# Estimation of Static Young’s Modulus for Sandstone Formation Using Artificial Neural Networks

^{1}

^{2}

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## Abstract

**:**

_{static}) for sandstone formation from conventional well logs. ANN design parameters were optimized using the self-adaptive differential evolution optimization algorithm. The ANN model was trained to predict E

_{static}from conventional well logs of the bulk density, compressional time, and shear time. The ANN model was trained on 409 data points from one well. The extracted weights and biases of the optimized ANN model was used to develop an empirical relationship for E

_{static}estimation based on well logs. This empirical correlation was tested on 183 unseen data points from the same training well and validated using data from three different wells. The optimized ANN model estimated E

_{static}for the training dataset with a very low average absolute percentage error (AAPE) of 0.98%, a very high correlation coefficient (R) of 0.999 and a coefficient of determination (R

^{2}) of 0.9978. The developed ANN-based correlation estimated E

_{static}for the testing dataset with a very high accuracy as indicated by the low AAPE of 1.46% and a very high R and R

^{2}of 0.998 and 0.9951, respectively. In addition, the visual comparison of the core-tested and predicted E

_{static}of the validation dataset confirmed the high accuracy of the developed ANN-based empirical correlation. The ANN-based correlation overperformed four of the previously developed E

_{static}correlations in estimating E

_{static}for the validation data, E

_{static}for the validation data was predicted with an AAPE of 3.8% by using the ANN-based correlation compared to AAPE’s of more than 36.0% for the previously developed correlations.

## 1. Introduction

_{static}) is an essential parameter required to develop the geomechanical earth model [2] which is required for fracture mapping and designing [3]. A complete description of the in-situ stresses which requires assessment of different petrophysical and mechanical parameters is also needed during the drilling operations to ensure wellbore stability [4]. Several previous studies confirmed the impact of the E

_{static}on both fracture design and wellbore stability [1,5].

_{static}. According to Howard and Fast [6] and Fjaer et al. [1], E

_{static}for shale ranges from 0.1–1.0 MPsi; for sandstone it is between 2 and 10 MPsi; and for limestone it is between 8 and 12 MPsi [6]. These ranges confirm the very large variation in E

_{static}in different formations, as well as the wide range for same lithology type. These facts indicate the necessity to estimate E

_{static}along the different sections of the drilled well.

_{dynamic}), Equation (1):

^{3}, V

_{S}and V

_{P}denote the shear and compressional wave’s velocities in km/s, and E

_{dynamic}is the dynamic Young’s modulus in GPa.

_{dynamic}is significantly greater than E

_{static}[9,10,11]. E

_{dynamic}could be 1.5–3 times greater than E

_{static}[12,13], and in some cases E

_{dynamic}could be up to ten times larger than E

_{static}[14,15,16]. The difference is attributed to the strain amplitude between the two testing techniques, and it decreases with the increase in the strength of the rock [17].

_{static}from the E

_{dynamic}, every correlation is restricted for a specific formation type.

_{static}as a function of both Edynamic and formation density. The authors developed this correlation (Equation (2)) based on the regression analysis of 76 tests, with data collected from different sources, and they found that considering the formation density improved the predictability of the E

_{static}considerably:

_{static}and E

_{dynamic}are in Gpa, and γ is the formation density in g/cm

^{3}.

_{static}for any rock type. This correlation enabled prediction of the E

_{static}where only E

_{dynamic}is known, the results of the E

_{static}predicted with (Equation (3)) was compared to previously available correlations and it found to be well correlated to these models:

_{static}and E

_{dynamic}are in GPa.

_{static}for Sarvak and Asmari limestone based on only the compressional velocity (Vp). This model is very useful when the shear velocity (Vs) is not available:

_{static}= 0.169 × V

_{P}

^{3.24}

_{static}is in Gpa, and Vp is in km/s.

_{static}from E

_{dynamic}especially for sandstone formation. The developed equation (Equation (5)) is based on the triaxial tests conducted on 22 sandstone core samples:

_{static}and E

_{dynamic}are in GPa.

