3D simulations for evaluation of location factors in an urban environment: application of a novel methodology to calculate external exposure doses for evacuees from Pripyat

Abstract

This article presents a methodology for assessing the radiation doses in an urban environment due to external irradiation from radionuclides deposited on the ground and other surfaces as well as from a passing radioactive cloud. The approach was developed and applied to assess individual doses of residents of the town of Pripyat who were evacuated shortly after the Chernobyl accident. Typically, the so-called location factor is defined as the ratio of the dose rate at a point of exposure and the dose rate at an undisturbed lawn far from any buildings. The present study used a new definition of the location factor as a regular four-dimensional grid of ratios of air kerma rates indoors and outdoors distributed in space and time. The location factors were calculated for two scenarios: outdoor and indoor values for typical apartments and buildings in Pripyat. Indoor location factors varied within two orders of magnitude depending on the floor of residence and place of staying inside the apartment. Values of the indoor location factor differed during the daytime and night by a factor of 30–40 depending on the behaviour of an individual within the apartment. Both, outdoor and indoor location factors decreased with decreasing distances between buildings. It was shown that during the first 4 days after the accident, air kerma rates in Pripyat were governed by the radionuclides deposited on the ground surface, and not by radionuclides in the cloud. Specifically, the contribution of the radioactive cloud to air kerma rate was maximal (i.e., 2.3%) on the morning of 28 April 1986. The methodology and results of this study are currently being used to reconstruct the radiation gonadal dose for the subjects of the American–Ukrainian study of parental irradiation in Chernobyl cleanup workers and evacuees for investigating germline mutations in their offspring.

Appendix: Gamma radiation sources in urban environment of Pripyat

Release from the Chernobyl accident had two components: fuel particles and radioactive gases and aerosols. Explosion at Unit 4 resulted in the release of coarse fuel particles into the atmosphere to a height up to 1.5 km (Izrael 1996). Fuel particles dominated contamination of Pripyat and other areas up to 20 km around Chernobyl NPP immediately after the explosion (IAEA 2006). Table 6 shows the composition of radionuclides according to their activity in the Unit 4 of the Chernobyl NPP before the accident (Kryuchkov et al. 2011). It should be noted that refractory radionuclides, including fuel particles, did not contribute significantly (except ⁹⁵Zr + ⁹⁵Nb) to air kerma rate (see last column in Table 6). The smallest fuel particles, consisting of one granule of crystalline nuclear fuel, had a size of at least 10 µm and a high density, 8–10 g cm⁻³. Larger aggregate particles were detected at several km from the destroyed reactor. Later, radioactive gases and aerosol particles began to play a leading role. As the intensity of throwing Unit 4 with ballast materials increased in May–June 1986, the leading role again returned to fuel dust with higher melting point, but this period of the accident is beyond the scope of this article.

The impact of an urban environment on the formation of radiation fields has two aspects: the shielding impact of buildings and the effect of wind on radioactive aerosol transfer. Accounting for the shielding impact of buildings is a routine task for the Halden Planner software. The effect on wind transport of radioactive aerosols is considered in the framework of microscale meteorological simulations.

Microscale urban meteorological models, such as M2UE, and several other software codes, such as ADERA, STAR-CD, CFX, allow to simulate microscale atmospheric meteorological transport. Nuterman et al. (2008) showed that the distribution of impurities throughout the city under the impact of wind was extremely heterogeneous because of the peculiarities of wind fields in the urban environment. However, calculations based on microscale meteorological models use point sources of release with a height, typically, of several meters, and not higher than two heights of multi-story buildings, i.e. transport of released radionuclides is defined by wind fields that are influenced by the urban environment.

The nature of the dry deposition of fuel particles can be described by the value of the non-dimensional particle relaxation time, ${\tau }_{+}$, using Eq. (4) (Williams and Crane 1983):

$${\tau }_{+}=\frac{{a}_{p}^{2}\cdot {\rho }_{p}\cdot {C}_{c}}{18\cdot {\mu }_{a}}\cdot \frac{{u}_{*}^{2}}{{\nu }_{a}},$$

(4)

where ${a}_{p}$ is the particle diameter (m); ${\rho }_{p}$ is the particle density (kg m⁻³);${C}_{c}$ is the Cunningham slip coefficient (unitless); ${u}_{*}$ is the friction speed (m s⁻¹); ${\mu }_{a}$=1.82 × 10⁻⁵ (kg m⁻¹ s⁻¹) is the aerodynamic viscosity; and ${\nu }_{a}$=1.51 × 10⁻⁵ (m² s⁻¹) is the kinematic viscosity of air.

