A Novel Inversion Method for Permeability Coefficients of Concrete Face Rockfill Dam Based on Sobol-IDBO-SVR Fusion Surrogate Model

Abstract

The accurate and efficient inversion of permeability coefficients is significant for the scientific assessment of seepage safety in concrete face rockfill dams. In addressing the optimization challenge of permeability coefficients with few samples, multiple parameters, and strong nonlinearity, this paper proposes a novel intelligent inversion method based on the Sobol-IDBO-SVR fusion surrogate model. Firstly, the Sobol sequence sampling method is introduced to extract high-quality combined samples of permeability coefficients, and the equivalent continuum seepage model is utilized for the forward simulation to obtain the theoretical hydraulic heads at the seepage monitoring points. Subsequently, the support vector regression surrogate model is used to establish the complex mapping relationship between the permeability coefficients and hydraulic heads, and the convergence performance of the dung beetle optimization algorithm is effectively enhanced by fusing multiple strategies. On this basis, we successfully achieve the precise inversion of permeability coefficients driven by multi-intelligence technologies. The engineering application results show that the permeability coefficients determined based on the inversion of the Sobol-IDBO-SVR model can reasonably reflect the seepage characteristics of the concrete face rockfill dam. The maximum relative error between the measured and the inversion values of the hydraulic heads at each monitoring point is only 0.63%, indicating that the inversion accuracy meets the engineering requirements. The method proposed in this study may also provide a beneficial reference for similar parameter inversion problems in engineering projects such as bridges, embankments, and pumping stations.

1. Introduction

A concrete face rockfill dam is a novel dam type that gradually emerged in the 1980s, and it is favored within the dam engineering field because it shows the advantages of economic safety, convenient construction, and good adaptability [1]. After decades of vigorous development, modern concrete face rockfill dams characterized by “stratified filling and thin-layer vibratory compaction” techniques are becoming more and more mature. Based on incomplete statistics, by the end of 2021, approximately 360 concrete face rockfill dams with heights exceeding 30 m were completed in China, and around 100 are currently under construction, with an additional 80 in the planning stage. With the accumulation of dam construction experience and improvements at the theoretical level, China has successively built several ultra-high concrete face rockfill dams with heights exceeding 200 m and is gradually transitioning from the 200 m grade to the 300 m grade [2]. At the same time, with the increasing dam heights, concrete face rockfill dams will inevitably encounter more complex technical challenges and more severe service conditions [3]; among these, the seepage safety issue of dams is the key issue restricting their long-term stable operation [4,5]. Therefore, it is of important engineering significance to accurately predict the seepage behavior of concrete face rockfill dams based on fully understanding the permeability characteristics of the materials.

The rational selection of permeability coefficients is not only the core aspect of seepage behavior analysis but also a prerequisite for reliable seepage calculation results [6]. Currently, the methods for the determination of the permeability coefficients of materials primarily include the geotechnical test method [7], empirical formula method [8], and inversion analysis method [9]. Among them, the penetration test method is constrained by the test conditions and scales, and the permeability coefficients of specimens often exhibit significant dispersion, making it difficult to obtain macroscopically accurate results. Regarding the empirical formula method, due to its lack of comprehensive theoretical support and susceptibility to subjective factors in parameter selection, it leads to lower accuracy in permeability coefficient calculation, thereby limiting its practical application in engineering. In fact, during the long-term service of concrete face rockfill dams, the permeability coefficients of the materials evolve due to the coupled effects of complex factors such as seepage, stress, and temperature [10,11,12], but the above two methods cannot reflect the time-varying characteristics of the permeability coefficients. Fortunately, the seepage monitoring data obtained during operation can be used to characterize the seepage behavior of the dam in real time [13]; because of this, the inversion analysis method based on the measured seepage monitoring data has become the main means to determine the permeability coefficients of materials [14].

By minimizing the mean square error between the measured values and inversion values at seepage monitoring points, the inversion calculation of permeability coefficients can be transformed into solving a complex nonlinear least squares optimization problem. To address this challenge, some classical optimization algorithms were initially applied in parameter inversion studies and achieved relatively satisfactory optimization results [15], but the implementation of such methods requires the large-scale repetition of numerical simulation calculations, resulting in high inversion costs but low efficiency [16]. Subsequently, some scholars began to introduce surrogate models based on machine learning algorithms to replace the cumbersome and time-consuming numerical models, which strongly promoted the intelligent development of parameter inversion. For instance, Nan [17] combined the multi-island genetic algorithm with the Kriging surrogate model to establish a seepage parameter inversion method for earth-rock dams considering the coupling effect of seepage and thermal transfer. Li [18] utilized the non-dominated sorting genetic algorithm and the extreme learning machine surrogate model to achieve the high-precision inversion of unsaturated seepage parameters for earth-rock dams driven by multi-source data. Tong [13] proposed a rock mass permeability coefficient inversion method based on the particle swarm optimization algorithm and the support vector machine algorithm surrogate model, which was successfully applied to the seepage safety evaluation of the LJH arch dam.

Although the above studies have achieved positive results, some issues still remain to be resolved. First, most existing parameter inversion surrogate models have largely overlooked the impact caused by the selection of datasets, resulting in significant fluctuations in the computational accuracy of the surrogate models. This is precisely why it is essential to introduce an efficient sampling method to properly select the dataset for the surrogate model. In addition, traditional surrogate models such as the Kriging models [19], radial basis functions [20], response surface models [21], and artificial neural networks [22] still have shortcomings. This stems from the theoretical basis of these models, relying on the law of large numbers based on traditional statistical theory, and they take empirical risk minimization as a criterion. In practical applications, a sufficient number of samples is required to ensure the accuracy of the parameter inversion results, which is often challenging to achieve in real-world problems. Therefore, the question of how to establish a surrogate model with excellent learning generalization ability under limited sample conditions remains to be further studied.

At present, concrete face rockfill dams are the preferred dam type in the dam design stage in China, but the existing dam parameter inversion analysis research is still mainly focused on conventional earth-rock dams and concrete dams. The research on the intelligent inversion method for the calculation of the permeability coefficients for concrete face rockfill dams is still obviously weak. Song [23] established a fast seepage parameter inversion method for core earth-rock dams based on the fusion surrogate model. Zhou [24] presented a seepage parameter inversion approach for earth-rock dams based on the radial basis function and improved sparrow algorithm. Kang [25] proposed a response surface model for the parameter inversion of concrete dams based on the kernel extreme learning machine. Zhang [26] developed a seepage field inversion model for concrete dams by combining a finite element model with an artificial neural network. Because the material partitioning and anti-seepage structures of different dam types vary, the seepage parameters that need to be inverted are diverse. Therefore, it is necessary to develop a novel inversion method for permeability coefficients tailored to the structural characteristics of concrete face rockfill dams.