_{static}estimation for different rock types. The developed correlations do not require the knowledge of E

_{dynamic}and they directly evaluated E

_{static}based on the bulk density, shear, and compressional time data.

_{static}required retrieve cores from specific depth of the well which is costly and time consuming which required to perform the laboratory analysis. In addition, the analysis will be performed for specific well which cannot easily generalized through the entire field while using the developed empirical correlations had their own limitations such as core type, data range and accuracy. The main objective of this study is to develop an ANN model to predict E

_{static}from the well logs using a real field data (592 core and log data points) which were collected from the whole sandstone field. Furthermore, a new empirical correlation will be developed for estimating E

_{static}for sandstone reservoirs; the correlation is developed based on the extracted weights and biases of the optimized ANN model.

## 2. Uses of Artificial Intelligence in Estimating Rock Mechanical Parameters

## 3. Methodology

_{static}. A new empirical correlation for estimating E

_{static}for sandstone reservoirs will be developed based on the extracted weights and biases of the optimized ANN model.

#### 3.1. Data Preparation

_{static}as an output. The input well log data has been selected based on their correlation coefficient with E

_{static}, the importance of the input parameters considered in this study in estimating E

_{static}is reported by several previous studies. Eissa and Kazi [20] confirmed the ability of improving E

_{static}prediction by incorporating the formation density, and the necessity of E

_{dynamic}, which is dependent on the compressional and shear transit times as reported by several previous studies [21,22,23,24].

_{static}and the log data, gamma ray log was considered to perform the data matching. Then, statistical analysis was performed on the input and output parameters to remove data outliers. For the purpose of outlier removal, all parameter values without a range of ±3.0 standard deviation are considered an outlier and not considered to develop the ANN model. Six data points (outliers) from Well-A were removed in this process.

#### 3.2. Training the ANN Model

_{static}were considered as valid data to build the ANN model. Sixty-nine percent (409 data points) of Well-A data, were randomly selected to train the ANN model.

_{b}) for the input dataset ranges from 2.312–2.968 g/cm

^{3}; the compressional time (ΔT

_{C}) ranges between 44.3 and 77.8 μsec/ft; the shear time (ΔT

_{S}) is between 73.2 and 136.1 μsec/ft; and E

_{static}ranges from 7.5–92.8 GPa.

_{static}with correlation coefficients of 0.724 and −0.815, respectively, while the E

_{static}dependence on the shear time is moderate with a correlation coefficient of 0.439.

_{static}with the lowest average absolute percentage error (AAPE), as well as highest correlation coefficient (R) and coefficient of determination (R

^{2}). During the optimization process, we evaluated the performance of different training functions such as Levenberg–Marquardt backpropagation (trainlm), gradient descent with momentum backpropagation (traingdm), Gradient descent with adaptive learning rate backpropagation (traingda), Bayesian regularization backpropagation (trainbr), and conjugate gradient (traincgf); different transfer functions such as tan-sigmoid (tansig), log-sigmoid (logsig), and pure line (purelin); number of hidden layers from 1–3; and the number of neurons per each hidden layer from 5 to 30 on estimating the E

_{static}.

_{static}prediction. As listed in Table 2, trainbr is the best training function that optimizes the E

_{static}predictability of the ANN model. trainbr is a network training function that updates the weight and bias values of for the ANN model based on Levenberg–Marquardt optimization, and it determines the correct output variable after minimizing a combination of weights and squared errors in a process called Bayesian regularization [33]. logsig is the optimum transfer function. The use of a single hidden (training) function (i.e., a single layer) with 20 neurons also optimized predictability of the ANN model for E

_{static}. Figure 2 shows the structure of the suggested ANN model for E

_{static}prediction.