Parameter ${C}_{c}$ is calculated according to (Seinfeld and Pandis 1998) as Eq. (5):

$${C}_{c}=1+\frac{{\lambda }_{a}}{{a}_{p}}\left(2.514+0.8\cdot \text{exp}\left(-\frac{0.55{\cdot a}_{p}}{{\lambda }_{a}}\right)\right),$$

(5)

where ${\lambda }_{a}$= 0.067 × 10⁻⁶ (m) is the mean free path in air.

The following effects are observed depending on the value of the non-dimensional particle relaxation time, ${\tau }_{+}$, (Giardina and Buffa 2018):

• If ${\tau }_{+}$<0.1 (very small particles), the Brownian diffusion becomes significant, and the combination of Brownian and vortex diffusion affects the deposition;

• If 0.1 < ${\tau }_{+}$<30, the particle movement is highly dependent on turbulent oscillations in the fluid flow; and

• For ${\tau }_{+}$>30, particles have a high inertia and the effect of turbulence on particle motion is significantly reduced; moreover, gravitational effect becomes dominant.

Table 6 Composition of radionuclides according to their activity in the Unit 4 of the Chernobyl NPP before the accident (Kryuchkov et al. 2011)

Assuming the following parameter values for the Chernobyl release (IAEA 2006), ${a}_{p}$ = 10 µm, ${\rho }_{p}$ = 8000 kg m⁻³ and ${u}_{*}$ = 0.1–1.5 m s⁻¹, it follows from Eqs. (4) and (5) that ${C}_{c}$ = 1.0 and ${\tau }_{+}$ ≈ 160–2400. Such values of non-dimensional particle relaxation times indicate that the effect of turbulence on the dry deposition of fuel particles was negligible and the gravitational effect was dominant. Therefore, the conditions of the urban environment in Pripyat, i.e., the specific location of buildings and the height of the buildings, did not significantly affect the formation of radioactive deposition within the territory of the town during the first 4 days after the Chernobyl accident.

Components of gamma radiation source in Pripyat

General description

Two components of the radiation source resulting in an external irradiation should be considered for residents of Pripyat: (1) a semi-infinite cloud with a uniformly distributed activity concentration in the half-space above the surface of the earth, and (2) a plane source on the ground surface or source uniformly distributed to some depth in soil. The contribution of each of these components to the total exposure is described in this section.

The air kerma rate ($\text{AKR})$ from radionuclides deposited on the ground surface was calculated as:

$$\dot{K}{}_{\text{air}}^{\text{ground}} ={A}_{\text{ground}}\cdot {d}^{\text{ground}},$$

(6)

where ${\dot{K}}_{\text{air}}^{\text{ground}}$ is the $\text{AKR}$ at a height of 1 m above the ground from radionuclides deposited on the ground surface (nGy h⁻¹); ${A}_{\text{ground}}$ is the ground surface deposition density (kBq m⁻²); ${d}^{\text{ground}}$ is the conversion coefficient from surface activity to the $\text{AKR}$ (nGy h⁻¹ per kBq m⁻²). The value of ${d}^{\text{ground}}$ is equal to 1.20 and 1.17 nGy h⁻¹ per kBq m⁻², respectively, for fuel matrix and for aerosol particles with a mean energy of gamma radiation photons corresponding to photons emitted by ¹³¹I (ICRP 2020).

The air kerma from radionuclides deposited on the ground surface during the time when the radioactive cloud passes was calculated as:

$$K_{{{\text{air}}}}^{{{\text{ground}}}} = \int\limits_{0}^{{\Delta t}} {A_{{{\text{ground}}}} (t) \cdot d^{{{\text{ground}}}} {\text{d}}t} = \frac{1}{2} \cdot \Delta t \cdot A_{{{\text{ground}}}} (t = \Delta t) \cdot d^{{{\text{ground}}}} ,$$

(7)

where ${K}_{\text{air}}^{\text{ground}}$ is the air kerma from radionuclides deposited on the ground surface during time of cloud passing (mGy); ${A}_{\text{ground}}(t)$ is the ground surface deposition density as a function of time (kBq m⁻²). It was assumed that the ground surface deposition density increased with constant deposition rate during the time when the radioactive cloud passed, $\Delta t$ (h); ${A}_{\text{ground}}(t=\Delta t)$ is the maximal ground surface deposition density at time $\Delta \text{t}$ (kBq m⁻²).

