Review Paper ON THE AIR-FILLED EFFECTIVE POROSITY PARAMETER OF ROGERS AND NIELSON’S (1991) BULK RADON DIFFUSION COEFFICIENT IN UNSATURATED SOILS Zakaria Saâdi* Abstract—The radon exhalation rate at the earth’s surface from soil or rock with radium as its source is the main mechanism behind the radon activity concentrations observed in both indoor and outdoor environments. During the last two decades, many subsurface radon transport models have used Rogers and Nielson’s formula for modeling the unsaturated soil bulk radon diffusion coefficient. This formula uses an “air-filled effective porosity” to account for radon adsorption and radon dissolution in the groundwater. This formula is reviewed here, and its hypotheses are examined for accuracy in dealing with subsurface radon transport problems. The author shows its limitations by comparing one dimensional steady-state analytical solutions of the two-phase (air/water) transport equation (Fick’s law) with Rogers and Nielson’s formula. For radon diffusion-dominated transport, the calculated Rogers and Nielson’s radon exhalation rate is shown to be unrealistic as it is independent of the values of the radon adsorption and groundwater dissolution coefficients. For convective and diffusive transport, radon exhalation rates calculated using Fick’s law and this formula agree only for high values of gas-phase velocity and groundwater saturation. However, these conditions are not usually met in most shallow subsurface environments where radon migration takes place under low gas phase velocities and low water saturation. Health Phys. 106(5):598–607; 2014 Key words: diffusion; dose; health effects; modeling; dose assessment; radon
INTRODUCTION SINCE DEVELOPMENT of the empirical formulas of Rogers and Nielson (R&N) in 1991 (Rogers and Nielson 1991a and b) for modeling effective and bulk diffusion coefficients of radon in unsaturated soils, the latter has been used extensively
*Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PRPDGE/SEDRAN/BRN, 31 Avenue de la Division Leclerc, B.P. 17, 92262, Fontenay-aux-Roses, Cedex, France. The author declares no conflicts of interest. For correspondence contact the author at the above address, or email at [email protected]. (Manuscript accepted 12 September 2013) 0017-9078/14/0 Copyright © 2014 Health Physics Society DOI: 10.1097/HP.0000000000000034
in many transport models of radon in the subsurface (e.g., Andersen 1992, 2000; Ferry et al. 2001, 2002; Fournier et al. 2005). Rogers and Nielson (1991b) pointed out that adsorption and dissolution effects on radon diffusion are not taken into account in older diffusion models (e.g., Buckingham 1904; Millington and Quirk 1961). They proposed an alternative model accounting for these processes by introducing an effective air-filled porosity parameter to enhance calculation of bulk diffusion in unsaturated soils. Their conceptual model has been tested against experimental data for some specific soil and boundary conditions, but up to now, their hypotheses have never been verified. Van der Spoel (1998) showed experimentally that the R&N diffusion model is not appropriate for coarse-grained sand at low moisture content and proposed an alternative empirical one to better fit radon bulk diffusion at these low water contents. Meslin et al. (2010) used the R&N approach as well as many other correlation functions and physically based laws for calculating the bulk diffusion of radon in two reconstructed porous media, namely consolidated and unconsolidated soils (typical of undisturbed and repacked soils). They demonstrated that the R&N-approximation was not appropriate for modeling radon diffusion in different soil structures compared to physically based or powerlaw models (Buckingham 1904; Moldrup et al. 2001). Even if the R&N approximation has been shown to be accurate for some non-structured soils within a specific range of soil porosity, hypotheses behind its development have never been verified in the transport equation. It is then the purpose of this work to demonstrate its limited application in transport models of radon in the unsaturated zone.
GENERAL RADON TRANSPORT EQUATION MODEL Assuming isothermal and stationary conditions, the one-dimensional equation for vertical radon transport accounting for decay, emanation from a radium source, adsorption, and dissolution in an unsaturated www.health-physics.com
598
Copyright © 2014 Health Physics Society. Unauthorized reproduction of this article is prohibited.
