Two -Well Tracer Test in 

Fractured Crystalline 

Rock

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1544r-I

ff^ork done in cooperation with the 

U.S. Atomic Energy Commission
Two-Well Tracer Test in 

Fractured Crystalline 

Rock

By D. S. WEBSTER, J. F. PROCTOR, and I. W. MARINE

GENERAL GROUND-WATER TECHNIQUES 

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1544-1

tf^ork done in cooperation with the 

U.S. Atomic Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 - Price 25 cents (paper cover)
CONTENTS

Page

Symbols________________________________________________ iv

Abstract_______________________________________________________ II

Introduction____-___-_______________________________ 1

Tracer test_________________________________________ 2

Transit times between wells for nondispersed flow___________________ 5

Tritium concentration at discharge well for nondispersed flow___________ 11

Model for dispersed flow__________________________________________ 13

Procedure for calculating the tritium concentration at the discharge well.. 16

Conclusions_____________________________________________________ 20

References__ ---________--___--___--_--__-___-_____-_____________ 22

ILLUSTRATIONS

Page 

FIGURE 1. Geologic section showing the wells and fracture zone used in

the two-well tracer test.__________________________________ 13

2. Photograph of tritium injection equipment at well DRB 5____ 4

3. Photograph of tritium sampling equipment at well DRB 6____ 5

4. Graph showing flow rates and water levels in the pumping and

injection wells________________________________________ 6

5. Graph showing tritium concentration as observed at discharge

well and as predicted by three calculated curves.___-__--_- 7

6. Diagram showing streamlines between wells in a dischargerecharge system in a homogeneous and isotropic medium. _ _ _ 8

7. Construction diagram for determining the relation between 

selected flow paths and the shortest flow path in a dischargerecharge system in a homogeneous and isotropic medium. _ _ _ 9 

8-10. Graphs showing:

8. Relative transit times along streamlines plotted on

log-log scales._________________________________ 12

9. Curves of the ratio of exit concentration to the initial 

concentration versus the ratio of selected times to 

the transit time for nondispersed flow for step 

addition of tracer___----_----__--__------------ 15

10. Curves of the ratio of exit concentration to the reference concentration versus the ratio of selected times 

to the transit time for nondispersed flow for pulse 

addition of tracer______________ ________________ 16

TABLES

Page 

TABLE 1. Example of calculations for computing tritium concentration

in discharge well from pulse injection of tracer.____________ 118

2. Example of calculations for computing tritium concentration

in discharge well from step injection of tracer.______-._--__ 19

in
SYMBOLS

C____-_--..-. Concentration of tracer from the discharge well

C*__________ Concentration of tracer during pulse injection

C0 _ _______ Concentration of tracer in step addition

CT ----------- Reference concentration for pulse addition of tracer

D___________ Axial dispersion coefficient; units are length squared divided by

time 

I- ___________ Characteristic dispersion length, units are length (also termed

"medium dispersion constant" by Orcutt and others, 1957,

p. 793; and "mixing length" by Hennico and others, 1963, p. 10) 

DRB_ _______ Deep rock boring, a cored well in crystalline rock

e ------------ Porosity

gpm_________ Gallons per minute

gpd/per sq ft__ Gallons per day per square foot, a unit of permeability 

h-----_------ Head difference between the discharge and recharge wells

K----------- Coefficient of permeability

L___________ Straight line distance between the injection and discharge wells,

or length of column in an experiment 

m- __________ Thickness of aquifer

pC/I- ________ Microcuries per liter

N- __________ Number of characteristic dispersion lengths in a column length or

flow path 

NO ---------- Number of dispersion lengths along the direct path between the

two wells 

P______-____ Peclet number, a dimensionless number that characterizes

dispersion 

^____________ Angle at which streamline leaves injection well (or enters discharge

well) measured from line connecting the two wells 

g;_ __________ Pumping-injection rate while tracer was being injected

go----------- Average pumping-injection rate during entire test

r- ___________ Radius of the arc that defines a streamline between the injection

and discharge wells 

r«,___________ Radius of the pumping well

<S_______---_ Length of flow path

*__________ Time

ta ___________ Transit time for nondispersed flow

tao---------- Transit time for nondispersed flow along the direct path between

the injection and the discharge wells. 

tai----------- Transit time for a specific streamline defined by angle fa

f*___________ Duration of pulse injection of tracer

9- __-__.-____ Dimensionless time, the ratio of a selected time to the time

required to fill completely a column with a displacing fluid 

U---------.- Average interstitial velocity

F______---__ Volume

F/___________ Volume of fluid

IV
GENERAL GROUND-WATER TECHNIQUES

TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE

ROCK

By D. S. WEBSTER,1 J. F. PROCTOR,* and I. W. MARINE

ABSTRACT

A pulse injection of tritium (300 curies) was made to flow from an injection 

well to a discharge well through fractures in crystalline rock buried beneath 

about 1,000 feet of coastal plain sediments at the Savannah River Plant near 

Aiken, S.C. The wells were 1,765 feet apart, and the duration of the test was 

2 years. The concentration of tritium arriving at the discharge well can be 

duplicated by calculations based on fluid dispersion in a homogeneous medium. 

