An Inversion Study of Reservoir Colluvial Landslide Permeability Coefficient by Combining Physical Model and Data-Driven Models

Abstract

The saturated permeability coefficient (ks) is a key parameter for evaluating the seepage and stability of reservoir colluvial landslides. However, ks values obtained from traditional experimental methods are often characterized by large variations and low representativeness. As a result, there are significant deviations from actual observations when used in seepage field calculations for reservoir landslide analysis. This study proposes an intelligent inversion method that combines a physical model and a data-driven model for reservoir landslide ks based on actual groundwater level (GWL) monitoring data. This method combines Latin Hypercube Sampling (LHS), unsaturated flow finite element (FE) analysis, particle swarm optimization algorithm (PSO), and kernel extreme learning machine model (KELM). Taking the Hongyanzi landslide in Sichuan Province, China, as the research object, the GWL of the landslide under different ks was first obtained by LHS and transient seepage FE analysis. Then, a nonlinear functional relationship between ks and the landslide GWL was fitted based on the PSO-KELM model. Finally, the optimal landslide ks was obtained by minimizing the root-mean-squared error between the predicted and actual GWL using the PSO. A global sensitivity analysis was also conducted on the ks of different rock and soil layers to reveal their control rules on the calculation of landslide GWL. The research results demonstrate the feasibility of the proposed method and provide valuable information for similar landslides in practice.

1. Introduction

As of 2014, the number of reservoirs in China with a dam height of more than 30 m reached 6539 [1]. Reservoir colluvial landslides are widely distributed in major reservoirs in Southwestern China, such as the Three Gorges Reservoir Area (TGRA), the Baihetan Reservoir Area, and the Dagangshan Reservoir Area [2,3,4,5]. These landslides pose a significant threat to the safety of people and property in the reservoir areas and to the normal operation of hydropower plants [6,7].

Numerous studies have shown that changes in reservoir water levels and precipitation are two critical factors influencing the stability of reservoir landslides [8,9,10,11,12,13]. Reservoir water and rainwater infiltrate into the landslide mass through soil and rock layers with different permeability characteristics, weakening the soil and rock mass parameters, altering the seepage field within the landslide, and subsequently affecting the stability of the landslide. ks plays a key role in determining the response of reservoir colluvial landslides to dynamic water actions, such as reservoir water levels and rainfall [14,15]. Therefore, an accurate evaluation of ks is critical to reliably assess the stability of reservoir colluvial landslides [16].

The landslide ks is usually obtained from permeability tests, such as double-ring permeability tests, hydraulic tests, laboratory permeability tests, etc. [17]. Since the reservoir colluvial landslide mass is typically composed of soil and rock mixtures with strong spatial variability, the ks obtained by traditional test methods have significant dispersion, making it difficult to reflect the overall permeability characteristics of the landslide mass. The calculated results of the landslide seepage field based on the experimentally obtained parameters are often significantly different from the actual monitoring data. ks can also be obtained using inversion methods. Inversion methods can neglect the influence of ks heterogeneity, resulting in ks that closely match the actual monitoring information [18,19,20,21,22,23]. For example, in the study [16], the ks of the reservoir colluvial landslide was inverted through the GA-GRNN model, and in the research work [24], the ks of the Xietan landslides was obtained through the GA-BP model. The research results demonstrate that the ks obtained by inversion methods are more representative compared to experimental methods. However, although inversion methods have shown promising prospects for obtaining the ks of reservoir colluvial landslides, there is still a lack of relevant case studies. The proposed inversion algorithms have not been validated by other cases of reservoir landslides, and there is still room for further improvement in the precision of inversion.

A major bottleneck in the development of inversion research for reservoir landslide ks is the lack of abundant GWL monitoring data. Sufficient GWL data are a prerequisite for inversion research. However, in the past, such data were scarce due to immature monitoring technologies and high monitoring costs. Fortunately, with the continuous development of monitoring technologies in recent years and the gradual reduction in monitoring costs, comprehensive and detailed professional monitoring has been carried out for many reservoir landslides that pose significant hazards. In these monitoring activities, the GWL is a key monitoring object. For example, professional GWL monitoring has been conducted for reservoir landslides such as the Hongyanzi landslide in the Pubugou reservoir [25] and the Shuping landslide in the TGRA [26], accumulating a large amount of time-series data of landslide GWL. These data provide abundant resources for the inversion study of landslide ks and inject new momentum into the development of this field.