In order to explore the beneficial integration of the sampling method, optimization algorithm, and surrogate model, as well as to enhance the application value of intelligent inversion technology in the seepage analysis of concrete face rockfill dams, this study provides corresponding improvements to address the deficiencies in existing inversion methods. On the one hand, an intelligent sampling technology based on the Sobol sequence is creatively introduced, which can quickly extract high-quality samples with uniform distribution characteristics, fully reflect the data information of the complex high-dimensional parameter space, and enhance the generalization ability of the surrogate model. On the other hand, the support vector regression (SVR) machine model, with excellent data mining abilities, is applied to replace the cumbersome and time-consuming finite element numerical model, thereby improving the optimization efficiency of the parameter inversion. Compared with the traditional surrogate models, the SVR model is based on modern statistical theory and takes structural risk minimization as a criterion, which can better solve the regression prediction problem under limited sample conditions. Meanwhile, inspired by the novel dung beetle optimization (DBO) algorithm, we further fuse multiple strategies from various perspectives to improve the optimization performance of the DBO algorithm. Building upon this improved dung beetle optimization (IDBO) algorithm, we have successfully achieved the optimization and adjustment of the SVR model parameters and the intelligent inversion of the best permeability coefficient combination.

In summary, in order to accurately and efficiently master the seepage characteristics of concrete face rockfill dams, this paper proposes a novel permeability coefficient inversion method based on the Sobol-IDBO-SVR fusion surrogate model, which gives full play to the triple advantages of intelligent sampling technology, meta-heuristic optimization algorithms, and machine learning algorithms. Taking the Altash concrete face rockfill dam in the northwestern part of China as an example, we carried out the intelligent inversion of the material permeability coefficients based on the engineering seepage monitoring data, thereby realizing the scientific evaluation of the dam’s seepage safety. It is worth noting that the method proposed in this study may also provide a theoretical reference for similar parameter inversion problems in bridges, embankments, pumping stations, and other projects.

2. Methodology

2.1. Sobol Sequence Sampling Method

Scientific and efficient parameter spatial sampling techniques are of great significance in improving the accuracy of the surrogate model. Accurately selecting a few samples with homogeneity and representativeness can effectively reduce the calculation costs and improve the calculation efficiency of the surrogate model based on fully exploring the parameter space. The Monte Carlo sampling method [27] is a commonly used sampling method based on pseudo-random sequences, but it is difficult to obtain a uniformly distributed sample set when the sampling times are smaller, leading to issues such as sample aggregation and gaps. In order to overcome the limitations of large-scale sampling required by the traditional Monte Carlo method, McKay proposed the Latin hypercube sampling (LHS) method based on the stratified sampling idea [28]. The LHS method can obtain high sampling accuracy with a small sampling scale, and the extracted samples are relatively uniform without obvious aggregation phenomena. Nevertheless, it still exhibits insufficient uniformity when sampling in the complex high-dimensional space.

In recent years, quasi-Monte Carlo sampling methods based on low-discrepancy sequences, such as the Hammersley sequence, Halton sequence, and Sobol sequence, have gained widespread application in the field of engineering. Among these methods, the Sobol sequence stands out due to its shorter computation period, faster sampling speed, higher efficiency in dealing with high-dimensional sequence problems, and more uniformly traversed distribution of sampling points, which offer obvious advantages for popularization [29,30]. In order to visually reflect the distribution status of sample points extracted using pseudo-random sequences, LHS sequences, and Sobol sequences in the parameter space, a two-dimensional sample point set with a size of 100 is generated within the [0, 1] range. As depicted in Figure 1, compared to the pseudo-random sequence and LHS sequence, the sample points generated by the Sobol sequence exhibit higher spatial coverage, wider distribution traversal, and the best uniform distribution characteristics. Therefore, this work uses the Sobol sequence sampling method to construct the training sample set and the test sample set for the SVR surrogate model.

2.2. Dung Beetle Optimization Algorithm

The dung beetle optimization algorithm is a novel swarm intelligence optimization algorithm proposed by Jiankai Xue and Bo Shen in 2022 [31] that is inspired by the rolling, dancing, spawning, foraging, and stealing behaviors of dung beetles in nature. Through a series of well-known numerical simulation experimental tests, the outstanding optimization performance of the DBO algorithm has been validated. Compared with some current popular optimization algorithms, the DBO algorithm shows more competitive performance in terms of convergence speed, computational accuracy, and robustness, which indicate its broad application prospects in the field of engineering optimization. A detailed introduction to the DBO algorithm can be found in Reference [31]. Herein, we briefly describe the mathematical modeling process of the DBO algorithm.

2.2.1. Rolling Dung Beetles

Dung beetles act as decomposers in nature; they feed on animal dung and use celestial cues to roll dung balls as quickly and efficiently as possible to avoid competition with other dung beetles. The DBO algorithm adopts Equation (1) to describe the rolling behavior of dung beetles in the unobstructed mode:

$$\begin{array}{c}{X}_{\mathrm{roll}}(t+1)=X_{\mathrm{roll}}(t)+\alpha \times k\times X_{\mathrm{roll}}(t-1)+b\times \Delta X\\ \Delta X=\left|X_{\mathrm{roll}}(t)-X^w\right|\end{array}$$

(1)

where $X_{\mathrm{roll}}(t)$ represents the position information of the rolling dung beetles at the tth iteration; $X^w$ denotes the global worst position; $\Delta X$ is used to simulate changes in light intensity; $\alpha$ is a coefficient assigned as −1 or 1; $k\in (0,0.2]$ indicates the deflection coefficient, which is taken as 0.1; $b\in (0,1)$ is a constant, which is taken as 0.3.

