OQ

I"S

&

fei

a

H-H

5: 

1

S

g

§8

S

CO

Temperature Rise Within 

Radioactive Liquid Wastes 

Injected Into Deep 

Formations

GEOLOGICAL SURVEY PROFESSIONAL PAPER 386-A

Prepared in cooperation with the 

U.S. Atomic Energy Commission
Temperature Rise Within 

Radioactive Liquid Wastes 

Injected Into Deep 

Formations

By HERBERT E. SKIBITZKE

CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE 

DISPOSAL

GEOLOGICAL SURVEY PROFESSIONAL PAPER 386-A

Prepared in cooperation with the 

U.S. Atomic Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

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

Page 

Abstract_________________________________.____!__ A-1

Introduction. ________________________________ 1

Influence of flow paths and adsorption in temperature 

rise_________________________________________ 1

Flow paths of the liquid elements_ _____-_--_____ 1

Influence of flow paths, etc. Continued

Concentration of thermal energy by adsorption

Other parameters controlling temperature rise 

References. ______________________________

Page

A-3 

4 

8

ILLUSTRATIONS

FIGURE 1 Diagram showing flow lines of liquid waste 

being introduced into a homogeneous and 

isotropic aquifer through an injection well__

2 Diagram showing effects of adsorption on flow 

of wastes away from well_____________ 

3 Approximate temperature rise in 3,650 days 

per gram of fission products per gallon of 

waste.._________________________________

Page

A-2

Page

FIGURE 4 Approximate amount of fission products, in

grams per gallon of waste, producing a 

temperature rise of 100° F, for different 

cooling periods ___ _____________ ___ _ _ A-6

5 Graphs illustrating variations of erf ;

with thickness, thermal diffusivity of aquifer, 

and cooling period____________________

TABLES

TABLE 1. Thermal properties of some common substances.

Page 

A-7

CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

TEMPERATURE RISE WITHIN RADIOACTIVE LIQUID WASTES INJECTED INTO DEEP

FORMATIONS

By HERBERT E. SKIBITZKE

ABSTRACT

The data and method of analysis presented here provide for 

an approximate determination of expected temperature rises. 

They show that, in general, when large masses of liquid waste 

are injected into an aquifer through wells, heat conduction is 

not very significant, and as an approximation the temperature 

rise within the aquifer, since injection began, is equal to the

total heat generated in a unit volume of aquifer divided by the 

product of the specific gravity of the aquifer and contained 

waste and its heat capacity. The generalizations outlined here 

should lead to an understanding of the relative importance of 

heat storage as compared to thermal conduction away from the 

region.

INTRODUCTION

The operation of a nuclear reactor results in the 

fission of the nucleus of heavy atoms into atomic particles of smaller mass. The resulting particles, or 

fission products, inhibit the fission process; consequently, they are periodically removed from the reactor. These particles, together with the solvent and 

other material in solution, constitute the radioactive 

wastes under discussion.

Because of radiation, the radioactive wastes are 

dangerous and must be handled accordingly. The 

fission products in the wastes start decaying from the 

time they are produced; energy is released in the form of 

beta particles (/3) and gamma rays (7), which produce 

heat. In considering the disposal of these wastes by 

injecting them through wells into the earth, the amount 

of heat they release is of considerable importance, for, 

in general, earth material is a poor thermal conductor.

The temperature rise within earth material due to 

the injection of heat-emitting radioactive liquid wastes 

is difficult to determine analytically. Because of the 

complexity of the boundary conditions involved, the 

temperature rise can be determined only by elaborate 

numerical analysis. It is possible, however, to establish a simple relationship, which is valid for certain conditions, and from this relationship to determine the 

rise in temperature. The relationship to be established 

is that thermal conduction is unimportant with relation 

to heat storage for the conditions outlined here.

INFLUENCE OP PLOW PATHS AND ADSORPTION IN 

TEMPERATURE RISE

The total rate of heat generation from a given quantity of fission products has been calculated from theory

586456 61 2

and determined by experiment. The results have been 

described by many authors, such as Coryell (1947); 

Coryell and Sugarman (1951); and Culler (1955).

