Bayesian-Based Standard Values of Effective Friction Angle for Clayey Strata

Abstract

In this study, a methodology using probabilistic distribution techniques to determine the parameters of the soil’s effective internal friction angle ($\phi ’$) was proposed. The method was grounded in quantitative survey information extracted from geotechnical reports. Extensive equivalent samples were estimated using Markov chain Monte Carlo (MCMC) simulations and probability density functions ($PDFs$). The effective internal friction angle ($\phi ’$) of silty clay layers was probabilistically characterized using the plasticity index ($P_I$), in situ static cone penetration test ($q_c$), and standard penetration test ($N_{SPT}$). A systematic quantitative analysis integrated prior information from different sources was systematically integrated with sampling data. By establishing a Bayesian framework that incorporated the regression relationship and uncertainties associated with the effective internal friction angle ($\phi ’$), the model ensured balance and symmetry in the treatment of prior information and observed data. The model was then transformed into equivalent sample values based on three models, reflecting the symmetrical consideration of different data sources. Further considerations involved correcting the three different analysis methods. A comparison of equivalent sample values with the mean values of the sampling data, along with the parameter optimization updates, was performed by combining the three models. Using three sets of sampling data, a linear relationship model for the new soil parameters was derived. The analysis results demonstrated that the proposed method could obtain equivalent samples for the effective internal friction angle.

1. Introduction

Research on soil uncertainty has always been a popular research topic. In geotechnical engineering, uncertainty and heterogeneity [1,2,3,4] are constantly occurring, such as the application of loads and movements in the Earth’s crust. These are crucial factors within the realm of uncertainty, influencing the design and analysis of geotechnical engineering. When developing numerical models, such factors are often simplified or even ignored, resulting in numerical models that may not meet the required engineering standards. Consequently, correctly developing models and simplifying the uncertainty of soil behavior are important problems to address in geotechnical engineering research [5].

In assessing soil stability, the effective internal friction angle ($\phi ’$) has consistently been a crucial parameter [6]. However, in practical engineering, it often requires a substantial volume of data to better represent the value of ${\phi }^{’}$ for a specific measurement point or area. In other words, the value of ${\phi }^{’}$can be significantly influenced by the sample volume. In situations where data acquisition is relatively limited, it can be challenging to generate meaningful statistical data based solely on field survey data. Therefore, the introduction of probabilistic distribution methods [7,8] to determine the effective internal friction angle ($\phi ’$) of the soil is a critical problem in need of resolution.

In response to the aforementioned problems, the use of probability distribution has become a popular research topic, with most scholars [9,10,11] considering the determination of the probability distribution of the effective internal friction angle as a key step in validating model reliability. The Bayesian method employing the parameter bootstrap technique has been used to characterize the joint probability density functions (PDFs) of $c’$ and $\phi ’$, demonstrating its feasibility through real case studies and sensitivity analyses [10]. Statistical methods have been applied to cone penetration test data obtained from 26 locations. By calibrating the mean, standard deviation, and correlation length of the effective internal friction angle, its distribution characteristics in rocky sediments could be determined. A comparison between semi-deterministic and probabilistic results has been conducted, providing insights into the practical use of slope stability regulations [9]. The Copula-based bivariate distribution model and Bayesian model comparison method have been employed to select the most likely bivariate distribution model for $c’$ and $\phi ’$, the proposed method being explained and validated using real data from the clay core of the Xiaolangdi rockfill dam in China [11].

The scholars [9,10,11] mentioned above had access to abundant data for probabilistic characterization in parameter analysis. However, in studies involving soil parameters or regions with limited data, the use of probability distribution methods becomes essential. When collecting soil parameters in such regions, it is crucial to adhere to the normal distribution or log-normal distribution in line with the central limit theorem [12]. Consequently, establishing the probability distribution of general soil properties becomes a highly challenging task [13]. In comparison to research conducted under data scarcity, Bayesian methods have proven to be effective in integrating prior information with limited survey data [14], transforming this information into a large number of equivalent samples, aiding in determining the statistical representation, probability distribution, standard deviation, and other soil characteristics.

The Bayesian method combines prior knowledge from different sources with project information and has found extensive applications in geotechnical engineering [15,16,17]. Using the Bayesian framework for inverse analysis and updating soil parameters based on field observations has been successfully employed to achieve responses to observations, such as maximum wall deflection or settlement. By extending the Bayesian framework, it becomes possible to simultaneously update two types of response observations [15]. In one application, a probability model—combined with the expectation–maximization algorithm, Bayesian estimation, and relevance vector machine ($RVM$)—was used for predicting convergence. This method demonstrated higher predictive accuracy compared to backpropagation neural networks and Gaussian process regression [16]. In a novel probability-based inverse analysis method for slope failure, it was found that the correlation between soil cohesion and the internal friction angle had no major impact on the posterior distribution of the slope. However, the type of prior distribution had a substantial influence [17]. Consequently, this study evolves Bayesian analysis by leveraging algorithms and the relationships between parameters, enabling effective Bayesian analysis based on given soil parameters and survey data.

