1 Introduction

Young’s modulus is one of the key parameters, which are required to build geomechanical earth models [1]. Young’s modulus is defined as the measure of resistance by an object against being deformed elastically [2]. The static Young’s modulus is used to estimate the in situ stresses and to construct geomechanical earth model, which in turn is used for fracture mapping and fracture design [3]. Many factors affecting the value of Young’s modulus such as lithology and rock properties (rock consolidation, pore pressure, pore structure, porosity) [4, 5]. Howard and Fast [2] stated that static Young’s modulus for shale ranges from 0.1 to 1.0 MPsi, while it ranges from 2 to 10 MPsi for sandstone and from 8 to 12 Mpsi for limestone. These ranges confirmed that there is no specific value for Young’s modulus and determination of the static Young’s modulus in necessary to build the geomechanical models.

Rock elastic parameters estimated from the sonic and compressional wave velocities coupled with density log are used to determine dynamic elastic parameters. There are two methods to measure rock elastic parameters in the laboratory; (i) the dynamic method which measures the ultrasonic body wave velocities including compressional (p-wave) and shear (s-wave), and (ii) the static method, which implies the measurement of deformation by the application of a known force, usually done by uniaxial and triaxial compressional tests [6]. In oil and gas fields, the direct method to measure dynamic elastic moduli is from the wireline logging tools that measure the shear and compressional wave velocities [7]. The dynamic elastic modulus can be determined from acoustic velocities using Eq. 1.

$$E_{\text{dynamic}} = \frac{{\rho V_{\text{S}}^{2} \left( {3V_{\text{P}}^{2} - 4V_{\text{S}}^{2} } \right)}}{{V_{\text{P}}^{2} - V_{\text{S}}^{2} }}$$
(1)

whereas ρ is the bulk density in g/cm3, VP is the compressional wave velocity in km/s and VS is the shear wave velocity in km/s. Equation 1 gives dynamic elastic moduli in gigapascals (GPa).

For the same rock, the static and dynamic moduli are always different from each other; usually dynamic Young’s modulus is 1.5–3 times greater than static Young’s modulus [8, 9]. The reason behind this difference is because of the deformation (strain) amplitude between two types of the tests. For the static tests, the strain rate is 10−2 s−1 or even lower while for the dynamic tests, the rate of strain varies from 1 to 10−4 s−1 while the strain amplitude in dynamic tests is around 10−6–10−7 and for static tests the amplitude is around 10−2–10−3 [10]. The difference between static and dynamic moduli also depends on the rock minerals and physical origin of the rock, the difference is greater in weak rocks, and this difference gets reduced by increasing rock strength [11]. In steel, which is a pure a homogenous material, the dynamic and static elastic moduli are the same [12].

Static elastic parameters more truly represent the reservoir in situ stress–strain condition [13]; however, retrieving cores throughout the reservoir section and conducting compressional test is very expensive and time-consuming [9, 14]. Because of high capital investment on retrieving core samples and conducting laboratory tests, quite often few core samples are taken from the depth of interest and then correlation is developed between log and the core samples. These log-derived correlations are used to calibrate dynamic elastic modulus to give static modulus throughout the depth of reservoir section, also from where the core samples have not been retrieved [3, 9]. In heterogeneous formations, because of complex lithology and due to the limitations of these log-derived correlations, these correlations cannot capture the trend of the formation. To avoid these limitations, various models and correlations were built to obtain the static Young’s modulus from the dynamic Young’s modulus. The developed models were built for specific rock type and for specific conditions.

Belikov et al. [15] stated that Estatic of microcline and granite rocks can be estimated from Edynamic using a specific correlation. Gorjainov [16] developed another two empirical correlations to estimate Estatic based on Edynamic for clays and for wet soils. McCann and Entwisle [17] equation is valid for Jurassic granites. Morals and Marcinew [18] empirical’s equation can be used for rocks with high permeability values. Bradford et al. [19] determined Estatic from Edynamic using Eq. 2.

$$E_{\text{static}} = 0.4145 E_{\text{dynamic}} - 1.0593$$
(2)

King [20] established a correlation for Estatic using Edynamic for igneous and metamorphic rocks, Eq. 3

$$E_{\text{static}} = 1.26 E_{\text{dynamic}} - 29.5$$
(3)

