Computer simulation of sound fields in rock samples with inclusions of pseudofractal structure

This paper examines mathematical and computer models and methods for studying rock samples containing pseudofractal inclusions and the potential for using computer modeling to investigate their structure. Such structures are found, for example, in fractured rock massifs. These structures, in turn, can serve as hydrocarbon reservoirs, and the study and investigation of their properties makes this research relevant. The article proposes a method for computer modeling pseudofractal inclusions in rock samples in the form of parallelepipeds and ellipsoidal dimples, with anisotropic elastic parameters. A calculation method based on volumetric integral representations and integral field equations in the frequency domain is used to mathematically model sound wave propagation in a geological rock sample saturated with anisotropic fractal inclusions (microanisotropy). The article presents the results of computational experiments on a comprehensive study of fractured reservoirs, with the possibility of expanding the study area and modeling the propagation of various types of fields within it. The materials of the article are of practical value for the study of fractured rocks at great depths by applying the obtained algorithm for generating anisotropic pseudofractal inclusions and calculating seismic fields in their presence.

1. Nefedov Yu., Gribanov D., Gasimov E., et al. Development of Achimov Deposits Sedimentation Model of One of the West Siberian Oil and Gas Province Fields. Reliability: Theory & Applications. 2023;18(S5):441–448. https://doi.org/10.24412/1932-2321-2023-575-441-448

2. Dvoynikov M.V., Sidorkin D.I., Yurtaev S.L., Grokhotov E.I., Ulyanov D.S. Drilling of Deep and Ultra-Deep Wells for Prospecting and Exploration of New Raw Mineral Fields. Journal of Mining Institute. 2022;258:945–955. https://doi.org/10.31897/PMI.2022.55

3. Sinitsa N.V., Prishchepa O.M. A Conceptual Model for the Formation of Oil and Gas Accumulation Zone Within the Paleozoic Basement of the Southeastern West Siberian Basin. Actual Problems of Oil and Gas. 2023;(1):14–26. (In Russ.). https://doi.org/10.29222/ipng.2078-5712.2023-40.art2

4. Martynov A.V. Results of Geological Exploration Activity for Oil and Gas Within the Chernyshev Ridge and the Western Pre-Polar Ural Slope. Petroleum Geology. Theoretical and Applied Studies. 2020;15(1). (In Russ.). https://doi.org/10.17353/2070-5379/9_2020

5. Prishchepa O.M., Borovikov I.S., Grokhotov E.I. Oil and Gas Content of the Understudied Part in the Northwest of the Timan-Pechora Oil and Gas Province According to the Results of Basin Modeling. Journal of Mining Institute. 2021;247:66–81. https://doi.org/10.31897/PMI.2021.1.8

6. Bol'shakova N.V., Danil'ev S.M., Danil'eva N.A. Hydrocarbon Resources Potential and Development Prospects of the Behring Sea Shelf and the Pacific Ocean Joining the Eastern Kamchatka Onshore Area. Petroleum Geology. Theoretical and Applied Studies. 2020;15(4). (In Russ.). https://doi.org/10.17353/2070-5379/34_2020

7. Mardashov D., Duryagin V., Islamov Sh. Technology for Improving the Efficiency of Fractured Reservoir Development Using Gel-Forming Compositions. Energies. 2021;14(24). https://doi.org/10.3390/en14248254

8. Vladov M.L., Struchkov V.A., Sudakova M.S., Shmurak D.V. Limitations of Crosshole Tomography at Large Distances: Geometry Factors. In: Inzhenernaya seismorazvedka i seismologiya-2020. Georadar-2020. Teper' vmeste: Sbornik tezisov nauchno-prakticheskoi konferentsii, 16–22 October 2020, Moscow, Russia. Penza: Izdatel'skijj Dom "Akademija Estestvoznanija"; 2020. P. 124–128. (In Russ.).

