Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 2. P. 68 – 105. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2026-54-2-68-105
Review Article
Full text in English
MSC 76R50, 86A60, 35R11

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Dedicated to the 85th anniversary of Professor P.P. Firstov

Mathematical Modeling of Radon Transport in Geological Media: Classical and Non-Classical Approaches

R. I. Parovik¹²^{\ast}

¹Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7 Mirnaya str., Paratunka, 684034, Russia
²Institute of Applied Mathematics and Automation KBSC RAS, 89a Shortanov str., Nalchik, 360000, Russia

Abstract. The article provides a review of mathematical models describing the diffusion-convective transport of radon (²²²Rn) in porous geological media within the framework of the emanation method. The core of these mathematical models is the linear non-stationary diffusion-convection equation for radon, used to describe its transport in homogeneous porous soil or in a “soil–atmosphere”system under various initial and boundary conditions. Due to their linearity, the mathematical models are studied using the Laplace integral transform, which makes it possible to find an analytical form of the solution to the problem. The calculated curves obtained from the analytical solution allow one to establish the patterns of radon transport and to model it depending on various physical and geological parameters. The obtained solutions can be applied to practical problems. In particular, the article describes an algorithm for estimating the radon flux density from the Earth’s surface, which has an increased sensitivity to deformation disturbances in the geological medium. This makes it possible to study, for example, the seismic activity of Southern Kamchatka. Further, generalizations of previously developed mathematical models are discussed; these are based on the concept of a fractal structure of the geomedium and the mathematical apparatus of fractional differentiation. In the stationary case, models of superdiffusion (1 < \alpha < 2) and subadvection (0 < \alpha < 1) are constructed, and their ability to describe anomalously fast radon migration towards the Earth’s surface is demonstrated. A model with a variable order of the fractional derivative is considered, allowing the dynamics of radon volumetric activity to be correlated with seismic activity, as exemplified by Southern Kamchatka. Here, the key role is played by the order of the fractional derivative, which is related to the permeability (fractality) of the geomedium. The results obtained can be used in monitoring geodynamic processes, mineral exploration, or in solving environmental problems. It should be noted that the results described in the article were obtained jointly with Professor P.P. Firstov (06.04.1941–20.07.2021), a leading geophysicist who organized a network of observation sites for radon monitoring in Kamchatka. In 2026, P.P. Firstov would have turned 85. This article is dedicated to this significant date, and is also a tribute to the memory and respect for a colleague and an outstanding scientist whom P.P. Firstov was to the author.

Key words: radon, diffusion, convection,superdiffusion, seismic activity

Received: 30.07.2026; Revised: 12.08.2026; Accepted: 13.08.2026; First online: 14.09.2026

For citation. Parovik R. I. Mathematical modeling of radon transport in geological media: classical and non-classical approaches. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 2, 68-105. EDN: JJYGSB. https://doi.org/10.26117/2079-6641-2026-54-2-68-105.

Funding. The work was carried out within the framework of the state assignment of IКIR FEB RAS No. 124012300245-2.

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

Contribution and Responsibility. The author contributed to this article and is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author.

^{\ast}Correspondence: E-mail: parovik@ikir.ru

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

© Parovik R. I., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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Information about the author

Parovik Roman Ivanovich – D. Sci. (Phys.-Math.), Chief Researcher, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia; Leading Researcher, Institute of Applied Mathematics and Automation KBSC RAS, Nalchik, Russia, ORCID 0000-0002-1576-1860.

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