Transverse Dispersion in Liquid Flow 

Through Porous Media

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-C
Transverse Dispersion in Liquid Flow 

Through Porous Media

By EUGENE S. SIMPSON

FLUID MOVEMENT IN EARTH MATERIALS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-C

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEW ART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C.
CONTENTS

	Page 

Symbols and glossary-- _____________________________ iv

Abstract.__________________________________________ C-l

Introduction _______________________________________ 1

Acknowledgements. ___ _ __-_-___-______-.-____________ 2

The'oretical analysis._-___-_--__---_-___-__________-_ 2

Description of apparatus and experimental procedure, _ _ 7

Page 

Experimental results.____-_____-_-____-_____-_--_-__ C-13

Comparison between theory and experiment-____________ 23

Discussion _________________________________________ 26

References. ________________________________________ 28

Appendix.______________________-____-_----__----__ 29

ILLUSTRATIONS

FIGURE 1. Diagram of porous medium showing line 

source and coordinate axes______________

2. Diagram illustrating concepts of tortuosity 

and circuity____________________________

3. Diagram illustrating set of idealized flow 

paths_ _______-____-__-_-___________-_ 

4. Diagram illustrating motion of tracer elements between adjacent flow paths______

5. Diagram of idealized porous medium_______

6. Diagram of idealized pore space ___________

7. Graph of probability q plotted against 1/X__

8. Graph of a/a VdL plotted against 1/X______

9. Flow diagram of dispersometer____________

10. Diagram of experimental porous block______

11. Flow diagram of apparatus to supply deaired water_ __________________________

12. Photograph of dispersometer, front view____

13. Photospectrograph of tracer solution.______

14. Mechanical analysis of sand used in porous 

block_ ________________________________

15. Concentration distribution, run 1, suites A, 

B, and C______-_-_.______-___.______-

16. Concentration distribution, run 2, suites A, 

B, and C_____-_-_-_-________________-

17. Concentration distribution, run 3, suites A, 

B, and C______-_--______-__-__-____

Page

C-3

4

5

6

8

8

9

10

11

12

12

13

14

14

15

Page

18. Concentration distribution, run 4, suites A

and B_____-_________---___------ C-15

19. Concentration distribution, run 5, suites A

and B____________________________ 16

20. Concentration distribution, run 6, suites A

and B___-__-__-___-_-__--__________ 16

21. Concentration distribution, run 7, suites A

and B_________-_____---_- _ _- 17

22. Concentration distribution, run 8, suites A,

B, and C___----------------_-___----- 17

23. Concentration distribution, run 9, suites A

and B______-_-_--_-__-_-_____-_- 18

24. Concentration distribution, run 10, suites A

and B________________________ 18

25. Concentration distribution, run 11, suites A,

B, and C________________-_-__-----_ 19

26. Concentration distribution, run 12, suites A,

B, and C- .__ _- ------ 19

27. Concentration distribution, run 13, suites A

and B_____-___-__--_______-----__--_- 20

28. Graph of dispersion versus distance in

hypothetical aquifer__________________ 24

29. Graph of a* plotted against H*, theoretical

and experimental ________-____---__- 25

TABLE

TABLE 1. Summary of experimental results.--_________ C-21
SYMBOLS

a 

C

Co

d 

D

f

H 

H* 

L 

P

P

half-diameter of idealized pore space

concentration of tracer elements

initial concentration of tracer elements

characteristic grain diameter

coefficient of bulk dispersion of tracer elements

coefficient of molecular diffusivity of tracer

elements in host fluid 

lithologic factor accounting for the effect of a

porous medium on the diffusion of tracer

elements in a fluid that saturates the medium, 

dimensionless number, H=vpd/DM 

dimensionless number, H*=(a/2)2H 

length of porous medium 

permeability of porous medium, or porosity of

porous medium 

probability that a tracer element will move from

one flow path to an adjacent flow path in an

idealized pore space 

q=l p 

time

bulk velocity of host fluid 

average pore velocity of host fluid; vp=vb/P

x spatial flow coordinate parallel to direction of

bulk flow 

y spatial flow coordinate transverse to direction

of bulk flow, and direction in which dispersion

was measured 

z spatial flow coordinate transverse to direction

of bulk flow, and direction wherein dC/dz=Q

and dvt,/dz=0 

at dimensionless constant related to distance

between idealized flow paths 

otg dimensionless constant related to distance

between idealized pore spaces_ 

a dimensiom^ss constant; a=a1 /A/a2 

X dimensionless number; \=DMt/a2 

H mean or expected position of tracer element on

y axis

v kinematic viscosity of water 

a standard deviation of dispersion of tracer

elements along y axis 

<r* dimensionless measure of dispersion along y

axis <r* = <r

GLOSSARY

Diffusion: The spreading of particles, molecules, atoms, 

or ions into a vacuum, a fluid, or a fluid-filled porous 

medium, in a direction tending to equalize concentrations in all parts of a system; it is understood to 

occur as a result of the thermal-kinetic energy of the 

particles, including that of the particles of the host 

fluid, but in the absence of fluid convection.

Dispersion: Same as above, but in the presence of 

fluid convection. (In many cases fluid convection 

may be so important that diffusion is negligible 

by comparison).

Diffvsivity: The property or properties of matter that 

result in diffusion.

Coefficient of molecular diffusion (or molecular diffusivity): A measure of the property or properties of 

a given system causing diffusion, having the dimensions of length squared divided by time.

Coefficient of dispersion (or bulk dispersion, or eddy 

diffusion): A measure of the property or properties 

of a given system causing dispersion, having the 

dimensions of length squared divided by time.
FLUID MOVEMENT IN EARTH MATERIALS

TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA

BY EUGENE S. SIMPSON

ABSTRACT

It is assumed that a line source of tracer elements exists in a 

porous medium through which an incompressible fluid is flowing. 

Each tracer element is assumed (a) to be completely miscible 

in the fluid, (b) to be unaffected by the solid medium or by any 

other tracer element, and (c) to have a path dependent both on 

the motion of the fluid and on its own molecular diffusivity. As 

the tracer elements move away from the source they will disperse. 

A mathematical model, describing the dispersion transverse to 

the direction of bulk flow, is formulated under the following 

assumptions: (a) each path has an average direction parallel to 

the direction of bulk flow, herein termed "zero circuity"; and (b) 

tracer elements may transfer from one flow path to another by 

virtue of the molecular diffusivity of the elements. It is then 

found that the standard deviation of the dispersion is proportional 

to the square root of the product of grain size, length of flow, and 

probability that a tracer element will transfer from one flow line 

to another in a given unit distance. An estimate of the probability is obtained by equating it to that fraction of a large number 

of tracer elements that would diffuse across the centerline of an 

idealized pore space. The fraction will depend on (a) size of 

pore space; (b) velocity of fluid; and (c) coefficient of diffusivity 

of tracer elements.

A curve is plotted giving the relation between a dimensionless 

parameter, tr*, which is a measure of the dispersion in a given 

medium, and a dimensionless parameter, H*, which is a measure 

of fluid velocity and tracer-element diffusivity in the same 

medium. It is predicted that, within the range of Darcy's law, 

as velocity increases dispersion decreases.

A series of laboratory experiments designed to measure 

transverse dispersion in a relatively uniform medium, and performed in the range of Reynolds number from 0.04 (approx) 

to 1.0 (approx), showed that as velocity increased dispersion 

decreased. The measured standard deviation of the dispersion 

ranged from 1.57 cm to 0.89 cm (with some scatter) in a bulkflow distance of 119 cm, and the individual concentration 

distributions were found to be normal except near the end points.

Experimental data were reduced to dimensionless form by the 

assigning of an arbitrary but reasonable value to a dimensionless 

constant, and in this way the measured dispersion is found to 

be roughly double that predicted by theory. By analysis of 

other possible mechanisms of dispersion one concludes that the 

observed dispersion is best explained by assuming that a velocitydependent dispersion (as predicted by theory) is superposed on 

a constant dispersion, independent of velocity, which results 

from wandering of stream paths (that is, nonzero circuity).