9. Chugaev A.V., Kuznetsov A.I. Sravnenie optovolokonnoi sistemy registratsii seismoakusticheskikh signalov i gidrofonov pri mezhskvazhinnykh issledovaniyakh. Gornoe ekho. 2022;(3):42–49. (In Russ.). https://doi.org/10.7242/echo.2022.3.7

10. Shchekin A.I., Vasiliev V.A., Nikolaychenko A.S., Kolomiytsev A.V. Field Classification of Fractured Reservoirs of Crystalline Basement. Georesources. 2021;23(3):90–98. (In Russ.). https://doi.org/10.18599/grs.2021.3.12

11. Ilyinov M.D., Petrov D.N., Karmanskiy D.A., Selikhov A.A. Physical Simulation Aspects of Structural Changes in Rock Samples Under Thermobaric Conditions at Great Depths. Mining Science and Technology (Russia). 2023;8(4):290–302. https://doi.org/10.17073/2500-0632-2023-09-150

12. Chernyshov S.E., Popov S.N., Varushkin S.V., Melekhin A.A., Krivoshchekov S.N., Ren Sh. Scientific Justification of the Perforation Methods for Famennian Deposits in the Southeast of the Perm Region Based on Geomechanical Modelling. Journal of Mining Institute. 2022;257:732–743. https://doi.org/10.31897/PMI.2022.51

13. Kochnev A.A., Kozyrev N.D., Krivoshchekov S.N. Estimation of the Influence of Fracture Parameters Uncertainty on the Dynamics of Technological Development Indicators of the Tournaisian-Famennian Oil Reservoir in Sukharev Oil Field. Journal of Mining Institute. 2022;258:1026–1037. https://doi.org/10.31897/PMI.2022.102

14. Vakhitova G.R., Dyudbina A.A., Shaybekova G.F. Petrophysical Model for the Frasnian D3fr Stage with Hard-to-Recover Reserves in the Pre-Caspian Basin Sediments. Herald of the Academy of Sciences of the Republic of Bashkortostan. 2020;36(3):5–15. (In Russ.).

15. Kireev S.B., Litvinenko V.S., Telegin A.N. The Modern Technology of Seismic Prospecting with the Use of Reflection Method Applied to Oil and Gas Exploration. In: 6th EAGE Saint Petersburg International Conference and Exhibition: Volume 2014, 07–10 April 2014, Saint Petersburg, Russia. European Association of Geoscientists & Engineers; 2014. Р. 1–5. https://doi.org/10.3997/2214-4609.20140208

16. Yakovleva A.A., Movchan I.B., Medinskaia D.K., Sadykova Z.I. Quantitative Interpretations of Potential Fields: From Parametric to Geostructural Recalculations. Bulletin of the Tomsk Polytechnic University. Geo Assets Engineering. 2023;334(11):198–215. (In Russ.). https://doi.org/10.18799/24131830/2023/11/4152

17. Bayuk I.O., Dubinya N.V., Tikhotskiy S.A. Some Problems of Rock Physics Modeling of Fractured Carbonate Reservoir Rocks. PROneft. Professionals About Oil. 2019;(3):11–17. (In Russ.).

18. Uzyanbaev R.M., Poveshchenko Yu.A., Podryga V.O., et al. Ispol'zovanie parallel'nykh tekhnologii dlya raschetov flyuidodinamicheskikh protsessov v kollektore treshchinovato-porovogo tipa s uchetom neizotermichnosti. In: Parallel computational technologies (PCT'2023), 28–30 March 2023, Saint Petersburg, Russia. Chelyabinsk: Izdatel'skii tsentr YuUrGU; 2023. P. 246. (In Russ.).