#### 3.3. Evaluation and Validation of the Developed ANN-Based Empirical Correlation

_{static}. The ability of the developed ANN-based empirical correlation in evaluating the E

_{static}for the validation data collected from Well-B will be compared with four of the available correlations, namely, Eissa and Kazi [20], Canady [21], Najibi et al. [22], and Fei et al. [23] correlations are presented earlier by Equations (2)–(5).

#### 3.4. Evaluation Criterion

_{static}will be evaluated based on the AAPE, R

^{2}, R, and visualization check.

## 4. Results and Discussion

#### 4.1. Training the ANN Model

_{static}as output. The training data were collected from Well-A. The optimization process was conducted using the SaDE algorithm. The optimized design parameters of ANN model are summarized in Table 2. Figure 3 shows the well log data and their depth corresponding core-derived and predicted E

_{static}values for the training dataset. As shown in Figure 3 the ANN model predicted the E

_{static}with very high accuracy where the AAPE is only 0.98% and R equals 0.999. A visual check of the plot confirms the excellent matching between the core-derived and the predicted E

_{static}.

_{static}of the training dataset. The ANN model is highly accurate in estimating the E

_{static}as confirmed by the very high R

^{2}of 0.9978.

#### 4.2. Developing the ANN-Based Empirical Correlation

_{sn}is the normalized E

_{static}, N represents the number of neurons in the hidden layer (N = 20 neurons), i is the index of each neuron in hidden layer, w

_{1i}denotes the weight associated with input and hidden layers for each input parameter, b

_{1i}is the bias associated with hidden and input layers, w

_{2i}represents the weight associated with hidden and output layers, b

_{2}is the bias associated with hidden and output layers (b

_{2}= −1.0767). All weights and biases associated with the hidden and output layers are summarized in Table 3.

_{bn}, ΔT

_{cn}, and ΔT

_{sn}are the normalized input parameters calculated from Equations (7), (8), and (9) respectively:

_{sn}in Equation (6) is in the normalized form and should be converted to real value by using Equation (10):

#### 4.3. Testing the Developed ANN-Based Empirical Correlation

_{static}and the estimated E

_{static}. As shown in Figure 5, the ANN-based empirical correlation predicted the E

_{static}for the unseen data with very high accuracy, where the AAPE is only 1.46% and R is 0.998. A visual check of the plot in Figure 5 confirms the excellent matching between the core-derived and predicted E

_{static}.

_{static}of the testing dataset. The ANN-based empirical correlation is highly accurate in estimating the E

_{static}as confirmed by the very high R

^{2}of 0.9951.

#### 4.4. Validation of the Developed ANN-Based Empirical Correlation

_{static}of Well-B and the predicted E

_{static}that developed using ANN-based empirical correlation.

_{static}; which is validated by the low AAPE of 3.8% and high R of 0.991 in addition to the visual check of the plot. Similarly, the core-derived and the predicted E

_{static}values for Well-C and Well-D were plotted in Figure 7. Due to the limited number of data points in Well-C and Well-D, it was enough to check the predictive accuracy of the ANN-based empirical correlation visually. In all three wells used in validation, the ANN-based empirical correlation was able to provide a continuous profile of the predicted E

_{static}that conforms to the available core-derived values.

_{static}of the 38 data points of Well-B, which are used in validating the ANN-based empirical correlation. This plot affirms the high accuracy of the developed ANN-based empirical correlation in estimating E

_{static}as confirmed by the very high R

^{2}of 0.9816.

#### 4.5. Comparing the Developed ANN-Based Empirical Correlation to the Available Correlations

_{static}for the validation dataset of Well-B. Eissa and Kazi [20], Canady [21], Najibi et al. [22], and Fei et al. [23] correlations are presented earlier by Equations (2)–(5). As indicated in Figure 9, the ANN-based correlation overperformed all the four correlations in evaluating = E

_{static}with an AAPE of 3.80% compared to an AAPE of 37.2%, 36.3%, 61.7%, and 68.5% for the E

_{static}values predicted using Eissa and Kazi [20], Canady [21], Najibi et al. [22], and Fei et al. [23]. These results confirm the high accuracy of the developed ANN-based empirical correlation for E

_{static}estimation.

