Analysis of equivalent thickness of geological media for lab-scale study of radon exhalation

Geological media are omnipresent in nature. Lab-scale tests are frequently employed in radon exhalation measurements for these media. Thus, it is critical to find the thickness of the medium at an experimental scale that is equivalent to the medium thickness in a real geological system. Based on the diffusion–advection transport of radon, theoretical models of the surface radon exhalation rate for homogeneous semi-infinite and finite-thickness systems were derived (denoted as Jse and Jfi, respectively). Analysis of the equivalency of Jse and Jfi was subsequently carried out by introducing several dimensionless parameters, including the ratio of the exhalation rates for the semi-infinite and finite-thickness models, ε, and the number of diffusion lengths required to achieve a desired ε value, n. The results showed that when radon transport in geological media is dominantly driven by diffusion effect, if n > 3.6626, then ε > 95%; if n > 5.9790, then ε > 99.5%. When radon migration is dominantly driven by advection effect, if n > 2.5002, then ε > 95%; if n > 4.0152, then ε > 99.5%. Therefore, if the thickness of the geological media (x0) is greater than a certain n times the radon diffusion length of the media (L), the media can be modeled as semi-infinite. To validate the model, a pure radon diffusion experiment (no advection) was developed using uranium mill tailings, laterite, and radium-bearing rocklike material with different thicknesses (x0). The theoretical model was demonstrated to be reliable and valid. This study provides a basis for determining the appropriate thickness of geological media in lab-scale radon exhalation measurement experiments with open bottom.