The factors controlling the rate of heat transmission 

in an aquifer away from an injection well are the 

geometrical configuration formed by the heat-emitting 

waste material, the paths of heat conduction away 

from that material, and the thermal diffusivity of the 

saturated porous medium. The shape of that portion 

of the aquifer occupied by the injected waste changes 

in a complex manner with time and other factors and 

thus is difficult to define analytically. The establishment of the principles involved requires some simplifying approximations from which the main features of 

those principles may be derived. First, however, it is 

desirable to describe the dynamics of the flow of waste 

away from the site of injection. With the exception 

of liquid viscosity the flow of the waste is controlled, 

in general, by parameters independent of temperature.

FLOW PATHS OF THE LIQUID ELEMENTS

The flow lines for a liquid entering a homogeneous 

and isotropic aquifer from an injection well are shown 

in figure 1. The flow lines are the result of compounding the radial flow from the well with the linear flow 

that existed in the aquifer prior to the injection of the 

liquid waste; thus, the flow system shown in figure 1 

is linearly homogeneous. The maximum width to be 

reached by the radioactive waste is shown in figure 1 

by the dashed lines A A' and BB''. This width, which 

may serve as a scale factor, is proportional to the rate 

of injection divided by the flow per unit strip of aquifer.

A-l
A-2 CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

B

Flow lines of ground water moving in the aquifer before injection starts

Radial flow from injection well assuming that there is no movement of 

ground water in the aquifer

*" Flow lines resulting from injection of waste in a system of regional 

ground-water flow

FIGURE 1. Diagram showing flow lines of liquid waste being introduced into a homogeneous and isotropic aquifer through an injection well.

Many deep-lying aquifers have a low transmissibility, resulting in low flow rates. The width of the 

contaminated zone in such aquifers could thus be very 

large; for example, an injection rate of 10,000 gallons 

a day could give a maximum width of several thousand 

feet in an aquifer having a transmissibility of a few 

hundred gallons a day per foot. For such an injection 

rate that part of the flow system within the circle 

lettered "C" in figure 1 would be largely radial and is 

the most significant. The region "C" would be the 

region in which the contaminant would be confined by 

adsorption.

The movement of the fission products and other 

solutes in the waste does not necessarily coincide with 

that of the solvent. The rate of motion of the dissolved matter is retarded, as the solute spends a portion 

of its time in transit adsorbed on the solid surfaces of 

the porous medium after the waste is injected. The 

solvent, however, is not adsorbed; consequently, the 

solute moves at a slower rate. The adsorption process 

depends upon the chemical nature of the dissolved 

matter as well as upon the chemical nature of the surface of the porous medium. Although it is not the

purpose of this discussion to analyze in detail the effects 

or process of adsorption, it is of significance to consider 

in a general way the effects of adsorption on retardation of the motion of the waste.

With relation to adsorption within the radial-flow 

system three different situations may appear: (a) no 

adsorption, (b) adsorption uniform for all materials, 

and (c) nonuniform adsorption; these are represented 

schematically in figure 2. The most strongly adsorbed 

materials will tend to occupy most of the available absorption sites, thereby permitting the less strongly 

adsorbed ions to move on. Because some of the strongly 

adsorbed materials are fixed temporarily by adsorption, 

these materials move most slowly. This allows the 

weakly adsorbed components to move to a part of the 

aquifer where adsorption sites are available and where 

they become adsorbed. Thus, a selective banding of 

differing chemicals would be produced circumferentially about the well.

The first illustration in figure 2, case A, represents a 

condition in which no adsorption occurs and the solute 

moves outward at the same rate as the solvent. This 

is an idealized condition that would not occur in reality.
TEMPERATURE RISE WITHIN RADIOACTIVE LIQUID WASTES A-3

In case B all solutes are assumed to be adsorbed at the 

same rate and the solvent moves outward at a faster 

rate than the solutes. This second situation also is idealized and probably would not occur in reality. The 

third illustration, case C, represents a multicomponent 

system wherein one of the components is strongly adsorbed whereas the remainder of the components are 

less strongly adsorbed. With respect to the disposal of 

reactor wastes, the aluminum nitrate common to those 

wastes would be more strongly adsorbed then the fission 

products; consequently a large cylinder of adsorbed 

aluminum nitrate would occur close to the well. There 

could be little adsorption of fission products within 

this cylinder; then they would move through the cylinder into an uncontaminated part of the aquifer where 

they would be adsorbed. The solvent would move 

outward beyond the area occupied by the fission products. This third condition is more like the actual 

conditions; moreover, further complications would occur 

because of the large number of different elements present, which are adsorbed at different rates, and also 

because of anisotropy and nonhomogeneity of the porous 

medium. The saturated sediments would behave as a 

chromatographic column; hence, a series of rings 

corresponding to different fission products and other 

solutes would occur. The illustrations shown in figure 

2 depict, in simplified form, the configuration of those 

parts of the aquifer occupied by heat-generating material that must be described analytically in order to 

determine the temperature rise due to radioactivity. 