When dung beetles encounter obstacles preventing forward movement, they redirect themselves through dancing behavior to acquire a new rolling path. The DBO algorithm utilizes the tangent function to describe the dancing behavior of dung beetles in the obstructed mode:

$$X_{\mathrm{roll}}(t+1)=X_{\mathrm{roll}}(t)+\tan (\theta )\times \left|X_{\mathrm{roll}}(t)-X_{\mathrm{roll}}(t-1)\right|$$

(2)

where $\theta \in [0,\pi ]$ denotes the deflection angle, while $\theta$ equals 0, $\pi /2$, or $\pi$, and the position of the dung beetle remains unchanged.

2.2.2. Spawning Dung Beetles

Dung beetles roll their dung balls to a secure area and subsequently bury them, providing resources for the spawning and nurturing of the next generation by female dung beetles. The dung balls serve not only as developmental sites for larvae but also provide essential sustenance for their livelihood. Therefore, selecting an appropriate spawning area is crucial for dung beetles. The DBO algorithm proposes a boundary selection strategy to simulate the spawning area for female dung beetles:

$$\begin{array}{l}L{b}^{*}=\max (X^{*}\times (1-R),Lb),\\ U{b}^{*}=\min (X^{*}\times (1+R),Ub)\end{array}$$

(3)

where $X^{*}$ denotes the current local optimal position; $L{b}^{*}$ and $U{b}^{*}$ indicate the lower and upper bounds of the spawning area, respectively; $R=1-t/T_{\max }$ represents the convergence factor; $T_{\max }$ is the maximum iteration number; $Lb$ and $Ub$ indicate the lower and upper bounds of the optimization problem, respectively.

Once spawning dung beetles determine the optimal spawning area, they proceed to lay their eggs within this designated region. The DBO algorithm assumes that each spawning dung beetle produces only one brood ball in each iteration. The spawning area is dynamically adjusted with the convergence factor $R$, indicating that the position of the brood balls also changes dynamically in the iteration process. The DBO algorithm defines Equation (4) to depict the spawning behavior of female dung beetles:

$$X_{\mathrm{spawn}}(t+1)=X^{*}+b_1\times (X_{\mathrm{spawn}}(t)-L{b}^{*})+b_2\times (X_{\mathrm{spawn}}(t)-U{b}^{*})$$

(4)

where $X_{\mathrm{spawn}}(t)$ represents the position information of spawning dung beetles at the tth iteration; $b_1$ and $b_2$ denote two independent random vectors by size $1\times D$; $D$ denotes the dimension of the optimization problem.

2.2.3. Foraging Dung Beetles

As small dung beetles grow into adults, they will crawl out of the ground to forage for food. Similarly, the foraging behavior of small dung beetles still requires the selection of a safe area. The DBO algorithm designs a boundary selection strategy to simulate the best foraging area for small dung beetles:

$$\begin{array}{l}L{b}^b=\max (X^b\times (1-R),Lb),\\ U{b}^b=\min (X^b\times (1+R),Ub)\end{array}$$

(5)

where $X^b$ denotes the global best position; $L{b}^b$ and $U{b}^b$ indicate the lower and upper bounds of the best foraging area, respectively.

Similar to the spawning area, the foraging area is dynamically adjusted with the convergence factor $R$, indicating that the position of the small dung beetles also changes dynamically in the iteration process. The DBO algorithm adopts Equation (6) to describe the foraging behavior of small dung beetles:

$$X_{\mathrm{forage}}(t+1)=X_{\mathrm{forage}}(t)+C_1\times (X_{\mathrm{forage}}(t)-L{b}^b)+C_2\times (X_{\mathrm{forage}}(t)-U{b}^b)$$

(6)

where $X_{\mathrm{forage}}(t)$ represents the position information of foraging dung beetles at the tth iteration; C₁ denotes a random number that follows a normal distribution; C₂ indicates a random vector of size $1\times D$ within the interval (0, 1).

2.2.4. Stealing Dung Beetles

It is a common phenomenon in nature that, despite the efforts of dung beetles to protect their acquired dung balls, some stealing dung beetles still choose to compete with other dung beetles for food and consistently attempt to steal dung balls from the vicinity of the best food source. The DBO algorithm defines Equation (7) to depict the stealing behavior of dung beetles:

$$X_{\mathrm{steal}}(t+1)=X^b+S\times g\times (\left|X_{\mathrm{steal}}(t)-X^{*}\right|+\left|X_{\mathrm{steal}}(t)-X^b\right|)$$

(7)

where $X_{\mathrm{steal}}(t)$ represents the position information of stealing dung beetles at the tth iteration; $S$ is a constant, which is taken as 0.5; $g$ denotes a random vector by size $1\times D$ that follows a normal distribution.

2.3. Improving the DBO Algorithm

The DBO algorithm requires few parameter adjustments and has strong universality. However, similarly to other traditional meta-heuristic algorithms, the DBO algorithm also suffers from the shortcomings of reduced convergence performance due to the initial population aggregation, the imbalance between global exploration in the early stage and local exploitation in the later stage, and the lack of escape abilities, which makes it prone to becoming trapped in local optima [32]. Therefore, in order to further enhance the optimization performance of the DBO algorithm, this paper attempts to improve it by fusing multiple strategies from multiple perspectives, so that it can better serve the optimization problem of permeability coefficient inversion.

2.3.1. Population Initialization Based on Sobol Sequence

For meta-heuristic algorithms, the position distribution of the initial population significantly affects the algorithm’s convergence performance [33]. While dealing with optimization problems where the position of the optimal solution is unknown, the initial position of the population should be distributed as uniformly as possible in the parameter space to ensure high traversability and diversity, thereby improving the optimization efficiency. The DBO algorithm adopts a pseudo-random sequence with strong randomness and weak robustness to initialize the population. This may lead to the sparse distribution of individuals near the optimal solution, which slows down the convergence speed of the algorithm but also leads to the aggregation of individuals near the local optimal solution, thus reducing the computational accuracy of the algorithm. As previously mentioned, the adoption of the Sobol sequence can generate a more uniformly traversed initial population in the high-dimensional parameter space, expand the initial global exploration range, reduce the possibility of falling into local optima, and thereby effectively improve the optimization performance of the DBO algorithm.

The Sobol sequence is a low-discrepancy sequence with a minimum prime number of 2 as its base, generating quasi-random numbers ranging between 0 and 1 [34]. The principle of constructing a one-dimensional Sobol sequence [35] and the implementation method for the initialization of the dung beetle population are briefly described as follows.

