The Spread of a Dye 

Stream in an Isotropic 

Granular Medium

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-G

Prepared in cooperation with the 

U.S. Atomic Energy Commission, 

Division of Isotope Development and 

Division of Reactor Development
The Spread of a Dye 

Stream in an Isotropic 

Granular Medium

By AKIO OGATA

FLUID MOVEMENT IN EARTH MATERIALS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-G

Prepared in cooperation with the 

U.S. Atomic Energy Commission^ 

Division of Isotope Development and 

Division of Reactor Development

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

The U.S. Geological Survey Library catalog cards for this publication appear after page Oil.

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

Washington, D.C. 20402 - Price 20 cents (paper cover)
PREFACE

The ground-water hydrologist seeks to identify and use the physical parameters controlling 

ground-water flow. Scientists are seldom required to define a more heterogeneous or complex 

region than that involved in an underground flow system. In this respect, the hydrologist's 

problem is different from that of, say, the electrical engineer or the physicist, who normally 

analyzes regions constructed with homogeneous characteristics and definable geometries. The 

heterogeneity of the flow region involved in ground-water studies is so great that it is difficult 

to find physical characteristics adequate to describe the region. In effect, then, the region 

must be described by more complex parameters than are generally used in other scientific 

fields.

Two such parameters are particularly significant. The first is the factor of heterogeneity, 

which somehow must be identified. But the nature of the heterogeneous region can hardly 

be described through reference to the individual geometric discontinuities. Such a description 

would require an endless compendium of individual descriptions, a device so obviously impractical that it renders the region not amenable to description by measurement of any of the 

characteristics visible or accessible from the surface of the region. The second factor embraces 

the flow relationships at any given point and these relations often reflect an angular discrepancy 

between potential gradient and direction of flow. Thus tensor description is required because 

any scalar factor is inadequate.

The measurement of the two factors heterogeneity and flow relations has been one of 

the principal difficulties in accomplishing an engineering analysis of ground-water flow. 

Laboratory experiments and study have illuminated some of the basic principles involved in 

the measurement of these factors. The effects of heterogeneity have been studied by hydraulic 

experiments on two types of laboratory models, one type constructed of heterogeneous materials 

and the other of homogeneous materials. A comparison of the experimental results for flow 

through the two model types shows that for heterogeneous porous material a dissolved constituent is dispersed widely; conversely, for material that is simply homogeneous a dissolved 

constituent is not dispersed in the moving fluid. This paper discusses results of experiments 

on the latter type material. The paper was prepared and should be read as a companion to 

USGS Professional Paper 411-F, by H. E. Skibitzke, in which experiments on heterogeneous 

materials are discussed.

Because the effects of heterogeneity cannot be expressed quantitatively, experimental 

procedures must entail the use of a more suitable concept. The observation of the flow is 

generally accomplished by following a tracer element in its progress downgradient. However, 

as tracer elements flow downstream they are dispersed to a degree depending on the conditions 

met in the porous medium assuming the flow and fluid properties are unchanged. If it 

were possible to delineate a given series of streamlines, as they would be defined by Darcy's 

law, the need for describing the flow might be circumvented provided some measure of heterogeneity could be established. Generally, however, establishment of such measure is impossible 

because of physical limitations in experimental techniques.

Because it is physically impossible to trace out any given streamline, the description of 

the dispersive mechanism may be of importance. Dispersion in porous media has been studied 

to some extent, and a general description of the spreading of the flowlines can be predicted for 

isotropic media. This paper describes, for such a medium, an experiment on dispersion 

transverse to the direction of flow. Good correlation was obtained between analytical and 

experimental results. The distribution of concentration about the centerline thread of flow 

was almost symmetrical.

The tracer elements illustrate the history of events as the fluid progresses downstream. 

This seems to imply that if the dispersive effects can be separated the flow system can be

m
IV PREFACE

completely defined within the region. Although the tracer elements show the historical 

progress of each fluid element and even if it were possible to describe geometrically any flowline, 

the process of integrating this description throughout a region is extremely formidable more 

likely impossible; hence, there is the need to establish a new outlook in describing the flow 

system, as discussed in some detail in Professional Paper 411-F.

Further research is required to determine the utility of a macroscopic concept of the flow 

system to replace the prevailing microscopic concept. The use of the microscopic analysis 

has led to the conclusion that the Navier-Stokes equation is valid; however, the integration 

of the point function throughout the region becomes impossible. In other words, each streamline needs to be described as a separate entity. Further, the microscopic concept has led to 

a statistical analysis of flow employing an ordered porous body and a random-flow system, 

which is in direct contradiction to the conclusion that the Navier-Stokes equation is valid. In 

essence, then, the microscopic analysis has produced a method of describing a point, whereas 

the description of a large-scale aquifer is the goal of the hydrologist.

The implications of studies already accomplished are that further experiments should be 

tried in larger regions and that a few small-scale supporting laboratory experiments should be 

added. Because in many of the tests made to date only dissolved constituents were observed, 

some different and more suitable type of tracer must be used to reveal the components of the 

dispersion process. Radiotracers have proven especially well suited for this purpose in most 

laboratory experiments.

The analysis of dispersion processes will necessarily require the use of some tracer 

techniques, simply because dispersion is the product of the motion of the dissolved component. 

In a macroscopic sense, of the order involved in ground-water field studies, the dispersion of 

contaminant does not depend upon any measurable hydraulic factor. Therefore, part of the 

experimental technique must involve addition of dissolved components to the fluid. Because 

the dispersion is also dependent upon the location and geometry of the injection site, no naturally 

occurring tracer element will suffice. Furthermore, once a tracer is decided upon, it will, in 

general, need to be a radiotracer because the problem of remote sensing is involved in groundwater studies. It seems necessary, then, that techniques be developed for extending the 

laboratory use of radiotracers into the field.