Bulk radon diffusion coefficient in unsaturated soils c Z. SAÂDI
homogeneous soil column of length H can be written in the following form:
DFe
∂2 C ∂C −υ − λ C − C ∞Þ ¼ 0; 2 ∂z ∂z
(1)
with:
DeF ¼ DbF =Rg
(2a)
DFb ¼ ϕ S g Deg þ K Ost S l Dle
(2b)
Dβe ¼ τ 0 τ β d β
ðβ ≡ l; g Þ
(2c)
υ ¼ q=Rg
(2d)
q ¼ qg þ K Ost ql
(2e)
Ra C ∞ ¼ C 226 ρd E=Rg s
(2f )
Rg ¼ ϕ S g þ S l KOst þ ρd Kd g ;
(2g)
where z is the vertical space coordinate (m), positive downward, and z-axis origin (z = 0) is assumed to be at the soil surface (upper boundary of the soil column; Fig. 1); C is the radon activity concentration in the soil gas-phase (Bq m −3); C∞ is the infinite or maximum radon activity concentration in the soil-gas that can occur in the pore space (Bq m −3); λ is the radioactive decay constant of radon (2.1 10 −6 s −1); De and Db are total effective and bulk radon diffusion coefficients (m2 s −1), respectively (De is defined here as the
599
ratio between the flux density of a species per unit air-filled pore area and its concentration gradient in air-filled pores); the superscript F in DFb refers to Fick’s law; Dβe is the effective diffusion coefficient in phase β (m2 s −1), with β being liquid (l) or gas (g); ϕ is porosity; τ0τβ is the tortuosity that includes a porous medium dependent factor τ0 and a coefficient τβ that depends on saturation of phase β, Sβ; dβ is the diffusion coefficient of radon in free fluid phase β (m2 s −1); v is the total effective pore velocity (m s −1); qβ is Darcy’s flow velocity or flux density of the fluid phase β (m s −1); ρd is the soil dry bulk density (kg m −3);Cs226Ra is the soil radium activity mass content (Bq kg −1); E is the dimensionless emanation coefficient that varies between 0 and 1; KOst is the Ostwald coefficient, a dimensionless form of Henry’s law coefficient, which is the proportionality constant between equilibrium liquid- and gas-phase concentrations; Kdg is the distribution or adsorption coefficient between solid- and gas-phase (m3 kg −1), which is the proportionality constant between equilibrium solid- and gas-phase concentrations; and Rg is the effective air-filled soil porosity (−), defined also as the bulk gas-phase partition coefficient. Eqn (1) is a combination of the radon mass conserving equation with Fick’s law for radon flux density F(z) (Bq m −2 s −1). The latter is given by:
F ðzÞ ¼ −DFe Rg
∂C ðzÞ þ υ Rg C ðzÞ: ∂z
(3)
Eqn (1) is a second order, nonhomogeneous differential equation with constant coefficients. Hence, it can be solved easily for constant radon concentration at both boundaries of a homogeneous soil column (Fig. 1), as given below:
C ðzÞ ¼ C ∞ þ C 1 ðzÞ þ C 2 ðzÞ C 1 ðzÞ ¼ ðC 0 −C ∞ Þ eð l Þ z
C 2 ðzÞ ¼ ðC H −C ∞ Þ e½
−ðH−zÞ l
(4a)
sinh½α ðH−zÞ sinhðαH Þ
(4b)
sinhðαzÞ ; sinhðαH Þ
(4c)
with:
2DeF l¼ υ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 λ α¼ þ F; 2 De l Fig. 1. A sketch of the geometry used for the soil column configuration along with z-axis origin position and orientation, top and boundary conditions, and transport parameters for solving eqn (1).