In developing the theoretical curve, the only unknown variables were (1) the 

transit time for nondispersed flow along the line connecting the two wells and 

(2) the characteristic dispersion length in the medium (fractured crystalline 

rock). These two values were obtained by curve fitting. The results of the tracer 

test agree with computations based on the assumption that flow was through a 

homogeneous medium consisting of numerous intersecting fractures.

INTRODUCTION

At the Savannah River Plant near Aiken, S.C., as at other locations where there are chemical separation plants for the processing 

of nuclear fuels, practically all the high-level radioactive wastes are 

stored in concrete and steel tanks buried just beneath the surface of 

the ground. This waste is of such activity and longevity that it cannot be dispersed into the environment, but it must be contained for 

periods of time extending at least into hundreds of years. One concept 

for the terminal containment of this waste is to store it in chambers 

excavated within crystalline basement rock, which is covered by 

about 1,000 feet of coastal plain sediments at the plant site.

As part of the safety evaluation of this concept, the hydrology of 

the buried crystalline rock has been intensively studied and was 

reported by Christl (1964), Siple (1964), Proctor and Marine (1965), 

and Marine (1966, 1967a, b). The present paper reports on the analysis of a 2-year tracer test in which a pulse injection of tritium was 

made to move from an injection well to a pumping well.

1 E. I. Du Pont de Nemours & Co. (retired); present address, Argonne National Laboratory, Argonne 

HI.

2 Savannah River Laboratory, E. I. Du Pont de Nemours & Co., Aiken, S.C. 

s U.S. Geological Survey, Aiken, S.C.

II
12 GENERAL GROUND-WATER TECHNIQUES

The buried crystalline basement rock at the Savannah River 

Plant is predominantly chlorite-hornblende schist and hornblende 

gneiss with lesser amounts of quartzite. Water occurs under confined conditions in the small fractures that exist within the crystalline 

rock. The nature of the fractures in the buried crystalline rock was 

discussed by Marine (1966); from water injection and removal tests 

on packed-off sections of the rock, he concluded that there were two 

types of fractures. The first type consists of minute fractures that 

pervade the entire rock mass but transmit water extremely slowly. 

Rock that consists only of this type of fracture is termed "virtually 

impermeable" in this report. The other type of fracture is restricted 

to definite zones, and consists of larger openings that transmit water 

at a faster rate. Rock that includes this type of fracture is termed 

"hydraulically transmissive" in this report. Some zones of hydraulically transmissive rock are tapped by two or more wells.

The permeability of both types of rock was discussed by Marine 

(1967b). The apparent permeability of the hydraulically transmissive 

zones was estimated from pumping tests to average about 1 gallon 

per day per square foot, even though some fractured sections within 

these zones locally exceed this value.

The computation of hydraulic constants from pumping tests was 

necessarily based on the assumption that fractures were so uniformly 

distributed and interlaced that the idealized conditions of homogeneity and isotropy were reasonably well satisfied. However, it was 

realized that, if the hydraulically transmissive fracture zones did 

not consist of numerous interlacing fractures, the estimated hydraulic 

constants could depart significantly from values that would be 

applicable to actual water movement. A tracer test using tritium 

was designed to provide information on the nature of the flow through 

fractures in a known fracture zone.

TRACER TEST

The previously inferred "inclined fracture zone" (Marine, 1966 

p. D225) which is tapped by wells DRB 5 and DRB 6 was used for 

the test (fig. 1). A submersible pump was set in well DRB 6, the 

discharge well, at which power was available from an electric line. 

A 1-inch diameter plastic pipe was laid across the land surface to 

conduct the discharge water to well DRB 5, the injection well, where 

it was injected by a drop pipe that extended below the normal static 

water level. A semipermanent packer was set at a depth of 1,500 feet 

(1,212 ft below sea level) in DRB 5, so that all the water, as well as 

all the tritium tracer, would be injected into fractures that were 

known to be connected to the fractures near the bottom of DRB 6.
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 13

Injection we 

DRB5

rtP SEA 

LEVEL

FIGURE 1. Geologic section showing the wells and fracture zone used in the two-well tracer test. Arrows 

show the generalized direction of water now.

On August 12, 1964, the discharge-injection system was started and 

pumped at about 10 gallons per minute for 6 days. It was assumed 

that hydraulic equilibrium was approximated, and a condition of near 

stabilization was achieved for the cones of depression and elevation 

in the area between the two wells.

On August 18-19, 1964, 297.44 curies of tritium was added to the 

system in 28% hours at a uniform rate, giving a tritium concentration 

of 4,750 microcuries per liter in the injected water. Tritium from a 

plastic jug was added through a plastic tube by a metering injection 

pump driven by electricity supplied from a portable generator, as 

shown in figure 2. Also shown, are a standby generator, the packer 

pipe extending about 2% feet above the well casing, and the injection 

system itself with flow meter and pressure gage.

At DRB 6, the discharge well, there were flow control valves, seven 

sampling taps, a pressure gage, and a flow meter (fig. 3). The sampling 

taps were equipped with solenoid valves connected to electric timers. 

A sample of about 10 gallons was collected daily, and every week a 

part of each sample was transferred to a 1-liter polyethylene bottle 

for storage and analysis.

