A gated recurrent unit model to predict Poisson’s ratio using deep learning

Fahd Sad Alakbari ,Mysara Eissa Mohyaldinn* ,Mohammd Abdalla Ayoub ,Ibnlwald A.Hussin ,Ali Samr Muhsan ,Syahrir Ridha ,Abdullah Abduljabbar Salih

a Petroleum Engineering Department,Universiti Teknologi PETRONAS,32610,Bandar Seri Iskandar,Perak,Malaysia

b Institute of Hydrocarbon Recovery,Universiti Teknologi PETRONAS,32610,Bandar Seri Iskandar,Perak,Malaysia

c Gas Processing Center,College of Engineering,Qatar University,P.O.Box 2713,Doha,Qatar

d Department of Chemical Engineering,College of Engineering,Qatar University,P.O.Box 2713,Doha,Qatar

e Mechanical Engineering Department,Universiti Teknologi PETRONAS,32610,Bandar Seri Iskandar,Perak,Malaysia

Keywords: Static Poisson’s ratio Deep learning Gated recurrent unit (GRU)Sand control Trend analysis Geomechanical properties

ABSTRACT Static Poisson’s ratio (νs) is crucial for determining geomechanical properties in petroleum applications,namely sand production.Some models have been used to predict νs;however,the published models were limited to specific data ranges with an average absolute percentage relative error(AAPRE)of more than 10%.The published gated recurrent unit (GRU) models do not consider trend analysis to show physical behaviors.In this study,we aim to develop a GRU model using trend analysis and three inputs for predicting νs based on a broad range of data,νs (value of 0.1627-0.4492),bulk formation density(RHOB) (0.315-2.994 g/mL),compressional time (DTc) (44.43-186.9 μs/ft),and shear time (DTs) (72.9-341.2 μs/ft).The GRU model was evaluated using different approaches,including statistical error analyses.The GRU model showed the proper trends,and the model data ranges were wider than previous ones.The GRU model has the largest correlation coefficient (R) of 0.967 and the lowest AAPRE,average percent relative error (APRE),root mean square error (RMSE),and standard deviation (SD) of 3.228%,-1.054%,4.389,and 0.013,respectively,compared to other models.The GRU model has a high accuracy for the different datasets: training,validation,testing,and the whole datasets with R and AAPRE values were 0.981 and 2.601%,0.966 and 3.274%,0.967 and 3.228%,and 0.977 and 2.861%,respectively.The group error analyses of all inputs show that the GRU model has less than 5% AAPRE for all input ranges,which is superior to other models that have different AAPRE values of more than 10% at various ranges of inputs.

1.Introduction

Poisson’s ratio(ν)is a crucial input in determining the minimum and maximum horizontal stresses that assist in building a geomechanical earth model(Abdulraheem,2019).The ν can be utilized in maximizing reservoir productivity,wellbore stability,hydraulic fracturing,and sand production control.This ν could be affected by changing rock bulk density (Kumar,1976;Phani,2008;Ameen et al.,2009;Wang et al.,2009;Al-Anazi and Gates,2010;Miah,2021).Generally,the soft rocks have the ν value between 0.1 and 0.3.The ν values of medium rocks,such as sandstone,are 0.2-0.3.The hard rocks have a ν value of 0.3-0.4 (Gercek,2007).

Two approaches are generally used to obtain ν.First,the ν is determined using a dynamic method called dynamic νdyn:

where vpis the compressional velocity,km/s;vsis the shear velocity,km/s;and νdynis the dynamic Poisson’s ratio.

The second method is a static method used in laboratory measurements,and ν is called static Poisson’s ratio(νs)(Canady,2011):

where εyis the radial strain iny-direction,and εxis the axial strain inx-direction.

Basically,the νsshows the actual behavior in the reservoir(Canady,2011);however,the lab measurements to acquire νsare costly.In addition,the lab measurements require a long time to measure νs(Abdulraheem et al.,2009;Khaksar et al.,2009).Therefore,some studies have used alternative methods to predict the νsvalue.

Kumar et al.(2003) collected 83 datasets to obtain the correlation used to determine νs.Their correlation can be limited to isotropic rocks (Kumar et al.,2003).Khandelwal and Singh (2009)used 11 sandstone,shale,and coal datasets from various coal mines in India.They showed that νsis governed by a power relation with the P-wave velocity or compressional time(DTc)(Khandelwal and Singh,2009).Ranjbar-Karami et al.(2014) determined νsas a function of νdyn.They stated that the correlation has anR2value of 0.3.In addition,they applied a fuzzy inference system (FIS) to obtain νswith theR2of 0.983(Ranjbar-Karami et al.,2014).Brand?s et al.(2012) developed an empirical correlation between νsand shear transit time(DTs)at the range of 130-210 μs/ft based on νdyn.Feng et al.(2019) used the piecewise linear method to predict νsusing 18 core samples and well logs from 12 wells from the low permeability reservoirs in the western Ordos Basin,China.Gowida et al.(2019) applied an artificial neural network (ANN) to determine νsusing 692 data points from Saudi Arabia Fields.Table 1 shows the used parameters,number of datasets,data range,and authors’ accuracy for the previous correlations used to predict νs.