Finally, it is shown that at very low fluid velocity (R<0.005 

for the experimental porous medium) the effect of molecular

diffusivity of the tracer elements overshadows the effects of 

other possible mechanisms causing dispersion. However, it is 

believed that dispersion caused by nonzero circuity increases 

with nonuniformity of the medium. The experimental medium 

was relatively uniform; hence, the relation between nonzero 

circuity and nonuniformity of the medium (as it occurs in most 

natural aquifers) remains to be investigated.

INTRODUCTION

Recent theoretical and experimental work on the 

mechanics of dispersion in liquid flow through porous 

media (Rifai and others 1956, p. 100) recognizes that 

the molecular diffusivity of tracer ions or of tracer 

particles does have an effect on dispersion. The importance of this effect, however, was considered to be 

relatively slight except at very low fluid velocity and 

generally was ignored in the analysis of experimental 

data. The data so treated, almost without exception, 

pertained to longitudinal dispersion (dispersion parallel to the direction of bulk flow), and the experiments 

were performed in tubular columns of sand or beads. 

In this kind of dispersion the effect of molecular diffusivity undoubtedly is small, but the flow conditions, 

particularly in nature, under which it is small enough 

to be neglected remain to be identified.

In the case of transverse dispersion (dispersion perpendicular to the direction of bulk flow) different 

mechanisms are'mvolved, and it is not at all permissible 

to ignore molecular diffusivity. In the longitudinal 

case the mechanisms related to dispersion are (1) fluidvelocity gradients within the individual pore spaces,

(2) variations in length of individual flow paths, and

(3) molecular diffusivity of tracer ions. In the transverse case the mechanisms seem to be (1) the intertwining or "wandering" of flow paths and (2) the 

molecular diffusivity of tracer ions. Within the realm 

of Darcy's law the intertwining-paths mechanism is 

independent of velocity and may be treated analytically 

by use of a random-walk model or some modification 

thereof (Scheidegger, 1958). The molecular-diffusivity 

mechanism, however, is dependent on flow velocity.

C-l
C-2 FLUID MOVEMENT IN EARTH MATERIALS

Most analyses of transverse dispersion that have 

appeared in the literature indicate that the flow-path 

mechanism is thought much more important than 

molecular diffusivity, particularly in the upper range 

of flow velocities that accord with Darcy's law. Beran, 1 

for example, gives a rough estimate, based on mathematical analysis alone, of the range in which diffusivity 

may be neglected entirely. On the other hand, C. V. 

Theis of the U.S. Geological Survey, in various discussions and in unpublished memoranda, has proposed an 

opposite view; that is, that for uniform media diffusivity 

is the more important mechanism and that intertwining 

of flow paths either does not exist or is so minor that 

it may be neglected. Theis points out that the distances separating individual flow paths within pore 

spaces are so small that there is ample time available, 

even at velocities in the upper range of Darcy's law, 

for tracer ions to diffuse from one flow path to another.

For this study I devised a mathematical model for 

transverse dispersion based on the mechanism of molecular diffusivity, and tested the validity of the mathematical analysis by experiments designed for that 

purpose. So far as I know, these laboratory experiments are the first of their kind, except for those performed by Kitagawa (1934) in Japan. Kitagawa used 

a salt solution as tracer in a horizontal bed of sand and 

made no allowance for possible density effects. He also 

averaged the results of 15 individual experimental 

runs, each having significant differences from the others 

in distribution of concentration and in position of 

mean value. He then used the average distribution 

to compute the standard deviation of the dispersion. 

Consequently, his measurement was more than 20 

times larger than mine. No other laboratory data 

seem to be available with which results might be 

compared.

ACKNOWLEDGMENTS

The theoretical investigation and the experiments 

described in this report were carried out at Columbia 

University in New York, 1957-58, under the sponsorship of the U.S. Geological Survey, at a time when I 

was a graduate student in the Department of Geology. 

Prof. Richard Skalak, Sidney Paige, and A. N. Strahler 

acted as advisors. Throughout the course of the work 

these men gave freely of their time for discussion and 

counsel. In addition, I had the advantage of many 

helpful discussions with various colleagues in the Geological Survey, in particular with C. V. Theis, R. R. 

Bennett, Wallace de Laguna, and H. E. Skibitzke. 

De Laguna also suggested the tracer dye. used in the 

experiments. S. K. Aditya of the Chemical Engineer1 Beran, M. J., 1955, Dispersion of soluble matter in slowly moving fluids: doctoral 

dissertation, Harvard Univ., p. 4:12.

ing Department of the university kindly permitted the 

use of his laboratory for the measurement of molecular 

diffusivity of the tracer solution and helped in other 

ways.

THEORETICAL ANALYSIS

An analysis is made of the transverse dispersion from 

a continuous line source of tracer elements dissolved 

or suspended in a host fluid that flows through a porous 

medium. The length and breadth of the medium are 

large compared to its thickness (fig. 1). For this 

purpose a tracer element is defined as an identifiable 

ion, or molecule, or colloidal particle, dissolved or suspended in the host fluid; a fluid element is defined as a 

"point volume" of the host fluid whose diameter is 

small compared to the pore diameter of the porous 

medium, but large compared to molecular dimensions. 

The line source is considered to extend along the z axis 

across the full thickness of the medium, and the tracer 

elements enter the host fluid in such manner as to 

cause no change in its velocity field. The direction of 

bulk flow of the host fluid is parallel to the positive 

direction of the x axis. There is no bulk flow in either 

the y or the z direction, because the fluid is incompressible and the medium is homogeneous and of constant 

thickness, and there is no flow across the boundaries 

of the medium except in the x direction. It is assumed 

that motion of any tracer element in the z direction 

will be compensated for by an equal and opposite 

motion of some other element; hence, there is no change 

in concentration in the z direction. It is assumed also 

that steady-state motion of both host fluid and tracer 

elements is established. Under these conditions the 

concentration distribution of the tracer elements may 

be discribed in terms of the space coordinates x and y. 

It is the present purpose to describe transverse dispersion only that is, dispersion in the y direction, in terms 

of distance from origin, fluid velocity, diffusivity of 

tracer elements, grain diameter and porosity of medium, 

and certain dimensionless numbers related to the 

geometry of the medium.

It is useful now to define conceptually two characteristics of a flow path: tortuosity and circuity. Tortuosity is a measure of the wiggles in a flow path and is 

the difference between the real length of flow path and 

the straight-line length per unit length of bulk flow. 

Circuity is a measure of the difference between the 

average direction of an individual flow path and the 

direction of bulk flow (see fig. 2). Both tortuosity 

and circuity are measured over distances that are 

large compared to grain size.

In a disordered porous medium, such as may be 

obtained with randomly packed, irregularly shaped 

grains of sand, only cases C and D of figure 2 are con-
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-3

Line source of tracer element

^i

FIGURE 1. Diagram of porous medium showing line source and coordinate axes; A, Cross-sectional view, B, Plane view. 

Porous medium is assumed infinite in y direction (width), semiinfinite in x direction (length), and finite in ? direction 

(thickness). Line source of tracer elements is coincident with z axis across full thickness of medium. Bulk flow of host 

fluid is in positive direction of x axis.

sidered possible; that is, tortuosity is always greater 

than zero, but circuity may either be everywhere zero 

or everywhere greater than zero. When one assumes 

case C, one says that, as the length of path increases, 

the tortuosity of each path approaches the same finite 

value greater than zero, and the circuity of each path 

approaches zero. If one assumes case D, one says that, 

as the length of path increases, the tortuosity of each 

path approaches the same finite value, but the circuity 

of each path is in general greater than zero and randomly changes with total path length. Although the 

latter case, which implies that stream paths intertwine, 

is briefly treated in a later section, this report mainly 

considers case C.

The motion of tracer elements will depend on the 

motion of the host fluid in which the tracer is carried 

and on the molecular diffusivity of tracer elements 

within the host fluid. By definition, when circuity is 

zero, there can be no transverse dispersion due to motion 

of host fluid. It will be shown that the molecular
C-4 FLUID MOVEMENT IN EARTH MATERIALS

FIGURE 2. Diagram illustrating concepts of tortuosity and circuity. Solid lines 

indicate flow (not necessarily in plane of flgure) of fluid elements; dashed lines 

indicate direction of bulk flow (in plane of flgure). A, Tortuosity=0; circuity=0. 

P, Tortuosity=0; circuity=>0. C, Tortuosity>0; circuity=0. D, Tortuosity 

>0; circuity>0.

diffusivity of tracer elements, when combined with 

the tortuosity of fluid stream paths, provides a mechanism for transverse dispersion. The fluid paths 

coalesce and diverge repeatedly. In a pore space 

where two paths coalesce, a very small motion of a 

tracer element, which is due to molecular diffusivity, 

will carry the element from one path into the other. 

The paths then diverge and coalesce again with other 

paths and the same opportunity for transfer occurs. 

In this way the relatively small total distance traveled 

by a tracer element because of molecular diffusivity

< I Arbitrary flow path

.Average position

of flow path

\ 

x

FIGUBE 3. Diagram illustrating set of idealized flow paths.

is amplified by the nature of fluid movement in the 

medium through which it is being carried. The 

analysis that follows seeks to put this line of reasoning 

in mathematical terms.

It is necessary to idealize flow conditions in a manner 

that will permit mathematical treatment and at the 

same time be related to measurable flow and media 

variables. The first idealization will be to assume that 

all possible positions of the tracer elements are restricted to an arbitrary set of flow paths. The flowpath set is arranged in an orderly manner as shown on 

figure 3. The spacing between successive paths in the 

y direction is determined by a characteristic grain 

diameter, d, multiplied by a dimensionless coefficient, 

ax , which depends on gram shape and packing and on 

porosity in other words, on the geometry of the 

medium. The spacing between paths in the z direction 

is of no consequence because there is no change in 

concentration hi that direction. It is assumed also 

that the motion of each tracer element, which is due 

to molecular diffusivity, is independent of the motion 

of every other tracer element and that for each unit 

distance a2d (fig. 4) each tracer element will have a 

probability, p, of moving to an adjoining flow path, 

and a probability, g, of staying in the same flow path, 

where «2 is a dimensionless constant also depending on 

the geometry of the medium, and where p + 2 = 1. In 

effect, each tracer element will experience a succession

FIGURE 4. Diagram illustrating motion of tracer elements between adjacent flow 

paths. A, Relation between flow paths and grains of idealized medium. B, In 

bulk-flow distance aid each tracer element has probability p of moving to an 

adjacent flow path and probability g of staying in same path.
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-5

Arbitrary flow path

Pore space-?-

\ /

FIGTJKE 5. Diagram of idealized porous medium.

of Bernoulli trials (Feller 1950, pp. 104-110), with the 

probability p for "success" and g for "failure" in each 

trial.

In order to preserve symmetry, it will be necessary 

to modify the idealized flow pattern shown on figures 

3, 4, and 5. According to the figures, if the first trial 

of the tracer element permits a transfer to the left, 

then the next trial permits only a transfer to the right, 

and the next left, and so on, and two different distributions would result, depending on which way the first 

trial went. If, however, it is assumed that the direction at each trial is uncertain and is independent of the 

preceding trial, then symmetry is preserved and only 

one solution is possible. Since in a real three-dimensional porous medium the orientation of the grains and 

the tortuosity of the flow paths are random and not 

regular, this departure from the idealized model probably is toward real conditions. Under this assumption, 

at each trial there is a probability of %p to move to the 

right (positive y direction), a probability of %p to move 

to the left (negative y direction), and a probability of q 

to remain in the original path.

If the zero point of the y axis (the intersection of the 

x and y axes) is defined by the flow path initially occcupied by the tracer element, then the y ordinate of the 

mean, or expected, or "average," position of the element for each trial is (Feller, 1950, p. 180):

=0,

and in n trials

The variance, <r2, of each trial is

and in n trials

(la)

(Ib)

(2a)

(2b)

The number, n, of trials must equal the total length, 

L, of bulk flow in the x direction, divided by the 

distance between successive trials (see fig. 5):

(3)

Substituting equation 3 in 2b and dropping the subscript, we get

a2

Let c?=-^-} then

The standard deviation, <r, is

(4a)

(4b)

(4c)

It is now necessary to obtain an estimate of the 

probability, p, under various flow conditions. For this 

purpose, flow conditions are further idealized by our 

assuming a rectangular pore space of width ^d and 

length a2d. (See figs 5 and 6.; Thus in figure 5, the 

idealization consists of (1) a set of arbitrary flow paths 

along which, alone, tracer elements may move at a 

velocity equal to average pore velocity, and (2) rectangular pore spaces in which tracer elements are 

assumed to transfer from one flow path to another. 

Within the context of the idealization we may regard 

the probability with which an individual tracer element 

will transfer within time, t, in a given pore space, to 

equal the fraction of a large number of elements that 

would transfer because of molecular diffusivity within 

the same time and in the same space. This type of

618550O-62 2
C-6 FLUID MOVEMENT IN EARTH MATERIALS

mass transfer one-dimensional molecular diffusion in 

an isotropic medium is described by the equation

where C= concentration of tracer elements, and 

DM= coefficient of molecular diffusivity.

Particular solutions of equation 5 depend on choice 

of boundary conditions. For the present purpose it 

is sufficient to assume the simplest possible. The 

boundary is defined as a rectangle of width aid and 

length a2d; at time t=G the tracer elements are assumed 

to occupy one-half this space in concentration C0 

(fig. 6) the time available for molecular diffusion of 

the elements is assumed to equal the length of the 

pore space divided by average pore velocity.

It will be noted that, by this method of getting an 

estimate of p, the tracer elements are not restricted to 

the arbitrary flow paths, but by virtue of molecular 

diffusivity they may occupy any position within the 

idealized pore rectangle. This is a departure from the 

original idealization but is closer to real conditions. 

However, the elements cannot move out of the pore

2a

 *" y

FIGURE 6 Diagram of idealized pore space. A uniform concentration Co of tracer 

elements is assumed to occupy the region to the left of centerline at time=<o.

rectangle except by motion of the host fluid. The 

limit of validity of this latter assumption is considered 

under "Discussion."

With the aforementioned boundary conditions, two 

equivalent solutions of equation 5 may be obtained:

n-l (-i)-r mry cos-^ exp

and

(7)

Equation 6 was obtained by the method of separation 

of variables (see, for example, Miller, 1953, p. 93-108) 

and equation 7 was taken from Crank (1956, p. 15). 

By definition p=l & where g is both (1) the probability that a given tracer element remains in the same 

flow path in going through a pore rectangle and (2) the 

fraction of tracer elements that remain in the same 

semipore space after the diffusion of tracer elements 

occurs. By the latter definition

2=77T (8)

By substituting equations 6 and 7 in 8 and integrating 

each with respect to y over the distance y=a,v?e obtain 

two equivalent solutions for g, where g will be a function 

of the semipore diameter, a, the time, t, and the coefficient of diffusivity of tracer elements, Dm . As will be 

seen, one solution is convenient to use for small values 

of t and the other for large.

First substituting equation 6 in 8, and-integrating 

term by term, we get (see app., p. C-29):

-; exD I ^ I Z?M<

+4 ±1 2 7T OT = 1,3,5.

exp-ArX (9a)

where

,

A=

DMt

or

(9b)

Equation 9b converges satisfactorily for moderate to 

large values of t. To obtain a solution that will con-
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-7

verge satisfactorily for small values of t, we use 7 in 

conjunction with 8 and obtain the following (see app., 

p. C-29).

- C (

. |(exp

+

(10a)

where X= ^- as before. a2

To a very close approximation, when XfSO.Ol, equation 

lOa simplifies to

and

By figure 5

2=1 »/-= 1-0.5642 Vx>

V*

(lOb) 

(lOc)

(Ha)

where VP = average pore velocity 

v6 = average bulk velocity

= PvP, where P = porosity of medium.

By figure 6

(lib)

and

X=DMt

where

a2 2

or

a2= as in equation 4,

<X2

=i=( £ ) H

where

(12b)

(12c)

2 To my knowledge, the dimensionless number H was first explicitly used in flow 

through porous media by Beran (see footnote 1, p. C-2), although Taylor (1953, p. 190) 

used it implicitl y in his treatment of dispersion in flow through tubes. It is analogous 

to the Peclet number (Pai, 1956, p. 98) In which thermal diffusivity replaces molecular 

diffusivity. The H number is derivable also by dimensional analysis.

By equation 4c

and by 9b and lOc p is a known function of X. Hence, 

a curve can be plotted giving the relation between the 

dimensionless parameter (T/o/-\fdL} which is a measure 

of dispersion in a given medium, and the dimensionless

parameter ( ~ YH, which is a measure of fluid velocity

and tracer-element diffusivity through the same 

medium. The relation between q and 1/X as computed 

by means of 9a and lOb is shown on figure 7. Finally, 

figure 8 shows the relation between 1/X and a

DESCRIPTION OF APPARATUS AND EXPERIMENTAL 

PROCEDURE

The name given to the principal apparatus herein 

described is "dispersometer." Additional apparatus 

also was built to supply de-aired water. For purposes 

of description the dispersometer may be separated into 

three parts: (A) reservoirs and overflow (constanthead) tanks to provide water and tracer solution at 

constant head, (B) porous block (and associated flow 

controls) through which the fluid moves and in which 

the dispersion takes place, and (C) connections for 

taking samples and measuring flow gradiant and discharge. The apparatus is shown in the following 

illustrations: figure 9 is a diagram showing the arrangement of the various parts; figure 10 is a drawing of the 

porous block; figure 11 is a diagram of the apparatus 

to supply de-aired water; and figure 12 is a photograph 

of the assembled apparatus.

Measurements of dispersion were made at various 

flow velocities. Each dispersion measurement made 

at a given flow velocity and involving all the following 

steps is referred to as a "run." In the tabulation of 

data (table 1, next section) runs are consecutively 

numbered.

Measuring dispersions, or runs

1. The two overhead reservoirs were filled with de-aired water 

and allowed to stand for at least 24 hours to permit equalization of temperatures.

2. Indigo carmine dye, used as a tracer, was dissolved in one 

reservoir in the proportion of 50 parts per million (ppm) for 

the first 5 runs and 20 ppm for the remaining 8 runs.

3. The constant-head overflow tanks were adjusted to provide 

the desired head. The clamps controlling discharge from 

the overhead reservoirs into the overflow tanks were opened 

and then clamps between overflow tanks and porous block 

were opened.

4. Discharge was maintained through the porous block until 

steady-state conditions were attained with respect to 

dispersion. The time necessary for this was estimated by 

calculation but determined by visual inspection of the dye 

front as it moved through the block. The block was
C-8 FLUID MOVEMENT IN EARTH MATERIALS

1.0 I

0.8

0.6

0.4

0.2

o.i 0.5 1.0 5.0 10 50 100 500 1000 5000 10,000

FIGURE 7. Graph of probability q plotted against -

1.0

0.8

OT6

0.4

0.2

0.1 0.5 1.0 5.0 10 50 100 500 1000

FIGURE 8. Graph of - -<JdL plotted against ;: 

5000 10,000
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-9

>»

Q. 

Q.

3

OT

k_

1i

1

?

£

11

^ ^

Overhead

s

4 : )

reservoir, Overhea

water plus dye tracer v

7?/////////////

i

"8

0)

i

i 

 * 

8

£

* 

^

x/1 /1/

,

^^^^B ^^^^H

/////S/////////S //////////////s

1 I

 x

''

y^ >.

/ /// /7/y/

L funnels

To waste To w

_ r>ie-^KorrTA erwMitc

water

////////////x

/////> / 

To wa

Flow-d 

cha

'SSS

^-^

ste 

:5

"Adjusta 

ft

Discha 

(not

tubes/ Manometer /

!/ 

\

/

stribution^. 

mbers

s+s* tfy

f:::f.

;: ;'.> ;

'.' ' ".

[v.v/:

;£y;';

 r-'-v-:

1

'Constant-head 

tank

/

 >< 

FIGURE 9. Flow diagram of dispersometer.
C-10 FLUID MOVEMENT IN EARTH MATERIALS

. Holes connected to tubuldr 

rubber discharge spouts

Upper flow-separation plates, 23 in all

Note: Lucite frame not shown

Water in

Water plus dye tracer in

FIGURE 10. Diagram of porous block.
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-ll

To waste

Low-pressure steam in

Copper tubing 50 ft by Vz inch

T i I f f f f i//r/rf/i/tf / f f r / f ft f //

il 111 f

To water reservoirs

FIGURE 11. Diagram of apparatus to supply de-aired water.

mounted vertically; flow entered at the bottom and discharged at the top through a series of horizontal spouts.

5. After steady-state dispersion was established, the rack of 

test tubes was inserted under the discharge spouts to obtain 

a sample suite from all spouts simultaneously. Either 2 or 

3 such suites were sampled per run, at intervals of 1 to 2 

hours.

6. Total rate of discharge of fluid was measured; it together 

with manometer readings of flow gradient and known 

dimensions of the porous block, permitted calculation of 

permeability and bulk-flow velocity. The rate was measured several times during each run. Temperature of influent and effluent fluid also was measured.

7. Relative concentration of dye in each test tube was determined colorimetrically.

The rate of discharge from each spout could be 

regulated by raising or lowering the high point of a 

rubber discharge tube. Each discharge tube had a hole 

cut at its high point to break suction. At the beginning 

of each run the tubes were adjusted as required to make 

the discharge through each proportional to the cross-sectional area of the porous block that the tube drained 

(tubes near the centerline of the block were on 0.75 cm 

centers, whereas tubes farther out were on wider centers; 

see fig. 10). For runs at relatively high velocities this

was easily accomplished, but for runs at low velocities 

precise adjustment was not possible, presumably because surface-tension forces within the tubes were large 

enough compared with other forces to affect discharge. 

The possible effect on dispersion of this non-uniformity 

of discharge will be discussed later.

For each run 2 or 3 suites of samples were obtained 

in 50-ml test tubes. Each suite contained 24 samples. 

It was early determined that only the nearest 4 or 5 

samples on each side of the interface between the dye 

solution and the water needed measurement. The concentration change, if any, in the others could not be 

detected. The procedure in measuring the distribution 

of concentration in each suite of samples was as follows:

Measuring distribution of concentration

1. The sample on the dye side farthest from the interface was 

considered to contain the original dye concentration C= C0 . 

(The concentration of this sample and that of the overhead 

reservoir should, perhaps, have been identical but there 

usually was a very small, unexplained, difference.) Similarly, the sample on the water side farthest from the interface was considered to contain the original concentration 

C=0. By successive dilutions of one in the other, and
C-12 FLUID MOVEMENT EST EARTH MATERIALS

-, '*v So m pie. I 

rock

FIGUHE 12. Photograph of dispersometer, front view.

measuring each in the colorimeter, a rating curve of 

relative concentration C/C0 plotted against colorimeter 

reading was obtained.

2. Colorimeter readings were then taken of the samples of unknown concentration and converted into relative concentration by means of the rating curve.

3. The relative concentrations of the samples in each suite were 

plotted on arithmetic probability paper, and the value of 

the standard deviation of the dispersion was obtained 

graphically (figs. 14-26).

Measuring the concentration of dilute dye solutions 

with a colorimeter turned out to be more time consuming than had been expected. So-called precision pyrex 

tubes had to be carefully matched and positioned for 

each suite of measurements, and frequent checks against 

the standard solution had to be made to avoid errors 

due to drift of the instrument.

The coefficient of thermal diffusivity of the indigo 

carmine dye was measured by following the procedure 

described by McBain and Liu (1931). A dye solution 

of 20 ppm by weight in distilled water was placed in 

the upper cell of the diffusion apparatus, and distilled 

water was placed in the lower cell. The two cells were 

separated by a sintered-glass membrane.

After 3 days 19 hours at 20 °C the liquids were 

withdrawn and the relative concentration in each cell 

was measured. A calculation based on the concentration in the lower cell gave a coefficient DM=Q.57 

X10~5 cm2/sec, and a calculation based on the concentration in the upper cell gave DM=Q.67 X 10~s cm2/sec. 

For purposes of calculation in this report the coefficient 

will be taken as DM=O.G2 x 10~5 cm2/sec.

A photospectrogram of the dye solution (20 ppm by 

weight) was obtained and is shown in figure 13.

1.5

i.o

 0.5

2000 4000 

WAVELENGTH, A

6000

FIGUBE 13. Photospectrograph of tracer solution. Tracer solution consisted of 

indigo carmine dye in de-aired water, 20 ppm by weight.

The porous block deserves special mention. It was 

made by mixing an epoxy resin with crushed quartz. 

The quartz grams were angular, with sharp edges, 

and some were platy or rod shaped. Sieving for 10 

minutes in a ro-tap machine resulted in the size distribution shown on figure 14. As indicated by this 

figure, the sieve diameter of about 95 percent of the 

grains was between 0.40 mm and 0.60 mm, and by 

inspection 0.48 mm is taken to be average. For purposes of calculation in this report, a characteristic 

grain diameter of d=OA8 mm will be used.

The resin-sand mix was prepared in batches in the 

proportion of 2 parts resin plus hardener to 98 parts 

crushed quartz by weight. First the resin and hardener 

were mixed together in a container; then the quartz 

was added slowly and thoroughly stirred into the resin. 

The finished mixture had the appearance and feel of 

a damp beach sand. The batch then was poured into 

the plexiglass frame and vibrated into place to get 

maximum density and uniformity. The vibrator con-
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-13

100

80

ICJ I-Z

az

60

20

0.03 0.04 0.05 

SIEVE OPENING, IN CENTIMETERS

0.06 0.07

FIGURE 14. Mechanical analysis of sand used in porous block.

sisted of a tamp-rod attached to a commercial Vibrotool. Unfortunately the batch method resulted in 

horizontal zones of contact that could be visually 

identified. The porosity of the finished block was 

computed to be 0.37, on the basis of measured volume 

of the plexiglass container and computed volume 

(based on measured weight and specific gravity) of 

the sand and resin.

The advantages of a porous block cast in a plexiglass 

frame over a medium of nonbonded grains are that (1) 

permeability of the block is constant or nearly so, 

whereas permeability of nonbonded grains tends to 

decrease with time and with use, presumably because 

grains continue to settle; (2) the porous block may be 

drilled at any place for manometer connections and 

other needs, whereas for nonbonded grains special' 

connections must be machined into the container at 

the time of fabrication; and (3) a wire-mesh support 

for the porous block is unnecessary. The chief disadvantage of a porous block (and this is not an inherent 

feature) appears to be the "bedding planes" where one 

batch is in contact with the next. However, these 

planes are in themselves quite uniform and are at 

right angles to the flow direction and should not affect 

the validity of the dispersion measurements.

De-aired water was obtained in the following manner 

(fig. 11): (1) low-pressure steam from a vent supply 

in the laboratory was directed through a 50-foot coil 

of K-inch copper tubing; (2) the tubing was immersed 

in a bath of cold water which caused the steam to condense and cool; (3) the resulting water contained air 

which showed up in large bubbles; (4) the water plus 

air-bubble mixture was fed into the midsection of a 

standpipe; the standpipe was open at the top and 

virtually all the air with some water rose to the top of 

the pipe and overflowed to waste; (5) the remainder 

of the water was withdrawn from the base of the pipe 

and discharged into the dispersometer reservoirs. 

Samples of the de-aired water were examined for air 

content by slow heating in a closed vessel provided with 

an expansion tube. Tiny air bubbles began to appear 

only after water temperature exceeded 90°C, and these 

bubbles redissolved when the sample was cooled. 

Apparently, a little air was dissolved by the condensed 

steam in the time required for it to flow through the 

coil and into the standpipe. The de-aired water 

system was operated as required to fill the overhead 

reservoirs before each run.

EXPERIMENTAL RESULTS

Thirteen runs were made in accordance with the 

procedures outlined in the previous section. The 

distribution of concentration for the several suites of 

each of the 13 runs is plotted on arithmetic probability 

paper, figures 15 through 27. When so plotted the 

points of each suite fall on a straight line, except for 

the extreme end points. Hence, to a close approximation the distribution of concentration in each suite 

is normal. That is, the concentration distribution in 

each case is defined by the following equation

where, <r= standard deviation of distribution,

m=distance to mean value; that is, where C/C0= %, 

y=distance to value C/C0.

If, in each case, the origin of coordinates is placed 

at the mean value, then m=0 and equation 14 becomes

618550 O-62 3
C-14 FLUID MOVEMENT IN EARTH MATERIALS

99.99 99.9 90 80 70 60 50 40 30 20 

C/C0 , IN PERCENT

10 0.1 0.01

FIGURE 15. Concentration distribution, run 1, suites A, B, and C.

99.99 99.9 99 90 80 70 60 50 40 30 20 

C/C0 , IN PERCENT

10 0.01

FIGURE 16. Concentration distribution, run 2, suites A, B, and C.
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-15

17

16

15

14

13

12

10

99.99 99.9 99 90 80 70 60 50 40 30 20 

, IN PERCENT

10 0.1 0.01

FIGUEE 17. Concentration distribution, run 3, suites A, B, and C.

20

19

18

B

16

15

A

Xl

14

13

 ~^~

12

99.99 99.9 99 90 80 70 60 50 40 30 20 10 

C/C0 , IN PERCENT

FIGURE 18. Concentration distribution, run 4, suite^ A, and B.

0.1 0.01
C-16 FLUID MOVEMENT IN EARTH MATERIALS

19

18

17

16

B

15

14

13

12

99.99 99.9 99 90 80 70 60 50 40 30 20 

C/C0 , IN PERCENT

10 0.1 0.01

FIGTJKE 19. Concentration distribution, run 5, suites A and B.

16

15

14

13

10

99.99 99.9 99 90 80 70 60 50 40 30 

C/Ca , IN PERCENT

20 10 0.1 0.01

FIGTJKE 20. Concentration distribution, run 6, suites A and B.
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-17

12

99.99 99.9 90 80 70 60 50 40 30 20 

CIC0 , IN PERCENT

10 0.1 0.01

FIGURE 21. Concentration distribution, run 7, suites A and P.

18

17

16

15

\

14

13

\

12

11

10

\

99.99 99.9 99 90 80 70 60 50 40 30 20 10 

C/C0 , IN PERCENT

0.1 0.01

FIGURE 22. Concentration distribution, run 8, suites A, B, and C.
C-18 FLUID MOVEMENT IN EARTH MATERIALS

18

17

16

15

14

13

N

12

11

99.99 99.9 99 90 80 70 60 50 40 30 20 

CIC0 , IN PERCENT

10

FIGURE 23. Concentration distribution, run 9, suites A and B.

0.1 0.01

17

16

15

14

13

12

11

10

99.99 99.9 99 90 80 70 60 50 40 30 20 

C/CD , IN PERCENT

10 0.1 0.01

FIGUBE 24. Concentration distribution, run 10, suites A and Ji.
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-19

18

17

16

\

14

13

12

99.99 99.9 99 90 80 70 60 50 40 30 20 

C/Co, IN PERCENT

10

FIGURE 25. Concentration distribution, run 11, suites A, fi, and C.

0.1 0.01

17

16

15

« 

S

3

i12

10

99.99 99.9 99 90 80 70 60 50 40 30 20 10 

C/C0 , IN PERCENT

0.1 0.01

FIGURE 26. Concentration distribution, run 12, suites A, fi, and C.
C-20 FLUID MOVEMENT IN EARTH MATERIALS

15

14

13

12

10

99.99 99.9 99 90 80 70 60 50 40 30 20 

C/C0 ,IN PERCENT

10 0.1 0.01

FIGURE 27. Concentration distribution, run 13, suites A and B.

According to the geometry of the experiment (fig. 10) 

the mean value should occur midway between spouts 

12 and 13. Actually, the mean value (the interface 

between dye solution and water) tended to migrate 

sideways slightly during the course of each run. Also, 

the vertical line of the interface as seen through the 

lucite frame was not perfectly straight but was gently 

sinuous. However, the sinuousity was fixed and was 

the same at all velocities. Therefore, it can be presumed to have resulted from a nonuniformity of the 

medium and did not represent an instability in the 

flow field.

The migration of the interface probably resulted 

from a nonuniform, gradually increasing resistance to 

flow at the base of the porous block that occurred 

during the course of each run. This, in turn, probably 

was caused by the presence of fine lintlike sediment 

found suspended in the feed fluids. As the fluids moved 

through the block, this sediment must have become 

trapped in the pores of its base. (The concentration 

of the sediment was low and its presence was not 

noted until the experiments were well under way. No 

remedial measures were taken.) In any case, it was 

found that the position of the interface could be 

changed at will; that is, it could be made to migrate 

laterally simply by varying the entrance-head relationship between water and dye solution. Except for the 

first few runs which were begun with entrance heads 

equal, the heads were adjusted to put the interface 

reasonably close to the centerline of the block. The 

possible effect, if any, on test results of interface 

migration will be discussed farther on.

As was noted, the extreme end points of the concentration distribution curve do not, in general, fall on 

the straight lines of figures 15 and 26. There are 

various possible explanations for this: (1) the colorimetric method of measuring concentration is less 

accurate for the extremes of any given range of concentrations than for the intermediate concentrations 

(the filters, light intensity, etc., were adjusted to provide full-scale deflection of the instrument for the range 

of concentrations measured). (2) According to the 

theory presented in this report, the distribution of the 

concentrations is binomial rather than normal. For 

distributions involving a large number of independent 

events, the normal and binomial distributions are 

practically identical, except that the limits of the 

binomial are finite, whereas those of the normal are 

infinite. On normal probability paper a normal distribution would plot as a straight line, whereas a 

binomial distribution would plot as a straight line 

except for the extreme values, which would tend to
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-21

plot off the line in a direction away from the mean. 

This is seen to be the case for most of the extreme 

values plotted. 

If, in spite of the slight deviations, the distributions 

are treated as normal, it becomes possible to measure 

the standard deviation, a, in each case from the 

straight-line plots. In a normal distribution at +0-, 

<7/<70=0.841, and at a, <7/<?0 =0.159. The concentraTABLE 1. Summary c 

Time Veloc

Run Suite

1 1A

IB

1C

2 2A

2B

2C

3 3A

3B

3C

4 4A

4B

5 5A

5B

6 6A

6B

7 7A

7B

8 8A

8B

8C

9 9A

9B

10 10A

10B

11 11A

11B

11C

12 12A

12B

12C

13 13A

13B

Date

1/22/58

do.

1/24/58

1/30/58

do.

1/31/58

2/3/58

do.

do.

2/10/58

do.

2/11/58

do.

2/14/58

do.

2/19/58

do.

2/26/58

do.

8/27/58

3/12/58

do.

3/19/58

do.

3/28/58

do.

do.

4/3/58

do.

do.

4/30/58

do.

Start

of run

?

?

?

12:15P

do.

11:30 A

12:30P

do.

do.

11:30A

do.

10:45A

do.

1:30P

do.

11:40 A

do.

12:OON

do.

1:OOP

12:OON

do.

12:10P

do.

11:OOA

do.

do.

11:15A

do.

do.

2:30P

do.

1 Temperature shown is that of

never more

2 Riilk VP

than a few

Innit.v J)ii=

tenths of a

OIA Pnrf

Suite Temoerasampled ture i

4:OOP

6 :OOP

3:40P

3:30P

5:OOP

2:05P

3:25P

4:50P

7:40P

3:35P

7:10P

1:10P

2:30P

4:10P

6:05P

2:40P

4:15P

2:40P

4:25P

3:25P

2:OOP

3:25P

2:30P

4:05P

3:40P

6:25P

7:15P

3:35P

5:OOP

6:OOP

3:55P

5:18P

influent water;

degree different

> \7plni»it.v n _^=-

(°C)

26

26

27

28

28

27

25

25

25

24

24

24

24

24

24

23

23

28

28

27

29

29

28

28

29

29

29

28

28

28

28

28

tion distributions are plotted against spout numbers. 

Hence, 2o- equals the distance (determined by the spout 

numbers and fractional distances thereof) between 

(7/^=0.841 and <7/<70=0.159. The figures given in 

the final column of table 1 were obtained in this way. 

To evaluate the validity of these figures, it is necessary 

to attempt to identify and evaluate possible sources 

of error.

/ Experimental Results 

ity'. Permeability* Coefficient of ..Standard? (cm per see)

V* vp

0. 0141 0. 0381

.0139

.0139

.0245

.0244

.0238

.0184

.0181

.0176

.00811

. 00731

.0320

.0312

. 01365

.0128

. 0223

.0220

.0183

.0174

. 0164

.0605

.0594

.0324

.0316

. 00383

. 00306

. 00293

. 00387

. 00327

. 00322

. 0859

.0837

effluent water

Vb Vb

.0376

.0376

.0663

.0660

.0644

.0497

.0489

.0476

.0219

.0198

.0865

.0843

. 0369

.0346

.0603

.0595

.0495

.0470

.0443

. 164

. 160

.0875

.0855

. 01035

. 00828

. 00792

. 01045

. 00884

. 00870

. 232

. 226

Gradient *

0. 0655

.0655

.0645

. 114

. 113

. 112

.0900

.0885

.0880

.0427

.0378

. 160

. 157

.0700

. 0662

. 1170

. 1153

.0858

.0812

.0785

. 291

. 286

. 162

. 157

.0185

. 0146

. 0140

.0202

.0172

.0168

. 3940

. 3835

5 Coefficient of

formula

(em per sec) Diffusi vitv n w >

P,

0. 215 0.

. 212

. 216 .

. 215 .

. 216 .

. 212 .

. 204 .

. 204 .

. 200 .

. 190 .

. 193 .

. 200 .

. 199 .

. 195 .

. 193 .

. 191 .

. 191 .

. 213 .

. 214 .

. 209 .

. 208 .

. 208 .

. 200 .

. 201

. 207 .

. 209 .

. 209 .

. 192 .

. 190 .

. 192 .

. 218 .

. 218 .

diffusivity

T DM=7fTD'M where,

1 20

Pjo (em* per sec)

19 0. 63X10~5

18 .63

18 . 63

18 . 64

18 .64

18 .63

18 .63

18 . 63

18 . 63

17 . 63

18 . 63

18 .63

18 . 63

18 . 63

18 . 63

18 .63

18 .63

18 . 64

18 . 64

18 . 63

17 . 64

17 . 64

17 . 64

17 . 64

17 . 64

17 .64

17 .64

16 . 64

16 . 64

16 .64

18 . 64

18 .64

H- Vpd

DM

290

290

290

500

490

490

380

370

360

170

150

660

640

280

260

460

450

370

350

340

1230

1200

660

640

78

62

59

78

66

65

1740

1700

deviation

(cm)

1.04

1.06

1. 09

1.34

1. 28

1.33

1. 20

1. 21

1. 19

1. 14

1. 28

1. 16

1. 14

1. 11

1. 11

1. 14

1. 13

1. 12

1. 14

1. 23

1.01

1.01

1.06

1.-03

1.31

1.42

1.57

1. 24

1. 31

1. 35

.89

.89

adjusted for temperature change by

T is absolute temperature of water

. p

where Q is discharge in cm3 /sec. A is area of cross section = 100 

cm2 .

3 Figure shown is average of gradient measured by manometers 

on each side of porous block.

4 Permeability computed by relationship P=Q/AI. Pt Permeability at temperature of experiment. 

P2o=Permeability at temperature 20°C

__ p Kinematic viscosity of water at temperature t 

1 Kinematic viscosity of water at 20° C

T2o = 293°K ( = 20°C) D'M = coefficient of molecular diffusivity 

at 293°K=0.62X10-5 cm2 /sec.

6 H=-£- is dimensionless parameter discussed in text as equaL>M

tion 12c; d=0.048 cm.

7 Figure shown is standard deviation taken from graphs, 

figs. 15-27.
C-22 FLUID MOVEMENT IN EARTH MATERIALS

At the outset, it should be pointed out that most 

uncontrolled variables, such as the small change with 

time in the permeability of the medium and in the 

hydraulic gradient, or undetected nonuniformities 3 of 

the medium, or adsorption of tracer on the medium, 

either will have no effect on dispersion or will act to 

increase the dispersion over that which would cocur 

in an ideal experiment. In other words, the measured 

dispersion may be thought of as the sum of two parts: 

(1) the dispersion that would occur under ideal experimental conditions, and (2) the dispersion caused by 

experimental conditions that departed from ideal and 

that may not be identifiable or, if identifiable, may not 

be measurable.

The question of uniformity of the medium was 

perhaps the most perplexing, and in attempting to 

assess it, an experiment was performed as follows: 

(1) The block was completely saturated with clear 

water; (2) a dye solution was introduced simultaneously 

on both sides of the centerline along the entire base of 

the block; (3) the progress of the dye front through 

the block was then observed. If the block were 

absolutely uniform, all parts of the dye front would 

travel at the same average velocity and the front would 

appear as a horizontal line moving upward through the 

block. In point of fact, a bulge developed at one side 

and the fluid in that portion of the block moved about 

50 percent faster than the overall average. In a repetition of the experiment, at a different velocity, the 

bulge developed in exactly the same place, showing 

that the flow was stable, but that the block was not 

completely uniform. Also, as was previously pointed 

out, the block was built up in a series of layers so that 

a number of "bedding planes" existed perpendicular 

to the flow direction.

According to the theory developed herein, transverse 

dispersion is velocity-dependent, but a velocity difference of 50 percent in the range studied is not enough 

to produce a measurable change in dispersion. However, if the nonuniformity produced a circuity greater 

than zero (contrary to the original premise), then the 

dispersion could be affected measurably, but independently of velocity. Therefore, the nonuniformity of the 

medium either produced no measurable effect on dispersion or it affected dispersion measurements at all

3 Uniform is here used in the sense that the properties, such as porosity, grain size, 

grain orientation, and permeability, of any small region (containing, say, 10 to 100 

grains) have values nearly identical to those of any other small region, so that (1) the 

standard deviation of any property of a large number of small regions, taken along any 

line through the medium, will be close to zero, and (2) the average of any property 

taken along a line will approach the same value as the average of the same property 

taken along any other parallel line, as the length of the lines increases. A medium is 

also isotropic if the average of any property will approach thf same value regardless of 

direction.

Since absolute uniformity is a mathematical convenience having no real counterpart 

in nature, the term nonuniform used herein merely implies a gross or readily measurable departure from the. conditions described above.

velocities to the same degree. Furthermore, if the 

effect of the nonuniformity on one side of the block 

was substantially different from that on the other side, 

a skewed concentration distribution should result. 

This did not occur.

The 24 spouts through which the fluid was discharged 

were individually adjustable. At relatively high flow 

velocity, it was possible to adjust the spouts so that 

individual departures from average rate of discharge 

did not exceed 15 percent. At low velocity, however, 

individual departures from average were as much as 

100 percent. Since the forces producing these variations were associated with the spouts, the effect on flow 

through the medium would be confined to the immediate 

vicinity of the upper flow-separation plates. At most, 

this type of error would cause a scattering of the points 

on the concentration-distribution curve. As is seen, 

except for the extreme values, all points fit the curves 

rather closely.

The migration of the interface during the course of 

a run would tend to skew the dispersion. In run 13, 

where velocity of flow was maximum of all runs, the 

rate of sideways migration at the spouts was about 

0.02 mm per minute; in run 11, where velocity was 

minimum, the rate of migration was about 0.10 mm 

per minute. The reason for the difference is that, 

when flow velocity is low, small changes in entrance 

resistance (which caused the migration) are large compared to total gradient; when velocity and gradient 

are high, then the changes in entrance resistance are 

almost negligible by comparison. In run 11, the 

average time required for fluid to move through the 

porous block was about 220 minutes. In this interval 

the sideways migration amounted to about 2 cm. In 

run 13 the average time for fluid movement through 

the block was 8.6 minutes; during this interval sideways migration was about 0.02 cm. Hence, it is concluded that (1) at high-flow velocity the maximum 

possible effect of interface migration is less than the 

error of measurement of the standard deviation and may 

be neglected; (2) at low-flow velocity the effect may not 

be neglected on this basis but may be treated as follows: 

To begin with, if the effect is significant, the distribution should be skewed; that is, the dispersion should 

be greater on the side away from which the interface 

migrated. An examination of the plotted points on 

suites 11B and 11C (fig. 25) show this to be the case. 

On the other hand, the dispersion on the side toward 

which the interface is migrating should be compressed. 

This is because the flow lines, which started at the base 

of the block by occupying half the width of the base, 

end by occupying less than half. Therefore, if only 

the latter points are considered, the standard deviation 

based on those points should give a value that is too
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-23

low. In order for one to be on the conservative side, 

insofar as this analysis is concerned, the latter points 

are taken as correct. Also, it should be pointed out 

that the interface migrated by moving parallel to 

itself except in the region immediately above the flowseparation plate at the base of the block. In other 

words, the flow adjustment required by the changing 

head relationship was accomplished close to the base 

of the block. . Also, no tendency to "smear" could be 

seen; that is, near the base the interface was always 

clear and sharp in spite of the slow migration. Farther 

along, of course, the interface became fuzzy independently of migration, as dispersion progressed. It is 

concluded that, in spite of the various departures from 

ideal experimental conditions mentioned above, the 

measured values of dispersion for various flow velocities 

are valid.

To provide some notion of the meaning of these data 

in terms of dilution and dispersion over long distances 

within an aquifer, the following idealized analogy is 

proposed. Assume an aquifer of finite thickness but of 

infinite area, whose physical characteristics are similar 

to those of the experimental medium; assume an injection well that penetrates the full thickness of the aquifer; and assume that a solution containing tracer 

elements in concentration Co is injected into the well. 

As a function of the distance downgradient from the 

injection well, compute the width of the band of tracer 

elements between points where relative concentration 

C/Co=0.05, and compute the change in maximum 

concentration of the tracer elements, under the following 

conditions: (1) ground water velocity is 22.5 ft per day 

(equivalent to pore velocity of run llC), and groundwater velocity is 658 ft per day (equivalent to pore 

velocity of run 13A), and (2) initial width within the 

aquifer of injected band of tracer elements is 1 foot, 

initial width is 2 feet, and initial width is 4 feet. The 

coefficient of bulk dispersion D is calculated for each 

case from values of standard deviation obtained in runs 

11C and 13A, according to the relationship D=<r2 /2t.

The resulting computations are shown graphically on 

figure 28. It is seen, for example, that 10 miles downgradient maximum concentration, which would occur 

along the centerline of the band, ranges from 44 percent 

of initial concentration for the faster velocity and 4-foot 

initial band width to 6.6 percent for the slower velocity 

and 1-foot initial width. The width of the band containing tracer elements at relative concentration of 5 

percent or higher, at 10 miles downgradient, ranges 

from 8.8 feet for the faster velocity and 1-foot initial 

width, to 22 feet for the slower velocity and 4-foot 

initial width. It is also seen that the change in maximum (centerline) concentration is sensitive to initial 

band width, and that the wider the band the longer the

distance of downstream travel before the maximum 

concentration begins to change.