## 5. Conclusions

_{static}) for sandstone formations using well log data of the bulk density, compressional time, and shear time. The ANN model was trained and tested using real field measurements of 592 data points. Based on the results of this study, the following points are concluded:

- The developed ANN model was capable of estimating E
_{static}for the training dataset with very high accuracy, as indicated by the low AAPE of 0.98%, very high R of 0.999, and R^{2}of 0.9978. The ANN-based empirical correlation was able to predict E_{static}for the testing dataset (unseen) accurately; the E_{static}values of the testing dataset were estimated with AAPE, R, and R^{2}values of 1.46%, 0.998, and 0.9951, respectively. - The developed empirical correlation was validated using a dataset composed of unseen 55 data points of three wells. Validating the developed correlation using the dataset of Well-B (38 points) revealed a highly accurate prediction of the developed correlation where AAPE, R and R
^{2}valueswere 3.8, 0.991 and 0.9816 respectively. - The ANN-based empirical correlation is useful in predicting continuous profile of static Young’s modulus for sandstone formation using conventional log data, bulk density, shear transit time and compressional transit time when there are no available cores.
- Comparing the predictive accuracy of the ANN-based correlation with four of the available empirical equations confirmed the high accuracy of the ANN-based correlation which was able to estimate the E
_{static}for the validation data of Well-B with an AAPE of 3.8% compared with an AAPE of more than 36.0% for all available correlations.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

