JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. B1, PAGES 567-580, JANUARY 10, 1993

Modeling Radon Transport in Dry, Cracked Soil

DIANA J. HOLFORD

Pacific Northwest Laboratory, Richland, Washington

STEPHEN D. SCHERY, JOHN L. WILSON, AND FRED M. PHILLIPS

New Mexico Institute of Mining and Technology, Socorro

A two-dimensional finite element code was used to investigate the effect of changes in surface air pressure on radon flux from soil with parallel, partially penetrating cracks. A sensitivity analysis investigates the effects of various crack dimensions, soil characteristics, and surface air pressure on radon flux from the soil surface to the atmosphere. Simulation restfits indicate that radon flux is most sensitive to soil properties; the diffusion coefficient is most important, followed by permeability and porosity. Radon flux is also sensitive to changes in barometric pressure, which cause variations in radon flux above and below the average diffusive flux. Sinusoidal variations in barometric pressure cause a net increase in the average radon flux from the soil, because increases in flux during periods of decreasing pressure are greater than the decreases in flux during periods of decreasing pressure of equal magnitude. Cracks were found to significantly increase radon flux from soils of low permeability.

INTRODUCTION

Radon 222 gas is a decay product of radium 226, which is found in soils that have weathered from rocks containing uranium 238. Radon transport in soil gas and groundwater has been studied for many years in connection with uranium exploration, as a tracer for moving air and groundwater, and as a possible predictor of seismic activity. Nazaroff [1992], Nazaroff and Nero [1988], and Tanner [1964, 1980] give comprehensive reviews of the extensive literature on the subject. Exposure to the decay products of radon gas, when they are allowed to accumulate in a closed space such as a uranium mine, has been linked to lung cancer. There is a possibility that high radon concentrations in buildings may also pose a risk to health. Factors that may increase the flux of radon out of the soil are therefore of great concern; cracks in soil are one of those factors and are investigated in this paper.

Radon fluxes from gravelly sandy loam soil were measured at a site near the New Mexico Institute of Mining and Technology (NMIMT) campus in Socorro, New Mexico [Schery et al., 1984]. The data, collected over 2-hour intervals, showed greater changes in surface radon flux than would be predicted by simple diffusion driven by the subsurface concentration gradients. Also, experiments that involved forcing air up through soil covered with a surface film of water showed the presence of discrete points where air exited rapidly [Schery et al., 1984]. It was hypothesized that soil cracks might be causing the rapid changes in the observed flux, because cracks have a higher permeability and diffusion coefficient than the surrounding soil [Schery et al., 1984] and because polygonal cracks were observed by the authors. The cracks were less than a millimeter wide, were spaced 0.05 to 0.1 m apart, and were probably not more than 0.1 m deep. At an outcrop near the site, cracks and plant roots extending from the surface to at least 5 m were observed.

Copyright 1993 by the American Geophysical Union.

Paper number 92JB01845. 0148-0227/92/92JB-01845505.00

In this paper, we discuss a two-dimensional finite element model developed to simulate the effects of transient surface pressure changes on radon flux from cracked soil. A complete derivation of the governing equations, the finite element equations, and verification of the model against available analytical solutions were presented in a previous report [Holford et al., 1989]. This paper presents the theory and development of the finite element equations used in the code. The model is used to investigate the sensitivity of radon flux to various soil and crack parameters and pressure perturbations. The model is then compared to field measurements of radon flux and atmospheric pressure.

PREVIOUS MODELING EFFORTS

Analytical Solutions

One of the earliest models of radon transport in soils was developed to simulate measured radon fluxes from soil at a site near the NMIMT campus [Clements, 1974]. Clements simulated the data collected over 12-hour intervals with an

analytical solution for constant gas velocity, diffusion of radon, radioactive decay, and production of radon in a one- dimensional, homogeneous, and isotropic soil. Clements' model has been used subsequently to show that subsurface pressure gradients measured in field soil could cause significant changes in subsurface radon concentrations and hence in the radon surface flux [Schery et al., 1984].

The effects of cracks in soil have been evaluated using simple analytical solutions for radon transport in soil with cracks penetrating to depths where concentrations of radon have reached a maximum [Schery and Siegel, 1986]. From these results, Schery and Siegel argued that cracks would affect surface fluxes more than they would affect the subsurface concentration gradients. Also, they concluded that cracks could cause a net enhancement of flux for sinusoidal variations

in pressure, such as those caused by fluctuations in wind speed or diurnal variations in atmospheric pressure. Our subsequent studies indicated the importance of cracks in enhancing advective transport through concrete slabs and from

567

568 HO•ORD ET AL.: Mor•m•o RADO• T•SPORT • DRY, CRACKED Sor[.

sufficiently deep and wide cracks in soil [Schery et al., 1988a,b].

An analytical solution has been derived for radon transport through cracks in concrete slabs overlying soil [Landman and Cohen, 1983]. This solution treated the crack not as an air-filled channel but as a line source whose strength depended on the width of the crack. Solutions were developed for both steady state diffusion and advection. Although this model is useful for estimating how much crack density will increase radon diffusion, in field situations the effects of transient barometric pressure fluctuations and heterogeneous soil properties are likely to be significant.

Several analytical solutions for solute transport in groundwater flowing in fractured rock have been derived [Neretnieks, 1980; Sudicky and Frind, 1982]. Typical assumptions for these solutions are that transport is by advection and dispersion in the crack and by diffusion in the soil transverse to the crack. However, radon concentrations

observed in soil exhibit strong vertical gradients, which will also cause diffusion parallel to vertical cracks.

MATI-IEMATICAL MODEL

General Considerations and Assumptions

The finite element model simulates pressure gradients in the soil and cracks caused by pressure changes at the ground surface. The resulting air flow drives the advective transport of radon gas, affecting the flux density of radon across the soil/air interface. The soil in the model may be heterogeneous and anisotropic, but it is isothermal and free of water down to the water table. The cracks in the model are evenly spaced and of a uniform width and depth.

Figure 1 is a schematic diagram of the model, showing various components affecting transport. Advection and diffusion are two-dimensional, because the cracks do not extend down to the water table and pressure changes will result in a complex, nonuniform flow field.

Molecular diffusion of 222Rn is assumed to be governed by Fick's law [Crank, 1975]. The advective flow of gas is assumed to be laminar and governed by Darcy's law. A constant source of radon-producing 226Ra is evenly distributed in the soil.

Numerical Models

Following up the one-dimensional modeling of Clements [1974], a finite difference model was developed that simulated transient radon transport and flux from a cylindrical coordinate system that is axisymmetric about a pinhole [Edwards and Bates, 1980]. Simulations were conducted to show how barometric pumping enhanced radon flux from pinholes in shotcrete, the upper surface of which is coated with an impermeable polymer and which overlies homogeneous, radium-bearing sandstone. Thousands of pinholes would be necessary to cause a significant radon flux through the polymeric barrier. However, planar cracks would open up more surface area to advective transport than pinholes and therefore would be more likely to enhance radon flux. This assertion is based on the assumption that air flow in cracks can be modeled by laminar Poiseuille flow between parallel plates [White, 1991] and that longitudinal velocities at the center of the crack would be proportional to pressure gradients by a permeability of w2/2, where w is the crack width. Assuming pinholes can be modeled as circular tubes, the longitudinal permeability at the center of the hole would be w2/4, half the value for planar cracks.

