# Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber

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## Abstract

**:**

## 1. Introduction

## 2. Existing Models and Mechanisms of ${}^{\mathbf{222}}$Rn Migration in the Emanation Method Theory

- Models of the formation of radon precursors based on mechanical concepts:
- Deformations that contribute to the squeezing of radon from the crystal lattice and an increase in the coefficient of emanation ${}^{222}$Rn from rocks into pore fluids [21];
- Ultrasonic vibrations, which promote the release of radon from the crystal lattice [23];
- Variations in vertical gas flow velocity due to changes in rock fracturing and porosity under the action of tectonic stresses [24];
- An increase in radon concentration due to its desorption from the surface of the pore space under the influence of elastic vibrations that occur at the last stage of the preparation of strong earthquakes [25].

- Model of a hydrothermal system as a resonator with a natural frequency of fluctuations in gas concentration [26].
- Physico-chemical model of periodic fractionation of impurity gases in gas reservoirs in the zone of phase separation of a hydrothermal solution [27].
- Geogas model. According to the concept of ${}^{222}$Rn migration in soil with complete moisture saturation, the flow of gases in the form of microbubbles is the main mechanism for transporting ${}^{222}$Rn to the day surface [28,29,30,31]. In this case, the mechanism of migration of endogenous gases is determined by the interaction of water in pores and cracks with the rock. It is assumed that the carrier gases (${H}_{2}$, $He$, $C{O}_{2}$, and $C{H}_{4}$) are in several states (flow in the gas phase, displacement of water by gas, gas plugs, and bubbles) and provide the main process of migration of heavier inert gases (Rn, Tn).

**Remark**

**1.**

- Diffusion by concentration gradient ${}^{222}$Rn;
- Effusion due to pressure gradient in the Earth’s crust;
- Heat-liquid convection due to the lifting force induced by the geothermal gradient;
- Gas lifting force in a porous medium when the pores are filled with water;
- Change in pore pressure under the influence of changing stresses in the rock mass;
- Turbulent effects in soil air when meteorological factors change.

**Remark**

**2.**

**Remark**

**3.**

- Zones of decompaction acting as conductive collectors for subsoil gases from great depths;
- The presence of vertical and horizontal inhomogeneities of the upper layer of soil;
- Groundwater level.

## 3. Hereditary $\mathbf{\alpha}$-Model RVA

**Definition**

**1.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

## 4. Hereditary $\mathit{\alpha}\left(\mathit{t}\right)$-Model RVA

**Remark**

**10.**

**Remark**

**11.**

**Remark**

**12.**

**Definition**

**5.**

**Remark**

**14.**

## 5. Subsoil Gas Monitoring Stations and Their Equipment

## 6. Simulation Results

**Example**

**1.**

**Remark**

**15.**

**Example**

**2.**

**Remark**

**16.**

**Example**

**3.**

**Definition**

**6.**

**Example**

**4.**

**Definition**

**7.**

## 7. Used Software

- Greater universality, as when working only with experimental data and modeling on them;
- Convenience of adding new variations of the model and experimental data for subsequent software processing;
- By orders of magnitude of increased speed, since it became possible to completely get away from symbolic calculations.

## 8. Discussion

## 9. Conclusions

## 10. Patents

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RVA | Radon Volumetric Activity |

Rn | Radon |

AER | Air Exchange Rate |

IFDS | Implicit Finite-difference Method |

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**Figure 1.**Block diagram (

**a**) and general view (

**b**) of the subsoil radon registration kit: (

**1**) perforated pipe in a borehole; (

**2**) compressor; (

**3**) accumulation chamber with the MR107 device.

**Figure 2.**Layout of sensors for recording the concentration of subsoil gases at the PRTR reference point. The numbers in the circles number the gas discharge meters: (

**1**) pressure sensor; (

**2**) temperature sensor; (

**3**) gas-discharge counters $\beta $-radiation; (

**4**) sensor $\gamma $ radiation; (

**5**) molecular hydrogen sensor; (

**6**) carbon dioxide sensor.

**Figure 8.**Data on the accumulations of radon in the chamber on YSSR (green circle) and model curves with different parameters.

**Figure 9.**Data on radon accumulation in the chamber on GLLR (green points) and model data with different model parameters.

**Figure 11.**Extracted from the data at the MRZR point Figure 6 burst RVA, lasting 22.5 h.

**Figure 12.**(a) RVA burst data from MRZR in 10 min increments; (b) simulation results for (12) with determination coefficient (${R}^{2}=0.91$) and Pearson correlation coefficient ($Corr=0.96$).

**Figure 13.**Dependence in time: (

**a**) on the value of the fractional derivative; (

**b**) model curve (12) for Example 4.

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**MDPI and ACS Style**

Tverdyi, D.; Makarov, E.; Parovik, R.
Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber. *Mathematics* **2023**, *11*, 850.
https://doi.org/10.3390/math11040850

**AMA Style**

Tverdyi D, Makarov E, Parovik R.
Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber. *Mathematics*. 2023; 11(4):850.
https://doi.org/10.3390/math11040850

**Chicago/Turabian Style**

Tverdyi, Dmitrii, Evgeny Makarov, and Roman Parovik.
2023. "Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber" *Mathematics* 11, no. 4: 850.
https://doi.org/10.3390/math11040850