The pumping rate for the system and hydrographs for the discharge 

well (DRB 6), the injection zone (843-1,212 ft below sea level), and 

the zone below the semipermanent packer in DRB 5 are shown in 

figure 4. Ordinarily, the discharge and injection rates were the same;
14 GENERAL GROUND-WATER TECHNIQUES

however, if a leak occurred in the plastic pipe, the discharge rate 

exceeded the injection rate, and water was removed from the system. 

As shown in figure 4, when water was removed from the system, the 

water level declined in all three zones.

Initially, daily samples of the discharge water from DRB 6 were 

analysed for tritium concentration by the liquid scintillation method. 

Tritium was first detected 73 days after it was added to the injection 

well. From the 112th day to the end of the test on the 739th day,

FIGURE 2. Tritium injection equipment at well DRB 5.
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 15

FIGURE 3. Tritium sampling equipment at well DRB 6.

every fifth sample was analysed and corrected for radioactive decay. 

Figure 5 shows the tritium concentration at the discharge well plotted 

against time for the entire test.

From the overall average pumping rate at DRB 6 of 7.9 gpm or 

43,000 liters per day, it was computed that the tritium recovered 

after adjustment for decay was about 108 curies; however, this includes some recirculated tritium (estimated to be 9.5 curies by methods 

presented below). With this estimate taken into account, about 33 

percent of the original tritium injection was recovered.

TRANSIT TIMES BETWEEN WELLS FOR NONDISPERSED FLOW

The flow paths from an injection to a discharge well, both of which 

have equal flow rates and fully penetrate a medium of great extent 

and of uniform thickness and flow properties, are arcs of circles that 

pass through the two wells and have their centers on the line perpendicular to, and bisecting, the line connecting the wells as shown in 

figure 6 (Muskat, 1937, p. 474; Jacob, 1950, p. 345; DeWiest, 1965, 

p. 249). Potential-flow theory permits calculation of the transit time 

for a packet of fluid moving from the injection to the discharge well 

along any streamline, in terms of the total flow, the porosity and 

thickness of the medium, and the distance between wells. Since the 

medium is assumed to be homogeneous and isotropic, the potential

367-850 70 2
16 GENERAL GROUND-WATER TECHNIQUES

SI

la 

Sm33 pj

s2

Ills 1^41 § CD CD CD 

Q:Q:Q:

QQQ

133J Nl '13A31 V3S 3A08V 30VJMPS M31VM JO NOI1VA313

SNomvs NI 

'aiva Nonoar

NI39«VHOSia
JULY AUG SEPT OCT NOV DEC ' JAN FEB MAR APR MAY JUNE JULY AUG SEPT OCT NDV DE

Curve calculated for dispersed flow 

from pulse and step injection

400 

TIME, IN DAYS

Figure 5. Tritium concentration as observed at the discharge well and as pre
18 GENERAL GROUND-WATER TECHNIQUES

Discharge well

Injection well

£=distance between wells 

S = length of flow path (arc) 

>s =time of transit

Flow crescent Range of ^, 

in radians I (avg)

1-- - ._- . .-.______

2. ............... .

3... ..........................

4....................... .

5................................

6... ....... ...... ..........

 . ......_.-- _. 0.0-0.3

- . .-,....._ .. .3-. 6

-. .......-.. - .6-. 9

--.--...-.. . . ..... .9-1.2

-.-- .--..___-.- 1.2-1.5

-.-.....-......... 1.5-1.8

1.005

1.034

1.10

1.21

1.39

1.64

1.010

1.087

1.265

1.60

2.20

3.45

FIGURE 6. Streamlines between wells in a discharge-recharge system in a homogeneous and isotropic medium and the division of the flow field into crescents used to calculate the concentration of tracer arriving 

at the discharge well.

gradient at a given radius around, and near, a well is equal in all 

directions, and consequently, the outward (or inward) flow velocities 

at that radius will be equal. With this statement as a postulate, an 

analytical expression for time of transit along any streamline can be 

derived. The procedure involves obtaining an expression for the pore 

volume of a very narrow section cut through the medium from well 

to well along a streamline and dividing this volume by an expression 

for the rate of fluid inflow to the section. The result is the time required to fill the section with fluid, or alternatively, the time of transit 

along the streamline for nondispersed flow.
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 19

The construction for this derivation is given in figure 7. A streamline is identified by the angle, $, at which it leaves the injection well 

or enters the discharge well, the angle being measured relative to the 

straight line connecting the wells (ordinate). Since the inner streamline 

shown in figure 7 is the between-wells arc of the circle of radius r, a

Ordinate

L ^distance between wells

^ = angle a streamline makes with 

line connecting the two wells

i- = radius of the arc of a streamline 

between the two wells

FIGURE 7. Construction diagram for determining the relation between selected flow paths and the shortest 

flow path in a discharge-recharge system in a homogeneous and isotropic medium.
110 GENERAL GROUND-WATER TECHNIQUES

tangent to the circle (dashed line) passes through the well at angle \j/ 

and forms a triangle with the axes; this triangle is similar to that 

formed by the radius and the axes. Consequently, the radius is at an 

angle \j/ with the abscissa, and the included angle of the between-wells 

arc is 2^.