A three-dimensional model was used to simulate steady state advection and diffusion through cracks in a basement [Loureiro et al., 1990]. The model includes generation and decay of radon in the soil, transport through a finite size crack at the wall-floor joint in the basement, and the calculation of indoor radon concentration. A sensitivity analysis indicated that the most important soil parameters under steady state pressure differentials were soil permeability, the magnitude of the applied pressure differential, and the radon source term (which is proportional to the radium concentration in the soil).

There are many numerical models for groundwater flow and contaminant transport in fractured rock [Grisak and Pickens, 1980a,b; Huyakorn et al., 1983a,b]. The available computer codes, such as FTRANS [INTERA Environmental Consultants Inc., 1983], which uses Huyakorn et al.'s approach, have many of the same limiting assumptions as the analytical solutions discussed previously.

Radon Transport

The diffusive flux density in soil (Fa), the mass of radon transported per unit time per unit bulk cross-sectional area of soil by molecular diffusion, is defined by Fick's law for molecular diffusion [Bear, 1979]:

Fdi = - Dij •(nC ) (1)

where Dij is the diffusion coefficient for radon in dry soil, i and j are subscripts indicating direction and are summed over the ranges i=1,2 and j=l,2, C is the concentration of radon gas in

Atmosphere

':..' '..".'..".'..".'..":..".'.. 0 ..".'..".'..", .'...• ....' .....' .....' .....' .....' ... • '...'.'...'.'...• :.; .'.:.'.:.'..'.'.:.'..'., • .'..'.'..'.;.

Model B ndary

Water Table

Planes of Symmetry

Diffusion Advection

Advection and Diffusion -->'

Fig. 1. Two-dimensional model of radon transport in cracked soil.

HOL•OP.D ST •.: MODELING RADON TRANSPORT IN DRY, CRACKED SOIL 569

air, defined as the mass of radon per unit volume of air, nis the total porosity of the soil (ratio of the volume of void space to the total volume of soil), and nC is the mass of radon transported in dry soil per unit bulk volume of soil.

The diffusion coefficient for radon in dry soil is defined as [Bear, 1979]

Dij = DA xij (2)

where DA is the diffusion coefficient for radon in pure air and xij is the coefficient of tortuosity of the soil. The diffusion of radon in pure air was calculated as a function of temperature and pressure [Bird et al., 1960]. The tortuosity can be defined as the ratio of the straight-line distance between two points to the distance between the same two points by way of the connected pores. The tortuosity has been empirically determined to be between 0.01 and 0.66 for most soils [Bear, 1979].

The advective flux density (Fa), the mass of radon transported per unit time per unit bulk cross-sectional area of soil by air flow, is defined as [Bear, 1979]

Fag = v i C (3)

where Vi is the Darcy velocity of soil air, which is defined as the volume of air flowing per unit bulk cross-sectional area of soil per unit time in the direction i =1,2.

Conservation of mass results in the following continuity equation

•(nC) _ 3F i + •(nC)- n• •t •xi (4)

where t is time, Fi is the total flux density (in some direction i = 1,2), • is the 222Rn decay coefficient (ln(2) / half-life of 222Rn), and the source term, qb is the production rate of radon per unit volume of soil pore space.

Substituting equations (1) and (3) into equation (4) and neglecting the effect of rock compressibility yields the governing equation for radon transport in dry soil:

•C - •)t 3X i •X i (5)

Cracks and holes are assumed to be perfectly dry, with a porosity of ! and no production of radon. Therefore the radon transport in cracks can be described by

a Io?cl a c)- xc •)t •Xi• •--•]-•ixi (6)

Air Flow

Radon transport and air flow are coupled by Darcy's law [Bear, 1979]'

) Vi =- 7 ta;l• + pgj (7) where vi is the Darcy velocity, kij is the intrinsic permeability of the dry soil, I.t is the dynamic viscosity of air at a given temperature, P is the absolute air pressure, p is the density of air at a given temperature, and gj is the gravitational acceleration vector.

Conservation of mass requires that the change of fluid mass stored within a unit volume of soil equal the net rate of fluid

flow into that volume. The resulting continuity equation takes the form [Bear, 1979]

•)t •x i (8)

Combining the continuity equation (8) and Darcy's law (7) gives

•)t - •x i •xj + pgj (9)

For an ideal gas, the equation of state is

p =MP RT (10)

where M is the molecular weight, R is the universal gas constant, and T is the absolute temperature. For isothermal flow, then

c3p _ M c3P 3t RT •t (11)

Substituting equations (10) and (!1) into (9) and assuming the effects of rock compressibility to be negligible gives an equation for isothermal air flow

• g •_• kij •P p l + ?• kij n -•-= OXi•T•j• ! OXi[ [1 PgJP (12) Poiseuille flow between parallel plates [White, 1991]. The permeability distribution is parabolic, with the maximum permeability at the center of the crack and an average vertical permeability of

w 2 kc2 =•-• (13)

where w is the width of the crack. The horizontal permeability in the crack, kcl, is arbitrarily assigned a value several orders of magnitude larger than that of the soil.

Boundary Conditions

The boundary conditions at the soil surface may vary in both space and time. The boundary conditions for the flow and transport equations for the two-dimensional soil model with cracks are shown in Figure 2. At the top boundary, the atmospheric pressure is prescribed but varies with time. The concentration at the top of the soil is prescribed as zero, but the concentration at the top of the crack is allowed to rise above zero by specifying no concentration gradient. Advective flux of radon occurs across the top crack boundary if the velocity is not equal to zero. The left and right boundaries of the model are assumed to be axes of symmetry, at the center of the crack and at the midpoint between two cracks, respectively. The bottom boundary, at the water table, is considered to be impervious to flow and transport. The boundary conditions for the flow and transport equations for a one-dimensional soil model without cracks are exactly the same as those for the soil in the two-dimensional case.

Verification of Model

The governing equations for air flow and radon transport are solved using the Galerkin finite element numerical method

570 HOLFORD ET AL.: MODFJ_.ING RADON TRANSPORT IN DRY, CRACKE• SOIL

P = Patm(t).•

•C/•x2 = 0 -' •.. C = 0 • -• • -. Atmosphere

x• I I I I

x2 I I • 5...'..::.:.•s oi•i'...."_..'.:...':?: I I

I I • "'.'.".'.'."....'...'.'........'.'......'............'.. ..'::...'...':i. I I I I I_ I__•

I I

c3P/c3x • = 0 c3P/c3x• = 0 I I c3C/c3x• = 0 • = 0 I I • I I I

•,•p/•) •-- Water Table x2=0 c3C/c3x: = 0

Fig. 2. Boundary conditions for a two-dimensional gas flow and radon transport simulation.