In this paper, a new method for inversion of ks of reservoir colluvial landslides is proposed by combining the physical model and data-driven models. The computational principles and procedures of this method are described in detail, and its application to the Hongyanzi landslide in the Pubugou reservoir, Hanyuan County, Sichuan Province, is presented to validate the feasibility and accuracy of the proposed method.

2. Methodology

The mapping relationship between the ks and the GWL of the landslide can be expressed as follows:

$$Y=f\left(ks\right)+E$$

(1)

where ks is the saturated permeability coefficient of the landslide, Y is the GWL monitored at different times, f is the mapping function, and E is the computational error.

In this study, a physics-based FE model was used to generate a sample dataset with ks as input and GWL as output. The data-driven based PSO-KLEM model was used to train this input–output dataset and obtain a neural network surrogate model for the FE model. This surrogate model serves as the mapping function f in the context. The objective of landslide ks inversion is to minimize the error between the calculated GWL and the actual monitored GWL. In this paper, the root-mean-squared error (RMSE) between the calculated values and the actual values was selected as the objective function, which is represented as follows:

$$\min \left(E\right)=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(y_i^{cal}-y_i^{mon}\right)}^2}{N}}$$

(2)

where $y_i^{cal}$ and $y_i^{mon}$ signify the ith calculated GWL and monitored GWL, respectively. N is the total number of dataset samples.

According to Equations (1) and (2), the PSO algorithm is used to invert the optimal ks.

2.1. Unsaturated Seepage Analysis

In this study, we employed the SEEP/W module, which is a component of Geostudio 2018, to perform the seepage analysis. SEEP/W is a powerful FE program that is widely used for soil seepage analysis, especially for studying transient seepage phenomena in landslides exposed to rainfall or reservoir water level fluctuations [27,28,29]. Within SEEP/W, the two-dimensional seepage fields in variably saturated slopes are governed by the following equation.

$$\nabla \cdot K\left(h\right)\nabla H=\frac{\partial \theta \left(h\right)}{\partial t}$$

(3)

where h is the pressure head; H is the total head; K(h) is the hydraulic conductivity function (HCF); θ(h) is the soil–water characteristic curve (SWCC); and t is the time.

The widely used van Genuchten model is adopted for the SWCC [30] and HCF. The models can be expressed as follows:

$$S_e=\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}=\left\{\begin{array}{cc}{\left[1+{\left|{\alpha }^{-1}h\right|}^n\right]}^{-m}& h<0\\ 1& h\ge 0\end{array}\right.$$

(4)

$$K=\left\{\begin{array}{cc}{K}_s{S_e}^{0.5}{\left[1-{\left(1-{S_e}^{1/m}\right)}^m\right]}^2& h<0\\ K_s& h\ge 0\end{array}\right.$$

(5)

where S_e is the effective saturation; θ_r and θ_s are the residual water content and the saturated water content, respectively; n and m are each fitting parameters related to the pore size distribution, m = 1 − 1/n; and α is a fitting parameter related to the air-entry value.

2.2. KELM

The KELM algorithm [31] is an advanced machine learning technique that combines the merits of extreme learning machines and kernel methods. KELM is primarily employed for tasks such as binary classification, regression, and unsupervised learning. By leveraging a kernel function, KELM maps the input data to a high-dimensional feature space, where a linear system of equations is solved. This approach enables rapid learning and exceptional generalization performance. KELM presents several advantages, including computational efficiency, simplicity, and strong generalization capability. Compared to traditional kernel methods, KELM significantly reduces computational burden by randomly assigning input weights. Additionally, it avoids time-consuming iterative optimization processes. The performance of KELM is heavily affected by the two hyperparameters of penalty parameter C and kernel bandwidth γ. This paper employs the widely utilized PSO algorithm to optimize the two hyperparameters.

2.3. PSO

The PSO algorithm [32] is a population-based optimization technique inspired by bird flocking or fish schooling. It iteratively updates the positions and velocities of particles to find the optimal solution. PSO balances exploration and exploitation by leveraging historical information and the best positions of neighboring particles. It efficiently explores the search space and converges towards the optimal solution by adjusting velocities based on personal and global best positions. PSO is advantageous because of its ease of implementation, compatibility with various optimization problems, and lack of reliance on derivative information. This paper used the PSO algorithm to optimize the KELM model and the inversion of ks.

2.4. Steps of ks Inversion

The proposed method can be divided into five detailed steps as follows:

(1)

Determine the range of ks values. Based on permeability test data and similar landslide ks values, the range of ks values is determined.

(2)

Sample extraction. The LHS method extracts ks samples within the defined range of values. The LHS method ensures the representativeness and randomness of the samples.

