Application of Non-Destructive Test Results to Estimate Rock Mechanical Characteristics—A Case Study
Abstract
:1. Introduction
2. Materials and Methods
2.1. Case Study
2.2. Materials
2.3. Methods
2.4. Data Normalization
2.5. The SVR Approach
2.6. The ANFIS Method
2.7. KNN Approach
- Select the optimal K value;
- Obtain the distances based on input specifications;
- Form the K class according to the closest distance (maximum similarity) and then calculate the distance of the new record from all educational records;
- Choose the nearest neighbor;
- Use the K category label of the nearest neighbor to predict the new record category.
2.8. Evaluating Criteria
3. Results
3.1. Laboratory Results
3.2. Correlation Heatmaps and Simple Regression Analysis
3.3. UCS and EM Estimation Using Multiple Linear Regression Method
3.4. Comparison with Previous Studies
3.5. The SVR Results
3.6. Estimation of UCS and EM Using BPNN
- The train set, with 70% of the total data for training the network;
- The test group, with 15% of the total data to test the network;
- The validation set, with 15% of the total data for preventing overfitting.
3.7. Results of ANFIS Approach
3.8. The KNN Results
3.9. Nonlinear Multivariate Regression Analysis
3.10. Comparison of Used Methods
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Li, S.; Wang, Y.; Xie, X. Prediction of Uniaxial Compression Strength of Limestone Based on the Point Load Strength and SVM Model. Minerals 2021, 11, 1387. [Google Scholar] [CrossRef]
- Ren, C.; Yu, J.; Liu, S.; Yao, W.; Zhu, Y.; Liu, X. A Plastic Strain-Induced Damage Model of Porous Rock Suitable for Different Stress Paths. Rock Mech. Rock Eng. 2022, 55, 1887–1906. [Google Scholar] [CrossRef]
- Yu, J.; Zhu, Y.; Yao, W.; Liu, X.; Ren, C.; Cai, Y.; Tang, X. Stress Relaxation Behaviour of Marble under Cyclic Weak Disturbance and Confining Pressures. Measurement 2021, 182, 109777. [Google Scholar] [CrossRef]
- Ulusay, R.; Tureli, K.; Ider, M.H. Prediction of engineering properties of a selected litharenite sandstone from its petrographic characteristics using correlation and multivariable statistical technique. Eng. Geol. 1994, 37, 135–157. [Google Scholar] [CrossRef]
- Yasar, E.; Ranjith, P.G.; Perera, M.S.A. Physico-mechanical behaviour of southeastern Melbourne sedimentary rocks. Int. J. Rock Mech. Min. Sci. 2010, 47, 481–487. Available online: http://pascal-francis.inist.fr/vibad/index.php?Action=getRecordDetail&idt=22570877 (accessed on 1 April 2010). [CrossRef]
- Jin, J.; Zhang, X.; Liu, X.; Li, Y.; Li, S. Study on Critical Slowdown Characteristics and Early Warning Model of Damage Evolution of Sandstone under Freeze-Thaw Cycles. Front. Earth Sci. 2023, 15, 18–25. [Google Scholar] [CrossRef]
- Lawal, A.I.; Kwon, S.; Aladejare, A.E.; Oniyide, G.O. Prediction of the static and dynamic mechanical properties of sedimentary rock using soft computing methods. Geotech. Eng. 2022, 28, 313–324. [Google Scholar]
- Armaghani, D.J.; Mamou, A.; Maraveas, C.; Roussis, P.C.; Siorikis, V.G.; Skentou, A.D.; Asteris, P.G. Predicting the unconfined compressive strength of granite using only two non-destructive test indexes. Geomech. Eng. 2021, 25, 317–330. [Google Scholar]
- Aladejare, A.E.; Akeju, V.O.; Wang, Y. Data-driven characterization of the correlation between uniaxial compressive strength and Youngs’ modulus of rock without regression models. Transp. Geotech. 2022, 32, 100680. [Google Scholar] [CrossRef]
- Rastegarnia, A.; Lashkaripour, G.R.; Sharifi Teshnizi, E.; Ghafoori, M. Evaluation of engineering characteristics and estimation of dynamic properties of clay-bearing rocks. Environ. Earth Sci. 2021, 80, 621. [Google Scholar] [CrossRef]
- Mahmoodzadeh, A.; Mohammadi, M.; Ibrahim, H.H.; Abdulhamid, S.N.; Salim, S.G.; Ali, H.F.H.; Majeed, M.K. Artificial intelligence forecasting models of uniaxial compressive strength. Transp. Geotech. 2021, 27, 100499. [Google Scholar] [CrossRef]