The air kerma from the passing cloud, which formed surface activity, can be determined as (UNSCEAR 2013):

$${K}_{\text{air}}^{\text{cloud}}=2.78\cdot {10}^{-4}\cdot {A}_{\text{ground}}/{V}_{m}\cdot {d}^{\text{cloud}},$$

(8)

where ${K}_{\text{air}}^{\text{cloud}}$ is the air kerma at a height of 1 m above the ground from activity concentration of the radionuclide in the cloud (nGy); 2.78 × 10^–4 is a unit conversion coefficient (s h⁻¹); ${V}_{m}$ is the deposition velocity of the m-th radionuclide for specific weather and surface conditions (m s⁻¹); ${d}^{\text{cloud}}$ is the conversion coefficient from activity concentration of the radionuclide in air to the air kerma rate (nGy h⁻¹ per kBq m⁻³). The value of ${d}^{\text{cloud}}$ is equal to 0.0879 and 0.825 (nGy h⁻¹ per kBq m⁻³), respectively, for fuel matrix and for aerosol particles with a mean energy of gamma radiation photons corresponding to photos emitted by ¹³¹I (ICRP 2020).

The ratio of air kerma from radionuclides in the cloud to air kerma from ground deposition of radionuclides was:

$$\frac{{K}_{\text{air}}^{\text{cloud}}}{{K}_{\text{air}}^{\text{ground}}}=2\cdot 2.78\cdot {10}^{-4}\frac{{d}^{\text{cloud}}}{{d}^{\text{ground}}\cdot \Delta t\cdot {V}_{m}}={4.27\cdot 10}^{-2}/({V}_{m}\cdot \Delta t)$$

(9)

where $\Delta t$ is the time interval between measurements of air kerma rate (h).

Variation with time

According to Eq. (9), the ratio of air kerma from radionuclides in the cloud to air kerma from radionuclides deposited on the ground surface depends on the product ${V}_{m}\cdot \Delta t$. To evaluate the variation with time the contribution of radionuclides in the cloud and those deposited on the ground surface to the $\text{AKR}$ in Pripyat, 16 grids of $\text{AKR}$ corresponding to different time periods, $\dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})$, were analysed in the present study, where $m$ is the grid number ($m=1, 2,\dots , 16)$, and ${x}_{i}$ and ${y}_{j}$ are the coordinates of the i–j grid node. Every grid was created as 61 × 100 matrix by interpolation with uniform step $\Delta x$ and $\Delta y$ along the $X$ and $Y$ axes of the $\text{AKR}$ measurements that were conducted by the radiation safety service of the Chernobyl NPP from time of the accident to the end of 29 April 1986 (Kryuchkov et al. 2009). Table 7 shows dates and time of the $\text{AKR}$ measurements in Pripyat during on 26–29 April 1986. Figure 6 shows 31 locations where $\text{AKR}$ was measured in Pripyat during 26–29 April 1986.

Table 7 Air kerma rate grids in Pripyat on 26–29 April 1986

Fig. 6

Locations (triangles) where air kerma rate was measured in Pripyat during 26–29 April 1986; $x$-axis and $y$-axis provide coordinates in longitude and latitude, respectively

The measured $\text{AKR}$ grids are called in this paper ‘initial’. Other $\text{AKR}$ grids, which were considered in this paper, represent the evolution of $\text{AKR}$ with time and were called ‘difference’. A ‘difference’ grid numerically equals to the difference of two successive ‘initial’ grids:

$$\Delta \dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})=\dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})-\dot{K}{}_{\text{air}}^{m-1}({x}_{i},{y}_{j}),$$

(10)

where $\Delta \dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})$ is the ‘difference’ $\text{AKR}$ grid (nGy h⁻¹); $\dot{K}{}_{\text{air}}^{m-1}({x}_{i},{y}_{j})$ and $\dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})$ are the two successive ‘initial’ $\text{AKR}$ grids (nGy h⁻¹).