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Rn (hereafter referred to as radon) is a major contributor to ionizing radiation, corresponding to about half of the total radiation exposure received by the human population. It is the second most frequent cause of lung cancer (3-14% of all lung cancer is attributable to radon) after smoking (Baskaran 2016). 226 Ra (hereafter referred to as radium) is the radon source which is always present in geological media, such as the ground (e.g., soil, sand, or bedrock), uranium ore stockpiles, and mill tailings piles, in various amounts (Girault et al. 2010). Such geological media are called radium-bearing porous media.
Radon transport in geological media includes two modes: diffusion and advection. The diffusive transport of radon is driven by a concentration gradient, described by Fick's law. The advective transport of radon is driven by pressure-induced flow or advection, described by Darcy's law (Fleischer 1997;Tanner 1980). To understand the radon diffusion-advection transport mechanism, extensive lab-scale studies of radon release from geological media have been carried out. The most widely studied subjects are the radon profiles in various media for geophysical purposes, indoor radon concentrations, radon exhalation from uranium ore stockpiles and uranium mill tailings (UMTs) piles, and the radioactive impacts of radon on the surrounding environment. For instance, van der Graaf et al. (1992van der Graaf et al. ( , 1994 and Van der Spoel et al. 1997; Van der Spoel 1998; Van der Spoel et al. 1998, 1999 studied the diffusive and advective radon transport in soil and sand using laboratory experiments with a stainless-steel vessel (height and diameter of 2 m). De Jong and Van Dijk (2005) tested radon-reducing measures by constructing a laboratory-scale house (dimensions of about 3.5 m × 2 m) with a living area (height 2.6 m) and a crawlspace beneath it (height 1 m) made from phosphogypsum blocks ( 222 Rn source strength of approximately 800 Bq·h −1 ), where the two spaces were separated by a concrete floor (thickness 15 cm). Catalano et al. (2015aCatalano et al. ( , 2015b carried out a laboratory study of radon transport through porous soil in a large cylindrical stainless-steel vessel (diameter 0.5 m, height 1.25 m). The height was on the order of the diffusion length of radon in dry soil (1-2 m). Wang et al. (2014) developed a columnar experimental device for exploring radon seepage migration and exhalation in uranium ore piles. The total height of this device was 2.76 m, and it was mainly composed of five identical cylinders filled with sample material, each with a height of 0.5 m and an internal diameter 0.1 m. Ye et al. (2016) analyzed the effects of the water level on the radon exhalation from fragmented uranium ore using a self-made apparatus with an ore loading barrel (diameter of 0.1 m, height of 0.5 m) and a water loading barrel (height of 0.70 m). Ferry et al. (2002) conducted an experimental study of the effect of a cover layer on the transient radon flux from UMTs. This study was conducted using a small artificial pond with dimensions of 14 m × 15 m and total height of 2.1 m (namely, a draining layer of 0.3 m, UMTs layer of 0.8 m, and cover layer of 1 m).
In addition to experimental studies, various theoretical models have been proposed for the study of radon transport.
The key difference between these approaches is that geological media is modeled using semi-infinite models, whereas uranium ore stockpiles and mill tailings piles are modeled using finite-thickness models with thicknesses of several meters or even greater than 20 m. Jönsson (1997) presented a model for radon transport in a homogeneous, semi-infinite medium. Yakovleva (2005) extended this model to examine the transport and equilibrium concentration of radon as well as its flux from the Earth's surface. Catalano et al. (2015a) presented a numerical model to examine the radon transport through porous media and compared the model with experimental results. Yakovleva and Parovik (2010) solved the diffusion-advection equations to examine radon transport in geological media with multiple layers using an integrointerpolation approach. Hafez and Awad (2016) used a numerical model to examine the transport of radon from deep soil to the Earth's surface. Several authors have examined the diffusion of radon through building materials using numerical models (Sabbarese et al. 2020;Szajerski and Zimny 2020). Muhammad et al. (2020) proposed a hybrid regression model and Monte Carlo simulation to describe seasonal surface radon variation and its depth profile in soil; if well fitted, the model can also be used to estimate parameters like emanation coefficient, diffusion rate, soil velocity, etc. With respect to earthquake occurrence, Külahcı and Şen (2014) proposed the joint application of absolute point cumulative semivariogram (APCSV) and perturbation method (PM) to depict spatial behaviors and statistical uncertainties in 13,000 222 Rn measurements in Turkey's Keban Reservoir along East Anatolian Fault System (EAFS).
The abovementioned radon studies of geological media based on lab-scale experiments have achieved significant conclusions. The equipment adopted for these studies was relatively well-designed and well-constructed, and the experimental parameters could be set and strictly controlled in the laboratory. However, the bases for selecting the dimensions (especially the height or thickness) of the sample materials were vaguely described or not specified. In general, due to the vertical migration of radon in geological media, the influence of a lab-scale sample's thickness on the comparability and reliability of the experimental results cannot be ignored. In order that the experiment is meaningful for real-world conditions, the sample material with a certain thickness at the lab-scale should be representative of a semi-infinite or finite-thickness medium in an actual environment in terms of radon exhalation. Although numerous models have been presented, a theoretical analysis of the sample thickness at the lab scale and the full-scale equivalent thickness has not been performed. Thus, analysis of the equivalent thickness of geological media for the lab-scale study of radon transport is presented in this paper.
In the "Derivation of theoretical formulas" section, the theoretical equations for the advection-diffusion transport of radon through semi-infinite and finite-thickness models are presented, and the conditions under which the surface fluxes of radon predicted by the models are equivalent are derived. The experiments used to validate the models are presented in the "Experiments" section. The results are presented in the "Results and discussion" section, and the paper is concluded in the "Conclusions" section.