The area of the circle-segment enclosed by the arc and the ordinate 

is a function of the radius and included angle:

ABeK=~ 1s (2^ sin 2^). (1) 2,

By inspection of figure 7

L/2 fcy \

T=~^ ' ( «) sin^ 

so that equation 1 becomes

A circle-segment on the left side of the ordinate is a mirror image of 

that shown in figure 7 and is also specified by \f/. Consequently, the 

total area enclosed by the paired arcs specified by \j/ is twice the value 

given in equation 3. Therefore, the pore volume of the solid bounded 

by the paired arcs in a medium of uniform thickness m and porosity 

e is

The pore volume of the two slices of infinitesimal width along the 

paired streamlines through the medium is, then

d r2^- "| 

J

The rate at which fluid enters this volume is obtained from the 

postulate that flow leaves the injection well equally at all angles. In 

the construction used, TT radians (180°) encompasses all possible angles 

of exit, hence represents the total flow ; therefore, the fraction of flow 

which leaves through the sector between zero and \j/ radians on both

sides of the ordinate is *-'

IT

For a pumping rate of q0 and a pumping period of t, the volume of 

fluid Vf) injected into the sector is q0 1-. Consequently, the volume of 

injection into the two infinitesimal slices defined by equation 5 is

dV^S&fy. (6)
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 111

If at time ta the pore volume to be filled (dVMKB) and the fluid volume 

injected (dVf) are equal, then ta is the time of transit along the streamline defined by \^. Equate the expressions on the right side of equations 

5 and 6, and solve for ta :

. ,_. £a=-r em -TT . , . (7) 4 ffo aiAL sin2 ^ J

The derivative of the bracketed expression is 4 . , . 

v L sm2 tA J

Therefore, equation 7 becomes

ri-

 

L s

, ,

(8) 

J

This expression becomes indeterminate for ^=0, the straight line 

between wells that gives the least time of transit, but by 1'Hospital's 

rule the limiting value can be determined from its derivative, which is

., i - « r- Since ., , .. 5 7 *% as tA^O, then . , . *}{ as 1+2 cos2 if/ 1+2 cos2 \l/ smr y

$ >0. Therefore, when the value of *A in equation 8 is allowed to go 

to zero, the expression becomes

2o

tr.\ > (9)

the transit time for the straight path between wells. This is the same 

relationship presented by Muskat (1937, p. 475) for the shortest path. 

The transit time for the general streamline in equation 8 can be 

expressed as a ratio to the shortest transit time in equation 9:

j (10) 

Equation 10 was used to construct the log-log plot of ( -^- I ) versus

VaO /

$ in figure 8. The log-log plot magnifies the region of interest in this 

study, that is, values of taltaQ less than 2, and will be used in subsequent 

calculations.

TRITIUM CONCENTRATION AT DISCHARGE WELL FOR NONDISPERSED

FLOW

The information of the preceding section on time of transit along 

the various streamlines can be used to calculate the concentration 

of tracer in the water being removed at any time from the discharge 

well, in the absence of dispersion effects. Traced water enters all
112 GENERAL GROUND-WATER TECHNIQUES

10

0.1

0.01

0.1 1.0 v 10

^,IN RADIANS

FIGCKE 8. Relative transit times along streamlines plotted on log-log scales.

streamlines from the injection, well at time zero, and continues to 

enter for a tune t* (1.2 days=28% hr). The traced water is assumed to 

move as a plug along each of a given pair of streamlines, the front 

eventually reaching the discharge well at time ta ; because of the 

symmetry of flow and the assumed absence of dispersion, a period t*
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 113

is also required for the plug to pass into the discharge well. At the 

same time that the tail end of this plug is entering the discharge well 

(tai+t#), the front of a plug having a transit time of (£ai+£*) is just 

starting to enter. If the streamlines with these two times of transit are 

designated fa and fa, respectively, then the fraction of flow (consisting 

of the original traced water) that they include at any moment is

 -, and the original concentration of the tracer is reduced cor7T

respondingly by mixing with the main flow out of the well. Thus,

cufeiifcc; (ii)

where C is the concentration of tracer from the discharge well, and 

C* is the concentration of tracer initially put into the injection well. 

Later analysis indicated that it takes about 440 days for the transit 

time, taQ, of tracer if no dispersion occurred. So in the period between 

440 days and 440+1.2 days, the concentration leaving the discharge 

well rises from zero to a maximum. The value of ta/tao for the end of

441 2 this period is - !--= 1.00273, and from a more detailed plot similar

to figure 8, the corresponding ^ is 0.0826 radians. The resulting 

concentration is given by equation 11 as X4,750 pC/l, or

T

125 fJiC/l; this is more than 25 times greater than the highest concentration observed. Twenty days later (that is, in the period from 

460 to 461.2 days) the bounding values of ^ are 0.337 and 0.347 

radians, and the concentration had fallen to about 15 nC/l. It is 

obvious that this kind of profile bears no resemblance to the observed 

profile, as illustrated by the curve labeled "curve calculated for 

nondispersed flow" in figure 5.

