tracer diffusion in the ground in radioactive leak location

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Tracer Diffusion in the Ground in Radioactive Leak Location Andrew Gemant Citation: Journal of Applied Physics 24, 93 (1953); doi: 10.1063/1.1721141 View online: http://dx.doi.org/10.1063/1.1721141 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Diffusion Pump Oil Deposition Measurements Utilizing Radioactive Tracers J. Vac. Sci. Technol. 2, 59 (1965); 10.1116/1.1492399 Radioactive Tracers Phys. Today 16, 78 (1963); 10.1063/1.3051103 Leak Location by Radioactive Gases in Buried Pipes J. Appl. Phys. 22, 460 (1951); 10.1063/1.1699984 Measurement by Radioactive Tracers of Diffusion in Liquids J. Appl. Phys. 19, 1160 (1948); 10.1063/1.1715037 A New Method of Measuring Diffusion Coefficient in Solids with Radioactive Tracers J. Appl. Phys. 19, 308 (1948); 10.1063/1.1715066 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 130.18.123.11 On: Mon, 22 Dec 2014 03:09:43

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Tracer Diffusion in the Ground in Radioactive Leak LocationAndrew Gemant Citation: Journal of Applied Physics 24, 93 (1953); doi: 10.1063/1.1721141 View online: http://dx.doi.org/10.1063/1.1721141 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Diffusion Pump Oil Deposition Measurements Utilizing Radioactive Tracers J. Vac. Sci. Technol. 2, 59 (1965); 10.1116/1.1492399 Radioactive Tracers Phys. Today 16, 78 (1963); 10.1063/1.3051103 Leak Location by Radioactive Gases in Buried Pipes J. Appl. Phys. 22, 460 (1951); 10.1063/1.1699984 Measurement by Radioactive Tracers of Diffusion in Liquids J. Appl. Phys. 19, 1160 (1948); 10.1063/1.1715037 A New Method of Measuring Diffusion Coefficient in Solids with Radioactive Tracers J. Appl. Phys. 19, 308 (1948); 10.1063/1.1715066

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JOURNAL OF APPLIED PHYSICS VOLUME 24, NUMBER 1 JANUARY. 1953

Tracer Diffusion in the Ground in Radioactive Leak Location

ANDREW GEMANT

Engineering Laboratory and Research Department, The Detroit Edison Company, Detroit, Michigan (Received August 18, 1952)

The diffusion in the ground of a radioactive tracer gas, used for locating leaks in buried pipes, has been calculated, and the results are presented in the form of graphs. It is shown how the information can be used to obtain the amount of radioactivity needed in a test for both beta-emitters (C-14 in carbon monoxide) and gamma-emitters (Br-82 in methyl bromide).

INTRODUCTION

A METHOD using radioactive tracer gases for locating leaks in buried pipes was recently de­

scribed by the author.1·2 In short succession a few other papers were published on the same subject, notably by Brock-Mannestad,3 Klein,4 and Gueron and Pages.~

One of the chief factors affecting the efficiency of the method is the diffusion of the tracer gas in the ground after it has escaped from the leak. This factor deter­mines the concentrations reached in the ground and the amounts of radioactivity required for specific cases.

In the present paper the equation that controls the diffusion of the gas in the ground is derived under cer­tain simplifying assumptions. Next, a few graphs are presented, showing the progress of diffusion as a func­tion of several essential variables, such as depth below the surface, time elapsed, and diffusion coefficient of the gas. Finally, a few numerical computations will illus­trate the usefulness of the information for calculating amounts of radioactivity needed.

TRANSIENT DIFFUSION EQUATION

The leak in the pipe is considered as a point source in an infinite medium. It is realized that when the gas reaches the ground surface, conditions, chiefly because of convection, will change. A high water table level will also affect the results. The derived equation is, therefore, approximate; the shorter the elapsed time since the start of diffusion, the better is the approximation. For an instantaneous point source of strength M i, one has in analogy of the thermal equation6 for the gas concen­tration c,

porosity P (ratio of air volume to total volume). If k is the diffusion coefficient of the tracer gas in air,. then the coefficient k. in soil is given by

k.=Pk, (2)

since the average free cross section per unit area is also P. The capacity c. is the amount of gas that can be stored in the unit volume; hence,

e.=P. (3)

Since the diffusivity IC= k./ c., it follows that

I(=k. (4)

Equation (1) is now applied to the time differential dt, for which M i= mdt, if m is the effiux of gas per unit time. If the gas escapes during the interval tl to t2, and if to is the time of observation, then

m ft2 e (to-t)-l exp-r2/[4k(to-t)Jdt. (5)

8(7rk)lp 11

With suitable substitutions, this integral can be solved and yields

e=~(erf r erf r ). (6) 47rkPr 2ki(to-t2)t 2k!(tO-tl)t

Integration of the amount of gas over the total space results in the total effiux, Le.,

47rP £"'Cr2dr=m(t2-t1), (7)

(1) as may be verified by means of Eq. (6),

where t=time, lC=diffusivity of gas, r=distance from source, and c.= gas capacity of soil.

