1 Introduction

For the subsurface layer, many researchers have indicated that accurate knowledge of the shear wave velocity data is considered very important and essential to estimate reliable mechanical rock properties with usage along with formation bulk density and compressional wave velocity [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Shear wave velocity is fundamental to achieve accurate reservoir characterization which leads to an effective and successful field development plan (FDP). Reservoir characterization requires precise estimation of rock properties which is essential for uncertainty reduction, assigning of reservoir rock properties, and geological information recognition. Moreover, trustable reservoir characterization is essential to enhance hydrocarbon recovery techniques [16,17,18,19,20,21,22].

In fact, compressional sonic data are the only available sonic data and are included in the set of acquired logging data. Therefore, estimating the shear wave velocity became a challenging topic. Shear wave velocity can be directly obtained in the laboratory by core analysis or by using the tools of dipole sonic imager (DSI); unfortunately, the discussed methods are considered time- and money-consuming [23,24,25,26]. Therefore, many researchers have studied petrophysical models, empirical correlations, and artificial intelligence methods for estimating the shear wave velocity (Vs).

It has been proved that the shear wave velocity Vs can be predicted from the compressional wave velocity Vp because Vs and Vp are both affected by the same factors [27]. Bastos et al. (1998) [28] established a technique to estimate shear waves depending on the compressional wave. A relationship was proposed between the shear wave and compressional wave. The applicability of the proposed technique was tested on 120 limestone samples. Oloruntobi and Butt [29] studied the shear wave prediction technique due to its importance for geomechanical analysis and reservoir characterization. The introduced technique was obtained by expressing the shear wave (km/s) as a function of both the formation bulk density (g/cc) and compressional wave velocity (km/s). Moreover, the produced model was developed to predict the shear wave that can be applied for more ranges of lithology types than the other conventional techniques.

1.1 Literature

Anemangely et al. [30] studied shear wave velocity prediction by applying machine learning on petrophysical wireline logs. A model was developed by using a set of wireline logging data which were acquired as follows: neutron porosity (NPHI), density (RHOB), photoelectric (PEF), gamma-ray (GR), caliper, true resistivity (RT), VP, and Vs. Hybrid algorithms of both Cuckoo Optimization Algorithm (COA) combined with the Least Square Support Vector Machine (LSSVM) were used to estimate the shear wave velocity. The results showed that the LSSVM-COA hybrid algorithm was reliable and accurate in estimating the shear wave velocity.

In the case of knowing the reservoir fluids, there are many introduced extensive studies that estimate the values of the shear wave velocity by studying the ratio between both shear wave velocity and compressional wave velocity [31,32,33,34,35,36,37,38,39,40].

Wantland et al. [41] studied a technique to assess the shear wave velocity based on assuming the value of Poisson’s ratio. Various researchers have studied pulse transmission techniques to use acoustic measurements of shear wave velocity (Vs) and compressional wave velocity (Vp) to determine elastic reservoir rock properties [42,43,44,45,46,47]. For the sake of estimating the shear wave velocity, many researchers have proposed wide studies to report empirical correlations that give higher reliable values by using physical rock properties [1, 48,49,50].

Nowadays, extended studies have been concentrated on using the high power of artificial intelligence (AI) to estimate geomechanical reservoir rock properties. A wide range of AI methods has been used such as support vector regression (SVR), fuzzy logic, artificial neural network (ANN), and neuro-fuzzy [23, 51,52,53,54,55,56,57]. Therefore, a lot of studies use the wide artificial intelligence methods for the aim of accurately estimate the shear wave velocity [58, 59]. Objectives of artificial neural network (ANN) are summarized in developing a biological event mathematical model, in order to simulate the capability of biological neural structures aiming to design a system for processing intelligent information. In recent years, back-propagation neural network (BPNN) has been considered as an active topic for modeling nonlinear dynamic systems due to its efficiency.

Maleki et al. [55] indicated that the accurate study of the mechanical properties of reservoir rock is considered very important during all reservoir production and development phases. The aim of the study was to replace the conventional method of using shear and compression sonic data for determining the mechanical properties of rock with a technique to estimate shear wave velocity by using artificial intelligence methods and empirical correlations. Two artificial intelligence methods were used which are back-propagation neural network (BPNN) and support vector regression (SVR); moreover, the empirical correlations of Castagna, Brocher, and Carroll were applied to determine the shear wave velocity. It was concluded that support vector regression (SVR) was the best approach for estimating the shear wave velocity; however, all applied methods did not show a reliable estimation of the shear wave velocity, thus indicating the importance of acquiring realistic shear velocity data.