It is important at this point to note also that the use of tracers to determine hydraulic 

parameters is at present a useless endeavour. This has been discussed by Skibitzke (1958) 

in an earlier paper.
CONTENTS

Preface.-._-__-__-__-______________________-___ ___

Abstract_ _ ___________ ___________________________

Introduction _______________________________________

Adsorption. ________________________________________

Mathematical analysis of transverse dispersion with 

radial symmetry..________________________________

Page

in

Gl

1

3

Experimental method. ______

Analysis of experimental data. 

Results-- _ _______-__-__--_-.

References. ________________

Page 

G7

8

9

11

ILLUSTRATIONS

FIGURE 1. Graph of the isotherm characteristic of multilayer adsorption___________________________________________ G4

2. Graph showing comparison between equations 11 and 15------___-__--_-__--_-__-_--_-----_--_-_-_--_-___ 8

3. Graph showing theoretical relationship ________________________________________________________________ 9

4-15. Graphs showing radial concentration distribution:

4. Block 11__________________________________________________________ 9

5. Block 10______ _ __________________________________________________ 9

6. Block 9__---__--_ __-_--_-_--_------__----__-_-.--_-__-_-._____________---.__.____ 10

7. Block 8-------_.--__-_______--.__-__-__.___-__-__-_--____-__-----._---.______ 10

8. Block 7-------__-------___--__._____--__-__-__._______.________-_.-_________ 10

9. Block 6-..____________________________________.________-_--__-_-_._______________ 10

10. Block 5--_----------__-_______-____-_-_-._____-__-_.________________________._ 10

11. Block 4_____-_____-_______-__-___-__-__--_-_._________-_-_-___-_-.-_-_-___---_---.-_ 10

12. Block 3-___-_---__---__-_-___.___.___________._________-__-_____-_--_--------_--. 11

13. Block 2__-_-_____.___________________________________________--.___-._________ 11

14. Block 1_____________________________________________________________-__-__-___-.___ 11

15. Block 14_______________________________________________._____________________ 11

TABLE

TABLE 1. Computation of ua?/4xDr from actual ratio Cr=a/Cr=0 for various distances x> 11.0 cm. G9

FLUID MOVEMENT IN EARTH MATERIALS

THE SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM

By AKIO OGATA

ABSTRACT

The mechanics of dispersion transverse to the direction of 

fluid flow was studied faoth analytically and experimentally. A 

discussion of the overall mechanism is presented and the analytical expressions obtained are compared with experimental data. 

Correlation between experimental and theoretical results for 

the spread of a tracer is fair. However, from the data available 

an initial approximation is made for the dispersion constant in 

the direction transverse to the direction of flow.

INTRODUCTION

When a region saturated by a given fluid (say, fresh 

water) is invaded by another fluid (say, a tracer or salt 

water), a process of mixing occurs if the two fluids are 

miscible. Of special import is the mixing of salt and 

fresh water along coastal regions (Cooper, 1959). 

Additional occurrences of economic importance are the 

injection of low-level fission products into ground-water 

systems and the secondary recovery methods utilized 

by oil companies. For recovery the fluids can be 

immiscible and a zone of mixing is not created.

In the other two processes, because the carrier is 

usually water, there is creation and movement of a 

zone called "the zone of diffusion." This is a region in 

which there exists a gradation from the concentration 

of the original fluid to that of the invading fluid. 

Recent investigations have been concerned about the 

zone of diffusion because the final disposition of the 

contaminant in the transient state is dependent on the 

creation and movement of this zone. The purposes of 

the investigations reported herein are to describe the 

mechanisms involved and to predict quantitatively the 

final disposition of the contaminant with the passage 

of time.

The dilution of the contaminant may be attributed 

to three major processes mechanical or hydrodynamic 

dispersion, adsorption, and dispersion due to heterogeneous aquifer conditions. Other minor mechanisms, 

such as ionic diffusion or discontinuous pore channels, 

have a tendency to cause spreading. However, the 

magnitude of spread due to these minor mechanisms is 

small in comparison to that due to the other processes 

except when the flow rate of the ground water is of the

same order as the mean flux due to diffusion. The rate 

of movement and the path traversed by this zone of 

dispersion are dependent on the existing flow condition 

and the method of injection. Thus, assuming that 

other things are equal such as density of fluid and 

external pressures and that Darcy's law is valid, the 

major mechanisms of dilution are classified as follows:

1. Mechanical or hydrodynamic dispersion due 

to microscopic velocity variation controlled 

by viscosity and pore-channel diameters; 

principle mechanism in homogeneous and 

isotropic media.

2. Adsorption due to mass transfer from liquid 

to solid phase controlled by chemical properties of contaminant and solid phase.

3. Heterogeneity due to regional or macroscopic 

changes in the properties of the granular 

media.

Most papers on dispersion in a porous medium 

published to date have dealt exclusively with "mechanical dispersion" in the direction of flow, which is of 

primary importance in the analysis of reactor columns. 

However, in ground-water systems, adsorption and 

variation in the permeability of the media may be the 

major causes of the apparent dispersion. Adsorption 

or ion-exchange processes may cause significant dilution 

in a short distance owing to the presence of a large 

surface area available for the transfer of mass. Heterogeneity on the other hand relates to the finite changes 

in the properties of the aquifer. The occurrences of the 

discontinuities cause additional spreading that is due 

to refraction of the streamlines.

There is a different type of problem in regions near 

the seacoast. Although the problems involved are not 

the same, the mechanism of mixing is similar. Studies 

have been initiated (Kohout, 1960; Henry, 1959) to 

analyze the mixing in response to the tidal oscillation 

in the ocean and the natural velocity with which the 

ground water flows through the system. These effects, 

however, have not been separated and the importance 

of each process cannot yet be evaluated. Investigation 

is as yet in its initial stage, and a great deal of field data

Gl
G2 FLUID MOVEMENT IN EARTH MATERIALS

is being accumulated, especially in Florida. However, 

the distribution of velocity within the two fluids and 

the effect of the change in densities along their interface 

remain to be determined either by in-place measurements 

or by model studies.

