Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 86-104. ISSN 2079-6641

MATHEMATICAL MODELING 

https://doi.org/10.26117/2079-6641-2023-44-3-86-104

Research Article

Full text in Russian

MSC 34A08, 34A34

Contents of this issue

Read Russian Version

Research of Stress-Strain State of Geo-Environment by Emanation Methods on the Example of \alpha\left(t\right)-Model of Radon Transport

D. A. Tverdyi¹²^\ast, E. O. Makarov²³, R. I. Parovik¹²

¹Institute for Cosmophysical Research and Radio Propagation FEB RAS, 684034, p. Paratunka, Mirnaya st., 7, Russia
²Vitus Bering Kamchatka State University 683032, Petropavlovsk-Kamchatsky, st. Pogranichnaya, 4, Russia
³Kamchatka branch Unified Geophysical Service of the RAS, 683006, Kamchatka Petropavlovsk-Kamchatsky, Piipa boulevard, 9, Russia

Abstract. Continuous monitoring of variations in the volumetric activity of radon in order to search for its anomalous values preceding seismic events is one of the effective techniques for studying the stress-strain state of the geosphere. We propose a Cauchy problem describing the radon transport taking into account its accumulation in the chamber and the presence of the memory effect of the geo-environment. The model equation is a nonlinear differential equation with non-constant coefficients with a derivative in the sense of Gerasimov-Kaputo of fractional variable order. In the course of mathematical modeling, in MATLAB environment, of radon transport by the ereditary \alpha\left(t\right)-model a good agreement with experimental data was obtained. This indicates that the ereditary \alpha\left(t\right)-model of radon transport is more flexible, which allows it to describe various anomalous variations in the values of volumetric activity of radon due to the stress-strain state of the geosphere. It is shown that the order of the fractional derivative can be responsible for the intensity of the radon transfer process associated with the characteristics of the geo-environment. It is shown that due to the order of the fractional derivative, as well as quadratic nonlinearity in the model equation, the results of numerical modeling give a better approximation of the experimental data of radon monitoring than by classical models.

Key words: mathematical modeling, nonlinear equations, saturation effect, fractional equations, fractional derivatives, hereditarity, memory effects, nonlocality in time, radon volumetric activity, stress-strain state, geo-environment, earthquake precursors

Received: 19.10.2023; Revised: 26.10.2023; Accepted: 28.10.2023; First online: 02.11.2023

For citation. Tverdyi D. A., Makarov E. O., Parovik R. I. Research of stress-strain state of geo-environment by emanation methods on the example of \alpha\left(t\right)-model of radon transport. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 44: 3, 86-104. EDN: AOBZGA. https://doi.org/10.26117/2079-6641-2023-44-3-86-104.

Funding. The research was carried out within the framework of the grant of the President of the Russian Federation
MD-758.2022.1.1 on the topic “Development of mathematical models of fractional dynamics in order to study oscillatory processes and processes with saturation”

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing
the final version of the article in print. The final version of the manuscript was approved by all authors.

^\astCorrespondence: E-mail: tverdyi@ikir.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Tverdyi D. A., Makarov E. O., Parovik R. I., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