AAPE | Average absolute percentage error |

ANN | Artificial neural networks |

ΔT_{C} | Compressional transit time |

ΔT_{S} | Shear transit time |

E | Young’s modulus |

E_{static} | Static Young’s modulus |

E_{dynamic} | Dynamic Young’s modulus |

logsig | Log-sigmoid |

R | Correlation coefficient |

R^{2} | Coefficient of determination |

ρ_{b} | Bulk density |

SaDE | Self-adaptive differential evolution |

trainbr | Bayesian regularization backpropagation |

## References

- Fjaer, E.; Horsrud, H.P.; Raaen, A.M.; Risnes, R. Petroleum Related Rock Mechanics; Elsevier B. V.: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Chang, C.; Zoback, M.D.; Khaksar, A. Empirical relations between rock strength and physical properties in sedimentary rocks. J. Pet. Sci. Eng.
**2006**, 51, 223–237. [Google Scholar] [CrossRef] - Gatens, J.M.; Harrison, C.W.; Lancaster, D.E.; Guldry, F.K. In-situ stress tests and acoustic logs determine mechanical properties and stress profiles in the devonian shales. SPE Form. Eval.
**1990**, 5, 248–254. [Google Scholar] [CrossRef] - Nes, O.M.; Fjaer, E.; Tronvoll, J.; Kristiansen, T.G.; Horsrud, P. Drilling time reduction through an integrated rock mechanics analysis. J. Energy Res. Technol.
**2012**, 134, 2802:1–2802:7. [Google Scholar] [CrossRef] - Meyer, B.R.; Jacot, R.H. Impact of stress-dependent Young’s moduli on hydraulic fracture modeling. In Proceedings of the 38th U.S. Symposium on Rock Mechanics, Washington, DC, USA, 7–10 July 2001. ARMA-01-0297. [Google Scholar]
- Howard, G.C.; Fast, C.R. Hydraulic Fracturing. In Doherty Memorial Fund of AIME, Society of Petroleum Engineers of AIME; Henry L.: New York, NY, USA, 1970; Monograph Volume 2 of SPE. [Google Scholar]
- Barree, R.D.; Gilbert, J.V.; Conway, M.W. Stress and rock property profiling for unconventional reservoir stimulation. In Proceedings of the SPE Hydraulic Fracturing Technology Conference, The Woodlands, TX, USA, 19–21 January 2009. SPE-118703-MS. [Google Scholar]
- Colin, C.; Potter, S.; Darren, F. Formation elastic parameters by deriving S-wave velocity logs. CREWES Res.
**1997**, 9, 1–10. [Google Scholar] - King, M.S. Wave Velocities in Rocks as a Function of Changes in Over burden Pressure and Pore Fluid Saturants. Geophysics
**1966**, 31, 50–73. [Google Scholar] [CrossRef] - Rinehart, J.S.; Fortin, J.-P.; Baugin, P. Propagation Velocity of Longitudinal Waves in Rock. Effect of State of Stress, Stress Level of the Wave, Water Content, Porosity, Temperature Stratification and Texture. In Proceedings of the 4th Symposium on Rock Mechanics, University Park, PA, USA, 30 March–1 April 1961. ARMA-61-119. [Google Scholar]
- Simmons, G.; Brace, W.I. Comparison of Static and Dynamic Measurements of Compressibility of Rocks. J. Geophys. Res.
**1965**, 70, 5649–5656. [Google Scholar] [CrossRef] - Abdulraheem, A.; Ahmed, M.; Vantala, A.; Parvez, T. Prediction of Rock Mechanical Parameters for Hydrocarbon Reservoirs Using Different Artificial Intelligence Techniques. In Proceedings of the Saudi Arabia Section Technical Symposium, Al-Khobar, Saudi Arabia, 9–11 May 2009. SPE-126094-MS. [Google Scholar]
- Larsen, I.; Fjær, E.; Renlie, L. Static and Dynamic Poisson’s Ratio of Weak Sandstones. In Proceedings of the 4th North American Rock Mechanics Symposium, Seattle, WA, USA, 31 July–3 August 2000. ARMA-2000-0077. [Google Scholar]
- Bai, P. Experimental research on rock drillability in the center of junggar basin. Electron. J. Geotech. Eng.
**2013**, 18, 5065–5074. [Google Scholar] - Ca, H.; Xi, L.; Guo, L. Rock mechanics study on the safety and efficient extraction for deep moderately inclined medium-thick orebody. Electron. J. Geotech. Eng.
**2015**, 20, 11073–11082. [Google Scholar] - Li, P.; Liu, X.; Zhong, Z. Mechanical Property Experiment and Damage Statistical Constitutive Model of Hongze Rock Salt in China. Electron. J. Geotech. Eng.
**2015**, 20, 81–94. [Google Scholar] - King, M.S. Static and Dynamic elastic moduli of rocks under pressure. In Proceedings of the 11th U.S. Symposium on Rock Mechanics, Berkeley, CA, USA, 16–19 June 1969. ARMA-69-0329. [Google Scholar]
- Wang, Z.; Nur, A.A. Dynamic versus static elastic properties of reservoir rocks. J. Seism. Acoust. Veloc. Res. Rocks