The discussion that follows is confined to dispersion 

of miscible fluids without density variations. It should 

be useful in determining the final disposition of radioactive waste in the event of spillage or leakage. The 

accumulated knowledge of the dispersion process in the 

ground-water system is directly applicable provided the 

geologic conditions can be described.

In essence, the final disposition of any contaminant 

is dependent on the path traversed. In ordinary circumstances the liquid waste first flows through the 

unsaturated zone overlying the ground-water system. 

Most of the contaminants are adsorbed in this region 

owing to the existing exchange potential of the soil. 

Because the adsorption depends on the types of 

materials present, all the contaminants may be fixed 

to the surface of the soil particles. A few of the 

fission products, however, may enter the saturated 

zone in which further dilution may occur.

In the saturated zone there^are two possible means 

of reducing the concentration of the contaminated 

fluid. These are dilution by spreading, caused by 

mechanical dispersion, and dilution by actual removal 

of the contaminant from the liquid phase by adsorption. 

The fate of the contaminant in this region is thus 

dependent on the rate of flow, the geologic conditions, 

and the types of materials present.

Most of the reported studies of the dispersion phenomena are based on a gross analysis in which is 

assumed a diffusion process described by a relationship 

similar to Fick's law. The magnitude of the constant 

of proportionality, or the dispersion coefficient, is 

determined experimentally. These investigations of 

longitudinal dispersion (for example, de Josselin de 

Jong, 1958; Orlob and Radhakrishna, 1958) indicate 

that the magnitude of the dispersion coefficient D is 

proportional to the magnitude of the velocity u that 

is, D mu. The constant m is dependent on the size 

and distribution of aquifer material and also on the 

degree of saturation; thus it varies according to the 

individual situations considered. This relationship has 

been obtained from studies of unconsolidated-sand 

models composed of fairly uniform material. Thus, for 

application of the results to the existing field condition, 

or to consolidated materials, correction factors must be 

introduced. In regions where materials are extremely 

heterogeneous, it may be possible to increase the 

constant m to describe the dispersion or spread of 

contaminant.

The mechanism that controls the lateral spread

has not yet been extensively correlated experimentally, 

de Josselin de Jong (1958) and Saffman (1960) concluded from the study of statistical models that the 

lateral spread is also dependent on the average longitudinal velocity. However, in extremely slow flow, 

ionic diffusion becomes important (Beran, 1957). 

Laboratory study has been reported by Simpson (1962), 

and some laboratory investigations of lateral dispersion 

have been reported in chemical engineering journals. 

The investigations reported in these journals deal 

with flow rates much higher than those found in 

ground-water reservoirs. In most studies, good correlation is obtained between the lateral-dispersion 

coefficient and the velocity in the longitudinal direction.

For both lateral and longitudinal dispersion in 

isotropic media, the spread is due primarily to microscopic controls or variations. But, in nonhomogeneous 

or consolidated materials, macroscopic variations may 

have a marked effect. When these variations constitute changes in geologic conditions, the term "heterogeneity" applies. Heterogeneous conditions exert a 

large influence on the breaking up and the spread of 

a stream of contaminant progressing through the 

saturated zone. There is little doubt that these 

macroscopic discontinuities exist in many aquifer 

systems and depend on environmental conditions at 

the time the aquifer materials were deposited. Of 

course, the objective is still to obtain quantitative 

measures of the spread of a contaminant. Because 

the prevailing environmental conditions at the time of 

deposition arenotknown, it would be virtually impossible 

to describe quantitatively the individual discontinuities 

or changes in a large region. But reason suggests 

that, as for dispersion in homogeneous media, all macroscopic variations may be treated separately by using an 

averaging process. By this means it may be possible 

to describe the spread or mixing due to heterogeneity, 

assuming that the constant m is dependent on the 

size and distribution of isolated discontinuities.

The mechanism of adsorption or ion exchange, 

which is of primary importance in predicting the fate 

of the contaminant, is usually not mentioned in studies 

of dispersion. Adsorption cannot be classified as 

a dispersion phenomenon; it is primarily a surface 

chemical process due to the existence of an exchange 

potential between the dissolved contaminant and the 

surface of the media through which the fluid flows. 

This exchange capacity is primarily dependent on the 

solid material and the type of contaminant present 

(Kaufman and Orlob, 1956). The dilution of the 

liquid phase is realized because of the transfer of 

dissolved solid from the liquid state to the solid state. 

Although the contaminant may not be permanently 

fixed, the rate of exchange from liquid to solid is
SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G3

usually large enough that considerable dilution is 

realized. In this way adsorption may be the most 

effective method of reducing contamination hazards. 

Adsorption is discussed further in the next section.

Analysis of dispersion with linear adsorption isotherm superimposed on diffusion has been attempted. 

However, major simplification is usually necessary to 

carry out analytical details. In laboratory experiments 

utilizing continuously injected organic dye tracers, 

a considerable travel delay due to adsorption or ion 

exchange is usually noted, but ordinarily no quantitative description is attempted. Investigations or Hougan and Marshall (1947) are an example of this attempt; 

however, the fluid was gas and the simplification utilized 

cannot be applied to liquid flows.

The lack of available data on transverse dispersion 

and effects of heterogeneity prompted the Phoenix 

research office of the U.S. Geological Survey to construct models for laboratory analysis of dispersion. 

Owing to time limitations and the urgency of other 

projects, laboratory-model studies of heterogeneity 

were intended to furnish qualitative rather than quantitative description of the process (Skibitzke and Robinson, 1963). Both mathematical and laboratory studies 

were made to analyze transverse diffusion. A value of 

the transverse-diffusion coefficient hag been obtained; 

however, inasmuch as only one test run was made, 

this value must be substantiated further.