- Siddig, O.; Gamal, H.; Elkatatny, S.; Abdulraheem, A. Applying Different Artificial Intelligence Techniques in Dynamic Poisson’s Ratio Prediction Using Drilling Parameters. J. Energy Resour. Technol. 2022, 144, 073006. [Google Scholar] [CrossRef]
- Zoveidavianpoor, M.; Samsuri, A.; Shadizadeh, S.R. Adaptive neuro fuzzy inference system for compressional wave velocity prediction in a carbonate reservoir. Appl. Geophys. 2013, 89, 96–107. [Google Scholar] [CrossRef]
- Chang, C.; Mark, D.; Zoback, M.B.; Khaksar, A. Empirical relations between rock strength and physical properties in sedimentary rocks. J. Pet. Sci. Eng. 2006, 51, 223–237. [Google Scholar] [CrossRef]
- Heidari, M.; Momeni, A.; Rafiei, B.; Khodabakhsh, S.; Torabi-Kaveh, M. Relationship between Petrographic Characteristics and the Engineering Properties of Jurassic Sandstones, Hamedan, Iran. Rock Mech. Rock Eng. 2013, 46, 1091–1101. [Google Scholar] [CrossRef]
- Wang, Z.; Li, W.; Chen, J. Application of Various Nonlinear Models to Predict the Uniaxial Compressive Strength of Weakly Cemented Jurassic Rocks. Nat. Resour. Res. 2022, 31, 371–384. [Google Scholar] [CrossRef]
- Shahani, N.M.; Zheng, X.; Liu, C.; Li, P.; Hassan, F.U. Application of Soft Computing Methods to Estimate Uniaxial Compressive Strength and Elastic Modulus of Soft Sedimentary Rocks. Arab. J. Geosci. 2022, 15, 384. [Google Scholar] [CrossRef]
- Cemiloglu, A.; Zhu, L.; Arslan, S.; Xu, J.; Yuan, X.; Azarafza, M.; Derakhshani, R. Support Vector Machine (SVM) Application for Uniaxial Compression Strength (UCS) Prediction: A Case Study for Maragheh Limestone. Appl. Sci. 2023, 13, 2217. [Google Scholar] [CrossRef]
- Abdelhedi, M.; Jabbar, R.; Said, A.B.; Fetais, N.; Abbes, C. Machine Learning for Prediction of the Uniaxial Compressive Strength within Carbonate Rocks. Earth Sci. Inform. 2023, 7, 1–15. [Google Scholar] [CrossRef]
- Asare, E.N.; Affam, M.; Ziggah, Y.Y. A Hybrid Intelligent Prediction Model of Autoencoder Neural Network and Multivariate Adaptive Regression Spline for Uniaxial Compressive Strength of Rocks. Model. Earth. Syst. Environ. 2023, 6, 1–17. [Google Scholar] [CrossRef]
- Wang, Y.; Rezaei, M.; Abdullah, R.A.; Hasanipanah, M. Developing Two Hybrid Algorithms for Predicting the Elastic Modulus of Intact Rocks. Sustainability 2023, 15, 4230. [Google Scholar] [CrossRef]
- Zhao, R.; Shi, S.; Li, S.; Guo, W.; Zhang, T.; Li, X.; Lu, J. Deep Learning for Intelligent Prediction of Rock Strength by Adopting Measurement While Drilling Data. Int. J. Geomech. 2023, 23, 04023028. [Google Scholar] [CrossRef]
- Rahman, T.; Sarkar, K. Empirical Correlations between Uniaxial Compressive Strength and Density on the Basis of Lithology: Implications from Statistical and Machine Learning Assessments. Earth Sci. Inform. 2023, 1, 1–25. [Google Scholar] [CrossRef]
- Weng, M.C.; Li, H.H. Relationship between the deformation characteristics and microscopic properties of sandstone explored by the bonded-particle model. Int. J. Rock Mech. Min. Sci. 2012, 56, 34–43. [Google Scholar] [CrossRef]
- Naresh, K.T.; Shuichiro, Y.; Suresh, D. Relationships among mechanical, physical and petrographic properties of Siwalik sandstones, Central Nepal Sub-Himalayas. Eng. Geol. 2007, 90, 105–123. [Google Scholar] [CrossRef]
- Ghobadi, M.H.; Heidari, M.; Rafiei, B.; Mousavi, S.D. Investigation of the relationship between mineralogical and physical properties of sandstones with their tensile strength. In Proceedings of the First National Conference on Geotechnical Engineering, Mashhad, Iran, 14 June 2013. Article COI Code: GEOTEC01_371 (In Persian). [Google Scholar]
- Qi, Y.; Ju, Y.; Yu, K.; Meng, S.; Qiao, P. The effect of grain size, porosity and mineralogy on the compressive strength of tight sandstones: A case study from the eastern Ordos Basin, China. J. Pet. Sci. Eng. 2022, 208, 109461. [Google Scholar] [CrossRef]