Under undrained conditions, the effective cohesion of clay is the primary shear strength parameter, while the effective internal friction angle is typically small. However, under drained conditions, the effective internal friction angle plays a critical role in long-term stability, making its analysis essential for engineering design. To address the determination of standard values for the effective internal friction angle ($\phi ’$), this study proposed a method based on survey information from geotechnical reports—that is, a comprehensive study of the probabilistic characterization of the effective internal friction angle ($\phi ’$) in silty clay layers, with a particular emphasis on the plasticity index ($P_I$), in situ static cone penetration test ($q_c$), and standard penetration test ($N_{SPT}$). Through systematic integration of prior information from different sources with sampling data and the application of the Markov chain Monte Carlo (MCMC) method and PDFs for estimating equivalent samples, the regression relationship of the effective internal friction angle ($\phi ’$) could be transformed into equivalent sample values using three models. Furthermore, parameter optimization updates were considered by combining the three models, and this methodology was subsequently applied to the Wuhan Xunsi River Basin Comprehensive Management Phase II (Wutai Sluice Sewage Treatment Plant) project.

2. Materials and Methods

2.1. Theory: Probability Modeling

2.1.1. Heterogeneity Modeling Probability

After thousands of years of crustal movement, the characteristics of soil have undergone many changes, including the soil properties [18], mechanical interaction [19], and drainage characteristics [20]. Nonetheless, investigating the distribution and probability characteristics of soil properties is crucial for engineering design and analysis. When studying the effective internal friction angle as a continuous variable, researchers typically assume that it follows a probability distribution [21] to better describe its statistical features. Considering the premise that the effective internal friction angle is a non-negative value, a common assumption is to use the log-normal distribution [22,23] with a mean of $\mu$ and a standard deviation of $\sigma$, which can be expressed as follows:

$${\phi }^{’}=exp({\mu }_N+{\sigma }_{N }z)$$

(1)

where z denotes a standard normal random variable, and ${\mu }_N=ln\mu -{{\sigma }^2}_N/2$,${\sigma }_N=\sqrt{ln\left[1+{\left(\sigma /\mu \right)}^2\right]}$ denotes the mean and standard deviation of this log-normal distribution.

The expression for $ln\left({\phi }^{’}\right)$ following a normal distribution [22,23] can be expressed as follows:

$$ln\left({\phi }^{\prime }\right)={\mu }_N+{\sigma }_{N}z$$

(2)

.

Whether it is the $\sigma$ of ${\phi }^{’}$ or the ${\sigma }_N$ of $ln\left({\phi }^{’}\right)$, both can represent the probability modeling of the heterogeneity of the effective internal friction angle ($\phi ’$). Using the effective internal friction angle ($\phi ’$) of clay layers as an example, it is a critical parameter reflecting the shear strength performance of the soil. Its value is a continuous variable and typically varies within the non-negative range. In the literature [24], it is commonly represented using the log-normal distribution.

2.1.2. Transformation of Heterogeneity Probability Modeling

The effective internal friction angle of clay layers under drained conditions can be obtained experimentally. However, in some regions with harsh conditions, it is impractical to conduct on-site tests, making direct testing of soil layers challenging. Consequently, indoor experiments are employed to supplement the limitations of on-site tests. The purpose of these experiments is to obtain shear strength parameters [25]. However, the data on the internal friction angle obtained from indoor experiments often lacks sufficiency, making it challenging to derive statistically significant characteristic values. Therefore, in some small- to medium-sized engineering projects, regression equations or calculations are frequently used to estimate ${\phi }^{’}$.

In geotechnical engineering surveys, the measured parameters are not necessarily the final design parameters used in the design process. Typically, survey data need to be transformed into computational models. Common computational models [26,27,28] include Young’s modulus, horizontal stress coefficient ($K_0$), etc. They can be expressed as follows:

$$\xi =aX\prime +b+\epsilon$$

(3)

where $\xi$ denotes the measured parameter; $X\prime$ denotes the design parameter $X$ or its logarithmic form $lnX$ (equivalent to $ln\left({\phi }^{\prime }\right)$); $a$ and $b$ denote the parameters of the transformation model; $\epsilon$ denotes a standard normal random variable, representing uncertainty in the modeling process, with an average value of ${\mu }_N$ and a standard deviation of ${\sigma }_N$. Depending on the form of $X\prime$, different computational models can be used to characterize the performance of the design parameter.

After combining Equations (2) and (3), when $\xi$ follows a log-normal distribution, the resulting expression is obtained:

$$\xi =\left(a{\mu }_N+b\right)+\left(a{\sigma }_{N}z+\epsilon \right)$$

(4)

Here, assuming the independence of z and ε, the random variable $\xi$ follows a log-normal distribution with a mean of $a{\mu }_N+b$ and a standard deviation of $\sqrt{{\left(a\sigma \right)}^2+{{\sigma }_{\epsilon }}^2}$.

2.2. Methods: Determining Bayesian Framework Under Limited Data Conditions

2.2.1. Probability Distribution of Soil Parameters

Building upon Equations (1) and (2), where ${\phi }^{\prime }$ denotes a random variable following a log-normal distribution with a mean of $\mu$ and a standard deviation of $\sigma$, for a given set of survey data (Data) and prior information (Prior), the probability distribution [22] of ${\phi }^{\prime }$ can be expressed as $f\left({\phi }^{\prime }|Data,Prior\right)$. Based on this probability distribution, $f\left({\phi }^{\prime }|Data,Prior\right)$ can be expressed as follows [23]:

$$f\left({\phi }^{\prime }|Data,Prior\right)=\int f\left({\phi }^{\prime }|\mu ,\sigma \right)f\left(\mu ,\sigma |Data,Prior\right)d\mu d\sigma ,$$

(5)

where $\mu$ and $\sigma$ denote the mean and standard deviation of the effective internal friction angle ($\phi ’$), $f\left({\phi }^{\prime }|Data,Prior\right)$ denotes the probability distribution of a given mean and standard deviation under the effective internal friction angle ($\phi ’$), and $f\left({\phi }^{\prime }|Data,Prior\right)$ denotes the joint probability distribution of the mean and standard deviation under the given conditions of survey data and prior distribution.