The laboratory experiment and the analysis of the 

experimental block was carried out by H. T. Chapman 

and J. M. Cahill of the Phoenix research office. Their 

diligence and ingenuity in overcoming many difficult 

technical problems resulted in the successful conclusion 

of the experimental phase.

ADSORPTION

Although adsorption is not a dispersion process, a 

short discussion is presented because the experiment 

reported requires knowledge of concentration relationships between the liquid and the solid phases. Generally, if the concentration of the tracer used in the 

experiment is low, a linear relationship can be assumed 

provided an equilibrium condition is reached. This 

relationship follows from the assumption that the 

Langmuir exchange isotherm is valid. All the isotherms reported are semiempirical in nature. Although 

there are many published reports of transfer from 

solutions to solid surfaces, the exchange isotherms 

utilized are valid only for the systems reported, inasmuch as the theoretical expressions are obtained only 

after a series of simplifications to construct a mathematical model.

First, consider the exchange isotherms for equilibrium 

conditions. The isotherms presented are based on

740-419 O 64 2

the assumption that only a monomolecular film is 

formed around the solid particles. This fact, in essence, 

tends to limit the use of the isotherms presented unless 

extremely low concentrations are used in the experiments.

Adsorption of particles from gaseous mixtures onto 

a solid surface depends primarily on temperature and 

pressure (Adamson, 1960). In transfer of mass from 

a solution, concentration of the solution replaces the 

pressure. The isotherms proposed by Langmuir and 

Freundlich are those most widely used, because of their 

simplicity. Freundlich's isotherm, at first thought to 

be an empirical relationship, is written

F=aCl/n

where

F= amount of adsorbate per unit amount of solid,

a and n=constants, and

C= concentration of liquid phase.

Langmuir's isotherm, on the other hand, can be written

F=abC!(l+bC)

where

a=a constant related to the specific surface area 

of the solid, and

6=a constant related to the heat of adsorption.

Roughly speaking, the constants appearing in the 

Freundlich isotherm may be interpreted to represent 

the same specific area and heat of adsorption. For 

both the Fruendlich and Langmuir isotherms, a curve 

of F versus C shows a convexity towards the C axis. 

However, there is one major difference: the Langmuir 

isotherm reaches a saturation value of a as C increases 

and bC becomes much greater than unity, whereas the 

Freundlich isotherm does not reach an upper limit. 

Furthermore, when the concentration is small, or 

6Cr«l, the Langmuir isotherm gives a straight-line 

relationship between the liquid concentration and the 

solid concentration, though the Freundlich isotherm 

does not show this effect and the straight-line approximation is obtained only when n >1. In this circumstance, the isotherm is a straight line regardless of 

concentration.

Investigations of adsorption on solids from solutions 

indicate that porous adsorbents usually obey 

Langmuir's equation because the pore volume limits 

the amount adsorbed (Condon and Odishaw, 1958). 

However, this evidence does not prove that Langmuir's 

expression is superior to Freundlich's expression. On
G4 FLUID MOVEMENT IN EARTH MATERIALS

the contrary, many laboratory data show better correlation with the Freundlich isotherm, but when concentrations are low Langmuir's isotherm seems better 

suited for a large number of adsorption systems.

Later researchers have reported a sigmoid isotherm 

(fig. 1), which is indicative of a multilayered formation 

for transfer from the vapor phase to the solid phase 

(Adamson, 1960). Although very little is known as 

to whether multilayers actually form, there seems to 

be a general acceptance of this possibility. The 

theoretical analysis involves the use of Langmuir's 

isotherm for n layers. The expression was developed 

by Brunauer, Emmett, and Teyerson and is generally 

known as the BET equation (Condon and Odishaw 

(1958) or Adamson (I960)).

LIQUID CONCENTRATION 

FIGUBE 1. Graph of the Isotherm characteristic of multilayer adsorption.

In utilizing the equilibrium isotherms discussed previously, it is assumed that there is instantaneous transfer of mass. The assumption of instantaneous adsorption does simplify to a great extent the analysis of the 

combined effects of adsorption and diffusion. However, for a fluid flowing through a stationary porous 

bed, this assumption may not be valid unless the time 

elapsed is large. In nonequilibrium conditions the expression that must be considered is

assuming that rate of transfer is a function of the 

liquid concentration only. Generally the function /(<7) 

is not described but an assumed relationship, usually 

linear, is superposed on the diffusion equation.

Two types of nonlinear expressions for f(O) were 

discussed by de Vault (1943).' Generally, /(O) is assumed to be linear; a typical relationship (for example, 

that given by Merriam and others (1952)) has the form

Here again it must be assumed that the exchange 

process is instantaneous. Thus, whether equilibrium 

is reached or not, the assumed relationship of F or 

b-F/bf to C is similar to Langmuir's or Freundlich's 

isotherm.

MATHEMATICAL ANALYSIS OF TRANSVERSE DISPERSION WITH RADIAL SYMMETRY

In the mathematical development of dispersion in 

granular media, various simplifications are made so 

that the problem becomes tractable. The first simplification is the assumption that a piston-type flow exists, 

that is, that every element of fluid is flowing at a 

velocity equal to its average velocity. The second assumption is that the apparent dispersion due to microscopic variation of velocity in the individual pore 

channels is described by a relationship similar to Fick's 

law of diffusion. The component of total mass transported in the x direction (say) for isotropic media is 

thus given by the relationship

(1)

where

g=the total mass flux across any given plane, 

Z?x =the constant of proportionality depending

only on direction x, 

C= tracer concentration in fluid, 

x=directional component, and 

u=average fluid velocity in the x direction.

If no chemical reaction occurs, the law of conservation of mass gives the differential equation

where

y g _

"*" d "*" ds Dt

(2)

e/i=mass flux due to dispersion only in the 

direction i, and

u, v, w=constant average velocity components 

in the x, y, and 2 directions, respectively.

In anisotropic media where the dispersion coefficient 

D depends on the direction, D takes on the properties
SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G5

5(7 5(7

+ 23

of a tensor. Hence, the components of dispersion at 

any given point are described by

(3)

In other words, the mass flux in any given direction is 

not only dependent on the concentration gradient in 

the same direction but also on the gradient in the other 

two directions. Owing to mathematical difficulties 

and the lack of experimental data available for anisotropic conditions, it is assumed that the principle 

direction of dispersion is along the axis of the coordinate 

system chosen.

Equation 2 has been expressed in terms of Cartesian 

coordinates; however, the equation may be written for 

any system of coordinates. Using appropriate transformations

x=r cos 6 

y=r sin 6

z=z,

equation 2 in cylindrical coordinates is

DC_l 5 / 5<7\ 1 5 / 5<7\ d 

Dt~r *r (r D' &r7* 50 (De W+te

(4)

The differential equation of diffusion may also be 

written in spherical coordinates, but it is not as useful 

as the expression involving Cartesian coordinates for 

problems of flow through granular media.

Generally in flow through porous media the velocity 

component can be considered to be unidirectional and 

the principal axes of diffusion are assumed parallel and 

transverse to the velocity. Thus for unidirectional 

flow, equation 2 is reduced to

I'l/'Y* 1/ / ^1

Utilizing either the substitution C=Y exp ^7^ 777

[_2£/ 4//z_|

or the moving coordinate z=x ut, equation 5 can be 

reduced to the standard diffusion equation

5C (6)

In the cylindirical coordinate system, where radial 

symmetry exists, the principal axes of dispersion are

assumed to be in the directions parallel and transverse 

to the direction of flow. Thus the dispersion equation 

can be written

5(7. 5(7 52<7

5r \ 5r (7)

Equation 7 can be reduced to the diffusion equation byemploying the moving-coordinate system, or i}=x ut; 

hence

-+Dr *_ f &C

r 5r \ br (8)

Equation 6 in its one-dimensional form has been 

solved for various boundary conditions in the investigations of both heat-flow and diffusion processes. 

Analytical and experimental correlation studies have 

been carried out by many investigators, as previously 

stated. These investigations have indicated that the 

dispersion coefficient is directly proportional to the 

average velocity of the fluid both transverse and parallel 

to the direction of flow, but there is no indication that 

the coefficients are of equal magnitude in both directions.

Industrial interest has prompted extensive study of 

longitudinal dispersion, in both the laminar range 

(Reynolds No. <1) and the so-called turbulent range 

(Reynolds No. >1), but few studies have been made 

either analytically or experimentally of dispersion 

transverse to the direction of flow. The remainder of 

this section will therefore be confined to the development 

of a mathematical description of transverse diffusion.

Analytical difficulties are inherent in equation 7, 

even for DT=DX , unless boundary conditions are simple. 

Thus, to simplify the analysis to a large degree, consider 

separately the two processes of longitudinal and transverse dispersion. Except in the frontal zone where 

longitudinal dispersion predominates, the system can 

be described approximately by

n 1 5 / 5<7\ 5(7 5(7 ,Q. Dr-^-i r ) u^ =3 (9) 

r 5r\ 5r/ 5z ot

Von Rosenburg (1956) indicated that the length of 

the frontal zone, over which the concentration varies 

about 80 percent, is proportional to the velocity and 

inversely proportional to the diffusion coefficient. 

Because the diffusion coefficient is proportional to 

velocity once the front is established, it remains nearly 

constant as the stream progresses through the media. 

Further, if no chemical reaction or adsorption of the 

contaminant takes place, there is no "holdup" time and 

the front progresses through the media at a rate equal 

to the average velocity. The foregoing observations 

are based on one-dimensional problems; thus, the 

simplifications may not be wholly justified physically.
G6 FLUID MOVEMENT IN EARTH MATERIALS

To simplify the mathematical model it is assumed 

that the contribution of the front to the overall lateral 

spread is small, especially in the zone x > ut. In other 

words, in the mathematical model it is assumed that 

a piston flow takes place; that is, a sharp front traverses 

the media with a velocity u, and diffusion occurs normal 

to the direction of flow. For this simplified model it is 

apparent that equation 9 holds.

By letting z=x ut and r=t, it is possible to simplify 

equation 9 further: that is,

dr

Physically, this equation means that an observer 

follows a slug of contaminant at a rate equal to the 

average velocity u as the contaminant progresses 

through the system. For the unsteady condition this 

approximation requires that the boundary conditions be

t=Q, C=CQ , r<a 

j=0, £7=0, r>a

r-O, g-0. ' dr

If the source emits at a constant rate, this problem is 

the same as diffusion from a semi-infinite cylinder of 

radius a into the surrounding media. In the steady 

situation, dC/d£=0; thus t can be replaced by x/u in 

the boundary condition specified above.

The solution of the above system has been obtained 

and discussed extensively (Goldstein, 1932). This 

solution written in equivalent form is

=exp -

m=l

(10)

where rj=a/r, i- -ra/2Dt, and Ji and J0 are Bessel 

functions of the first kind of order 1 and 0, respectively. 

The integral solution was written in its alternate form 

to facilitate computation of the general solution. 

The author showed this solution in graphical form in 

chapter B of this series (Ogata, 1961). As previously 

indicated, the solution of the steady-state system is

X

obtained by simply replacing t by Because the

\M

experiment that was performed is assumed to have 

reached steady state, the solution to equation 10 is 

hereafter written in its steady-state form.

Although equation 10 can be computed, the radialdispersion coefficient cannot be easily determined from 

the resulting expression. However, the integral can 

be integrated for the specific situation of either r=0 

or r=a. Consider first the distribution of concentration along r=0. Because «70 (0) = 1, equation 10 

can be written

c r

7^=a \

*^o Jo

CO 

e

Watson (1948, p. 394) gave the value of the above 

integral as

, . (U)

If the concentration distribution is experimentally 

determined along r=0, the radial component can 

easily be computed using equation 1 1 .

However, the radius of a finite source of tracer may 

be such that within the limits of the experimental 

model no noticeable change in concentration will 

occur along r=0. For this situation, it would be 

preferable to determine distribution along a known 

value of r, say a. When r=a, equation 10 can be 

written

7T=a c r

M) JO

-Do?t

Goldstein (1932) expressed the above integral in terms 

of the modified Bessel function of the first kind, that is,

C (12)

Inasmuch as the function e~x!0 (x) is well tabulated, 

the term D can be readily computed from any given 

experimental data. Here again the value of D can 

be obtained by a curve-fitting process.

Further reflection on the problem of radial dispersion 

led to the hypothesis that the physical assumptions 

may not be realistic and that this problem may best 

be described as diffusion from a moving-disk source 

in a semi-infinite medium. A second analysis was 

thus attempted based on the assumption that, owing 

to radial symmetry, the coefficient of dispersion depended only on the x and r directions. A summary of 

the results obtained is presented. It should be noted 

that the complete solution for the unsteady-state 

system was not computed because of the complexity 

of the resulting integral. But, for comparison purposes 

the steady-state system was computed for various 

values of the ratio Dx/Dr.

In approximating the process of transverse diffusion,
SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G7

it was assumed that the term Dx > §- was negligible in

\jJU

equation 9. It was also assumed that the conditions of 

the experiment justified the use of the steady-state 

system. These conditions cannot be checked experimentally although the theoretical solution can be correlated with experimental evidence.

Because radial symmetry exists, equation 7 would 

describe the process. For the instantaneous point 

source (Carslaw and Jaeger, 1959, p. 260), it is possible 

to write the solution of the problem immediately. The 

expression

c_ u'a' r r

Co 2-v/JrZ? Jo \

{I

W(t-r)

(13)

isobtained by integrating the expression for the continuous point source from zero to a, where a is the 

radius of the disk source. The terms used in equation 

13 are

P=r^jD/Dr

Inasmuch as the strength of the source is u'C0 , the 

expression for steady -state conditions is

7T=-r7= f a'</0 

M> W'T J° f" exp f-f

Jo L \

+

47)2 1/2 d\ (14)

Equation 14 cannot be integrated further, but comparison with equation 10 shows that when

4712

X2 «l,

the expressions are identical.

Consider the concentration distribution along r=0. 

Substituting p=0 in equation 14 and integrating yields 

the expression

a2 Dr

Note 1/2 that if -, ~ «1 and the approximation (1 + x)

X LJX

»(1 + 3/2) is used, the term under the radical in equation 15 can be simplified and equation 15 becomes

which is identical to equation 11. Thus it can be 

stated that for any given ratio DXIDT the approximate 

solution is valid for large distances from the source.

a2 D The value of the parameter -g jf at which the apX LJf

proximation is correct to the nearest thousandth is 

about 10~4 . Figure 2 shows a comparison between 

equations 11 and 15.

For r=a the solution based on the disk source gives

a2 D Again, as -^ =^ r becomes small,

X Lff

Dx

Hence

C

which is identical to the approximate solution, or 

equation 12, for r=a.

Above analysis of the disk source thus indicates that 

if the experimental column is sufficiently long the radialdiffusion coefficient can be computed by equation 11 

or 12. In addition, by utilizing equation 15 or 16, 

the longitudinal-dispersion coefficient can be readily 

computed provided reliable data are obtained near the 

source.

Experimental Method

The experiment on diffusion transverse to the 

direction of flow was conducted with an isotropic, 

artificially constructed sandstone. Kadioactive phosphorous was utilized as a tracer which was introduced 

into the flow system through a tube three-fourths of an 

inch in diameter. Except for the diameter of the source 

and the mode of controlling the quantity of tracer 

injected, the method utilized was exactly the same as 

that described by Skibitzke and others (1961). Physical data are as follows:

Length of block._________________ 43.5 cm

Diameter of source______________ .95 cm

Average fluid velocity________--__- 2.27X10"2 cm per min

Average sand-grain diameter _______ .44 mm

Uniformity coefficient _____________ 1.92

Total time of experiment_____--___ 383.5 hr

As in the previous experiment, the amount of tracer 

introduced was determined by the ratio of the crosssectional areas of the source and the block. By 

controlling the quantity of flow rather than the head 

of the source, the problem of measuring extremely small
G8 FLUID MOVEMENT IN EARTH MATERIALS

20 30 

LONGITUDINAL DISTANCE, IN CENTIMETERS

FIGTJKE 2. Graph showing comparison between equations 11 and 15 for various ratios of D»/Dr. Concentration distribution along r=0.

Curves, theoretical; O. experimental.

differences in head was eliminated. The quantity of 

flow was controlled by dropping glass beads into a 

storage tank which fed the tracer source. Although 

this method of quantity control does not satisfy the 

boundary condition calling for a continuous source, 

over the period of the experiment (17 days) the condition 

was averaged; the continuous source was therefore 

approximated.

After a steady flow of tracer was detected at the 

outflow end of the block, the experiment was terminated 

and the block dissected. The cross section of the 

block was then assayed by means of a needle proble 

and a photoemulsion method. The data reported were 

obtained using the needle probe. The emulsion method 

showed a distinctive pattern of spread; however, no 

correlation could be obtained between the counts and 

the density of light transmitted through the film.

ANALYSIS OF EXPERIMENTAL DATA

In analyzing the experimental data, the basic assumption is made that a multilayer of P32 was formed by 

adsorption, except in the lower reaches of the model. 