(5a)
(5b)
where C0 and CH are radon activity concentrations at the top (z = 0) and bottom (z = H ) boundaries of the soil column
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(Bq m −3) (Fig. 1); l is the diffusion length (m); and α is a scaling length parameter (m −1). Combining eqns (3) and (4a–c), the radon flux densities at the top and bottom boundaries are given, respectively, by:
F 0 ¼ a1 ðC ∞ −C 0 Þ þ a2 ðC ∞ −C H Þ þ vRg C 0
(6a)
F H ¼ b1 ðC∞ −C 0 Þ þ b2 ðC ∞ −C H Þ þ vRg C H ;
(6b)
with:
a1 ¼ 2Rg DeF =l−b2 −H a2 ¼ Rg DeF eð l Þ
(7a)
α
(7b)
sinhðα H Þ
2H b1 ¼ −a2 eð l Þ
b2 ¼ Rg DFe
(7c)
1 α þ : l tanhðαH Þ
(7d)
The analytical solution of eqn (1) given by eqns (4a–c) and (6a–b) can be extended easily to radon transport in layered soil columns (Saâdi et al. 2012). It has been used for verification of the numerical scheme used for time discretization of the radon production or emanation term in the newly developed module EOS7Rn (Saâdi et al. 2012) for the TOUGH2 (Transport Of Unsaturated Groundwater and Heat, version 2; Pruess et al. 1999) software to study radon transport in the subsurface. ROGERS AND NIELSON APPROXIMATIONS First approximation The first approximation consists in neglecting the liquidphase velocity (eqn 2e). This approximation can be legitimate for some conditions, such as in the absence of rainfall or irrigation events at the soil surface. However, Ho (2008) showed the importance of high rainfall events in predicting radon exhalation from Uranium Mill Tailings (UMT) landfill soils. Second approximation The unsaturated tortuosity factor for the soil gasphase can be estimated using the following correlation equation:
h
i
τ0 τg ¼ ϕ exp −6ϕS l −6ðS l Þ14ϕ :
(8)
Eqn (8) is an empirical fit to a set of more than a thousand laboratory diffusion measurements on earthen materials (recompacted soils ranging from sandy gravels to fine
May 2014, Volume 106, Number 5
clays) performed with a transient method (Nielson et al. 1982a). Van der Spoel (1998) showed that eqn (8) did not fit well to experimental radon bulk diffusion measurements on coarse sand at low moisture contents. Eqn (8) can only be expected to apply within the range of soil types tested. Moreover, Meslin et al. (2010) showed that eqn (8) was not able to reproduce bulk radon diffusion data generated by the lattice Boltzman approach for consolidated and unconsolidated porous media (undisturbed and repacked soils). Their conclusion is that power-law or physicallybased models like Buckingham (1904) or Millington and Quirk (1961) can be more precise than eqn (8). Third approximation The effective diffusion coefficient of radon in the gas g phase (DRN e ¼ De ) is linked to its bulk diffusion coefficient in liquid- and gas-phase (DRN b ) by the partition-corrected porosity factor Rg (eqn 2g); see also Andersen (1992), which takes into account radon solubility in groundwater (through the Ostwald coefficient, KOst) and radon adsorption at the soil solid phase (through the adsorption coefficient, Kdg): g DRN b ¼ Rg De ¼ Rg τ 0 τ g d g :
(9)
The hypothesis behind this approximation is that eqn (9) does not ignore pore air-water interactions, such as absorption and radon transfer between pore-air and pore-water, relative to older physically-based or power-law models of the simple air-filled porosity ϕSg. This hypothesis can be interpreted mathematically as the use of unique mean radon diffusion coefficient in gas phase by neglecting radon diffusion in liquid phase and changing the air-filled porosity ϕSg by Rg (see eqn 2g). However, eqn (9) for DbRN can only equal eqn (2b) forDFb if Kdg = 0 (no radon adsorption) and one of the three following conditions are realized: (1) single-phase gas conditions (Sl = 0), (2) effective radon diffusion coefficients in soil liquid-phase (Dle ) equal to that in soil gas-phase (Dge ) in eqn (2b), or (3) KOst = 0 in both eqns (2b) and (2g). The first condition can be true, but the two last ones are physically unrealistic in unsaturated conditions. Eqn (9) has been used blindly without any verification in many radon transport models in the subsurface (e.g., Andersen 2000; Ferry et al. 2001; Fournier et al. 2005). The purpose of this paper is to verify this third approximation in unsaturated soil conditions.
