%)	Maximum Dry Density Ρmax (G/Cm3)	Proportion of Clay Ccl (%)	Proportion of Gravel Cgr (%)	Hydraulic Conductivity Ksat (Cm/S)
CG-1	2.71	2.21	4.4	2.25	25.9	42.6	2.42 × 10−6
CG-2	2.71	2.21	5.1	2.25	21.9	48.5	4.33 × 10−6
CG-3	2.71	2.21	5.6	2.25	17	54.2	8.68 × 10−6
CG-4	2.71	2.23	4.9	2.28	18.7	53.4	3.89 × 10−6
CG-5	2.70	2.18	4.8	2.22	18.7	52.8	7.92 × 10−6
CG-6	2.65	2.08	7.0	2.12	16.7	45.3	8.93 × 10−6
CG-7	2.65	2.06	7.6	2.10	16.9	41.8	6.58 × 10−6
CG-8	2.66	2.02	8.5	2.06	24.8	38.5	8.24 × 10−6
CG-9	2.64	2.07	7.1	2.11	22.5	38.9	6.29 × 10−6
CG-10	2.69	2.14	6.4	2.18	26.1	37.3	1.66 × 10−6
CG-11	2.73	2.18	6.4	2.22	22.8	52.2	9.15 × 10−6
CG-12	2.72	2.21	5.3	2.25	19.1	54.4	9.87 × 10−6
CG-13	2.72	2.18	5.2	2.22	29.1	39.8	1.26 × 10−6

and Figure 4, respectively. A brief description of the seepage experiment conducted in this study is herein given, as the test procedure is officially standardized. A conventional variant-head permeameter TST-70 (Nanjing Soil Instrument Factory Co. Ltd., Nanjing, China), with a specimen dimension of Ø 100 × 400 mm, was used in our case (see Figure 5). Installed on the lateral side of the permeameter, two drilling holes were attached with plastic tubes, so that the water inlet and outlet of the specimen could be provided. This design enabled the water flow to continuously circulate throughout the permeability test, and the hydraulic conductivity was deduced using the following expression:

$$k=2.3{\displaystyle \frac{a{L}_1}{A\Delta t}}\log \left({\displaystyle \frac{h_1}{h_2}}\right)$$

(11)

where a = the cross-section of the measuring tube, L₁ = the seepage length within the specimen, A = the cross-section of the specimen, Δt = the duration of the permeability test, h₁ = the initial water head, and h₂ = the water head at the end of test.

The dry remolded soil was saturated to its optimum moisture state, after which a stepwise filling method was conducted and a static compaction process was performed. With the desired dry density of mixture being achieved, the vacuum saturation method was selected to fully saturate the specimen. Then variant-head permeability tests were carried out in a sufficient time period (24 h minimum). The test results unveiling the hydraulic conductivity of each soil sample are also listed in Table 2.

4.2. Comparative Analysis between Models

In order to assess the applicability of existing hydraulic models to engineering-adapted CGMs, the models focusing on cohesionless soil, as well as cohesive soil, were selectively applied. Both simple forms of the binary model (in-series and in-parallel models) were equally examined, and afterwards, the idealized binary model derived from structural characteristics was constructed and placed under evaluation.

By using the widely recognized empirical or semi-empirical permeability equations listed in Table 1, we made relevant predictions with an emphasis on the comparison of measured hydraulic conductivity, as displayed in Figure 6a. The classical model for non-cohesive soil offered an overestimated prediction, with the majority of the calculated permeability being larger than 10⁻⁴ cm/s. To the contrary, all predictive models for cohesive soil underestimated the permeability of test soil, with their calculation results ranging from 10⁻⁸ to 10⁻⁹ cm/s. None of the above formulas provided a satisfactory estimation, which is hardly surprising as a result of the irrational applicable scenarios. Specifically, these equations were developed from porous media with the uniform hydraulic characteristics, so that they appeared to inappropriately consider the admixing influence of clay and gravel content [39].