Owing to the fairly low concentration readings realized 

in the lower parts of the block, the relationship of the 

liquid concentration and the solid concentration was 

assumed to be almost linear. That is, Langmuir's 

isotherm was assumed to be valid with 6<7«1. This

basic assumption made possible the comparison of 

theoretical and experimental values of any given section 

of the experimental model.

This type of analysis was attempted because of large 

discrepancies observed along the centerline of the 

trace filament near the tracer source. The rapid 

decline of concentration from distances 0 through 10 

cm could not be correlated with any of the theoretical 

results. However, a good fit was obtained for distances 

greater than 11 cm from the source and the dispersion 

coefficient D was therefore computed from these data. 

The computations showed a range from 6.7X10"4 to 

11.8X10"4 cm2 per min. Similar computations for 

r a, using equation 12 gave a range in D from 6.0X10"4 

to 10.3X10~4 cm2 per min. For distances less than 11 

cm the value of the coefficient increased approximately 

tenfold, the largest values being near the source.

Because the concentration of the entering fluid was 

not measured, the assumption that the adsorption 

formed a multilayer except in the lower reaches made 

it necessary that this entering concentration be computed. Inasmuch as the ratios of the concentration 

Cr-alCr-o for various distances were available for both 

theoretical and experimental conditions, these ratios 

were used. Figure 3 shows the theoretical relationship 

between <7r=a/CUo and ua?/4xDr. After experimental 

values were obtained for Cr,alCr,o, it was possible to
SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G9

3.0 4.0 5.0

0.6 0.8 1.0

a2w/4xZ>r

7.0 

1.4

8.0 

1.6

FIOUEE 3. Graph showing theoretical relationship between CWCV-o and MO*/

4xDT.

compute values of D for each given section of the 

experimental model. The value of D obtained by this 

method ranged from 3.9X10~4 to 6.0X10~4 cm2 per 

min, as shown in table 1. The arithmetic average was 

computed to be 5.0X10"4 cm2 per min. By utilizing 

the average value of the dispersion coefficient, an 

average value of C0 was computed to be approximately 

140 counts per minute.

TABLE 1. Computation of ua*/4xDr from actual ratio 

Cr=a/Cr=o/or various distances x>J? J?.0 cm

[Average: 9=10.3 cm; D,=4.97X10-< cm* per min]

X

11.1 

14.8 _ 

18.1.. ..

21.3 

24.3. __ 

27.6 

30.6 ..

33.3 .

36.7 ..

G-o

83.0

72.0

69.2

64.0

46.2

38.6 

39.3

37.6

26.0

Cr-,

48.6

43.0

39.6

34.6

34.5

28.0 

30.3

30. 0

10 n

Cr-a/CV-0

O KQA

.597

.667

639

.747

.727 

.771

QflA

.760

Theoretical 

ua*l4xDr

0 81

.76

.53

A1

.35

.39

41

.26

oo

5=woV4Dr

9.G

H O

9.6

19. f\

Q R

10.8 

9.5

8 7

19 1

DrXlO*

4.66

6.32

Q QQ

6.01

4.73 

6.38

6.87

4 99

RESULTS

By utilizing the values <?0 =140 cpm and Z>=5X10~4 

cm2 per min, theoretical and experimental values 

were compared as shown in figures 4 through 15. 

Note that for most of the sections the correlation is 

good. However, in some sections all experimental 

points fall consistantly below the theoretical curve, 

perhaps primarily because of instrumental errors, 

though the exact causes are unknown.

The approximate analysis of the experimental data 

does seem to indicate that diffusion coefficients transverse to the direction of flow are of the same order of 

magnitude as ionic diffusion. A spread to approximately three times the width of the source at a dis-

[ I

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

RADIAL DISTANCE (r/a)

FIGURE 4. Graph showing concentration distribution along radius r, block 11. 

ar=7.5 cm, a =0.95 cm. Q, experimental.

tance of 43 cm seems indicated. Because only one test 

run was made, these figures should be taken as preliminary in nature.

Future experiments should take into account the 

nature of the adsorption isotherms. Generally, it is 

good practice to ascertain the range of concentration 

for which the linear part of the Langmuir isotherm can 

be used. Analysis of the data does show that transverse diffusion is small; thus, if future experiments are 

desired, the described method can be utilized provided 

the gross transfer of mass between the liquid and the 

solid phases is determined for the concentration and 

for the tracer used.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

RADIAL DISTANCE (r/a)

FIGURE 5. Graph showing concentration distribution along radius r, block 10. 

ar=ll.l cm, a=0.95 cm. Q, experimental.
G10 FLUID MOVEMENT IN EARTH MATERIALS

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4. 

RADIAL DISTANCE (r/a)

FIOUEE 6. Graph showing concentration distribution along radius r, block 9. 

a;=14.8 cm, a =0.95 cm. Q, experimental.

0.5 1.0 1.5 2.0 2.5 3.0 

RADIAL DISTANCE (r/a)

3.5 4.0

FIGURE 7. Graph showing concentration distribution along radius r, block 8. 

i=18.1 cm, a=0.95 cm. Q, experimental.

1.0 1.5 2.0 2.5 3.0 

RADIAL DISTANCE (r/a)

3.5 4.0

FIGURE 8. Graph showing concentration distribution along radius r, block 7 

1=21.3 cm, o=0.95 cm. Q. experimental.

1.0 1.5 2.0 2.5 3.0 

RADIAL DISTANCE (r/a)

3.5 4.0

FIGURE 9. Graph showing concentration distribution along radius r, block 6. 

£=24.3 cm, a=0.95 cm. Q, experimental.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.C

RADIAL DISTANCE (r/a)

FIGURE 10. Graph showing concentration distribution along radius r. block 5. 

£=27.6 cm, a=0.95 cm. Q. experimental.