References

1. Rudakov V. P. Emanacionnyj monitoring geosred i processov [Emanational monitoring of  geoenvironments and processes]. Moscow: Science World, 2009, 175 pp.  (In Russian).
2. Neri M., Giammanco S., Ferrera E., Patane G., Zanon V. Spatial distribution of soil radon as a tool to recognize active faulting on an active volcano: The example of Mt. Etna (Italy), Journal of environmental radioactivity, 2011, vol. 102(9), pp. 863–870. DOI:10.1016/j.jenvrad.2011.05.002.
3. Barberio, M. D., Gori, F., Barbieri, M., Billi, A., Devoti, R., Doglioni, C., Petitta, M., Riguzzi, F., Rusi, S. Diurnal and Semidiurnal Cyclicity of Radon (222Rn) in Groundwater, Giardino Spring, Central Apennines, Italy. Water, 2018, vol. 10(9), no. 1276. DOI:10.3390/w10091276.
4. Imme G., Morelli D. Radon as earthquake precursor, In book: Earthquake Research and Analysis – Statistical Studies, Observations and Planning, 2012, pp. 143–160. DOI: 10.5772/29917.
5. Hauksson E. Radon content of groundwater as an earthquake precursor: evaluation of worldwide data and physical basis, Journal of Geophysical Research: Solid Earth, 1981, vol. 86, no. B10, pp. 9397–9410. DOI: 10.1029/JB086iB10p09397.
6. Cicerone R. D., Ebel J. E., Beitton J. A systematic compilation of earthquake precursors, Tectonophysics, 2009, vol. 476, no. 3-4, pp. 371–396. DOI: 10.1016/j.tecto.2009.06.008.
7. Petraki E., Nikolopoulos D., Panagiotaras D., Cantzos D., Yannakopoulos P. et al. Radon222: A Potential Short-Term Earthquake Precursor, Earth Science & Climatic Change, 2015, vol. 6, no. 6. DOI: 10.4172/2157-7617.1000282.
8. Parovik R. I. Matematicheskoe modelirovanie neklassicheskoj teorii emanacionnogo metoda [Mathematical modeling of the non-classical theory of the emanation method]. Petropavlovsk-Kamchatsky, Vitus Bering Kamchatka State University, 2014, 80 pp. (In Russian).
9. Ponamarev A. S. Frakcionirovanie v gidroterme kak potencial’naya vozmozhnost’ formirovaniya predvestnikov zemletryasenij [Fractionation in hydrothermal fluid as a potential opportunity for the formation of earthquake precursors]. Geohimiya [Geochemistry], 1989, no. 5, pp. 714–724 (In Russian).
10. Barsukov V. L., Varshal G. M., Garanin A. V., Zamokina N. S. Znachenie gidrogeohimicheskih metodov dlya kratkosrochnogo prognoza zemletryasenij [Significance of hydrogeochemical methods for short-term earthquake prediction], In book: Gidrogeohimicheskie predvestniki zemletryasenij [Hydrogeochemical precursors of
    earthquakes], 1985, Moscow: Science, pp. 3–16.
11. Varhegyi A., Baranyi I., Somogyi G. A. Model for the vertical subsurface radon transport in «geogas» microbubbles, Geophys. Transactions, 1986, vol. 32, no. 3, pp. 235–253.
12. King C. Y. Gas-geochemical approaches to earthquake prediction, In: Isotopic geochemical precursors of earthquakes and volcanic eruption, Proceedings of an Advisory Group Meeting held in Vienna (Vienna, September 9–12), Vienna, International atomic energy agency, 1991, pp. 22–36.
13. Dubinchuk V. T. Radon as a precursor of earthquakes, In: Isotopic geochemical precursors of earthquakes and volcanic eruption, Proceedings of an Advisory Group Meeting held in Vienna (Vienna, September 9–12), Vienna, International atomic energy agency, 1991, pp. 9–22.
14. Novikov G. F. Radiometricheskaya razvedka [Radiometric intelligence]. Leningrad: Science, 1989, 407 pp. (In Russian).
15. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier Science Limited, 2006, 523 pp.
16. Nahushev A. M. Drobnoe ischislenie i ego primenenie [Fractional calculus and its application]. Moscow, Fizmatlit, 2003, 272 pp. (In Russian).
17. Uchaikin V. V. Fractional Derivatives for Physicists and Engineers. Vol. I. Background and Theory. Berlin, Springer, 2013, 373 pp. DOI: 10.1007/978-3-642-33911-0.
18. Tverdyi D. A., Parovik R. I., Makarov E. O., Firstov P. P. Research of the process of radon accumulation in the accumulating chamber taking into account the nonlinearity of its entrance, E3S Web Conference, 2020, vol. 196, no. 02027, pp. 1–6. DOI:10.1051/e3sconf/2020196020278.
19. Tverdyi D. A., Parovik R. I. Application of the Fractional Riccati Equation for Mathematical Modeling of Dynamic Processes with Saturation and Memory Effect, Fractal and Fractional, 2022, vol. 6(3), no. 163, pp. 1–35. DOI: 10.3390/fractalfract6030163.