The estimating results obtained by binary models in series (Equation (2)) and in parallel (Equation (3)) are also plotted in Figure 6a. Note that only the modified K-C formula was used in both synthetical models for graphical conciseness, as predictions provided by different cohesive-soil-related equations were of high similarity. It can be seen that the deviation of model predictions narrows to a certain extent compared to that determined from existing equations. However, an unacceptable margin of an order of magnitude, on average, larger or smaller could be indicative of an oversimplified misrepresentation from the granulometric perspective.

Prior to the utilization of Equation (9), the value of the anomaly coefficient α_A was crucial. The requisite number of samples for the calculation of α_A was a mere four. The most appropriate value of α_A was selected on the basis of the measured values of the four groups of samples in order to ensure that the four groups of calculated values were as closely aligned with the measured values as possible. This value of α_A could then be used for the hydraulic conductivity coefficient calculation of all the samples in the project. The α_A of the three models were 5.3 × 10⁻², 5.5 × 10⁻², and 6.4 × 10⁻², respectively.

The conceptual binary model, as is ideally interpreted in Equation (9), has great potential for the permeability description of CGMs. Three series of prediction results based on distinctive hydraulic models for cohesive soil are herein illustrated in Figure 6b, and a striking advancement in terms of model accuracy can be found throughout. The related determination coefficient (R²) embedded in Figure 6b further proves the rationality of this model.

5. Parameter Sensitivity Analysis

Parameter sensitivity analysis is an essential tool for identifying the key parameters of a model [40]. It helps in understanding the contributions of various parameters to the model’s output quantitatively, as well as in assessing the impacts of different parameters on the model’s estimation [41]. In our case, the hydraulic conductivity of clayey soil, the hydraulic conductivity of gravelly soil, and the anomaly coefficient are critical indexes for evaluating the permeability of CGMs. Both conductivity values are affected by the empirical model that is being selected, whilst the anomaly coefficient reflects the extent to which the CGM is well admixed. Therefore, a sensitivity analysis regarding the above three inputs is performed following a program in

Table 3. Parameter sensitivity analysis program.

Parameters Base State	αA	KGr	KCl
Anomaly coefficient, αA	2 αA, 5 αA, 10 αA, αA/2, αA/5, αA/10	KGr	KCl
Permeability of gravelly soil, KGr	αA	2 KGr, 5 KGr, 10 KGr, KGr/2, KGr/5, KGr/10	KCl
Permeability of clayey soil, KCl	αA	KGr	2 KCl, 5 KCl, 10 KCl, KCl/2, KCl/5, KCl/10

[42]. It is worth noting that the proportions of clay and gravel are also vital factors, yet they are determined by the sieve analysis. Soils with six different gradations are selected, so that the analysis results can be compared in the presence of variant grain-size distribution characteristics.

According to Table 3, the parameter being investigated is changeable, while the others are fixed, and the changing ratio varies from 0.1 to 10 for all three parameters. Figure 7 illustrates the impacts of three parameters on the calculated hydraulic conductivity K_mix of the CGM at different input levels. It is found that with the anomaly coefficient increasing, the permeability of CGM increases for all soil specimens, which is consistent with the definition of α_A.

Meanwhile, from a hydraulic perspective, a discrepant pattern can be revealed for two hydraulic conductivity parameters. Despite the similar trend of increasing hydraulic conductivity of both gravelly soil and clayey soil leading to an increase in K_mix, the impact of K_Gr is more decisive compared with that of K_Cl. Specifically, as K_Cl becomes ten times larger or smaller, the variation in K_mix is less than 5%, whereas as K_Gr follows the same changing level, the variation in K_mix remains at the same level. This can be ascribed to the significant divergence between K_Cl and K_Gr. The value of K_Cl is primarily responsible for the hydraulic behavior of the pure clay channel in the parallel arrangement (Figure 2a), as well as that of the clay–gravel channel in the series arrangement (Figure 2b). As the permeability of the clayey soil is several orders of magnitude smaller than that of gravelly soil, the flow rate induced by these channels is relatively small. On the contrary, the value of K_Gr is primarily responsible for the hydraulic behavior of the pure gravel channel in the parallel arrangement (Figure 2a). This type of channel could be regarded as the percolating flow path, and it has the potential for triggering a strikingly large flow rate. Therefore, the impact of the permeability of gravelly soil is higher than that of clayey soil on the permeability of CGM. This also suggests that when using this model to calculate hydraulic conductivity of CGM, attention should be paid to the determination of the anomaly coefficient and the hydraulic conductivity of gravelly soil.