[Huyakorn and Pinder, 1983] and a fully implicit time-weighting scheme. A complete derivation is given by Holford et al. [1989].

Four test cases were run to verify the accuracy of the numerical model by comparing it to analytical solutions found in the literature. The radon transport calculations were compared to a one-dimensional analytical solution of the advection-dispersion equation with radioactive decay and a linear source term [Clements, 1974]. If the concentration is zero at the surface, the surface flux density is given by

F(z=0) = • •- + + (14) while the concentration as a function of depth is given by

C:•{1-ex•z[2%+{(2%)2+D•---)l/2]]} (15) The transport calculations were tested in two dimensions for

the case of a single, vertical crack with vertical advection, dispersion, and decay in the crack and horizontal dispersion and decay in the soil matrix [Sudicky and Frind, 1982]. The flow equations were verified by linearizing equation (12) and comparing the results to one and two-dimensional transient heat conduction equations [Carslaw and Jaeger, 1959]. Complete results have been presented by Holford et al., [1989].

SENSITIVITY ANALYSIS

In the sensitivity analysis, the input parameters of the model were varied to better understand how they affected radon flux densities from soil. First, the crack dimensions, including crack depth, width, and spacing, were varied. Second, the soil properties, including porosity, intrinsic permeability, and soil diffusion coefficient, were perturbed for cracked and uncracked

TABLE 1. Base Case Parameters for Sensitivity Analysis

Parameter Value

Air dynamic viscosity g 1.8x10-5 kg/m s Radon diffusion coefficient in air DA 1.0X10-5 m2/s Radon decay coefficient •. 2.1x10-6 Radon production rate • 1.1x10-4 Bq/L s Depth to water table 10 m Crack depth 4 m Crack spacing 8 m Crack width 6x10-4 m

Soil porosity n 0.35 Soil intrinsic permeability k 2.7x10-1• m2 Tortuosity • 0.26 Radon diffusion coefficient in soil D 2.6x10-6 m2/s Initial atmospheric pressure Po 85 kPa Rate of change in atmospheric pressu• •atm/At -0.09 kPa/h Time increment At 15 min

From Schery et al. [ 1984]

Base Case

The base case parameters for the sensitivity analysis are listed in Table 1. The depth to the water table used is an estimate for the Socorro, New Mexico, field site, which is 1370 m above

mean sea level [Schery et al., 1984]. The initial atmospheric pressure is one that is common for the field site. The production rate of radon per unit volume of soil pore space, o•,

0.0

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f i,f l, f, fl, f ff It f ,,

: 1!) 20 - i - i

; 30 - I - , 40 - I

I I --

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0.0 2.0 4.0

Distance from Crack Center (m)

soils. Third, the effects of decreasing, increasing, and Fig. 3. Steady state concentration (Bq/L) profile for soil with cracks 4 m sinusoidally varying the atmospheric pressure conditions with deep, 6x10-4 m wide, and 8 m apart (dashed line is crack boundary, width time were tested. Fourth, the depth to the water table was exaggerated 160x). The concentration profile reaches an asymptotic value varied. of 52.5 Bq/L with depth, which is equal to q•/•,.

HOLFORD ET AL.' MODEI2NG RADON TRANSPORT IN DRY, CRACKED SOIL 571

was determined for the soil at the field site based on the

observation that the radon concentrations asymptotically reach a maximum value of 52.5 Bq/L with depth. Far below the surface, where 222Rn escape is negligible, the rate of radioactive decay must be balanced by the production rate. This requires that the 222Rn concentration deep in the soil approach •/•. The decay constant for radon, •, is 2.1x10-6 s-l, yielding a value of 1.1x10-4 Bq/L s for the production rate.

The quadrilateral finite element grid used for the simulations was very fine near the crack in the xl direction and near the surface in the x2 direction. This fine grid was necessary because concentration gradients are likely to be strong in those areas. All simulations were started from constant pressure and steady state concentration conditions. Initially, the soil and cracks are both at atmospheric pressure. The steady state concentration profile incorporates the equilibrium effects of diffusion, decay, production, and boundary conditions. Equilibrium conditions vary slightly based on the width and depth of the crack. Figure 3 shows a steady-state diffusive profile calculated using the base case parameters in Table 1. The concentration gradient is strongest near the surface and approaches an asymptotic value with depth.

Figure 4 shows that the total radon flux density increases with time as the atmospheric pressure deereases. The fraction of flux from the crack and the portion from the soil are also shown in Figure 4. The fraction of flux from the crack is much smaller than the fraction from the soil beeause the volume of the cracks

is very small compared to the volume of surrounding soil. The increase in flux is nonlinear for the first hour or two, while radon concentrations are still increasing.

Figure 5 shows that the subsurfacepressure gradients generally slope toward the surface and crack, but pressures are strongest near the bottom of the crack. As a result, radon is driven into the

0.06 i i i i i i i i i i i i i i i i i i i i i

t Total Flux from :

I Uncracked-Soil Model ' 0.03 - [ Total Flux from

0.02 •- Cracked-Soil Model - • =Advective+Diffusive •_ Advective Flux from _- 0.01

0

-0.01 0 1 2 3 4 5 6

Time (hours)

Fig. 4. Effect of decreasing atmospheric pressure at a rate of 90 Pa/hour on radon flux from soil with cracks 4 m deep, 6x10-4 m wide, and 8 m apart. The fraction of flux density from the crack was calculated by dividing the advective flux from the crack by the crack width.

-4.0

-6.0

-8.0

-10.0

$4.9800 --

84.9900-

0.0 2.0 4.0

Distance from Crack Center (m)

0.0

-6.0-

-

- --

-8.0- _

--

-10.0

--•-- 84.9'/25

--

--

--

0.0 2.0 4.0

Distance froln Crack Center

Fig. 5. Effect of decreasing atmospheric pressure at a rate of 90 Pa/hour on gas pressures (kPa) in soil with cracks 4 m deep, 6x!0-4 m wide, and 8 m apart (dashed line is crack boundary, width exaggerated !60x). Subsurface pressure gradients increase rapidly at first, eventually reaching a steady state. (a) 0.25 hour, (b) 0.5 hour (c) 6 hours.

572 HoLvo• ET AL.: MODEI•G RADON TIL•SPORT IN DRY, CRA(XED SOIL

-6.0 - -

-

-8.0- .

.

-!0.0

: 84 4625 .i -2.0 '

-4.0

------ 84.4825 '•-

rl'!i I • III J'• •"l I'i

0.0 2.0 4.0

Distance from Crack Center (m) Fig. 5. (continued)

crack, where velocities arc higher duc to the crack's higher ,

permeability. Because the prcssurc gradients around the bottom of the crack increase with time, radon concentrations in the crack also increase steadily with time (Figure 6). However, the difference in concentrations is much greater betwcen 0.5 and !.0 hour than between 1.0 and 2.0 hours. This indicates that changes in prcssurc gradients and radon concentrations are rapid at first but cvcntually reach some equilibrium. After radon concentra- tions have reached equilibrium, the increase in flux per unit time is proportional to thc decreasc in atmosphcric pressure.

Crack Dimensions

Cracks of different depths, widths, and spacings were modeled to see how much the total flux density could be enhanced over the flux predicted by a model of soil without cracks. All other parameters are the same as in Table 1.

The results of the cracked-soil model and a one-dimensional

uneracked-soil model were compared after 6 hours of decreasing atmospheric pressure. The one-dimensional uneracked-soil model uses the same governing equations and boundary conditions as the cracked-soil model, the only difference being that no cracks are present in the soil. Results are compared at a time of 6 hours because the high rate of decreasing pressure used in the model was never observed to persist for a period longer than 6 hours at the Sototto field site.

Crack depth. The effect of increasing crack depth is shown in Figure 7, where an increase in depth results in an increase in

• 0.0

-0.2

= -0.4-

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o -0.6

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IO ,

20

30

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0.0 0.2 0.4 0.0 0.2 0.4

Distance fi'om Crack Center (m) Distance from Crack Center (m)

-0.6

-0.8 - -

-1.0 i 'l''","'" 0.0 0.2 0.4

Distance from Crack Center (m)

Fig. 6, Effect of decreasing atmospheric pressure at a rate of 90 Pa/hour on radon concentrations (Bq/L) in soil with cracks 4 m deep, 6x104 m wide, and 8 m apart (dashed line is crack boundary, width exaggerated 16x). Concentrations in the crack increase with time, enhancing radon flux from the soil. (a) 0.25 hour, (b) 0.5 hour (c) 6 hours.

Hou:oP.v v.? nL: Movat•o Rn• T•,•Sl•nr • Dn¾, C•cxat) Son. 573

0.065

0.06

-• 0.055

0.05

0.045

Crack Depth 0.5

No Cracks

004 II ' ' ' I .... I .... I , , , , I .... I , , , , ß

0 1 2 3 4 5

Time (h)

2.5

0.5

' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' '

'• "' Crack Width = lx10 -2 rn

"•,/ 3x10-3 m • / lxlO -3 rn

3x10 -4 rn :

0.4 0.8 1.2 1.6 2

Crack Spacing (m)

Fig. 8. Effect of crack spacing and width on percent difference in flux density between soil with 0.3 m deep cracks and soil without cracks after 6 hours of decreasing atmospheric pressure at a rate of 90 Pa/hour. Radon flux from shallow cracks shows little sensitivity to variations in crack width and spacing.

coefficient are higher in the cracks. Closer spacing causes an Fig. 7. Effect of crack depth on radon flux from soil with cracks 4 m deep, increase in flux density only until pressure drawdowns between 6x104 m wide, and 8 m apart during 6 hours of decreasing atmospheric the cracks interfere, and then flux density decreases sharply. pressure at a rate of 90 Pa/hour. Radon flux increases as cracks penetrate This effect is similar to the way drawdowns between adjacent higher subsurface radon concentrations. wells can overlap and reduce the amount of water pumped by

flux density compared to that predicted by the uncracked-soil model. The increase in flux density is not proportional to crack depth but rather approaches an asymptotic value. As the crack deepens, it transports higher radon concentrations from depth, causing the radon concentrations near the surface and therefore the radon fluxes to increase. However, the velocities at the bottom of the crack become lower as the crack deepens. Therefore, even though the bottom of the crack may intercept higher radon concentrations, the rate at which the radon is brought to the surface decreases.

Crack width and spacing. The effects of crack width and spacing on a relatively shallow crack (0.3 m)can be seen in Figure 8. As cracks become wider and more closely spaced, the flux density increases. The increases in flux in comparison to those predicted by the uncracked-soil model are very small, less than 3%. A shallow crack does not penetrate far enough into the soil to intercept the higher radon concentrations at depth.

Flux density increases with crack width. Flux density is related to crack width because the permeability of the crack is proportional to the square of crack width (equation (13)). Close crack spacing results in having a greater number of cracks per unit area, causing an increase in total flux because of the higher permeability and diffusion coefficient in the cracks.

Figure 9 shows that effects of width and spacing for a relatively deep crack (4 m) differ substantially from those for a shallow crack (Figure 8). The increases in flux are much larger

eachß This result is consistent with those of Nilson et al. [1991], whose analytical solution for the sinusoidal pressure response of a medium containing identical vertical fractures equally spaced by slabs of permeable matrix material predicted a maximum rate of transport for an intermediate fracture spacing (typically a few meters). Another contributing factor is that, because there is no source of radon in the cracks, flux will

eventually decrease as cracks are spaced closer together.

20

Crack Width = lx10 -2 rn

lx10 -3 m •

•• ( 6x10-4 m Base Case • •

3x10 -4 m .

.

0 5 10 15 20

Crack Spacing (m) for a deep crack than for a shallow one; closer spacing causes an increase in flux density up to a point, but then with closer Fig. 9. Effect of crack spacing and width on percent difference in flux

density between soil with 4 m deep cracks and soil without cracks after 6 spacing flux density decreases sharply. Thus packing cracks hours of decreasing atmospheric pressure at a rate of 90 Pa/hour. Radon closer together enhances radon flux under transient conditions flux from deep cracks is significantly more sensitive to changes in width up to a point, because the permeability and diffusion and spacing than that from shallow cracks.

574 HOL•ORD ET AL.: MOD•..•O RADON TRA•S•RT • DRY, CRA• Sore

Soil Properties

Simulations with varying soil porosity, permeability, and diffusion coefficients are presented in this section. All other parameters for these simulations are the same as in Table 1.

Porosity. The porosity of the soil (n)was varied while holding the permeability (k) equal to 2.7x10-12 m2 and the diffusion coefficient in soil (D) equal to 2.6x10-6 m2/s. Figure 10 shows radon flux from uricracked and cracked soils of

varying porosity. The effect of variations in porosity is larger than that of addition of cracks to the soil. Because the diffusive

flux density is proportional to the porosity, the flux density from both homogeneous and cracked soils increases with an increase in porosity. Also, the greater the porosity, the more the flux density increases per unit time. This effect is more pronounced in cracked soils because the porosity of the cracks is defined as 1.0. The pressure gradient into the crack increases with increasing soil porosity, resulting in more advective flux out of the crack. The governing equation (12) for gas flow dictates that the pressure gradient be directly proportional to the porosity. Increasing soil porosity increases the gas storage capacity of the soil for changes in pressure, based on the continuity equation (8), so that pressure changes do not move as far down into the soil.

Permeability. The permeability of the soil was varied while

0.06

0.055

0.05 0.045

0.04

Permeability = • •' 1 1.OxlO -11 m 2 • •

!

- //• • // 2.7xlO'12m 2

1.OxlO -13 m 2

1 2 3 4 5

Time (h)

the porosity was held at 0.35 and the diffusion coefficient in Fig. 11. Effect of soil permeability on radon flux from soil with cracks 4 m soil was held at 2.6x10-6 m2/s. The permeabilities selected were deep, 6xl 0-4 m wide, and 8 m apart during 6 hours of decreasing one order of magnitude greater and less than the base-case atmospheric pressure at a rate of 90 PaPaour (dashed curves, cracked soil value. model; solid curves, uncracked soil model). For uncracked soils, radon flux

For the uncracked soils shown in Figure 11 (solid curves), is higher from higher permeability soil. For cracked soils, however, the intermediate permeability produces the largest flux. This seemingly radon flux is higher from higher-permeability soils. This result anomalous result is discussed in the text.

0.09

0.08

Poros = 0.5

0.07

0.06

0.05

0.04

0.03

0.02

0.35

0.2

0 1 2 3 4 5 6

Time (h) Fig. 10. Effect of soil porosity on radon flux from soil with cracks 4 m deep, 6x10-4 m wide, and 8 m apart during 6 hours of decreasing atmospheric pressure at a rate of 90 Pa/hour (dashed curves, cracked soil model; solid curves, uncracked soil model). Because the diffusive flux density is proportional to the porosity, the flux density from both homogeneous and cracked soils increases with an increase in porosity.

is to be expected, given that soil air velocities, and hence advective flux in the soil, are proportional to the permeability. However, for the cracked soils shown in Figure 11 (dashed curves), radon flux is lower from both lower-permeability (lx10-13 m2) and higher-permeability (lx10-l! m2) soils than it is from the base case (2.7x10-12 m2)soil. This seemingly anomalous result can best be explained by reviewing Figure 9, where radon flux varies as a function of crack spacing. The effect of increasing the soil permeability is similar to that of decreasing the crack spacing. Noting the location of the base case run in Figure 9, it can be seen that either increasing or decreasing the crack spacing can result in a lower radon flux. Assuming that increasing or decreasing the soil permeability would have the same effect, it is not surprising that the radon flux was lowered both by increasing and by decreasing the permeability.

It can also be seen in Figure 11 that the difference between the flux from a soil with no cracks and that from a soil with

cracks is greatest for the lowest-permeability soil. Decreasing soil permeability prevents pressure decreases from propagating into the soil, but the crack permeability remains the same, so that the pressure gradients from the soil to the crack increase. As a result, the concentrations in and close to the crack increase

significantly with a decrease in soil permeability. This causes a nonlinear increase in flux density, because the increase is due almost entirely to an increase in flux density from the crack while the flux density from the soil remains nearly the same. For the highest-permeability soil tested, the difference in flux from a soil with cracks and that from a soil without cracks was

HOL•Om:) ET AL.: MODELING RADON TRANSPORT IN D•¾, C•tACXSD Son. $75

0.12

0.1

• 0 08 • '

Diffusion Coefficient = 1.3x 10 -5 m2/s '

. 2.6x10 -6 m2/s 0.06 - __-,

0.04 - __ ........

0.02

0 ,,,,I,,,,I,,,,I,,,,I,,,,I,,,, 0 1 2 3 4 5 6

Time (h) Fig. 12. Effect of soil diffusion coefficient on radon flux from soil with cracks 4 m deep, 6x104 m wide, and 8 m apart during 6 hours of decreasing atmospheric pressure at a rate of 90 Pa/hour (dashed curves, cracked soil model; solid curves, uncracked soil model). Radon flux increases as the soil diffusion coefficient is increased.

negligible because the permeability contrast between the crack and the soil is not great enough to cause strong pressure gradients from the soil to the crack.

Soil diffusion coefficient. The diffusion coefficient in soil (D) was varied while the porosity (n) was held at 0.35 and the permeability (k) was held at 2.7x10-12 m2. The two values of soil diffusion coefficient chosen were 5 times greater and 5 times less than the value used in previous simulations. The higher value is close to the diffusion coefficient of radon ia pure air (lx10-5 m2/s).

For uncracked soils, Figure 12 shows that radon flux increaqcs as the soil diffusion coefficient is increased. This resuh is

consistent with the analytical solution for steady state diffusion from a semi-infinite soil, where the surface flux density (equation (14)) is proportional to the square root of the diffusion coefficient. The radon flux curves for cracked soils

with different diffusion coefficients also have the same slope at late times (Figure 12), but at early times the slope is higher for the soil with a small diffusion coefficient. The greater the contrast between the diffusion coefficient in the crack (lx10-5 m2/s) and the diffusion coefficient in the soil, the greater the difference in radon flux between the uncracked- and

cracked-soil models. The early time nonlinear increase in flux results from a rapid buildup of radon concentrations in the crack. Eventually, the concentrations in the crack reach equilibrium and the increase in flux is more gradual.

A smaller soil diffusion coefficient results in an increase in

concentrations in both the crack and the soil. With a longer diffusion path in the soil, the radon cannot escape quickly to the surface and concentrations in the soil increase. Also, the

crack brings up radon that cannot rapidly diffuse back into the soil, and concentrations build up in the crack. Conversely, a shorter diffusion path allows more radon to escape across the soil/air interface, so the flux density increases as the soil diffusion coefficient of the soil is increased.

Pressure Variations

This section examines the effect of increasing and decreasing atmospheric pressure, consistent with rates measured in the field [Schery et al., 1984]. The effects of short-term sinusoidal fluctuations in surface pressure are also examined. All other parameters for these simulations are given in Table 1.

Linear variations. The effect of changing atmospheric pressure at different rates on radon flux is shown in Figures 13 and 14 for uncracked and cracked soils, respectively. The rate of decrease in flux density for a certain rate of increasing pressure is not as great as the rate of increase in flux density for a decrease in pressure of the same magnitude. As the rate of pressure decrease at the surface increases, the subsurface pressure gradients in the soil and crack increase proportionately. The velocities in the soil and the crack then increase, bringing up radon from zones of greater concentration at depth, causing concentrations to increase. The faster the radon is advected, the

less radon is lost to radioactive decay before it reaches the surface, thus increasing concentration gradients near the surface. The combined increase in advective and diffusive flux

towards the soil surface causes the rate of increase in flux to be

disproportionately greater for a rate of decreasing pressure of- 270 Pa/h than for-30 Pa/h.

When the surface pressure is increased at a constant rate, the increase in downward subsurface gradient is proportional to the increase in the rate of pressure change. Increasing surface

0.12

0.1

0.08

0.06

0.04

Change in Atmospheric Pressure per unit time = -270 Pa/h

0.02 +270 Pa/h

0 1 2 3 4 5 6

Time (hours)

Fig. 13. Effect of linear atmospheric pressure variations on radon flux from soil with no cracks. The rate of decrease in flux density for a certain rate of increasing pressure is not as great as the rate of increase in flux density for a decrease in pressure of the same magnitude.

576 HoL•om> ET nL.' MODm.•NO Rnr•N T•d•SmaT • DaY, CanomD Sore

0.12

0.1

Change in Atmospheric Pressure per unit time = -270 Pa/h

0.08

0.06 -90 Pa/h

• -30 P•/h_ _-

0.04 •----•-•--- •,,,,.-- L-- L-- --- -+'•EP"•'/h- •

0.02 0 1 2 3 4 5 6

Time (h) Fig. 14. Effect of linear atmospheric pressure variations on radon flux from soil with cracks 4 m deep, 6x104 m wide, and 8 m apart. The flux from the cracked soil is greater than from uncracked soil during periods of both increasing and decreasing surface pressure.

pressure causes the direction of gas flow to reverse; air now flows from the surface down into the crack and into the soil.

Although the pressure gradients are downward for the case of increasing atmospheric pressure, the concentration gradients in both the soil and the crack remain upward because the concentration of radon in the atmosphere is assumed to be zero. For this reason, diffusion continues to bring radon to the surface of the soil.

The net flux from the cracked soil is greater than from uncracked soil during periods of rising pressure. During periods of rising pressure at the surface, in the one-dimensional homogeneous soil model, velocity vectors are completely vertical. In the two-dimensional cracked model, there is a

horizontal velocity component pointing away from the crack, that deflects the downward velocity gradients from the soil surface. The net effect is that the downward vertical velocity gradient at the soil surface is smaller in the cracked soil than in the uncracked soil, thus increasing the upward flux.

Sinusoidal variations. Sinusoidal variations in

atmospheric pressure around a mean value of 85 kPa were applied at the upper boundary of the model. The period of the sine function was varied while the amplitude remained constant.

Figure 15 shows the effect that a sinusoidal pressure variation with a long period (2 hours) has on radon flux density. Soil with cracks shows a greater variation in flux than an uncracked soil, and in either case the mean flux increases slightly with

0.12

0.09

0.06

0.03

1.

...... Atmospheric Pressure 1.

Flux from Uncracked Soil

........... Flux from Cracked Soil

0 ,,,• .... • .... i .... • .... •'''' 84.15 0 1 2 3 4 5 6

84.95

84.75

84.55

84.35

Time (h)

Fig. 15. Effect of sinusoidal atmospheric pressure variations (period = 2 hours, amplitude = 90 Pa) on radon flux from soil with cracks 4 m deep, 6x10-4 m wide, and 8 m apart. Soil with cracks shows a greater variation in flux than an uncracked soil, and in either case the mean flux increases slightly with time.

analytical solution for the sinusoidal pressure response of a medium containing identical vertical fractures equally spaced by slabs of permeable matrix material. They found that in a fractured permeable medium "the fractures will generally serve as breathing passages for all of the gas-filled porosity, greatly increasing the amplitude and nonuniformity of vertical motions" relative to a homogeneous porous medium.

0.15 .... [ .... ] .... ] .... ] .... [ .... 85.15

0.12

0.09

0.06

0.03

ß ß ß ß

...... ^tmosphcric Pressure

-.' '.. : • : \

Flux from Uncracked Soil

........... Flux from Cracked Soil

84.95

84.75

84.55

84.35

0 .... i .... I .... I .... • .... i .... 84.15 0 0.1 0.2 0.3 0.4 0.5 0.6

Time (h)

time. The increase in the mean flux can be explained by the Fig. 16. Effect of sinusoidal atmospheric pressure variations (period = 0.2 results of the previous section, where a decrease in pressure hour, amplitude = 90 Pa) on radon flux from soil with cracks 4 m deep,

6x10-4 m wide, and 8 m apart. Decreasing the period of the sinusoidal caused a greater increase in flux than the decrease in flux caused variation in surface pressure by a factor of 10 causes the mean flux and the by an increase in pressure of the same magnitude. This result is variation in flux to be significantly larger for cracked soil than for soil with consistent with that of Nilson et al. [1991], who derived an no cracks.

HOL•OR• ET AL.' MODELING RADON TRANSlaORT IN D•¾, CV. AOmD Son. 577

There appears to be very little phase lag between a change in surface pressure and the response of the radon flux density in Figure 15. The lag time between changes in surface pressure and changes in concentrations near the surface is very short. It may also be observed that the response of surface flux to changes in pressure equilibrates rapidly. In other words, the amplitude of the variation in the fluxes in Figure 16 increases only slightly with time. This result is not surprising, because both pressures and concentrations were observed to reach steady state quickly for constant changes in pressure (see discussion related to Figure 5).

Decreasing the period of the sinusoidal variation in surface pressure by a factor of 10 (Figure 16) increases the change in pressure per unit time. In this case, the mean flux and the variation in flux are much larger for cracked s0i ! than for soil with no cracks. Again, this relates to the discussion in the previous section, where the increase in flux from a cracked soil was much larger than for an uncracked soil during periods of decreasing pressure. Not surprisingly, increasing the amplitude of the sinusoidal pressure variation while holding the period constant yielded similar results [Holford et al., 1989].

Depth to Water Table

........... Field Measurements , , , Model without Cracks

...... Atmospheric Pressure (kPa) 0.25 ',,,i,,,i.., i,,.i.,', i•.• i..,i,',.i',',, i.,.

0.2 - ,,, ,,,

0.15

o

86.5

85.5

84.5

83.5 82.5

81.5 80.5

79.5

78.5 48 96 144 192 240

Time (h)

Figure 17 shows the effect of varying the depth to water table Fig. 18. Comparison of radon fluxes measured at the socorro field site to radon fluxes predicted by a model with no cracks. Flux predicted by one- on the base case simulation with parameters detailed in Table 1. dimensional model is not as variable as measured flux. Although the steady state diffusive flux increases slightly with increasing depth to water table, there is a significantly greater rate of increase in radon flux from the soil under the influence

of decreasing atmospheric pressure. Persistent pressure decreases

0.09

0.08

: - Depth to Water - Table = 30 m

0.07

0.06

0.05

0.04

!

!

!

!

/ 10m /

31TI

0 1 2 3 4 5 6

Time (hours)

or increases at the surface are able to propagate to great depths in the soil, much deeper than the diffusive path length. Also, as surface pressure decreases, the cracked soil model shows a greater rate of increase in radon flux relative to the uncracked soil model. This appears to be a very sensitive parameter, even at time periods of less than I hour.

COMPARISON OF MODEL RESULTS TO FIELD DATA

This section compares the results of the sensitivity analysis to data collected at the Socorro, New MeXico, field site [$chery et al., 1984]. The effects of long-term atmospheric pressure changes, shorter-term wind speed fluctuations, and diurnal temperature variations on radon fluxes from soil are considered.

Figures 18 and 19 show atmospheric pressure and radon flux density measured at 2-hour intervals. These figures also show the results of two model simulations: a model of soil with no

cracks (Figure 18) and a model of soil (Figure 19) with cracks 4 m deep, 8 rn apart, and 6x10-4 m wide. The atmospheric pressures shown were used as the top boundary condition for both the uncracked-soil and the Cracked-soil models. Typical pressure changes were between 30 ?a/h and 90 Pa/h, with a few changes of up to 240 Pa/h, The soil parameters in Table 1 were used.