Eissa and Kazi [21] used 714 published data points from the literature and developed the correlation for s Estatic using Edynamic, Eq. 4

$$\log_{10} E_{\text{static}} = 0.02 + 0.77 \log_{10} \left( {\gamma E_{\text{dynamic}} } \right)$$
(4)

Wang [22] mentioned that for hard rocks, Estatic can be determined from Edynamic by Eq. 5

$$E_{\text{static}} = E_{\text{dynamic}} - 15.2$$
(5)

Canady [13] developed a correlation for Estatic using Edynamic, Eq. 6

$$E_{\text{static}} = \frac{{\ln \left[ {\left( {E_{\text{dynamic}} + 1} \right)*\left( {E_{\text{dynamic}} - 2} \right)} \right]}}{4.5}$$
(6)

Based on 45 core data points from limestone formation, Najibi et al. [23] introduced a new empirical equation to determine Estatic from compressional wave velocity, Eq. 7

$$E_{\text{static}} = 0.169* V_{\text{P}}^{3.24}$$
(7)

where ρ is the bulk density in g/cm3 and VP is the compressional wave velocity in km/s. Equation 7 gives static Young’s modulus in gigapascals (GPa).

Elkatatny et al. [24] developed an empirical equation for estimating Estatic based on wireline log data (bulk density, compressional and shear time). They concluded that this equation can be used for different rock types. Mahmoud et al. [25] enhanced the accuracy of Elkatatny et al. [24] equation by applying clustering techniques. Mahmoud et al. [25] showed that data clustering improved the accuracy of Estatic prediction, where they developed 6 equations for 6 different clusters.

To avoid the need for clustering and to enhance the accuracy of static Young’s modulus profile, artificial intelligence techniques are applied to build a new model for estimating Estatic. The objectives of this research are to: (1) build a robust model based on artificial intelligence technique to estimate Estatic from the basic well log data (bulk density, compressional time, and shear time), (2) asses the best artificial intelligence technique in estimating the static Young’s modulus, (3) derive mathematical correlation from the optimized artificial intelligence model to convert the black box to a white box in order to predict static Young’s modulus, and (4) validate the new correlation with actual measured laboratory and published data.

1.1 Artificial intelligence techniques

Álvarez del Castillo et al. [26] stated that artificial neural network (ANN) is a robust statistical tool, and it can be used for many industrial applications. ANN does not require any physical phenomenon to characterize the system under study [27]. Any nonlinear complex function between input and output parameters can be approximated using ANN. ANN requires less formal statistical training [28]. Burbidge et al. [29] stated that ANN is capable of detecting all potential interactions between predictor parameters, in addition, ANN has many training algorithms.

Hinton et al. [30] stated that fewer number of neuron causes under-fitting and excessive number of neurons cause over-fitting, so optimization is required for the selecting the appropriate number of neurons. Increasing the size of the model by increasing the number of hidden layer neuron not only increases the computational time but can also cause data memorization which resulted in poor prediction during testing of unseen data [31].

Jang [32] mentioned that ANFIS uses Sugeno-type fuzzy interference system. Tahmasebi and Hezarkhani [33] stated that ANFIS has the strength to produce the advantages of both neural network and fuzzy logic in a single platform. Evolutionary algorithm is required to optimize the parameters of ANFIS [33].

ANFIS presents a much better learning ability, a much smaller convergence error is achieved, and although the convergence is slower, the smallness of the error in ANFIS can compensate that fact [34]. ANFIS requires lesser adjustable parameters than those required in other neural network structures and, specifically, backpropagation MPLs, [34].

If membership functions of ANFIS are large, system requires large computational time. If membership functions are small, it may not cover the entire input space adequately [34]. The major issue of the ANFIS system is the presence of exponential term in the number of consequent parameters; this could be modified to increase the efficiency of the system [35].

Guo [36] stated that support vector machine (SVM) stands on the kernel neuron function which allows projection to higher planes and able to solve more complicated and complex highly nonlinear problems. A good characteristic of SVM worth mentioning at this point is its “stability.” This is a characteristic that ensures that once the data set is fixed, running it several times will always produce the same results unlike the neural networks that generate different results for different runs even when the data set is fixed [37]. When SVM is run on different data set arrangement, it arrives at results with very small ranges of values [37]. SVM can be able to handle large features space; it can effectively avoid over-fitting by limiting the margin and identify automatically a small subsets of useful information that are called as support vectors [38].