The heterogeneity of the soil can be characterized using random variables, with commonly employed distribution types being normal distributions. For the effective internal friction angle ($\phi ’$) following a normal distribution, its probability distribution can be expressed as follows [22]:

$$f\left({\phi }^{\prime }|\mu ,\sigma \right)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}exp\left[-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\phi }^{\prime }-\mu }{\sigma }}\right)}^2\right],$$

(6)

where ${\mu }_N=ln\mu -{{\sigma }^2}_N/2$ and ${\sigma }_N=\sqrt{ln\left[1+{\left(\sigma /\mu \right)}^2\right]}$ represent the mean and standard deviation of ${ln\phi }^{\prime }$, respectively.

2.2.2. Bayesian Framework

According to the given survey data (Data) and prior information (Prior), under the Bayesian approach [29], $f\left(\mu ,\sigma |Data,Prior\right)$ can be expressed as follows [23]:

$$f\left(\mu ,\sigma |Data,Prior\right)=Kf\left(Data\right|\mu ,\sigma \left)f\right(\mu ,\sigma ),$$

(7)

where $K$ denotes a normalization constant independent of $\mu$ and $\sigma$, and $K={\left(\int P\left(Data\right|\mu ,\sigma \left)P\right(\mu ,\sigma )d\mu d\sigma \right)}^{-1}$, and $f(\mu ,\sigma )$ denotes the prior distribution of $\mu$ and $\sigma$, reflecting the prior information about $\mu$ and $\sigma$.

Rewriting the equation by substituting Equation (7) into Equation (5):

$$f\left({\phi }^{\prime }|Data,Prior\right)=K\int f\left({\phi }^{\prime }|\mu ,\sigma \right)f\left(Data|\mu ,\sigma \right)f\left(\mu ,\sigma \right)d\mu d\sigma .$$

(8)

Here, incorporating the prior distribution $f(\mu ,\sigma )$ and the likelihood function $f\left(Data\right|\mu ,\sigma )$ relies heavily on prior information. Common types of prior distributions [30] include uniform, normal, and gamma distributions, among others. For the prior distribution $f(\mu ,\sigma )$ in Equation (7), a uniform distribution can be adopted, expressed as follows [22]:

$$f\left(\mu ,\sigma \right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\left({\mu }_{max}-{\mu }_{min}\right)\left({\sigma }_{max}-{\sigma }_{min}\right)}}\\ 0 \end{array}\mu \in \left({\mu }_{min},{\mu }_{max}\right),\sigma \in \left({\sigma }_{min},{\sigma }_{max}\right),\right.$$

(9)

where ${\mu }_{min},{\mu }_{max}$ denote the minimum and maximum values of $\mu$, respectively, while ${\sigma }_{min},{\sigma }_{max}$ denote the minimum and maximum values of $\sigma$, respectively.

In accordance with Equation (4), it is evident that $\xi$ follows a normal distribution with a mean of $a\mu +b+{\mu }_{\epsilon }$ and a standard deviation of $\sqrt{{\left(a\sigma \right)}^2+{{\sigma }_{\epsilon }}^2}$. Therefore, the likelihood function for this normal distribution can be expressed as follows [22]:

$$f\left(Data|\mu ,\sigma \right)=\prod _{i=1}^n{\displaystyle \frac{1}{\sqrt{2\pi }\sqrt{{\left(a\sigma \right)}^2+{{\sigma }_{\epsilon }}^2}}}exp\left\{-{\displaystyle \frac{1}{2}}{\left[{\displaystyle \frac{ln\left[{\phi }^{\prime }\right]-\left(a\mu +b+{\mu }_{\epsilon }\right)}{\sqrt{{\left(a\sigma \right)}^2+{{\sigma }_{\epsilon }}^2}}}\right]}^2\right\}$$

(10)

After substituting Equations (6) and (11) into Equation (8), the joint PDF of ${\phi }^{’}$ can be obtained [31] as follows:

$$\begin{gathered} f\left({\phi }^{\prime }|Data,Prior\right) \\ =K\int {\displaystyle \frac{1}{\sqrt{2\pi {{\sigma }_N}^2{{\phi }^{\prime }}^2}}}exp\left[-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{ln\phi }^{\prime }-{\mu }_N}{{\sigma }_N}}\right)}^2\right]\prod _{i=1}^n{\displaystyle \frac{1}{\sqrt{2\pi }\sqrt{{\left(a\sigma \right)}^2+{{\sigma }_{\epsilon }}^2}}}exp\left\{-{\displaystyle \frac{1}{2}}{\left[{\displaystyle \frac{ln\left[{\phi }^{\prime }\right]-\left(a\mu +b+{\mu }_{\epsilon }\right)}{\sqrt{{\left(a\sigma \right)}^2+{{\sigma }_{\epsilon }}^2}}}\right]}^2\right\}f(\mu ,\sigma )d\mu d\sigma \end{gathered}$$

(11)

Upon obtaining Equation (11), this study employed the MCMC method [32,33] for simulation. Using Equation (10), a substantial volume of sample data for ${\phi }^{’}$ can be generated, allowing for the inference of the joint PDF of ${\phi }^{’}$. For any given joint PDF, the MCMC method is capable of generating numerous sample data, facilitating the characterization of the joint PDF. Particularly for complex forms of joint PDFs, such as Equation (11), the MCMC method proves to be highly applicable.

