Application of K-Nearest Neighbors (KNN) method to undrained shear strength prediction of bauxite tailing

Hugo Brandão ¹*, Guilherme Pinto² and Tatiana Santos³

1. 1. Pimenta de Ávila Consultoria, Minas Gerais – Brazil, Geotechnical Engineer, hugo.assis@pimentadeavila.com.br, +55 31 99912-1993
   2. Pimenta de Ávila Consultoria, Minas Gerais – Brazil, Geotechnical Engineer, guilherme.henrique@pimentadeavila.com.br, +55 31 99680-2136
   3. Postgraduation Program of Mineral Engineering, Universidade Federal de Ouro Preto, Minas Gerais – Brazil, Doctor Professor, tatiana.santos@ufop.edu.br, +55 31 3559-1735

ABSTRACT

Undrained shear strength of tailings is one of the most important parameters used to evaluate tailing’s dam stability conditions. Several methodologies (empirical and analytical) have been developed to evaluate the yield undrained shear strength based on piezocone penetration data (CPTu) using bearing capacity factor theory (N_kt). However, most of these methodologies or correlations consider assumptions proposed to natural soils and they can result in inconsistent values if directly applied to manufactured material such as bauxite tailing. Researchers have recently applied data science techniques in order to obtain more robust and accurate models, reducing empiricism and increasing the reliability of the results by developing models capable of predicting the undrained strength. In this article, K-Nearest Neighbors (KNN) method was applied to evaluate the yield undrained shear strength of a bauxite tailings based on CPTu data (independent variables) and field vane shear test (dependent variable). KNN method is a machine learning technique capable of classify or predict a value based on the distance of nearest values from the unknown data and it does not require assumptions to learn from previous cases/instances. To apply KNN method, a dataset was constructed by eleven pairs of CPTu and field vane shear test from a specifical site. The results obtained by KNN method were compared with the conventional approach by using the N_kt, N_ke and N_Δu values that results in the lowest error metrics. The results of this study clearly demonstrated that the KNN model can be applied to evaluate the undrained shear strength of the bauxite tailings.

INTRODUCTION

The correct determination of the strength parameters of tailings is highly important in the context of geotechnical engineering since they are crucial for evaluating the stability condition of a geotechnical structures. If this parameter is wrongly determined, there will be a risk of inaccurate design of geotechnical structures and it could not present enough resilience against the expected load that the structure will face throughout its life cycle, leading to its failure. Besides, the importance of the physical and chemical characterization of the tailings is highlighted in the Global Industry Standard on Tailings Management – GISTM (GISTM, 2020).

The yield undrained shear strength (S_u) is defined as soil/tailing strength in a saturated or nearly saturated condition, which is mobilized under a fast loading without allowing time for the soil to change its volume, dissipating the excess of the pore water pressure generated (Lunne et al., 1997). To evaluate the S_u is common to perform field investigations, such as Cone Penetration Test with Porepressure measurement (CPTu) and Vane Shear Test (VST).

The VST is the equipment used to determine Su values in saturated clay deposits, by the rotation of a set of cruciform rectangular blades pushed to pre-defined depths with velocity equal to 6±0.6°/min as described by ASTM D2573-08 standard (ASTM, 2015). The CPTu test consists of a 60° cone penetrometer pushing equipment, with a cross-sectional area of 10 cm2 and a 150 cm2 friction sleeve (commonly), and a data acquisition system. The cone penetration test is usually carried out with a speed of 2.0±0.5cm/s, with readings being recorded every 1cm to 5cm, as described by ASTM D-5778 standard (ASTM, 2020). The CPTu provides three independent measurements: i) the cone tip resistance (q_c), which characterizes the soil/tailings strength to cone penetration, ii) the sleeve friction (f_s), which represents the soil/tailings adhesion to friction sleeve and iii) the porewater pressure, commonly measured behind the cone tip (u₂). Also, it is common to perform dissipation test to evaluate the equilibrium porepressure (u₀), which consist in the fully stop of the cone penetration and the measurement of porewater pressure dissipation.

To obtain the Su values from the CPTu test, there are three independent equations based on the bearing capacity factor theory, by using N_kt, N_ke and N_Δu values, as described by Equations 1 to 3 (Lunne et al., 1997). Commonly, these parameters are obtained through specific empirical correlations, considering each studied site.

