Improved prediction of shear wave velocity for clastic sedimentary rocks using hybrid model with core data

Mohammad Islam Miah

Department of Petroleum and Mining Engineering, Chittagong University of Engineering and Technology, Chittagong, 4349, Bangladesh

Keywords:Machine learning Acoustic shear velocity Elastic constants Rock strength Geomechanics

ABSTRACT Accurate measurement of acoustic velocities of sedimentary rocks is essential for prediction of rock elastic constants and well failure analysis during drilling operations. Direct measurement by advanced logging tools such as dipole sonic imager is not always possible. For older wells, such data are not available in most cases.Therefore, it is an alternate way to develop a reliable correlation to estimate the shear wave velocity from existing log and/or core data.The objective of this research is to investigate the nature of dependency of different reservoir parameters on the shear wave velocity (Vs) of clastic sedimentary rocks, and to identify the parameter/variable which shows the highest level of dependency. In the study, data-driven connectionist models are developed using machine learning approach of least square support vector machine (LSSVM). The coupled simulated annealing (CSA) approach is utilized to optimize the tuning and kernel parameters in the model development. The performance of the simulation-based model is evaluated using statistical parameters. It is found that the most dependency predictor variable is the compressional wave velocity, followed by the rock porosity, bulk density and shale volume in turn.A new correlation is developed to estimate Vs,which captures the most influential parameters of sedimentary rocks. The new correlation is verified and compared with existing models using measured data of sandstone,and it exhibits a minimal error and high correlation coefficient (R2=0.96). The hybridized LSSVM-CSA connectionist model development strategy can be applied for further analysis to predict rock mechanical properties. Additionally, the improved correlation of Vs can be adopted to estimate rock elastic constants and conduct wellbore failure analysis for safe drilling and field development decisions, reducing the exploration costs.

1. Introduction

Rock elastic constants(i.e.Poisson’s ratio,modulus of elasticity,bulk modulus and shear modulus) play an important role in the evaluation of sedimentary formation, sanding potentiality, and wellbore stability for safe drilling operations. These properties are closely linked to the rock bulk density, compressional and shear wave velocities as well as other petrophysical parameters such as shale content, porosity and fluid saturation of rocks. The combination of both acoustic wave velocities is vital to identify the formation lithology and reservoir fluid types (Pickett, 1963; Eberli et al., 2003), reliable prediction of rock physical parameters and elastic constants(Fj?r et al.,2008;Miah,2020),formation strength properties for hydrofracturing as well as wellbore stability (Chang et al., 2006; Sayers, 2007; Ameen et al., 2009; Karacan, 2009;Rasouli et al., 2011; Rajabzadeh et al., 2012; Kassab and Weller,2015; Li et al., 2015; Gholami et al., 2016; Onalo et al., 2018;Onalo, 2019; Moussas and Diamantis, 2021).

The acoustic wave velocities can be measured experimentally using core samples through the ultrasonic test as per the suggested steps of the American Society for Testing Materials (ASTM,1986).But reliable core sample preparation is not only tedious and timeconsuming but also very expensive and vulnerable to stress unloading (Raaen et al.,1996). Additionally, it is difficult to collect high-quality core samples from complex sedimentary structures,including fractured and unconsolidated rock formations (Castagna et al.,1985).Alternatively,compressional and shear wave velocities can be obtained from seismic measurements and wireline logs combined with advanced tools like dipole sonic borehole imager(DSI). However, that may not be the case for many wells due to availability and cost. For older wells, such data are mostly unavailable, and new tools cannot be re-run in them. Therefore,many researchers attempted to develop empirical correlations/models using core and log data for estimating the shear wave velocity Vs. For instance, those correlations were developed for different lithologies such as volcanic and hard rocks(Carroll,1969;Wadhwa et al.,2010;Beltran et al.,2020),carbonate rocks(Pickett,1963; Eskandari et al., 2004; Koesoemadinata and McMechan,2004; Brocher, 2005; Mabrouk and Pennington, 2009;Anemangely et al., 2019), and clastic rocks (Han et al., 1986;Greenberg and Castagna,1992; Oloruntobi and Butt, 2020).