Some machine learning methods are applied to determine the νs(Lawal and Kwon,2021).Singh and Singh (2006) used uniaxial compressive strength (UCS) and tensile strength (T) as inputs to predict νsfor different rocks applying ANN and neuro-fuzzy methods.Nejati et al.(2019) determined νsbased on a single uniaxial compression test.Shalabi et al.(2007) employed linear regression to predict νsas a function of rock hardness and UCS with anRvalue of 0.87.Abdulraheem et al.(2009) applied ANN,fuzzy logic (FL),and fuzzy neural (FN) to predict νsbased on 77 data points.The ANN and FL models have an average absolute percentage error(AAPE)of 5.42% and 8.2% for the training dataset and 5.16% and 7.65% for the testing dataset,respectively(Abdulraheem,2019).Al-Anazi et al.(2011)used depth,porosity,in situ stress,pore pressure,bulk density,Vs,andVpto find νsusing 602 data points.The alternating conditional expectation (ACE) was used to create the model with anR2value of 0.994 (Al-Anazi et al.,2011).Tariq et al.(2017) applied ANN to determine the νsbased onVs,andVpusing 550 data points with anR2value of 0.82.Elkatatny et al.(2017) used ANN to find νsas a function of bulk density,Vs,andVp,using 610 data points.The ANN model has an average absolute error of 1.3%,and anRvalue of 0.98 (Elkatatny et al.,2017).Tariq et al.(2018) used gamma-ray,bulk density,porosity,Vs,andVsas inputs to predict νswith anR2value of 0.97 applying the FN method and 580 data points.

Deep learning methods,such as gated recurrent unit (GRU),have been successfully used in some petroleum engineering applications.Zhang et al.(2021) applied the GRU to predict the reservoir porosity using well-logging parameters.Their GRU model has anRvalue of 0.928,a mean absolute error (MAE) of 86.1089%,and an RMSE of 1.2643.Al-Shabandar et al.(2021) forecast petroleum production using the GRU.They indicated that the GRU has a better performance and is much easier than other models such as long short-term memory (LSTM) and recurrent neural network(RNN).TheR2values of the GRU model are 0.9687,0.83272,and 0.55078 for training,validation,and testing datasets,respectively.Sumarna et al.(2022) applied RNN,LSTM,and GRU to determine oil,gas,and water production as a function of surface and subsurface parameters.They proved that the GRU model is the best compared to the other models.Tian et al.(2019) applied the GRU,kernel-based hidden Markov model (HMM),and deep neural network (DNN) models to obtain the lithology classification.They stated that the performance of the GRU and HMM is higher than the DNN model(Tian et al.,2019).Zhang et al.(2022)used the GRU to determine the drilling rate of penetration.The GRU model hasR2,mean absolute error (MAE),and mean absolute percentage error(MAPE) of 0.96,0.15 m/h,and 3.57%,respectively.Azarafza et al.(2022) used the deep neural network (DNN) method to determine the rock’s mechanical properties,such as strength,rock material’s stiffness parameters,and uniaxial compressive strength.They stated that the model has the highest accuracy of 0.95,a precision of 0.97,and the lowest RMSE of 0.17.Zhu et al.(2022)applied a deep convolutional neural network (CNN) to classify rock blocks based on the block theory principles.The CNN model has an accuracy of 0.95 and a precision of 0.93.Furthermore,the CNN model decreases the loss function,RMSE,and MSE to less than 0.3,0.2,and 0.2,respectively(Zhu et al.,2022).Table A1 shows the advantages and disadvantages of the machine learning methods that are used for predicting νsand the deep learning techniques applied in petroleum engineering applications.The hyperparameters of the machine learning methods used for νsprediction and the deep learning techniques applied in petroleum engineering applications are displayed in Table A2.

As discussed previously,some models have been used to predict νs.However,the published models are developed based on the specific data ranges to show an average absolute percentage relative error (AAPRE) of more than 10% in wide data ranges.Furthermore,the published GRU models in other engineering calculations do not consider the trend analysis to show proper relationships between the inputs and outputs.Therefore,this study aims to develop a new,robust,and accurate GRU model for predicting νsbased on data from different parts of the world.The data are collected from the United States,Malaysia,India,Saudi Arabia,and Venezuela,and have a wider range.The GRU model for predicting νswas evaluated by applying different approaches: dividing the data into three subparts,trend analysis,cross plotting,error histograms,statistical error analyses,and group error analysis.This study provided an accurate and robust GRU model to determine the νs,which is one of the crucial parameters for determining geomechanical properties for different oil and gas applications,such as wellbore stability,hydraulic fracturing,and sand production control.

