Equation of Continuity in 

Geology with Applications 

to the Transport of 

Radioactive Gas

GEOLOGICAL SURVEY BULLETIN 1052-1

This report concerns work done on behalf 

of the U.S. Atomic Energy Commission 

and is published with the permission of 

the Commission

ERRATUM, BULLETIN 1052-1 

Insert on page 294, equation 3.1

On the right side of the equation, insert a Greek letter "lambda" 

immediately following the first minus sign.

INT.-DUP., D.

Equation of Continuity in 

Geology with Applications 

to the Transport of 

Radioactive Gas

By A. Y. SAKAKURA, CAROLYN LINDBERG, and HENRY PAUL

EXPERIMENTAL AND THEORETICAL GEOPHYSICS

GEOLOGICAL SURVEY BULLETIN 1052-1

This report concerns work done on behalf 

of the U.S. Atomic Energy Commission 

and is published with the permission of 

the Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1959
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.G. - Price 15 cents (paper cover)
CONTENTS

Page

Symbols.._ _______________________________________________________ iv

Abstract__ _______________________________________________________ 287

Introduction._ ____________________________________________________ 287

Acknowledgments. ----____-_--___-____-________________________-__ 289

Steady-state transport of radon by gases__-_--_-__-_____________-_-_- 289

Cylindrical source.____________________________________________ 289

Extended source_____________________________________________ 292

Transient-state transport of radon by gases_________________________ 294

Nearest boundary of source.-_______________________________________ 296

Application to the problem of radon flow in the Texas Panhandle______ 298

Discussion and conclusions________________________________________ 303

Literature cited-__________________________________________________ 305

ILLUSTRATIONS

FIGURE 89. Typical curve for computing distance of radon source from Page

well___________________________________ 299-

90. Dynamic test data for Thompson B-7 well._______________ 30Q

91. Dynamic test data for Masterson A-2 well._______________ 301

92. Dynamic test data for Thompson A-1 well_-____--_-______ 302

93. Dynamic test data for Kilgore A-l 1 well-____________.--__ 303

94. Dynamic test data for Thompson C-l well ________________ 304

TABLES

TABLE 1. Data for calculated values of nearest boundary of radon source

for five Colorado Interstate gas wells in the Texas Panhandle Page 

area. ____________________.__ 298 

2. Total calculated uranium source density for wells in the Texas

Panhandle gas field___________________________________ 300

m
SYMBOLS

A Substance carried

a Number of moles of A participating in a chemical reaction

D Pay-zone thickness

j Porosity of medium

7 Density of carrier

v Effective flow velocity of carrier

F Mass flux=70

ki Reaction constant of a chemical reaction

k2 Reaction constant of reverse reaction

k Permeability

LA Rate of production of substance A per unit volume of medium,

through chemical interaction or radioactive decay 

MQ Radon density (/iyuc/unit volume) at STP from cylindrical shell

source

p pressure =

pe Reservoir pressure

pw Well head pressure

Ap2 Squared pressure difference =^e2 Pw

p0 Standard pressure

Q Volume production at well-head conditions

Qo Volume production at STP

Q* Cumulative production at well-head conditions when radon first

	appears

Q* Cumulative production at STP when radon first appears

re Reservoir radius

rw Well radius

r0 Radius of cylindrical shell source

R Inner radius of extended source

.SA Rate of production of A by the medium per unit volume of the

	medium 

TO Standard absolute temperature

IV
EXPERIMENTAL AND THEORETICAL GEOPHYSICS

EQUATION OF CONTINUITY IN GEOLOGY WITH APPLICATIONS TO THE TRANSPORT OF RADIOACTIVE GAS

BY A. Y. SAKAKURA, CAROLYN LINDBERG, and HENRY FAUL

ABSTRACT

The transport of matter by fluids percolating through a porous medium is 

described by setting up a mass-conservation equation, analogous to that of 

hydrodynamics, for each of the substances in question. The sources and sinks 

of these equations serve to take into account the contribution from the medium 

and the interactions with other transported substances. The transport of radon 

by natural gases is treated in detail for the case of two important sources: the 

extended source and the cylindrical-shell source. From the solutions of the 

steady-state cases, the two source types can be distinguished, provided source 

density is uniform. From the solution of the transient case, the boundary of the 

source nearest to the gas well can be established. The results are applied to some 

selected data from the Texas Panhandle gas field.