Zhang et al. [60] proposed a machine learning method called (long short-term memory neural network, LSTM) to enhance the accuracy of estimating the shear wave velocity from wireline logging tools. By using LSTM, they established a good relation between shear wave velocity and data acquired from wireline logging tools. The study was tested on carbonate reservoir rock by using different six wireline logs as an input data while applying the LSTM model. The results showed that the proposed model is more accurate to estimate the shear wave velocity than the Xu–Payne model and considered to be applicable in carbonate reservoir rock.

Xu–Payne petrophysical model is a model used to quantitatively characterize reservoir and predict S-wave velocity. Clarifying that the m Xu–Payne model accuracy is strongly dependent on the accuracy of rock composition interpretation. The disadvantage of this model is that porosity and saturation data should be provided accurately which resulting in non-satisfying results.

There are many other previously introduced studies that used for the prediction of the shear wave velocity. Table 1 shows different proposed techniques illustrating the applied techniques, input parameters, coefficient of determination, and root-mean-square error.

Table 1 Different previous proposed techniques for shear wave velocity prediction
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In this paper, a new artificial neural network (ANN) model is developed by MATLAB for the accurate prediction of the shear wave velocity as a function of porosity, GR, bulk density, and compressional velocity with large data ranges. Other studies have provided models to calculate the shear wave velocity as a function of one, two, or three parameters; on the other hand, for more accurate calculation in the study here discussed, the new model employed four parameters to predict the shear velocity as shown in Fig. 2. Furthermore, the large data that were used in the model resulted in producing a highly accurate model compared with the past models. The validity of the new model was tested by applying the new model on 2350 data sets, where 1645 data sets were used to process the model, and the other 705 data sets were used to validate the applicability and performance of the new model. MATLAB is used in this study to integrate deep learning models by providing an open framework that is compatible with Python, PyTorch, TensorFlow, and other open-source deep learning frameworks; moreover, it allows the ability to create end-to-end programs.

2 Methodology

2.1 Artificial Neural Network

Artificial neural networks (ANNs) technique is considered one of the most popular machine learning techniques. ANNs consist of an input layer, one or more hidden layers, and an output layer, with many interconnected nodes (neurons) in each layer. Neurons use a nonlinear function to calculate their output based on the sum of their inputs, with weights on the edges and nodes that are adjusted during the learning process. ANNs have various applications in petroleum engineering, such as using them to estimate water saturation in different formations [63], calculating the minimum miscibility pressure (MMP) for CO2–reservoir oil [64], and predicting sweep efficiency for different injection patterns in reservoir engineering [65]. In drilling engineering, ANNs can be used to estimate drilling pressures for mixed lithologies [66] and to predict properties of invert emulsion mud, such as yield point and plastic viscosity [67]. Moreover, there are many studies based on applying ANNs to build a model to help in several real-life applications [68,69,70,71,72,73].

2.2 Data Description and Preparation

The data set considered in this study includes 2350 data points of shear velocity as a function of compressed velocity (Vc), gamma-ray (GR), porosity (\(\boldsymbol{\varnothing }\)), and bulk density \(({{\varvec{\rho}}}_{{\text{b}}})\). Figure 1 shows \(\boldsymbol{\varnothing },\boldsymbol{ }\boldsymbol{ }{{\varvec{\rho}}}_{{\text{b}}}, GR,\) Vc, and Vs versus depth for the studied intervals. Table 2 presents a summary of the statistical evaluation of all the data used in this paper. The bulk density (g/cc) ranged from 2.385 to 2.85, porosity (V/V) ranged from 5.6 to 45.6%, GR ranged from 12.089 to 176.8, compressed velocity (km/s) ranged from 2.702 to 5.391, and shear velocity (km/s) ranged from 1.381 to 2.929. The kurtosis and skewness for all input and output parameters ranged from − 0.5 to nearly 3, and this means that these variables are normally distributed (their normal values between − 3 and 3), and not need to outlier any values from them.

Fig. 1
figure 1

Petrophysical logs of the studied interval. Bulk density (g/cc) ranged from 2.385 to 2.85, porosity (V/V) ranged from 5.6 to 45.6%, GR ranged from 12.089 to 176.8, compressed velocity (km/s) ranged from 2.702 to 5.391, and shear velocity (km/s) ranged from 1.381 to 2.929

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Table 2 Statistical evaluation of all data sets used in this study
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Before developing the model, it was also very important to check the effect of each parameter on the required output. Figure 2 shows that the shear velocity is related to the input parameters (bulk density, porosity, GR, and compressed velocity). Shear velocity (km/s) is directly proportional to bulk density (g/cc) and compressed velocity (km/s) whereas it is inversely proportional to porosity and GR values. Figure 3 shows the correlation between input and output parameter.

Fig. 2
figure 2

Heatmap of the input and output parameters

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Fig. 3
figure 3

Correlation between input and output parameter

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2.3 Training Optimization Using Levenberg–Marquardt (LMT) Technique

To reach the optimum training model, an optimization method called Levenberg–Marquardt (LMT) is used to resolve least squares nonlinear problems. These miniaturization issues are particularly prevalent when fitting least squares curves. LMT mainly interpolates between the gradient color (GD) and Gauss–Newton algorithm (GNA) methods. Since LMT is more resilient than GNA, in many software programs, LMT is utilized to address generic curve fitting issues. It frequently converges more quickly with GNA than first-order techniques. In other words, Levenberg–Marquardt (LMT) is one of the built-in MATLAB software techniques. This is trial-and-error technique used to compare between the model predicted values and the measured values to reach the optimum accuracy. This technique is used to solve nonlinear least squares problems by minimizing the sum of the squares of the errors between the model predicted values and the actual ones. The Levenberg–Marquardt algorithm combines two numerical minimization algorithms: the gradient descent method and the Gauss–Newton method. In the gradient descent method, the sum of the squared errors is reduced by updating the parameters in the steepest descent direction. In the Gauss–Newton method, the sum of the squared errors is reduced by assuming the least square’s function is locally quadratic in the parameters and finding the minimum of this quadratic.

2.4 Model Evaluation

It is a vital tool to evaluate the model’s performance relative to the experimental data. In this paper, the developed model reliability was investigated based on coefficient of determination (R2), root-mean-square error (RMSE), and average absolute relative error (AE) in addition to the least optimum number of neurons in hidden layer to reduce the running time and required capacity of the computer memory. The following statistical expressions are used to compute these statistical parameters:

$${\text{RMSE}}=\sqrt{\frac{1}{n}\sum \left[{\left({{V}_{{\text{S}}}}_{{\text{true}}}-{{V}_{{\text{S}}}}_{{\text{cal}}}\right)}^{2}\right]}$$
(1)
$${\text{AE}}=\frac{1}{n}\sum \left(\frac{\left|{{V}_{{\text{S}}}}_{{\text{true}}}-{{V}_{{\text{S}}}}_{{\text{cal}}}\right|}{{{V}_{S}}_{true}}\right)\times 100$$
(2)
$${R}^{2}=1-\left[\sum_{i=1}^{n}{\left({{V}_{{\text{S}}}}_{{\text{cal}}}-{{V}_{{\text{S}}}}_{{\text{true}}}\right)}^{2}/\sum_{i=1}^{n}{\left({{V}_{{\text{S}}}}_{{\text{cal}}}-{{V}_{{\text{S}}}}_{{\text{avg}}}\right)}^{2}\right]$$
(3)

3 Results and Discussion

Before building the ANN model to estimate shear velocity, the collected data were divided into two sets. The first set was employed for the process of model training, which represents 1645 data sets out of 2350 (70%), while 705 data sets (30%) were used to validate the performance of the models.

In this work, an ANN model was developed to predict shear velocity as a function of bulk density, porosity, GR, and compressed velocity. The model was built based on three layers, where the first layer is the input layer, which has seven neurons for inputs. There are seven neurons that contribute to the hidden layer, which is the second layer. The output layer, which has one neuron to estimate the output parameter, shear velocity, is the third layer as illustrated in Fig. 4. To reach the optimum ANN model, the logistic-sigmoid versus tan-sigmoid was examined as a transfer function at different numbers of hidden neurons (5, 6, 7, 8, 9, and 10) as presented in Tables 3 and 4. It was found that the highest coefficient of determination (R2) and the lowest root-mean-square error (RMSE) were at n = 7 with sigmoid transfer function, where the coefficient of determination is 0.958, and RMSE is 0.043. In all cases, the LMT was used as an optimization training technique, and the pure-linear was examined to be the output function.

Fig. 4
figure 4

Schematic for the developed neural network

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Table 3 Check model accuracy using sigmoid function
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Table 4 Check model accuracy using tan-sigmoid function
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When the calculated data by the model are plotted versus the actual data and the points locate on the unit-slope straight line, this means that the model has excellent performance and, in this case, R2 = 1. As the R2 values are near the one-value, this means that the model performance is good. Also, the RMSE is a standard measure of model performance when the error is normally distributed. Case on cross-plot the optimum model was selected based on R2, RMSE, and AE in addition to the least optimum number of neurons in hidden layer to reduce the running time and required capacity of the computer memory. Because the model has high performance in case of 7 and 10 neurons and the error is nearly equal in our case. So, we selected seven neurons as optimum to reduce the running time and the required computer memory storage (Table 5).

Table 5 Characteristics of the ANN model of shear velocity
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We started with a small value like 7, and we noticed a good model performance. But we do not know if this number of hidden layer neurons is the most suitable or not. So, we examined the model performance at values more and less than this value to ensure that 7-case is the optimum one. The characteristics of proposed models are presented in Table 3.

Equations 1 and 2 represent the new developed model, where shear velocity can be calculated using the following ANN-based mathematical model with the coefficients presented in Tables 6 and 7.

Table 6 The values \({{\text{w}}}_{{\text{i}},{\text{j}}}\) and \({{\text{w}}}_{{\text{hi}}}\) used in Eqs. 1 and 2
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Table 7 The values of \({\mathbf{m}}_{\mathbf{i}}\) and \({\mathbf{a}}_{\mathbf{i}}\) used in Eq. 2
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$${V}_{s}=2.32889+0.77383\sum \left(\frac{{w}_{hi}}{1+{e}^{-{S}_{i,j}}}\right)$$
(4)
$${S}_{i,j}={b}_{i}+\sum \left({w}_{i,j}\left({m}_{j}{x}_{j}-{a}_{j}\right)\right)$$
(5)

Figures 5 and 6 show the predicted vs. actual plots for training and validation data using the proposed ANN model. These plots show reasonable correlation coefficients between the real and estimated values with coefficients of determination more than 0.95 for training and validation data.

Fig. 5
figure 5

Cross-plots of training data

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Fig. 6
figure 6

Cross-plots of validation data

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4 Conclusion

Shear wave velocity is considered as one of the most important parameters used to accurately estimate mechanical rock properties that lead to enhance reservoir characterization. In this work, a new model was provided based on an ANN study for the accurate prediction of shear wave velocity. The study was built on data sets that can be available and easily obtained in any field, validating the new model by applying it on 2350 data sets of gamma-ray (GR), bulk density \({{\varvec{\rho}}}_{{\varvec{b}}}\), compressional velocity Vc, and porosity \(\boldsymbol{\varnothing }\). Statistical analysis was applied on the data sets which showed that porosity ranged from 5.6 to 45.6%. The bulk density ranged from 2.385 to 2.85, GR ranged from 12.089 to 176.8, shear velocity ranged from 1.381 to 2.929, and compressed velocity ranged from 2.702 to 5.391. The collected data sets were divided into two sets, where 1645 data sets were used to process the model, and the other 705 data sets were used to validate the applicability and performance of the new model. The new model was built based on three layers which were an input layer, hidden layer, and output layer. For obtaining the maximum accuracy of the ANN model, logistic-sigmoid was examined versus tan-sigmoid as a transfer function at different numbers of hidden neurons (5, 6, 7, 8, 9, and 10). The results of the new model showed high accuracy at n = 7 with the highest coefficient of determination equal to 0.958, and the lowest RMSE was equal to 0.043. The training algorithm of the Levenberg–Marquardt was selected, and the pure-linear was tested to be the output function. It is concluded that the estimation of shear velocity data through the new model was valid and highly recommended within the studied data ranges.