2.Methodology

The flowchart of the methodology is presented in Fig.1.First,the data were collected from different regions to cover wide ranges used to develop the GRU model for predicting νs.After that,the collected data were cleaned using a box and whisker to remove the outliers.Then,the clean data were split into three parts: training,validation,and testing datasets.Next,the training and validation datasets were applied to build the GRU model for predicting νs.After the model was created,trend analyses were performed to evaluate the proposed GRU model.The optimum hyperparameters of the proposed GRU model were selected for all proper input trends with high accuracy to predict νs.When all inputs followed the proper relationships,the cross-plotting,error histograms,statistical error analyses,and group error analyses were conducted.The proposed GRU model was developed to accurately determine νs,and the testing dataset was utilized to evaluate the proposed and previous models.In addition,trend analyses were conducted.Finally,the previously published models were compared with the proposed GRU model.

2.1.Data collection and pre-processing

In total,1691 datasets were gathered from different locations:the United States,Malaysia,India,Saudi Arabia,and Venezuela,to predict the νsin a wide range of data using the GRU.The data were collected from the well logging measurements: bulk formation density (RHOB),shear time (DTs),compressional time (DTc),and core sample measurements νscorrelated to the well logging measurements.The statistical description of the collected data for the proposed νsmodels is shown in Table 2.Fig.2a-c displays the histograms of the input parameters (RHOB,DTs,and DTc) for the gathered data.The histogram of νsfor the collected data is represented in Fig.2d.

Fig.2.Histograms of the input parameters: (a) Bulk formation density,(b) Compressional time,(c) Shear time,and (d) Output parameter νs for the collected data.

Table 2The statistical description of the gathered data for the proposed νs models.

After the data were collected,all input and output parameters were cleaned using the box and whisker to select and eliminate outliers to obtain clean data.The box and whisker consist of the interquartile range (IQR),lower boundary (Q1-1.5IQR),lower quartile Q1,upper quartile Q3,and upper boundary (Q3+1.5IQR).Any values less than the lower and higher than the upper boundary are considered outliers(Alakbari et al.,2021).Table 3 shows the box and whisker parameters for the collected data for νsmodel.The statistical description,the box,and whisker parameters of the clean data for νsmodels are displayed in Table A3.

Table 3The box and whisker parameters for the gathered data for νs models.

Fig.A1a-c (Appendix A) shows the histograms of the inputs RHOB,DTs,and DTc for the clean data for νs.The histograms of the output νsare shown in Fig.A1d.Although there are some imbalanced distributions of the clean data,as shown in Fig.A1a-d,the clean data were used to cover wide ranges of data to build the proposed GRU model.In this study,the data were collected from different places to cover wide ranges compared to the previous studies that used specific data ranges from specific locations.

Fig.3 shows that νsis a moderate function of the RHOB with anRof-0.57 and a strong function of the DTc and DTs with anRof 0.693 and 0.868 for the collected dataset,respectively.As represented in Fig.4,νsis a moderate function of the RHOB with anRof-0.539 and a strong function of the DTc and DTs withRof 0.736 and 0.873,respectively for the clean dataset.

Fig.3.The relative importance of input parameters with νs for the collected dataset.

Fig.4.The relative importance of input parameters with Static Poisson’s ratio (νs) for the clean dataset.

Table 4 shows the published models and the inputs to obtain νsvalue.Khandelwal and Singh (2009) used only the DTc to determine νs.Brand?s et al.(2012)and Kumar et al.(2003)applied only the DTs to find νs.Ranjbar-Karami et al.(2014),Christaras et al.(1994),Feng et al.(2019),and Gowida et al.(2019) used the DTc and DTs to determine νs.All inputs and output parameters are normalized in the range(-1 to 1)after cleaning the collected data:

Table 4Used parameters for νs in previous models.

whereYis the parameter in the normalized form;Ymaxis the maximum value of the normalized form (1);Yminis the minimum value of the normalized form (-1);Xis the parameter to be normalized;Xminis the minimum parameter;andXmaxis the maximum parameter.

The normalized clean data for νsmodels are split into 60% for the training GRU model,while 20% and 20% datasets are used to validate and test the model.Data randomization is used by applying the shuffle method.The shuffle method is used to change the original list of data.Randomization is applied to ensure that each data set does not memorize the pattern to prevent model overfitting and generalization (Alakbari et al.,2021).

2.2.Deep learning technique: gated recurrent unit (GRU)

Deep learning can be defined as a machine learning method that utilizes deep neural networks(multi-layer neural networks).There are many types of deep learning,like simple DNN,CNN,and recurrent neural networks (RNN).The RNN method is effectively used in many engineering applications.However,the RNN technique has a problem of long-term dependencies or gradient vanishing problems.The long short-term memory network(LSTM),as a special RNN capable of learning long-term dependencies,can solve this problem.The LSTM is mainly applied to define the relationship between the current data and the previous input data,and its memory is utilized to save the internal information of the previously input data.Nevertheless,the LSTM has many parameters for its training to take more time to develop the model(Wang et al.,2021).