The Markov chain Monte Carlo (MCMC) method is a Monte Carlo integration approach that uses Markov chains to generate samples and perform statistical inference. By constructing a Markov chain with a stationary distribution [34], the samples eventually converge to the target distribution. This study integrates the MCMC method with Bayesian approaches, employing Metropolis–Hastings sampling to generate samples for the effective internal friction angle. This approach effectively combines field-measured data with prior distributions. When measured data are limited, the samples generated by MCMC can supplement the prior information, while their influence diminishes as more data become available. By generating a large number of $\phi ’$ samples and analyzing them through cumulative frequency plots and probability distribution functions, this method simplifies the computation of posterior joint probability density functions under complex conditions. It addresses the challenge of determining standard values for soil parameters in cases of data scarcity, thereby enhancing practicality and efficiency.

2.2.3. Analysis and Processing of Experimental Data

The experimental data used in this study were derived from the geotechnical investigation report of the Wuhan Xunsi River project. The sampled soil layer consisted of silty clay at depths of 10.8–18.8 m below the surface, with a thickness of 8 m. The investigation methods included drilling and sampling, laboratory geotechnical tests, and in situ field tests, which provided the relevant geotechnical parameters such as natural water content, density, cohesion, and internal friction angle.

In the geotechnical investigation report, the silty clay layer exhibited uniform soil characteristics and stable properties. To ensure the reliability of the experimental data, the provided data were screened, and the reliability of the geotechnical parameters was evaluated. This evaluation included statistical metrics such as sample size, maximum value, minimum value, mean, standard deviation, coefficient of variation, and standard value, as detailed in

Table 1. Statistical characteristics of experimental data for the silty clay layer.

Parameter	$\mathbf{Sample} \mathbf{Size} \left(\mathit{n}\right)$	$\mathit{m}\mathit{i}\mathit{x}$	$\mathit{m}\mathit{a}\mathit{x}$	Mean	Standard Deviation	$\mathbf{Coefficient} \mathbf{of} \mathbf{Variation} \left(\mathit{\%}\right)$	Standard Score
Natural Water Content$\left(\mathrm{\%}\right)$	30	22.4	30.8	26.6	2.5	9.33	26.0
Density$\left({\mathrm{g}/\mathrm{c}\mathrm{m}}^3\right)$	30	1.71	1.93	1.82	0.06	2.19	1.80
Cohesion $(°)$	30	18.2	26.4	22.3	2.3	9.36	22.0
Internal Friction Angle$\left(\mathrm{k}\mathrm{P}\mathrm{a}\right)$	30	18.5	32.3	25.4	3.3	13.57	24.0

.

From Table 1, it can be observed that the parameters demonstrate high concentration levels, particularly with coefficients of variation for the internal friction angle and cohesion of 9.36% and 13.57%, respectively. This indicates good stability of the soil layer. To further analyze the distribution characteristics of the parameters more intuitively, Figure 1 presents histograms and cumulative distribution function graphs for the internal friction angle and cohesion.

The concentration and variability of the data both indicate that the silty clay layer exhibits high uniformity, providing a reliable data foundation for the Bayesian framework. The mean and standard values, as important input parameters, are used for the evaluation of soil stability and bearing capacity. Calculations show that both the internal friction angle and cohesion approximately follow a normal distribution, with a positive correlation between the two. The correlation coefficient is about 0.72, which can be used to further optimize the joint distribution setting of the parameters in the probabilistic model.

2.3. Implementation Process

The diagram in Figure 2 illustrates the implementation process of equivalent samples, comprising the following five steps:

(1) Invert the basic information from the geological exploration report to derive the measured parameter $P_I$ value and transform it into data as follows: $Data=\left\{{\xi }_i=ln\left(N_i\right),i=\mathrm{1,2},3\cdots \right\}$.

(2) To determine the prior information and range for the mean (${\mu }_N$) and standard deviation (${\sigma }_N$), obtain their respective maximum (${\mu }_{max},{\sigma }_{max}$) and minimum (${\mu }_{min},{\sigma }_{min}$) values.

(3) Derive the interval size for $f(\mu ,\sigma )$ and $f\left(Data\right|\mu ,\sigma )$ at the mean (${\mu }_N$) and standard deviation (${\sigma }_N$) based on Equations (9) and (10).

(4) Choose the initial state of the effective friction angle (${\phi }^{’}$) on the Markov chain. Determine the mean (${\mu }_N$) and standard deviation (${\sigma }_N$) of prior knowledge based on Equations (9) and (10). Use the Metropolis–Hastings sampling method to generate a large number of samples for the effective friction angle (${\phi }^{’}$) according to Equations (5) and (8).

(5) After generating a large number of equivalent samples for ${\phi }^{’}$, estimate the mean (${\mu }_N$) and standard deviation (${\sigma }_N$) (or $\mu$ and $\sigma$). Use the equivalent samples to construct cumulative frequency plots and determine the probability distribution functions and cumulative distribution functions.

3. Case Study

3.1. Project Overview

Using the Wuhan Xunsi River Basin Comprehensive Management Phase II (Wutai Sluice Sewage Treatment Plant) project as an example, experimental data obtained from tests conducted in silty clay layers were used. In this region, the majority of soil layers consist of silty clay, as shown in Figure 3. A depth of 10.8–18.8 m below the ground surface was selected within the silty clay layer—that is, representing a thickness of 8 m. Statistical experiments were conducted on the silty clay layer based on the geological survey report, the fundamental information of which is summarized in

Table 2. Basic information of the silty clay layer.

Silty Clay	Number of Tests (n)	$\mathit{m}\mathit{a}\mathit{x}$	$\mathit{m}\mathit{i}\mathit{n}$	${\mathit{X}}_{\mathit{m}}$	$\mathbf{Standard} \mathbf{Deviation} \left(\mathit{\sigma }\right)$	$\mathbf{Coefficient} \mathbf{of} \mathbf{Variation} \left(\mathit{\delta }\right)$	$\mathbf{Statistical} \mathbf{Correction} \mathbf{Coefficient} \left(\mathit{\psi }\right)$	$\mathbf{Standard} \mathbf{Score} \left(\mathit{Z}\right)$
Plasticity Index	63	14.0	10.3	12.2	0.048	0.025	0.994	1.92
Cone Penetration Test	565	2.42	0.74	1.47	0.41	0.21	0.98	1.44
Standard Penetration Test	66	8	5	6.70	1.15	0.16	0.93	6.20

.

3.2. Equivalent Samples of Effective Internal Friction Angle ${\phi }^{’}$

If direct tests for the friction angle are not conducted, empirical formulas are often employed for assessment. Prior to this, numerous scholars have extensively researched the effective internal friction angle ($\phi ’$) and proposed conversion models, as summarized in

Table 3. Conversion models for the effective internal friction angle (${\phi }^{\prime }$).

Serial Number	Design Parameter	Measurement Parameter	Input Parameter	Conversion Model	$\mathit{a}$	$\mathit{b}$	${\mathit{\sigma }}_{\mathit{\epsilon }}$	Distribution Type
1	$Effective friction$ $angle, {\phi }^{’}$	$P_I$	$P_I$	$ln\left(P_I\right)=a{\phi }^{’}+b+\epsilon$	−10.638	8.511	0.745	Normal
2	$Effective friction$ $angle, {\phi }^{’}$	$q_c$	${\displaystyle \frac{q_c/p_a}{\sqrt{{{\sigma }^{’}}_{v0}/p_a}}}$	$ln\left({\displaystyle \frac{q_c/p_a}{\sqrt{{{\sigma }^{’}}_{v0}/p_a}}}\right)=a{\phi }^{’}+b+\epsilon$	0.209	−3.684	0.586	Normal
3	$Effective friction$ $angle, {\phi }^{’}$	$N_{SPT}$	${\left(N_1\right)}_{60}$	$ln{\left(N_1\right)}_{60}=a{\phi }^{’}+b+\epsilon$	0.161	−3.724	0.496	Normal

Note: The conversion models mentioned above are from the literature [23,35,36], where $P_{\mathrm{I}}$ denotes the plasticity index, $q_{\mathrm{c}}$ denotes the static cone penetration test, and $N_{\mathrm{S}\mathrm{P}\mathrm{T}}$ denotes the standard penetration test.

.

Applying the conversion models listed in Table 2 to analyze the plasticity index, cone penetration test, and standard penetration test data in silty clay layers reveals the trends in the data. After substituting the coefficients into the three respective conversion models, the results can be expressed as follows:

$$ln{P}_I=-10.638{\phi }^{\prime }+8.5106+\epsilon ,$$

(12)

$$ln\left({\displaystyle \frac{q_c/p_a}{\sqrt{{{\sigma }^{\prime }}_{v0}/p_a}}}\right)=0.209{\phi }^{\prime }-3.684+\epsilon ,$$

(13)

$$ln{\left(N_1\right)}_{60}=0.161{\phi }^{\prime }-3.724+\epsilon ,$$

(14)

where $\epsilon$ denotes a standard normal random variable, indicating the uncertainty in the modeling process.

Referring to Equation (12) and the information reflected in Table 2, using an average value of 12.2 and a standard deviation of 0.048, the plasticity index information provided in the geological survey report is detailed in Table 2. In 63 test instances, the values were inverted after testing, revealing some values with both randomness and repeatability. These values were then converted to obtain the effective internal friction angle ($\phi ’$) using Equation (12), as shown in the first plot of Figure 4. All 63 data points fall within the controllable range, but this still does not precisely determine the range of ${\phi }^{’}$, making it necessary to sample and use a normal distribution to characterize the performance of design parameters (as in Equation (4)). For the prior distribution of the average value ${\mu }_N$ and standard value ${\sigma }_N$ of ${\phi }^{’}$, a uniform distribution can be considered to be the prior distribution (as in Equation (9)). Therefore, the range for the average value was set as ${\mu }_Nϵ\left[{15}^{°},{35}^{°}\right]$, and the range for the standard deviation was set as ${\sigma }_Nϵ\left[{5.5}^{°},{10.5}^{°}\right]$.

For Equation (13), based on the conditions provided in Table 2, 565 tests were conducted using the on-site static cone penetration test. Although the number of tests was relatively large, several values exhibited repeatability. These values could be converted into the effective internal friction angle (${\phi }^{\prime }$) using Equation (13). The second plot shown in Figure 4 displays the results after 100 random simulations. However, this still does not precisely determine the range of ${\phi }^{\prime }$, making it necessary to sample these values and use a normal distribution to characterize the performance of design parameters (as in Equation (4)). For the prior distribution of the average value ${\mu }_N$ and standard value ${\sigma }_N$ of ${\phi }^{\prime }$, a uniform distribution can be considered to be the prior distribution (as in Equation (9)). From Table 2, the average value is 1.47, and the standard deviation is 0.41. Therefore, the range for the average value was set as ${\mu }_Nϵ\left[{20}^{°},40\right]$, and the range for the standard deviation was set as ${\sigma }_Nϵ\left[{5.5}^{°},{10.5}^{°}\right]$.