19. Belozerov I.P., Gubaydullin M.G. Concept of Technology for Determining the Permeability and Porosity Properties of Terrigenous Reservoirs on a Digital Rock Sample Model. Journal of Mining Institute. 2020;244:402–407. https://doi.org/10.31897/PMI.2020.4.2

20. Normatov J.S. Fractals and Their Application. Ekonomika i sotsium. 2020;(3):427–432. (In Russ.).

21. Banerjee S., Easwaramoorthy D., Gowrisankar A. Fractal Functions, Dimensions and Signal Analysis. Cham: Springer; 2021. 132 р. https://doi.org/10.1007/978-3-030-62672-3

22. Zhao J., Li Sh., Wang Ch., You T., Liu X., Zhao Y. A Universal Soil-Water Characteristic Curve Model Based on the Particle Size Distribution and Fractal Theory. Journal of Hydrology. 2023;622. https://doi.org/10.1016/j.jhydrol.2023.129691

23. Ming F., Zhang M., Pei W., Chen L. A New Hydraulic Conductivity Model of Frozen Soil Considering the Hysteresis Effect Based on Fractal Theory. Geoderma. 2024;442. https://doi.org/10.1016/j.geoderma.2024.116790

24. Zhang P., Zhang Ya., Huang Y., Xia Y. Experimental Study of Fracture Evolution in Enhanced Geothermal Systems Based on Fractal Theory. Geothermics. 2022;102. https://doi.org/10.1016/j.geothermics.2022.102406

25. Sadeghi B. Fractals and Multifractals in the Geosciences. Elsevier; 2024. 302 p. https://doi.org/10.1016/C2020-0-03441-9

26. Masjukov V.V., Yurchenko O.S. Multi-Scaled Lineament Analysis for Fractured Geological Medium Fractal Properties Detection. In: Geomodel 2020, 07–11 September 2020, Gelendzhik, Russia. Moscow: European Association of Geoscientists & Engineers; 2020. P. 56. https://doi.org/10.3997/2214-4609.202050080

27. Ivanov S.N., Kushnarev P.I. Assessment of the Complexity of the Geological Structure of Gold Deposits from the Standpoint of Fractal Geometry. News of the Tula State University. Technical Sciences. 2021;(2):251–258. (In Russ.).

28. Shaikhutdinova L.R. Fraktal'naya struktura dendritov v arkozovom peschanike saldamskoi svity Mezhegeiskogo ugol'nogo mestorozhdeniya (Vostochnaya Sibir') kak dokazatel'stvo samoorganizatsii geologicheskoi sredy. In: Metody, metody i snova metody v litologii: Materialy 4-i Vserossiiskoi shkoly studentov, aspirantov, molodykh uchenykh i spetsialistov po litologii, 19–23 October 2020, Yekaterinburg, Russia. Yekaterinburg: Zavaritsky Institute of Geology and Geochemistry; 2020. P. 153–155. (In Russ.).

29. Suleimanov B.A., Ismailov F.S., Dyshin O.A., Huseinova N.I. Analysis of Oil Deposit Exploration State on the Base of Multifractal Approach. Oil Industry. 2011;(12):111–115. (In Russ.).

30. Sentemov A.A., Dorfman M.B. Percolation Approach in Reservoir Simulation of Well Treatment Methods. Bulletin of the Tomsk Polytechnic University. Geo Assets Engineering. 2022;333(7):157–165. (In Russ.). https://doi.org/10.18799/24131830/2022/7/3612

31. Voevudko A.E. Fractal Dimension of the Kronecker Product Based Fractals. arXiv. URL: https://doi.org/10.48550/arXiv.1803.02766 [Accessed 20th October 2025].

32. Aleksandrov P.N., Krizsky V.N. Direct and Inverse Problems of Seismic Exploration of Anisotropic and Dispersive Elastic Media on Volume Integral Equations. Mathematical Models and Computer Simulations. 2023;15(6):976–986. https://doi.org/10.1134/S2070048223060042

33. White J.E. Underground Sound. Application of Seismic Waves. Moscow: Nedra; 1986. 264 p. (In Russ.).

34. Kech V., Teodoresku P. Vvedenie v teoriyu obobshchennykh funktsii s prilozheniyami v tekhnike. Moscow: Mir; 1978. 518 p. (In Russ.).