Two of these variables IC and c. involve the soil

1 Andrew Gemant, Conference on Electrical Insulation, (Pocono Manor, Nov. 1-3, 1950).

S Gemant, Hines, and Alexanderson, J. Appl. Phys. 22, 460 (1951). .

a L. Brock-Mannestad, Teleteknik 2, 28 (1951). • N. Klein, Bull. Research Council IsraelI, 110 (1951). i J. Gueron and A. Pages, Cables et Transmission 6, 96 (1952). S H. S. Carslaw and J. C. Jaeger, Operational Methods in A ppUed

Mathematics (Clarendon Press, Oxford, 1941), p. 107.

93

GRAPHS SHOWING PROGRESS OF DIFFUSION

For numerical calculations the amount of gas will be expressed in microcuries 1Jc. Since k is expressed in cm2/sec, and r in em, the effiux m is measured in IJC/sec, and the concentration is obtained in IJC/cm3•

In order to present the graphs in a rather general form, the quantity Cl is introduced by the equation

c= mel! P. (8)

This ;Iuantity el is plotted as the ordinate in the graphs, and if P and m are given, c can be calculated. The

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94 ANDREW GEMANT

4

5

6

K=0.2 .IIt=4hr

7~ ____ ~ ____ ~ __ ~~ ____ L-__ ~ 4 8 20

hours

FIG. ~. Co.ncentration of tracer gas in the ground vs time, wIth dIstance from the leak as variable parameter.

abscissa is the time (to - tI ) elapsed since the gas started leaking. There are then three parameters: the diffusivity k, the distance from the leak r, and the duration of the efflux At=(t2-tI). The expression (to-t2) in Eq. (6) is then given by (to-t I) -At.

Figure 1 shows - 10gcI as a function of (to -tI) in hours with the variable parameter r in the range from 1 to 10 feet. The duration of efflux is taken as 4 hours and k as 0.2. The concentration always rises for the first 4 hours, thence going through a maximum; the larger the distance the later this occurs.

In Fig. 2, the variable parameter is At, ranging from 1 to 8 hours, while k again is 0.2 and r is specified as 3 ft. They show the same characteristic as the previous curves; the longer the duration of the outflow the later the maximum occurs.

In Fig. 3, the diffusivity k varies from 0.06 to 0.6, r being 3 ft and At, 4 hours. With increasing diffusivity the maxima shift to shorter times; the concentrations for short times become larger, and for long times smaller.

THE CARBON MONOXIDE METHOD

The information can be used for the CO method, as developed by the author and Hines.I,2 Carbon-14 is a soft beta-emitter; labeled CO is introduced into the pipe, then pumped out from bore-holes in the ground and collected on filters. 7 Experiments showed that 1 p.C produces with a thin-window Geiger counter about 4000 counts per min.

7 For further experimental details, the paper quoted under reference 2 should be consulted.

Let it be required that a certain distance r from the leak C counts per min be obtained in order to permit location. This means that C/4000 micro curies must be removed by pumping. The volume of gas removed by pumping is designated by v; its magnitude will be es­timated below. If the concentration of the tracer is c, one has the requirement

C/4000=vc.

If Eq. (8) is introduced, one has

C/4000=vmCI/P.

(9)

(10)

The total activity M needed for the test is related to m by

M=nmAt, (11)

where n is a forcing factor, determined by an additional leak introduced into a distant man-hole in order to accelerate the travel of the tracer in the pipe. The factor n is defined as the ratio of the tracer gas arriving at the leak to the amount flowing into the ground. From (10) and (11) one has in microcuries

M = nPCAt/4000VCI. (12)

In order to use this equation, v must be known. Using a small pump for about 10 minutes, the amount of air pumped out of the soil (clay of a pore diameterS of about 10-3 em) can be estimated from our previous experimental data.2 The rate of effiux in one test was m= 6.3X 10-3 p.C/sec. A representative value for the porosity is 0.2, since dry soil has a porosity about 35 percent, and the moisture content of the soil is about 15 percent. Thus, from Eq. (8), c= 0.32X to-3CI. The pumping took place 5 hours after the tracer started to flow into the ground; hence, to-tI=At=5 hr.