(3)

Creation of a physical model. Based on the engineering geological profile of the landslide, a physically based FE model for landslide unsaturated seepage analysis is established in SEEP/W. The ks sample data are input while keeping other unsaturated parameters fixed. The FE model is used to calculate the GWL of landslides with different ks values.

(4)

Establishment of a data-driven model. Direct inverse calculations that use the FE model would be computationally inefficient. To accelerate computational efficiency, this study used a response surface model based on the PSO-KELM as a surrogate model for the FE model, facilitating the inverse calculation of ks. Based on the ks values and their corresponding GWL data, the PSO-KELM model is trained. The training and test datasets are divided in a ratio of 2:8, and the KELM is trained using a fivefold cross-validation method to ensure the generalizability of the model. Once the data-driven model is trained and validated, it can serve as a response surface model to replace the physical model to calculate the GWL of landslides under given ks.

(5)

ks inversion. Based on the PSO-KELM model and real GWL data of landslides, an objective function for inversion is constructed according to Equation (2). Then, the PSO algorithm is executed to invert the ks value. When the PSO algorithm satisfies the iteration conditions, the optimum permeability coefficient ks can be obtained.

3. Hongyanzi Landslide Case

3.1. Engineering Geology Overview

The Hongyanzi landslide is located in Hongyanzi Village, Hanyuan County, Sichuan Province, China, on the south bank of the Dadu River, approximately 23 km upstream of the Pubugou Dam (Figure 1) [25]. This landslide exhibits a semicircular plane shape, with a length of approximately 600 m and a width of 580 m, covering an area of approximately 34 × 10⁴ m² (Figure 2). The crown of the landslide is located at an elevation of 1050 m above sea level, while the toe of the surface of rupture is at 770 m above sea level. The thickness of the main body varies between 30 and 70 m, with a total volume estimated at around 750 × 10⁴ m³. The primary sliding direction of the landslide is approximately 338°. The composition of the main body mainly consists of debris soil, with a gravel content ranging from 60% to 80%. The bedrock of the Hongyanzi landslide is bounded by the Hanyuan-Zhaojue fault. The upper plate consists of Permian Yangxin formation dolomite, with a bedding strike of 305° and a dip angle of 50° (Figure 3). The lower plate comprises basalt of Permian Emeishan formation.

The water level of Pubugou Reservoir has exhibited periodic fluctuations between 790 and 850 m since November 2009. Consequently, the GWL and the deformation characteristics of the Hongyanzi landslide undergo dynamic changes in response to these variations in the reservoir water level. Figure 4 illustrates the fluctuations in the GWL recorded in the ZK5 monitoring hole relative to the reservoir water level. It can be observed that the variation of GWL in the borehole shows a positive correlation with the fluctuation of the reservoir water level, albeit with lag. During the decline in the reservoir water level, the GWL in the borehole is higher than the reservoir water level. This results in the development of a hydraulic gradient pointing toward the exterior of the slope, consequently amplifying the sliding force of the landslide and compromising its stability.

3.2. ks Inversion

This study simultaneously performed inversions of the ks for both the landslide mass and the bedrock, resulting in a total of two inversions. The period from 1 January 2013 to 1 August 2013 captured the complete fluctuation process of the reservoir water level. Therefore, the reservoir water level, rainfall, and GWL monitoring data from this timeframe were employed to train the proposed algorithms and invert the ks parameters for the landslide. Data between 2 August 2013 and 1 January 2014 were utilized for validating the inverted values of ks.

3.2.1. FE Model

Figure 5 shows the FE model of the Hongyanzi landslide with a 694 m length and a 221 m height established in SEEP/W. The FE model consists of 5674 quadrilateral and triangular elements with 5760 nodes. The front edge of the landslide mass acts as the boundary for reservoir water level fluctuations, while the surface of the landslide slope serves as the boundary for rainfall, utilizing real data recorded between 1 January and 1 August 2013. The left boundary was established as a fixed total head boundary at 880 m according to [25]. Homogeneity was assumed for both the landslide mass and the bedrock, disregarding spatial variability and anisotropy. Given that the majority of the bedrock lies beneath the GWL and is saturated, the analysis was simplified by neglecting the unsaturated characteristics of the bedrock and focusing solely on the unsaturated properties of the landslide mass. The ks for the landslide mass and the bedrock were determined to range from 0.01 m/day to 15 m/day and 0.001 m/day to 2 m/day, respectively, guided by relevant experimental data and similar landslide cases.

Table 1. Hydraulic parameters of the Hongyanzi landslide.