Recent studies have shown the superiority of machine learning approaches over empirical and statistical methods, including mining, oil and gas engineering related problems. A growing tendency is observed among scholars to adopt machine learning algorithms to solve problems of various fields,such as geomechanics and formation evaluation. Numerous studies have been investigated to analyze the performances of data-driven models using machine learning tools (Ceryan, 2014; Behnia et al., 2017;Zendehboudi et al.,2018; Miah et al.,2020a). The most commonly used supervised machine learning tools are artificial neural network (ANN), convolution neural network (CNN), general regression neural network (GRNN), fuzzy logic (FL), neuro-fuzzy(NF), support vector machine (SVM), genetic algorithm (GA), gene expression programming (GEP), and random forest (RF). Additionally,hybrid connectionist models such as adaptive neuro-fuzzy inference system (ANFIS), least squares support vector machine(LSSVM) with optimization techniques of particle swarm optimization (PSO), cuckoo optimization algorithm (COA) or coupled simulated annealing (CSA), ANN-GA, ANN with wavelet transform(WT),SVM-GA and ant colony-fuzzy inference system(ACOFIS)are also applied to different disciplines such as rock mechanics (Yang and Zhang, 1997; Meulenkamp and Grima, 1999; Yilmaz and Yuksek, 2009; Cranganu and Bautu; 2010; Ocak and Seker, 2012;Khandelwal and Monjezi, 2013; Kumar et al., 2013; Maleki et al.,2014; Yurdakul et al., 2014; Anemangely et al., 2019; Dumke and Berndt, 2019; Lawal and Kwon, 2021; Miah et al., 2021), mining(Zhang et al., 2020), and geotechnical engineering (Zhang et al.,2021a; Zheng et al., 2021). Due to the considerations of computational time and cost-effective manner,above tools are broadly used to develop the data-driven model for obtaining rock mechanical properties including wave velocities related to rock elastic constants,brittleness performance,rock failure analysis for safe drilling operations, as well as rock mass characterization (Gaviglio, 1989;Hoek and Brown,1997;Sonmez et al.,2004;Saadat et al.,2014;Kim et al.,2017;Zoveidavianpoor,2017;Onalo,2019;Bukar et al.,2019;Mews et al., 2019; Sulaimon and Teng, 2020; Miah et al., 2020a,b;Onalo et al., 2020; Rezaee et al., 2020; Zhang et al., 2021b).

ANN and LSSVM tools are becoming a more widespread strategy for model development and prediction in the fields of rock mechanics and mining and geological engineering using nonlinear and multi-dimensional predictor variables of measured core and log data. The ANN has several inherent limitations such as slow convergence speed, less generalization, reaching local minimum and over-fitting challenges due to the choice of excessive hidden neurons (Esene et al., 2020). Moreover, it does not perform very well with less number of data points to develop a robust, reliable and accurate model(Esene et al.,2020).By contrast to the ANN tool,the supervised machine learning-based LSSVM can be applied with a limited number of data points and is fast to train to obtain a reliable and accurate predictive model. For instance, it has more generalization and training efficiency.

Most of the researchers used machine learning tools with different inputs based on the available core and log data in their studies to obtain Vsfor carbonate formations.The existing literaturereveals that rock parameters such as formation gamma-ray (GR),deep resistivity(Rt),shale content(Vsh),bulk density(RB),neutron porosity(NPHI),compressional slowness(DTc),rock porosity(PHI),cementation factor (m), formation depth (Z) and permeability (k)are used in data-driven connectionist models to predict the shear wave velocity(Vs), compressional wave velocity (Vp), and Stoneley wave velocity (Vst). Contributions of some authors were critically reviewed by focusing on the corresponding research methodologies and limitations while obtaining the acoustic wave velocities(Miah,2020;Miah et al.,2021).A number of scholars(e.g.Eskandari et al., 2004; Rezaee et al., 2007; Rajabi et al., 2010; Tabari et al.,2011; Asoodeh and Bagheripour, 2012; Bagheripour and Asoodeh,2013; Zoveidavianpoor et al., 2013; Zoveidavianpoor, 2014)analyzed the performance of machine learning approach-based predictive models for carbonate sedimentary formations using data samples, but most of the authors did not perform feature ranking through the predictive models to obtain an accurate model of Vs.A list of selective studies is tabulated in Table 1 to capture the corresponding author’s research strategies and model performance to obtain acoustic wave velocities using machine learning tools.Only a few studies were performed to investigate the contributions of predictor variables such as log and core data while obtaining Vsfrom clastic sedimentary rocks.

Table 1 List of machine learning applications to predict acoustic wave velocities for rock mechanics and formation evaluation studies.

Rezaee et al. (2007) constructed a neural network model by adopting Levenberg-Marquardt training algorithm using log variables of clastic sedimentary formation.The model performed excellently with a correlation coefficient (R2) of 0.94 and was in good agreement with other data-driven models of NF and FL. However,Rezaee et al.(2007)did not investigate the individual predictor variables of the model while obtaining Vs.Maleki et al.(2014)studied the performance of a hybrid connectionist model using log data and compared it to the empirical correlations of carbonate reservoir.Both ANN-GA and SVM-GA based models performed excellently with least error and high values of correlation coefficient (0.97 and 0.94,respectively). They concluded that compressional slowness, bulk density,and gamma-ray are the vital input variables to predict Vsfor the studied carbonate reservoir of Iran oil field.Akhundi et al.(2014)also investigated data-driven models and developed correlations using multi-varaiate regression approach, and subsequently compared the model performance with that of neural network-based model. It was found that neutron porosity, rock bulk density and compressional wave velocity are vital input variables to predict Vsof Asamari formation of Iranoil field.Later,Aleardi(2015)presentedGAbased models and compared those with both neural network and multi-varaiate regression-based models. It was observed that gamma-ray has a minimal effect on Vsprediction,while bulk density and true resistivity are the most vital variables for clastic sedimentary formation with sand-shale sequence layers.Bagheripour et al.(2015)studied the support vector regression(SVR)-based predictive model for Vsusing 2879data samples collected fromthe gaswell fieldof Iran.According to their research output, it was concluded that sonic compressional slowness has relative significance with least R2of about 0.41. Additionally, they examined the data-driven model performance, and also compared with ANN and other empirical correlations for the carbonate rock formation. Furthermore, Al-Dousari et al. (2016) developed a data-driven GRNN-based model with 35 data samples from petrographic analysis and petrophysical rock properties of dolomite, limestone, and siliciclastic formations, and compared it with five empirical models of carbonate rock. It was concluded that rock porosity and grain density are the vital input variables,compared withother variablesof claycontent,permeability(k)and cementation exponent(m)to estimate Vs.Behnia et al.(2017)investigated GEP-based model performance using 516 data points,of which 80%are for training and 20%for testing.It was concluded that all three rock variables (input parameters) of Vp, PHI, and RB are essential to predict an accurate model of Vsfor limestone formation.However, Behnia et al. (2017) did not rank those parameters as per their significant contributions to the predictive model for sedimentary rocks. Later, Wang and Peng (2019) developed a model for Vsprediction with extreme machine learning-based tool by adopting 516 data samples of Ordos Basin in China. According to the mean impactvalueanalysis,itwasshownthat compressional wave velocity,bulk density and neutron porosity are the major input variables to predict Vsfor carbonate sedimentary formation.It was claimed that ELM is a more efficient tool with R2of 0.97 compared to the SVR,ANN and CNN models. Eventually, the authors did not propose a new correlationto estimate Vsusing available core and log data.Bukaret al.(2019) studied different predictive models by adopting Gaussian process regression(exponential and Matern 5/2),ensembles of trees(both boosted and bagged),and SVM using log variables,i.e.Vp,Rt,GR,NPHI and watersaturation(Sw)fromthe area of Browsebasin offshore northwest Australia. It was evaluated that the exponential-based Gaussian process regression performed better with less error(RMSE=119.21)thanothermodelsoflinearSVM(RMSE=171.04)and boosted trees(RMSE=173.22),where RMSE is the root mean square error.Alkinani et al.(2019)investigated the performance of machine learning-based classic ANN and dynamic RNN predictive models by adopting Vp, RB and NPHI as the predictor variables for carbonate formation. They remarked that the RNN model outperformed the neural network model. Also, Anemangely et al. (2019) assembled different machine learning models using log data of carbonate formation(such as GR,Rt,RB,NPHI and Vp),whereas they noticed that LSSVM-COA outperformed the hybrid models of LSSVM-PSO and LSSVM-GA. Also, they claimed that all predictive variables make contributions to Vsof carbonate formation,i.e.Vp,RB,NPHI,GR and Rt,from higher to lower ranking.