**2000**, 19, 531–539. [Google Scholar] - Khaksar, A.; Taylor, P.G.; Fang, Z.; Kayes, T.; Salazar, A.; Rahman, K. Rock strength from core and logs, where we stand and ways to go. In Proceedings of the EUROPEC/EAGE conference and exhibition, Amsterdam, The Netherlands, 8–11 June 2009. SPE-121972-MS. [Google Scholar]
- Eissa, E.A.; Kazi, A. Relation between static and dynamic Young’s moduli of rocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1988**, 25, 479–482. [Google Scholar] [CrossRef] - Canady, W.J. A Method for Full-Range Young’s Modulus Correction. Presented at the North American Unconventional Gas Conference and Exhibition, The Woodlands, TX, USA, 14–16 June 2011. Paper SPE-143604-MS. [Google Scholar] [CrossRef]
- Najibi, A.R.; Ghafoori, M.; Lashkaripour, G.R.; Asef, M.R. Empirical relations between strength and static and dynamic elastic properties of Asmari and Sarvak limestones, two main oil reservoirs in Iran. J. Pet. Sci. Eng.
**2015**, 126, 78–82. [Google Scholar] [CrossRef] - Fei, W.; Huiyuan, B.; Jun, Y.; Yonghao, Z. Correlation of Dynamic and Static Elastic Parameters of Rock. Electron. J. Geotech. Eng.
**2016**, 21, 1551–1560. [Google Scholar] - Mahmoud, M.A.; Elkatatny, S.A.; Ramadan, E.; Abdulraheem, A. Development of Lithology-Based Static Young’s Modulus Correlations from Log Data Based on Data Clustering Technique. J. Pet. Sci. Eng.
**2016**, 146, 10–20. [Google Scholar] [CrossRef] - Tariq, A.; Elkatatny, S.A.; Mahmoud, M.A.; Zaki, A.; Abdulraheem, A. A New Approach to Predict Failure Parameters of Carbonate Rocks using Artificial Intelligence Tools. In Proceedings of the SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition, Dammam, Saudi Arabia, 24–27 April 2017. SPE-187974-MS. [Google Scholar]
- Elkatatny, S.M.; Tariq, Z.; Mahmoud, M.A.; Abdulraheem, A.; Abdelwahab, A.Z.; Woldeamanuel, M. An Artificial Intelligent Approach to Predict Static Poisson’s Ratio. In Proceedings of the 51st US Rock Mechanics/Geomechanics Symposium, San Francisco, CA, USA, 25–28 June 2017. ARMA 17-771. [Google Scholar]
- Tariq, Z.; Elkatatny, S.M.; Mahmoud, M.A.; Abdulazeez, A. A Holistic Approach to Develop New Rigorous Empirical Correlation for Static Young’s Modulus. In Proceedings of the Abu Dhabi International Petroleum Exhibition & Conference, Abu Dhabi, UAE, 7–10 November 2016. SPE-183545-MS. [Google Scholar]
- Tariq, Z.; Elkatatny, S.M.; Mahmoud, M.A.; Abdulazeez, A. A New Artificial Intelligence Based Empirical Correlation to Predict Sonic Travel Time. In Proceedings of the International Petroleum Technology Conference, Bangkok, Thailand, 14–16 November 2016. IPTC-19005-MS. [Google Scholar]
- Tariq, Z.; Elkatatny, S.M.; Mahmoud, M.A.; Abdulraheem, A.; Abdelwahab, A.Z.; Woldeamanuel, I.M. Development of New Correlation for Unconfined Compressive Strength for Carbonate Reservoir Using Artificial Intelligence Techniques. In Proceedings of the 51st US Rock Mechanics/Geomechanics Symposium, San Francisco, CA, USA, 25–28 June 2017. ARMA 17-428. [Google Scholar]
- Tariq, Z.; Elkatatny, S.M.; Mahmoud, M.A.; Abdulraheem, A.; Abdelwahab, A.Z.; Woldeamanuel, M. Estimation of Rock Mechanical Parameters using Artificial Intelligence Tools. In Proceedings of the 51st US Rock Mechanics/Geomechanics Symposium, San Francisco, CA, USA, 25–28 June 2017. ARMA 17-301. [Google Scholar]
- Omran, M.G.H.; Salman, A.; Engelbrecht, A.P. Self-adaptive Differential Evolution. In Computational Intelligence and Security; Hao, Y., Liu, J., Wang, Y., Cheung, Y.-m., Yin, H., Jiao, L., Ma, J., Jiao, Y.-C., Eds.; CIS 2005. Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2005; Volume 3801. [Google Scholar]
- Al-Anazi, A.F.; Gates, I.D. A support vector machine algorithm to classify lithofacies and model permeability in heterogeneous reservoirs. Eng. Geol.
**2010**, 114, 267–277. [Google Scholar] [CrossRef] - MathWorks. Available online: https://www.mathworks.com/help/deeplearning/ref/trainbr.html;jsessionid=7cc70c77fdb3f0bb58ed870c69c7 (accessed on 1 April 2019).