(1)

For any decimal integer N, it can be uniquely represented as an expression related to base 2:

$$N={\displaystyle \sum _{j=0}^k{c}_j{2}^j}$$

(8)

where k denotes the smallest integer not less than ${\log }_{2}N$; $c_j$ denotes coefficients with values of 0 or 1.

(2)

To generate a Sobol sequence X₁, X₂, X₃,…, X_N containing N quasi-random numbers in the interval (0, 1), we first need to determine a primitive polynomial of degree d with a base of 2:

$$P=X^d+a_1{X}^{d-1}+a_2{X}^{d-2}+\cdots +a_{d-1}X+1$$

(9)

where $a_1$, $a_2$, …, $a_{d-1}$ are the primitive polynomial coefficients, which take the values 0 or 1.

(3)

Then, the direction numbers $v_i$ are calculated based on the determined primitive polynomial, which can be expressed as follows:

$$v_i=\frac{m_i}{2^i}$$

(10)

where $m_i$ denotes an odd integer, $0<m_i<2^i$. When d, $a_1$, $a_2$, …, $a_{d-1}$ and $m_1$, $m_2$, …, $m_d$ are given, the direction numbers $v_i$ can be calculated through the recurrence formula as follows:

$$v_i=a_1{v}_{i-1}\oplus a_2{v}_{i-2}\oplus \cdots \oplus a_{d-1}{v}_{i-d+1}\oplus v_{i-d}\oplus [v_{i-d}/2^d]$$

(11)

where $\oplus$ indicates a bit-by-bit exclusive or operation.

Equivalently, $m_i$ can be calculated through the recurrence formula as follows:

$$m_i=2a_1{m}_{i-1}\oplus 2^2{a}_2{m}_{i-2}\oplus \cdots \oplus 2^{d-1}{a}_{d-1}{m}_{i-d+1}\oplus 2^d{m}_{i-d}\oplus m_{i-d}$$

(12)

(4)

According to Equations (11) and (12), any kth quasi-random number in the Sobol sequence can be calculated as follows:

$$X_k=b_1{v}_1\oplus b_2{v}_2\oplus b_3{v}_3\oplus \cdots \oplus b_k{v}_k$$

(13)

where $b_1$, $b_2$, …, $b_k$ are binary representations of the positive integer k.

Up to this point, quasi-random numbers in the one-dimensional Sobol sequence can be obtained. To generate multi-dimensional Sobol sequences, one simply chooses multiple primitive polynomials and determines direction numbers for each dimension; then, one repeats the above algorithm.

(5)

On this basis, according to the parameter dimension $D$ of the optimization problem to be solved, the mapping expression of dung beetle population initialization by the Sobol sequence can be described as follows:

$$X_{\mathrm{initial}}(j)=L{b}_j+Sobol(j)\times (U{b}_j-L{b}_j)$$

(14)

where $X_{\mathrm{initial}}(j)$ denotes the jth column of the initial dung beetle population matrix; $Sobol(j)$ indicates the jth column of the Sobol sequence quasi-random number matrix; $L{b}_j$ and $U{b}_j$ represent the lower and upper bounds of the jth dimension in the parameter space, respectively.

2.3.2. Improved Nonlinear Convergence Factor

In the DBO algorithm, the convergence factor R directly controls the ranges of the spawning area and foraging area in the iterative process, thereby dynamically adjusting the global exploration and local exploitation of the algorithm. The DBO algorithm utilizes a linearly varying form to reduce the convergence factor R, but its effectiveness in balancing global exploration and local exploitation is limited. Therefore, this paper uses a nonlinear cosine varying form to improve the convergence factor R, and the modified convergence factor expression is as follows:

$$R^{\prime }=\{\begin{array}{l}\frac{1-\cos ((t-1)\pi /(T_{\max }-1))}{2},0\le t\le \frac{1}{2}{T}_{\max }\hfill \\ \frac{1+\cos ((t-1)\pi /(T_{\max }-1))}{2},\frac{1}{2}{T}_{\max }\le t\le T_{\max }\hfill \end{array}$$

(15)

where $R^{\prime }$ denotes the improved nonlinear convergence factor; $t$ indicates the current iteration number; $T_{\max }$ is the maximum iteration number.

As shown in Figure 2, compared with the original convergence factor $R$, the improved convergence factor $R^{\prime }$ decreases more slowly in the early stage and more rapidly in the later stage, which means that the convergence factor based on a nonlinear variation of the cosine law can better balance the global exploration and local exploitation of the DBO algorithm. On the one hand, by increasing the early convergence factor and expanding the ranges of the spawning area and foraging area, the global exploration ability of the DBO algorithm is strengthened. On the other hand, by decreasing the later convergence factor and narrowing the ranges of the spawning area and foraging area, the local exploitation ability of the DBO algorithm is improved.

2.3.3. Adaptive Levy–Brownian Mixed Variation

As the number of iterations continues to increase, the DBO algorithm will gradually focus on local exploitation, at which point the majority of the dung beetles will gather around the current elite solution and keep searching. However, if the current elite solution is not the global optimal solution, the algorithm is likely to fall into the local optimal trap or even experience premature convergence, resulting in a significant decrease in computational accuracy and the inability to converge to the global optimal solution. Aiming at the above problems, this paper proposes an adaptive Levy–Brownian mixed variation mechanism, which endows dung beetles with the ability to escape from the local optimal trap and return to other regions of the solution space to continue searching for the global optimal solution, thereby further improving the optimization accuracy of the DBO algorithm.

Levy flight [36] is a random walk characterized by primarily short-distance movements and occasional long-distance jumps. Its step size probability distribution follows a heavy-tailed distribution, allowing a random movement in any dimensional space with any step size, thereby facilitating the full exploration of the global unknown space. In contrast, Brownian motion [37] describes the irregular movement of suspended particles resulting from molecular collisions. Its step size probability distribution conforms to a Gaussian distribution, enabling the generation of small variable step sizes with high probability and possessing excellent local search capabilities. In order to fully combine the advantages of the different step size probability distribution characteristics of Levy flight and Brownian motion, we propose a novel adaptive Levy–Brownian mixed variation mechanism as follows:

$$X_{\mathrm{new}}^b(t)=X^b(t)\times (1+{\lambda }_1\times R_L+{\lambda }_2\times R_B)$$

(16)

where $X_{\mathrm{new}}^b(t)$ represents the new position of the elite dung beetle in the tth iteration after Levy–Brownian mixed variation; $X^b(t)$ represents the position of the elite dung beetle in the tth iteration; $R_L$ denotes the Levy flight mutation operator; $R_B$ denotes the Brownian motion mutation operator; ${\lambda }_1$ and ${\lambda }_2$ indicate the variation weight coefficients, where ${\lambda }_1=1-t/T_{\max }$ and ${\lambda }_2=t/T_{\max }$.