To avoid the empiricism of the bearing capacity factors correlation, data science techniques have been successfully applied, as presented by many authors, such as Zhang et al. (2021), Ly & Pham (2020), Abu-Farsakh & Mojumder (2020) and others. These techniques are based on algorithms that use mathematical, statistical, and/or computational knowledge, to detect database patterns and perform supervised and unsupervised training, allowing to reduce empiricism and development of models that use the knowledge of interrelationships between the variables. The use of these tools can provide greater assertiveness in the determination of the investigated parameters, since the elaborated models reduce the subjectivity attributed to the expert, increasing the reliability of the results obtained and ultimately, producing geotechnical projects safer.

Error Metrics

To evaluate the performance of the generated models, it was calculated the Coefficient of the Determination (R₂), Mean Absolute Error (MAE) and Mean Squared Error (MSE) by using the Equations 4 to 6. The R2 is commonly applied to evaluate the dispersion of the data, and the MAE/MSE is used to evaluate the distance of the predict value of the models from the real value (measured).

Bearing Capacity Factor Calibration (N_kt, N_ke and N_Δu)

To obtain N_kt, N_ke and N_Δu values from Equations 1 to 3, the train dataset was used to correlate the measurement of the CPTu data and S_u from VST test. The bearing capacity factors were calculated in order to obtain the highest accuracy of the model, being the highest R2 and the lowest MAE and MSE.

RESULTS AND DISCUSSION

As obtained by laboratory tests, the studied bauxite tailings were classified as silty-clayed, with a silt percentage varying between 60% and 80%, according with grain size classification of ASTM D422 (ASTM, 2007). The obtained grain density presented a mean value equal to 3.10g/cm³. After the pre- processing of the CPTu and the VST data, the results of the bauxite tailing dataset is presented in Figure 2. Is possible to observe in Figure 3 that q_{c and u2 show a tendency of increase over the depth, a common behaviour of a homogeneous material. Despite natural dispersion of the Su and fs values, the tendency to increase with depth is also notable.}

Before application of KNN algorithm, Bartllet’s test was performed. It was obtained a p-value equal 1.76 x 10^-178 by 4 Degrees of Freedom and a statistic of x² equal to 9.48. The p-value is lower than the significance levels of 5% (1.76×10^-178 << 0.005), thus KNN is applicable. After remove of the multivariate and split of the dataset into train and test subsets, bearing capacity factors were calibrated as shown in the Figure 3. The obtained values were N_kt = 11,47 N_Δu = 5,08 and N_ke = 10,93.

Based on the train data set used to calibrate the bearing capacity factors and the test dataset, KNN method was applied in order to obtain the best “K” neighbor value. In the KNN model calibration, it was necessary the use of the test dataset to choose of the best “K” value, in order to evaluate problems related to overfitting (R²_train >> R²_test) and underfitting (R²_train << R²_test). Using the same error metrics of the bearing capacity factor calibration, and varying the “K” values from 1 to 15, the best “K” value obtained is K = 3 as shown in Figure 4.

The calibrated models were applied into the test data set to obtain the predict S_u values and compared with the measured values from VST. Figure 5 shows the boxplot and the statistical summary of the models and the comparison with the measured VST data.

As can be seen, the mean values obtained from the models are almost the same of the measured value from the VST. However, is important to note that KNN model predicted S_u values with low dispersion as seen by the extremes of the boxplot and the lowest standard deviation (σ) if compared with the bearing capacity factors models and also the measured values from the VST. The next step was the evaluation of the error metrics (R², MAE and MSE) for bearing capacity factors and KNN models. Figure 6 shows the comparison of the train dataset error metrics to each model.

Comparing the results with literature, Abu-Farsakh & Mojumder (2020) reached to the same conclusion by the application of the Artificial Neural Network (ANN), another data science technique, to predict S_u values of a natural soil by using CPT data and measured values from unconfined triaxial test (UU). The author obtained R² equal to 0.84 and 0.77 to the ANN model and the N_kt model, respectively, by using N_kt=15.9. Besides the difference of the techniques and the material, the data science techinics has proved its value as an important tool to evaluate S_u.

CONCLUSION

Application of data science techniques in geotechnical problems can provide site specific models that can be used to predict tailings parameters more accurately than empirical correlations developed based on limited data from other material and conditions, especially with large datasets. Based on the calibrated KNN model with raw CPTu data presented, it is shown that data science technique was capable of provide reliable results, reaching better results with low dispersion than the empirical correlations when applied to predict the undrained shear strength. KNN is a simple ML algorithm and it can be easily calibrated, providing a great tool to geotechnical test analysis.