In Table I are first listed the distances from the artificial leak, and the observed counts per min, both at bore-holes along the pipe and away from the pipe. These data indicate a pumping volume in the clay soil of about 2 liters, as the following columns show. They list CI as calculated from Eq. (6), the concentration of the tracer in the soil, the microcuries pumped out in 2 liters, and the counts expected with those activities.

3.5,r--v-""-

4.0.

4.5'-----~-----:!:------:-~----L----l 4 16

hours 20.

FIG. ~. Conce?tration of tracer in the ground !IS time, WIth duratIOn of outflow as variable parameter.

----8 Arnold Eucken, Zahlenwerte und Funktionen (Verlag. Julius

Springer, Berlin, 1952), vol. III, p. 359.

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RADIOACTIVE LEAK LOCATION 95

It is seen from the last row that observed counts away from the pipe are more'in keeping with the trend of calculation, than those along the pipe, as is to be expected. The surface of the pipe provides a by-pass for the tracer, actually facilitating its detection in a manner not accounted for by the diffusion theory.

The value of v= 2X lOS cm3 is now used in Eq. (12), as well as P= 0.2. In this manner one has in millicuries

(13)

This last equation can be used for evaluating the amount of CO activity needed for locating a leak in a pipe containing 120-kv cables. If the original length of the tracer column is 10 ft, it will spread2 in 10 days to about 60 ft which, without forced drive, takes 24 hours to pass the leak. Hence, t:..t= 24 hr, and for the purpose of calculations we might take 10= 24 hr also. The actual value of to is obviously not known. It is now required that at a 10-ft distance, in undisturbed soil, pumping should yield 200 counts per min. With this requirement fulfilled, measurable counts can be expected even at 20 ft from bore-holes along the pipe. From Eq. (6), Cl is calculated for k = 0.2, the diffusion coefficient of CO in air, ,.=305 cm, (to-t2)=0 and (to-tl)=24X3600 sec. The result is l.4X 10--4, and with n= 1, Eq. (13) gives 3.1 millicuries.

GAMMA-EMITTER TRACERS

Some authors 3-5 recommend the use of gamma­emitters, particularly methyl bromide labeled by bromine 82, emitting betas of 1.04 Mev and having a half-life of 35 hours.

If C counts per min are wanted on the ground surface over the leak, then the gamma-radiation intensity at that point is approximately C/3000 milliroentgens per

TABLE I. Estimate of pumping volume from experimental data.

Observed counts

per min Away Calculated

r Along from c poC counts ft pipe pipe ct poC/em' pumped per min

1 1900 1900 lX10~ 3.2XI0-4 0,64 2500 5 60 24 2XlO-4 6,4XlO~ 1.3XlO~ 52

10 19 0 4XlO-7 1.5XlO-s 3XI0-6 <1

/r.=3ft .<It = 4 hr

6'L-~ __ ~ ____ ~ ____ ~~ ____ ~ _____ ~ 4 8 12 16 20

hours

FIG. 3. Concentration of tracer in the ground vs time, with difIusivity as variable parameter.

hr. If the contamination of the soil surface is assumed to be uniform, this contamination is given (in J..LC) by 0.5 X 10-6 C per cm3 soil. This last relation follows from data9 relating water contamination to intensity at the surface.

This last figure may be identified with mCl (P cancels out); hence,

and, as before, in millicuries

nAtC M=0.5Xl0-9--.

Cl (14)

Because of the short half-life of Br-82, forced drive is advantageous; say, n=3. C, as before, is taken as 200. With the tracer column traveling fast, t:..t and to may be taken as 1 hr each. If the pipe (telephone cable, for instance) is buried 3 ft below ground, ,.=92 cm. The diffusion coefficient of methyl bromide is somewhat less than 0.2. With these data cl=6X1O-6, and M= 18 millicuries. Gases like methyl bromide may be partially adsorbed by the grains of the soil, a circumstance not considered by the diffusion theory as presented.

9 Atomic Energy Commission, The Effect of Atomic Weapons (U. S. Govt. Printing Office, Washington, D, C., 1950) p. 262.

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