MODEL FOR DISPERSED FLOW

The most likely explanation for the shape of the observed curve 

in figure 5 is dispersion of the tracer by water movement through 

the inclined fracture zone. Dispersion occurs when flow paths for 

individual particles are not identical, and some elements of tracer 

move ahead of the rest. The result is a profile of tracer concentration 

that decreases in the direction of flow. When the mixing of tracer 

with water already in a pore or fracture is rapid and when the flushing time for all openings is nearly equal, the concentraction gradient 

is steep. When the mixing is slow or the rapidity of flushing varies 

widely, however, a "finger" of tracer can move much farther before 

attaining a dilution equivalent to that in other openings, and the 

concentration gradient is more gentle.
114 GENERAL GROUND-WATER TECHNIQUES

The whole process of "fingering" and mixing is highly complex, 

and can only be approximated by models that are mathematically 

tractable. A frequently used model is one obtained by assuming that 

mechanical dispersion is analogous to molecular diffusion and can 

be described with identical equations. Complete solutions to the 

diffusion model equations were obtained by Brenner (1962), and 

numerical results were computed. The results were specifically for 

"bounded" diffusion (Brenner 1962, p. 230), a situation which occurs 

in a column with entrance and exit holes so small relative to the 

column cross section that tracer is transferred in (and out) only by 

bulk flow that is, transfer by dispersion at these points is negligible. 

This criterion is met by two wells in a discharge-recharge system, 

such as the one described for this test.

Other workers have also presented solutions for the general diffusion model equation (for example: Ogata and Banks, 1961; Shamir 

and Harleman, 1967); however, the work of Brenner (1962) was 

used here because the solution was for "bounded" diffusion (that is, 

a finite column length) and because of the tabular presentation of 

the results.

Brenner computed results for the "step" addition of a second fluid 

to a packed column. In step addition, flow of the second fluid is 

started abruptly and continued until displacement of the first-traced 

fluid from the column is acceptably complete. Miscibility and similar 

viscosities of the fluids are assumed. Brenner tabulated C/C0 , the 

ratio of tracer concentration at the column exit to tracer concentration 

in the original liquid in the column, for various Peclet numbers and at 

various times for each. Brenner (1962, p. 230) used the parameter 

UL/4D which he referred to as the "Peclet number," where U is the 

average interstitial velocity, L is the length of bed, and D is the axial 

dispersion coefficient. The extraneous factor of 4 was introduced in 

the denominator to simplify some of his equations. For the present 

purposes it is advantageous to remove the extraneous factor of 4 ; thus, 

JV=4P where N is the dimensionless column length or the number of 

dispersion lengths in the column (Hennico and others, 1963, p. 10) and 

is the Peclet number as usually defined. The axial dispersion coefficient, 

D, is equal to the product of the characteristic dispersion length in 

the medium, I, and the interstitial velocity, U (Orcutt and others, 

1957, p. 793). Thus,

The units of I are those of length, and N is dimensionless.

Time is used dimensionlessly by Brenner as 0, the ratio of the 

volume of second fluid that has entered the column by a given time, t,
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 115

to the pore volume of the column. For a constant flow rate, 6 is the 

ratio of the given time to the time required to fill completely the 

column. C/CQ and 6 are almost always measured quantities in an 

experiment; and N is the sought quantity. Because of the complexity 

of the relationship, N cannot be solved for explicitly, but is obtained 

by curve fitting.

Brenner tabulated values of the ratio of tracer concentration in the 

effluent to tracer concentration in the first fluid. The primary interest 

in the test reported in this paper is in the complementary situation 

one in which the tracer is in the displacing liquid so that C/C0 is the 

ratio of effluent concentration to concentration in the displacing 

liquid. Such values are obtained simply by subtracting Brenner's 

values from 1.0, and typical curves of C/C0 versus 0 for fixed values of 

N are plotted in figure 9. It is apparent that as displacement continues, the effluent concentration approaches the concentration in the 

displacing liquid that is, most of the first liquid is displaced. Plots 

similar to those of figure 9 were used to calculate the amount of 

tracer contributed to the discharge well after several hundred days 

by the recycling of traced water through the fracture zone.

i.o

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

FIGURE 9. Curves of the ratio of exit concentration to the initial concentration of tracer (C/Co) versus the 

ratio (0) of selected times (t) to the transit time for nondispersed flow (« ) for selected numbers of dispersion 

lengths (N) plotted for step addition of tracer from data given by Brenner (1962).

The initial passage of traced water through the formation in this 

experiment is of a different nature than that described. Addition of 

the tracer fluid was not continued indefinitely, once started; but 

consisted of a pulse of high concentration (4,750 nCfi) injected for 

only 1.2 days as the front of the "second fluid." As displacement progresses, the pulse is spread by dispersion.
116 GENERAL GROUND-WATER TECHNIQUES

To allow calculation of effluent concentrations for this case, a 

reference concentration is needed, similar to the concentration (70 for 

step addition of tracer. The reference concentration, CT , is that which 

would exist if the pulse of tracer were distributed uniformly throughout 

the column. As explained by Levenspiel (1962, p. 249), C/Cr (for pulse 

addition) equals the derivative of C/C0 (for step addition) with respect 

to 6. Accordingly, values of C/CT for a given number of dispersion 

lengths, N, were calculated from pairs of entries in Brenner's table 

by dividing A((7/(7o) by A0. Tables giving values for the differentiated 

equations (Brenner 1962, eq 20, 33) were presented by G. A. Latinen 

and F. D. Stockton in 1959, but were never published. However, use 

of these values gives the same results as the use of finite differences 

from Brenner's tables. Typical results are shown in figure 10 on 

rectangular coordinates. Similar plots were used for the calculations 

described in the following section.