Derivation of theoretical formulas
For simplicity, geological media are typically modeled using with two types of homogeneous porous media models: (1) homogeneous semi-infinite models and (2) homogeneous finite-thickness models. In a semi-infinite model, the medium has only a single boundary surface, and the opposite boundary is considered to be infinitely far away. In a finite-thickness model, the medium has two boundary surfaces separated by a finite distance. For a homogeneous porous medium, the values of the density, porosity, tortuosity, and radium content are constant. Therefore, the emanation coefficient, diffusion coefficient, and permeability coefficient of radon are determinable if the temperature and moisture content of the medium are known. In this case, the free radon production rate of geological media can be expressed as follows (IAEA 2013): where A is the free radon production rate (Bq·m −3 ·s −1 ), λ is the radon decay constant (s −1 ), C Ra is the radium content (Bq· kg −1 ), f is the radon emanation coefficient, and ρ is the density of the medium (kg·m −3 ). The density and other parameters (namely the radon concentration, radon diffusion coefficient, and radon seepage velocity) used in the subsequent analysis are the bulk values.
It is assumed that the radium, radon and its progeny reach a state of radioactive equilibrium. The flux of radon is assumed to be steady and one-dimensional exhalation from inside the geological medium to the exposed surface. The radon concentration in the medium is a time-invariant function of the thickness from the exposed surface, and radon advection occurs with a stable seepage velocity and direction. In this study, we consider the case in which the direction of radon advection is opposite to the apparent direction of point upwards driven from radon diffusion. In a Cartesian coordinate system ( Fig.  1), the radon flux from the surface of the homogeneous semiinfinite or finite-thickness model can be treated as a onedimensional migration process. Based on the theory of the radon diffusion-advection transport mechanism, the radon migration process for the semi-infinite and finite-thickness models can be described as follows (Antonopoulos-Domis et al. 2009;Rogers and Nielson 1991;Tanner 1964): where x is the depth from exposed surface to inside the medium (m), C is radon concentration at the depth x (Bq· m −3 ), D is radon diffusion coefficient in the medium (m 2 · s − 1 ), and v is the radon seepage velocity (m·s − 1 ). Substituting Eq. (1) into Eq. (2) yields the following: For the convenience of deriving theoretical formulas of the radon exhalation rate, it is assumed that the studied geological medium is placed in open environment with a very low background radon concentration. Moreover, in a state of radioactive equilibrium, the pore radon concentrations inside the geological medium are much higher than the radon concentration of the surrounding environment close to the exposed surface of the medium. Therefore, the environmental radon concentration is theoretically set to zero (see Appendix A).

Homogeneous semi-infinite model
The radon migration and exhalation processes from a homogeneous semi-infinite medium are depicted in Fig. 1. The upper surface is the only free surface for radon exhalation. The boundary conditions are as follows: x = 0, C = 0; x→∞, C is bounded (see Appendix A). Zhang et al. (2010) have given the solution of Eq. (3) as follows: The surface radon exhalation rate (at x = 0, J se , Bq·m −2 ·s −1 ) can be expressed as follows: Substitution of Eq. (4) into Eq. (5) yields the following: Homogeneous finite-thickness model The radon migration and exhalation processes from a homogeneous finite-thickness medium are depicted in Fig. 2. Radon exhalation occurs at two free surfaces, namely the upper and lower surfaces. Therefore, the boundary conditions are as follows: x = 0, C = 0; x = x 0 , C = 0 (see Appendix A). Zhang et al. (2010) have given the solution of Eq.

Derivation of equivalency conditions
In this section, the conditions for which the surface exhalation of radon from the semi-infinite and finite-thickness models are equivalent are derived. The surface exhalation expressions given by Eq. (6) for the semi-infinite model and Eq. (8) for the finite-thickness model are highly similar, differing only by the coefficient of the term. In both models, the direction of radon advection is opposite to the apparent direction of point upwards driven from radon diffusion. We introduce two dimensionless parameters, Ψ and ζ, which are defined as follows: As ζ 1 and ζ 2 are greater than zero, and ζ 1 ≥ ζ 2 , we can conclude the following: (1) exp (ζ 1 x 0 ) > 1, and 0 < exp (−ζ 2 x 0 ) < 1; (2) exp (ζ 1 x 0 ) + exp (−ζ 2 x 0 ) -2 ≥ exp (ζ 1 x 0 ) + exp (−ζ 1 x 0 ) -2 ≥ 0. Thus, 0 < Ψ < 1. In addition, ζ is not less than 1. We define a dimensionless parameter (m) as the ratio of the radon seepage velocity (v, m·s −1 ) to the radon diffusion velocity (v diff , m·s −1 ) in the geological medium (see Appendix B). Based on Eq. (10), m can be expressed as follows: Using Eq. (11), we can determine the following criteria: Furthermore, where L is the radon diffusion length in the geological medium (m).
We assume that where n is a real number greater than zero. Specifically, it is the number of radon diffusion lengths in the geological medium. By substituting Eqs. (10), (13), and (14) into Eq. (9), we arrive at the following: Finally, to facilitate the equivalency analysis, ε is defined as the ratio of J fi to J se . Based on the equations above, we determined that and 0 < ε < 1.