The ‘initial’ grids contain only positive $\text{AKR}$ values, while the ‘difference’ grids can be either positive or negative depending whether $\text{AKR}$ was increasing or decreasing between two measurements. Therefore, the difference $\text{AKR}$ grid can be presented as sum of positive and negative difference grids:

$$\Delta \dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})=\dot{\Delta K}{}_{\text{air}}^{m+}({x}_{i},{y}_{j})+\Delta \dot{K}{}_{\text{air}}^{m-}({x}_{i},{y}_{j}),$$

(11)

where $\dot{\Delta K}{}_{\text{air}}^{m+}({x}_{i},{y}_{j})$ > 0 and $\dot{\Delta K}{}_{\text{air}}^{m-}({x}_{i},{y}_{j})$ < 0.

Table 8 summarizes the definition of different quantities derived from $\text{AKR}$ values measured in Pripyat. The first four quantities were described above. The other four quantities represent integral characteristics of the difference $\text{AKR}$ grid, $\Delta \dot{K}{}_{\text{air}}^{m}({x}_{i},{y}_{j})$. Values of ${\dot{B}}_{m}$ and ${B}_{m}$ per unit of grid area characterize mean $\text{AKR}$ over the $m$-th grid and mean air kerma since time of the accident until time of measurement of $m$-th grid.

Table 8 Definition of different quantities derived from $\text{AKR}$ values measured in Pripyat

Table 7 also provides the values of ${PV}_{m}$ and ${NV}_{m}$ calculated for 16 $\text{AKR}$ grids measured in Pripyat during 26–29 April 1986. The ${PV}_{m}$ and ${NV}_{m}$ values defined significance of ‘difference’ grids for evaluation of the contribution of radionuclides in the cloud and radionuclides deposited on the ground surface to air kerma rate. A detailed evaluation of these contributions is provided in the section below.

Evaluation of the contribution of radioactive components to air kerma rate in Pripyat

Figure 7 shows an example of the contribution of airborne radionuclides in the passing cloud (upper part) and radionuclides deposited on the ground surface (middle part) to air kerma rate, ${\dot{K}}_{\text{air}}^{\text{cloud}}$ and $\dot{K}{}_{\text{air}}^{\text{ground}},$ respectively, as well as an evaluation of the values of ${PV}_{m}$ and ${NV}_{m}$ (bottom part). The example is given for air kerma rates measured in three grids for the same grid node. The following equations can be written:

$$\dot{K}{}_{\text{air}}^{\text{cloud}, m}\Delta x\Delta y={NV}_{m},$$

(12)

$$\sum_{k=1}^{m}\dot{K}{}_{\text{air}}^{\text{ground}, k}\Delta x\Delta y={\dot{B}}_{m}=\sum_{k=1}^{m}({PV}_{k}-{NV}_{k})=\sum_{k=1}^{m}{PV}_{k}-{NV}_{m}$$

(13)

Fig. 7

Contribution of radionuclides in the passing cloud and radionuclides deposited on the ground surface to air kerma rate, $\dot{K}{}_{\text{air}}^{\text{cloud}}$ and $\dot{K}{}_{\text{air}}^{\text{ground}},$ respectively, and evaluation of the values of ${PV}_{m}$ and ${NV}_{m}$

It is obvious that Eqs. (12) and (13) will be correct for any $m$, not only for $m$ = 3. The ${NV}_{m}$ values correspond to the values of air kerma rate from the $m$-th cloud, $\dot{K}{}_{\text{air}}^{\text{cloud},m},$ while the difference between the sum of ${PV}_{m}$ value and ${NV}_{m}$ value from the last cloud matches, with accuracy $\Delta x\Delta y$, the value of air kerma rate from radionuclides deposited from all clouds on the ground, $\dot{K}{}_{\text{air}}^{\text{ground}}.$ It should be noted that the decrease of air kerma rate and ${NV}_{m}$-values resulted not only from the cloud passing away from the point of measurement but also from radioactive decay of the radionuclides deposited on the ground.