PROCEDURE FOR CALCULATING THE TRITIUM CONCENTRATION AT 

THE DISCHARGE WELL

The flow field was divided into crescents 0.3 radians "wide," as 

illustrated in figure 6. The crescents were considered as parallelconnected columns of different lengths, but with cross sections chosen

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

FIGURE 10. Curves of the ratio of exit concentration of tracer to the reference concentration (C/Cr) to the 

ratio (0) of selected times (t) to the transit time for nondispersed flow (<«) for selected numbers of dispersion 

lengths (N) plotted for a pulse addition of tracer from values obtained from those given by Brenner (1962).
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 117

(that is, equal A^'s), so that flow is partioned equally among the 

columns. Only the 6 crescents that were short enough to become 

involved during the 2-year test, according to this analysis, are numbered. The time of transit for nondispersed flow, ta , between wells is 

listed in figure 6 for each crescent, expressed as a ratio to the time ta0 

for nondispersed movement directly between wells; these values of 

ta/ta0 were read from figure 8 for the central value of if/ in each crescent. 

The average path length, S, of each crescent is also listed, and was 

obtained as follows:

Inspection of figure 7 shows that the length of a streamline or arc (S) 

is simply the product of the radius, r, of the arc and the included angle

L/2 2if/. Since equation 2 states that r= . .> then sin $

_ 

IT sin*

If the time of transit, or filling time, for a crescent column is ta , 

then from equation 6 the pore volume of the crescent is

The tracer solution of concentration C* is injected as feed for a time 

t* (1.2 days) at a rate <?*. The fraction that enters each crescent is

 > therefore the quantity of tracer in each is C^q_t > the reference

7T 7T

concentration, Cr) is this quantity divided by the crescent pore volume 

from equation 13, or

Ideally, q{ would equal q0, and the ratio would be unity. However, in 

the test reported here they were not equal (fig. 4), and it was essential 

to recognize this fact in the analysis.

Equation 14 gives the proper reference concentration for calculating 

the tracer concentration in effluent from any one crescent. But the 

tracer-test effluent concentrations were measured only after mixing 

together in the pipe of the discharge well the effluents from all crescents. Therefore, the concentration, C, from one crescent must be

reduced (diluted) by It is convenient to include this multiplier

T

with the reference concentration. With 0^=4750 pCfl, ^ = 1.2 days,
118 GENERAL GROUND-WATER TECHNIQUES

g,=9.63 gpm, g0=7.9 gpm, and A^=0.3 radians-,

(15)

A likely value for ta0 must be estimated from the plot of observed 

effluent concentrations versus time (fig. 5). For example, 400 days can 

be taken for this time of transit for nondispersed flow along the direct 

line between wells and adjusted in the direction indicated by curve 

fitting. Also, a value should be selected for N0t the number of dispersion lengths along the direct line. To illustrate the method of calculation, N0 will be taken as 8. For a given crescent, then,

ta==tao (-f-)' withl -2-) from figure 8,

>/

663

N=N0 IT)' with ( T ) from equation 12. \L/ \L/

For a specific time, t days,

(C/Cr) is obtained as a function of N and 6 from a plot like figure 10,

For convenience, the calculations can be arranged as in table 1.

TABLE 1. -Example of calculations for computing tritium concentration in discharge wettfrom puke injection of

tracer

<«o=400 days, 2Vo=8 t= 100 days <=200days £=350 days

Crescent No.

1.... ............

2................

3... .

4.. -- . .

5... .............

Effluent concentration. .

days

. 404

. 435

506

640

. 880

1,380

V

1.64

1.52

1.31

1.03

.75

.44

2V

s n

as

Q Q

9 7

ILl

13.1

0

0.25

.23

.20

.15

.11 .

.07 ..

(C/CV)

0.053

.030

009

.001

MCA

0.087

.046

.012

.001

.146

0 (

0.50

.46

.40

.31

.23

.15 ..

(C/CV)

0.76

.62

.38

.06

.007

c,

1.W

.94

50

06

01

2.76

0 (

0.87

.80

.69

.55

.40

.25

;c/cr)

0.99

1.06

1.07

.83

.26

.006

c, MCA

1.62

1.61

1.40

.85

.19

.00

5.67
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 119

Comparison of these points with the data in figure 5 shows that 

agreement is good at 100 days, but that the calculated value at 200 

days is below the observed concentration, whereas the calculated value 

at 350 days is above. Other trial values for taQ and N0 may be selected, 

and the calculated concentrations compared with the observed concentrations at various times. The best fit resulted for ^0=4.0 and Za0=440 

days; this fit is shown as the dashed line labeled "curve calculated for 

dispersed flow from a pulse injection" in figure 5.

This calculated curve drops below the observed curve after about 

400 days. The difference is the result of ignoring tracer that has been 

through the system once and is appearing a second time because of 

recycling of water from the discharge well back to the injection well. 