INTRODUCTION

By the term "transport phenomena" we mean those situations in 

which the materials of interest are present in such small quantities 

that their individual diffusive properties are overwhelmed by the 

motion of the carrier. Such situations are common in geology. There 

are many instances where fluids flow, picking up, carrying, and depositing substances. Waters may transport and deposit various 

minerals. Gases and liquids may be tagged artificially with stable or 

radioactive tracers so that their flow may be observed. Sometimes 

gas may pick up radon, the gaseous decay product of radium contained 

in the pore space, and carry it as a natural tracer.

Under these conditions we can solve for the behavior of the carrier 

separately, and then set up an equation (or equations) of continuity 

for the substance of interest. These problems are greatly simplified 

by the availability of a large number of solutions to the various flow 

problems (Muskat, 1946). In the first part of this paper the equation 

for a general situation will be derived; the remainder of the paper will 

be devoted to the specific problem of radon transport.

Let A=substance carried,

a=moles of A participating in a chemical reaction, ->

F=m&ss flux of carrier (grams per unit area per second),

 TIA =concentration of substance A per gram of carrier, 

/= porosity of the medium,

287
288 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

ki= chemical-reaction constant, 

k2 re verse-reaction constant, 

7= density of the carrier (grams per unit volume), 

/SrA=rate of production of substance A by the porous medium

per unit volume of the medium, 

LA =rate of production (or loss) of substance A per unit volume

of the porous medium through chemical interaction or

radioactive decay,

\A= decay constant of A (per second) ; 

then the equation of continuity (mass conservation) becomes

and so on for other substances.

If the rate determining chemical reaction is of the form

the L's can be shown easily to be

_b 

a '

LD =-^. (1.2) 

In the radioactive chain

X, Xl _ \4 AI

the L's become

etc. (1-3)

We may well have a combination of both processes. The flux F and 

the density 7 of the carrier (if gases) are well known from solutions 

of Muskat's phenomenological equations of flow through porous 

media. The equations in general are nonlinear and will present great 

difficulty in solution except by approximate methods. Of course, if 

the importance of the problem warrants it, they may be solved by 

modern high-speed computers.
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 289

It should be noted that the reaction constants ki and k2 in equation 

1.2 are not necessarily the ones encountered in the laboratory, for 

the surface effects of the porous medium might be considerable.

ACKNOWLEDGMENTS

The writers wish to acknowledge the friendly cooperation of the 

personnel of various gas companies operating in the Panhandle area: 

Colorado Interstate Gas Co., Panhandle Eastern Pipeline Co., and 

Phillips Petroleum Co. This paper concerns work done by the U.S. 

Geological Survey on behalf of the Division of Research of the U.S. 

Atomic Energy Commission.

STEADY-STATE TRANSPORT OF RADON BY GASES

CYLINDRICAL SOURCE

We shall consider only sources possessing cylindrical symmetry 

about a gas well and extending throughout the vertical dimension of 

the gas pay zone. This problem is of paramount interest in a study 

of the radioactivity associated with some natural gases (Faul and 

others, 1954).

The gas sample tapped at the well will represent the source averaged 

over the surface of a cylinder centered about the well. The equation 

of continuity then becomes

r0 ) (2.1)

where (Muskat, 1946)

~* k Q /v,\ If r 1 " = " V* = -'

M

D=pay-zone thickness,

(pressure),

p e=reservoir pressure, 

pw= well-head pressure, 

re=reservoir radius, 

r«,=well radius, 

Q=volume production at well-head conditions
290 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

and where

<r=micromicrocuries of radon per unit area emitted by a cylindrical source 

of radius r0, height D. The total source strength is \2-aroD<r,

»;=radon concentration (micromicrocuries per gram of carrier), 

The equation of continuity,

is integrated to

where

_^L> _a\_ roe-xncr0)sf (r_ro ), 

V *y p»

I

 

P»Jr

(2.3)

(2.4)

As no radon can flow backwards, 77=0 when r>r0. Thus, we find 

^)=^^^v (2.5)

The significance of the exponential term can be seen if we rewrite 

equation 2.4,

* (2'6)

which is merely the time elapsed while the radon travels from r0 to 

rw. Thus, the exponential factor is the decay factor as radon travels 

from r0 to rw. Q can be calculated through integration by parts, and 

we find

3

n=Q

where the remainder, RN, after N terms satisfies

SAT _

which is generally of the order oAme. Thus, if

Ap2 ,j

t' e *

2pw2 log 

TV, j

(2.7)
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 291

the series is semiconvergent. We find then that the following expression is an adequate approximation.

f A ^.t. 1 I

7ZT (2-8)

The expression is exact at r=rw.

It is most convenient to express the measured radon content in 

terms of the gas sample tapped at the well head. Then the density 

(Af0), in micromicrocuries per unit volume at standard conditions 

(STP), of the radon due to a cylindrical shell source of radius r0 and 

surface density c is

M0=y(T0, poh(rw )=-° ^e'^rf (2.9) 

Vo

where

Qo^productioQ at STP, 

po= standard pressure, 

TO = standard temperature, 

Tw <= well-head temperature.

It should be noted that a differs from the actual surface source density 

by a factor owing to the incomplete sweeping by the gas. The degree of sweeping is a function of the emanating power of the radium 

or uranium mineral present in the pore space. In the Panhandle gas 

field the radioactive parents are probably present in what is described 

as "asphaltic petroleum residues." The emanating power of this 

material is estimated at 10 percent (F. J. Davis, Oak Ridge National 

Laboratory, and J. N. Rosholt, Jr., U.S. Geological Survey, oral 

communications) .

We shall now examine the behavior of solution 2.9. An examination of equation 2.8 reveals that for r0 ^100 feet the first term in 

square brackets can be neglected, although for all cases the second 

term is essentially zero. Thus, we rewrite equation 2.9 as

-Q ,

This can be rewritten using the relation

i * =* '-© ©^«' (2-n)

4i93765 59 2
292 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

and upon taking the square root and retaining the first two terms

(2.12) 2 k Tn v2 irD 

Equation 2.10 finally becomes

,, 27rDr0 . f X/irD T0 p e _~| fX/ M »o /1 , , r.\~| , 0 1Q . MO ~ 7^ X«r exp -^ ffr ro exp -~f- T ( "o+log ) (2.13)

Several points are now immediately evident. If we multiply M0 

(measured at series of steady states) by corresponding values of Q0 

and plot against 1/Q0 on a semilogarithmic paper, the result will be 

a straight line. If the measured values yield a straight line when 

treated in the foregoing manner, the slope will yield the radius of the 

equivalent shell source. Moreover Xcr27rr0Z? is the total effective 

source strength which will be equal to the vertical intercept of the plot.

EXTENDED SOURCE

We are also interested in radon distribution originating from a 

source extending from R to the edge of the reservoir. We then 

replace a by f dr0 (in micromicrocuries per unit volume) and integrate 

M0 from. R to re ,

M't'0 = f 1 "' £ Modro^ 2^- Xf ('' e-* QW radro. (2.14)

The integration can be readily carried out to first order in 

- To this order, equation 2.8 gives sufficiently accurate

value of 0(r0 ), with ^w2 substituted for ^?2 (r0) in the denominator of the 

first square bracketed term. Upon expanding the ratio ^-^- to first

Pw

order, we have

7rZ)/X

where

Ap2

2pw2 log 
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 293

with the additional restriction that

Ar>2 A

The logarithm is expanded as

, 7*0 i O

log «log -2

/ M) '10

(2.16)

Note that the expansion has very large error near rw and re . However, the absolute magnitude of the exponent is very small at rw so 

that the exponential is substantially one, whereas at re the exponent 

is so large that the exponential does not contribute to the integral. 

Moreover, the value of the logarithm is closely matched near R0 , the 

most significant portion of the integration interval. Thus

(2.17)

(g-Xn<ro1_g-Xn(r.)) _j_ "C ^ (2.18)

Although the integral can be expressed in terms of erj or erfc functions, 

it is much more useful to expand it. It is only necessary to use the 

first term of the expansion as the entire term is first order in co. We 

distinguish several cases.

(2.19)

(2.20)

B.

g-XQ(fl)

C. R0«R<r e
294 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

From C one can infer for an extended source of inner radius R and 

outer radius RI such that RQ<g.Ri<^R<re, that the equivalent shell 

source located at R has the surface density

_Xf/ cA W-fl2 )/ . «g0 \ , x<r ~^V 27 21? V I+^R) (2 '22)

A semilogarithmic plot of measured values of M0 ' times Q0 versus 

l/Qo will not yield a straight line as with the finite source. Therefore, 

it may be possible to determine whether the radon can be attributed 

to a finite cylindrical or an extended source, assuming the source 

concentration remains constant for the latter source.

TRANSIENT-STATE TRANSPORT OF RADON BY GASES

Considerable geologic information can be obtained from artificially induced transient motion of a gas. For example, a well may 

be shut down for 2 or 3 weeks, or long enough to allow dynamic 

and radioactive equilibria to establish themselves. The well is then 

opened and produced at a known (and fairly constant) rate. Radon 

content is measured at suitable intervals for several days.

The tune-dependent transport equation takes the following form :

«(r-ro) (3.1)

where the quantities have the same meaning as before, except that 

the gas properties are now functions of tune. We shall solve the 

problem formally to show the general procedure for solving equations 

of this type. Let

(3.2) 

where if

= _

dt f

with solution G(r,t) = constant, then t0 and t' satisfy the following 

boundary conditions,

G(r,t0 ) = G(r0,Q), G(r,t) = G(r0,t') (3.4)

t0 is the tune at which the radon, originating at r0 at t=G, first appears 

at r; the step function states the physical condition that the radon 

concentration is zero up to that tune, t' is the tune at which radon 

must start from r0 in order to arrive at r at time t. Then the exponential merely denotes the decay of radon en route. The step
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 295

function in r0 denotes the fact that radon cannot drift backwards. 

Inserting equation 3.2 now into equation 3.1 we find,

(3.5) 

The first term is zero because

dt /

dr v(r, t)

The second term also can be shown to be zero by the following 

argument,

G(r,0 = GW) or t'=H[r0 ,G(r0,t)] 

5G .&G

The remaining terms contain singularities of different orders, so 

they must vanish separately. The vanishing of the third term gives 

us the boundary condition 011 <j>, as when r r0, to=0 and t'=t. 

Thus, when r=r0,

v(r0, t) 

The remaining term finally becomes the differential equation for <£.

We solve equation 3.7 by successive substitution. On the right 

hand side of equation 3.7 we set <j> equal to , -r^> which satisfies the

V(T} t)

boundary condition in equation 3.6. Then we find, upon solving 

equation 3.7,

(3-8)

This process, of course, can be repeated indefinitely, that is, substitute 

equation 3.8 into the right-hand side of equation 3.7 and integrate. 

However, if the second term in equation 3.8 is small compared to the 

first, then we can conclude that equation 3.8 furnishes a sufficiently 

good solution. It may be pointed out that the first approximation 

yields an exact answer when t approaches infinity, as the steady state 

is approached wherein the velocity is no longer time dependent. We
296 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

shall now show that the equation 3.8 is an adequate approximation. 

It has been shown previously (Ritchie and Sakakura, 1956) that

1 k

(3.9)

where

2lrt Jy Zc

and T is large for almost all values of t. It further has been shown 

that the resulting expressions for the quantities in question are semiconvergent series in r, and cannot be differentiated term by term. 

Thus, the contour integrals must be differentiated first, then integrated. If we retain only the dominant terms, we find the second

I / 2 _ - 2\

term in the brackets of equation 3.8 to be =(- ) which is small, as

the validity of the semi convergent series requires. Thus, we can state 

that equation 3.8 yields an adequate solution. The radon density 

(at STP), from equations 3.2 and 3.8, becomes

exp

Transient gas-flow-phenomena are strongly dependent on the 

characteristics of the reservoir. These parameters are never sufficiently well known to warrant numerical evaluation of equation 3.10. 

The principal purpose of this derivation is to determine the nearest 

boundary of the radioactive source from which the radon may come.

NEAREST BOUNDARY OF SOURCE

We shall now discuss the most important aspect of the transient 

problem, common to the cylindrical source problem and to the extended source problem, which results from integrating the cylinder 

over r0 from R to re . When radon first appears, it must necessarily 

come from the point nearest to the well. Thus, we are interested in 

the solution to the equation

dr_ vr(,t) ,.-.. dt T ( }
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 297

with the boundary conditions,

t=Qt r =r0

that is, we are looking for tQ . Using the substitutions for T and p 

(equation 3.9) and letting

,__1_ f T+»« Ko(Vzp) e>

? o 

a--^2 (4.2)

equation 4.1 can be rewritten into a dimension!ess form

^~dz (4.3)

with p=p0 . when T=T (eventually we shall set r=0) and p 1 when 

T TO . If we consider values of r0 sufficiently small and (or) T sufficiently

fry

large, the radical can be se b equal to its steady state value and only

Pe

the first term of the expansion of the Be»sel functions need be considered in the Laplace inversion integral. It has been shown that

1 TY+z' 00 KI(^ZP) e" , 1 P 0 T . N "I -. .. 2~~- I _. ^ , -' -=.dz= \ 2/i,_i(T)+. . . I (4.4)

where

1 ro+ z"e"

P==eT,'Y = Euler's constant. 

Thus equation 4.3 becomes

^?««P2/M (4.5)

0 )]. (4.6)

We could obtain t from the above equation by fixing T and solving 

for r0 . We shall not do so for the same reason given for not evaluating equation 3.10. Returning now to finding the limiting case 

when £=0, T in equation 4.3 represents some earlier value of T, as the 

series diverge for r 0. However, it is known also that the cumulative 

production is

(4.7)
298 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

so we can write equation 4.6 as

\ Q*(TO)> (4.8)

when we have dropped the cumulative production at T as it is zero 

when t is equal to 0. Furthermore, we rewrite this in terms of the 

f unction Jn_equationf2.4,

n*

(4.9)

which relates the function, £2, to Q*, the cumulative production at 

which radon first appears. This must be corrected for the volume of 

gas standing in the well before opening. If V were the volume of the

p well, we replace Q* by Q* V^-> and upon correcting to STP, we find

 L in

as the expression which will yield the nearest boundary of the source.

APPLICATION TO THE PROBLEM OF RADON FLOW IN 

THE TEXAS PANHANDLE

A series of calculations were made to find the source density (equation 2.19) for five wells in the Texas Panhandle gas field. The exponential term in the equation is a function of the source radius R and 

of the pressure at R, and these quantities cannot be measured. If 

our assumption that R is small enough compared to the reservoir 

boundary radius re and that R can be set equal to the well radius rw is 

valid, calculations of the source density can be made with data obtained on the surface. To test the validity of this assumption, we 

investigated the location of the nearest boundary of the radioactive 

source. Equation 4.10 yields the boundary in terms of the time fl(r0) 

it takes the radon to travel from the source to the well and can be 

calculated from well data. In order to determine the actual distance 

to the source, we made use of equation 2.8 which expresses & in terms 

of rQ . For each well various values are assigned to r0) the corresponding & is calculated (table 1) and plotted against r0 giving a curve of

TABLE 1. Data for calculated values of nearest boundary of radon source for five 

Colorado Interstate gas wells in the Texas Panhandle area

Well

Kilgore A-ll._ ________ . __ .......

QS 

(10W)

2.1

3.5

1.7

1.9

2.5

Qo

(10W 

per day)

2.50

2.66

2.17

1.84

2.61

F(ft')

1300

1210

1120

1440

1270

Reservoir 

pressure,

Pe (PSI)

302

252

273

317

313

fl(ro) 

(days)

_ n (vii

.006

 001

-.005

.0002

ro (ft)

fw

7

r«

r»
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 299

the type shown in figure 89. The r0 corresponding to J2, calculated 

from equation 4.10 is found from these curves. No ambiguity arises 

from using the equations 't,o find r0 to justify setting R equal to re ; 

the location of a cylindrical source will serve equally well in locating 

the nearest boundary of an extended source.

The graphs of J2(r0) versus r0 were calculated from equation 2.8. 

 Here, p(r<>) was found from the relation

(5.1)

where pw is measured at the well, Ap2 is determined from Ap2=p2 Pi 

where pe was the measured shutdown pressure in October 1951, re 

was taken equal to 500 feet, and rw equal to % foot. The values 

assigned to r0 were rw, 1, 2, 5, 10, 25, and 50 feet. The pay-zone 

thickness D was taken equal to 50 feet, as no exact data were available on the actual thickness. Q0 is the production measured at STP 

at time of sampling, and Tu was taken as 300° K. The porosity of 

the pay zone / is known to vary between 8 and 13 percent in the 

Panhandle area; 10 percent was used in the calculations. The 

various values assumed are reasonable, and the error involved is 

negligible in the overall results.

FIGURE 89. Typical curve fir computing distance of radon source from well.
300 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

The nearest boundary Q(r0) was computed from equation 4.10, 

where V r^irh where h is the total depth of the drill hole and rw, 

Pet PQ> Tw, and T9 are taken as before. Q$ was taken from the 

graphs ((?o versus M'9) of dynamic tests on the wells (figs. 90, 91, 92, 

93, 94) at the point at which radon first appears; Q0 was taken from 

the production curve at the same time as Q% was chosen. The 

results are summarized in table 2. Graphs, similar to figure 89, for 

each well were used to find r0.

TABLE 2. Total calculated uranium source density for wells in the Texas Panhandle

gas field

Company and well

Panhandle Eastern: Kilgore 1-16 _ _____ ____ _.

Colorado Interstate: 

Thompson B-2 » ___________

Thompson B-7. . ___ _________ . ... _ _

Thompson A-1.. _____ _ ______________

Masterson A-2 ____________

Kilgore A-11-- .................. ..... ..

Thompson C-l _______________________

Well-head 

pressure, 

Pw (psi)

232

216

204

215

206

94n

223

M0 

We peri)

522

475

226

163

71

23

10-9 gc per g

9.08

7.54

3

AR

1.10

01

.38

i Average of 10 samples.

t 200

10"

O JDlyl951

July 1952

10s 10s 

CUMULATIVE PRODUCTION, IN CUBIC FEET

II

. Dynamic test data for Thompson B-7 well giving points at which data were taken for 

calculation.
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 301

-1951-

 1952 

. VI

0 July 1951

July 1952

2.4

22 lg

2;° z^ 

IS ly

"is

1.4

10' 10s 10s 

CUMULATIVE PRODUCTION, IN CUBIC FEET

10"

FIOTJBE 91. Dynamic test data for Masterson A-2 well giving points at which data were taken for

calculation.

The negative values of 12 can be attributed to various sources of 

error: (1) assumed values* of the parameters; (2) neglect of higher 

order terms in the derivation of the formulas; (3) assumption that 

the gas in the well expands isothermally, which is not true in general; 

(4) the uncertain effect of acidizing on the well radius and volume 

caliper logs in the Panhandle area show that the well diameter 

increases about an inch after acidizing; it seems logical that acidizing 

could produce near-cavernous regions at the well bottom and (5) 

experimental error involved in measuring pw, pe, Q0, M0' and h. We 

have taken these negative values to mean zero time, giving r0=rw. 

These negative values also give us a rough measure of the error 

involved that is, ±0.005. Thus, even though the Q (r0) versus 

r0 plot for Thompson A-l gives r0=7 feet, it would be more appropriate to assume r0=rw here.

The effective source density r, computed from

. I/tn JL 0 f tr i

f=r- 7p-fM0' 

Po 1» (5.2)

as the other factor in equation 2.19, is always very nearly equal to 

one for the wells in question. M0f, the measured radon content at 

the well, was chosen after the radon content reaches equilibrium 

(figs. 90, 91, 92, 93, 94) and pw is the pressure at the time of sampling.
302 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

200 

£ 180

£ 160 

a.

E 14° 

° 120 z

Radon content

103 105 106 

CUMUUTIVE PRODUCTION, IN CUBIC FEET

FIGURE 92. Dynamic test data for Thompson A-l well giving points at which data were taken for

calculation.

The remaining parameters were chosen as before. The results were 

converted from micromicrocuries per liter to grams of uranium per 

gram of the medium. As the effective source is assumed to have an 

emanating power of 10 percent, £ was corrected to give total source 

density £'. These source densities are shown in table 2.