**Figure 3.**From left to right, bulk density, compressional transit time, shear transit time, and their corresponding predicted and core-derived static Young’s modulus values of the training set of Well-A.

**Figure 7.**Plot of the predicted and the core-derived E

_{static}for the validation datasets collected from Well-B, Well-C, and Well-D.

Statistical Parameter | ρ_{b}, g/cm^{3} | ΔT_{C}, μsec/ft | ΔT_{S}, μsec/ft | E_{static}, Gpa |
---|---|---|---|---|

Minimum | 2.312 | 44.3 | 73.2 | 7.5 |

Maximum | 2.968 | 77.8 | 136.1 | 92.8 |

Range | 0.656 | 33.4 | 62.9 | 85.3 |

Standard Deviation | 0.106 | 4.69 | 8.39 | 13.93 |

Sample Variance | 0.011 | 22.0 | 70.3 | 194.0 |

Kurtosis | 0.569 | 4.262 | 1.673 | 0.167 |

Skewness | 0.011 | 1.569 | 0.564 | 0.186 |

Parameter | Value |
---|---|

Learning function | trainbr |

Transfer function | logsig |

Number of hidden layers | 1 |

Number of neurons | 20 |

i | w_{1i,1} | w_{1i,2} | w_{1i,3} | b_{1i} | w_{2i} |
---|---|---|---|---|---|

1 | −4.368 | 20.303 | −14.485 | −3.638 | 5.437 |

2 | −0.216 | 1.507 | 2.017 | 4.178 | −2.494 |

3 | 1.792 | −2.322 | −21.160 | −3.877 | −4.132 |

4 | 0.269 | 0.057 | −0.949 | 1.777 | 6.325 |

5 | 1.498 | −19.164 | 3.620 | 6.726 | 9.261 |

6 | 13.802 | 12.907 | −0.662 | 2.170 | 0.640 |

7 | 3.466 | 8.897 | 1.549 | −3.110 | 4.174 |

8 | −4.369 | −0.142 | 17.692 | 1.732 | 5.813 |

9 | −1.604 | 21.932 | −1.059 | −9.030 | 5.608 |

10 | 10.803 | −12.301 | 24.520 | 16.752 | 9.748 |

11 | 14.298 | 13.932 | 0.070 | 0.895 | −5.642 |

12 | −49.173 | −25.972 | −1.332 | 15.768 | 9.929 |

13 | −3.062 | 15.989 | −11.554 | −3.115 | −4.766 |

14 | −18.124 | −17.674 | 0.206 | −1.307 | −3.944 |

15 | 40.409 | 23.746 | 0.690 | −14.273 | −11.118 |

16 | 6.280 | 3.526 | −8.930 | 2.199 | −0.945 |

17 | 7.010 | 3.251 | −11.579 | 1.502 | 1.544 |

18 | 19.888 | 7.137 | 0.149 | −8.787 | −1.596 |

19 | −32.100 | −16.426 | −1.400 | 11.426 | −23.697 |

20 | −3.053 | 1.125 | 19.499 | 2.853 | −9.194 |

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## Share and Cite

**MDPI and ACS Style**

Mahmoud, A.A.; Elkatatny, S.; Ali, A.; Moussa, T.
Estimation of Static Young’s Modulus for Sandstone Formation Using Artificial Neural Networks. *Energies* **2019**, *12*, 2125.
https://doi.org/10.3390/en12112125

**AMA Style**

Mahmoud AA, Elkatatny S, Ali A, Moussa T.
Estimation of Static Young’s Modulus for Sandstone Formation Using Artificial Neural Networks. *Energies*. 2019; 12(11):2125.
https://doi.org/10.3390/en12112125

**Chicago/Turabian Style**

Mahmoud, Ahmed Abdulhamid, Salaheldin Elkatatny, Abdulwahab Ali, and Tamer Moussa.
2019. "Estimation of Static Young’s Modulus for Sandstone Formation Using Artificial Neural Networks" *Energies* 12, no. 11: 2125.
https://doi.org/10.3390/en12112125