1.0 1.5 2.0 2.5 3.0 

RADIAL DISTANCE (r/a)

3.5 4.0

FIGURE 11. Graph showing concentration distribution along radius r, block 4. 

z=30.6 cm, a=0.95 cm. Q. experimental.
SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM

REFERENCES

^-Theoretical

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4 

RADIAL DISTANCE (r/a)

FIGURE 12. Graph showing concentration distribution along radius r, block 3. 

1=33.3 cm, a=0.95 cm. O» experimental.

0 0.5 1.0 1-5 2.0 2.5 3.0 3.5 ' 

RADIAL DISTANCE (r/a)

FIGURE 13. Graph showing concentration distribution along radius r, block 2. 

1=36.7 cm, a=0.95 cm. Qi experimental.

i i i i r

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 

RADIAL DISTANCE (r/a)

FIGURE 14. Graph showing concentration distribution along radius r, block 1. 

z=39.8 cm, o=0.95 cm. O, experimental.

I I

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 A

RADIAL DISTANCE (r/a)

FIGURE 15. Graph showing concentration distribution along radius r, block 14. 

a-=43.3 cm, o=0.95 cm. O, experimental.

Gil

Adamson, A. W., 1960, Physical chemistry of surfaces: New

York, Intersci. Publishers, 629 p. 

Beran, M. J., 1957, Dispersion of soluble matter in flow through 

granular media: Jour. Chem. Physics, v. 27, no. 1, p.

270-274. 

Carslaw, H. S., and Jaeger, J. C,, 1959, Conduction of heat in

solids: Oxford Univ. Press, 510 p. 

Condon, E. IL, and Odishaw, Hugh, 1958, Handbook of physics:

New York, McGraw Hill Book Co., 172 p. 

Cooper, H. H., Jr., 1959, A hypothesis concerning the dynamic

balance of fresh water and salt water in a coastal aquifer:

Jour. Geophys. Research, v. 64, no. 4, p. 461-467. 

de Josselin de Jong, G., 1958, Longttudinal and transverse diffusion in granular deposits: Am. Geophys. Union Trans., v.

39, no. 1, p. 67-74. 

De Vault, Don, 1943, Theory of chromatography: Am. Chem.

Soc. Jour., v. 65, p. 532-540. 

Goldstein, S., 1932, Some two dimensional diffusion problems

with circular symmetry: London Math. Soc. Proc., ser. 2,

v. 34, p. 51-88. 

Henry, H. R., 1959, Salt intrusion in fresh-water aquifers: Jour.

Geophys. Research, v. 64, no. 11, p. 1911-1920. 

Hougen, O. A., and Marshall, W. R., 18*7, Adsorption from a

fluid stream flowing through a stationary granular bed:

Chem. Eng. Process, v. 43, no. 4, p. 197. 

Kaufman, W. J., and Orlob, G. T., 1956, An evaluation of

ground-water traces: Am. Geophys. Union Trans., v. 37,

no. 3, p. 297-306. 

Kohout, F. A., 1960, Cyclic flow of salt water in the Biscayne

aquifer of southeastern Florida: Jour. Geophys. Research,

v. 65, no. 7, p. 2133-2142. 

Merriam, C. N., Jr., Southworth, R. W., and Thomas, H. C.,

1952, Ion exchange mechanism and isotherms for deep bed

performance: Jour. Chem. Physics, v. 20,no. 12,p. 1842-184&. 

Ogata, A., 1961, Transverse diffusion in saturated isotropic

media: U.S. Geol. Survey Prof. Paper 411-B, 8 p. 

Orlob, G. T., and Radhakrishna, J. N., 1958, The effects of entrapped gases on hydraulic characteristics of porous media:

Am. Geophys. Union Trans., v. 39, p. 648-659. 

Saffman, P. G., 1960, Dispersion due to molecular diffusion and

macroscopic mixing in flow through a network of capillaries:

Jour. Fluid Mech., v. 7, no. 2, p. 194-208. 

Simpson, E. S., 1962, Transverse dispersion in liquid flow through

porous media: U.S. Geol. Survey Prof. Paper 411-C, 30 p. 

Skibitzke, H. E., 1958, The use of radioactive tracers in hydrologic field studies of ground-water motion: Internat. Assoe.

Sci. Hydrology, General Assembly, Toronto, Canada, 1957,

1957, v. 2, p. 243-252.

 1964, Extending Darcy's concept of ground-water motion: U.S. Geol. Survey Prof. Paper 411-F Cm press). 

Skibitzke, H. E., Chapman, H. T., Robinson, G. M., and

McCullough, R. A., 1961, Radio tracer techniques for the

study of flow in saturated porous material: Internat. Jour.

Appl. Radiation and Isotopes, v. 10, no. 1, p. 38-46. 

Skibitzke, H. E., and Robinson, G. M., 1963, Dispersion in

ground water flowing through heterogeneous materials:

U.S. Geol. Survey Prof. Paper 386-B, 3 p. 

Von Rosenburg, D. U., 1956, Mechanics of steady-state single

phase displacements from porous media: Am. Inst. Chem.

Eng. Jour., v. 2, p. 55. 

Watson, G. N., 1948, Treatise on the theory of Bessel functions:

Cambridge Univ. Press, 804 p.

O
The U.S. Geological Survey Library has cataloged this publication as follows:

Ogata, Akio, 1927-

The spread of a dye stream in an isotropic granular medium. Washington, U.S. Govt. Print. Off., 1964.

v, 11 p. diagra, table. 30 cm. (U.S. Geological Survey. Professional paper 411-G)

Fluid movement in earth materials.

Prepared in cooperation with the U.S. Atomic Energy Commission, Division of Isotope Development and Division of Reactor 

Development.

Bibliography: p. 11.

(Continued on next card)

Ogata, Akio, 1927- ,The spread of a dye stream in an isotropic granular medium. 1964. (Card 2)

1. Dispersion. 2. Water, Underground. 3. Fluids. I. U.S. Atomic 

Energy Commission. II. Title. (Series)