The analytic representations so derived become the 

boundary conditions for the differential equations for 

heat conduction. The solution of such differential 

equations in closed form is very difficult; however, 

certain limiting approximations illustrate the effects of 

the various factors controlling the temperature rises.

CONCENTRATION OP THERMAL ENERGY BY 

ADSORPTION PROCESSES

The amount of heat produced by fission products is 

proportional to the amount of j8 particles, 7 radiation 

and the respective energies that they release during 

their decay. The rate of heat production can be computed from the sum of the energies emitted by the 

different fission products or can be computed approximately by considering the fission products as a statistical assembly. The first method is, of course, more 

precise, especially after the short-lived products have 

decayed completely.

Experimental data furnished by various authors (see 

Coryell and Sugarman, 1951, p. 457) show that the total 

energy released by the fission products, and thus the 

total heat produced, follow an exponential law of the 

form CT1 - 0 to G't~lM depending on the cooling period. 

Where G and C' depend upon units of power desired.

An exponential law representing the energy release 

after a cooling period of a year probably would be of the

P-1.35. Watts

Grams of fission products

form where Ci is the number of Watts per Gram being 

emitted hi one day. This is the law corresponding to 

the longest cooling period recorded, 16 to 340 days, and 

probably would be valid for a period of about 6 or 7 

years. (From the experimental data of Brady and 

Turkevich, quoted by Coryell and Sugarman, 1951, 

p. 457).

The heat emission per unit volume of aquifer would 

be higher near the well for case A (fig. 2) because the 

waste hi the outer region would have a longer tune to 

decay, the magnitude of the effect being controlled by 

the tune of travel and the exponential law governing the 

heat release. If the same amount of heat-generating

Solutes Solute

Solvent

I I 

I I

Solvent

Aluminum byproducts 

and other strongly 

adsorbed solutes

Fission products and 

other less strongly 

adsorbed solutes*

Solvent

Case A

Homogeneous distribution 

of solute within the 

solvent; no adsorption

Case B . Nonhomogeneous distribution 

of solute within the solvent; 

uniform adsorption

Case C

Nonhomogeneous distribution 

of solute within the 

solvent; nonuniform 

adsorption

FIGURE 2. Diagram showing effects of adsorption on flow of wastes away from well.
A-4 CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

material per unit volume of injected liquid were involved in case B as in case A, the heat emission per 

unit volume in the central cylinder would be higher 

in case B. Case C is more complex because the fission 

products traveling from the well to the cylinder of 

fission products would have the least time to decay. 

However, the concentration of heat produced per unit 

volume in the inner cylinder would be relatively low 

because the adsorption process concentrates the material 

emitting heat in the outer cylinder and therefore the total 

heat being emitted per unit volume in the cylinder of 

fission products would be substantially higher.

OTHER PARAMETERS CONTROLLING TEMPERATURE

RISE

The factors controlling temperature rise, other than 

the rate of heat emission by the fission products, are 

the geometry of the heat-emitting zone, the thermal 

diffusivity of the aquifer, and the temperature of the 

injected water. In case A, the effects of the temperature 

of the injected water have importance only in the sense 

of setting the allowable temperature rise before the 

liquid changes to vapor. In cases B and C, the 

injected water flows through that part of the aquifer 

occupied by heat-emitting material and thereby could 

have a cooling effect.

The remaining factors of geometry and thermal diffusivity may be determined for the simple condition 

condition shown in case A, wherein the flow has ceased 

and heat emission continues. The relative importance 

of various effects may be determined for case A and from 

these data the effects for cases B and C may be evaluated.