Type	ks (m/day)	α (kpa)	n	m	θs	θr
Mass	0.01~15	10	1.5	0.33	0.45	0.1
Bedrock	0.001~2	/	/	/	/	/

presents the various hydraulic parameters of the landslide, and Figure 6 is the volumetric water content function diagram of the landslide mass.

The FE analysis was divided into two sections: the first section corresponds to steady-state seepage calculations, where the initial reservoir water level at time 0 serves as the boundary condition to compute the initial GWL of the landslide. The second section involves transient seepage calculations, which simulate the GWL between 1 January and 1 August 2013, using subsequent real reservoir water level and rainfall data, based on the initial GWL. The transient seepage calculations were executed over 200 steps, preserving the daily GWL calculation results during the specified period.

3.2.2. Sensitivity Analysis

Using LHS, 1000 samples were obtained for the ks corresponding to both the landslide mass and the bedrock. These samples were used as inputs in the FE model to compute the GWL, resulting in 1000 sets of calculated GWL values. Figure 7 indicates that the GWL computed by the FE model covers the actual GWL, implying that the ks corresponding to the simulated GWL closest to the real GWL falls within the selected range. This underscores the rationality of the determined ks range.

To reveal the sensitivity of GWL changes in response to ks in the landslide mass and bedrock, this study performed a global sensitivity analysis (GSA) using the PAWN method based on 1000 simulated samples [11]. The PAWN method is a distribution-based approach used for GSA [33,34]. It aims to assess the sensitivity of a mathematical model’s uncertain input factors by analyzing the cumulative distribution functions of the model’s outputs. Compared to other moment-independent methods, PAWN offers advantages in characterizing output distributions by their cumulative distribution functions rather than probability density functions. This characteristic makes the numerical approximation of PAWN sensitivity indices (SIs) easy and robust. The SIs range from 0 to 1. The higher the sensitivity index, the more sensitive the system output is to the parameter, and vice versa.

Figure 8 shows the temporal evolution of the SIs for the ks values of the landslide mass and the bedrock in response to the varying GWL. The following key observations can be made: (1) The minimum, maximum, and average SI values for the ks of the landslide mass are 0.41, 0.78, and 0.63, respectively. These changes in SI are negatively correlated with fluctuations in reservoir water levels. (2) For the bedrock ks, the minimum, maximum, and average SI values are 0.19, 0.56, and 0.33, respectively. These SI variations show a positive correlation with reservoir water level fluctuations. (3) The global sensitivity of the ks of the landslide mass is consistently greater than that of the bedrock. The GWL have a significant influence on the ks values of the landslide mass, while they have a relatively weaker influence on the ks values of the bedrock. Therefore, special attention should be paid to the determination of ks values for the landslide mass during inversion analysis.

3.2.3. PSO-KELM Model

The PSO-KELM model was trained using the 1000 input–output samples generated by the FE model. In the PSO algorithm, the swarm consists of 20 particles, and the maximum number of iterations was set to 1000. The iteration process terminates when, for 40 consecutive iterations, the difference in fitness values between two neighboring iterations is less than 10⁻⁴. In KELM, the ranges of C and γ were [2⁵, 2¹⁵] and [2⁻⁵, 2⁵], respectively. Figure 9 depicts the iteration curve of the PSO algorithm that optimizes the KELM model. It can be observed from the graph that the fitness value reaches a stable state after around 80 iterations. The optimal values for C and γ were determined to be 2852.9 and 0.1422, respectively. To evaluate the predictive performance of the optimized KELM model, both the training and testing datasets were used, and the results are presented in Figure 10. As shown in Figure 10, the predicted values of the KELM model closely align with the values computed by FE model for both the training and testing datasets. In the training dataset, the RMSE of the predicted values is 0.3422, with a coefficient of determination (R²) of 0.9745. In the testing dataset, the corresponding values are 0.4220 for RMSE and 0.9719 for R². These results demonstrate the favorable predictive performance of the PSO-KELM model, affirming its ability to effectively replace the FE model in conducting ks inverse calculations.

3.2.4. Validation of Inversion Results

Figure 11 presents the iterative convergence curve of the PSO algorithm for the ks inversion task. The graph indicates that the fitness function reaches a steady state around the 30th iteration. Upon completion of the iterative of PSO computation, the final inversion values of ks were determined: 9.27 m/day for the landslide mass and 1.15 m/day for the bedrock.