Experiments
An experimental setup was developed, as shown in Fig. 3. The radon exhalation rates for various media with different thicknesses were measured. In particular, the radon exhalation rates were measured for UMTs (height 0.1−2.0 m), laterite (height 0.1 −2.5 m), and radium-bearing rocklike material (height 0.1−1.0 m) (Hong et al. 2018). The stainless-steel vessel of the experimental setup had a diameter of 0.2 m, and the height was adjustable as required by a flange connection. The top cover of the vessel was a radon-collecting hood, and the bottom was unrestricted. The closed chamber method is utilized, and RAD 7 is adopted to monitor cumulative radon concentrations in the radon-collecting hood (Zhang et al. 2012). The monitoring time is set as 1 h (5 min for a sampling cycle) for avoiding the considerably growing radon concentrations in the top cover, and the radon concentrations are quite less than those in pore space of the media. The sample of the geological medium in the vessel was separated from the experimental table by gauze screen (100 mesh size). The parameters of the abovementioned geological media are summarized in Table 1.

Results and discussion
To intuitively show the dependence of Ψ on ζ and n from the "Derivation of theoretical formulas" section, seven typical cases are listed in Table 2, and plots Ψ versus n and ε are shown in Figs. 4 and 5, respectively. Based on Eq. (15), Table 2, and Fig.  4, we determined that Ψ first increases rapidly with increasing n, then slowly, and eventually approaches 1. As stated in the "Homogeneous finite-thickness model" section, ε is defined as the ratio of the surface fluxes. Based on Eq. (16), Table 2, and Fig. 5, we determined that ε is linearly proportional to Ψ, and the slope increases gradually from case 1 to case 7; the value interval of ε is 0-1. The intuitive corresponding relations among ε, Ψ and n are shown in Table 3.
In most previous studies, radon exhalation from geological media considered cases in which the radon migration was dominantly driven by diffusion. With respect to this scenario (namely cases 1 and 2), as shown in Table 3, to achieve an ε > 95%, x 0 must be at least 3.6626L; to achieve an ε > 99.5%, x 0 must be at least 5.9790L. In case 3, where the radon migration is jointly driven by diffusion and advection effects, the appropriate thickness of the medium (x 0 ) for a lab-scale experiment to achieve an equivalent percentage (ε) of 95% or greater is at least 2.5002L. For an ε of 99.5% or greater, x 0 needs to be at least 4.0152L. For the cases 4−7, where radon migration is dominantly driven by advection, to achieve an ε > 95%, x 0 must be at least 2.3607L, and to achieve an ε > 99.5%, x 0 must be at least 3.7789L.
Case 1 was considered for three geological media (i.e., UMTs, laterite, and radium-bearing rocklike material) for different thicknesses (x 0 ). Experiments were performed to measure the surface radon exhalation rates from these media, as described in the "Experiments" section. As shown in Figs. 6, 7, and 8, the models derived above were fit to the data for each medium (J fi = J se ·ε) with a high goodness of fit (adjusted coefficients of determination R 2 = 0.97563, 0.95803, and 0.89125 for UMTs, laterite, and radium-bearing rocklike material, respectively, and J se = 5.49, 0.0185, and 0.079 Bq·m −2 ·s −1 , respectively). The high fitting degrees indicated the reliability of the theoretical models. However, in order to achieve approximately ε = 1, the columns for UMTs and laterite need to be within several meters long for pure diffusive transport. Thus, lab-scale models with scaleddown radon exhalation rates maybe more applicable to the studies of radon exhalation from these media.

Conclusions
The issue of radon exhalation from radium-bearing geological media (e.g., soil, sand, bedrock, uranium ore, and its mill tailings) is of worldwide concern, and lab-scale tests are frequently employed to study these media. However, the influence of a lab-scale sample's thickness on the comparability to a real environment and the reliability of the experimental  200Ψ−199) results cannot be ignored. Based on the theory of radon diffusion-advection transport, theoretical and experimental analyses of the equivalent thicknesses of geological media for lab-scale studies of radon exhalation were conducted. The following were concluded: (1) Geological media can be modeled by semi-infinite and finitethickness models, and the former can be equivalent to the latter in terms of the rate of radon exhalation from the media's exposed surface (namely J fi = εJ se , where 0 < ε < 1).
(2) By introducing several dimensionless parameters (i.e., ε, m, n, Ψ, and ζ), dimensionless models of homogeneous semi-infinite and finite-thickness media were derived,  and the equivalency of J se and J fi was subsequently analyzed.
Only case 1 was examined experimentally using three geological media to validate the theoretical model, and the maximum thicknesses of the UMTs, laterite, and radium-bearing rocklike material were restricted to 2.0, 2.5, and 1.0 m, respectively. Moreover, more complicated finite-thickness models (Type A.2 and Type A.3) as shown in Appendix A should be analyzed in the future.