VERIFICATION EXAMPLE OF THE THIRD R&N—APPROXIMATION Analytical simulations Let us now consider the simple exercise of vertical radon transport in an unsaturated soil column of length H=10 m, submitted to constant concentration (Dirichlet condition) at both ends [C(z = 0) = C0; C(z = H) = CH; Fig. 1]. Isothermal
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Bulk radon diffusion coefficient in unsaturated soils c Z. SAÂDI
Table 1. Physical properties of the Uranium Mill Tailings (UMT) soil (Ferry et al. 2002). Parameter
UMT
ϕ (—)
0.4 1,370 4.0856 10−13 60,000 0.05 0.32 0.15 3.4218 10−3 12
ρd (kg m−3) kpa (m2) Cs226Ra (Bq kg−1) Ea (—) Ew (—) S* (—) kdg0 (m3 kg−1)b b (—)b a
kp is the intrinsic permeability. Estimated from Rogers and Nielson (1991b).
b
conditions are assumed at 25°C such that: KOst = 0.2263 (Clever 1979), dg = 1.1 10 −5 m2 s −1 and dl = 1.4 10 −9 m2 s −1. The adsorption and emanation coefficients are calculated as functions of liquid-saturation [i.e., Kdg(Sl) and E (Sl)] using the equations of Rogers and Nielson (1991b) and Nielson et al. (1982b), respectively:
Kd g ¼ Kd 0g expð−bS l Þ
E¼
(10a)
n
E w S l =S þ E a ð1−S l =S Þ if S l S Ew elsewhere ;
(10b)
where Kdg0 is the distribution coefficient at dryness (m3 kg −1); b is a dimensionless correlation constant, generally lying between 10 and 15, but it can be significantly higher; Ew
601
and Ea are the emanation coefficients at full saturation and at dryness (−), respectively; and S* is the minimum water saturation on the plateau of an emanation–water saturation curve. The physical properties of the studied soil are given in Table 1 (Ferry et al. 2002). The soil, classified as sandy silt according to the USDA soil classification, corresponds to the uranium mill tailings from the Lavaugrasse (France) landfill with high radium activity concentration. Six analytical simulations (Table 2) were applied to the soil column at unsaturated conditions corresponding to three values of Sl (0.25, 0.5, and 0.75) and two values for gas pressure difference between the column boundaries ΔPg (0 and 1,000 Pa). The latter was chosen so as to produce an upward convective transport to the soil surface (negative gas-phase velocity value). The gas-phase velocity value is calculated using Darcy’s law, which is equal to vg = −2.23 10 −6 m s −1 for ΔPg = 1,000 Pa. Calculations were performed with C0 = 0; two values of CH, 0 and C∞ (eqn 2f); two values of KOst, 0 as a hypothetical value (i.e., all radon dissolved in liquidphase is volatilized into the gas phase) and 0.2263; and two values of Kdg0, 0 and 3.4218 10 −3 m3 kg −1. The effective diffusion coefficient in liquid-phase has been assumed negligible (Dle
INTRODUCTION SINCE DEVELOPMENT of the empirical formulas of Rogers and Nielson (R&N) in 1991 (Rogers and Nielson 1991a and b) for modeling effective and bulk diffusion coefficients of radon in unsaturated soils, the latter has been used extensively
*Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PRPDGE/SEDRAN/BRN, 31 Avenue de la Division Leclerc, B.P. 17, 92262, Fontenay-aux-Roses, Cedex, France. The author declares no conflicts of interest. For correspondence contact the author at the above address, or email at [email protected]. (Manuscript accepted 12 September 2013) 0017-9078/14/0 Copyright © 2014 Health Physics Society DOI: 10.1097/HP.0000000000000034
in many transport models of radon in the subsurface (e.g., Andersen 1992, 2000; Ferry et al. 2001, 2002; Fournier et al. 2005). Rogers and Nielson (1991b) pointed out that adsorption and dissolution effects on radon diffusion are not taken into account in older diffusion models (e.g., Buckingham 1904; Millington and Quirk 1961). They proposed an alternative model accounting for these processes by introducing an effective air-filled porosity parameter to enhance calculation of bulk diffusion in unsaturated soils. Their conceptual model has been tested against experimental data for some specific soil and boundary conditions, but up to now, their hypotheses have never been verified. Van der Spoel (1998) showed experimentally that the R&N diffusion model is not appropriate for coarse-grained sand at low moisture content and proposed an alternative empirical one to better fit radon bulk diffusion at these low water contents. Meslin et al. (2010) used the R&N approach as well as many other correlation functions and physically based laws for calculating the bulk diffusion of radon in two reconstructed porous media, namely consolidated and unconsolidated soils (typical of undisturbed and repacked soils). They demonstrated that the R&N-approximation was not appropriate for modeling radon diffusion in different soil structures compared to physically based or powerlaw models (Buckingham 1904; Moldrup et al. 2001). Even if the R&N approximation has been shown to be accurate for some non-structured soils within a specific range of soil porosity, hypotheses behind its development have never been verified in the transport equation. It is then the purpose of this work to demonstrate its limited application in transport models of radon in the unsaturated zone.