From the literature review, it appears to be an attractive research topic for variable ranking and developing an improved model for clastic sedimentary rocks, while predicting shear wave velocity using real field data for reservoir geomechanics. In this study, the author attempted to develop data-driven connectionist models with the LSSVM using a global optimization of CSA. Additionally,the author attempted to evaluate model performance,and selected the best performing predictor variables to carry out the improved prediction of Vsfor clastic reservoir.

The main objectives of this study are: (i) to examine the performance of the data-driven hybrid connectionist model;(ii)to achieve the parametric sensitivity analysis and find the relative importance of rock parameters in the model; and (iii) to verify the proposed model for Vswith existing correlations using statistical analysis.

2. Research methodology

2.1. Hybrid connectionist model development strategy

The LSSVM is one of the effective connectionist models adopted in data classification as well as regression analysis to predict the desired variable. The classic SVM was firstly introduced by Vapnik(1999). Furthermore, Suykens and Vandewalle (1999) recommended a modified version of the SVM algorithm, i.e. LSSVM,which is less complex than the classic SVM (Miah et al., 2020a).Compared to SVM strategy, the LSSVM learning approach is less time-consuming,more generalization and robustness(Esene et al.,2020).More information concerning the theory and algorithm with different features of LSSVM would be retrieved in the literature(Smola and Sch?lkopf, 2004; Mahdevari et al., 2014; Miah et al.,2020a). From the literature review, it is found that the Levenberg-Marquardt training algorithm-based ANN approach is more efficient for a large number of data sets,whereas LSSVM can be adopted for the least number of data samples, while obtaining water saturation for reservoir characterization (Esene et al., 2020;Miah et al., 2020a).

Based on classic SVM formulations, the following function can be expressed (Suykens et al., 2002):

Fig.1. Flowchart for LSSVM-CSA model development using field data.

where the nonlinear function φ(·)denotes the primal space Rnto a feature space Rniwith higher dimensions, and the dimension niof this space is only identified in an implicit way; ω?Rniintroduces the weight factor; and b is a bias term in the data-driven model.

In the primal space,the optimization problems can be expressed as follows:

where ekrepresents the error term,J(·)is the cost function,and γ is the tuning parameter.

The Lagrangian function(L )can be written as follows using Eq.(2):

where Ω is the kernel matrix,and Ω = K(x，xk) = [φ(xk)]Tφ(xk)，in which K(x， xk)represents the kernel function that should satisfy the Mercer’s condition (Pelckmans et al., 2002).

The final expression can be written for the LSSVM function estimation as follows:

where αiand b are the weight factor and bias term for the connection, respectively.

Some major steps are shown in Fig.1 by the flowchart to capture the major outlines for the development of predictive model with LSSVM-CSA.

In the kernel function-based predictive model with the LSSVM,the tuning and kernel parameters such as γ and σ2are optimized with a global optimization technique of CSA(Xavier-de-Souza et al.,2010; Miah et al., 2020a). The CSA is a more expert and authentic optimization method compared to multi-start gradient descent approach (Suykens et al., 2002; Miah et al., 2020b). A data-driven hybrid model structure with LSSVM applied in the study is illustrated in Fig. 2.

The radial basis kernel function (RBF) is mostly applied in the data-driven LSSVM learning scheme to find the best possible output compared to other kernel functions including polynomial and sigmoidal functions (Suykens et al., 2002; Samui, 2008; Miah,2020) due to its more simplicity of computation as well as capability of solving nonlinear problems (Miah, 2020). The RBF can be expressed as follows (Samui, 2008; Miah, 2020):

The databank is randomly divided into two groups to construct the LSSVM models with RBF. The available core and log data samples are saturated bulk density(RB),porosity(PHI),compressional wave velocity(Vp)and shear wave velocity(Vs)of rocks,which are adopted to examine model performance and conduct parametric sensitivity analysis in the study.All data samples of sandstone were collected from the available literature (Han et al., 1986; Yusuf,2019). Based on the trial and error process, total data samples are classified into two groups, 75% for training, and 25% for testing in the hybrid model of LSSVM with the CSA optimization tactic.