From Equation (16), it can be seen that the introduction of the variation weight coefficients effectively fuses the advantages of the two stochastic motions. At the early stages of iteration, when the value of ${\lambda }_1$ is relatively large, the DBO algorithm primarily carries out global large-amplitude mutation perturbation through Levy flight to expand the exploration range of the algorithm and improve the speed of optimization. At the later stages of iteration, when the value of ${\lambda }_2$ is relatively large, the algorithm mainly carries out local small-amplitude mutation perturbation through Brownian motion to improve the accuracy of the optimization search. Meanwhile, in order to avoid significantly increasing the complexity of the DBO algorithm, we only mutate the elite dung beetle in each iteration. Additionally, a greedy selection strategy is introduced to determine whether to update the position of the elite dung beetle by comparing the fitness values before and after the mutation, so as to effectively enhance the solution performance of the DBO algorithm.

2.4. Support Vector Regression Machine Learning Algorithm

The support vector regression machine is a new generation of machine learning algorithms that has attracted much attention in the field of engineering optimization in recent years [38]. The SVR model is based on modern statistical theory and takes structural risk minimization as a criterion; it can effectively solve complex regression prediction problems with multiple parameters, high dimensionality, and strong nonlinearity [39]. The core idea of the SVR model is to minimize the error between all training samples and the optimal regression decision hyperplane. Similarly to the support vector classification (SVC) machine classification decision function derivation process, the solution of the SVR model regression decision function is also essentially a convex optimization problem.

Firstly, the SVR model regression decision function optimization problem is constructed by introducing an insensitive loss function and a non-negative relaxation factor to modify the optimization objective and constraints of the original SVC model problem. Then, the optimization problem to be solved is converted into a dual problem by utilizing the Lagrange duality theory, and the kernel function is used to map the low-dimensional input space to the high-dimensional feature space to realize the linearization of the nonlinear problem. Finally, the transformed dual problem is solved based on the quadratic programming method, so as to establish the SVR model optimal regression decision function (Equation (17)). The SVR model structure is schematically shown in Figure 3, where the intermediate nodes in the structure represent support vectors and the final output is the linear combination of each intermediate node.

$$f(x)={\omega }^{*}\Phi (x)+b^{*}={\displaystyle \sum _{i=1}^l({\alpha }_i-{{\alpha }_i}^{*})}K(x_i,x)+b^{*}$$

(17)

where $l$ denotes the total number of training samples; $x_i$ denotes the input vector of the ith training sample; ${\alpha }_i$ indicates the Lagrange coefficient; ${{\alpha }_i}^{*}$, ${\omega }^{*}$, and $b^{*}$ indicate the best parameter solution; $\Phi (x)$ represents the nonlinear mapping function; $K(x_i,x)$ represents the kernel function and satisfies $K(x_i,x)=\Phi (x_i)\Phi (x)$.

When solving the dual problem, the SVR model reduces the algorithmic complexity and overcomes the “curse of dimensionality problem” by introducing the kernel function to replace the extensive inner-product computations in the high-dimensional feature space. Apparently, the type of kernel function directly determines the mapping method from the low-dimensional space to the high-dimensional space, which serves as a critical factor affecting the prediction performance and generalization ability of the SVR model. In this paper, we choose the radial basis kernel function, with strong nonlinear feature mapping capabilities, a wide range of applicability, and simple parameter adjustment [40]. Its function expression is shown in Equation (18).

$$K(x_i,x)=\exp (-\frac{{\Vert x_i-x\Vert }^2}{2{\sigma }^2})$$

(18)

where $\sigma$ denotes the radial basis kernel function variance, which determines the scope of the function.

2.5. Constructing the Sobol-IDBO-SVR Fusion Surrogate Model

To summarize, this paper elaborates on the principles of the Sobol sequence sampling method, improved dung beetle optimization algorithm, and support vector regression machine. On this basis, a novel inversion model of the permeability coefficients of concrete face rockfill dams based on Sobol-IDBO-SVR is constructed. The specific steps for the inversion of the permeability coefficients based on the fusion surrogate model are as follows. (1) According to the hydrogeological conditions of the engineering area and combined with the results of the permeability test, a comprehensive analysis is conducted to determine the reasonable range of the permeability coefficients of the material to be inverted. (2) The uniformly distributed permeability coefficient combination samples in the parameter space are selected based on the Sobol sequence sampling method. Then, the training sample set and the test sample set of the surrogate model are constructed through the forward simulation of the finite element equivalent continuum model. (3) In order to make up for the shortcomings of the DBO algorithm, multiple strategies are fused to improve it from multiple perspectives, thereby establishing the improved dung beetle optimization (IDBO) algorithm with superior optimization performance. (4) The key parameters (penalty factor, insensitive loss factor, and kernel function variance) of the SVR model are optimized by the IDBO algorithm to accurately establish the nonlinear mapping relationship between the permeability coefficient and hydraulic head calculated by a finite element model. (5) The fully optimized and trained SVR surrogate model is employed to replace the finite element calculation model, and the mean square error between the observed and predicted hydraulic head at the monitoring point is used as the optimization objective function to determine the optimal combination of permeability coefficients. The flowchart of the Sobol-IDBO-SVR fusion surrogate model is shown in Figure 4.

3. Case Study

3.1. Project Overview

The Altash hydraulic project is located in the Xinjiang Uygur Autonomous Region of China, serving as a key control project in the downstream mountainous section of the mainstream of the Yarkand River. With the premise of meeting the ecological water supply goals of the Tarim River, it possesses comprehensive functions including flood control, irrigation, power generation, and so on. The primary structure of the project is a concrete face rockfill dam with a maximum dam height of 164.80 m. The normal water level of the Altash dam is 1820.00 m and the dead water level is 1770.00 m. The riverbed overburden at the dam foundation is very thick (with a maximum depth of 94.00 m), demonstrating strong permeability and poor anti-seepage stability. For this reason, the Altash dam employs an anti-seepage system consisting of “concrete face slab-toe slab-connecting plate-anti-seepage wall-curtain grouting” to meet the requirements of seepage control and stability. Meanwhile, in order to master the seepage behavior of the Altash dam in real time, there are several seepage pressure monitoring points inside the dam. A typical cross-section of the Altash dam (0 + 305.00) is shown in Figure 5, which displays the distribution positions and buried elevations of the piezometers in normal operation.