GENERAL RADON TRANSPORT EQUATION MODEL Assuming isothermal and stationary conditions, the one-dimensional equation for vertical radon transport accounting for decay, emanation from a radium source, adsorption, and dissolution in an unsaturated www.health-physics.com
598
Copyright © 2014 Health Physics Society. Unauthorized reproduction of this article is prohibited.
Bulk radon diffusion coefficient in unsaturated soils c Z. SAÂDI
homogeneous soil column of length H can be written in the following form:
DFe
∂2 C ∂C −υ − λ C − C ∞Þ ¼ 0; 2 ∂z ∂z
(1)
with:
DeF ¼ DbF =Rg
(2a)
DFb ¼ ϕ S g Deg þ K Ost S l Dle
(2b)
Dβe ¼ τ 0 τ β d β
ðβ ≡ l; g Þ
(2c)
υ ¼ q=Rg
(2d)
q ¼ qg þ K Ost ql
(2e)
Ra C ∞ ¼ C 226 ρd E=Rg s
(2f )
Rg ¼ ϕ S g þ S l KOst þ ρd Kd g ;
(2g)
where z is the vertical space coordinate (m), positive downward, and z-axis origin (z = 0) is assumed to be at the soil surface (upper boundary of the soil column; Fig. 1); C is the radon activity concentration in the soil gas-phase (Bq m −3); C∞ is the infinite or maximum radon activity concentration in the soil-gas that can occur in the pore space (Bq m −3); λ is the radioactive decay constant of radon (2.1 10 −6 s −1); De and Db are total effective and bulk radon diffusion coefficients (m2 s −1), respectively (De is defined here as the
599
ratio between the flux density of a species per unit air-filled pore area and its concentration gradient in air-filled pores); the superscript F in DFb refers to Fick’s law; Dβe is the effective diffusion coefficient in phase β (m2 s −1), with β being liquid (l) or gas (g); ϕ is porosity; τ0τβ is the tortuosity that includes a porous medium dependent factor τ0 and a coefficient τβ that depends on saturation of phase β, Sβ; dβ is the diffusion coefficient of radon in free fluid phase β (m2 s −1); v is the total effective pore velocity (m s −1); qβ is Darcy’s flow velocity or flux density of the fluid phase β (m s −1); ρd is the soil dry bulk density (kg m −3);Cs226Ra is the soil radium activity mass content (Bq kg −1); E is the dimensionless emanation coefficient that varies between 0 and 1; KOst is the Ostwald coefficient, a dimensionless form of Henry’s law coefficient, which is the proportionality constant between equilibrium liquid- and gas-phase concentrations; Kdg is the distribution or adsorption coefficient between solid- and gas-phase (m3 kg −1), which is the proportionality constant between equilibrium solid- and gas-phase concentrations; and Rg is the effective air-filled soil porosity (−), defined also as the bulk gas-phase partition coefficient. Eqn (1) is a combination of the radon mass conserving equation with Fick’s law for radon flux density F(z) (Bq m −2 s −1). The latter is given by:
F ðzÞ ¼ −DFe Rg
∂C ðzÞ þ υ Rg C ðzÞ: ∂z
(3)
Eqn (1) is a second order, nonhomogeneous differential equation with constant coefficients. Hence, it can be solved easily for constant radon concentration at both boundaries of a homogeneous soil column (Fig. 1), as given below:
C ðzÞ ¼ C ∞ þ C 1 ðzÞ þ C 2 ðzÞ C 1 ðzÞ ¼ ðC 0 −C ∞ Þ eð l Þ z
C 2 ðzÞ ¼ ðC H −C ∞ Þ e½
−ðH−zÞ l
(4a)
sinh½α ðH−zÞ sinhðαH Þ
(4b)
sinhðαzÞ ; sinhðαH Þ
(4c)
with:
2DeF l¼ υ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 λ α¼ þ F; 2 De l Fig. 1. A sketch of the geometry used for the soil column configuration along with z-axis origin position and orientation, top and boundary conditions, and transport parameters for solving eqn (1).