2.2. Model performance indicator

In the study, five statistical parameters, i.e. average absolute percentage relative error (APRE), maximum absolute percentage error (MAPE), RMSE, correlation coefficient (R2) and performance index (PI) are used to assess model performance, which are presented below:

where n is the total number of core data samples, Ytis the experimentally measured variable, Yt,meanis the mean value of Yt, Ypis the predicted (output) variable, and VAF stands for the variance account factor in the study.The accuracy and reliability of the datadriven model are evaluated on the basis of high or low magnitude of statistical performance indicators. The model is considered the best when it has low statistical errors of RMSE,APRE and MAPE,and high magnitudes of R2and PI (close to 1).

2.3. Parametric sensitivity evaluation and correlation development strategy

A systematic scheme is adopted to accomplish the parametric sensitivity evaluation with data-driven models for assessing Vs.Furthermore, it is utilized to find the relative significance of the predictor variables in the hybrid model of LSSVM-CSA to obtain shear wave velocity,as shown in Fig.3.In the study,the optimized connectionist data-driven model is used to predict the formation Vs, which is assessed using the standard statistical performance indicators.

Fig. 2. A generalized LSSVM structure to obtain Vs in the study.

Fig. 3. Generalized major steps to conduct parameter sensitivity analysis and to rank them for Vs model.

Table 2 Summary of the statistical parameter values from the real field log data used.

Considering the relative impact of input variables on the connectionist models,the predictor(input)variables are ranked.In the input parameter ranking through the studied model, if the model results in high APRE,MAPE and RMSE,and low R2and PI,it means that the selected input variable has a low contribution to the predictive model. It is mentioned that only the most prominent parameters are adopted to develop the new correlation to obtain Vsprofile using multi-variable regression analysis (MVRA) and real field data samples of sandstone (Miah et al., 2020b). Meanwhile,MVRA is an extension of the normal regression analysis that includes more independent variables in the predictive correlation(Balan et al.,1995).

It fit really well, and he was really focused intently as he carefully made a double square knot to keep it secure (those Scouting13 skills really are handy)

Fig. 4. Variation of rock shear wave velocity with bulk density, compressional wave velocity and porosity depending on the number of data points.

Fig. 5. A graphical comparison between predicted and target values of Vs using the LSSVM-CSA approach.

3. Results and discussion

3.1. Data collection and quality analysis

The measured shear wave velocities of 74 sandstone samples with varying compressional wave velocity, clay content and porosity at a confining pressure of 30 MPa were used. It is noteworthy that the properties of rock samples vary due to the changes of rock depositional environments. Some samples are collected from the tight gas sandstones with very low porosity,while the rest are collected from the well-consolidated sandstone. For instance,four samples are clean sandstone with zero shale volume. The descriptive statistics of all the studied data are tabulated in Table 2.Relations between the input variables and output of shear wave velocity are shown in Fig.4.The magnitudes of the studied dataset considerably vary with each sample due to complex behavior of sedimentary composition and diagenesis.

3.2. Predictive model performance

In the study, the hybrid connectionist model of LSSVM with global CSA optimization system is utilized as an iterative random search scheme to obtain Vs. The hyper-parameters or kernel parameters are adjusted using CSA through the RBF with LSSVM connectionist approach. The LSSVM model has a better performance in terms of model accuracy and consistency, and it contributes to the lowest errors of APRE, MAPE and RMSE, and high correlation coefficient (R2). A graphical presentation of the connectionist model performance is shown in Fig. 5. The statistical performances of the LSSVM-CSA based data-driven model are listed in Table 3.

The hybrid connectionist model behaves excellently with R2of 0.9667 and 0.8871 for training and testing phases, respectively.Meanwhile, the optimized annealing parameters of σ2and γ are8.18 × 105and 6.99 × 104, respectively, with CSA technique. The scatter plots with 45°slope line between the target(measured)and predicted results are shown in Fig. 6.

Table 3 Statistical performance of hybrid LSSVM-CSA based data-driven predictive model.

Fig. 6. A graphical representation of predicted and measured Vs for (a) training(R2 = 0.97) and (b) testing (R2 = 0.96) phases in the studied model.