This second appearance follows the mathematics of step addition of 

tracer, figure 9. For ease of explanation, this step addition will be 

considered to consist of the introduction of a steady flow of 4.5 pC/l 

water into the injection well at 200 days. (Because of the gradual, 

instead of abrupt, introduction of recycled tracer into the injection 

well, the calculations were actually carried out as three step additions: 

1.75 nC/l at 150 days, 1.75 at 200 days, and 1.00 at 300 days. These 

three steps add together to give a total effect that is practically identical with that calculated for a single step addition of 4.5 nC/l at 200 

days.)

The concentration in the step addition will be diluted to Co=4.5

M/

 =0.43 nC/1 for purposes of calculation, as was done with the

IT

reference concentration in pulse addition. Values for ta , N, and 0 are 

computed as before, but with ia0=440 days and JV0=4.0; (C/C0) 

comes from plots similar to those in figure 9. An example of these 

calculations is given in table 2.

This concentration of 0.72 pC/l from recycling tritium is added to 

the previously calculated concentration from the pulse to give the

TABLE 2. Example of calculations for computing tritium concentration in discharge 

well from step injection of tracer

1

?,

3

4

5

6

i_o=440 days, -Vo=4.0

Crescent No. « , days **. Co.jiC/l

.___..____.__.____.__ 444 0 43

-_-_._..___ __ _______ 478 43

_____________________ 556 43

-_-__-._________.____ 704 43

. -__ ___ ___ _ ___ 968 43

_____________________ 1,520 43

N e

4. 0 0 90

4.2 84

4.4 72

4.9 57

5.6 41

6.6 26

t=400 days

(C/Co) M^/l

0. 54 0 23

.49 21

.38 16

. 22 09

.07 03

Concentration_________________________________________________ 0. 72
120 GENERAL GROUND-WATER TECHNIQUES

total concentration at 400+200=600 days since start of the pulse 

test. That is, the 200 days of delay before starting the step addition 

must be added to the 400 days of flow to put the two tracer additions 

on the same time scale.

The solid line labeled "curve calculated for dispersed flow from 

pulse and step injection" in figure 5 shows the results of combining 

the two types of tracer addition. Agreement with the data is surprisingly good. The relatively high calculated concentrations at the 

start of the breakthrough curve, as well as other deviations, may be 

a result of an inexact model. For example, a portion of the first traced 

water passing through the system may be held in relatively dead 

pores, and reappears only slowly in the effiuent at much later times. 

Lateral dispersion, which has been neglected in this analysis, may 

account for the early observed concentrations of tritium being lower 

than the calculated values. Tritium may be lost laterally from the 

earliest arriving central flow path, an action that is apparent only 

in the early stages before transfer back from adjacent crescents 

occurs. Finally, crescents are probably not identical, as assumed in 

the model.

CONCLUSIONS

The purpose of the two-well tracer test using tritium was to determine hydraulic constants from the physical movement of fluid 

through fractures with which the hydraulic constants developed from 

induced head changes within the crystalline rock could be compared. Figure 5 shows the concentration of tritium arriving at the 

discharge well. This observed curve can be closely duplicated from 

dispersion theory and the assumption that the fracture zone behaves 

as a homogenous and isotropic medium with respect to fluid transmission and dispersion.

In fitting the theoretical tritium arrival curve to the observed 

curve, only two unknown variables occur. The first of these variables 

is the transit time for nondispersed flow along the direct path between the wells, tao- Variations as small as 3 percent in the value of 

tao cause a noticeably poorer fit for the theoretical curve.

The second unknown variable, the characteristic dispersion length 

(the number of dispersion lengths between the two wells, N, was 

determined), is a convenient mathematical quantity with which to 

describe dispersion, but it is difficult to assign physical significance to 

this quantity. Mathematically, it is equivalent to the mean free path 

in the diffusion equations. The analysis showed that there were 4.0 

dispersion lengths along the line between the two wells, 1,765 feet, 

making a dispersion length about 440 feet. Variations of 10 percent 

in NO cause a noticeably poorer fit for the theoretical curve.
TWO-WELL TRACER TEST IN FRACTURED CRYSTALLINE ROCK 121

The results of the tracer test agree with computations based on the 

assumption that flow was through a homogeneous medium consisting 

of numerous intersecting fractures. Any particular well in this area, 

however, is apt to penetrate individual fractures within the fracture 

zone that show large differences in permeability as concluded by 

Marine (1967b, p. B211). However, the larger, more permeable fractures that may be penetrated are probably connected with each other 

by less permeable fractures as shown schematically in figure 1. The 

tracer test suggests that large planar fractures do not form direct 

hydraulic connections between the wells used in the test and that the 

fractures in the "inclined fracture zone" in this area form an interlacing network.

From the hydraulic data obtained during the tracer test (that is, 

the calculation is not based on the physical movement of tracer but 

on the head relationships), the average permeability of the aquifer 

may be calculated.

In (DeWeist, 1965 p. 249),

where

!£= coefficient of permeability, 

q0= discharge-recharge rate (7.9 gpm), 

A^= difference in head between the discharge and recharge wells

(120 ft),

m thickness of the aquifer (250 ft), 

i=distance between the two wells (1,765 ft), and 

rw =radius of the pumping well (0.23 ft).

The average coefficient of permeability so calculated is 1.1 gpd per 

sq ft. From pumping tests Marine (1967b, p. B210) estimated the 

average coefficient of permeability of fracture zones in the area to be 

0.77 gpd per sq f t («1 gpd per sq ft).