Several types of radon exhalation media in theory
(1) The type and implication of radon exhalation media In nature, without deviating from reasonable assumptions, there are several types of radon exhalation media in theory (see Fig. 9 in the Appendix), such as homogeneous semi-infinite media with single exposed surface (e.g., crustal surface), homogeneous finite-thickness media with single exposed surface or double exposed surfaces (e.g., multi-level underground mine, underground laboratory and air-raid shelter).
As shown in Fig. 9, by taking the apparent direction of point upwards of radon migration in the media into consideration, we can find that: Layer 6 belongs to the homogeneous semi-infinite media with single exposed surface (since the exposed surface is confined to finite air space with adequate ventilation, radon concentration in the air space close to the exposed surface can be regarded as zero). Here we classify this layer as Type A.1.
Layer 1 belongs to the homogeneous finite-thickness media with single exposed surface (since the exposed surface is open to infinite air space, radon concentration in the air space close to the exposed surface can be regarded as zero). Here we classify this layer as Type A.2.
Layer 3 belongs to the homogeneous finite-thickness media with double exposed surfaces (since the upper or lower surface is confined to finite air space with poor ventilation or non-ventilation, radon concentration in each air space close to the corresponding exposed surface can be regarded as much higher than zero). Here we classify this layer as Type A.3.
Layer 5 belongs to the homogeneous finite-thickness media with double exposed surfaces (since the upper or lower surface is confined to finite air space with adequate ventilation, radon concentration in each air space close to the corresponding exposed surface can be regarded as zero). Here we classify this layer as Type A.4.
(2) The calculating formula for radon exhalation rate of each type First of all, radon concentrations of upper and lower surfaces are denoted as C 1 and C 2 , respectively.
As shown in Fig. 9, Layer 6 (Type A.1) is the homogeneous semi-infinite media with single exposed surface (radon concentration in the air space close to the exposed surface can be regarded as zero), radon exhalation rate of the exposed surface is determined by 2) is the homogeneous finite-thickness media with single exposed surface (radon concentration in each air space close to the corresponding exposed surface is regarded as much higher than zero), radon exhalation rate of the exposed surface is determined by 3) is the homogeneous finite-thickness media with double exposed surfaces (radon concentration in each air space close to the corresponding exposed surface can be regarded as zero), radon exhalation rate of the upper exposed surface is determined by As shown in Fig. I, Layer 5 (Type A.4) is the homogeneous finite-thickness media with double exposed surfaces (radon concentration in each air space close to the corresponding exposed surface can be regarded as zero), radon exhalation rate of the upper exposed surface is determined by As to Eqs. (A.1)-(A.4), they are within similarities in form, so they have certain foundation for being of equivalence with each other. In this study, Type A.1 and Type A.2 are our studied types. As shown in Eqs. (A.1) and (A.4), if the thickness x 0 , where x 0 ∈{x 0,1 , x 0,2 , x 0,3 , x 0,4 , x 0,5 , x 0,6 }, is independent variable, J 4 is dependent variable, and other parameters are constants, we can easily find that J 1 is constant, and J 4 first increases rapidly with increasing x 0 , then slowly, and eventually approaches J 1 . As to Eqs. (A.2) and (A.3), that is more complicated, and they will be studied further in the future.

The derivation procedures for complex equations
The Eqs. (10), (11), and (16) are relatively intricate, so the derivation procedures for these equations are shown as follows: (1) As to Eq. (10): (2) As to Eq. (11): In this study, C is bulk concentration of radon, D is bulk diffusion coefficient of radon, L is bulk diffusion length of radon. According to the radon literature (Van der Graaf and Cozmuta 2001), L can be determined expressed as follows: In the field of nuclear physics, it is commonly known that average lifetime (τ) of radon atom is the reciprocal of radon decay constant (λ), namely τ ¼ 1 λ Therefore, we can obtain bulk diffusion velocity (v diff ) of radon as shown in the following equation: So derivation of Eq. (11) can be seen as follows: