U.S. NUCLEAR REGULATORY COMMISSION June 1989 

REGULATORY GUIDE 

OFFICE OF NUCLEAR REGULATORY RESEARCH 

REGULATORY GUIDE 3.64 

(Task WM 503-4) 

CALCULATION OF RADON FLUX ATTENUATION BY EARTHEN 

URANIUM MILL TAILINGS COVERS

USNRC REGULATORY GUIDES 

Regulatory Guides are issued to describe and make available to the pub

lic methods acceptable to the NRC staff of Implementing specific parts 

of the Commission's regulations, to delineate techniques used by the 

staff In evaluating specific problems or postulated accidents, or to pro

vide guidance to applicants. Regulatory Guides are not substitutes for 

regulations, and compliance with them is not required. Methods and 

solutions different from those set out In the guides will be acceptable if 

they provide a basis for the findings requisite to the Issuance or continu

ance of a permit or license by the Commission. 

This guide was issued after consideration of comments received from 

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experience. 

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TABLE OF CONTENTS 

Page 

A. INTRODUCTION ..................................................... 3.64-1 

B. DISCUSSION ....................................................... 3.64-1 

1. Determination of Parameters .................................. 3.64-2 

2. Cover Thickness Calculations ................................. 3.64-3 

C. REGULATORY POSITION .............................................. 3.64-4 

1. Determination of Parameters .................................. 3.64-5 

1.1 Fundamental Parameters ...................................... 3.64-5 

1.1.1 Layer Thicknesses ......................................... 3.64-5 

1.1.2 Layer Densities, Specific Gravities, and Porosities ....... 3.64-7 

1.1.3 Long-Term Average Moistures ............................... 3.64-8 

1.1.4 Radium Activities ......................................... 3.64-10 

1.1.5 Radon Diffusion Coefficients .............................. 3.64-10 

1.1.6 Radon Emanation Coefficients .............................. 3.64-11 

1.1.7 Other Fundamental Parameters .............................. 3.64-11 

1.2 Calculated Parameters ....................................... 3.64-11 

2. Calculations of Cover Thickness .............................. 3.64-12 

D. IMPLEMENTATION ................................................... 3.64-14 

APPENDIX A, Examples of Uranium Tailings Cover Thickness Calculations. 3.64-15 

APPENDIX B, The RADON Program ......................................... 3.64-19 

REFERENCES ............................................................ 3.64-41 

VALUE/IMPACT STATEMENT ................................................ 3.64-42 

LIST OF TABLES 

Table Page 

1 Cover Design Parameters, Symbols, and Reference Values ...... 3.64-6 

1B RADON Program Listing ....................................... 3.64-27 

2B RADON Program Sample Problem Output ......................... 3.64-39

iii
A. INTRODUCTION 

The Uranium Mill Tailings Radiation Control Act (UMTRCA) of 1978 (Public 

Law 95-604) gives the NRC responsibility to ensure, through the licensing proc

ess, that final disposal of uranium byproduct material (tailings) is conducted 

in a way that will protect the public health and safety and the environment. 

Public Law 95-604 also requires that uranium tailings disposal conform to stan

dards promulgated by the Environmental Protection Agency (EPA). The NRC staff 

is required to analyze the adequacy of uranium tailings covers proposed in li

cense applications to meet the EPA rules. The EPA rules in 40 CFR Part 192 

require that a cover be designed to produce reasonable assurance that the radon

222 release rate would not exceed 20 pCi m-2 s- 1 for a period of 1000 years to 

the extent reasonably achievable and in any case for at least 200 years when 

averaged over the disposal area over at least a one-year period. NRC regulations 

in 10 CFR Part 40 also require that the radon-222 release rate not exceed 20 pCi 

m-2 s- 1 for active (UMTRCA Title II) sites. Alternatively, for inactive (UMTRCA 

Title I) sites, the EPA rules permit the Department of Energy (DOE) to choose to 

meet an optional standard for radon concentration of less than 0.5 pCi per liter 

over background. 

This regulatory guide describes methods acceptable to the NRC staff for 

calculating radon fluxes through earthen covers and for calculating the result

ing minimum cover thickness needed to meet NRC and EPA standards. The guide 

also suggests methods for obtaining the various parameters used in calculating 

the radon fluxes and earthen cover thicknesses and suggests default values for 

certain parameters. 

This regulatory guide is applicable to active uranium tailings sites. The 

NRC staff is using the methods stated in this guide as a basis for review and 

concurrence of DOE remedial action plans for inactive sites. The guidance is 

intended to be used for calculating radon flux attenuation by earthen uranium 

mill tailings covers. The parameter values and examples presented are limited 

to earthen cover materials, but the diffusion theory and the methods presented 

are also applicable to man-made materials. Detailed supporting information for 

calculating minimum cover thickness is published separately in the "Radon Atten

uation Handbook For Uranium Mill Tailings Cover Design," NUREG/CR-3533 (Ref. 1). 

Any information collection activities mentioned in this regulatory guide 

are contained as requirements in 10 CFR Part 40, which provides the regulatory 

basis for this guide. The information collection requirements in 10 CFR Part 40 

have been cleared under OMB Clearance No. 3150-0020. 

B. DISCUSSION 

The design of a cover to reduce radon releases from uranium tailings de

pends on the values of a variety of fundamental parameters that characterize 

the tailings and cover materials. Once determined, the values of these param

eters may be used to calculate the thickness of cover that is required to

3.64-1
reduce the flux of radon from the tailings to any prescribed limit. This guide presents guidance on (1) determining appropriate values of the parameters and 

(2) exact and approximate methods for calculating radon fluxes for most cover 

configurations and conditions. The approximate calculation methods give results 

similar to those from the exact methods and thus provide for flexibility in 

performing the cover design calculations. For multilayer covers, an exact hand

calculated solution is not available. Either an approximate method or a com

puter solution must be used for systems with more than two layers. 

1. DETERMINATION OF PARAMETERS 

Because the radon flux limit in the EPA standard is given as an average 

over the entire disposal area over a period of at least one year (40 CFR 

Part 192), anomalies in position and time may be ignored and estimates of the 

long-term spatial average for all parameters should be used. Because the EPA 

limit on radon flux is to apply for a period of at least 200 and up to 1000 

years, parameter values should be selected conservatively taking into account 

potential degradation of the cover over time. Parameters needed to characterize 

the tailings and cover materials include thickness, density, specific gravity, porosity, moisture, radium activity, radon diffusion coefficient, and radon 

emanation coefficient (Refs. 1 and 2). 

Values of some of the parameters such as tailings pile thickness and aver

age radium activity can be reasonably estimated for initial license applica tions from mill plans, anticipated average ore grades, and tailings dam design. The most significant parameter affecting the thickness of earthen cover needed 

to meet the EPA radon flux criterion is the radon diffusion coefficient of the 

cover. The value of the radon diffusion coefficient is very sensitive to the 

availability of interconnected air-filled pores and therefore, at moderate to 

high moisture contents, to the cover moisture content and porosity. The param eter that introduces the greatest uncertainty into the calculation of earthen 

cover thickness for radon attenuation is the cover moisture content. The values 

for the cover and tailings moisture contents, densities, and radon diffusion 

coefficients are difficult to estimate or measure because long-term moisture 

changes and settling may occur after the installation of a cover system. There

fore, conservative values of these parameters measured under similar long-term conditions or conservative predictive correlations of these parameters with the 

significant variables may be needed. Specific guidance with respect to measur

ing each parameter is detailed in the regulatory position of this guide. For 

each parameter, the applicant will provide information describing the test 

method, its precision and accuracy, and its applicability for representing a 

long-term, large-area conservative average. 

Although accurate site-specific measurements of all parameters would be ideal, the uncertainties, costs, and reliability of such measurements may make 

the use of default values or conservative correlation predictions a reasonable 

and satisfactory alternative. However, the applicant should recognize that 

careful measurement of parameter values may be justified by savings in the cost 

of covering the tailings. The NRC staff has selected default values with the 

intent that minimum cover thicknesses calculated using default parameters will 

be equal to or greater than thicknesses calculated using measured parameters. If reasonable evidence indicates that default values or correlation predictions 

may not be conservative or realistic for a site, the applicant should measure 

the parameter values for which default values could be nonconservative.

3.64-2
An important factor in the long-term performance of a tailings cover system 

is its ability to remain free of defects over long time periods. Centimeter

scale defects caused by soil shrinkage, erosion cracks, erosion piping, animal 

burrows, and former root channels that are deeply penetrating and relatively 

frequent could cause a significant loss in cover performance (Ref. 3). To cause 

a factor of two increase in radon flux, cracks must be at least 2 cm wide, must 

be spaced less than 1 m apart, and must penetrate at least 75% of the cover 

thickness (Ref.3). Such defects would be easily detected by visual inspection 

except when covered by riprap (Ref. 3). To promote long-term effectiveness, 

the staff recommends that smectite clays or other swelling clays, if used, 

should be well compacted and protected from excessive wetting and drying by 

additional soil layers. The staff further recommends that well-compacted soils 

generally be utilized to minimize biointrusion as well as to minimize shrinkage 

and formation of other defects. Proper surface covering and contouring are 

also important to reduce erosion. Cover layers such as riprap or topsoil that 

are used solely for erosion control should not be included in radon flux cal

culations. Likewise, cover material subject to erosion should not be included. 

However, the effects of riprap or topsoil on the long-term moisture content of 

the earthen cover need to be considered. 

2. COVER THICKNESS CALCULATIONS 

The basis for the radon flux and minimum cover thickness calculations 

presented here is one-dimensional steady-state gas diffusion theory (Ref. 1). 

Only vertical diffusion is considered because the horizontal dimensions of 

tailings piles are large compared to the typical mean radon diffusion length 

of at most 1 to 2 meters. Short-term variations are ignored because the 

regulation addresses the long-term average radon flux. Advective transport, 

the externally forced movement of radon, also affects radon fluxes, but pri

marily over short time periods. For tailings covers, advective effects usually 

have negligible impacts in comparison to diffusion when averaged over the 

natural long-term cycles in thermal, barometric, and other advective driving 

forces (Ref. 4). Advective transport may thus be ignored unless local anomalies 

are known to cause sustained directional transport of soil gases. 

The thickness of earthen cover required to reduce radon fluxes to accept

able levels depends on the radon source strength of the tailings and on the 

efficiency of the cover material in reducing the flux. Radioactive decay of 

radium-226 in tailings and soil produces radon-222, which is an inert, short

lived, radioactive gas. Radon diffuses through the soil pore space over average 

distances defined by its 3.8-day half-life and by its diffusion coefficient. 

The flux of radon reaching the atmosphere is reduced by delaying its release 

because a greater fraction decays in the cover. The delay may be accomplished 

by increasing the cover thickness, employing a cover material with a lower 

diffusion coefficient, increasing the cover compaction, or increasing the 

long-term moisture content of the cover. Thus the diffusion coefficient for 

radon in the cover material is a key parameter determining its efficiency. 

The one-dimensional steady-state diffusion equation appropriate for radon 

flux determinations (Ref. 1) is: 

D 2C -XC + RpEXn 0 ()

3.64-3
where 

D = diffusion coefficient for radon in the total pore space (cm2s-') 

C = radon concentration in the total pore space (pCi cm- 3 ) 

X = radon decay constant (2.1x1O- 6 s-') 

R = specific activity of radium-226 (pCi g-1) 

p = dry bulk mass density of soil or tailings (g cm- 3 ) 

E = radon emanation coefficient (dimensionless) 

n = soil or tailings porosity (dimensionless) 

Radon flux is related to the radon concentration gradient by: 

J = -10 4 D dnCx (2) 

where 

J = radon flux (pCi m-2 s-') 

104 = units conversion (cm2/m2 ) 

Solutions to Equation 1 are obtained by applying boundary conditions for the system being analyzed and solving for the surface radon flux. For a thick bare tailings source, boundary conditions are typically (1) a specified or zero radon concentration at the air surface and (2) zero or negative radon flux at the base of the tailings. For tailings with covers or other systems consisting of layers of different materials, additional boundary conditions for each inter face between layers are (3) continuity in radon concentration and (4) continuity in radon flux. Individual layers should be defined by the occurrence of distinct changes in radium content, soil texture, compaction, or moisture. 

The exact solutions to Equations I and 2 can be arranged to calculate di rectly the radon flux for a given set of tailings and cover parameters. The thickness of cover needed to achieve a specified radon flux can also be deter mined directly. Specific guidance on the cover thickness calculations is given in the regulatory position. 

C. REGULATORY POSITION 

The parameter values and equations by which the NRC staff will estimate 

radon flux and minimum thicknesses for uranium tailings covers are presented 

below. These equations are appropriate for the design criteria that the staff 

routinely considers in its evaluations.

3.64-4
1. DETERMINATION OF PARAMETERS 

The design of an adequate tailings cover system depends on the values of 

several fundamental parameters of the tailings and cover materials. These 

include the thicknesses, densities, specific gravities, moistures, radium 

activities, radon diffusion coefficients, and radon emanation coefficients of 

the materials. In addition, several secondary parameters are calculated from 

the fundamental parameters for simplicity in performing the necessary calcula

tions. Examples of secondary parameters are the fractions of moisture satura

tion and the radon flux from the uncovered (bare) tailings source. Table 1 

lists all of the cover design parameters, their symbols, and reference values 

or sources of data. Table I and other sections of this guide provide default 

values that are, by their very nature, conservative. If the applicant does 

not elect to use actual measured values, the default values may be used. The 

applicant may use other values if it can be demonstrated that the use of such 

values is appropriate. On the other hand, default values should not be used if 

the applicant has information or indications that their use would lead to 

underestimation of the resultant radon flux. An example in point would be to 

use the default value for E, the radon emanation coefficient, for the case of 

windblown tailings that may have higher radon emanation rates. 

1.1 Fundamental Parameters 

Because of the difficulty and possible ambiguity in determining represen

tative values for the fundamental parameters, sections 1.1.1 through 1.1.7 

provide guidance on methods acceptable to the NRC staff for determining their 

values. 

1.1.1 Layer Thicknesses 

The thickness of the tailings source, xt, will be determined from the 

applicant's estimates of total tailings production and areal extent of the 

pile. Because a tailings thickness greater than about 100-200 cm is effectively 

equivalent to an infinitely thick radon source (Ref. 1), a value of xt = 500 cm 

represents an equivalent infinitely thick tailings source of radon that may be 

used in the absence of more specific smaller values. Cover layer thicknesses,

3.64-5
TABLE 1 Cover Design Parameters, Symbols, and Reference Values 

Symbol Parameter and Units Reference Value 

Fundamental ,xt, Xc Thickness of tailings (t) and cover (c) xt = 500 

Parameters layers (cm) 

Pt' Pc Dry bulk mass densities of tailings and Pt = 1.6 

cover (g cm- 3 ) Pc - measured 

Pw Mass density of water (g cm- 3 ) Pw= 1 

Gt, Gc Specific gravities of tailings and cover Gt = Gc = 2.65 

(dimensionless) 

wý W c Long-term average moisture content of w t 6 

tailings and cover (dry wt. percent) w -measured or 

estimated 

Rt, Rc Specific activities of radium-226 in the R = 2812 • U3 08 tailings and cover (pCi g-1) p~rcentage 

Dt, D Diffusion coefficients for radon in the Equation 7 

total pore space of the tailings and cover (cm2 s-') 

E Radon emanation coefficient for the E = 0.35 

tailings and cover (dimensionless) 

A Radon-222 decay constant (s-1) A = 2.1x1O- 6 

k Equilibrium distribution coefficient for k = 0.26 

radon in water and air (pCi cm- 3 water 

per pCi cm- 3 air) 

Jc Radon flux criterion from the cover into J = 20 the atmosphere (pCi m-2 s-1 ) c 

Calculated nt, nC Porosities of the tailings and cover measured or Parameters (dimensionless) Equation 4 

mt, mc Moisture saturation fractions in tailings Equation 8 

and cover (dimensionless) 

J 

t Radon flux from the bare tailings source Equation 9 (pCi m-2 s-') 

bt, bc Inverse relaxation lengths for tailings Equation 10 

and cover (cm-') 

at, ac Interface constants for tailings and Equation 11 

cover (cm s-1)

3.64-6
Xc, must be calculated to satisfy the radon flux criterion but must be suffi

ciently thick for the intended application and compaction techniques and for 

maintaining physical integrity. 

1.1.2 Layer Densities, Specific Gravities, and Porosities 

Bulk dry mass densities for the tailings, pt, and cover, Pc' are related 

to the specific gravities for the tailings, Gt, and cover, GC, the mass density 

of water, pw' and the porosities for the tailings, nt, and cover, nc, by the 

equations 

Pt = GtPw(1 - nt) PC = GcPw(1 - nc) (3) 

Likewise, the porosities may be calculated by the equations 

p P 

t_ c(4) 

nt =1 

n = 

G tPw c GcPw 

Thus only two of the three variables need be determined because the third 

can be calculated. However, for greater confidence and reliability, the NRC 

staff suggests measuring all three variables and cross checking for consistency. 

The density of water, pw' is equal to unity and is generally ignored in calcu

lations but is required to make the equations dimensionally consistent. The.. 

staff will use a default tailings density of Pt = 1.6 g cm- 3 unless acceptable 

documented alternative values are provided by the applicant. This dry bulk 

mass density is equivalent to a porosity of 40% at a specific gravity of 2.65, 

which is the density of quartz. The value of 2.65 for specific gravity is 

conservative because other common tailings and cover minerals have densities 

less than or equal to the density of quartz. The use of reference values for 

tailings parameters is acceptable because radon flux and cover thickness cal

culations are more sensitive to cover parameters than to tailings parameters. 

The staff recommends that cover materials be compacted to approximately 95% 

of the maximum dry density as determined by the standard Proctor density test. 

The staff will accept cover densities properly determined by the standard 

Proctor test (ASTM-D-698, Ref. 5) for the candidate soils. The staff will use a 

reference specific gravity of 2.65 for all quartzose tailings and cover materials.

3.64-7
If materials with specific gravities significantly different from quartz are 

used, acceptable documented alternative values should be provided by the 

applicant. 

Porosities may be measured by mercury porosimetry or other reliable method 

or determined by Equation 4, which, if reference values for bulk density and spe

cific gravity are used, is equivalent to using a default porosity of 40%. 

1.1.3 Long-Term Average Moistures 

The NRC staff considers several methods for predicting the long-term soil 

moisture content to be acceptable. An appropriate method is to measure the 

actual long-term moisture content at the cover material borrow site and to make 

adjustments, if needed, for differences between the borrow site and the dis

posal site. The staff recommends that soil moistures for the candidate cover 

materials, Wc, be measured from samples obtained from depths of 120 to 500 cm. 

The term w is the long-term average dry weight percent moisture content of the 

material. It is calculated by dividing the weight of free water by the weight 

of a dried sample. Shallow samples of the soil should be excluded because of 

the high seasonal variability in their moisture content. Samples close to a 

water table should also be excluded to avoid biasing the moisture estimate for 

the tailings cover. 

Moisture contents may be calculated by computer models of unsaturated flow 

provided that the models are fully documented and are validated for the range 

of possible site conditions. Parameter values used in modelling must be 

either accurate measured values or reasonable conservative estimates. Measured 

and conservative values must be applicable to periods of drought. 

Bayer (Ref. 6) has indicated that soil moisture reduced by drainage and 

evapo-transpiration will eventually approach and may even go beyond the perma

nent wilting point. The wilting point is the soil moisture content at which 

soil can no longer supply water at a rate sufficient to maintain plant life. 

The tension of the soil water when permanent wilting occurs is about 15 at

mospheres (Ref. 6). The NRC staff will accept the moisture content at which 

permanent wilting occurs as a reasonable value of the long-term moisture con

tent. This value may be determined from actual laboratory testing or from 

estimated empirical relationships such as those determined by Rawls and 

Brakensiek (Ref. 7). Rawls and Brakensiek conducted a study on 1323 soils with

3.64-8
approximately 5350 horizons and from 32 states. From their data, relationships 

were derived for predicting soil water retention volumes at matric potentials 

ranging from 0.04 to 15 bars based on percent sand, silt, and clay, on percent 

organic matter, on bulk density, and on the 0.33-bar and 15-bar soil water 

retention values. The accuracy of these equations increases as a greater number 

of these soil properties are identified. 

The empirical relationship established in Reference 7 that predicts 

volumetric moisture content of the soil corresponding to 15 bars is 

0 = 0.026 + 0.005z + 0.0158y (5) 

where 

z = % of clay in the soil 

y = % of organic matter in the soil. 

Because the 15-bar water retention value, 6, is the permanent wilting 

point of the soil, the NRC staff considers that this value is a reasonable 

lower bound for the soil moisture content over the long term. 

The long-term average dry weight percent moisture of the candidate cover 

material, Wc, is related to e by the following equation 

1 0 0 0 Pw (6) 

c PC 

where pw is generally unity. 

The applicant may use a reference value for wt because the calculation of 

cover thickness is not nearly as sensitive to the value of the moisture content 

of the tailings as it is to the moisture content of the cover. If acceptable 

documented alternative information is not furnished by the applicant, the staff 

will use a reference value of wt = 6% for the tailings moisture content because 

6% is a lower bound for moisture in western soils (Ref. 8).

3.64-9
1.1.4 Radium Activities 

For well-mixed tailings, the average specific radium activity of the tail

ings can be determined from the average uranium ore grade of the parent mate

rial, assuming secular equilibrium between uranium and radium. In this case, 

the radium activity should be estimated by multiplying the ore grade by 

2812 pCi g-1 radium per percent U3 08 . The basis for estimating the average 

uranium ore grade in units of percent U3 08 should be documented. Many tailings 

piles are layered, so the average radium activity of the tailings may not be 

adequate for determining the radon flux from the uncovered tailings. Layered 

tailings that have slimes on top will generally have higher radon fluxes than 

equivalent well-mixed tailings because tailings slimes generally have much higher 

radium contents than tailings sands. Therefore, the NRC staff advises in situ 

measurement of radium activity for nonuniform tailings. Because the criteria 

of 40 CFR Part 192 deal only with radon generated by the tailings, the radium 

activity in the cover soils may be neglected (Rc = 0) for cover design purposes 

provided the cover soils are obtained from background materials that are not 

associated with ore formations or other radium-enriched materials. 

1.1.5 Radon Diffusion Coefficients 

If measurements are not available, the staff estimates the radon diffusion 

coefficients of the tailings, Dt, and cover materials, Dc) from their moisture 

saturations, m, and porosities, n, using the correlation function 

D = 0.07exp[-4(m - mn2 + M5 )] (7) 

The moisture saturation fraction, m, is defined in Table 1 and in section 1.2. 

This correlation is based on numerous radon diffusion measurements in clays, 

silts, sands, gravels, and mixed earthen materials with compactions generally 

in the range of 80-105% of standard Proctor maximum dry density (Ref. 2). 

The effects of long-term fluctuations of the values of the diffusion coefficients 

should be factored into the values ultimately used. The staff will accept 

properly measured radon diffusion coefficients for candidate cover soils if 

adequate documentation of experimental procedures, including documentation of 

precision and accuracy, is provided.

3.64-10
1.1.6 Radon Emanation Coefficients 

The radon emanation coefficient, E, is the fraction of radon that is re

leased from the tailings or soil matrix into the pore space. The reference 

value of the radon emanation coefficient used by the NRC staff is 0.35 for all 

materials. The staff will accept measured or other substantiated values of E 

provided the applicant uses proper experimental procedure or provides clear 

documentation. Nielson et al. (Ref. 9) describe methods of measuring E that, if 

properly implemented and documented, would be acceptable. 

1.1.7 Other Fundamental Parameters 

The accepted value for the radon decay constant is 2.1xi0- 6 s-1. The value 

of the equilibrium distribution coefficient for radon between air and water 

that should be used is k = 0.26 pCi cm- 3 water per pCi cm- 3 air. This is the 

value of the distribution coefficient at a temperature of 20'C (Ref. 10). 

1.2 Calculated Parameters 

Values of several parameters that occur frequently in cover design and 

radon flux calculations are generally calculated separately for use in subse

quent design calculations. These include the moisture saturation fraction, 

m, and the bare source flux, J t" 

The moisture saturation fraction, m, is the volumetric fraction of satura

tion of pore space for the tailings or cover soil. It is calculated by con

verting weight percent moisture, w, to the percent of water-filled porosity 

with the equations

10-2 pt~wt, 

mt - ntpw

10-2 P cwc 

mc ncpw

where pc' pt, Pw' n, and w are defined in Table 1. A reference value of 1 for 

the density of water, Pw' is generally used, but for saline pore fluids, higher 

values would be appropriate. 

The radon flux from the bare tailings source and the other calculated 

parameters are also defined here for use in calculating radon flux, Jc' and

3.64-11

(8)
cover thickness, x . The radon flux from the bare (homogeneous) tailings 

source is calculated as 

it = 104 R tPt EtV•-Xttanh(xtVX-Dt) (9) 

where Rt, Ptý Et Dt0 xt, and X are defined in Table 1 and 104 changes the 

radon flux units from pCi cm- 2 s- 1 to pCi m-2s- 1 . The inverse relaxation 

lengths for the tailings, bt, and cover soils, bc, are calculated as 

bt = 1X_/t, bc =X7 (10) 

where X and D are defined in Table 1. The respective interface constants at 

and ac for the tailings and cover soils are calculated as (Ref. 1) 

2 2 at n2D [1 - (1 - k)mt] or ac =n 2 D [ - (1 - k)m] (11) t t t' c cc c 

where n, D, and k are defined in Table 1, and m is defined by Equation 8. 

2. CALCULATIONS OF COVER THICKNESS 

The applicant should determine the minimum necessary cover thickness by 

utilizing appropriate estimates of all parameters in one of several equivalent 

calculation methods. For simple single-layer covers, the radon flux pene

trating the cover can be calculated as 

2 Jtexp(-bcxc) (12) 

I + ýat/actanh(btxt) + [1 - 4i•ci7tanh(btxt)]exp(-2bx) 

where J t is defined by Equation 9, bt and bc are defined by Equation 10, at and 

ac are defined by Equation 11, and xt and xc are defined in Table 1 and tanh 

is the hyperbolic tangent. For thick tailings sources, the hyperbolic tangent 

term, tanh(btxt), is equal to unity and may be ignored. However, if the tailings 

are less than 200 cm thick, Equation 12 should be used. 

For practical applications for which the tailings are more than 200 cm 

thick and the cover achieves a flux reduction greater than a factor of ten, 

the following simplified equation is acceptable.

3.64-12
2Jtexp(-bc x) (13) 

1 + qIa-t/a~ 

Equation 12 can be solved for xc 

xc=1 •n Jc /J t(1 - 4_5actanh(btxt0)) 

b c 1 - [1 - (Jc/Jt)2 (l + V-acThI-tanh(btxt)) - aTactanh(btxt))]1⁄2 

(14) 

The term tanh(btxt) is approximately equal to 1 for uranium tailings 

thicknesses of 200 cm or more. Also, for almost all uranium tailings applica

tions, the desired reduction of radon flux from the cover is significant, so 

that Jc is much less than the bare tailings flux Jt" In these cases, Equation 

14 may be adequately approximated by solving Equation 13 for xc. 

xc = 4D 7 _ X lni t j (15) 

Further, Equation 15 can be expressed by a nomograph presented in Reference 1. 

Cover thicknesses determined accurately from the nomograph in Reference 1 are 

acceptable to the staff if the applicant shows that the tailings are more than 

2 meters thick and the required flux attenuation ratio, Jc/it' is less than 

0.1. Examples of the cover thickness calculations for a single-layer cover are 

presented in Appendix A. 

Although flux and thickness calculations may be performed for multilayer 

covers using computer programs, the following approximate method will be 

accepted as an alternative provided the results agree with staff calculations. 

This method utilizes successive applications of Equation 13 to individual cover 

layers with modifications in the source parameters to account for any under

lying cover layers. Because Equation 13 does not allow for radon sources 

(radium) in the cover layer, this approximate method may be applied to multi

layer covers, but not to multilayer tailings sources. For multilayer covers, 

the subscripts cl and c2 are used to denote cover layers 1 and 2. For example, 

Jcl denotes the radon flux out of layer 1 and J c2 the flux out of layer 2. If

3.64-13
the tailings are less than 2 meters thick, Equation 12 should be used instead 

of Equation 13. The specific steps in the approximate multilayer cover calcu

lation are: 

1. Calculate the radon flux through the first cover layer, J c1' 

using Equation 13 as if there were no other cover layers. 

2. Calculate an equivalent source diffusion coefficient, Dt0l for 

use in analyzing the second cover layer by the following 

equation: 

Dti = Dtexp(-bclxcl) + Dcl [1 - exp(-b clXcl)] (16) 

3. For two-layer covers, calculate the required thickness of 

the second layer, Xc2, using Jcl as Jt in Equation 15. 

Use D as Dt, ncl as nt, and mcl as mt in Equation 11 

to define at for Equation 15. The new source term 

thickness is Xci + xt

Examples of the cover thickness calculations for a three-layer system with 

two cover layers are presented in Appendix A. Should the user wish to use this 

iterative method for more than three layers, Reference 1 provides the general 

formulae. If the user wishes to calculate radon flux and cover thickness for 

multiple tailings layers for more than three layers total, the NRC staff advises 

using a computer program. The NRC staff has developed an interactive version of 

the RAECOM computer program, which has been named the RADON program. The RADON 

program is documented in Appendix B. 

D. IMPLEMENTATION 

The purpose of this section is to provide information to applicants and 

licensees regarding the NRC staff's plans for using this regulatory guide. 

Except in those cases in which an applicant proposes an acceptable alter

native method for complying with specified portions of the Commission's and EPA's 

regulations, the methods described in this guide will be used in the evaluation 

of the adequacy of uranium tailings covers proposed in license applications.

3.64-14
APPENDIX A 

EXAMPLES OF URANIUM TAILINGS COVER THICKNESS CALCULATIONS 

Two examples are presented to illustrate the use of the equations and 

other guidance in this regulatory guide. The first example is for a single

layer cover system, and the second example is for a three-layer cover system. 

The uses of Equations 13 and 15 are illustrated by the first example. Applica

tions of the approximate multilayer method defined by Equations 12 through 16 

are illustrated by the second example. 

EXAMPLE 1. SINGLE-LAYER COVER THICKNESS CALCULATION 

For the first example, the tailings pile has the following typical values: 

Rt = 400 pCi g-1 

Pt = 1.5 g cm- 3 

E = 0.2 

Dt = 1.3xI0- 2 cm2 s-' 

n = 0.44 

wt = 11.7% 

xt = 300 cm 

First, the radon flux from the surface of the uncovered tailings is 

calculated from Equation 9: 

J t = (104 cm2m-2)(400)(1.5)(O.2)[(2.1x1O-6)(O.013)])tanh(3.8) 

= 198 pCi m-2 s-'

3.64-15
Cover material that can be compacted to have the following properties is 

available: 

DC = 7.8x10- 3 cm2s-1 

nc = 0.3 

w c= 6.3% 

Furthermore, it is assumed that mc = mt' 

Next, the flux attenuation with 2 meters of cover material is calculated 

from Equation 13: 

J - 2(198)exp[-(2.1 x 10-6/7.8 x 10-3)1⁄2(200)] 

1 + [((0.44)2(0.013))/((0.3)2(0.0078))]1⁄2 

j - 2(198)(3.76x10- 2 ) 

2.893 

Jc : 5.1 pCi m-2 s- 1 

If a greater flux is acceptable, the same calculation can be repeated 

with smaller cover thicknesses, or the required thickness can be directly 

calculated from Equation 15. For example, what cover thickness would yield 

a radon flux of 20 pCi m-2s- 1? This is determined from Equation 15 as: 

: 1 l 19.8 

c 61In 2.893 - 0.009 

xc = 118 cm 

EXAMPLE 2. THREE-LAYER COVER THICKNESS CALCULATION 

The tailings pile described in the first example is to be covered with 

0.5 meter of a good quality clay and sufficient overburden to achieve a surface 

radon flux of 20 pCi m-2 s-'. What thickness of overburden should be used? 

The basic material parameters are:

3.64-16
D n w m 

Tailings 0.013 cm2 s- 2 0.44 11.7 0.4 

Clay 0.0078 cm2s- 2 0.30 6.3 0.4 

Overburden 0.022 cm2s- 2 0.37 5.4 0.25 

First, the bare tailings flux is the same as before: 

it = 198 pCi m-2 s- 1 

Then, calculate the attenuation through the clay component using Equation 13 

in the same way as in Example 1. The equation is simplified to: 

= 2(198)(0.440) Jcl 2.893 - (0.173) 

Jcl = 64.1 pCi m-2 s- 1 

Now, determine the diffusion coefficient for the source term to the 

overburden (the source is now the tailings and clay) using Equation 16: 

D ti = D texp(-b cl Xcl) + Dcl[1 - exp(-bclXcl)] 

Dti = (0.013)(0.440) + (0.0078)(1 - 0.440) 

Dti = 0.0101 cm2 s- 1 

The value of Dtl is then substituted for Dt, and Jcl = 64.1 is substituted 

for Jt in Equation 15, giving: 

S = 102 In 6.41 

Xc2 1.475 + 0.051 

Xc2 = 146 cm = overburden thickness 

The total cover thickness is therefore 146 cm + 50 cm = 196 cm. In this 

calculation the effective source thickness is assumed to be large enough that 

tanh(atxt) is unity. 

The parameters specified in the above example were also used as example 

input to the RADON program. The RADON calculations, shown in Appendix B, 

yield an overburden thickness of 149 cm. Thus the approximate solution of 

146 cm is 3 cm less than the exact calculation. This typical example demon

strates that the approximate procedure yields acceptable results.

3.64-17
EXAMPLE 3. COMPUTATION OF SOIL MOISTURES AND DIFFUSION COEFFICIENTS AT THE 

PERMANENT WILTING POINT 

As was mentioned previously, the 15-bar soil moisture retention value is 

estimated using Equation 5, which is taken from Reference 7: 

e = 0.026 + O.005z + 0.0158y 

where 

z = % of clay in soil 

y = % of organic matter in soil 

A candidate soil being considered for a cover material contains 

approximately 16% clay and 0.5% organics. Using the above equation: 

0 = 0.114 

The associated moisture saturation fraction is computed to be: 

0.114 ~ m = = 0.29 

where we use the reference value of n = 0.40. 

The radon diffusion coefficient could then be estimated using Equation 7: 

D = 0.07exp[-4(m - mn2 + MS)] 

D = 0.026 cm2 s-I 

The user then proceeds in the manner shown in the previous examples.

3.64-18
APPENDIX B 

THE RADON PROGRAM 

1. INTRODUCTION 

The RADON computer program, shown in Table iB, is an interactive BASIC 

version of the FORTRAN computer program RAECOM listed in Reference 1. Refer

ence 1 describes the procedures for implementing the RAECOM program and provides 

examples of its input and output for those who wish to use a FORTRAN program. 

The RADON program was designed specifically for interactive use on personal 

computers. The program is written for the IBM PC and compatibles. Minor 

modifications in input/output parameters may be required for use on other 

personal computers. The RADON program calculates the radon concentration and 

flux at the boundaries of defined layers and calculates the thickness of a 

cover layer needed to meet a defined flux limit. This appendix is structured 

to follow the order of operations that is followed in using the RADON program. 

1.1 Hardware Requirements 

The RADON program assumes that the user has, as a minimum, a monochrome or 

color graphics monitor, a printer, and one disk drive. Input may be entered in

teractively or from a diskette file named RNDATA on disk drive A. The inter

active input sets up an input file that is recorded as RNDATA on drive A so 

that, following an initial interactive run, the line editor may be used to vary 

input parameters without going through the interactive program. 

2. CONSTANTS USED 

The RADON program uses a radon decay constant of 2.1 x 10.6 s-1, a water/ 

kL. ir partition coefficient for radon of 0.26, and a specific gravity for tail

s and cover materials of 2.65. If the materials have specific gravities 

'rntially different from 2.65, both porosity and dry bulk mass density 

3.64-19

* . - -r 

- &
should be entered into the program as measured values, in which case the 

specific gravity value is not used. 

3. INTERACTIVE INPUT PARAMETERS 

3.1 Title of Data Set 

When prompted, enter the title of the data set, not to exceed 254 char

acters, exactly as you want it to appear in the printed output. 

3.2 Interactive or Direct Input 

When prompted, type Y to input interactively or type N to read input from 

drive A. Do not hit the return key. Either upper or lower case letters may 

be used. The user is given a choice of using the interactive input program, 

which gives a detailed printed record of input, or proceeding directly to the 

calculational part of the program with input that is taken from a diskette file. 

The interactive program will not accept erroneous input data that violate param

eters defined in the program. For example, the program will not allow less 

than two layers total of cover plus tailings or more than ninety-nine total 

layers, and it will prompt the user to input again if unacceptable values are 

entered. However, if erroneous data are entered on the diskette file, they will 

not be screened again by the program, and the program will either "crash" or 

produce meaningless results. 

3.3 Input Total Number of Layers 

Following the prompt, enter the total number of layers of cover plus 

tailings. 

3.4 Input Desired Radon Flux Limit 

Following the prompt, enter the value of radon flux i 

top boundary. The default value for the radon flux limits . " 

which is the EPA flux limit for uranium mill tailings. The 

ti¢

3.64-20
be selected by entering -1. For problems for which no flux limit is desired, 

enter 0 (zero). Note that entering zero is not entering a flux of zero. It 

is entering a flag to the program that there is no flux limit and that the 

thickness of a cover layer will not be optimized to meet a flux limit. 

3.5 Input the Number of the Layer To Undergo Thickness Optimization 

Enter the number of the layer for which the layer thickness is to be 

optimized in order to achieve a desired flux value. The layers are numbered 

starting from the bottom, with the tailings as layer number 1. Therefore, 

layer 1 cannot be optimized, and optimization cannot be done if a flux limit 

has not been entered. Enter 0 (zero) if optimization is not desired. 

3.6 Input Radon Concentration Above Top Layer 

When prompted, enter the radon concentration above the top layer (in pico

curies per liter). This concentration is generally the value of the soil/air 

interface, and a default value of 0 (zero) is assigned by entering any value 

less than or equal to 0. If a value for the radon concentration at the soil/air 

interface were measured, it could be entered, but for tailings covers, a value 

of 0 is conservative and is recommended. 

3.7 Input Lower Boundary Radon Flux 

When prompted, enter the value for radon flux (in picocuries per square 

meter per second) into the soil layer beneath the tailings. For most tailings 

situations, this value can be considered to be zero, but for the case of a thin 

tailings layer, this downward flux would reduce the amount of radon available 

for upward diffusion. Enter -1 to calculate the radon flux out of the tailings 

into an infinitely thick subsoil with no radon source and to adjust the source 

of radon available for upward diffusion accordingly. 

3.8 Input Surface Flux Precision 

When prompted, enter the value for the acceptable level of numerical 

precision in the radon surface flux computations (in picocuries per meter

3.64-21
squared per second). The number that should be entered is the level of com

putation error that is acceptable to the user between 0 (zero) and 1. The 

program assumes that values greater than 1 are unacceptably high. The user is 

cautioned that computation time increases as the value approaches zero. 

4. LAYER INPUT PARAMETERS 

The RADON program will enter a loop that will prompt for input parameter 

values that define each layer in the tailings and cover system starting from 

the bottom layer of the system. 

4.0 Input Material Type 

Name the layer. 

4.1 Input Thickness 

Enter the thickness of the layer in centimeters. 

4.2 Input Porosity 

Enter the measured porosity (unitless). The measured porosity must be 

between 0 (zero) and 1. Enter -1 to choose a default value of 0.40 or enter a 

value greater than one to calculate porosity based on a specific gravity of 

2.65 and a measured value of bulk density. 

4.3 Input Mass Density 

If the dry bulk mass density was measured, enter a density value between 0.5 

and 3 g cm- 3 . Otherwise enter -1 to calculate the mass density of the layer 

based on the porosity and a specific gravity of 2.65. If the dry bulk density 

was measured in engineering units in terms of pounds (weight) per cubic foot, 

the bulk density value should be divided by 62.43, which is the weight in pounds 

of 1 cubic foot of water.

3.64-22
4.4 Type S for Source Term, G For Ore Grade, or R for Radium Activity 

The user is given a choice of ways to enter the radon source term for the 

layer. Type S, G, or R to choose to enter the source term, to calculate it 

from the uranium ore grade, or to calculate it from the radium activity, 

respectively. 

4.4.1 Input Ore Grade Percentage 

If G is selected, a prompt to enter the ore grade percentage of U308 will 

follow. The value entered must be between 0 and 100. 

4.4.2 Input Radium Activity 

If R is selected, a prompt to enter the radium activity of the layer in 

picocuries per gram of tailings or soil will follow. 

4.4.3 Input Source Term 

If S is selected, a prompt to enter the radon source term for the layer in 

picocuries per cubic centimeter per second will follow. This input option 

would be used when measurements of the radon source term from the layer are 

available. 

4.5 Input Emanation Coefficient 

Following the prompt, enter the value of the radon emanation coefficient 

(unitless). Enter -1 to use the default value, which is 0.35. 

4.6 Input Weight Percent Moisture 

Following the prompt, enter the value of the weight percent moisture. 

This value is the weight of "free" water divided by the total weight of a dried 

sample. Negative values and values greater than 100 will not be accepted 

because they are meaningless. A calculation will be made to determine the 

moisture saturation fraction (the fraction of available pore space occupied by 

water). If the moisture saturation fraction is greater than 100%, which is 

physically impossible, a prompt will appear to enter an acceptable weight per

cent moisture value. The weight percent moisture of the layer must be entered 

whether the diffusion coefficient is measured or calculated.

3.64-23
4.7 Input Diffusion Coefficient

Following the prompt, enter either the measured diffusion coefficient or 

-1 to use the default calculation of the diffusion coefficient using equation 

7 as described in Regulatory Position 1.1.5. 

The diffusion coefficient is the final input parameter needed for the layer. 

The program will prompt for input for layer 2 and so on up to the maximum layer 

number in the same way as for layer 1. 

5. INPUT DATA STORAGE 

5.1 Save Data File, Erase Backup Data File 

The program has a built-in procedure to save the input data file on disk 

drive A. This feature allows the program to be rerun without going through the 

data entry steps. If an error is made in data entry, the data may be easily 

corrected using the EDLIN program or other editing programs. Likewise, if the 

user wishes to analyze the effects of varying a parameter, the EDLIN program 

can be used to modify the input data without going through the data entry 

procedures. A formatted floppy diskette with available storage space must be 

placed in disk drive A for the program to proceed. If the user wishes to write 

to a hard disk, the user will need to change "A:" to "C:" in lines 1550, 1600, 

1620, 1720, and 1750. The file saved on the floppy diskette will appear like 

the data in Table 2B under the heading "DATA SENT TO THE FILE 'RNDATA' ON DRIVE 

A:" None of the headings are recorded on the saved file nor are the layer 

numbers under the heading "LAYER." The headings in this portion of Table 2B 

are explained as follows: 

N = number of layers 

FOI = radon flux into layer 1 

CN1 = surface radon concentration 

ICOST = the number of the layer to be optimized 

CRITJ = desired radon flux limit 

ACC = surface flux precision 

DX = layer thickness 

D = layer diffusion coefficient

3.64-24
P = layer porosity 

Q = layer source term 

XMS = layer moisture saturation fraction 

RHO = layer density 

If any key other than N is typed, the input data set will be saved. If 

N is typed, the program will go directly to the calculations without saving the 

data set on disk drive A. The data set is filed under the name RNDATA. If a 

second data set is entered, the first data set is renamed RNDATA.BAK, and the 

second set is named RNDATA. Entering more than two data sets will result in 

erasing RNDATA.BAK, so the user should rename RNDATA.BAK if it is to be saved. 

Once the RNDATA file is created, minor changes in input values can be made by 

using EDLIN or an editing program. A "hard" copy of the contents of the RNDATA 

file is part of the printed output. 

6. METHODS OF COMPUTATION 

The RADON program uses the computational procedures of the RAECOM program, 

which is listed in Reference 1 and documented in Reference 11. The diffusion 

equation is solved for a system of simultaneous equations for radon flux and 

concentration by the Gaussian elimination method. A recursion method is used 

to optimize the cover thickness to meet the desired radon flux. 

7. OUTPUT 

Table 2B displays the printed output of the RADON program for the three

layer sample problem in Appendix A and Reference 1. The printed output displays 

the constants used by the program, the general input parameters, the layer 

input parameters, including the default or calculated layer input parameters, 

the data saved on the floppy disk file, the calculated bare source term flux 

from layer 1, and the results from the radon diffusion calculations. The data 

for many of the output parameters are printed in the exponential format in 

which D is used to denote the base 10 exponent rather than the usual E because 

D is the notation in double-precision BASIC. Note that, for the sample problem, 

the input thickness for layer 3 is 100 cm, but the output thickness is 149 cm 

layer 3 was specified to be adjusted so that the output radon flux S>because 

would be 20 pCi m-2 s-'.

3.64-25
A very slight difference exists between the sample problem results shown 

here and those in Reference 1 because the specific gravity in Reference 1 was 

specified to be 2.7 whereas 2.65 is used in the RADON program. Exact agreement 

between Reference 1 and the RADON program would result if the same specific 

gravities were used. 

Modifications such as changing the specific gravity may be made very 

easily by the user because the BASIC interpreter may be used and because items 

in the interactive part of the program are labeled.

3.64-26
TABLE 1B RADON Program Listing

10 KEY OFF : COLOR 15,1,0 : CLS 

20 PRINT " ***** RADON !***** ...... " :PRINT 

30 PRINT "Version 1.2 - May 22, 1989 - G.F. Birchard - tel.# (301)492-7000" 

40 PRINT "U.S. Nuclear Regulatory Commission Office of Research" : PRINT 

50 PRINT " RADON FLUX, CONCENTRATION AND TAILINGS COVER THICKNESS", 

60 PRINT " ARE CALCULATED FOR MULTIPLE LAYERS" : PRINT 

70 ON ERROR GOTO 90 

80 GOTO 120 

90 PRINT "CHECK THE PRINTER & PRESS ANY KEY TO CONTINUE" 

100 V$=INPUT$(1) 

110 RESUME NEXT 

120 LPRINT " *****! RADON !***** ----- " : LPRINT 

130 LPRINT "Version 1.2 - May 22, 1989 - G.F. Birchard tel.# (301)492-7000" 

140 LPRINT "U.S. Nuclear Regulatory Commission Office of Research" : LPRINT 

150 ON ERROR GOTO 0 

160 LPRINT " RADON FLUX, CONCENTRATION AND TAILINGS COVER THICKNESS", 

170 LPRINT " ARE CALCULATED FOR MULTIPLE LAYERS" : LPRINT 

180 INPUT "TITLE OF THIS DATA SET";TITLE$ : LPRINT :LPRINT TITLE$ :LPRINT 

190 DEFDBL A-H,O-Z : DEFINT I-N 

200 DIM D(50),DX(50),P(50),Q(50),XM(99) 'INPUT VARIABLES 

210 DIM AB(50),AIPZMI(50),ALP(50),RHO(50),X(50),XMS(99) 'COMPUTATION VARIABLES 

220 DIM A4(50),DDX(50),RC(50),RF(50) 'OUTPUT VARIABLES 

230 DIM A(245),AT(50),B(99),BS(50),BU(99),G(245),R(50),RR(50),T(50),U(50) 

'MATRIX VARIABLES 

240 XL=.0000021# : PC=.26# : GCT=2.65# 'CONSTANTS 

250 PRINT "TYPE Y TO INPUT INTERACTIVELY OR N TO READ INPUT FROM DRIVE A:" 

260 W$=INPUT$(1) 

270 IF W$="y" OR W$"Y" GOTO 290 

280 IF W$="N" OR W$"n" GOTO 1750 ELSE 250 

290 LPRINT :LPRINT " ","CONSTANTS" :LPRINT 

300 PRINT :PRINT " ","CONSTANTS" :PRINT 

310 LPRINT "RADON DECAY CONSTANT"," ".,XL,"s--1" 

320 PRINT "RADON DECAY CONSTANT", ",XL,"s--1" 

330 LPRINT "RADON WATER/AIR PARTITION COEFFICIENT",PC

3.64-27
340 PRINT "RADON WATER/AIR PARTITION COEFFICIENT",PC 

350 LPRINT "SPECIFIC GRAVITY OF COVER & TAILINGS",GCT :LPRINT 

360 PRINT "SPECIFIC GRAVITY OF COVER & TAILINGS",GCT :PRINT 

370 PRINT " ","GENERAL INPUT PARAMETERS" : PRINT 

380 LPRINT LPRINT " ","GENERAL INPUT PARAMETERS" : LPRINT 

390 PRINT "ENTER -1 TO USE DEFAULT VALUES WHICH ARE SPECIFIED IF THEY EXIST." 

400 INPUT "NUMBER OF LAYERS OF COVER AND TAILINGS";N 

410 IF N<2 GOTO 420 ELSE IF N>99 GOTO 430 ELSE 440 

420 PRINT "TWO LAYERS MINIMUM PLEASE" : GOTO 400 

430 PRINT "NINETY-NINE LAYERS MAXIMUM PLEASE" : GOTO 400 

440 LPRINT "LAYERS OF COVER AND TAILINGS",N : GOTO 460 

450 PRINT "THE LAYER THICKNESS CANNOT BE OPTIMIZED WITHOUT A FLUX LIMIT" 

460 INPUT "RADON FLUX LIMIT (pCi m_-2 s_-1) default=20, no limit=O";CRITJ 

470 IF CRITJ<O# THEN CRITJ=20# : PRINT "DEFAULT FLUX LIMIT ASSIGNED" 

480 IF CRITJ=O THEN LPRINT "NO LIMIT ON RADON FLUX" : GOTO 510 

490 LPRINT "DESIRED RADON FLUX LIMIT"," ",CRITJ,"pCi m--2 s_-i" :GOTO 510 

500 PRINT "THE LAYER NUMBER CANNOT EXCEED THE NUMBER OF LAYERS. 

510 INPUT "THE LAYER NUMBER FOR THICKNESS OPIMIZATION O=no optimization";ICOST 

520 IF CRITJ=O AND ICOST>O GOTO 450 

530 IF ICOST=O THEN LPRINT "LAYER THICKNESS NOT OPTIMIZED" :GOTO 570 

540 IF ICOST<=I THEN PRINT "THE LOWEST LAYER CANNOT BE OPTIMIZED" :GOTO 510 

550 IF ICOST>N GOTO 500 

560 LPRINT "NO. OF THE LAYER TO BE OPTIMIZED",ICOST 

570 INPUT "RADON CONCENTRATION ABOVE TOP LAYER (pCi 1f-1) default=O";CN1 

580 IF CN1<0 THEN CN1ZO 

590 IF CN1ZO THEN LPRINT "DEFAULT SURFACE RADON CONCENTRATION"',CN1,"pCi f--i" 

600 IF CNI>O THEN LPRINT "MEASURED SURFACE RADON CONCENTRATION",CN1,"pCi 1f-i" 

610 INPUT "LOWER BOUNDARY RADON FLUX (pCi m_-2 s_-1) default-calculation";FO1 

620 IF F01=-1 GOTO 630 ELSE 650 

630 PRINT "LOWER BOUNDARY FLUX TO BE CALCULATED ASSUMING AN INFINITE SUBSOIL 

640 GOTO 660 

650 LPRINT "RADON FLUX INTO LAYER 1"," ",FO1,"pCi m_-2 s_--" 

660 INPUT "SURFACE FLUX PRECISION (pCi m--2 s--1) This number is the acceptable 

level of computation error.";ACC 

670 IF I<ACC OR ACC<O GOTO 680 ELSE 700

3.64-28
680 PRINT "THE SURFACE FLUX PRECISION SHOULD BE BETWEEN 0 AND 1" 

"690 GOTO 660 

700 LPRINT "SURFACE FLUX PRECISION"," ",ACC,"pCi m--2 s"-1" 

710 PRINT PRINT " ","LAYER INPUT PARAMETERS" : PRINT 

720 LPRINT LPRINT : LPRINT " ","LAYER INPUT PARAMETERS" :LPRINT 

730 FOR I=1 TO N 

740 PRINT USING "LAYER #";I PRINT :PRINT : INPUT "MATERIAL TYPE"; MT$ 

750 LPRINT USING "LAYER #";I, : LPRINT " " MT$ : LPRINT LPRINT 

760 INPUT "THICKNESS (cm)";DX(I) 

770 IF DX(I)<O THEN PRINT "THE THICKNESS CANNOT BE NEGATIVE" GOTO 760 

780 LPRINT "THICKNESS"," ",DX(I),"cm" 

790 INPUT "POROSITY default =.40. Enter a value >1 to calculate porosity 

";P(I) 

800 IF P(I)>1 GOTO 820 ELSE IF P(I)=-1 GOTO 880 ELSE IF P(I)<O GOTO 890 

810 GOTO 910 

820 INPUT "MASS DENSITY(G CM--3)";RHO(I) 

830 IF RHO(I)<.5 OR RHO(I)>3 GOTO 840 ELSE 860 

840 PRINT "ACCEPTABLE DENSITY VALUES ARE BETWEEN 0.5 AND 3. PLEASE REENTER" 

850 GOTO 820 

860 P(I)=1-RHO(I)/GCT: LPRINT "CALCULATED POROSITY "," "," "" 

870 LPRINT USING "#.###"; P(I) :GOTO 980 

880 P(I)=.4# : LPRINT "DEFAULT POROSITY"," ",P(I) : GOTO 920 

890 PRINT "THE POROSITY VALUE CANNOT BE NEGATIVE. PLEASE REENTER." 

900 GOTO 790 

910 LPRINT "POROSITY "," "," ",P(I) 

920 INPUT "MASS DENSITY (G/CM--3) default-calculated";RHO(I) 

930 IF RHO(I)=-1 GOTO 940 ELSE IF RHO(I)<.5 OR RHO(I)>3 GOTO 960 ELSE 980 

940 RHO(I)=GCT*(1-P(I)) 

950 LPRINT "CALCULATED MASS DENSITY"," ",RHO(I),"g cm--3" : GOTO 990 

960 PRINT "ACCEPTABLE DENSITY VALUES ARE BETWEEN 0.5 AND 3. PLEASE REENTER" 

970 GOTO 920 

980 LPRINT "MEASURED MASS DENSITY"," ",RHO(I),"g cm -3" 

990 PRINT "CHOOSE TO ENTER EITHER THE RADON SOURCE TERM CONCENTRATION 

THE ORE GRADE % OR RADIUM ACTIVITY" 

1000 PRINT "TYPE S FOR SOURCE TERM G FOR ORE GRADE OR R FOR RADIUM ACTIVITY"

3.64-29
1010 QQ$=INPUT$(1) 

1020 IF QQ$="S" GOTO 1230 ELSE IF QQ$="s" GOTO 1230 . 

1030 IF QQ$="G" GOTO 1060 ELSE IF QQ$="g" GOTO 1060 

1040 IF QQ$="R" GOTO 1100 ELSE IF QQ$="r" GOTO 1100 ELSE 1000 

1050 PRINT "THE ORE GRADE % MUST BE BETWEEN 0 AND 100. PLEASE REENTER" 

1060 INPUT "ORE GRADE PERCENTAGE";OG :IF OG<O OR OG>100 GOTO 1050 

1070 LPRINT "ORE GRADE PERCENTAGE"," ",OG,"%" 

1080 RA=OG*2812# 

1090 LPRINT "CALCULATED RADIUM ACTIVITY"," ",RA,"pCi g--l" :GOTO 1130 

1100 INPUT "RADIUM ACTIVITY (pCi g--1)";RA 

1110 IF RA<O THEN PRINT "THE RADIUM ACTIVITY CANNOT BE NEGATIVE" :GOTO 1100 

1120 LPRINT "MEASURED RADIUM ACTIVITY"," ",RA,"pCi/g--l" 

1130 INPUT "EMANATION COEFFICIENT default=.35";E 

1140 IF E=-1 GOTO 1150 ELSE IF E<O OR E>I GOTO 1160 ELSE 1180 

1150 E=.35# : LPRINT "DEFAULT LAYER EMANATION COEFFICIENT",E : GOTO 1190 

1160 PRINT "THE EMANATION COEFFICIENT MUST BE BETWEEN 0 AND 1." 

1170 GOTO 1130 

1180 LPRINT "MEASURED EMANATION COEFFICIENT",E 

1190 Q(I)=XL*RA*E*RHO(I)/P(I) 

1200 PRINT "CALCULATED SOURCE TERM CONCENTRATION",Q(I) 

1210 LPRINT "CALCULATED SOURCE TERM CONCENTRATION", 

1220 LPRINT USING "##.###---- ";Q(I), :LPRINT "pCi cm--3 s--l" :GOTO 1270 

1230 INPUT "SOURCE TERM (pCi cm--3 s--l)";Q(I) 

1240 IF Q(I)<O GOTO 1250 ELSE 1260 

1250 PRINT "THE SOURCE TERM CANNOT BE NEGATIVE." :GOTO 1230 

1260 LPRINT "MEASURED SOURCE TERM CONCENTRATION",Q(I),"pCi cm--3 s--l" 

1270 INPUT "WEIGHT % MOISTURE";XM(I) 

1280 IF XM(I)<O OR XM(I)>100 GOTO 1290 ELSE 1310 

1290 PRINT "THE WEIGHT % MOISTURE MUST BE BETWEEN 0 AND 100. " 

1300 GOTO 1270 

1310 LPRINT "WEIGHT % MOISTURE"," ", XM(I), "I%"i 

1320 XMS(I)=.O1*XM(I)*RHO(I)/P(I) 

1330 LPRINT "MOISTURE SATURATION FRACTION", :LPRINT USING " .###";XMS(I) 

1340 IF XMS(I)>1! GOTO 1350 ELSE 1370 

1350 PRINT "THE MOISTURE CONTENT IS >100% SATURATION. PLEASE REENTER."

3.64-30
1360 LPRINT "MOISTURE SATURATION >100%. NEW VALUE REQUESTED." :GOTO 1270 

1370 INPUT "DIFFUSION COEFFICIENT (cm&-2 s--l) default=calculated ";D(I) 

1380 IF D(I)=-I GOTO 1410 ELSE IF D(I)<O OR D(I)>I GOTO 1390 ELSE 1440 

1390 PRINT "REENTER DIFFUSION COEFFICIENT VALUES BETWEEN 0 AND 1. 

1400 GOTO 1370 

1410 D(I)=.07*EXP(-4*(XMS(I)-XMS(I)*P(I)*P(I)+XMS(I)-5)) 

1420 LPRINT "CALCULATED DIFFUSION COEFFICIENT", 

1430 LPRINT USING "##.### .. ";D(I), :LPRINT "cm-2 s--l" :GOTO 1450 

1440 LPRINT "MEASURED DIFFUSION COEFFICIENT",D(I),"cm-2 s--l" 

1450 PRINT : PRINT : LPRINT : LPRINT 

1460 IF I=2 OR I=5 AND N=5 OR I=6 OR 1>8 THEN LPRINT CHR$(12) 

1470 NEXT I 

1480 ON ERROR GOTO 1500 

1490 GOTO 1510 

1500 IF ERR=53 GOTO 1540 

1510 PRINT "THE BACKUP FILE RNDATA.BAK WILL BE REPLACED IF IT EXISTS UNLESS N 

IS TYPED. TYPE ANY OTHER KEY TO CONTINUE. IF N IS TYPED THE INPUT 

DATA WILL NOT BE SAVED." 

1520 V$=INPUT$(1) 

1530 IF V$="N" OR V$="n" GOTO 1980 ELSE GOTO 1550 

1540 RESUME NEXT 

1550 KILL "A:RNDATA.BAK" 

1560 ON ERROR GOTO 1580 

1570 GOTO 1600 

1580 IF ERR=53 THEN PRINT "BACKING UP RNDATA" 

1590 RESUME NEXT 

1600 NAME "A:RNDATA" AS "A:RNDATA.BAK" 

1610 ON ERROR GOTO 0 

1620 OPEN "A:RNDATA" FOR OUTPUT AS #2 

1630 PRINT #2,USING " ##.# ";N; 

1640 PRINT #2,USING " ##-.### .. ";FO1;CN1; 

1650 PRINT #2,USING " ##.# ";ICOST; 

1660 PRINT #2,USING " ##.### ---- ";CRITJ;ACC 

1670 FOR 1=1 TO N 

1680 PRINT #2,USING " ##.### ... ";DX(I);D(I);P(I);Q(I);XMS(I);RHO(I) 

1690 NEXT I

3.64-31
CLOSE #2 

LPRINT :LPRINT :LPRINT 

LPRINT " ","DATA SENT TO THE FILE "RNDATA' ON DRIVE A:" :LPRINT 

GOSUB 1850 

GOTO 1980 

OPEN "A:RNDATA" FOR INPUT AS #3 

INPUT #3,N,F01,CNI,ICOST,CRITJ,ACC 

FOR I=1 TO N 

INPUT #3,DX(I),D(I),P(I),Q(I),XMS(I),RHO(I)

1700 

1710 

1720 

1730 

1740 

1750 

1760 

1770 

1780 

1790 

1800 

1810 

1820 

1830 

1840 

1850 

1860 

1870 

1880 

1890 

1900 

1910 

1920 

1930 

1940 

1950 

1960 

1970 

1980 

1990 

2000 

2010 

2020 

2030

NEXT I 

LPRINT 

RETURN 

' END

;CN1;

ACC" 

XMS

: IF N=4 OR N=8 OR N>11 THEN LPRINT CHR$(12) 

'END PRINT SUBROUTINE 

DATA INPUT MODULE BEGIN RAECOM MODULE

N03=O 

IF ICOST>O THEN NO3=ICOST : ICOST=I 

ITHK=0 

NO2=N03-1 : NO1=N02-1 : NO4=N03+1 

NM1=N-1 : NM2=N-2 : JTST=I

3.64-32

NEXT I 

CLOSE #3 

LPRINT :LPRINT :LPRINT 

LPRINT " ","DATA INPUT TO DIFFUSION CALCULATIONS" : LPRINT 

GOSUB 1850 

GOTO 1980 

' PRINT SUBROUTINE 

LPRINT " N F01 CN1 ICOST CRITJ 

LPRINT USING "### ";N; LPRINT USING " ##.### ... ";F01 

LPRINT USING " ### ";ICOST; 

LPRINT USING " ##.### ---- ";CRITJ;ACC : LPRINT 

LPRINT "LAYER DX D P Q 

RHO " 

FOR I=1 TO N 

LPRINT USING "### ";I; 

LPRINT USING " ##.### ---- ";DX(I);D(I);P(I);Q(I);XMS(I); 

LPRINT USING "###.###";RHO(I)
2040 

2050 

2060 

2070 

2080 

2090 

2100 

2110 

2120 

2130 

2140 

2150 

2160 

2170 

2180 

2190 

2200 

2210 

2220 

2230 

2240 

2250 

2260

2270 

2280 

2290 

2300 

2310 

2320 

2330 

2340 

2350 

2360 

2370 

2380 

• 2390

TANH=-1 :RETURN 'END OF SUBROUTINE TANH 

RF(1)=(Q(1)*P(1)/AB)*TANH 

IF F01=-1 GOTO 2300 ELSE 2320 

RF(1)=RF(1)/(1+(.5*TANH)/.5/(EXP(V)+EXP(-V))) 

FO =-.5*RF(1)*TANH 

DDX(1)=DX(1) 

RC(1)=CN/A4(1) 

JO#=-RF(1)*10000 

PRINT : LPRINT 

PRINT USING "BARE SOURCE FLUX FROM LAYER 1: ##.### ---- pCi m--2 s'-1";JO# 

LPRINT USING "BARE SOURCE FLUX FROM LAYER 1: ##.### ---- pCi m-2 s--l";JO# 

PRINT "calculation in progress" 

FOR I=1 TO N

3.64-33

FO=FO1/10000 

CN=CN1/1000 

CRITJ=CRITJ/10000 

FOR I=1 TO N 

A4(I)=1-. 74*XMS(I) 

NEXT I 

AB=SQR(XL/D(1)) 

V=AB*DX(1) 

GOSUB 2140 

GOTO 2280 

IF V<-9.01 GOTO 2260 'THIS SUBROUTINE CALCULATES TANH 

IF V<-.7 THEN GOTO 2240 

IF ABS(V)<=2--12 THEN GOTO 2200 

IF ABS(V)<=.7 THEN GOTO 2210 

IF V<=9.01 THEN GOTO 2220 

IF V>9.01 THEN GOTO 2230 

TANH:V :RETURN 

TANH=V*(1-V-2*(.0037828+. 8145651/(V"2+2.471749))) :RETURN 

TANH=1-2/(EXP(V)*EXP(V)+I) :RETURN 

TANH=1 :RETURN 

V=-V 

TANH=-(1-2/(EXP(V)*EXP(V)+1)): RETURN 

V=-V
2400 Q(I)=Q(I)*P(I) 

2410 P(I)=F(!)*A4(1) 

2420 D(I)=D(I)*P(I) 

2430 NEXT I 

2440 DDX(1)=DX(1) 

2450 ALP(1)=SfR(XL*P(1)/D(1)) 

2460 FOR I=2 TO N 'THICKNESS OPTIMIZATION FEEDS BACK HERE 

2470 DDX(I)=DX(I) 

2480 NEXT I 

2490 ' MODIFY PARAMETERS FOR PROGRAM LIMITS 

2500 SUMX=O : SUMA=O : SUMAX=O : XRED=O : XCHG=O : X(1)=O XO=O 

2510 SUMMAX=ALP(1)*DX(1) 

2520 IF SUMMAX>4.61 THEN XRED=4.61/ALP(1) ELSE GOTO 2550 

2530 FO=FO*EXP(4.61-SUMMAX) 

2540 SUMMAX=ALP(1)*(DX(1)-XRED) 

2550 IF XRED>O THEN XCHG=DX(1)-XRED 

2560 FOR I=1 TO N 

2570 ALPI=SQR(XL*P(I)/D(I)) 

2580 SUMX=SUMX+DX(I) 

2590 X(I+1)=SUMX-XCHG 

2600 SUMA=SUMA+ALPI 

2610 ALSUMALPI*X(I+I) 

2620 IF ALSUM>SUMMAX THEN SUMMAX=ALSUM 

2630 SUMAX=SUMAX+ALSUM 

2640 ALP(I)=ALPI 

2650 NEXT I 

2660 IF SUMMAX>174 GOTO 2670 ELSE 2690 

2670 LPRINT "THE LAYER THICKNESS OR DIFFUSION COEFFICIENT EXCEEDS LIMITS" 

2680 LPRINT 

2690 IF SUMMAX>87 THEN XO=SUMAX/SUMA : ELSE GOTO 2730 

2700 FOR I=0 TO N 

2710 X(I+1)=X(I+1)-XO 

2720 NEXT I 

2730 ' CALCULATE PARAMETERS FOR MATRIX 

2740 FOR 1=1 TO NM1 

2750 RDUM:SQR(P(I+1)*D(I+I)/(P(I)*D(I)))

3.64-34
R(I)=-.5*(l-RDUM) 

RR(I)=-.5*(l+RDUM) 

QP=(Q(I+1)/P(I+1)-Q(I)/P(I))/XL 

T(I)=QP*EXP(-ALP(I)*X(I+1)) 

U(I)=QP*.5*EXP(ALP(I)*X(I+1)) 

AIPIMI(I)=SQR(XL)*(SQR(P(I+1)/D(I+1))-SQR(P(I)/D(I) 

NEXT I 

ALP(N)=SQR(XL*P(N)/D(N)) 

'SPECIFY MATRIX ELEMENTS AND SOLVE 

FOR I=l TO NM1 

J=5*I-4 

K=2*I-I 

A(J)=EXP(-2*ALP(I)*X(I+1)) 

A(J+1)=-EXP(AIPlMI(I)*X(I+1)) 

A(J+2)=-EXP(-(ALP(I+1)+ALP(I))*X(I+1)) 

A(J+3)=R(I)*EXP((ALP(I+1)+ALP(I))*X(I+l)) 

A(J+4)=RR(I)*EXP(-AIPlMI(I)*X(I+1)) 

B(K)=T(I) 

B(K+1)=U(I) 

NEXT I 

-N5M4--5*N-4 

A(N5M4)=EXP(-2*ALP(N)*X(N+1)) 

N2Ml=2*N-1 

B(N2Ml)=(CN-Q(N)/(P(N)*XL))*EXP(-ALP(N)*X(N+I)) 

'UPPER TRIANGULARIZE MATRIX 

G(1)=A(1)+EXP(-2*ALP(1)*X(l)) 

G(2)=A(2)/G(l) 

G(3)=A(3)/G(l) 

BU(1)=(B(1)+FO*EXP(-ALP(1)*X(l))/(D(1)*ALP(l)))/G(l) 

FOR I=l TO NM2 

J=5*I-I 

K--2*I 

G(J)=A(J)-G(J-2) 

G(J+1)=(A(J+1)-G(J-1))/G(J) 

BU(K)=(B(K)-BU(K-1))/G(J) 

G(J+2)=A(J+2)-G(J+l)

2760 

2770 

2780 

'7790 

2800 

2810 

2820 

2830 

2840 

2850 

2860 

2870 

2880 

2890 

2900 

2910 

2920 

2930 

2940 

2950 

2960 

2970 

2980 

2990 

3000 

3010 

3020 

3030 

3040 

3050 

3060 

3070 

3080 

3090 

3100 

3110

3.64-35
3120 G(J+3)=A(J+3)/G(J+2) 

3130 G(J+4)=A(J+4)/G(J+2) 

3140 BU(K+1)=(B(K+1)-BU(K))/G(J+2) 

3150 NEXT I 

3160 N5M6=5*N-6 

3170 G(N5M6)=A(N5M6)-G(N5M6-2) 

3180 G(N5M6+1)=(A(N5M6+1)-G(N5M6-1))/G(N5M6) 

3190 N2M2=2*N-2 

3200 BU(N2M2)=(B(N2M2)-BU(N2M2-1))/G(N5M6) 

3210 G(N5M6+2)=A(N5M6+2)-G(N5M6+1) 

3220 BS(N)=(B(N2M1)-BU(N2MI-1))/G(N5M6+2) 

3230 AT(N)=BU(N2MI-1)-G(N5M6+1)*BS(N) 

3240 FOR I=1 TO NM2 

3250 J=5*(N-I)-3 

3260 K=2*(N-I)-I 

3270 L=N-I 

3280 BS(L)=BU(K)-G(J)*AT(L+1)-G(J+1)*BS(L+I) 

3290 AT(L)=BU(K-1)-G(J-2)*BS(L) 

3300 NEXT I 

3310 BS(1)=BU(1)-G(2)*AT(2)-G(3)*BS(2) 

3320 AT(1)=(BS(1)*EXP(-ALP(1)*X(1))'FO/(ALP(1)*D(1)))*EXP(-ALP(1)*X(l)) 

3330 'COMPLETE MATRIX SOLUTION 

3340 FOR I=1 TO N 

3350 ALPI=ALP(I)*X(I+I) 

3360 ASI=AT(I)*EXP(ALPI) 

3370 BSI=BS(I)*EXP(-ALPI) 

3380 RC(I)=ASI+BSI+Q(I)/(P(I)*XL) 

3390 RF(I)=-D(I)*ALP(I)*(ASI-BSI) 

3400 NEXT I 

3410 RC(N)=CN 

3420 IF ICOST=O GOTO 3800 

3430 FOP=FO 

3440 IF FO<I THEN FOP=1 

3450 IF RF(1)<O OR RF(1)=O THEN RF(1)=1 

3460 'BEGIN COVER THICKNESS OPTIMIZATION 

3470 IF JTST=O OR CRITJ>99 OR NOT CRITJ>O GOTO 3800

3.64-36
348( 

3490 

3500 

3510 

3520 

3530 

3540 

3550 

3560 

3570 

3580 

3590 

3600 

3610 

3620 

3630 

3640 

3650 

3660 

3670 

3680 

3690 

3700 

3710 

3720 

3730 

3740 

3750 

3760 

3770 

3780 

3790 

3800 

3810 

3820 

•~ 3830

T7=(RF(N)-CRITJ)/CRITJ 

ABT7=ABS(T7) 

IF ABT7<ACC GOTO 3800 

NTST=N03 

IIJ=N02 

I IF NTST=NSAVE GOTO 3560 

) DXMAX=O : DXMIN=O : RFMAX=O : RFMIN=O 

NSAVE=NTST 

'SET LIMITS AND SEARCH FOR DX 

IF T7=0 GOTO 3800 

IF T7<0 THEN DXMAX=DX(NTST) ELSE 3600 

RFMIN=RF(N) : GOTO 3610 

DXMIN=DX(NTST) : RFMAX=RF(N) 

IF DXMAX=O GOTO 3620 ELSE 3630 

DX(NTST)=DX(NTST)*(I+.5*T7) : GOTO 3640 

DX(NTST)=DXMIN+(DXMAX-DXMIN)*(RFMAX-CRITJ)/(RFMAX-RFMIN) 

IF RFMAX=O THEN DX(NTST)=.5*DXMAX 

IF NOT DX(NTST)>1 THEN DX(NTST)=O: JTST=O 

IF ITHK=O GOTO 2460 

T2T=O 

IF N04>N GOTO 3720 

FOR IJ=N04 TO N 

T2T=T2T+DX(IJ) 

NEXT IJ 

IF.NO1<2 GOTO 3760 

FOR IJ=2 TO NOI 

T2T=T2T+DX(IJ) 

NEXT IJ 

T23=DX(NTST) 

T2T=T2T+DX(NTST) 

IF NOT DX(NTST)=T23 THEN CRITJ=-1 

GOTO 2460 

'PRINT RESULTS 

PRINT PRINT " RESULTS OF RADON DIFFUSION CALCULATIOI 

PRINT " LAYER THICKNESS EXIT FLUX EXIT CONC.", 

PRINT " (cm) (pCi m--2 s--1) (pCi 1-i) "

3.64-37

I

r

NS" : PRINT 

: LPRINT

I
3840 

3850 

3860 

3870 

3880 

3890 

3900 

3910 

3920 

3930 

3940 

3950 

3960 

3970 

3980 

3990 

4000 

4010 

4020 

4030

3.64-38

FOR I=1 TO N 

RXYZ=RF(I)*10000 

CXYZ=RC(I)*1000*A4(1) 

PRINT USING " ### ";I; 

PRINT USING " ##.### ... ";DDX(I);RXYZ;CXYZ 

NEXT I 

LPRINT :LPRINT 

LPRINT " ","RESULTS OF THE RADON DIFFUSION CAL4 

LPRINT :LPRINT 

LPRINT " ","LAYER THICKNESS EXIT FLUX 

LPRINT " "," (cm) (pCi m--2 s--l) 

LPRINT 

FOR I=1 TO N 

RXYZ=RF(I)*10000 

CXYZ=RC(I)*1000*A4(I) 

LPRINT " ", : LPRINT USING "### ";I; 

LPRINT USING " ##.### ... ";DDX(I);RXYZ;CXYZ 

NEXT I 

LPRINT CHR$(12) 

END

CULATIONS" 

EXIT CONC 

(pCi 1--1

' I 

)"
TABLE 2B RADON Program Sample Problem Output

------------- RADON !** 

Version 1.2 - May 22, 1989 - G.F. Birchard tel. # (301)492-7000 

U.S. Nuclear Regulatory Commission Office of Research 

RADON FLUX, CONCENTRATION, AND TAILINGS COVER THICKNESS ARE 

CALCULATED FOR MULTIPLE LAYERS 

THREE-LAYER SAMPLE PROBLEM 

CONSTANTS

RADON DECAY CONSTANT 

RADON WATER/AIR PARTITION COEFFICIENT 

SPECIFIC GRAVITY OF COVER & TAILINGS

.0000021 

.26 

2.65

s^-1

GENERAL INPUT PARAMETERS

LAYERS OF COVER AND TAILINGS 

DESIRED RADON FLUX LIMIT 

NO. OF THE LAYER TO BE OPTIMIZED 

DEFAULT SURFACE RADON CONCENTRATION 

RADON FLUX INTO LAYER 1 

SURFACE FLUX PRECISION 

LAYER INPUT PARAMETERS 

LAYER 1 TAILINGS 

THICKNESS 

POROSITY 

CALCULATED MASS DENSITY 

MEASURED SOURCE TERM CONCENTRATION 

WEIGHT % MOISTURE 

MOISTURE SATURATION FRACTION 

MEASURED DIFFUSION COEFFICIENT

3 

20 

3 

0 

0 

.001

pCi m'-2 s^-1

pCi 

pCi 

pCi

500 

.44 

1.484 

.000573 

11.7 

.395 

.013

l^-1 

m^-2 s^-1 

m^-2 s--1

r 

cm 

g cm^-3 

pCi cm^-3 s^-1 

cm^-2 s--1

LAYER 2 CLAY COVER

THICKNESS 

POROSITY 

CALCULATED MASS DENSITY 

MEASURED SOURCE TERM CONCENTRATION 

WEIGHT % MOISTURE 

MOISTURE SATURATION FRACTION 

MEASURED DIFFUSION COEFFICIENT

50 

.3 

1.855 

0 

6.3 

.390 

.0078

cm 

g cm^-3 

pCi cm'-3 s--1 

cm^-2 s--1

3.64-39
LAYER 3 SOIL COVER

THICKNESS 

POROSITY 

CALCULATED MASS DENSITY 

MEASURED SOURCE TERM CONCENTRATION 

WEIGHT % MOISTURE 

MOISTURE SATURATION FRACTION 

MEASURED DIFFUSION COEFFICIENT

100 

.37 

1.6695 

0 

5.4 

.244 

.022

cm 

g cm^-3 

pCi cm"-3 s^-I 

cm^2 s^-i

DATA SENT TO THE FILE 'RNDATA' ON DRIVE A:

N 

3

LAYER 

1 

2 

3

F01 

0. O00D+00 

DX 

5.000D+02 

5. OOOD+01 

1 000D+02

CN1 

0. O00D+00 

D 

1.300D-02 

7.800D-03 

2.200D-02

ICOST 

3

P 

4.400D-01 

3.OOOD-01 

3.700D-01

CRITJ 

2. OOOD+01 

Q 

5.730D-04 

0. O00D+00 

0.O00D+00

ACC 

1. OOOD-03 

XMS 

3.946D-01 

3.895D-01 

2.437D-01

RHO 

1.484 

1.855 

1.670

BARE SOURCE FLUX FROM LAYER 1: 1.984D+02 pCi m^-2 s^-I

RESULTS OF THE RADON DIFFUSION CALCULATIONS

EXIT FLUX 

(pCi m^-2 s^-l) 

7.691D+01 

4.524D+01 

2.001D+01

EXIT CONC. 

(pCi I^-1) 

1.670D+05 

4.430D+04 

O.O00D+00

3.64-40

LAYER 

1 

2 

3

THICKNESS 

(cm) 

5.000D+02 

5.000D+01 

1.490D+02

I
REFERENCES 

1. V.C. Rogers and K.K. Nielson, "Radon Attenuation Handbook for Uranium 

Mill Tailings Cover Design," NUREG/CR-3533,* U.S. Nuclear Regulatory 

Commission, 1984. 

2. V.C. Rogers, K.K. Nielson, and G. B. Merrell, "Engineering Guides for 

Estimating Cover Material Thickness and Volume," Report UMTRA-DOE/ALO-193, 

U.S. Department of Energy, Albuquerque, New Mexico, 1982. 

3. D.R. Kalkwarf and D.W. Mayer, "Influence of Cover Defects on the 

Attenuation of Radon with Earthen Covers," U.S. Nuclear Regulatory 

Commission, NUREG/CR-3395,* 1983. 

4. V.C. Rogers, K.K. Nielson, and G.B. Merrell, "The Effects of Advection on 

Radon Transport Through Earthen Materials," U.S. Nuclear Regulatory 

Commission, NUREG/CR-3409,* 1983. 

5. American Society for Testing and Materials, "Standard Test Methods for 

Moisture-Density Relations of Soils and Soil-Aggregate Mixtures Using 

5.5 lb (2.49 kg) Rammer and 12 in. (305 mm) Drop," ASTM-D-698-78, 

Philadelphia, 1978. 

6. L.D. Baver, Soil Physics, John Wiley & Sons, Inc., New York, 1956. 

7. W.J. Rawls and D.L. Brakensiek, "Estimating Soil Water Retention From 

Soil Properties," Journal of the Irrigation and Drainage Division, American 

Society of Civil Engineers, Vol. 108, No. IR2, pp. 166-171, June 1982. 

8. U.S. Nuclear Regulatory Commission, "Final Generic Environmental Impact 

Statement," NUREG-0706,* Washington, DC, 1980. 

9. K.K. Nielson et al., "Radon Emanation Characteristics of Uranium Mill 

Tailings," in Proceedings of the Symposium on Uranium Mill Tailings 

Management, Colorado State University, Fort Collins, December 9-10, 1982. 

10. K.K. Nielson and V.C. Rogers, "A Mathematical Model for Radon Diffusion 

in Earthen Materials," U.S. Nuclear Regulatory Commission, NUREG/CR-2765,* 

1982. 

11. V.C. Rogers, G.M. Sandquist, and K.K. Nielson, "Radon Attenuation Effective

ness and Cost Optimization for Uranium Mill Tailings and Composite Covers," 

UMTRA-DOE/ALO 165, U.S. Department of Energy, July 1981. 

*NUREG-series reports are available at current rates from the Superintendent 

of Documents, U.S. Government Printing Office, Post Office Box 37082, 

Washington, DC 20013-7082; or from the National Technical Information 

Service, Springfield, Virginia 22161.

3.64-41
VALUE/IMPACT STATEMENT

A draft value/impact statement was published with the draft version of 

this regulatory guide (Task WM 503-4) when the draft guide was published for 

public comment in May 1987. No changes were necessary, so a separate value/ 

impact statement for the final guide has not been prepared. A copy of the 

draft value/impact statement is available for inspection and copying for a fee 

at the Commission's Public Document Room at 2120 L Street NW., Washington, 

DC, under Task WM 503-4.

3.64-42
UNITED STATES 

NUCLEAR REGULATORY COMMISSION 

WASHINGTON, D.C. 20555

FIRST CLASS MAIL 

POSTAGE & FEES PAID 

USNRC 

PERMIT No G-67

OFFICIAL BUSINESS 

PENALTY FOR PRIVATE USE, $300

IA. 2055-'

I

t