Although the cracked-soil model matches some of the observed peaks in radon flux more closely than the uricracked- soil model does, some variations in the observed fluxes do not correlate with the changes in atmospheric pressure. $chery et al. [1984] found that just within the limits of detection, a

Fig. 17. Effect of varying the water table depth on radon flux from soil correlation between wind and subsurface radon could be found. with cracks 4 m deep, 6x10-4 m wide, and 8 m apart (dashed curves, Schery and Siegel [1986] measured subsurface pressure cracked soil model; solid curves, uncracked soil model). Although the gradients between 5 and 56 cm below the surface, atmospheric steady state diffusive flux increases slightly with increasing depth to water table, there is a significantly greater rate of increase in radon flux from the pressure, and wind speed at the same Socorro field site. They soil under the influence of decreasing atmospheric pressure. found that the subsurface pressure gradients were correlated to

578 Hot.tORn st AL.: Mor•m.t•a RADON TRAnSPORt: • DRY, CRACKED SOIL

........... Field Measurements ......... Model with Cracks

...... Atmospheric Pressure (kPa) 0.25 ., i, •.,,.,, i ,",, / 'i,', •', ',', •,',' ,'

0.2

0.15

0.05 lo _ / • ,-'

0 f ' :

86.5

85.5

84.5

83.5 82.5

81.5

80.5 79.5

78.5 40 80 120 160 200 240

Time (h)

Fig. 19. Comparison of radon fluxes measured at the socorro field site to radon fluxes predicted by a model with cracks 4 m deep, 6x10-4 m wide, and 8 m apart. Cracked-soil model matches some of the peaks in measured radon flux more closely than the uncracked-soil model. However, some variations in the measured fluxes do not correlate with the changes in atmospheric pressure.

wind speed as well as atmospheric pressure. Pressure gradients of 0.1 Pa/cm were typical, while a few were as high as 0.5 Pa/cm.

A correlation between wind and radon flux has been observed

by other workers. At the Nevada Test Site, Kraner et al. [1964] found that the average observed flux on days of high wind was nearly twice the diffusion theory prediction. The effects of the diurnal pressure variation and the variation in surface pressure fluctuations due to atmospheric turbulence have been modeled [Duwe, 1976]. The surface pressure fluctuations produced by atmospheric turbulence were parameterized as a function of wind speed and produced soil air velocities which increased the transport of radon from the soil.

Diurnal variations in temperature could affect radon flux from the soil, by causing buoyant movement of air or increasing evaporative flux of water vapor from the soil. However, Schery et al. [1984] and $chery and Petschek [1983] did not find a detectable correlation between subsurface radon

concentrations and atmospheric or soil temperature at the Socorro field site.

CONCLUSIONS

then with closer crack spacing flux drops sharply. With close crack spacing, pressure decreases in adjacent cracks interfere with each other, resulting in reduced pressure gradients in the soil toward the surface and crack.

3. Increasing the width of a crack causes radon flux density to increase as atmospheric pressure decreases, up to some limit governed by the spacing and depth.

4. The effect of variations in porosity is larger than that of addition of cracks to the soil. Because the diffusive flux

density is proportional to the porosity, the flux density from both homogeneous and cracked soils increases with an increase in porosity.

5. For uncracked soils, radon flux is proportional to soil permeability. However, for cracked soils, radon flux may be higher from lower-permeability soils than from higher- permeability soils with the same size cracks. Stronger pressure gradients occur in the soil because the decreased permeability does not allow pressure drops at the surface and crack to move into the soil. Stronger pressure gradients in the crack increase concentrations and surface flux. Therefore, when measuring the permeability of a soil, it is important to collect a sample large enough to include cracks.

6. The diffusive flux across the soil boundary is proportional to the diffusion coefficient in soil. Increasing the diffusion coefficient of the soil causes an increase in radon flux density, because the larger the value of diffusion, the shorter the diffusion path through the soil, which in turn enables radon to reach the surface faster without decaying. The greater the contrast between the diffusion coefficient in the crack and the

diffusion coefficient in the soil, the greater the difference in radon flux between the uncracked- and cracked-soil models.

7. The decrease in flux density caused by rising pressure is not as great as the increase in flux density caused by dropping pressure by the same magnitude. Therefore, periods of alternately dropping and rising pressure (such as a sinusoidal variation) show a net enhancement of flux. Decreasing the period (or increasing the amplitude) of a sinusoidal variation in atmospheric pressure causes an increase in mean radon flux densities and also in the deviation about the mean.

8. Under conditions of decreasing atmospheric pressure, radon flux increases significantly with increasing depth to the water table.

9. Although the two-dimensional model with cracks matches some of the observed peaks in radon flux better than the one-dimensional model, there are high-frequency variations in the observed fluxes that do not correlate to the changes in atmospheric pressure. Soil cracks and short-term variations in pressure attributed to wind may be sufficient to explain the difference between observed radon fluxes and those predicted by a one-dimensional radon transport model driven by barometric pressure changes.

The following major conclusions can be drawn from the modeling results presented in this paper.

1. During decreasing atmospheric pressure, increasing the depth of a crack causes an increase in radon flux density up to an asymptotic value. This asymptotic value is approached because the velocity near the bottom of the crack decreases as the crack gets deeper and because the radon concentration reaches an asymptotic value with depth.

2. Decreasing crack spacing causes radon flux density to increase as atmospheric pressure decreases, up to a point, but

RECOMMENDATIONS FOR FWI•RE WORK

Based on the results of the sensitivity analysis, radon flux appears to be sensitive to the values of the soil properties selected, with the diffusion coefficient being most important, followed by the permeability and porosity. Radon flux is also very sensitive to changes in barometric pressure, both long- term (e.g., 2000 Pa over 4 days) and short-term (e.g., 200 Pa over several minutes). For the relatively high-permeability soil examined herein, the predicted radon fluxes seem least sensitive

HOLFORD ET AL.: MODELING RADON TRANSPORT IN DRY, CRACKED SOIL 579

to the crack dimensions. However, the limited sensitivity analysis of soil properties suggests that cracks would play an important role in enhancing radon flux from low permeability, high porosity soils under transient pressure conditions. Cracks would be very likely to occur in soils with a swelling clay component. Validation of the model at a field site with lower permeability soils is needed. If cracks or other macropores are suspected, measurement of air permeabilities at scales smaller and larger than the scale of the cracks would be necessary to determine the degree to which permeabilities are enhanced by the presence of cracks.

The diffusion coefficient of radon in soil has been proven to be correlated to the water content of the soil [Rogers et al., 1979]. The soil considered in this study had a water saturation of 0.06 [Schery et al., 1984]. This value indicates that only 17% of the soil porosity (which is equal to 0.35) is filled with water, and this water content would have a small effect on the

diffusion coefficient. However, in soils with higher water contents, the transport of radon could be significantly affected by the reduction in air-filled void space [Kraner et al., 1964; Owczarski et al., 1990]. Also, movement of soil water could enhance or diminish the flux of radon from the soil surface. For

these reasons, the effects of water content on radon transport should be included in future field and modeling studies of radon transport in soils.