The RBF-based LSSVM connectionist model is also adopted for further analysis to investigate the most significant predictor variables while obtaining the shear wave velocity with different model schemes. The RBF-based LSSVM model shows values of MAPE,APRE and PI equal to 2.1509%,8.1969%and 0.9226 for the training,and 2.5484%, 9.9536% and 0.8871 for the testing phase,respectively.

3.3. Variable sensitivity and new correlation for shear wave velocity

The correlation matrix between the shear wave velocity(Vs)and other rock properties (e.g. compressional wave velocity, saturated bulk density, porosity and shale volume) is listed in Table 4. The shear wave velocity of the studied rock sample is linked with rock parameters such as compressional wave velocity and porosity as well as saturated bulk density. Based on the studied samples, the relationships between the shear and compressional wavevelocities,rock(saturated)bulk density,porosity,and shale content are depicted in Figs. 7-10, respectively.

Table 4 Correlation between shear wave velocity and other rock properties.

Fig. 7. Relationship between shear and compressional wave velocities of sandstone.

Fig. 8. Relationship between shear wave velocity and rock (saturated) bulk density.

Fig. 9. Relationship between shear wave velocity and porosity of rock samples.

Table 6 Comparison of statistical error performance for different models.

Fig.10. Relationship between shear wave velocity and shale volume of sandstone.

Fig.11. Comparison of error performance for various model schemes.

Linear regression shows a strong correlation between the shear and compressional wave velocities,with a correlation coefficient of 0.9406(Fig.7).For instance,Vsrises with increases in Vp(Fig.7)and rock(saturated)bulk density due to the higher density of electrons(Fig.8).Meanwhile,Vsdecreases with increasing porosity of clastic sedimentary rock samples, as illustrated in Fig. 9. Also, it is found that there is no significant relationship between Vsand the shale volume of sandstone samples (Fig.10). Based on the classic linear regression, the most important predictor variable is the compressional wave velocity, followed by the bulk density and porosity,with correlation coefficients of 0.9406, 0.4996 and 0.4813, respectively. In the meantime,the shale volume is less significant with a correlation coefficient of 0.1353.

Also, the hybrid connectionist model is adopted to perform parametric sensitivity analysis while predicting Vsusing the same datasets. To find out the relative importance of predictor variable(s),four model schemes are achieved using input variable(s)for Vspredictions.To further recognize the combined effect of multiple input variables,model schemes 5 and 6 are considered to predict Vs.The performance of different model schemes is listed in Table 5.The graphical presentation of the predictive model performance is illustrated in Figs.11 and 12.

Due to the low contribution of shale volume in the data-driven connectionist model, the model scheme1 shows less performance(such as higher statistical error, and lower R2and PI). Compared to other input variables, Vphas a greater contribution to Vsprediction(model scheme 4). The model scheme 4 leads to high performance with high values of R2and PI,as well as low statistical errors of APRE,MAPE and RMSE.It is demonstrated that the rock porosity and bulk density have an intermediate contribution to Vsprediction.According to different testing and generalization methods adopted in the current study, the most significant rock parameter (input variable,from higher to lower order)is Vp,followed by PHI,RB and Vshin turn,for estimating Vsof sandstone. Considering only the two major significant variables,the model scheme 5 shows excellent performance with high R2and PI of about 0.955 and 0.9113 for the training, and 0.9567 and 0.9264 for testing dataset,respectively.Meanwhile, this model exhibits lower statistical errors compared to other model schemes in the study. Moreover, these two variables are vital in capturing real scenarios of rock void space and particle motion in thedirection of propagation, while predicting accurate shear wave velocity or slowness of sedimentary rocks.

Table 5 Relative performance of the predictor variables in the hybrid model to predict Vs.

Fig. 12. Comparison of R2 and PI for various model schemes to predict Vs using the LSSVM models.

Above correlation has shown close results with measured values of Vsbecause it has high performance,and can capture the two vital rock parameters.The performance of the proposed model(Eq.(17))is compared with other correlations using the same dataset of sandstone, as shown in Table 6.

Fig.13. Evaluation of residual performance between the proposed model and other correlations in the study.

Fig.14. A cross-plot for Vs predictive models and comparison of R2 for different models.