On the other hand, the above parametric finding shows its consistency regardless of the particle size distribution characteristics of the test soil. With the proportion of gravel ranging from 37.3% to 54.4%, it can be seen that the prediction capability is consistent. The proposed model, which is both of high accuracy and high stability, shows its potential to be applied to a wide range of engineering-adapted CGMs.

6. Discussion

Hydraulic conductivity reflects the structures of pore channels through which seeping water passes on a global scale. Regarding severely non-uniform earth materials such as CGMs, their hydraulic conductivity levels are significantly dependent on gravel-involved pores, which are of larger size, provided the existence of interactions between coarse particles [43]. In this scenario, the permeability of fine content can be deemed neglected due to the drastic discrepancy in seepage behavior. Clay models established under variant microstructural generalizations, therefore, have a marginal influence on the credibility of our proposed binary model. This also indicates that no strict criterion shall be obeyed in terms of clay model application, as long as the necessary parameters can be reasonably determined. Such flexible applicability could potentially increase the practicality of this model, especially in the case of in-field estimation of hydraulic conductivity.

Throughout the simplification process, the only parameter additionally assumed is the anomaly coefficient α_A. It shares an identical numerical indication of weight coefficient, yet the proposition of this coefficient remains part of the generalized method to delineate structural anisotropy. Irrespective of the selection of the clay model, the value of α_A derived from regression analysis is small in a similar manner (see the embedment in Figure 6b). This indicates that the undesirable condition under which the infiltration channel is thoroughly constituted by coarse particles rarely prevails. The artificially blended CGM, as displayed in Figure 1, further proves this explanation. As is interpreted above, during the preparation of CGMs in engineering projects, implementation equipment such as a large-scale mechanical agitator and a roller are adopted to decrease the inhomogeneity of the soil mixture. It is, therefore, reasonable to witness the commonness with respect to low anomaly probability.

On the other hand, the microscopic configuration of CGMs is reflected in this binary model. Depending on the amount of coarse fraction content in this particular soil, the related microstructures can be identified as different inter-granular matrix phases [44]. The classification proposed by Park and Santamarina [32] is adopted, according to which the description of the soil state varies from fine-dominant and transitional to coarse-dominant with gravel content increasing. On the aspect of stress–strain behavior, the transitional state is particularly preferred as a result of the efficiency on deformation compatibility control [7]. Under this circumstance, contacts are formed among coarse particles, and the inter-particle voids consequently exist [45]. This implies that the hydraulic conductivity of such foundation soil is inherently governed by both fractions, which is consistent with the establishing procedure of our model. The permeability of mixtures following the other two states, as a consequence, cannot be estimated using the binary model. In other words, the permeability prediction of this model may be biased either when the clayey part prevails within the soil matrix or when the gravel content is dominant enough that macro-pores among gravel particles remain unfilled with clay. Another imperfection related to the incapability of considering the unsaturated state can also be found in our model. The introduction of soil water retention characteristics would complicate the model, and it can be challenging given the difficulty of ensuring the precise control of the degree of saturation in the case of coarse-grained CGMs [46]. Our model also shows its practicality by avoiding the use of stress-related parameters. The porous state, which is measurable on site, is herein selected as an alternative, so that the model expression follows the cardinal form of Equation (1).