The GRU architecture is similar to the LSTM architecture but has fewer parameters to speed up the training process (Gers et al.,2000).The LSTM was proposed in 1997 by Hochreiter Sepp and Jürgen Schmidhuber (1997).However,the GRU was established in 2014 by Cho et al.(2014a).The GRU has two gates,the update gate,and the reset gate,to manage the information flow (Wang et al.,2021).Fig.5 shows the architecture of the GRU(Wang et al.,2021).

Fig.5.GRU architecture (Wang et al.,2021).

First,the reset gate (rt) can be used to control the information that passes from the previous hidden state to the current hidden state:

where σgis a sigmoid function,andxtis the input parameters.

The new memory candidate (?ht) is determined as

After that,the update gate (zt) can be employed to update the hidden state with a new hidden state:

Finally,the hidden state can be determined (Dey and Salem,2017;Wang et al.,2021) by

wherehtis the output vector;rtis the reset gate vector;W,U,b are weights matrices and bias vectors;⊙is the Hadamard product;and ?his a tangent function.

The GRU was selected in this study because it can provide a better performance,solve the vanishing gradient problem in the RNN,and decrease the training time for the LSTM(Cho et al.,2014b;Chen et al.,2019;Hu et al.,2018).Training transformer models can take some time due to a large number of parameters and the complexity of the self-attention mechanism.In contrast,training a GRU model is relatively faster (Vaswani et al.,2017).Furthermore,the GRU can be used for non-time series (numeric features) by applying a feature input layer function in Matlab(Xue et al.,2021).

2.3.Trend analysis

The trend analysis is achieved to assess the robustness of the model in the presence of uncertainty.The trend analysis shows the model’s relationships between inputs and outputs.The trend analysis can find model errors to show unexpected relationships between inputs and outputs.Moreover,trend analysis can detect and eliminate unnecessary parameters of the model (Pannell,1997).Therefore,trend analyses were conducted in this research.The trend analyses of the inputs RHOB,DTc,and DTs were assessed to prove the correct relationships between the RHOB,DTc,and DTs and output,that is,νs.First,the experimental measurements that have constant values for all inputs with changing one input were used as inputs for the trend analyses.After that,each input was changed between the minimum and maximum values,while the other features were fixed at their constant mean values (Al-Shammasi,1999;Osman et al.,2005;Ayoub et al.,2020).Then,νswas determined for the experimental measurement inputs,and the values changed between the minimum and the maximum using the proposed GRU model.Finally,graphs were plotted for each parameter(RHOB,DTc,or DTs) asx-axis against νsasy-axis.

2.4.Statistical error analysis

The statistical error analysis is applied to present the models’accuracy.In this study,different statistical error analyses were used: average percent relative error (APRE),average-absolute percentage relative error(AAPRE),minimum absolute percent relative error(Emin),maximum absolute percent relative error(Emax),RMSE,and SD,andR.The statistical error analyses’equations are provided in Appendix B.

2.5.Group error analysis

The group error analyses indicate the model’s performance at various data ranges.The group error analyses demonstrate the model’s errors in different data groups.First,the data were sorted based on the minimum to maximum values of the studied inputs.The data were split into five groups,and the number of data in each group was kept at 20%.Then,these parameters were utilized to determine νs.Finally,the AAPRE was determined to show the model’s AAPRE for the five groups.The AAPRE was selected because it was the primary indicator of the accuracy in this study for all models.Furthermore,group error analyses were performed for all previously published models and compared with the proposed GRU model.

3.Results and discussions

3.1.Gated recurrent unit (GRU) model

Table 5 represents the optimum parameters of the GRU model to predict νs.The RHOB,DTc,and DTs were applied as inputs to predict νs.The shuffle method was applied to ensure that each dataset does not memorize the pattern to avoid generalization and model overfitting or under-fitting.The Shuffle method was used to change the original list of the data and does not return a new list of the data(Alakbari et al.,2021).The activation function to update the cell and hidden state (StateActivationFunction) was tanh.The activation function used to apply to the gates (GateActivationFunction) was sigmoid.Training options were stochastic gradient descent with momentum (sgdm).A function that was applied to initialize input weights (InputWeightsInitializer) was the glorot initializer.However,an orthogonal function was used to initialize recurrent weights(RecurrentWeightsInitializer).The hardware resource used for the training network(ExecutionEnvironment) was auto.

Table 5Optimized parameters for the proposed GRU model.

3.2.GRU model evaluation

The trend analysis,cross plotting,and statistical error analyses,namely R,APRE,AAPRE,RMSE,and SD,were used to evaluate the GRU model to predict νs.The trend analysis was applied to represent the relationships between the inputs and output to prove the physical behaviors.The cross-plotting and statistical error analyses were used to show the proposed GRU model’s accuracy.In addition,error histograms and group error analysis were employed to show the proposed GRU model’s accuracy.