(5a)
(5b)
where C0 and CH are radon activity concentrations at the top (z = 0) and bottom (z = H ) boundaries of the soil column
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Health Physics
(Bq m −3) (Fig. 1); l is the diffusion length (m); and α is a scaling length parameter (m −1). Combining eqns (3) and (4a–c), the radon flux densities at the top and bottom boundaries are given, respectively, by:
F 0 ¼ a1 ðC ∞ −C 0 Þ þ a2 ðC ∞ −C H Þ þ vRg C 0
(6a)
F H ¼ b1 ðC∞ −C 0 Þ þ b2 ðC ∞ −C H Þ þ vRg C H ;
(6b)
with:
a1 ¼ 2Rg DeF =l−b2 −H a2 ¼ Rg DeF eð l Þ
(7a)
α
(7b)
sinhðα H Þ
2H b1 ¼ −a2 eð l Þ
b2 ¼ Rg DFe
(7c)
1 α þ : l tanhðαH Þ
(7d)
The analytical solution of eqn (1) given by eqns (4a–c) and (6a–b) can be extended easily to radon transport in layered soil columns (Saâdi et al. 2012). It has been used for verification of the numerical scheme used for time discretization of the radon production or emanation term in the newly developed module EOS7Rn (Saâdi et al. 2012) for the TOUGH2 (Transport Of Unsaturated Groundwater and Heat, version 2; Pruess et al. 1999) software to study radon transport in the subsurface. ROGERS AND NIELSON APPROXIMATIONS First approximation The first approximation consists in neglecting the liquidphase velocity (eqn 2e). This approximation can be legitimate for some conditions, such as in the absence of rainfall or irrigation events at the soil surface. However, Ho (2008) showed the importance of high rainfall events in predicting radon exhalation from Uranium Mill Tailings (UMT) landfill soils. Second approximation The unsaturated tortuosity factor for the soil gasphase can be estimated using the following correlation equation:
h
i
τ0 τg ¼ ϕ exp −6ϕS l −6ðS l Þ14ϕ :
(8)
Eqn (8) is an empirical fit to a set of more than a thousand laboratory diffusion measurements on earthen materials (recompacted soils ranging from sandy gravels to fine
May 2014, Volume 106, Number 5
clays) performed with a transient method (Nielson et al. 1982a). Van der Spoel (1998) showed that eqn (8) did not fit well to experimental radon bulk diffusion measurements on coarse sand at low moisture contents. Eqn (8) can only be expected to apply within the range of soil types tested. Moreover, Meslin et al. (2010) showed that eqn (8) was not able to reproduce bulk radon diffusion data generated by the lattice Boltzman approach for consolidated and unconsolidated porous media (undisturbed and repacked soils). Their conclusion is that power-law or physicallybased models like Buckingham (1904) or Millington and Quirk (1961) can be more precise than eqn (8). Third approximation The effective diffusion coefficient of radon in the gas g phase (DRN e ¼ De ) is linked to its bulk diffusion coefficient in liquid- and gas-phase (DRN b ) by the partition-corrected porosity factor Rg (eqn 2g); see also Andersen (1992), which takes into account radon solubility in groundwater (through the Ostwald coefficient, KOst) and radon adsorption at the soil solid phase (through the adsorption coefficient, Kdg): g DRN b ¼ Rg De ¼ Rg τ 0 τ g d g :
(9)
The hypothesis behind this approximation is that eqn (9) does not ignore pore air-water interactions, such as absorption and radon transfer between pore-air and pore-water, relative to older physically-based or power-law models of the simple air-filled porosity ϕSg. This hypothesis can be interpreted mathematically as the use of unique mean radon diffusion coefficient in gas phase by neglecting radon diffusion in liquid phase and changing the air-filled porosity ϕSg by Rg (see eqn 2g). However, eqn (9) for DbRN can only equal eqn (2b) forDFb if Kdg = 0 (no radon adsorption) and one of the three following conditions are realized: (1) single-phase gas conditions (Sl = 0), (2) effective radon diffusion coefficients in soil liquid-phase (Dle ) equal to that in soil gas-phase (Dge ) in eqn (2b), or (3) KOst = 0 in both eqns (2b) and (2g). The first condition can be true, but the two last ones are physically unrealistic in unsaturated conditions. Eqn (9) has been used blindly without any verification in many radon transport models in the subsurface (e.g., Andersen 2000; Ferry et al. 2001; Fournier et al. 2005). The purpose of this paper is to verify this third approximation in unsaturated soil conditions.