Duwe [1976] simulated the effect of variations in wind speed on radon flux in the soil by variations in an effective diffusion coefficient. With our model it would be possible to directly simulate the effect of high-frequency subsurface pressure variations, caused by atmospheric turbulence, on advective radon transport from soils.

Bq

C

DO DA Fd

Fa F•

gj i

J k

kcl

I•c2 M

P

Farm R

At

T

vi

Xl

x2

NOTATION

one atomic particle disintegration per second (1 Bq = 27 pCi = 4.87 x 105 atoms). concentration of radon gas in air, Bq/L. diffusion coefficient for radon in dry soil, m2/s. diffusion coefficient of radon in pure air, m2/s. diffusive flux density in soil, Bq/m2/s advective flux density in soil, Bq/m2/s. total flux density in a given direction, Bq/m2/s gravitational acceleration vector, m/s2. index of the Cartesian coordinate axes = 1..2, see x. index of the Cartesian coordinate axes = 1..2, see x. intrinsic permeability of the dry soil, m2. horizontal intrinsic permeability of the crack, m2. vertical intrinsic permeability of the crack, m2. molecular weight, kg/mole. total porosity of soil, m3/m3. absolute air pressure, Pa. atmospheric pressure, Pa. universal gas constant, J kg/K. time, s.

current time step increment, s. absolute temperature, K. darcy velocity of soil air, the volume of air flowing per unit bulk cross-sectional area of soil per unit time, m/s. horizontal Cartesian coordinate, m. vertical Cartesian coordinate, m.

production rate of radon per unit volume of soil pore space, Bq/L s.

p

222Rn decay coefficient, s-1. = ln(2) / half-life of 222Rn (half-life of 222Rn = 3.8 days). dynamic viscosity of air at a given temperature, kg/m s. density of air at a given temperature, kg/m3. tortuosity coefficient of the soil = D ij/DA.

Acknowledgments. The assistance of Glendon Gee and Kyle Roberson in reviewing this article is gratefully acknowledged. This work was funded in part by the U.S. Department of Energy under contract DE- FG04-86ER60432. Funding for the computer time needed for simulations was provided by the New Mexico Institute of Mining and Technology computer center. Preparation of this article was funded by the U.S. Department of Energy under contract DE-AC06-76RLO 1830.

REFERENCES

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University Press, New York, 1959. Clements, W. E., The effect of atmospheric pressure variation on the

transport of 222radon from soil to the atmosphere, Ph.D. dissertation, N. M. Inst. of Min. and Technol., Socorro, 1974.

Crank, J., The Mathematics of Diffusion, Oxford University Press, New York, 1975.

Duwe, M.P., The diurnal variation in radon flux from the soil due to atmospheric pressure change and turbulence, doctoral thesis, Univ. of Wisc.-Madison, 1976.

Edwards, J. C., and R. C. Bates, Theoretical evaluation of radon emanation under a variety of conditions, Heath Phys., 39, 263-274, 1980.

Grisak, G. E., and J. F. Pickens, Solute transport through fractured media, 1, The effect of matrix diffusion, Water Resour. Res., 16, 719-730, 1980a.

Grisak, G. E., and J. F. Pickens, Solute transport through fractured media, 2, Column study of fractured till, Water Resour. Res., 16, 731-739, 1980b.

Holford, D. J., S. D. Schery, J. L. Wilson, and F. M. Phillips, Radon Transport in Dry, Cracked Soil: Two-Dimensional, Finite Element Model, Rep. PNL-7116, Pacific Northwest Laboratory, Richland, Washington, 1989.

Huyakom, P.S., and G. F. Pinder, Computational Methods in Subsurface Flow, Academic, San Diego, CaliL, 1983.

Huyakom, P.S., B. H. Lester, and C.R. Faust, Finite element techniques for modeling groundwater flow in fractured aquifers, Water Resour. Res., 19, 1019-1035, 1983a.

Huyakom, P.S., B. H. Lester, and J, W. Mercer, An efficient finite element technique for modeling transport in fractured porous media: 1. single species Transport, Water Resour. Res., 19, 841-854, 1983b.

INTERA Environmental Consultants Inc., FTRANS: A Two- Dimensional Code for Simulating Fluid Flow and Transport of Radioactive Nuclides in Fractured Rock for Repository Performance Assessment, Rep. 0NWI-426, Battelle Memorial Institute, Office of Nuclear Waste Isolation, Columbus, Ohio, 1983.

Kraner, H. W., G. L. Schroeder, and R. D. Evans, Measurements of the effects of atmospheric variables on radon-222 flux and soil-gas concentrations, in The Natural Radiation Environment, pp. 161-190, University of Chicago Press, Chicago, Ill., 1964.

Landman, K. A., and D. S. Cohen, Transport of radon through cracks in concrete slab, Health Phys., 44, 249-257, 1983.

Loureiro, C. d. O., L. M. Abriola, J. E. Martin, and R.G. Sextro, Three- dimensional simulation of radon transport into houses with basements under constant negative pressure, Environ. Sci. Technol., 24, 1338- 1348, 1990.

Nazaroff, W. W., Radon transport from soil to air, Rev. Geophys., 30, 137-160, 1992.

Nazaroff, W. W., and A. V. Nero, Radon and Its Decay Products in Indoor Air, Radon and Its Decay Products in Indoor Air, John Wiley, New York, 1988.

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Nilson, R. H., E. W. Peterson, K. H. Lie, N. R. Burkhard, and J. R. Hearst, Atmospheric pumping: a mechanism causing vertical transport

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of contaminated gases through fractured penncable media, Journal of Geophysical Research, 96, 21933-21948, 1991. Owczarski, P. C., D. J. Holford, H. D. Freeman, and G. W. Gee, Effects

of changing water content and atmospheric pressure on radon flux from the surfaces of five soil types, Geophys. Res. Lett., 17, 8i7-820, 1990.

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Schery, S.D., and A, G. Petschek, Exhalation of radon and thoron: The question of the effect of thermal gradients in soil, Earth Planet. Sci. Lett., 65, 57-60, 1983,

Schery, S. D., and D. Siegel, The role of channels in the transport of radon from the soil, J. Geophys. Res., 91, 12,366-12,374, 1986.

Schery, S. D., D. H. Gaeddert, and M. H. Wilkening, Factors affecting exhalation of radon from a gl-avelly sandy loam, J. Geophys. Res., 89, 7299-7309, 1984.

Schery, S. D., D. J. Holford, J. L. Wilson, and F. M. Phillips, The flow and diffusion of radon isotopes in f•'actured porous media, Part 1, Finite slabs, Radiat. ?rot. Dosira., 24, 185-189, 1988a.

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D. J. Holford, Battelle, Pacific Northwest Laboratories, P.O. Box 999, Richland, WA 99352 .

F. M. Phillips, S. D. Schery, and J. L. Wilson, New Mexico Institute of Mining and Technology, Socorro, NM 87801 .

(Received July 21, 1991, revised August 5, 1992,

accepted August 5, 1992.)