3.2.1.Trend analysis

The trend analyses of the RHOB are shown in Fig.6a-d.Fig.6a displays the trend analysis of the experimental RHOB for the proposed GRU model and some previous models.From the experimental measurements,νsis decreased by increasing the RHOB,Fig.6a.The GRU model shows that the νsis dropped by increasing RHOB to follow the correct relationships between RHOB and νs.On the other hand,the previous models show that νsis constant with changing the RHOB because they have not considered RHOB as input in their models (see Fig.6a and b).Fig.6c shows the trend analysis of the changing RHOB between the minimum and maximum values for the previous models.νsis also constant for all previous models.The trend analysis of changing the RHOB between the minimum and maximum for the proposed GRU model is shown in Fig.6d.Fig.6d shows that the GRU model successfully obeys the correct relationship between RHOB and νs.Therefore,the proposed GRU model has the proper physical behavior.

Fig.6.(a)Trend analysis of the experimental bulk formation density for the proposed GRU model and some previous models,(b)Trend analysis of the experimental bulk formation density for the models in Khandelwal and Singh(2009)and Brand?s et al.(2012),(c)Trend analysis of the bulk formation density for the previous models,and(d)Trend analysis of the bulk formation density for the proposed GRU model.

Fig.7a-d and Fig.A2a,b (Appendix A) display the compressional time(DTc)trend analyses for the previous and proposed GRU models.The trend analysis of the experimental DTc for the previous and the proposed GRU models is shown in Fig.7a,b and Fig.A2a,b.The experimental measurements show that νsis decreased by increasing the DTc.The proposed GRU model shows that the νsis also decreased by increasing the DTc to present the correct relationships between the DTc and νs,as shown in Fig.7a and Fig.A2a.In addition,νsvalues for the proposed GRU model are close to the experimental measurements.Models in Christaras et al.(1994),Feng et al.(2019),Gowida et al.(2019),and Ranjbar-Karami et al.(2014) followed the proper relationship between the DTc and νs.However,the νsvalues are not close to the experimental measurements as shown in Fig.7a and Fig.A2a.Khandelwal and Singh(2009) model displayed that νsis increased by increasing the DTc,Fig.7b and Fig.A2b.Models of Brand?s et al.(2012)and Kumar et al.(2003)showed that νsis constant with changing DTc,as illustrated in Fig.7a,b and Fig.A2a,b.Fig.7c and d shows the trend analysis of the DTc with changing DTc between the minimum and maximum values for the previous models and the proposed GRU model.Models in Christaras et al.(1994),Feng et al.(2019),Gowida et al.(2019),and Ranjbar-Karami et al.(2014) also followed the correct trend for these inputs (see Fig.7c).The proposed GRU model also has the proper trend for the values between minimum and maximum for DTc.The proposed GRU model’s physical behavior is shown in Fig.7d.

Fig.7.(a) Trend analysis of the experimental compressional time for the proposed GRU model and some previous models,(b) Trend analysis of the experimental compressional time for the models in Khandelwal and Singh (2009)and Brand?s et al.(2012),(c)Trend analysis of the compressional time for the previous models,and(d)Trend analysis of the compressional time for the proposed GRU model.

The trend of DTs is demonstrated in Fig.8a-e and Fig.A3a,b(Appendix A).Fig.8a,b and Fig.A3a,b show the trend analysis of experimental DTs for the proposed GRU model and some previous models;it shows that increasing DTs can increase νs.The proposed GRU model obeys the relationship between DTs and νs.Christaras et al.(1994),Feng et al.(2019),Gowida et al.(2019),Ranjbar-Karami et al.(2014),and Kumar et al.(2003) represented that increasing the DTs can increase νs(see Fig.8a and Fig.A3a).The model in Brand?s et al.(2012) showed that increasing the DTs decreases νs(see Fig.8b and Fig.A3b).Khandelwal and Singh(2009) model displayed that νsis constant with changing the DTs(see Fig.8b and Fig.A3b).Fig.8c shows the trend analysis of the DTs for models in Christaras et al.(1994),Feng et al.(2019),Gowida et al.(2019),Ranjbar-Karami et al.(2014),and Kumar et al.(2003).These models have the proper relationship between DTs and νs.The model of Brand?s et al.(2012)followed the proper trend at the range of 138.64-196.51 μs/ft as they developed their model at the range of 130-210 μs/ft (see Fig.8d).The proposed GRU model also has the correct trend with changing the DTs between the minimum and maximum,as shown in Fig.8e.Therefore,the proposed GRU model shows a proper physical behavior.

Fig.8.(a)Trend analysis of the experimental shear time for the proposed GRU model and some previous models,(b)Trend analysis of the experimental shear time for the models in Khandelwal and Singh (2009) and Brand?s et al.(2012),(c) Trend analysis of the shear time for some previous models,(d) Trend analysis of the shear time for the models in Khandelwal and Singh (2009) and Brand?s et al.(2012),and (e) Trend analysis of the shear time for the proposed GRU model.

The trend analyses using RHOB,DTc,and DTs as inputs for the proposed GRU model represent the correct relationships between these inputs and νs.Therefore,it suggests that the proposed GRU model could follow the proper physical behavior for all inputs.

3.2.2.Cross plotting

The cross-plotting of the testing,training,validation,and the whole datasets is demonstrated in Figs.9 and 10,and Fig.A4a-b(Appendix A).Fig.9 displays the cross-plotting of the testing dataset,and most data points are closer to the 45°line to show the high accuracy of the proposed model for the testing dataset.As shown in Fig.10,most datasets are also close to the straight line to show an excellent match between the measured and predicted values for the training dataset.Fig.A4a,b shows that the most measured νsmatched with the predicted νsfor validation and whole datasets.It displays that the proposed GRU model can accurately determine νsfor the validation and the whole datasets.The cross-plotting proves that the proposed GRU model has a high accuracy for training,validation,testing,and the whole datasets.

Fig.9.Cross-plot of the GRU model for the testing dataset.

Fig.10.Cross-plot of the GRU model for the training dataset.

3.2.3.Statistical error analysis

Statistical error analyses,namely APRE,AAPRE,Emax.,Emin,RMSE,R,and SD of the proposed GRU model for the training,validation,testing,and the whole datasets are shown in Table 6.TheRand AAPRE were applied as the major indicators to show the accuracy of the proposed GRU model.The training,validation,testing,and whole datasets have AAPRE of 2.601%,3.274%,3.228%,2.861%,andRof 0.981,0.966,0.967,and 0.977,respectively (see Table 6).The training,validation,testing,and whole datasets have similar values of AAPRE andR.It confirms that the proposed GRU model can accurately predict νswithout any over-or under-fitting issues.The RMSE values are 3.566,4.559,4.389,and 3.954 for the training,validation,testing,and whole datasets,as tabulated in Table 6.The statistical error analyses show that the proposed GRU model has a high accuracy in determining νsfor different datasets:training,validation,testing,and the whole datasets collected from different regimes.Therefore,the proposed GRU model can be used for generalizing the model’s applicability.

Table 6Statistical error analyses of the GRU model.

Fig.11 shows the measured and predicted νsfor the training,validation,and testing.In Fig.11,most data points for the measuredνsmatch the predicted νsfor the training,validation,and testing datasets to demonstrate the highest accurate proposed GRU model for predicting νs.

Fig.11.Measured/predicted static Poisson’s ratio of the GRU model for (a) Training,(b) Validation,and (c) Testing datasets.

3.2.4.Error histograms

Fig.12 a-c presents the histograms of the proposed GRU model’s errors (relative percentage error) for the testing,training,and validation datasets.In Fig.12a,the proposed GRU model has almost zero percentage error for most testing data points.Most training and validation data points have almost zero percentage error (see Fig.12b and c).The error histograms of the testing,training,and validation datasets show that the proposed GRU model can accurately predict νs.

3.2.5.Group error analysis

Fig.13a-c and Fig.A5a-c (Appendix A) show group error analysis of RHOB,DTc,and DTs that are performed to present the accuracy of the proposed GRU model with previous models in different ranges of RHOB,DTc,and DTs.

Fig.13.Group error analysis of(a)bulk formation density,(b)compressional time,and(c) shear time for the proposed GRU model and some previous models.

The group error analysis of RHOB is shown in Fig.13a and Fig.A5a.As shown in Fig.13a,the proposed GRU model has an AAPRE value of less than 5% for all ranges of RHOB.The proposed GRU model surpassed all models at different ranges of RHOB.Kumar et al.(2003)model has an AAPRE of less than 10% and more than 35% at 1.831-2.227 g/mL and 2.6-2.993 g/mL (see Fig.13a).The model of Christaras et al.(1994)has an AAPRE of less than 10% at 2.6-2.993 g/mL.However,the model of Christaras et al.(1994)has an AAPRE of more than 10% at 1.831-2.590 g/mL.The models of Feng et al.(2019),Gowida et al.(2019),and Ranjbar-Karami et al.(2014) have an AAPRE of 10%-16%,20%-25%,and 25%-30%,respectively (see Fig.13a).The model of Khandelwal and Singh(2009) has an AAPRE of 25% and 100% at 1.831-2.227 g/mL,and 2.2697-2.993 g/mL (see Fig.A5a).Brand?s et al.(2012) model has an AAPRE of less than 25% and more than 100% at 2.6-2.993 g/mL and 1.831-2.590 g/mL (see Fig.A5a).

Fig.13b and Fig.A5b present the group error analysis of DTc.The proposed GRU model has an AAPRE of less than 5% for all ranges of the DTc (see Fig.13b).The model of Kumar et al.(2003) had an AAPRE of less than 5% at 114.6-146.6 μs/ft(see Fig.13b).The model of Christaras et al.(1994) has an AAPRE of less than 10% at 47.4-56.8 μs/ft(see Fig.13b,model constructed at DTc of 46.59-87.08 μs/ft).However,the model of Christaras et al.(1994)has an AAPRE of more than 10% at 56.9-146.6 μs/ft(see Fig.13b).The model of Feng et al.(2019)has an AAPRE of 10%-15% at different ranges,and they built their model using DTc at 54.9-72.55 μs/ft (see Fig.13b).The models of Gowida et al.(2019) and Ranjbar-Karami et al.(2014)have an AAPRE of 20%-25% and 25%-30% at different ranges (see Fig.13b).The model of Gowida et al.(2019)was developed based on DTc at 44.34-80.49 μs/ft.The model of Brand?s et al.(2012)has an AAPRE of less than 25% and more than 100% at 47.4-56.8 μs/ft and 56.9-146.6 μs/ft,respectively (see Fig.A5b).The model of Khandelwal and Singh(2009)has an AAPRE of less than 25%,25%-50%,and more than 50% at 106.6-114.6 μs/ft,114.6-146.6 μs/ft,and 47.4-106.6 μs/ft,respectively(see Fig.A5b).Khandelwal and Singh(2009) created the model using DTc at 144.4-174.1 μs/ft.

The group error analysis of DTs is represented in Fig.13c and Fig.A5c.Fig.13c shows that the proposed GRU model has an AAPRE of less than 5% at different ranges of the DTs.The model of Kumar et al.(2003) has an AAPRE of less than 8% and more than 15% at 242.9-319.9 μs/ft and 81.8-242.9 μs/ft,respectively (see Fig.13c).Kumar et al.(2003) developed the model using DTs at 38.582-627 μs/ft.The model of Christaras et al.(1994)has an AAPRE of less than 10% and more than 10% at 81.8-104.8 μs/ft and 105.1-319.9 μs/ft,respectively (see Fig.13c).Christaras et al.(1994) built the model based on DTs at 79.93-143.972 μs/ft.The model of Feng et al.(2019) has an AAPRE of 10%-15% at different ranges (see Fig.13c),because they created the model using the DTs at 101.56-133.15 μs/ft.The models of Gowida et al.(2019)and Ranjbar-Karami et al.(2014)have an AAPRE of 20%-25% and 25%-30% at different ranges(see Fig.13c).Gowida et al.(2019)and Ranjbar-Karami et al.(2014)developed the models with the DTs at 73.19-145.6 μs/ft and 40-75 μs/ft,respectively.As shown in Fig.A5c,the model of Brand?s et al.(2012)has an AAPRE of less than 25% and more than 75% at 81.8-104.8 μs/ft and 105.1-319.9 μs/ft,respectively.Brand?s et al.(2012) created the model using the DTs at 130-210 μs/ft.Khandelwal and Singh (2009) model has an AAPRE of approximately 25% and more than 50% at 242.9-319.9 μs/ft and 81.8-242.9 μs/ft,respectively,as shown in Fig.A5c.

3.3.The proposed GRU and previously published models’comparison

3.3.1.Cross plotting

The cross-plotting for the proposed GRU model showed a high accuracy of the model,as explained previously.The cross-plotting of the previously published models is presented in Fig.14.As shown in Fig.14,most of the points for the models of Christaras et al.(1994) and Feng et al.(2019) are closer to the straight line with anRvalue of 0.939.The models of Gowida et al.(2019) and Ranjbar-Karami et al.(2014) have points close to the straight line,but most points are not in line withRof 0.939.The models of Christaras et al.(1994),Feng et al.(2019),Gowida et al.(2019),and Ranjbar-Karami et al.(2014) have anRvalue of 0.939 as they all develop the models based on the dynamic Poisson’s ratio.Most of the Khandelwal and Singh (2009) model’s points are far from the straight line as they built the model based on the DTc at the small range of 144.4-174.1 μs/ft.Brand?s et al.(2012)created the model using the DTs in the range of 130-210 μs/ft.Therefore,most of the points for the Brand?s et al.(2012)model are not in a straight line with anRvalue of 0.775.This study confirmed that it is crucial to develop the new proposed GRU with a wide input range to predict the νsvalue accurately.

Fig.14.Cross-plotting for the previous models.

3.3.2.Statistical error analysis

The proposed GRU and previously published models’ performances were proven by statistical error analysis using the testing datasets.TheRand AAPRE were used as the major indicators to rank the models.In addition,other statistical error analyses,such as APRE,Emax,Emin,RMSE,and SD,were computed for the proposed GRU and previously published models to show their performances.The proposed GRU model is the first-rank model with the highestRvalue of 0.967 and the lowest AAPRE of 3.228% (see Fig.15).Furthermore,the proposed GRU has APRE,Emax,Emin,RMSE,and SD of-1.054%,16.98%,2.39×10-4%,4.389,and 0.013,respectively(see Table 7).The second rank model is the model of Christaras et al.(1994) with AAPRE andRvalue of 10.65% and 0.939 (see Fig.15).The model of Christaras et al.(1994) has APRE,Emax,Emin,RMSE,and SD of-10.42,43.95%,0.4681%,11.78,and 0.02,respectively(see Table 7).The model of Feng et al.(2019)is the third with AAPRE,R,APRE,Emax,Emin,RMSE,and SD of 13.89%,0.939,-13.88%,66.9%,0.7099%,15.32,and 0.018，respectively (see Fig.15 and Table 7).The models of Kumar et al.(2003),Gowida et al.(2019),Ranjbar-Karami et al.(2014),Khandelwal and Singh (2009),and Brand?s et al.(2012) have an AAPRE of more than 20% (see Table 7).As shown in Table 7,the models of Christaras et al.(1994) and Feng et al.(2019) have an AAPRE of more than 10%.On the other hand,the proposed GRU model has the lowest AAPRE of 3.228%,surpassing all previously published models,which can determine the static Poisson’s ratio robustly and accurately.In addition,the statistical error analysis for the proposed GRU and previous models using the training and validation datasets were determined to compare the performance of the proposed GRU with previously published models as shown in Tables A4,A5.

Fig.15 . R and AAPRE comparison of the proposed GRU and all previous models.

Table 7The statistical error analysis for the proposed GRU and previous models using the testing dataset.

4.Conclusions

This study incorporates a GRU model to predict νsand validate the model using different techniques.The proposed GRU model was developed using the same datasets used for previously published models to make a fair comparison between the proposed and previous models.The comparison of the proposed model with the published model was conducted using different methods,such as group error analysis,and statistic error analysis trend analysis.The previous and proposed GRU models were compared based on theRvalue and AAPRE.

The proposed GRU model was developed based on 1691 datasets collected from different places: the United States,Malaysia,India,Saudi Arabia,and Venezuela.The evaluation of the proposed model showed a high level of accuracy compared to the measured data,indicating the effectiveness of the GRU approach with the trend analysis.The proposed GRU model has the proper trends for all inputs.The GRU model shows the best performance of νscompared to all previously published models.The testing dataset of the proposed GRU model has the highestRvalue of 0.967 and the lowest AAPRE,APRE,RMSE,SD,Emax,andEmi.The statistical error analyses show that the proposed GRU model has a high accuracy in determining the νsfor the different datasets.The cross-plotting figures confirm that the proposed GRU model has a high accuracy for training,validation,testing,and whole datasets.The group error analyses of RHOB,DTc,and DTs show that the proposed GRU model has an AAPRE of less than 5% for all RHOB,DTc,and DTs ranges.The proposed GRU model surpasses all models in the different ranges of RHOB,DTc,and DTs.There are limitations in the proposed GRU model as follows:

(1) The proposed GRU model was created using three inputs:RHOB,DTc,and DTs because they were considered inputs and their availability in the literature.

(2) The data ranges are νsof 0.1627-0.4492,RHOB of 0.315-2.994 g/mL,DTc of 44.43-186.9 μs/ft,and DTs of 72.9-341.2 μs/ft.

(3) The proposed GRU model was compared with only the previous correlations because of the unavailability of the previously published machine learning models to apply.

Future studies with more diverse data collection,considering other parameters that affect νs,and applying other deep learning methods could provide more robust models of νsprediction.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank the Yayasan Universiti Teknologi PETRONAS(YUTP FRG Grant No.015LC0-428) at Universiti Teknologi PETRONAS for supporting this study.

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2023.04.012.

Symbols and abbreviations

ν Poisson’s ratio

νsStatic Poisson’s ratio

νdynDynamic Poisson’s ratio

GRU Gated recurrent unit

RHOB Bulk formation density,g/mL

DTs Shear time,μs/ft

DTc Compressional time,μs/ft

VpCompressional velocity,km/s

VSShear velocity,km/s

εxStrain inx-direction

εyStrain iny-direction

UCS Unified compressive strength

T Tensile strength

FIS Fuzzy Inference System

NF Neuo-fuzzy

FL Fuzzy logic

ANN Artificial neural network

DNN Deep neural networks

CNN Convolution neural networks

LSTM Long short-term memory

RNN Recurrent neural network

HMM Hidden Markov model

ACE Alternating Conditional Expectation

IQR Interquartile range

APRE Average percent relative error

AAPRE Average absolute percent relative error

MAE Mean absolute error

MAPE Mean absolute percentage error

RCorrelation coefficient

RMSE Root mean square error

SD Standard deviation

Emax.Maximum absolute percent relative error

Emin.Minimum absolute percent relative error

tanh Hyperbolic tangent

rtReset gate vector

ztUpdate gate

xtInput parameters

htOutput vector

W,U,b Weights matrices and bias vectors

⊙ Hadamard product

σgA sigmoid function

?hA tangent function