To estimate the contribution of radioactive decay to the ${NV}_{m}$ values, the known empiric time dependence of the $\text{AKR}$ for fuel particles from Unit 4 normalized to 6 May 1986 (Kryuchkov et al. 2012) was used. The results are shown in Table 9 (5th column) as the ratio of the difference between the $\text{AKR}$ values at the end and the beginning of the time interval between measurements and the $\text{AKR}$ value at the beginning of the interval.

Table 9 Contribution of radioactive decay of radionuclides to ${NV}_{m}$-values and estimates of contribution of radionuclides in the passing cloud to air kerma rate

Table 9 also shows the ratio, ${NV}_{m}/{PV}_{m}$, that characterizes the ratio of the $\text{AKR}$ from the m-th cloud and the $AKR$ from ground deposition resulting from this cloud, $\dot{K}{}_{\text{air}}^{\text{cloud}, m}/\dot{K}{}_{\text{air}}^{\text{ground}, m}.$ Figure 8 shows the ratio $\dot{K}{}_{\text{air}}^{\text{cloud}, m}/\dot{K}{}_{\text{air}}^{\text{ground},m},$ for all of the 16 grids. During the first day after the accident, the $\dot{K}{}_{\text{air}}^{\text{cloud}, m}/\dot{K}{}_{\text{air}}^{\text{ground}, m}$ ratio was close to zero, while during the evening of April 27 (11th grid) values of the $\dot{K}{}_{\text{air}}^{\text{cloud}, m}/\dot{K}{}_{\text{air}}^{\text{ground}, m}$ ratio were significantly higher. This can be explained by the fact that after the explosion, the radioactive cloud was formed by coarse fuel dust particles, which have a high deposition rate, while in the subsequent period, condensation aerosols with a low deposition rate prevailed in the cloud. The diameter of the condensation aerosols was approximately two orders of magnitude smaller than the diameter of the fuel dust particles. According to the Stokes law, this ensured a decrease in the deposition rate by four orders of magnitude (Reist 1984).

Fig. 8

Dependence of the ratio of air kerma rate from cloud to that from ground deposition from the cloud on ${V}_{m}\cdot \Delta t$ value. The numbers on the graph indicate the numbers of air kerma rate grids built according to the measurements done by the radiation safety service of the Chernobyl NPP

The increase of the $\dot{K}{}_{\text{air}}^{\text{cloud}, m}/\dot{K}{}_{\text{air}}^{\text{ground},m}$ ratio from 0.001 to 30 correlated with changes in the values of parameter V_m·Δt. The mean value of the $\dot{K}{}_{\text{air}}^{\text{cloud}, m}/\dot{K}{}_{\text{air}}^{\text{ground},m}$ ratio was 0.4 (Fig. 8). One could conclude that the $\text{AKR}$ from the cloud contributed around 29% to the total $\text{AKR}$ during the first 4 days after the accident. However, the exposure from the cloud stopped when the cloud had passed Pripyat, while the exposure from ground deposition from the passage of this cloud continued in the follow-up period. This should be considered for assessing $\text{AKR}$ from ground deposition in subsequent periods of observation.

Knowledge of the contribution from the cloud and from ground deposition to the original $\text{AKR}$ grids is important for calculation of the location factors. The ratio of ${NV}_{m}/{\dot{B}}_{m}$ (Table 9, column 7) represents the contribution of ground deposition to the original $\text{AKR}$ as an accrued sum, while the contribution of the cloud to $\text{AKR}$ is not summarized. In such form, the ratio ${NV}_{m}/{\dot{B}}_{m}$, which is the ratio of the $\text{AKR}$ from the m-th cloud and the $\text{AKR}$ from ground deposition resulting from all the passed clouds, was significantly lower than shown in Table 9, column 6. The ratio ${(NV}_{m}\cdot \Delta {t}_{m})/{B}_{m}$, which is the ratio of air kerma from the m-th cloud and air kerma from ground deposition due to all passed clouds, is shown in Table 9, column 8. Data from columns 5 and 8 in Table 9 allow to compare the contributions to exposure from the clouds including radioactive decay of radionuclides deposited on the ground. Only for one grid out of the 16, the 12th grid, the relative contribution to the total exposure from the cloud component (0.058) exceeded the relative contribution from the radioactive decay of radionuclides (0.035). So, for the 12th grid, the relative contribution of the cloud component to the total $\text{AKR}$ was around 0.023.