Elkatatny et al. [39, 40] confirmed that ANN once optimized, it can be used in many applications in petroleum industry. They concluded that artificial neural network can be used to predict the rheological properties of the drilling fluid in a real time with a high accuracy. Elkatatny et al. [41,42,43] concluded that the black box of ANN can be changed to a white box by extracting the weights and biases of the optimized ANN model, and they developed a new robust correlation to determine the reservoir permeability, oil formation volume factor, and bubble point pressure.

1.2 E static prediction using artificial intelligence techniques

Abdulraheem et al. [9] used artificial neural network technique to predict Estatic. They used 77 data points to build the model. The input parameters for this model were: bulk density and compressional time only. Bandar et al. [44] estimated Estatic using artificial neural network technique, the input parameters of his model were porosity, bulk density, compressional time, shear time, pore pressure, minimum horizontal stress, and overburden stress. He built his model using 600 data points. The main disadvantage of his method is that his model depends on pore pressure, minimum horizontal stress, and over burden stresses which are calculated from Estatic.

Al-Anazi and Gates [5] used support vector machine to predict Estatic. The input data for this model consist of porosity, bulk density, minimum horizontal stress, compressional wave velocity, and shear velocity. They used 600 multidimensional data points to build their model. They implemented fuzzy ranking methodology to rank influential input parameters to predict Estatic. The main disadvantage of this method is the dependency on the minimum horizontal stress, as minimum horizontal stress normally derived from the elastic parameter.

Madhubabu et al. [45] predicted the UCS and Estatic from index parameters such as porosity, density, P-wave velocity, Poisson’s ratio, and point load index, using ANN and multivariate linear regression (MVLR) technique. They found that ANN predicted the two parameters more accurately than the MVLR. Beiki et al. [46] used regression technique optimized by genetic algorithm to predict Estatic and UCS in carbonate rocks of Asmari limestone. They found that this technique works better than other regression models to predict Estatic and UCS for Asmari formation. Dehghan et al. [47] predicted UCS and Estatic by ANN and nonlinear regression technique for travertine samples. The inputs of their model were P-wave velocity, the point load index, the Schmidt hammer rebound number, and porosity. They also found that on their data ANN model works better than regression technique. Yin et al. [48] developed new method for estimating static Young’s modulus and Poisson’s ratio in tight interbedded clastic reservoirs. Ghasemi et al. [49] used decision tree approach to predict UCS and Estatic of carbonate rocks. The input parameters of their model were Schmidt hammer, effective porosity, dry unit weight, P-wave velocity, and slake durability index. Aboutaleb et al. [50] used ANN and SVM to predict Estatic and UCS of carbonate rocks of Asmari formation using dynamic parameters as an input.

The main scope of this paper is to build a new robust ANN model to predict the static young’s modulus from wireline logs and convert the black box of ANN to a white box by extracting the weights and biases of the optimized ANN model and develop a new correlation for estimating Estatic.

2 Methodology for E static prediction

2.1 Data acquisition and processing

Data management and preparation is an essential step for the success of any AI modeling. The performance of the entire artificial intelligence model depends mainly upon the quality of the data [5]. Core and well log data serves as the input for these models. These data sets are anticipated to be very confidential in petroleum industry furthermore obtaining core elastic parameters from laboratory using triaxial tests are very rare due to high sophistication and cost. The first step is the depth matching between the obtained core data and wireline porosity and gamma ray. The wireline log data consist of gamma ray, bulk density, neutron porosity, compressional and shear wave velocities.

2.2 Data description and analysis

Actual field measurements and core data (600 data points) were collected from several wells located in different fields. Log data include: gamma ray, neutron porosity, bulk density, compressional time, and shear time. Core data consist of Estatic which were measured in the laboratory using triaxial compressional tests. Table 1 lists the statistical analysis of the data. Gamma ray range is: 7–114 API, neutron porosity range is: 0.0–0.23 (v/v), bulk density range is: 2.0–3.0 g/cm3, compressional wave travel time range is: 40 and 81.9 µs/ft, while shear wave travel time range is: 73–146 µs/ft. The set of data randomly divided into two subparts, 70% of the data are used for training, while 30% of the data are used for testing the accuracy and generalization capabilities of the model.

Table 1 Statistical description of the data (600 data points)
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The objective of this paper is to determine the best combination of the available logs to be used as inputs for the AI models to predict the static Young’s modulus with high accuracy (highest correlation coefficient and lowest average absolute percentage error). Different AI models (ANN, ANFIS, and SVM) were used to estimate the static Young’s modulus and the one which will yield the highest accuracy will be used to develop a new empirical correlation for Estatic. The new correlation was developed based on the weights and the biases of the optimized model. The set of data was randomly divided into two subparts, 70% of the data were used for training, while 30% of the data were used for testing the accuracy and generalization capabilities of the model.

3 Building a robust E static model using AI techniques

Artificial intelligence models are data driven and inserting the available parameters to serve as an input does not always guarantee good results. The best practice is to find which input parameter is contributing positively and which input contributes negatively. Multivariate linear regression correlation coefficient feature selection system is used to estimate the individual relationship in terms of correlation coefficient between input and output parameter as shown in Fig. 1. Correlation coefficient (R) between input and output can be determined by Eq. 8

$${\text{CC}} = \frac{{k\sum xy - \left( {\sum x} \right)\left( {\sum y} \right)}}{{\sqrt {k\left( {\sum x^{2} } \right) - \left( {\sum y} \right)^{2} } \sqrt {k\left( {\sum b^{2} } \right) - \left( {\sum b} \right)^{2} } }}$$
(8)
Fig. 1
figure 1

Relative importance of input parameters with static Young’s modulus

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Figure 1 shows that static Young’s modulus has a correlation coefficient of 0.32, − 0.68, 0.70, 0.77, and 0.8 with gamma ray, neutron porosity, bulk density, compressional time, and shear time, respectively. To make the model simple, the parameters with highest correlation coefficient with Estatic are selected. These parameters are: bulk density, compressional time, and shear time.

The first step is training the AI models; three AI techniques are implemented: ANN, ANFIS, and SVM. Figure 2 shows that ANN predicted Estatic with a coefficient of determination (R2) of 0.92 and AAPE of 5.35% between actual and predicted static Young’s moduli. ANFIS model yielded R2 of 0.89 and AAPE of 7.71%, while SVM model gave R2 of 0.87 and AAPE of 4.35%, Fig. 2. ANN model has the highest coefficient of determination during the training phase as shown in the cross-plot between actual and predicted data, Fig. 3.

Fig. 2
figure 2

Estimation of static Young’s modulus using different AI models: ANN, ANFIS, and SVM

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Fig. 3
figure 3

Coefficient of determination for the actual and predicted static Young’s modulus for the training data

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On set of 30% random unseen data, which was used for testing the accuracy and the generalization capabilities of the developed model, ANN yielded R2 of 0.922 with AAPE of 5.2%, ANFIS yielded R2 of 0.92 with AAPE of 6.31%, while SVM yielded R2 of 0.86 with AAPE of 7.25%, Fig. 4. Figure 5 shows the cross-plot of three models performances during testing on unseen data.

Fig. 4
figure 4

Estimating the static Young’s modulus using different AI models for unseen data

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Fig. 5
figure 5

Coefficient of determination for the actual and predicted static Young’s modulus for unseen data

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Based on the obtained results, it can be concluded that ANN model is the best AI model to estimate Estatic based on log data. The complete architecture and anatomy of ANN model is described in Sect. 3.1.

3.1 Neural network architecture

A backpropagation neural network algorithm is implemented to model Estatic. The Estatic ANN model consists of three input parameters, which are: bulk density, compressional time, and shear time, one hidden layer and one output parameter which is Estatic. The neurons number in the hidden layer varied from 5 till 50. In this study, the optimum number of neurons is 20 since it produced the highest correlation coefficient for the unseen data simultaneously with high correlation coefficient for the seen data.

Tan-sigmoidal-type activation function is used as a transfer function between input and hidden layer and linear-type activation functions is used between hidden and output layers. The general structure of Estatic ANN model is shown in Fig. 6. Table 2 shows the complete anatomy of proposed Estatic ANN model.

Fig. 6
figure 6

Architecture of the static Young’s modulus using ANN model

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Table 2 Estatic neural network model complete description
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Levenberg–Marquardt backpropagation algorithm is implemented as the training algorithm to obtain weights and bias, in order to avoid the model to stuck on local minima 10,000 realizations has been performed with the initialization of different weights and bias during training and cross-validation phases of the modeling. After training, the weights and biases, from the optimum model with minimum error (AAPE and RMSE) and highest coefficient of determination on both seen and unseen data are extracted as listed in Table 3.

Table 3 Weights and biases of Estatic ANN model
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3.2 Development of mathematical model using artificial neural network

Based on the weights and the biases of the optimized ANN model, an empirical equation for estimating Estatic was developed. The weights between input and hidden layers termed as w1 and weights between hidden and outer layers termed as w2 are given in Table 3. The proposed Estatic ANN-based empirical correlation is given by:

$$E_{{{\text{static}}_{n} }} = \left[ {\mathop \sum \limits_{i = 1}^{N} w_{{2_{i} }} \left( { \frac{2}{{1 + e^{{ - 2 \left( {w_{{1_{i,1} }} \rho_{n} + w_{{1_{i,2} }}\Delta t_{{C_{n} }} + w_{{1_{i,3} }}\Delta t_{{S_{n} }} + b_{{1_{i} }} } \right)}} }}} \right) } \right] + b_{2}$$
(9)

The black box of the ANN model was converted to a white box by developing Eq. 9. Users can use Eq. 9 and the data available in Table 3 to estimate Estatic from the log data without the need to run the ANN models. This is the first time to extract this equation for estimating the static Young’s modulus based on ANN model.

3.2.1 Steps to use the new empirical equation for E static

Step 1 Normalize the input parameters between the range [− 1 1] before using Eq. 9. Normalization was done by two points slope form given by Eqs. 10 and  11

$$\frac{{Y - Y_{ \hbox{min} } }}{{Y_{ \hbox{max} } - Y_{ \hbox{min} } }} = \frac{{X - X_{ \hbox{min} } }}{{X_{ \hbox{max} } - X_{ \hbox{min} } }}$$
(10)

where Ymin = − 1, Ymax = 1, X is the input parameters, Xmin is the input parameter minimum value, and Xmax is the input parameter maximum value.

$$Y = 2 \times \left( {\frac{{X - X_{ \hbox{min} } }}{{X_{ \hbox{max} } - X_{ \hbox{min} } }}} \right) - 1$$
(11)

Step 2 Use Eq. 9 to calculate Estatic in normalized form. Equation 9 can be applied by using the weights and biases given in Table 3. The input data should be organized in the following order: bulk density, compressional time, and shear time.

Step 3 The value of Estatic obtained from Eq. 9 is in the normalized form. Estatic (Mpsi) can be obtained by de-normalizing Eq. 9 as follows:

$$E_{\text{static}} = \frac{{\left( {13.456 - 1.090} \right)\left( {{\text{Estatic}}_{n} + 1} \right)}}{2} + 1.090$$
(12)

4 Validation of the developed empirical correlation

To validate the accuracy and of the ANN-based empirical correlation, it was compared with the real field data from two wells. These wells are located in limestone formation. Figures 7 and 8 show the wireline log input data for these wells, which have certain laboratory-measured values of static Young’s modulus.

Fig. 7
figure 7

Input data of well 1

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Fig. 8
figure 8

Input data of well 2

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Figure 7 shows the wireline log data of well 1, which consisted of bulk density, compressional wave travel time, and shear wave travel time data for an interval of 1000 ft. The range of bulk density is 2.0–2.9 g/cm3, the range of compressional wave travel time is 40–80 µs/ft, while for shear wave travel time the range is 80–120 µs/ft.

Figure 8 shows the input data for well 2 for an interval of 1000 ft. The minimum and maximum values of bulk density range are 2.4–2.9 g/cm3, compressional wave travel time range is 45–70 µs/ft, while shear wave travel time range is 90–138 µs/ft.

Figure 9 showed that Eq. 12 can predict the static Young’s modulus profile with high accuracy for well 1. The Estatic profile gives an image like when compared with the dynamic Young’s modulus, where the static modulus changes with the variation of the dynamic modulus. In addition, the value of the dynamic Young’s modulus is in the range of 1.5–3 times that of the static Young’s modulus. The predicted profile matched the core data at different depths. The cross-plot of the predicted data of static Young’s modulus and the core data confirmed the accuracy of the developed correlation, where the coefficient of determination was 0.956.