The character of the graph of radon content versus cumulative 

production for the Masterson A-2 well is different from the other four 

graphs. Although the radon first appears relatively shortly after 

opening, thus giving r0=rw, the intensity continues to rise slowly and 

the equilibrium value is not reached until about 2 million cubic feet 

have been produced. This behavior was thought anomalous, and the 

tests were repeated a year later with the same result. The slow rise 

could be explained by assuming that the producing zone has several 

layers of varying permeability and that the less permeable rock contains more radioactive parent elements. An alternative, and possibly 

better, explanation would be a lack of symmetry in the radon source 

about the well. It is plausible that the radium concentration could 

increase gradually in any direction away from the hole. Similarly, 

the double knee in the graph for Thompson A-l could be interpreted 

as evidence of a localized, sharply bounded concentration of radium. 

The value for £' shown in table 2 would then be a weighted average of 

the two sources, the farther (and stronger) source being less effective. 

It must be emphasized that all the source strengths in table 2 are 

minimum values. It is known from geological studies (J. W. Mytton,
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 303

Radon content

Sfc

103 10s 106 

CUMULATIVE PRODUCTION, IN CUBIC FEET'

10s

FIGTJBE 93. Dynamic test data for Kilgore A-ll well giving points at which data were taken for

calculation.

oral communication) in this area that the radioactive asphaltic 

residues generally tend to obstruct the permeability. Consequently, 

the more concentrated sources are likely to be swept less effectively 

by the gas.

DISCUSSION AND CONCLUSIONS

To recapitulate, we list here the various formulas derived in this 

paper under the restriction that «< 

I. Steady state, cylindrical source at r0 :

 Mo (/ «,) = -^ A<r exp [ AQ(r0)] 

Qo

2irr0D |~ Atf/D fsl T\ A<T exp I 7^ P^oj^o

yo L yp»

II. Steady state, extended source from R to r«

A -e- n-e- T> i/r / * w Po A. r0 ^R^R0, Mo' ^ g- -f

Ap2

(2.9) 

(2.10)

(2.19)

(2.20)

C. (2.21)
304 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

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CUMULATIVE PRODUCTION, IN CUBIC FEET

FIGURE 94. Dynamic test data for Thompson C-l well giving points at which data were taken for

calculation.

III. Steady state, effective surface density of an extended source 

from E to Bv where R0 «R<R,

ei \<r

j,

IV. Transient state, cylindrical source:

Cn 1 50,-| . w . .,, I T; 7\ =vT dr I exp [ \(t V. 

;r»«2 (r,f) of J

V. Nearest boundary of radon source:

(3.10)

. yo PO i v vo

Geologic conditions are never so simple and clear cut as indicated 

here. Geologic structures certainly are not homogeneous or isotropic, 

nor are they in a convenient geometric configuration. The greatest 

source of error in the evaluation of source densities is the value of the 

emanating power. Moreover, it should be noted that the source 

distribution obtained will be an average, and localized high concentrations are likely to go unnoticed. The calculation of source densities is made on the basis of extended source distribution, assuming that 

a uniform distribution would be the most probable one. In the
EQUATION OF CONTINUITY, TRANSPORT OF RADIOACTIVE GAS 305

Texas Panhandle gas field, from which examples were selected the 

edge of the source can be considered to be at the well according to the 

calculations on dynamic test data, although there is some ambiguity 

in determining the production at which radon first appears.

It is interesting to note that three of the wells (Masterson A-2, 

Kilgore A-ll, and Thompson C-l) exhibit normal uranium density, 

whereas the others show abnormally high uranium content.

LITERATURE CITED

Faul, Henry, Gott, G. B., Manger, G. E., Mytton, J. W., and Sakakura, A. Y., 

1954, Radon and helium in natural gas: Internat. Geol. Cong., 19th, 

Algiers 1953, Comptes rendus, sec. 9, fasc. 9, p. 339-348.

Muskat, M., 1946, The flow of homogeneous fluids through porous media: Ann 

Arbor, Edwards, Inc.

Ritchie, R. H., and Sakakura, A. Y., 1956, Asymptotic expansions of solutions of 

the heat conduction equation in internally bounded cylindrical geometry: 

Jour. Appl. Physics, v. 27, no. 12, p. 1453-1459.

O