From the values obtained from the tracer test (tfo0 =440 days), a 

value for the porosity due to fractures may be calculated from a

rearrangement of equation 9: e= t°2 a°-

The value calculated is 0.08 percent.

The average velocity for nondispersed flow along the direct line 

between the wells was 1,765 feet-^440 days=4.0 feet per day.

If it is assumed that Darcy's law is valid for fluid movement in this 

medium, an average value for the coefficient of permeability can be 

calculated from the velocity (4.0 ft per day), the porosity (0.08 percent), and the average gradient between the wells (360 ft per mile). 

This value is 0.35 gpd per sq ft and is derived entirely from the 

analysis of the physical movement of fluid through the medium.
122 GENERAL GROUND-WATER TECHNIQUES

This value may be compared to the value of permeability of about 

1 gpd per sq ft estimated from head changes in the medium resulting 

from the removal and injection of water.

REFERENCES

Brenner, Howard, 1962, The diffusion model of longitudinal mixing in beds of

finite length. Numerical values: Chem. Eng. Sci., v. 17, p. 229-243. 

Christl, R. J., 1964, Storage of radioactive wastes in basement rock beneath the

Savannah River Plant: E. I. du Pont de Nemours & Co., Savannah River

Lab., DP-844, 106 p.

DeWiest, R. J. M., 1965, Geohydrology: New York, John Wiley & Sons, 366 p. 

Hennico, A., Jacques, G., and Vermeulen, T., 1963, Longitudinal dispersion in

packed extraction columns: Univ. California, Lawrence Radiation Lab.,

UCRL-10696, 190 p. 

Jacob, C. E., 1950, Flow of ground water, chap. 5 in Rouse, Hunter, Engineering

hydraulics: New York, John Wiley & Sons, p. 321-386. 

Levenspiel, Octave, 1962, Chemical reaction engineering, an introduction to the

design of chemical reactors: New York, John Wiley & Sons, 501 p. 

Marine, I. W., 1966, Hydraulic correlation of fracture zones in buried crystalline

rock at the Savannah River Plant near Aiken, South Carolina, in Geological

Survey Research 1966: U.S. Geol. Survey Prof. Paper 550-D, p. D223-

D227, 5 figs. 

 1967a, The use of a tracer test to verify an estimate of ground-water

velocity in fractured crystalline rock at the Savannah River Plant near

Aiken, South Carolina, in Stout, G. E., Isotope techniques in the hydrologic

cycle: Am. Geophys. Union, Geophys. Mon. 11, p. 171-179.

1967b, The permeability of fractured crystalline rock at the Savannah

River Plant near Aiken, South Carolina, in Geological Survey Research 1967: 

U.S. Geol. Survey Prof. Paper 575-B, p. B203-211.

Muskat, Morris, 1937, The flow of homogeneous fluids through porous media: 

New York, McGraw-Hill, 763 p.

Ogata, Akio, and Banks, R. B., 1961, A solution of the differential equation of 

longitudinal dispersion in porous media: U.S. Geol. Survey Prof. Paper 

411-A, p. A1-A7.

Orcutt, R. G., Rifai, N. M.E., Klein, Gerhard, and Kaufman, W. J., 1957, Underground movement of radioative wastes: Sewage and Indus. Waste, v. 29, p. 

791-804.

Proctor, J. F., and I. W. Marine, 1965, Geologic, hydrologic, and safety considerations in the storage of radioactive wastes in a vault excavated in 

crystalline rock: Nuclear Sci. and Eng., v. 22, p. 350-365.

Shamir, U. Y. and Harleman, D. R. F., 1967, Numerical solutions for dispersion 

in porous mediums: Water Resources Research, v. 3, no. 2, p. 557-581.

Siple, G. E., 1964, Geohydrology of storage of radioactive waste in crystalline 

rocks at the AEC Savannah River Plant, South Carolina, in Geological 

Survey Research 1964: U.S. Geol. Survey Prof. Paper 501-C, p. C180-C184.

U. S. GOVERNMENT PRINTING OFFICE : 1970 O - 367-850
General

Ground-Water

Techniques

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1544

This volume was published 

as separate chapters A I
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

Library of Congress catalog-card No. 77-604794
CONTENTS

[Letters designate the separately published chapters]

(A) Filter-press method of extracting water samples for chloride analysis, by 

Norbert J. Lusczynski.

(B) An application of thermometry to the study of ground water, by Robert 

Schneider.

(C) Methods of flow measurement in well bores, by Eugene P. Patten, Jr., and 

Gordon D. Bennett.

(D) Application of electrical and radioactive well logging to ground-water

hydrology, by Eugene P. Patten, Jr., and Gordon D. Bennett. 

(E) A thermal flowmeter for measuring velocity of flow in a well, by Howard T.

Chapman and Albert E. Robinson.

(F) A field method for measurement of infiltration, by A. I. Johnson. 

(G) Calculation of resistance and error in an electric analog of steady flow through

nonhomogeneous aquifers, by Robert W. Stallman. 

(H) Methods of collecting and interpreting ground-water data, compiled by

Ray Bentall. 

(I) Two-well tracer test in fractured crystalline rock, by D. S. Webster, J. F.

Proctor, and I. W. Marine.