According to the Altash dam’s seepage monitoring data, the initial storage of the reservoir was conducted in two stages. The first stage began on 26 November 2019, with an initial water level of 1670.00 m. By 8 January 2020, the water level had risen to 1721.00 m and was maintained until 5 June 2020. The second stage commenced on 6 June 2020 and continued until 17 August 2021. During this period, the water level fluctuated due to the water used for irrigation and the insufficient amount of incoming water from upstream, but the reservoir level still rose to 1799.60 m. The reservoir water level variation curve and the hydraulic head variation curves of the monitoring points are depicted in Figure 6.

While performing the permeability coefficient inversion analysis, in order to eliminate the error effect caused by the seepage hysteresis effect on the observation values of the piezometers as much as possible, it is necessary to select a period with a small variation and a long duration of the upstream and downstream water levels of the Altash reservoir as the calculation period. According to the variation curve of the reservoir water level, it can be seen that, during the period from 8 November 2020 to 9 December 2020, the upstream water level of the Altash reservoir basically remained stable at the elevation of 1758.00 m. Thus, it can be considered that the seepage field of the dam reached a quasi-stable state in this period. Therefore, this study selects the hydraulic boundary conditions and seepage observation data from this period for the quasi-stable seepage field inversion analysis. Specifically, the upstream water level is determined to be 1758.00 m, the downstream water level is set to the groundwater level of the riverbed at 1662.20 m, and the observed seepage pressure head values at various monitoring points on 24 November 2020 are chosen as the known water level constraint conditions.

3.2. Finite Element Computation Model

Based on the typical cross-section of the Altash dam (0 + 305.00), a two-dimensional finite element equivalent continuum seepage model is established to calculate the simulated hydraulic head values at each monitoring point in the concrete face rockfill dam. The model incorporates the simulation of overburden I, overburden II, strongly weathered rock, weakly weathered rock, and fresh bedrock according to the topographic and geological conditions. For the dam filling zones, the weighted cover zone, upstream blanket zone, concrete face slab, cushion zone, transition zone, sand gravel zone, salvaging zone, blasting zone, and drainage zone are considered. Additionally, for the anti-seepage structure of the dam foundation, the anti-seepage wall and curtain grouting are simulated.

The Altash concrete face rockfill dam has a simple structural shape for each material partition. Therefore, we divide most regions into structured quadrilateral grids and some transitional regions into triangular grids. The meshing results and boundary conditions of the finite element model are shown in Figure 7, where the number of model nodes is 4516 and the number of mesh elements is 4449. When performing forward simulations for the quasi-stable seepage field, the known hydraulic head boundaries are set at the upstream reservoir water level and the downstream riverbed groundwater level in the finite element model. The dam slope between the upstream and downstream water lines serves as the overflow boundary, while the base of the dam foundation is designated as an impermeable boundary.

3.3. Fusion Surrogate Model Construction

According to the results of indoor penetration tests and on-site surveys of the Altash hydraulic project, the permeability coefficients of the materials in each zone of the dam can be determined (

Table 1. The values of the permeability coefficients for materials.

Symbol	Material	Permeability Coefficient (m/s)	Symbol	Material	Permeability Coefficient (m/s)
k1	Weighted cover zone	1.81 × 10−5	k9	Drainage zone	8.17 × 10−4
k2	Upstream blanket zone	2.64 × 10−6	k10	Overburden I	1.00 × 10−5~1.00 × 10−4
k3	Concrete face slab	1.00 × 10−9	k11	Overburden II	1.00 × 10−4~1.00 × 10−3
k4	Cushion zone	4.98 × 10−6	k12	Strongly weathered rock	1.00 × 10−6~3.00 × 10−6
k5	Transition zone	6.10 × 10−6	k13	Weakly weathered rock	5.00 × 10−7~1.00 × 10−6
k6	Sand gravel zone	6.00 × 10−5	k14	Fresh bedrock	1.00 × 10−7~5.00 × 10−7
k7	Salvaging zone	6.20 × 10−5	k15	Anti-seepage wall	1.00 × 10−9~1.00 × 10−8
k8	Blasting zone	3.06 × 10−4	k16	Anti-seepage curtain	1.00 × 10−7~1.00 × 10−6

). However, due to the coarse particles and lack of fine-grained fillings in the riverbed sand gravel layer, slurry leakage often occurs in the overburdened borehole samples, making it difficult to determine the permeability coefficients of the dam foundation materials. Furthermore, during the construction processes of the anti-seepage wall and curtain grouting, it is also likely that the phenomenon of collapsing holes will be triggered, leading to initial defects in the impermeable structures locally, which may weaken the anti-seepage ability of the dam foundation and threaten the seepage safety of the dam. Therefore, this paper focuses on inverting the permeability coefficients of the dam foundation overburden, strongly weathered rock, weakly weathered rock, fresh bedrock, anti-seepage wall, and anti-seepage curtain based on the proposed Sobol-IDBO-SVR fusion surrogate model, so as to provide reliable parameters for the subsequent accurate prediction of the seepage characteristics of the Altash dam. The reasonable range of values for the permeability coefficients of the materials to be inverted can be preliminarily determined by considering the hydrogeological conditions of the dam site area and the results of borehole pressurized water tests (Table 1).

The Sobol sequence sampling method was employed to extract 100 sets of uniformly distributed permeability coefficient combination samples within the specified ranges for the materials to be inverted. These samples were then substituted into the finite element model to obtain the simulated hydraulic head values at each monitoring point through forward calculations, and the computed results are shown in

Table 2. Simulated values of the hydraulic head of each sample.