Based on Equation (14), Table 2 indicates an average value of 6.70 and a standard deviation of 1.15. Under the various conditions provided in Table 2 and using the number of standard penetration tests $\left(n=66\right)$, all values were inverted. These values exhibit strong randomness and repeatability. Values for the effective internal friction angle (${\phi }^{\prime }$) were obtained through the conversion using Equation (14), as shown in the third plot of Figure 4. However, these scattered data points do not precisely determine the range of ${\phi }^{\prime }$ in the soil layers of the region, making it necessary to sample these values and use a normal distribution to characterize the performance of design parameters (as in Equation (4)). For the prior distribution of the average value ${\mu }_N$ and standard value ${\sigma }_N$ of ${\phi }^{\prime }$, a uniform distribution can be considered to be the prior distribution (as in Equation (9)). Consequently, the range for the average value was set as ${\mu }_Nϵ\left[{15}^{°},{35}^{°}\right]$, and the range for the standard deviation was set as ${\sigma }_Nϵ\left[{5.5}^{°},{10.5}^{°}\right]$.

Based on the plasticity index data, on-site static cone penetration test data, and standard penetration test data, combined with Equations (9), (11), (14)–(16), and prior knowledge, the effective internal friction angle (${\phi }^{\prime }$) values could be determined. Using the MCMC method, 10,000 sets of equivalent samples for the effective internal friction angle (${\phi }^{\prime }$) could be obtained, as shown in Figure 5.

The above three conversion models all used the MCMC method to obtain 10,000 sets of equivalent samples for the effective internal friction angle ($\phi ’$). Observing the scatter plots (Figure 5) and histograms (Figure 6) of the equivalent samples for the three models, it is evident that after 63 inversions, Equation (12) produces all values, and through sampling, a large number of equivalent samples can be obtained. Although the average values are close, there is considerable dispersion under different empirical relationships. It may be beneficial to consider merging the data from the three conversion models to further enhance the reliability of their formulas. Consequently, it can be inferred that with an increase in the number of equivalent samples, a more intuitive determination of the range of the effective internal friction angle ($\phi ’$) can be achieved. The specific results are shown in

Table 4. Values of effective internal friction angle (${\phi }^{’}$) from three model formulas.

Silty Clay	Model Formulas	Mean Value	Equivalent Sample Values	Error
Plasticity Index	$ln{P}_I=-10.638{\phi }^{’}+8.5106+\epsilon$	28.74	31.26	2.52
Cone Penetration Test	$ln\left({\displaystyle \frac{q_c/p_a}{\sqrt{{{\sigma }^{’}}_{v0}/p_a}}}\right)=0.209{\phi }^{\prime }-3.684+\epsilon$	30.39	33.94	3.55
Standard Penetration Test	$ln{\left(N_1\right)}_{60}=0.161{\phi }^{’}-3.724+\epsilon$	29.49	26.55	2.94

, indicating that under these three models, there is some error between the actual values and equivalent sample values. However, the error is not significant. To obtain a more accurate model, it may be beneficial to consider a method that combines the three conversion formulas to obtain a better parameter analysis.

3.3. Update of Formula Coefficient Values

The above three models can be employed in the MCMC method to obtain the effective internal friction angle ($\phi ’$) equivalent samples. A comparison between the actual and equivalent values reveals discrepancies. Consequently, a method was proposed that combined the three transformation formulas for a more comprehensive analysis of the parameters, aiming for greater accuracy. Typically, the coefficients $a$ and $b$ of the model equations are optimized using equivalent samples. Based on the three sets of data obtained from Figure 4, it can be challenging to derive a fully effective model owing to insufficient data and the inability to extensively use the same set of scattered points. To derive one set of model equations, adjustment values from the other two sets of original data (without introducing the ε random error term) are required. New data values can be generated by appropriately scaling $\epsilon$ and then substituting it into the respective formulas. By establishing a Bayesian framework that incorporates the regression relationship of the effective internal friction angle ($\phi ’$) and uncertainty and adjusting it to suitable coefficient values, the characteristics can be observed to precisely determine the effective internal friction angle ($\phi ’$) for a specific site. As shown in Figure 7, the newly derived linear relationship model formulas among soil parameters can be updated in Equations (12)–(14), the corresponding formulas for each of which can be expressed as follows:

$$ln{P}_I=-0.2449{\phi }^{\prime }+10.2697+\epsilon ,$$

(15)

$$ln\left({\displaystyle \frac{q_c/p_a}{\sqrt{{{\sigma }^{\prime }}_{v0}/p_a}}}\right)=0.5028{\phi }^{\prime }-5.0871+\epsilon ,$$

(16)

$$ln{\left(N_1\right)}_{60}=0.4021{\phi }^{\prime }-4.0324+\epsilon ,$$

(17)

where ε denotes a standard normal random variable, indicating the uncertainty in the modeling process.

After deriving the new model formulas, it is necessary to validate their practicality. For Equation (15), the plasticity index data, Equation (9), Equation (11), and prior information are still used. The effective internal friction angle ($\phi ’$) values can be obtained, and using the MCMC method, 10,000 sets of equivalent samples for the effective internal friction angle ($\phi ’$) were generated. As shown in Figure 8, it is evident that approximately 30% of the scattered points fall within the range of 25–30°, with the majority of points clustering around the corresponding data. This indicates that after adjusting the coefficients $a$ and $b$, the values more accurately reflect the effective internal friction angle ($\phi ’$) in the region.