The proposed model (correlation) exhibits lower statistical errors(MAPE=9.55%,APRE=2.28%,and RMSE=0.59)compared to other correlations. In contrast with the models of Han et al.(1986), Greenberg and Castagna (1992), and Oloruntobi and Butt(2020), the proposed model has lower residual errors, as shown in Fig.13. Additionally, a graphical representation of the correlation coefficient(R2)is illustrated in Fig.14.Based on the Vsprofile shown in Fig.14,the new model predictions are most close to the measured data with R2of 0.96. For instance, the models of Han et al. (1986) and Lee (2006) give slightly lower values of R2(0.94) than the actual core data. Oloruntobi and Butt (2020) and Greenberg and Castagna (1992) underestimated the Vsvalues with R2of 0.88 and 0.93, respectively. Additionally, the models proposed by Lee(2006),Greenberg and Castagna(1992),and Han et al. (1986) do not include the combined effect of rock porosity and bulk density to capture the effect of both shale lithology and electron density in the correlations. Moreover, the model of Oloruntobi and Butt (2020) does not consider the effect of pore space on sandstone rocks.

Rock engineers and/or drilling engineers can utilize the new model to estimate more reliable Vsvalues for the sandstone,compared to the models proposed by other researchers such as Han et al.(1986),Greenberg and Castagna(1992),Lee(2006),and Oloruntobi and Butt(2020).The new model takes into account the most significant rock parameters to figure out the effects of rock porosity and compressional wave velocity(inverse of slowness)of the clastic sedimentary formation,which has improved predictive performance.Furthermore, the obtained Vsprofile can be used to estimate elastic constants for investigating the formation in situ stress profile, rock failure criterion, wellbore stability, formation evaluation and drilling performance evaluation,and reducing the exploration costs with a timely manner during the field development and exploration phases.

4. Conclusions and recommendations

In this study,the hybrid machine learning technique of LSSVMCSA is proposed to predict shear wave velocity(Vs)using measured core and log data such as saturated bulk density(RB),porosity(PHI)and compressional wave velocity (Vp) of sandstone samples. The statistical parameters such as AAPE, MAPE, RMSE, R2and PI are suggested to evaluate the studied model performance. The key findings of this investigation are as follows:

(1) Based on the LSSVM-CSA optimization strategy, the connectionist models are capable of efficiently estimating the shear wave velocity with low statistical errors and high correlation coefficient (R2).

(2) Based on the novel implementation of the hybrid models,the compressional wave velocity is the most siginifcant parameter for predicting Vsof sandstone.

(3) A correlation is introduced to capture the compressional wave velocity and porosity of rocks to obtain Vsthrough regression analysis.Similar to existing models for sandstone,the developed model is verified using real data samples,and it exhibits a high performance with R2equal to 0.96.

For future research, scholars can adopt the developed correlation for more robustness and generous with different machine learning techniques(such as RF,GEP,ANN-GA and ANFIS-GA)using a large number of datasets (big data) of sedimentary formation.Additionally, different case studies can be conducted for more validation purpose with big data samples. The deterministic tools,model development and variable ranking strategy can be beneficial for field specialists, researchers, and rock engineers to deal with rock elastic constants estimation, rock mechanics problems, wellbore failure analysis, drilling penetration rate as well as sedimentary formation evaluation.

Declaration of competing interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The author would like to thank the Chittagong University of Engineering & Technology, Bangladesh, for providing research facilities to accomplish the study. Additionally, the author wants to express the gratitude to Drs. Stephen Butt, Salim Ahmed, Sohrab Zendehboudi(Memorial University of Newfoundland,Canada),and Mohammed Mahbubur Rahman (Bangladesh University of Engineering and Technology,Bangladesh)for their technical guidelines and research support during the study periods.

List of symbols

bBias term

RBRock bulk density (g/cm3)

R2Determination of coefficient (correlation coefficient)

NPHINeutron porosity

PHIRock porosity

VpCompressional wave velocity (km/s)

VsShear wave velocity (km/s)

VshShale volume (clay content)

VstStonely wave velocity(km/s)

xiInput variables

YpPredicted value

YtMeasured(target) variable

ZFormation depth

αWeight factor for connectionist model

γTuning parameter

σKernel bandwidth of LSSVM model