No.	k10 (10−5m/s)	k11 (10−4m/s)	k12 (10−6m/s)	k13 (10−7m/s)	k14 (10−7m/s)	k15 (10−9m/s)	k16 (10−7m/s)	P20 (m)	P21 (m)	P22 (m)	P23 (m)	P24 (m)	P25 (m)
1	1.00	1.00	1.00	5.00	1.00	1.00	1.00	1670.33	1669.38	1667.85	1665.89	1664.43	1664.00
2	5.50	5.50	2.00	7.50	3.00	5.50	5.50	1667.66	1667.24	1666.56	1665.32	1663.86	1663.25
3	3.25	7.75	1.50	8.75	2.00	7.75	3.25	1666.58	1666.25	1665.73	1664.84	1663.57	1663.15
4	7.75	3.25	2.50	6.25	4.00	3.25	7.75	1669.18	1668.57	1667.63	1666.00	1664.17	1663.32
5	2.13	6.63	2.75	9.38	3.50	2.13	4.38	1667.15	1666.75	1666.29	1664.99	1663.73	1663.21
…	…	…	…	…	…	…	…	…	…	…	…	…	…
96	9.79	2.76	1.98	8.32	4.28	1.35	9.51	1670.11	1669.37	1668.24	1666.42	1664.32	1663.35
97	1.21	4.59	1.89	9.02	1.84	8.80	4.59	1669.38	1668.72	1667.81	1666.26	1664.19	1663.32
98	5.71	9.09	2.89	6.52	3.84	4.30	9.09	1666.04	1665.77	1665.35	1664.56	1663.37	1662.98
99	3.46	6.84	1.39	5.27	2.84	2.05	2.34	1665.62	1665.41	1665.00	1664.28	1663.27	1662.93
100	7.96	2.34	2.39	7.77	4.84	6.55	6.84	1672.08	1671.14	1669.65	1667.23	1664.70	1663.42

. According to the results of the sample calculations, seven permeability coefficients are taken as input data for the SVR surrogate model, while the hydraulic head values at six monitoring points are taken as output data. In addition, 80 groups of samples are randomly selected as the training set to train the SVR model, and the remaining 20 groups of samples are used as the test set to test the SVR model. Meanwhile, during the training process of the SVR model, the IDBO algorithm is invoked to optimize and adjust the key parameters of the model, so as to obtain the best parameter combination and establish an SVR model with the optimal predictive performance. Among them, the main parameters of the IDBO algorithm are set as follows: the population size of dung beetles is 30, the maximum number of iterations is 300, and the optimization ranges of the penalty factor, insensitive loss factor, and kernel function variance are [1, 100], [0.001, 0.1], [0.1, 10], respectively.

The optimal parameter combination of the SVR model determined by the IDBO algorithm is as follows: the penalty factor is set to 7.82, the insensitive loss factor is set to 0.04, and the kernel function variance is set to 2.56. In order to further evaluate the prediction accuracy and validate the effectiveness of the SVR model, the optimized trained SVR model was used to predict the test samples (Figure 8), and the four prediction performance evaluation indexes for each monitoring point were calculated, as shown in

Table 3. The prediction performance of the surrogate model.

Monitoring Point	MAE (m)	MAPE (%)	RMSE (m)	R-Squared
P20	0.3264	0.0196	0.3753	0.9814
P21	0.2489	0.0149	0.3018	0.9843
P22	0.2790	0.0167	0.3250	0.9707
P23	0.1563	0.0094	0.1850	0.9733
P24	0.0512	0.0031	0.0765	0.9755
P25	0.0437	0.0026	0.0504	0.8957

. The results indicate that, for all monitoring points, the maximum mean absolute error (MAE) is 0.3264 m, the maximum mean absolute percentage error (MAPE) is 0.0196%, the maximum root mean square error (RMSE) is 0.3753 m, and the minimum coefficient of determination (R-squared) is 0.8957. Hence, it can be concluded that the prediction results of the SVR model are close to the calculation results of the finite element model, meeting the accuracy requirements of engineering applications. The surrogate model accurately establishes the nonlinear mapping relationship between the permeability coefficients of the materials to be inverted and the hydraulic head values at each monitoring point. On this basis, it can effectively replace the cumbersome and time-consuming finite element calculations in the subsequent permeability coefficient inversion process, thereby significantly improving the computational efficiency of intelligent inversion analysis.

3.4. Inversion Results of Permeability Coefficients

Based on the established SVR surrogate model with excellent predictive performance, a physically meaningful and straightforward equation, Equation (19), is formulated as the optimization objective function. The IDBO algorithm is employed to perform a global search within the specified range of permeability coefficients for the materials to be inverted, so as to find the best permeability coefficient combination. Herein, the population size of dung beetles is set to 30, the maximum number of iterations is set to 500, and the values for other relevant parameters of the IDBO algorithm can be found in Section 2.2 and Section 2.3.

$$f_{\min }=\frac{1}{N}{\displaystyle \sum _{i=1}^N{(H_i-{H_i}^{\prime })}^2}$$

(19)

where $f_{\min }$ denotes the optimization objective function; $N$ indicates the total number of monitoring points; $H_i$ represents the measured value of the hydraulic head at the ith monitoring point; ${H_i}^{\prime }$ denotes the predicted value of the hydraulic head at the ith monitoring point.

After full iterative calculation via the Matlab R2020a software, the permeability coefficients obtained by the intelligent inversion analysis are as presented in

Table 4. Inversion values of permeability coefficients for materials.

Material	Permeability Coefficient Value Range (m/s)	Inversion Value (m/s)
Overburden I	1.00 × 10−5~1.00 × 10−4	7.71 × 10−5
Overburden II	1.00 × 10−4~1.00 × 10−3	2.65 × 10−4
Strongly weathered rock	1.00 × 10−6~3.00 × 10−6	2.67 × 10−6
Weakly weathered rock	5.00 × 10−7~1.00 × 10−6	8.34 × 10−7
Fresh bedrock	1.00 × 10−7~5.00 × 10−7	2.49 × 10−7
Anti-seepage wall	1.00 × 10−9~1.00 × 10−8	7.87 × 10−9
Anti-seepage curtain	1.00 × 10−7~1.00 × 10−6	6.21 × 10−7

. As can be seen from Table 4, the permeability coefficients determined by the final inversion are all within the specified ranges, which proves the rationality and feasibility of the proposed inversion method. Meanwhile, in order to further elucidate the effectiveness of the improvement strategies of the DBO algorithm, the original DBO algorithm was also applied to optimize the parameters of the SVR model and invert the permeability coefficients. Subsequently, the permeability coefficients determined by the IDBO algorithm and DBO algorithm were, respectively, input into the finite element model for forward calculations and error analysis. The relative error between the measured and computed hydraulic head values at each monitoring point is shown in Figure 9.

$$R_{\mathrm{e}}=\frac{H_{\mathrm{m}}-H_{\mathrm{c}}}{H_{\mathrm{u}}-H_{\mathrm{d}}}\times 100\%$$

(20)

where $R_{\mathrm{e}}$ denotes the relative error; $H_{\mathrm{m}}$ indicates the measured value of the hydraulic head at each monitoring point; $H_{\mathrm{c}}$ represents the calculated value of the hydraulic head at each monitoring point; $H_{\mathrm{u}}$ denotes the upstream water level, which is 1758.00 m; $H_{\mathrm{d}}$ indicates the downstream water level, which is 1662.20 m.