VERIFICATION EXAMPLE OF THE THIRD R&N—APPROXIMATION Analytical simulations Let us now consider the simple exercise of vertical radon transport in an unsaturated soil column of length H=10 m, submitted to constant concentration (Dirichlet condition) at both ends [C(z = 0) = C0; C(z = H) = CH; Fig. 1]. Isothermal
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Copyright © 2014 Health Physics Society. Unauthorized reproduction of this article is prohibited.
Bulk radon diffusion coefficient in unsaturated soils c Z. SAÂDI
Table 1. Physical properties of the Uranium Mill Tailings (UMT) soil (Ferry et al. 2002). Parameter
UMT
ϕ (—)
0.4 1,370 4.0856 10−13 60,000 0.05 0.32 0.15 3.4218 10−3 12
ρd (kg m−3) kpa (m2) Cs226Ra (Bq kg−1) Ea (—) Ew (—) S* (—) kdg0 (m3 kg−1)b b (—)b a
kp is the intrinsic permeability. Estimated from Rogers and Nielson (1991b).
b
conditions are assumed at 25°C such that: KOst = 0.2263 (Clever 1979), dg = 1.1 10 −5 m2 s −1 and dl = 1.4 10 −9 m2 s −1. The adsorption and emanation coefficients are calculated as functions of liquid-saturation [i.e., Kdg(Sl) and E (Sl)] using the equations of Rogers and Nielson (1991b) and Nielson et al. (1982b), respectively:
Kd g ¼ Kd 0g expð−bS l Þ
E¼
(10a)
n
E w S l =S þ E a ð1−S l =S Þ if S l S Ew elsewhere ;
(10b)
where Kdg0 is the distribution coefficient at dryness (m3 kg −1); b is a dimensionless correlation constant, generally lying between 10 and 15, but it can be significantly higher; Ew
601
and Ea are the emanation coefficients at full saturation and at dryness (−), respectively; and S* is the minimum water saturation on the plateau of an emanation–water saturation curve. The physical properties of the studied soil are given in Table 1 (Ferry et al. 2002). The soil, classified as sandy silt according to the USDA soil classification, corresponds to the uranium mill tailings from the Lavaugrasse (France) landfill with high radium activity concentration. Six analytical simulations (Table 2) were applied to the soil column at unsaturated conditions corresponding to three values of Sl (0.25, 0.5, and 0.75) and two values for gas pressure difference between the column boundaries ΔPg (0 and 1,000 Pa). The latter was chosen so as to produce an upward convective transport to the soil surface (negative gas-phase velocity value). The gas-phase velocity value is calculated using Darcy’s law, which is equal to vg = −2.23 10 −6 m s −1 for ΔPg = 1,000 Pa. Calculations were performed with C0 = 0; two values of CH, 0 and C∞ (eqn 2f); two values of KOst, 0 as a hypothetical value (i.e., all radon dissolved in liquidphase is volatilized into the gas phase) and 0.2263; and two values of Kdg0, 0 and 3.4218 10 −3 m3 kg −1. The effective diffusion coefficient in liquid-phase has been assumed negligible (Dle