20. Makarov E. O. Firstov P. P., Voloshin V. N. Instrumental complex for registration concentration of subsurface gas to find precursory anomalies strong earthquake of Southern Kamchatka, Seismic instruments, 2012, vol. 48, no. 2, pp. 5–14.
21. Firstov P. P., Makarov E. O. Dinamika podpochvennogo radona na Kamchatke i sil’nye zemletryaseniya [Dynamics of subsoil radon in Kamchatka and strong earthquakes]. Petropavlovsk-Kamchatsky, Vitus Bering Kamchatka State University, 2018, 148 pp. (In Russian).
22. Firstov P. P., Makarov E. O., Gluhova I. P., Budilov D. I., Isakevich D. V. Poisk predvestnikovyh anomalij sil’nyh zemletryasenij po dannym monitoringa podpochvennyh gazov na Petropavlovsk-Kamchatskom geodinamicheskom poligone [Search for predictive anomalies of strong earthquakes according to monitoring of subsoil gases at PetropavlovskKamchatsky geodynamic test site]. Geosystems of Transition Zones, 2018, vol. 2, no. 1, pp. 16–32. DOI: 10.30730/2541-8912.2018.2.1.016-032,(In Russian).
23. Firstov P. P., Rudakov V. P. Rezul’taty registracii podpochvennogo radona v 1997–2000 gg. na Petropavlovsk-Kamchatskom geodinamicheskom poligone [Results of registration of subsoil radon in 1997–2000 at the Petropavlovsk-Kamchatsky geodynamic test site]. Vulkanologiya i sejsmologiya [Volcanology and seismology], 2003, no. 1, pp. 26–41 (In Russian).
24. Vasilyev A. V., Zhukovsky M. V. Determination of mechanisms and parameters which affect radon entry into a room, Journal of Environmental Radioactivity, 2013, vol. 124, pp. 185–190. DOI: 10.1016/j.jenvrad.2013.04.014.
25. Parovik R. I., Shevtsov B. M. Radon transfer processes in fractional structure medium, Mathematical Models and Computer Simulation, 2010, vol. 2, no. 2, pp. 180–185. DOI: 10.1134/S2070048210020055.
26. Pskhu A. V. Uravneniya v chastnyh proizvodnyh drobnogo poryadka [Fractional Partial Differential Equations]. Moscow, Science, 2005, 199 pp. (In Russian).
27. Parovik R. I., Mathematical modeling of radon sub diffusion into the cylindrical layer in ground, Life Science Journal, 2015, vol. 11, no. 9, pp. 281–283.
28. Volterra V. Functional theory, integral and integro-differential equations. Moscow, Science, 1982.(In Russian).
29. Gerasimov A. N. Generalization of linear deformation laws and their application to internal friction problems, Applied Mathematics and Mechanics, 1948, vol. 12, pp. 529–539. (In Russian).
30. Caputo M. Linear models of dissipation whose Q is almost frequency independent – II, Geophysical Journal International, 1946, vol. 13, no. 5, pp. 529–539. DOI: 10.1111/j.1365-246X.1967.tb02303.x3.
31. Rekhviashvili S. S., Pskhu A. V. Drobnyj oscillyator s eksponencial’no-stepennoj funkciej pamyati [Fractional oscillator with exponential-power memory function], Pis’ma v ZHTF [Letters to ZhTF], 2022, vol. 48, no. 7. DOI: 10.21883/PJTF.2022.07.52290.19137, (In Russian).
32. Patnaik S., Hollkamp J. P., Semperlotti F. Applications of variable-order fractional operators: a review, Proceedings of the Royal Society A, 2020, vol. 476, no. 2234, pp. 20190498. DOI: 10.1098/rspa.2019.0498.
33. Coimbra C. F. M. Mechanics with variable-order differential operators, Annalen der Physik, 2003, vol. 12, no. 11-12, pp. 692–703. DOI: 10.1002/andp.200310032.
34. Ortigueira M. D., Valerio D., Machado J. T. Variable order fractional systems, Communications in Nonlinear Science and Numerical Simulation, 2019, vol. 71, pp. 231–243. DOI: 10.1016/j.cnsns.2018.12.003.
35. Tverdyi D. A., Parovik R. I. Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation, Fractal and Fractional, 2022, vol. 6(1), no. 23, pp. 1–27. DOI: 10.3390/fractalfract6010023.
36. Tvyordyj D. A. Hereditary Riccati equation with fractional derivative of variable order, Journal of Mathematical Sciences, 2021, vol. 253, no. 4, pp. 564–572. DOI: 10.1007/s10958-021-05254-0.
37. Rzkadkowski G., Sobczak L. A generalized logistic function and its applications, Foundations of Management, 2020, vol. 12, no. 1, pp. 85–92. DOI: 10.2478/fman-2020-0007
38. Johnston F. R., Boyland J. E., Meadows M., Shale E. Some properties of a simple moving average when applied to forecasting a time series, Journal of the Operational Research Society, 1999, vol. 50, no. 12, pp. 1267–1271. DOI: 10.1057/palgrave.jors.2600823

---

Information about authors

---

---

---