The inverted ks values were incorporated into SEEP/W to calculate the GWL of the Hongyanzi landslide between 1 January 2013 and 1 January 2014, considering the effects of reservoir water levels and rainfall. The comparison between the calculated GWL at ZK5 and the actual monitored GWL is shown in Figure 12. It can be observed that the overall trend of the simulated GWL using the FE model is in good agreement with the actual monitored GWL, which proves the reliability of the inversion results and the applicability of the proposed method. The material of the Hongyanzi landslide is gravel soil, with a gravel content between 60% and 80%. Due to the high gravel content and large porosity of the landslide mass, the landslide mass has high permeability, and the inverted ks 9.27 m/day is consistent with the high permeability characteristics of the landslide mass, indicating the reliability of the inversion ks value of the landslide mass. The bedrock in the landslide area is mainly basalt. This type of rock mass has well-developed joints and fissures, which makes it have good permeability. However, because its density is higher than that of the landslide gravel soil, its ks is smaller than that of the landslide mass. The inverted ks of bedrock in this article is 1.15 m/day, which not only conforms to the permeability characteristics of this type of rock mass, but is also smaller than the ks of the landslide mass, which also proves the reliability of the inversion results.

It can be seen from Figure 12 that there are variations in the accuracy of the simulations in different periods: (1) During the period of declining reservoir water levels from 1 January to 1 April 2013, the accuracy of the GWL simulation is high, with absolute errors ranging from −2 to 2 m. (2) From 2 April to 1 July, when the reservoir water level is low, the accuracy of the GWL simulation is relatively poor, with a maximum absolute error of approximately 13.5 m and an average absolute error of 10.3 m. (3) From 2 July to 31 December, during the rising period and the high water operation period of reservoir water levels, the accuracy of the GWL simulation improves, with a significant decrease in absolute errors ranging from −6 m to 2 m and an average absolute error of −2.7 m. The simulation errors may arise from three main aspects: (1) The isotropic assumption of ks neglects its spatial variability. (2) The two-dimensional simulation cannot fully capture the actual three-dimensional flow conditions of the landslide. (3) It was assumed that the unsaturated hydraulic parameters and the total head boundary conditions of the left side within the FE model of the landslide were fixed, without considering the effect of their variations on the calculation of the GWL. Given the complexity of groundwater flow within the landslide, achieving an accurate simulation of GWL under the influence of rainfall and reservoir water levels is a significant challenge. For future research, addressing the above issues could improve the accuracy of the inversion results.

The ks obtained from the traditional experimental methods for the landslide mass and bedrock are 0.45 m/day and 0.001 m/day, respectively [25]. By incorporating the ks values obtained from the traditional methods into the FE model, the landslide GWL are calculated and compared with the GWL obtained from the proposed method, as shown in Figure 13. It can be observed that the GWL obtained from the proposed method is more consistent with the actual values and exhibits smaller errors compared to the GWL calculated using the traditional method. The RMSE of the GWL calculated by the traditional method is 7.86, while the RMSE of the GWL calculated by our proposed method is 6.05, representing a reduction of 23%. This reduction signifies the superiority of our proposed method.

By utilizing an inverse estimation of ks in conjunction with accurate seepage calculation models such as the FE model, it is possible to accurately predict the dynamic changes in GWL of reservoir landslides under the combined effects of reservoir water level and rainfall. The predicted GWL can then be incorporated into corresponding landslide stability evaluation models, such as the limit equilibrium method and the strength reduction method, to accurately forecast the dynamic changes in landslide stability. This approach provides robust support for landslide prediction.

4. Conclusions

This paper presents an intelligent method for the inversion of the ks of reservoir colluvial landslides by combining physical and data-driven models. Taking the Hongyanzi landslide in Sichuan as an example, the inversion analysis of both the ks of the landslide mass and the bedrock was performed using the proposed method, which led to the following conclusions:

(1)

The GWL has a high sensitivity to the variation of ks of the landslide mass, with an average SI of 0.63, while it is less sensitive to the variation of ks of the bedrock, with an average SI of only 0.33. The SI of the ks of the landslide mass shows a negative correlation with changes in the reservoir water level, while the SI of the ks of the bedrock shows a positive correlation.

(2)

By integrating LHS and an unsaturated seepage FE analysis and using the PSO-KELM model, an accurate implicit mapping relationship between ks and GWL time series data was established. This mapping relationship can be used to replace time-consuming unsaturated-flow FE calculations.

(3)

The GWL calculated by the corrected FE model obtained by the proposed inversion method shows a consistent overall trend with the measured data and shows good agreement. Therefore, the results of the inversion analysis are considered reliable.