- Yilmaz, N.G.; Goktan, R.M. Comparison and combination of two NDT methods with implications for compressive strength evaluation of selected masonry and building stones. Bull. Eng. Geol. Environ. 2019, 78, 4493–4503. [Google Scholar] [CrossRef]
- Daoud, H.S.D.; Rashed, K.A.R.; Alshkane, Y.M.A. Correlations of uniaxial compressive strength and modulus of elasticity with point load strength index, pulse velocity and dry density of limestone and sandstone rocks in Sulaimani Governorate, Kurdistan Region, Iraq. J. Zankoy Sulaimani-A 2018, 19, 57–72. [Google Scholar] [CrossRef]
- Mishra, D.A.; Basu, A. Estimation of uniaxial compressive strength of rock materials by index tests using regression analysis and fuzzy inference system. Eng. Geol. 2013, 160, 54–68. [Google Scholar] [CrossRef]
- Selçuk, L.; Yabalak, E. Evaluation of the ratio between uniaxial compressive strength and Schmidt hammer rebound number and its effectiveness in predicting rock strength. Nondestruct. Test. Eval. 2015, 30, 1–12. [Google Scholar] [CrossRef]
- Armaghani, D.J.; Amin, M.F.M.; Yagiz, S.; Faradonbeh, R.S.; Abdullah, R.A. Prediction of the uniaxial compressive strength of sandstone using various modeling techniques. Int. J. Rock. Mech. Min. 2016, 85, 174–186. [Google Scholar] [CrossRef]
- Abdi, Y.; Khanlari, G.R. Estimation of mechanical properties of sandstones using P-wave velocity and Schmidt hardness. New Find. Appl. Geol. 2019, 13, 33–47. [Google Scholar]
- Eremin, M. Three-dimensional finite-difference analysis of deformation and failure of weak porous sandstones subjected to uniaxial compression. Int. J. Rock Mech. Min. Sci. 2020, 133, 104412. [Google Scholar] [CrossRef]
- Bejarbaneh, B.Y.; Bejarbaneh, E.Y.; Amin, M.F.M.; Fahimifar, A.; Jahed Armaghani, D.; Majid, M.Z.A. Intelligent modelling of sandstone deformation behaviour using fuzzy logic and neural network systems. Bull. Eng. Geol. Environ. 2018, 77, 345–361. [Google Scholar] [CrossRef]
- Moradian, Z.A.; Behnia, M. Predicting the uniaxial compressive strength and static Young’s modulus of intact sedimentary rocks using the ultrasonic test. Int. J. Geomech. 2009, 9, 14–19. [Google Scholar] [CrossRef]
- Kılıç, A.; Teymen, A. Determination of mechanical properties of rocks using simple methods. Bull. Eng. Geol. Environ. 2008, 67, 237. [Google Scholar] [CrossRef]
- Çobanoğlu, İ.; Çelik, S.B. Estimation of uniaxial compressive strength from point load strength, Schmidt hardness and P-wave velocity. Bull. Eng. Geol. Environ. 2008, 67, 491–498. [Google Scholar] [CrossRef]
- Hebib, R.; Belhai, D.; Alloul, B. Estimation of uniaxial compressive strength of North Algeria sedimentary rocks using density, porosity, and Schmidt hardness. Arab. J. Geosci. 2017, 10, 383. [Google Scholar] [CrossRef]
- Bolla, A.; Paronuzzi, P. UCS field estimation of intact rock using the Schmidt hammer: A new empirical approach. In IOP Conference Series. Earth Environ. Sci. 2021, 83, 012014. [Google Scholar]
- ISRM. Rock characterization testing and monitoring. In ISRM Suggested Methods; Brown, E.T., Ed.; Pergamon Press: Oxford, UK, 1981; Volume 211. [Google Scholar]
- Designation D2845; Test Methods for Ultra Violet Velocities Determination. ASTM: West Conshohocken, PA, USA, 1983.
- Chen, H.; Liu, M.; Chen, Y.; Li, S.; Miao, Y. Nonlinear Lamb Wave for Structural Incipient Defect Detection with Sequential Probabilistic Ratio Test. Secur. Commun. Netw. 2022, 2022, 9851533. [Google Scholar] [CrossRef]
- Yang, J.; Fu, L.; Fu, B.; Deng, W.; Han, T. Third-Order Padé Thermoelastic Constants of Solid Rocks. J. Geophys. Res. Solid Earth 2022, 127, e2022J–e24517J. [Google Scholar] [CrossRef]
- ASTM D2938-95; Standard Test Method for Unconfined Compressive Strength of Intact Rock Core Specimens. ASTM: West Conshohocken, PA, USA, 2002.
- Chen, H.; Li, S. Multi-Sensor Fusion by CWT-PARAFAC-IPSO-SVM for Intelligent Mechanical Fault Diagnosis. Sensors 2022, 22, 3647. [Google Scholar] [CrossRef] [PubMed]
- Maleki, M.A.; Emami, M. Application of SVM for investigation of factors affecting compressive strength and consistency of geopolymer concretes. J. Civ. Eng. Mater. Appl. 2019, 3, 101–107. [Google Scholar] [CrossRef]
- Kookalani, S.; Cheng, B. Structural analysis of GFRP elastic gridshell structures by particle swarm optimization and least square support vector machine algorithms. J. Civ. Eng. Mater. Appl. 2021, 8, 12–23. [Google Scholar]
- Zhou, Q.; Herrera-Herbert, J.; Hidalgo, A. Predicting the risk of fault-induced water inrush using the adaptive neuro-fuzzy inference system. Minerals 2017, 7, 55. [Google Scholar] [CrossRef]
- Shirnezhad, Z.; Azma, A.; Foong, L.K.; Jahangir, A.; Rastegarnia, A. Assessment of Water Resources Quality of a Karstic Aquifer in the Southwest of Iran. Bull. Eng. Geol. Environ. 2021, 80, 71–92. [Google Scholar] [CrossRef]
- Hassanzadeh, R.; Beiranvand, B.; Komasi, M.; Hassanzadeh, A. Investigation of Data Mining Method in Optimal Operation of Eyvashan Earth Dam Reservoir Based on PSO Algorithm. J. Civ. Eng. Mater. Appl. 2021, 5, 125–137. [Google Scholar]
- Rastegarnia, A.; Ghafoori, M.; Moghaddas, N.H.; Lashkaripour, G.R.; Shojaei, H. Application of Cuttings to Estimate the Static Characteristics of the Dolomudstone Rocks. Geomech. Eng. 2022, 29, 65–77. [Google Scholar] [CrossRef]
- Folk, R.L. Petrology of Sedimentary Rocks; Hemphill Publishing Company: Hemphill, Austin, 1974; 600p. [Google Scholar]
- Anon, O.H. Classification of rocks and soils for engineering geological mapping, Part 1: Rock and soil materials. Bull. Int. Assoc. Eng. Geol. 1979, 19, 364–437. [Google Scholar]
- Deere, D.U.; Miller, R.P. Engineering Classification and Index Properties for Intact Rock; Technical Report AFWLTR; University of Illinois at Urbana-Champaign: Champaign, IL, USA, 1966; pp. 65–116. [Google Scholar]
- Mokhberi, M.; Khademi, H. The use of stone columns to reduce the settlement of swelling soil using numerical modeling. J. Civ. Eng. Mater. Appl. 2017, 1, 45–60. [Google Scholar] [CrossRef]
- Rastegarnia, A.; Alizadeh, S.M.S.; Esfahani, M.K.; Amini, O.; Utyuzh, A.S. The Effect of Hydrated Lime on the Petrography and Strength Characteristics of Illite Clay. Geomech. Eng. 2020, 22, 143–152. [Google Scholar] [CrossRef]
- Wu, Z.; Xu, J.; Li, Y.; Wang, S. Disturbed State Concept–Based Model for the Uniaxial Strain-Softening Behavior of Fiber-Reinforced Soil. Int. J. Geomech. 2022, 22, 4022092. [Google Scholar] [CrossRef]
- Arman, H.; Abdelghany, O.; Saima, M.A.; Aldahan, A.; Mahmoud, B.; Hussein, S.; Fowler, A.R. Petrological control on engineering properties of carbonate rocks in arid regions. Bull. Eng. Geol. Environ. 2021, 80, 4221–4233. [Google Scholar] [CrossRef]
- Rastegarnia, A.; Lashkaripour, G.R.; Ghafoori, M.; Farrokhad, S.S. Assessment of the engineering geological characteristics of the Bazoft dam site, SW Iran. Q. J. Eng. Geol. Hydrogeol. 2019, 52, 360–374. [Google Scholar] [CrossRef]
- Zhang, X.; Wang, Z.; Reimus, P.; Ma, F.; Soltanian, M.R.; Xing, B.; Dai, Z. Plutonium Reactive Transport in Fractured Granite: Multi-Species Experiments and Simulations. Water 2022, 224, 119068. [Google Scholar] [CrossRef]
- He, M.; Dong, J.; Jin, Z.; Liu, C.; Xiao, J.; Zhang, F.; Deng, L. Pedogenic Processes in Loess-Paleosol Sediments: Clues from Li Isotopes of Leachate in Luochuan Loess. Geochim. Cosmochim. Acta 2021, 299, 151–162. [Google Scholar] [CrossRef]
- Xu, Z.; Li, X.; Li, J.; Xue, Y.; Jiang, S.; Liu, L.; Sun, Q. Characteristics of Source Rocks and Genetic Origins of Natural Gas in Deep Formations, Gudian Depression, Songliao Basin, NE China. ACS Earth Space Chem. 2022, 6, 1750–1771. [Google Scholar] [CrossRef]
- Zheng, Z.; Zuo, Y.; Wen, H.; Zhang, J.; Zhou, G.; Xv, L.; Zeng, J. Natural Gas Characteristics and Gas-Source Comparisons of the Lower Triassic Jialingjiang Formation, Eastern Sichuan Basin. J. Pet. Sci. Eng. 2022, 221, 111165. [Google Scholar] [CrossRef]
- Xiao, D.; Hu, Y.; Wang, Y.; Deng, H.; Zhang, J.; Tang, B.; Li, G. Wellbore Cooling and Heat Energy Utilization Method for Deep Shale Gas Horizontal Well Drilling. Appl. Therm. Eng. 2022, 213, 118684. [Google Scholar] [CrossRef]
- Wang, G.; Zhao, B.; Wu, B.; Wang, M.; Liu, W.; Zhou, H.; Han, Y. Research on the Macro-Mesoscopic Response Mechanism of Multisphere Approximated Heteromorphic Tailing Particles. Lithosphere 2022, 2022, 1977890. [Google Scholar] [CrossRef]
- Xu, J.; Lan, W.; Ren, C.; Zhou, X.; Wang, S.; Yuan, J. Modeling of Coupled Transfer of Water, Heat and Solute in Saline Loess Considering Sodium Sulfate Crystallization. Cold Reg. Sci. Technol. 2021, 189, 103335. [Google Scholar] [CrossRef]
- Peng, J.; Xu, C.; Dai, B.; Sun, L.; Feng, J.; Li, C.; Liu, Y.; Huang, Q. Numerical Investigation of Brittleness Effect on Strength and Microcracking Behavior of Crystalline Rock. Int. J. Geomech. 2022, 22, 4022178. [Google Scholar] [CrossRef]
- Xu, Z.; Wang, Y.; Jiang, S.; Fang, C.; Liu, L.; Wu, K.; Chen, Y. Impact of Input, Preservation and Dilution on Organic Matter Enrichment in Lacustrine Rift Basin: A Case Study of Lacustrine Shale in Dehui Depression of Songliao Basin, NE China. Mar. Pet. Geol. 2022, 135, 105386. [Google Scholar] [CrossRef]
- Zhang, X.; Ma, F.; Dai, Z.; Wang, J.; Chen, L.; Ling, H.; Li, C.; Soltanian, M.R. Radionuclide Transport in Multi-Scale Fractured Rocks: A Review. J. Hazard. Mater. 2022, 424, 127550. [Google Scholar] [CrossRef]
- Shayesteh, A.; Ghasemisalehabadi, E.; Khordehbinan, M.W.; Rostami, T. Finite element method in statistical analysis of flexible pavement. J. Mar. Sci. Technol. 2017, 25, 15. [Google Scholar]
- Zhan, C.; Dai, Z.; Soltanian, M.R.; de Barros, F.P.J. Data-Worth Analysis for Heterogeneous Subsurface Structure Identification with a Stochastic Deep Learning Framework. Water Resour. Res. 2022, 58, e2022W–e33241W. [Google Scholar] [CrossRef]
- Li, R.; Wu, X.; Tian, H.; Yu, N.; Wang, C. Hybrid Memetic Pretrained Factor Analysis-Based Deep Belief Networks for Transient Electromagnetic Inversion. IEEE Trans. Geosci. Remote Sens. 2022, 60, 1–14. [Google Scholar] [CrossRef]
- Liu, Y.; Zhang, Z.; Liu, X.; Wang, L.; Xia, X. Efficient Image Segmentation Based on Deep Learning for Mineral Image Classification. Adv. Powder Technol. 2021, 32, 3885–3903. [Google Scholar] [CrossRef]
- Lerman, N.; Aronofsky, L.; Aghili, B. Investigating the Microstructure and Mechanical Properties of Metakaolin-Based Polypropylene Fiber-Reinforced Geopolymer Concrete Using Different Monomer Ratios. J. Civ. Eng. Mater. Appl. 2021, 5, 115–123. [Google Scholar] [CrossRef]
- Al-Anazi, A.F.; Gates, I.D. Support vector regression to predict porosity and permeability: Effect of sample size. Comput. Geosci. 2012, 39, 64–76. [Google Scholar] [CrossRef]
Equation | References | Lithology | Equation No. |
---|---|---|---|
UCS = 0.00021 × SN33.55 | Yilmaz and Goktan [28] | Different rocks | (1) |
UCS = 0.00004 SN4.164 | Daoud et al. [29] | Limestone and sandstone | (2) |
UCS = 287.7 − 615.90 | Mishra and Basu [30] | Sandstone rocks | (3) |
UCS = 0.05 PW − 126.40 | Mishra and Basu [30] | Sandstone rocks | (4) |
Mishra and Basu [30] | Sandstone rocks | (5) | |
UCS = 22.18 PW − 30.32 | Selçuk and Yabalak [31] | Various rocks, including sandstones | (6) |
UCS = 17.783 PW1.099 (MPa) | Armaghani et al. [32] | Sandstone rocks | (7) |
UCS = 0.041 PW − 15.40 | Abdi and Khanlari [33] | Sandstone rocks | (8) |
EM = 0.005 PW + 0.621 | Abdi and Khanlari [33] | Sandstone rocks | (9) |
UCS = 1.41 + 17.98exp(−19.01n) | Eremin [34] | Sandstone rocks | (10) |
EM = 11.237 PW − 6.894 | Bejarbaneh et al. [35] | Sandstone rocks | (11) |
EM = 2.06 PW2.78 | Moradian and Behnia [36] | Various rocks, including sandstone | (12) |
UCS = 2.304 PW2.43 | Kılıç and Teyman [37] | Various rocks, including sandstone | (13) |
UCS = 56.71 PW − 192.93 | Cobanoglu and Celik [38] | Sandstone and limestone | (14) |
UCS = 2.56EXP(0.063SN) | Hebib et al. [39] | Sedimentary rocks | (15) |
UCS = 0.007 × SN3.443 | Bolla and Paronuzzi [40] | Sedimentary rocks | (16) |
Q (%) | Fl (%) | Fr (%) | D (g/cm3) | UCS (MPa) | EM (GPa) | WA (%) | PW (km/s) | n (%) | SN (MPa) | |
---|---|---|---|---|---|---|---|---|---|---|
Mean | 11.15 | 38.04 | 48.66 | 2.58 | 63.87 | 16.41 | 4.05 | 4.20 | 6.56 | 37 |
Standard Error | 0.24 | 0.36 | 0.65 | 0.02 | 3.41 | 0.76 | 0.33 | 0.06 | 0.56 | 0.79 |
Standard Deviation | 1.95 | 2.85 | 5.18 | 0.13 | 27.31 | 6.10 | 2.66 | 0.50 | 4.47 | 6.35 |
Variance | 3.79 | 8.12 | 26.82 | 0.02 | 745.60 | 37.22 | 7.09 | 0.25 | 20.01 | 40.32 |
Kurtosis | (0.43) | (0.09) | (0.24) | 0.33 | (0.74) | (0.58) | (1.05) | (0.38) | (1.25) | (0.74) |
Skewness | 0.14 | 0.31 | (0.12) | (0.85) | 0.59 | 0.38 | 0.35 | (0.14) | 0.06 | 0.59 |
Minimum | 7.00 | 31.32 | 37.38 | 2.20 | 25.10 | 5.13 | 0.08 | 3.00 | 0.10 | 28 |
Maximum | 15.24 | 44.80 | 59.60 | 2.79 | 120.00 | 32.00 | 9.50 | 5.10 | 14.25 | 50 |
Samples number | 64.00 | 64.00 | 64.00 | 64.00 | 64.00 | 64.00 | 64.00 | 64.00 | 64.00 | 64.00 |
Regression Equation | %R2 | DW | RMSE | VAF% | Equation No. |
---|---|---|---|---|---|
UCS = −0.02 + 19.06 SN | 89.75 | 1.50 | 6.25 | 88.95 | (25) |
UCS = 100.88 − 5.642 n | 85.43 | 1.57 | 6.95 | 84.69 | (26) |
UCS = −148.8 + 50.69 PW | 84.80 | 1.5 | 8.89 | 84.01 | (27) |
UCS = 100.20 − 8.976 WA | 76.62 | 1.52 | 10.56 | 75.02 | (28) |
UCS = −330.9 + 152.9 D | 55.80 | 1.5 | 18.96 | 54.69 | (29) |
UCS = 291.8 − 4.685 Fr | 78.97 | 1.89 | 9.2 | 78.32 | (30) |
UCS = −255.6 + 8.397 Fl | 76.78 | 1.90 | 10.11 | 75.39 | (31) |
UCS = −76.50 + 12.592 Q | 80.54 | 2.10 | 8.12 | 80.12 | (32) |
EM = −10.59 + 2.422 Q | 59.71 | 1.51 | 16.03 | 58.62 | (33) |
EM = 58.30 − 0.861 Fr | 62.33 | 1.29 | 14.39 | 62.30 | (34) |
EM = −47.90 + 1.691 Fl | 53.38 | 1.34 | 26.35 | 52.6 | (35) |
EM = 4.75 + 2.202 SN | 59.96 | 1.5 | 15.90 | 58.95 | (36) |
EM = 23.518 − 1.083 n | 63.02 | 1.35 | 13.02 | 62.85 | (37) |
EM = −22.99 + 9.39 PW | 58.31 | 1.60 | 17.62 | 57.39 | (38) |
EM = 23.643 − 1.786 WA | 60.75 | 1.51 | 14.36 | 59.86 | (39) |
EM = −68.5 + 32.91 D | 51.77 | 1.52 | 28.36 | 50.29 | (40) |
Class of Inputs | Equation | R2% | DW | Equation No. |
---|---|---|---|---|
Petrography, physical and mechanical | UCS = 25.7 + 1.58 Q − 0.44 Fl − 1.18 Fr + 11.90 D + 0.41 WA + 10.19 PW − 0.92n + 8.9 SN | 93.18 | 1.59 | (41) |
EM = −75.6 + 0.81 Q + 1.07 Fl + 0.41 Fr + 9.16 D−0.29 WA + 0.57 PW − 0.22 n − 7.11 SN | 72.21 | 1.34 | (42) | |
Petrography and physical | UCS = 7.0 + 3.76 Q + 0.24 Fl − 1.04 Fr + 27.9 D − 0.22 WA − 2.28 n | 90.44 | 1.65 | (43) |
EM = −73.9 + 9.07 D − 0.31 WA − 0.24 n + 0.84 Q + 1.06 Fl + 0.42 Fr | 72.19 | 1.63 | (44) | |
Petrography and mechanical | UCS = 29.3 + 13.31 PW + 5.61 SN + 1.62 Q − 0.202 Fl − 1.261 Fr | 93.77 | 1.58 | (45) |
EM = −71.3 + 2.65 PW + 0.397 SN + 0.745 Q + 1.257 Fl + 0.37 Fr | 68.65 | 1.52 | (46) | |
Mechanical and physical | EM = −11.0 + 1.17 PW + 0.408 SN + 9.41 D − 0.318 WA − 0.41 n | 67.51 | 1.50 | (47) |
UCS = 5.9 + 9.73 PW + 6.37 SN − 1.5 D − 0.216 WA − 1.833 n | 92.79 | 1.63 | (48) | |
Petrography | EM = −72.4 + 1.382 Q + 1.469 Fl + 0.359 Fr | 65.93 | 1.65 | (49) |
UCS = −5.7 + 6.59 Q + 1.85 Fl − 1.522 Fr | 84.65 | 2.2 | (50) | |
Physical | UCS = 59.4 + 15.4 D − 1.39 WA − 4.535 n | 86.20 | 1.52 | (51) |
EM = −5.4 + 10.65 D − 0.414 WA − 0.617 n | 66.72 | 1.54 | (52) | |
Mechanical | UCS = −53.3 + 17.37 PW + 8.35 SN | 91.24 | 1.56 | (53) |
EM = −7.74 + 4.07PW + 1.334 SN | 61.06 | 1.53 | (54) |
UCS | EM | |
---|---|---|
Train data | 75% of whole data | 75% of whole data |
Test data | 25% of whole data | 25% of whole data |
Epsilon | 0.0022 | 0.0016 |
C | 35 | 26 |
Gamma | 0.90 | 0.40 |
Optimum BPNN | Activation Functions | Training Functions | R% (for Test Data) | RMSE (for Test Data) | ||
---|---|---|---|---|---|---|
UCS | EM | UCS | EM | |||
8*4*2 | {tansig, Purlin} | LM | 98.43 | 94.20 | 0.17 | 0.24 |
8*4*2 | {tansig, Purlin} | SCG | 97.25 | 93.19 | 0.18 | 0.26 |
8*5*2 | {tansig, Purlin} | BR | 97.01 | 93.00 | 0.19 | 0.28 |
Parameters | EM | UCS |
---|---|---|
Train data | 75% | 75% |
Test data | 25% | 25% |
FIS Generation approach | Genfis2 | Genfis2 |
Influence radius | 0.58 | 0.62 |
Number of epochs | 1500 | 1200 |
Error goal | 0 | 0 |
Type | Sugeno | Sugeno |
Rules | 7 | 7 |
Number of MFs | 7 | 7 |
Input MF type | GM | GM |
Output MF type | Linear | Linear |
Equation | R2 | Type of Equation | Equation No. |
---|---|---|---|
UCS = 0.43 Fl2 − 24.30 Fl + 367.77 | 0.76 | Polynomial | (55) |
UCS = 106.91 e−0.09n | 0.91 | Exponential | (56) |
UCS = 0.16 Fr2 − 20.50 Fr + 669.86 | 0.83 | Polynomial | (57) |
UCS = 106.43 e−0.15WA | 0.83 | Exponential | (58) |
UCS = 484.46 D2 − 2295.24 D + 2752.06 | 0.71 | Polynomial | (59) |
UCS = 1.25 Q2 − 15.43 Q + 75.59 | 0.82 | Polynomial | (60) |
UCS = 1.84 e 0.82PW | 0.89 | Exponential | (61) |
UCS = 0.03 SN2 − 4.60 SN + 250 | 0.91 | Polynomial | (62) |
EM = 0.06 FL2 − 3.22 Fl + 44.87 | 0.60 | Polynomial | (63) |
EM = 0.04 Fr2 − 4.61 Fr + 148.67 | 0.53 | Polynomial | (64) |
EM = 24.47 e−0.07n | 0.66 | Exponential | (65) |
EM = 24.82 e−0.12WA | 0.65 | Exponential | (66) |
EM = 1.10 e0.63PW | 0.62 | Exponential | (67) |
EM = 0.19 Q2 − 1.88 Q + 12.59 | 0.58 | Polynomial | (68) |
EM = 76.45 D2 − 353.40 D + 417.94 | 0.59 | Polynomial | (69) |
EM = 0.01 SN2 − 0.20 SN + 9.42 | 0.59 | Polynomial | (70) |
Developed Equations | R2 | RMSE | Condition | Equation No. |
---|---|---|---|---|
EM = 0.31 Fl1.2 − 6.71 Fl + 135.15 + 24.03Exp(−0.07n) + 24.92Exp(−0.12WA) + 1.08Exp(0.63PW) + 0.13 Fr1.57 − 6.31 Fr + 148 + 0.24 Q1.95 − 2.08 Q + 13 + 106.32 D1.83 − 414.82 D + 418 + 0.01 SN2.00 − 0.20 SN + 14.2 | 0.78 | 172 | For all inputs | (71) |
EM = 0.30 Fl1.47 − 5.54 Fl + 67.78 + 24.08Exp(−0.07n) + 24.69Exp(−0.12WA) + 1.13Exp(0.62PW) | 0.79 | 51 | For inputs with R2 > 60% | (72) |
Methods | R | MAPE% | RMSE | VAF% | ||||
---|---|---|---|---|---|---|---|---|
UCS | EM | UCS | EM | UCS | EM | UCS | EM | |
SVR | 0.996 | 0.971 | 13.64 | 6.75 | 0.051 | 0.11 | 98.87 | 93.87 |
ANFIS | 0.996 | 0.99 | 1.69 | 3.22 | 0.054 | 0.103 | 98.96 | 98.88 |
KNN | 0.98 | 0.84 | 6.06 | 17.58 | 0.11 | 0.25 | 95.89 | 70.22 |
PBNN | 0.98 | 0.92 | 5.48 | 5.69 | 0.17 | 0.25 | 95.96 | 84.00 |
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Fang, Z.; Qajar, J.; Safari, K.; Hosseini, S.; Khajehzadeh, M.; Nehdi, M.L. Application of Non-Destructive Test Results to Estimate Rock Mechanical Characteristics—A Case Study. Minerals 2023, 13, 472. https://doi.org/10.3390/min13040472
Fang Z, Qajar J, Safari K, Hosseini S, Khajehzadeh M, Nehdi ML. Application of Non-Destructive Test Results to Estimate Rock Mechanical Characteristics—A Case Study. Minerals. 2023; 13(4):472. https://doi.org/10.3390/min13040472
Chicago/Turabian StyleFang, Zhichun, Jafar Qajar, Kosar Safari, Saeedeh Hosseini, Mohammad Khajehzadeh, and Moncef L. Nehdi. 2023. "Application of Non-Destructive Test Results to Estimate Rock Mechanical Characteristics—A Case Study" Minerals 13, no. 4: 472. https://doi.org/10.3390/min13040472