In relation to Equation (16), where the symbol $\epsilon$ denotes the uncertainty in the modeling process and is a standard normal random variable, to validate the practicality of the new model formula, a combination of in situ static cone penetration test data, Equation (9), Equation (11), Equation (16), and prior information can be used. Using this information, the effective internal friction angle ($\phi ’$) values can be determined; consequently, 10,000 sets of equivalent samples for the effective internal friction angle ($\phi ’$) were generated using the MCMC method. Notably, as is evident from Figure 9, approximately 28% of the samples fall within the range of 30–35°, indicating that after moderately adjusting the coefficients $a$ and $b$, the distribution of scatter points becomes more concentrated, providing a more accurate reflection of the effective internal friction angle ($\phi ’$) in the region. This enhancement enables the model to be more widely applicable to engineering projects, even across different projects within the same region, with significant implications for engineering practice.

In Equation (17), the symbol ε denotes the uncertainty in the modeling process, and it is a random variable following a standard normal distribution. To validate the practicality of the new model formula, comprehensive consideration must be given to the standard penetration test, Equation (9), Equation (11), Equation (17), and prior information. Using this information, the numerical values for the effective internal friction angle ($\phi ’$) are obtained. In this study, 10,000 sets of equivalent samples for the effective internal friction angle ($\phi ’$) were generated using the MCMC method. From Figure 10, it is evident that approximately 35% of the samples are distributed within the range of 25–30°, indicating that by moderately adjusting the values of coefficients $a$ and $b$, the distribution of scatter points becomes more concentrated, providing a more accurate reflection of the effective internal friction angle ($\phi ’$) in the region. This enhancement allows the model to be more widely applicable to engineering projects, even across different projects within the same region.

After updating the model formulas, the previous method can continue to be used to obtain equivalent samples. The only difference is that owing to the limited volume of data and the inability to extensively use scattered points, when deriving one set of model formulas, adjustment values from the other two sets of original data (without introducing the ε random error term) are required. New data values are generated by appropriately scaling ε and then substituting it into the formula to obtain a suitable model applicable to different engineering projects in the same region (

Table 5. Updated values of the effective internal friction angle (${\phi }^{’}$) for the three models.

Silty Clay	Model Formulas	Mean Value	Equivalent Sample Values	Error
Plasticity Index	$ln{P}_{\mathrm{I}}=-10.638{\phi }^{\prime }+8.5106+\epsilon$	28.74	28.96	0.22
Cone Penetration Test	$ln\left({\displaystyle \frac{q_c/p_a}{\sqrt{{{\sigma }^{\prime }}_{v0}/p_a}}}\right)=0.209{\phi }^{\prime }-3.684+\epsilon$	30.39	30.11	0.28
Standard Penetration Test	$ln{\left(N_1\right)}_{60}=0.161{\phi }^{\prime }-3.724+\epsilon$	29.49	30.14	0.65

). By observing the scatter plots and histograms of equivalent samples for the three models (Figure 8, Figure 9 and Figure 10), it is evident that these updated model formulas better reflect the concentration of scatter points, greatly improving the applicability and efficiency of the models. This provides a more intuitive understanding of the range of scatter points, facilitating the estimation of the effective internal friction angle ($\phi ’$).

From Table 5, it is evident that after updating the model formulas, the error between the actual values and equivalent samples is small, indicating the feasibility of model updating. This adjustment represents continuous improvement and optimization of the model, making it more suitable for various soil conditions and more accurately estimating the effective internal friction angle ($\phi ’$). After thorough parameter adjustments and validation work, the reliability and practicality of the new model can be ensured, which contributes to a more accurate prediction of soil parameters, enhancing the feasibility and efficiency of engineering projects. In summary, this process plays a crucial role in improving the accuracy and reliability of the model.

4. Results and Discussion

4.1. Results

The starting point of this study was the extensive application of the MCMC method to probabilistically characterize the effective internal friction angle ($\phi ’$) of silty clay layers. Through the optimization of three model formulas, a new linear model relationship for soil parameters was derived, which significantly improved the accuracy and reliability of the effective internal friction angle ($\phi ’$) during the process of integrated optimization. However, the limited experimental data provided in the engineering geological survey reported only basic information, leading to uncertainties [1,2,3,4]. To address this uncertainty, the MCMC method and PDFs were employed, which substantially reduced the uncertainty by estimating many samples. Detailed quantitative information [37,38], including the distribution of the estimated effective internal friction angles, posterior probability density functions, and comparisons with observed data, has been included to demonstrate the robustness of the results. This approach resolves the challenge of accurate prediction under complex geological conditions, owing to insufficient survey data, and provides a solid quantitative basis for future applications of the method.

The essence of this research lies in the precise estimation of equivalent sample values using the transformation model formulas listed in Table 3. This process aims to determine the clustering range of histograms and scatter plots and ultimately establish the numerical value of the effective internal friction angle ($\phi ’$). Obtaining high-quality original samples was crucial for this procedure. However, owing to the absence of laboratory data during this project, multiple in situ datasets were employed to construct original samples through empirical relationships. The use of empirical methods to estimate original parameters introduces a certain level of uncertainty and errors.

According to the 10,000 equivalent samples of the effective internal friction angle ($\phi ’$), the mean value was 32.15° with a standard deviation of 5.36°. In comparison, the effective internal friction angle ($\phi ’$) measured through the in situ static cone penetration test (CPT) had a mean value of $30.15°$ with a standard deviation of $5.92°$. The mean difference was $2.0°$, and the standard deviation difference was $0.56°$. The relatively small differences between the two sets of data suggest that the method used in this study is feasible.