As depicted in Figure 9, the relative errors between the measured values and the calculated values of the hydraulic head at each monitoring point based on the IDBO algorithm are all within 1.00%. The maximum relative error is only 0.63% at the P20 measuring point, which satisfies the engineering requirements for the calculation accuracy and further verifies the reliability of the inversion results. In addition, comparing the relative errors of the IDBO algorithm and DBO algorithm, it is evident that the relative errors of the IDBO algorithm are significantly lower than those of the DBO algorithm at all monitoring points. The improved IDBO algorithm with multi-strategy fusion is superior to the original DBO algorithm in terms of convergence speed, optimization accuracy, and stability, which proves the validity of the improved strategies.

3.5. Prediction of Seepage Characteristics during Dam Operation

Since the completion of the Altash water conservancy project, the dry climate in the northwestern part of China has resulted in a decrease in rainfall in the Yarkand River basin and a shortage of water coming from the upper reaches of the reservoir area, leading to a delay in restoring the reservoir to the normal water level. At present, there are no monitoring data that can directly reflect the seepage behavior of the Altash dam during normal operation, but the complex geological conditions of the dam site may have adverse effects on the storage of the reservoir. Therefore, it is essential to predict the seepage characteristics of the project during the operation stage based on the permeability coefficients obtained through intelligent inversion analysis, further analyze the permeability stability of the key parts of the dam, and evaluate the service effect of the anti-seepage and drainage system, so as to provide a guarantee for the long-term stable operation of the hub project.

The finite element seepage calculation results of the Altash dam under a normal water level (1820.00 m) are depicted in Figure 10. The distribution law of the seepage field of the dam under a normal water level is clear and reasonable. On the one hand, the phreatic line in the dam forms a sudden drop at the concrete face slab, and the highest head of the phreatic line in the cushion zone below the face slab is 1676.93 m. The anti-seepage system cuts down the head by 143.07 m, accounting for 90.67% of the total head difference between the upstream and downstream regions, which indicates that the anti-seepage system of the dam has a significant seepage control effect. On the other hand, the phreatic line has a gentle trend in the sand gravel zone and drainage zone, indicating that the drainage system of the dam has effective drainage performance. Moreover, the equipotential lines in the dam are relatively dense at critical parts such as the concrete face slab, anti-seepage wall, and curtain grouting. The maximum hydraulic gradients at these parts are 154.71, 112.41, and 18.63, respectively, all of which are lower than the permissible hydraulic gradients for the materials. This affirms that the dam structure meets the stability requirements for seepage. The maximum hydraulic gradients of the overburden, cushion zone, transition zone, sand gravel zone, and drainage zone are 0.032, 0.031, 0.028, 0.059, and 0.023, respectively, which are all lower than the allowable hydraulic gradients determined by the penetration tests. As a result, both the dam foundation overburden and the dam filling zones also satisfy the stability requirements for seepage.

In general, the optimal combination of permeability coefficients determined through intelligent inversion based on the Sobol-IDBO-SVR surrogate model can accurately reflect the seepage characteristics of the Altash dam during the normal operation stage. The anti-seepage and drainage system of the dam is arranged reasonably and effectively, demonstrating that the seepage control effect is remarkable. The maximum hydraulic gradients at the key parts of the dam are all lower than the permissible hydraulic gradients, indicating that the seepage stability meets the requirements.

4. Conclusions

The accurate and efficient determination of material permeability coefficients provides a critical foundation in mastering the seepage characteristics of concrete face rockfill dams. In response to the deficiencies in existing studies on seepage parameter inversion, this paper proposes a novel method for the permeability coefficient inversion of concrete face rockfill dams driven by multi-intelligence technologies, which is successfully applied to the Altash concrete face rockfill dam with perfect seepage monitoring data. On this basis, we carry out the permeability coefficient inversion analysis of the dam foundation rock mass and anti-seepage structure, which realizes the scientific evaluation of the dam seepage safety and also provides a new perspective for similar parameter inversion problems. The main conclusions are briefly summarized as follows.

(1)

By introducing the Sobol sequence sampling method, it is possible to construct a sample set with high coverage of the parameter space and optimal characteristics such as a uniform distribution, which effectively improves the prediction accuracy of the surrogate model. Meanwhile, adopting Sobol sequences instead of pseudo-random sequences to initialize the dung beetle population can not only help to improve the traversability and diversity of the dung beetle population but also reduce the possibility of the DBO algorithm falling into the local optimum trap, which reasonably improves the convergence performance of the DBO algorithm.

(2)

The DBO algorithm is improved from multiple perspectives by fusing the three strategies of Sobol sequence initialization population, nonlinear convergence factors, and an adaptive Levy–Brownian mixed variation mechanism, which leads to the establishment of the IDBO algorithm with excellent optimization performance. By employing the IDBO algorithm to optimize and adjust the key parameters of the SVR model, we construct a surrogate model with excellent prediction performance, which can replace the cumbersome and time-consuming finite element model and greatly improve the efficiency of inversion optimization.

(3)

The application analysis of engineering examples shows that the Sobol-IDBO-SVR fusion surrogate model can efficiently invert the optimal permeability coefficient combination. The maximum relative error between the measured and inversion values of the hydraulic head at each monitoring point is only 0.63%, indicating that the inversion accuracy meets the requirements of engineering applications. The permeability coefficients determined through the intelligent inversion analysis can accurately reflect the seepage characteristics of the Altash concrete face rockfill dam during the normal operation stage, demonstrating that the seepage control and stability of the dam meet the requirements.