The heat emission from a region, such as that described by case A, can be determined by Fourier's method (Carslaw and Jaeger, 1947). This method permits 

the determination of the temperature rise at only one 

point in the heat-conduction system under study. For 

the problem under consideration, the temperature rise 

at the center of the cylinder containing waste that is, 

at the injection well would be the greatest and therefore the most critical. This point is chosen as the 

center of the coordinate system in the following analysis. The temperature at this point may be determined 

by summing the effects of all infinitesimal elements 

emitting heat. The effect on the coordinate center 

owing to an infinitesimal element at r', 0', z' at time t 

for heat emitted at tune t' would be (Carslaw, p. 218)

m^ }

. v_P(r'e',z't)r'dr'de'dz'dtf -

V -'V* e

Where

/3(r', 6', zf, £')=rate of heat emission in the infinitesimal element 

A V= temperature increase at the coordinate center 

rf= radial distance from origin about the

z axis 

£'=time of emission of a given quantity

of heat 

6'= angular distance from origin about

the z axis 

z'= distance above the origin along the

z axis 

t time at which the temperature is to be

determined 

p= specific gravity of the aquifer and

liquid waste combined 

c=heat capacity of the aquifer material

and contained waste 

k= thermal diffusivity of the aquifer

The summation of all such elements would be

(r'2+2'2)

o o o o

 r »20(r',g',g', t'Y

8pc(*k(t-t')r*

Where

 dt'dz'dr'dd'

.A = half the thickness of the heat-emitting cylinder

of wastes

a=the radius of the heat-emitting cylinder of wastes 

V= temperature increase after a period of time (t tf)

The first three integrations may be made by assuming 

that the waste has spread much farther radially than the 

vertical thickness of the invaded cylinder, and that the 

geometrical distribution of heat emission is of no significance so that each element is emitting heat at the same 

rate throughout the cylinder. With these limitations 

in mind, the integration of (2) yields

(3) dt'

As further integration would be possible only by numerical methods, which would further complicate the problem, an approximation is necessary to put the integral 

into closed form. The approximation is that 

, A

is roughly equal to unity for most values of A and k
TEMPERATURE RISE WITHIN RADIOACTIVE LIQUID WASTES A-5

throughout times of interest and for the type of 

aquifers likely to be used as waste disposal sites. When 

this is valid, then

(4) 7= 

Jo 

r M

pc

-dt'

Equation (4) states simply that the temperature rise at 

the center of that part of the aquifer invaded by the 

waste is the total of all heat emitted in a unit volume 

at the center, since time zero, divided by the product 

of the heat capacity of the aquifer and contained waste 

and its specific gravity.

This conclusion could be arrived at qualitatively 

from analyzing the temperature response at the center 

of the cylinder. At time t Q the temperature rises 

in each element at the same rate. Thus, all the elements start to emit heat at the same rate, except for 

the quantity of heat being transported away by conduction. The transport of heat by conduction requires 

a thermal gradient to exist. The gradient appears at 

the boundary instantly, but at the center all the surrounding elements are rising in temperature at the 

same rate. Therefore, the thermal gradient migrates 

inward at a very slow rate because of the low thermal 

diffusivity of earth material. The effects of migration 

of the thermal gradient into the center of the contaminated mass are represented by the erf term of equation 

(3). The approximation made depends on this erf 

term being essentially unity. The validity of this approximation is considered later in this paper; at this 

point, however, equation (4) is assumed to be valid.

The maximum temperature rise at the center of 

the contaminated zone may be determined by computing the total amount of heat emitted in a unit volume of aquifer and dividing it by the product of its 

specific gravity and its heat capacity. As an example 

of this effect, it will be assumed that a mass of 1 gram 

of gross fission products is dissolved in each gallon of 

waste and that the fission products have cooled for 

1 year prior to injection into a sandstone aquifer having a porosity of 20 percent and a factor of heat capacity times specific gravity of 30. Assuming that the 

 1.35 decay law holds, the total heat emitted per 

cubic foot of aquifer in 10 years will be about 30,000 

BTU. This divided by the specific heat per unit 

volume corresponds to a temperature rise of 1,000 degrees Fahrenheit. The effect of different waste-cooling periods on the value of the temperature rise is 

shown in figure 3. The maximum temperature rise 

before vaporization occurs determines the maximum 

concentration allowable in grams of fission product

6000

5000

5 F 4000

3000

2000

1000

°F = temperature rise in degrees 

Fahrenheit

/, = time which temperature determination is being made 

in days after material was 

removed from reactor = 

3650

f= cooling period, in days

_ 5.3x10 /y-0.35 ,> -0.35\

0.35 (f° ' )

200 400 600 800

FIGURE 3. Approximate temperature rise in 3,650 days per gram of fission 

products per gallon of waste.

per cubic foot of aquifer if the wastes are to remain 

in liquid form in the aquifer. The concentrations 

allowable for a 100° F. rise of temperature for various 

cooling periods are shown in figure 4. The response 

for a different maximum temperature is directly proportional to the ratio of that temperature to 100° F.