To minimize errors, additional data points were integrated, and advanced probabilistic methods, such as MCMC simulations, were employed to enhance the reliability of the original sample assessments. However, this integration also resulted in an increased variability in the samples, which can influence the precision of the results. Therefore, this study faces a challenge in addressing the issue of data discretization, which requires a balance between high-quality original samples and the impact of multiparameter estimation on sample variability. By carefully adjusting the model parameters, it is possible to achieve more accurate results while minimizing uncertainty and maintaining robustness in the final estimation of $\phi ’$.

4.2. Discussion

In practical engineering, field engineers typically rely directly on laboratory test results to determine key parameters such as the effective internal friction angle. However, when field conditions are complex or laboratory data are limited, traditional methods may have limitations. The Bayesian method proposed in this study serves as a supplementary tool by integrating prior information and empirical relationships to provide preliminary estimates for the effective internal friction angle. The Bayesian approach can narrow down the possible range of the internal friction angle while quantifying its variability and uncertainty, thereby supporting more reliable design, especially in cases where laboratory data are insufficient or traditional methods are constrained. Therefore, the Bayesian method offers significant advantages in uncertainty management and data integration. When combined with laboratory results, this method can provide valuable references for soil property evaluation and foundation design under complex conditions.

In this study, due to the absence of stress–strain data for each soil layer in the geotechnical engineering report, it was not possible to comprehensively analyze the influence of stress–strain behavior on the effective internal friction angle. The research primarily relies on existing geotechnical parameters to derive the internal friction angle, a common approach in engineering practice when stress–strain data are unavailable. However, we recognize its limitations in reflecting the mechanical behavior of soils. Future studies will incorporate stress–strain data through supplementary laboratory testing or more detailed field investigations to more rigorously explore the relationship between stress–strain behavior and the internal friction angle. This approach will not only improve the accuracy of the research methodology but also further expand its applicability.

The results of Bayesian analysis may vary depending on the choice of prior distributions; therefore, addressing this variability is crucial to ensuring the robustness of the method. In future research, at least three different prior distributions will be used for comparative analysis. By incorporating a wider range of prior distributions and employing methods such as sensitivity analysis, the impact of prior selection on the results can be quantified. Additionally, the use of hierarchical Bayesian methods will be considered to optimize the selection of prior distributions, thereby reducing subjectivity.

5. Conclusions

Within a Bayesian framework, this study proposed determining standard values for soil parameters based on prior information and a small volume of on-site survey data. The following conclusions were drawn:

(1)

This study employed an empirical relationship to indirectly assess the effective internal friction angle of soil and obtain equivalent samples. Based on survey information from geological reports, particular emphasis was placed on the probabilistic characterization of the effective internal friction angle ($\phi ’$) in silty clay layers using three methods—namely, the plasticity index ($P_I$), in situ static cone penetration test ($q_c$), and standard penetration test ($N_{SPT}$). Through the MCMC method and PDFs, a large number of equivalent samples were estimated. Different sources of prior information were systematically integrated with sampled data, and—in combination with the regression relationship and uncertainty of the effective internal friction angle ($\phi ’$)—a Bayesian framework was developed. The framework ensured a balance and symmetry in incorporating prior information and observed data, effectively reflecting the unbiased treatment of varying data sources. This framework was then transformed into effective internal friction angle equivalent samples for three model formulas, maintaining symmetry in the processing of different models and data inputs. This approach effectively addressed the problem of accurately predicting the internal friction angle in survey processes with limited data, particularly under complex geological conditions.

(2)

The MCMC method was effective in probabilistically characterizing the effective internal friction angle ($\phi ’$) of silty clay layers. Using information from the geological survey report of the Wuhan Xunsi River Basin Comprehensive Management Phase II (Wutai Sluice Sewage Treatment Plant) project, a comparison and analysis of measured data and equivalent sample values were conducted. Through the optimization and updating of parameters using the three models, a new linear relationship model for soil parameters could be derived. The new effective internal friction angle\ ($\phi ’$) closely approximated the actual values, demonstrating high accuracy and reliability, permitting its application in practical engineering projects in the region.

(3)

To further adjust the model formulas—particularly in geologically complex areas with limited data, where the extensive use of scattered points was not feasible—adjustments were made when deriving one set of model formulas. This involved using the other two sets of original data (excluding the $\epsilon$ random error term) as adjustment values. New data values were generated by appropriately scaling $\epsilon$ and then substituting it into the formula to obtain a suitable model formula applicable to different engineering projects in the same region. This process, combined with the regression relationship and uncertainty of the effective internal friction angle ($\phi ’$), established a Bayesian framework. The result was a model formula tailored to a specific engineering project or even region, ensuring the precise determination of the effective internal friction angle ($\phi ’$). The three model formulas mentioned above could thus better fulfill their roles in engineering practice.

(4)

The Bayesian equivalent analysis in this study is based on silty clay from a specific region; however, soil mineral composition varies by region, which may affect its mechanical properties and the analysis results. Although the model is not universally applicable to all soils, it is designed to integrate prior knowledge and flexibly adapt to specific datasets. As long as reliable soil data are available, the method can be extended to other soil types. This study uses silty clay as a case study, and future research should further verify the applicability and generalizability of this method across different regions and soil types.

(5)

The Bayesian equivalent sample method integrates field investigation data with statistical models to validate its applicability under data-scarce conditions. However, a comprehensive comparison with laboratory test results has not yet been conducted. Such a comparison is crucial to providing stronger support for the accuracy and applicability of this method. Future research plans include incorporating laboratory test data to enhance its reliability across different soil types and engineering applications, further refining the method, and expanding its scope of application.