A direct engineering implication of our method, as a consequence, can be summarized as the more precise estimation of the hydraulic characteristics of engineering-adapted CGMs. This can be notably realized with the construction records and limited boring samples, instead of extensive large-scale seepage experiments. As specific impervious materials used in hydraulic engineering, CGMs play a vital role in seepage safety. A reasonable permeability calculation of this soil could increase credibility when determining the phreatic line. A comparison between this theoretical value and the measured one could give reference to the habitual assessment of structure seepage safety behavior. In addition, as a parameter to delineate the structural anisotropy, α_A is capable of evaluating the mixing quality and, thus, possesses the potential to manage construction quality in a quantitative manner.

7. Semi-Empirical Method to Predict the Hydraulic Conductivity of Engineering-Adapted CGMs

Based on the aforementioned interpretation, a semi-empirical method for predicting the hydraulic conductivity of CGMs under a transitional soil state was developed. The calculation procedure composed of multiple steps is summarized as follows:

The first priority remains the segmentation of cohesive (fine) soil and cohesionless (coarse) soil with a separatrix grain size of 0.075 mm. Then, the portion of coarse particles is verified to ensure that this method is applicable to the designated earth material.

Proper permeability formulas are applied to calculate K_Gr and K_Cl separately. The K-C non-cohesive formula is recommended for K_Gr because of its uncomplicated form, wide range of applicability, and high accuracy, while no preference is given to formulas determining the conductivity of the cohesive fraction. The calculation of K_mix-s and K_mix-p is then followed.

Limited permeability tests regarding different soil samples are conducted as a calibration to determine the anomaly coefficient α_A. The number of tests depends on the in situ condition, and it is recommended that no less than four tests are carried out to fulfill the statistical requirements.

The semi-empirical model for the estimation of the hydraulic conductivity of CGMs under transitional state is, thus, constituted without the existence of undetermined parameters. It can significantly elevate the estimating precision of soil dominated by both clay and gravel fractions, and only a limited number of calibration tests are necessitated.

8. Conclusions

A CGM is classified as a non-conventional engineering material that involves the dual participation of clay and gravel content from a hydraulic perspective. A semi-empirical model based on the typical binary granular fabric of CGM was introduced in this paper to predict its permeability coefficient, and a parametric analysis was implemented to evaluate the impact of three fundamental factors in our model.

The existing estimation methods for soil permeability, with emphasis placed on either cohesive soil or non-cohesive soil, are outranged in the prediction of artificial CGM. Predictive values obtainable by multiple previous methods were calculated, and a significant error was produced in comparison with experimental results. By taking into account the inherent structural characteristics of CGMs, a theoretical binary model of blended mixtures was proposed, with further mathematical simplifications being conducted for practicality. Despite the poor accuracy resulting from the basic models (in-parallel and in-series models), the generalized binary model demonstrates its high precision for permeability prediction.

The universally high R² and the commonly low level of α_A shows that no strict restriction on the selection of a cohesive-related permeability model is needed. This is further proved by the sensitivity analysis of the three parameters, where rather than K_cl, the anomaly coefficient α_A and the hydraulic conductivity of gravel K_Gr have substantial impacts on the model prediction results. The reason appears to be the transitional state identified as the inter-granular matrix phase within CGMs, together with the huge hydraulic discrepancy between clay and gravel particles. It is demonstrated that the proposed model, being of convincing parametric stability regardless of the variant particle size distribution characteristics, could be applicable to a wide range of engineering-adapted CGMs. In addition, as an indication of mixing homogeneity, low α_A suggests that such an artificial earth material is relatively well blended on the construction site, which is essential in embankment quality control. A practical application of this binary hydraulic model was, thus, introduced, by fixing the necessary parameters, to provide a more intuitive illustration. The implications of this work may include a credible assessment of structure seepage safety behavior in the presence of CGMs and a quantitative method to evaluate its mixing quality.