The foregoing data are based on the approximation

^ of the function erf -= being equal to unity.

The function is dependent upon A, the half thickness 

of the aquifer; k, the thermal diffusivity of the aquifer; 

and the factors tr , the time when heat emission begins, 

and t, the time for which the temperature is to be 

determined.

The effects of these variables on the value of the erf 

are shown in the graphs of figure 5. Each of the three
A-6 CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

0.0016,

0.0014 -

0.0012

0.0010

fg 0.0008

0.0006

0.0004

0.0002

f =maximum fission products, in

grams per gallon of waste 

f= cooling period, in days

800

FIGURE 4. Approximate amount of fission products, in grams per gallon of waste, 

producing a temperature rise of 100° F, for different cooling periods.

columns of graphs corresponds to a different value of A. 

The first column is for A=5 that is, for an aquifer 10 

feet thick. The second column is for .4=50, for an 

aquifer 100 feet thick, and the last column is for A 

= 100, for an aquifer 200 feet thick. The test of the 

validity of the approximation made is the closeness with 

which the function approaches a value of unity a 

straight line along the top of the graph. The graphs 

are arranged vertically so that the shortest cooling period (365 days) is in the uppermost graphs. The next 

lower set of graphs is for a cooling period of 10 years 

and the bottom set for a period of 30 years. Each 

series of graphs has three curves, corresponding to different thermal diffusivities. The uppermost curve is 

for a thermal diffusivity of 0.02 ft2 per day, the next 

lower for a thermal diffusivity of 0. 2 ft2 per day, and 

the bottom curve for a thermal diffusivity of 0. 94 ft2 

per day. These data show that only one curve of the 

first group (aquifer thickness 10 feet) approaches the

approximation used here. This curve is for a small coefficient of thermal diffusivity (poor conductivity) and 

a short peroid of time. For other curves, representing 

higher thermal diffusivities, the thermal gradients have 

time to move through the short distance A=5 feet, and 

thus conduction in these conditions is of large significance.

The second column of graphs, for aquifers 100 feet 

thick (-4=50 feet), represents conditions greatly different from those expressed by the graphs in the first 

column. The upper graph, which represents a cooling 

time of 1 year, shows that the approximation is valid, 

being almost exact for all the three values of thermal 

diffusivity. The next lower set of curves (for a 10-year 

cooling period) shows that for all except the largst coefficient of thermal diffusivity, the approximation is 

very nearly exact; such a diffusivity, however, probably 

is to be expected in many consolidated sandstones. 

The graph shows that the approximation is, in general, 

more than 50 percent correct even for the largest thermal diffusivity. Because, at present, it is of interest 

to establish the order of magnitude of the problem, the 

curve for the highest thermal diffusivity may be considered a satisfactory approximation for as much as 10 

years after heat emission begins. The bottom graph 

shows that even for a 30-year cooling period the approximation is satisfactory except for the highest thermal diffusivity.

The third column of graphs, which is for an aquifer 

200 feet thick (-4=100), shows that the approximation 

is essentially valid for cooling periods of 1 year and 10 

years for all the thermal diffusivities considered and 

the approximation is valid also for a cooling period of 

30 years except for the highest thermal diffusivity. 

Even for that diffusivity, however, the approximation 

is more than 50 percent valid.

The graphs in figure 5 illustrate that the effect of 

conduction is felt much earlier in thin aquifers because 

of the lesser distance for the thermal gradient to migrate 

and also that the effect of conduction is felt quicker for 

larger thermal diffusivities. The thermal diffusivity 

of various earth materials is shown in table 1. The 

conclusion may be drawn that for thick sandstone 

aquifers having a thermal diffusivity of about 0.94 

(calculated from the thermal-diffusivity value of 1.0 

given for sandstone, table 1, and corrected for water 

saturation), the approximation is satisfactory with respect to a functional relationship. However the question of geometric distribution of wastes due to flow 

and adsorption conditions still remains.