Fig. 9
figure 9

Field verification of the developed ANN model using actual data from well 1

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Figure 10 shows the complete profile of the predicted Estatic using the developed empirical correlation based on the optimized ANN model. It is clear that the estimated values of the Estatic matched the actual measured values for well 2, which were obtained from laboratory tests. The cross-plot between the actual and predicted values of Estatic confirmed the accuracy of the developed correlation for estimating Estatic (R2 of 0.895).

Fig. 10
figure 10

Field verification of the developed ANN model using actual data from well 2

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4.1 Comparison with the commonly used correlations in oil and gas industry

For further validation of the accuracy and the performance of the proposed empirical correlation, it was compared with the commonly used empirical correlations in the oil industry to predict Estatic [13, 20,21,22,23]. The comparison was made on well 1 data whose complete input wireline log data is given in Fig. 7.

Figure 11 shows that the Estatic prediction from Eissa and Kazi [21] using Eq. 4 showed very poor match because only few data points matched with actual laboratory-measured Estatic values, also Eissa and Kazi [21] correlation results did not fully capture the nonlinearity of the data. Eissa and Kazi [21] equation underestimated the value of the static Young’s modulus which was obtained only using the dynamic Young’s modulus using logarithmic relationship. In addition, the predicted profile of static Young’s modulus did not reflect the change in the dynamic Young’s modulus. Equation 4 yielded a profile with an average and constant value of the static young’s modulus through the entire reservoir section.

Fig. 11
figure 11

Comparison of developed correlation for static Young’s modulus with the published correlation on well 1 interval (200–400) ft

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The Estatic prediction from Canady [13] correlation using Eq. 6 also showed very poor match, only two laboratory-measured triaxial test matched with the predicted profile. Canady [13] estimated the static Young’s modulus from the dynamic Young’s modulus using logarithm equation, Eq. 6, which represented an average value throughout the reservoir section. This method did not capture the change in the dynamic Young’s modulus and underestimated the value of static Young’s modulus. Canady [13] yielded very close results to Eissa and Kazi [21] equation.

King [20] model overestimated the static Young’s modulus value which was calculated using liner relationship with the dynamic modulus. Equation 3 results did not match the core data even it captured the changes in the dynamic Young’s modulus. Similar results were obtained using Wang [22] correlations. Equation 5 overestimated the static Young’s modulus values, and there was no match with the laboratory measurements. These results illustrated that using only one parameter to determine the static Young’s modulus will not yield an accurate estimation of the static Young’s modulus.

The Estatic predicted using Najibi et al. [23] correlation from Eq. 7 yielded only one match data point. Equation 7 overestimated the static Young’s modulus which was predicted using only one parameter, compressional velocity. These results confirmed that using one log only is not enough to predict the static Young’s modulus with high accuracy.

The static Young’s modulus predicted using proposed ANN correlation (Eq. 12) yielded the best match with almost all the data points along with fully captured the nonlinear trend of the data.

Table 4 lists the complete statistical comparison of estimating Estatic using the developed correlation based on ANN model and other correlations on well 1. It is clear that the new proposed empirical correlation yielded the lowest average absolute percentage error of 6.2, lowest root-mean-square error 0.353 and the highest coefficient of determination of 0.956 for predicting the static Young’s modulus.

Table 4 Comparison of proposed Estatic ANN model with widely used Estatic empirical correlations
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Based on these results, we can conclude that the developed empirical correlation for estimating Estatic can be used for any well that has a data range within the data range of the developed model.

5 Conclusions

Artificial intelligence techniques were used to predict Estatic from log data using more than 600 core data points. The following conclusions can be drawn:

  1. 1.

    AI techniques can be used to predict the Estatic from well logs (bulk density, compressional time, and shear time).

  2. 2.

    ANN is the best AI technique to predict Estatic as compared with ANFIS and SVM.

  3. 3.

    The developed empirical correlation based on the optimized ANN model can predict Estatic with high accuracy. (R2 was 0.96, and AAPE was 6.2%.)

  4. 4.

    The developed empirical correlation outperformed the published correlations for estimating Estatic.

  5. 5.

    It is not recommending to use only one log parameter to predict the static Young’s modulus.

The developed technique will help the geomechanical and reservoir engineers estimate the static Young’s modulus based on log data for any reservoir that have data range within the data range of the developed model. No need to have the ANN model, the developed equation with the optimized weights and biases can be programmed using any software, and it can be applied direct without the need for the ANN model this what is called the white box of the ANN model.