It has been assumed that the liquid has moved radially for greater distances than the aquifer half-thickness 

A. The amount of water required to accomplish this 

would be as follows:
TEMPERATURE RISE WITHIN RADIOACTIVE LIQUID WASTES A-7

A cylinder of an aquifer having a thickness of 10 

feet (.4=5), a radius of 5 feet, and a porosity of 20 

percent contains about 1,200 gallons of liquid. If the 

aquifer were 100 feet thick and the radius of the cylinder 50 feet, the cylinder would contain about 1,200,000 

gallons of liquid; and if the aquifer were 200 feet thick, 

and the radius 100 feet, it would contain about 8 times 

this latter amount. If there were less than these 

amounts of liquid for any of the above aquifer thicknesses, the calculated temperature rises would be too 

large for the long periods of time.

Because of adsorption the dimensions of the contaminated body would decrease, but the amount of heat 

being emitted per unit volume would increase. These 

two effects tend to cancel one another.

The effects of decay of thermal energy released per 

unit of time would be to institute a small thermal gradient radially away from the well. The magnitude of 

this effect is dependent upon the rate of growth of the 

cylindrical volume away from the well. If large quantities were injected, so that the growth would be rapid,

f=365

TABLE 1. Thermal properties of some common substances 

[Adapted from Carslaw and Jaeger, 1947, p. 382]

Substance

Air____ .___._.________.

Sandstone __ __________

Average rock___________

Ice

Concrete (1:2:4) _____

Brick (building) ________

Soil (average) __________

Soil (sandy, dry) _ _____

Soil (sandy, moisture content 8 percent) _______

Water _ _ ________

Specific 

gravity 

(P) lb/f t«

0.08

162.

156.

143.

57.

144.

162.

156.

103.

109.

62.5

Heat 

capacity (c) 

Btu/lb oF

0.24

.21

.22

.23

.50

.23

.20

.2

. 19

.24

1.0

Thermal 

Conductivity 

(K) 

Btu 

(day ft") 

(°F/ft)

0.34

35.

23.

35.

24.

31.

13.

12.

14.

3.7

8.23

8.5

Thermal 

Diffusivity 

ft2/day i

17.5

1.0

.65

1.0

1. 1

1. 1

.39

.35

.43

. 19

.31

. 13

1 Values computed from the table values for p, c, and K.

400

1.0 

0.5 

0

r-°"J *=0.02, 0.2,0.94,

_

3 100 200 300 4C

1.0 

0.5

°(

,4 = 100 

'=365 >=0.02, 0.2, 0.94

t=

) 100 200 300 4C

^=3650 ^=3650 -_ f=3650 i r

1.0

0.5

0

(

 -

 -i ' 

) 10

nO /c=0^__-

k 0

- " """" 

00 20

^

_^^ ^*~~

?

00 30

/i -M

/i

/ 1 ^ /

00

htt

1.0

0 *i

0

XX) (

-jc-0.2^

r £=0.94

) 10

K \J.\J£.

T 

_ - ^^

00 20

^s^

^-^

00 30

7-^ '

00 4(

1.0

0.5

0

XX) (

-k-0.2 .

-r^_o.y^

3 10

A \J.\J £.

^ >

l_ "

00 20 00 30

 

00

H

-

4C100

t= 10,950 ^=10,950 >= 10,950

1.0

0.5

1.0 *=0.02

0 3000 6000 9000 112,000 0 3000 6000 9000 12,000 0 3000 6000 9000 12,000 / /

t' in days

FIGURE 5. Graphs illustrating variations of erf =- with thickness, thermal diffusivity of aauifer, and cooling period.

2 "\ k\f t}
A-8 CONTRIBUTIONS TO PROBLEMS OF RADIOACTIVE WASTE DISPOSAL

this effect would be greatly reduced and there would be 

little or no thermal gradient.

With respect to hydraulics, the dynamic state of the 

region invaded by the waste would further modify the 

preceding conditions because of the cooling effect caused 

by the injected liquid flowing through the contaminated 

region, and by the porous solid which absorbs some heat. 

When injection ceases, these modifying effects lose their 

importance.

REFERENCES

Carslaw, H. S., and Jaeger, J. C., 1947, Conduction of heat in

solids: Clarendon, England, Oxford Press. 

Coryell, C. D., and Sugarman, Nathan, 1951, Radiochemical

studies: The fission products: New York, McGraw-Hill. 

Coryell, C. D. and others, 1947, The science and engineering

of nuclear power, Book 1: Cambridge, Mass., AddisonWesley Press. 

Culler, F. L., Jr., 1955, Notes on fission product wastes from

proposed power reactors: Oak Bidge National Laboratory

ORNL Rept. no. 55-4-25.