In addition to the cases shown on figure 28, it may 

be calculated that where ground-water velocity is 658 

feet per day, a band of 24 feet initial width would 

travel 10 miles before its centerline concentration would 

be affected; where ground-water velocity is 22.5 feet 

per day, a band of 42 feet initial width would travel 10 

miles before its centerline concentration would be 

affected. In this connection it should be noted that in 

practical cases of fluid injection into permeable formations, the head on the injected fluid is apt to be considerably higher than that of the ground water in the 

formation. Consequently, the initial width of band 

leaving an injection well is apt to be many times the 

diameter of the injection well.

COMPARISON BETWEEN THEORY AND EXPERIMENT

In a previous section (see equation 4c) the following 

was derived

For a given porous medium the denominator of the 

term on the rignt side is constant, and the term as a 

whole is dimensionless. This term may be used to 

transform the measured standard deviation of dispersion, a, into dimensionless form. Or,

r*_. (16a)

where <r* replaces yp f°r convenience in terminology. 

The length L is obtained by direct measurement of 

the porous block. The characteristic grain diameter 

d is, in general, not well defined, but for artificial 

media of nearly constant grain size it may be taken 

as the average grain size (fig. 14). A first approximation of the dimensionless constant a can be made by 

assuming values that seem intuitively reasonable. 

Referring to figures 4 and 5 and recalling that a=a^a2, 

one assumes that

that is,

that is,

aid (average distance between=0.5</; 

adjacent flow paths)

«i = 0.5;

a2d (average distance between=0.7c?; 

successive pore spaces 

along a flow path)

«2 =0.7.
C-24 FLUID MOVEMENT IN EARTH MATERIALS

0.01

DISTANCE DOWNGRADIENT, IN 

0.1

MILES 

1.0

FIGURE 28. Dispersion plotted against distance in imagined aquifer. Computations are based on formula:

where, w= initial width of band

<=time required to reach downgradient point indicated 

y= transverse distance from centeriine of band 

vp ^ground- water velocity 

D = coefficient of bulk dispersion =(r2f2t, /where t= 119 cm/t>/.
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-25

Then,

By measurement Z,= 119 cm and d=.048 cm. These 

values are substituted in 16a.

=0.7<r.

In a previous section a relationship was developed 

between <r* (=Vz>) and (o) H (see fig. 8 and equations 

9b, lOc and 13), where

Let

H*=(^}H (17a)

(I7b)

The last two columns of table 1 give the experimental 

values of <r and H, and <r* and H* are computed according to equations 16b and 17b. It is now possible to 

make a comparison between theory and experiment. 

The theoretical curve previously derived (fig. 8) is replotted on figure 29, and on the same figure values for 

<r* experimental are plotted against H* experimental. 

The scale for Reynolds number is shown (see app., 

p. C-30). It is seen that the values of <r* experimental 

are roughly double those of <r* theoretical.

Perhaps most significant of the experimental results 

is the apparent functional relation between dispersion 

and fluid velocity. As H increases, dispersion tends to 

decrease. By the method of least squares, and the 

assumption that within the range of the measurements 

ff is a linear function of the logarithm of H, one computes 

that

a= 1.3-0.28 log ff/100.

If it is further assumed that, corresponding to each 

value of H, the measured values of a are normally 

distributed, the product moment correlation coefficient 

7 is computed to be

7=-0.74

REYNOLDS NO. =

0.001 0.01 0.1 1.0

1.2 

1.1 

1.0 

0.9 

0.8

0.4 

0.3 

0.2 

0.1 

0

1

\

\

\

r*

\

^ MI

ur

d 

 , b

\

19

tri 

ffi

VP

\

N

of 

jn 

isi 

or

^

 n

"i

re! 

on 

ou

^

iei 

fc 

d 

fi

«i

Stl 

V

s

 V

3T 

W 

ff

ca

l

jsonable fit' 

ricted molec 

nth amplific 

medium

*«.

^^^^^ *«^

\

\

etical curve 

unrestricted 

usion with n 

tion by pore

ular 

ation

v^

N, 

\

calcula 

molec 

oampl 

us met

^

^x

tedular

-

bum

X

X,

0 

c

'>

Sk

I

(

V

-

i*

i

s

x

-.

1

^Experimental curve 

of best fit

"^

.

^X

"-

^.

00

o 

8

^

^^,

»

£ <

^^^^

s

^

*""^

-4

-^ 

c

0

^

^N.

 »

-

'T

*N,

^

^

x fc

i

Note:

0

Points 

experir 

«=0.6 

L = ll 

«/=0.(

calct 

nenti

9cm 

48 ci

leoretical curve calculated 

for restricted molecular 

diffusion with amplification

b

 

 ,

y

 **

^

K

^

^

trousmediu

^^^

"* -

m

^"^

- 

* ~^

jlated from 

al data for

m

 ». ^ ^ ^ ^

0.1 0.5 1.0 5 10 

// /=/$/*//=0.09*}

50 100 500 1000

FIGURE 29. Graph of a* plotted against H*, theoretical and experimental.
C-26 FLUID MOVEMENT IN EARTH MATERIALS

and by Student's t test

£=5.9 (for 30 degrees of freedom).

The 7 and t values indicate that the probability of the 

correlation arising by chance is extremely small.

Converting a and H to a* and H* by equations 16b 

and 17b, the line of best fit is plotted on figure 29. 

It is seen that the slope of this line is approximately 

parallel to the theoretical line within the same range 

of H. This inverse relation between a and H supports 

the assumption that transverse dispersion is dependent 

on molecular diffusivity. The effect is measurable into 

the upper range of Darcy'flow. However, other mechanisms may operate along with diffusivity to produce 

dispersion; this is further discussed in the next section 

where an attempt is made to account for the difference 

in magnitude between <r* experimental and a* theoretical.

DISCUSSION

In the previous section a dimensionless measure of 

dispersion, calculated by analysis of an idealized 

model, was found to be roughly half that calculated by 

analysis of experimental data. It is possible to bring 

the two sets of numbers closer together by adjusting 

the dimensionless constant, a, and this would be proper 

if the assumptions underlying the theoretical development were completely in accord with physical fact. 

However, as was stated in the beginning, the assumptions are simplifications and approximations. It does 

not seem fruitful, at this point, to try to refine the 

analysis by adjusting a dimensionless number, without 

being reasonably sure that this is where the error lies.

On the other hand, by making other assumptions and 

exploring other mechanisms, one might shed additional 

light on the significance of the experimental results. 

With this in mind, various possible idealized mechanisms are listed as follows:

1. Unrestricted molecular diffusion of tracer elements with no 

amplification by the medium and with zero circuity.

2. Unrestricted molecular diffusion of tracer elements with 

amplification by the medium and with zero circuity.

3. Zero molecular diffusion of tracer elements and nonzero 

circuity.

4. Unrestricted molecular diffusion of tracer elements with 

amplification by the medium and with nonzero circuity.

5. Restricted molecular diffusion of tracer elements with 

amplification by the medium and with zero circuity (which is the 

theoretical model previously developed).

"Restricted" molecular diffusion refers to the idealization whereby tracer elements are considered to 

diffuse within a pore space, but cannot move out of it 

except by transport in the host fluid. The idealization 

is good for R > 0.04 for the porous block used in the 

experiments. It can be shown by use of equation 18a 

(below) that, when R > 0.04, the standard deviation of

the dispersion is less than 0.003 cm in the average time 

required for fluid to move through a pore 0.01 cm long. 

Hence, the bulk of tracer elements would be carried out 

of a pore space by fluid flow before they could move out 

by molecular diffusion. "Unrestricted" molecular diffusion considers that tracer elements may move both 

within and out of pore spaces by molecular diffusion.

In the first idealization listed above, that of dispersion caused by unrestricted molecular diffusion with 

no amplification by the medium, diffusion would 

occur indpendently of fluid flow. Consider a porous 

medium (such as that used in the experients) half of 

which is occupied by fluid containing an initially uniform distribution of tracer elements flowing parallel to 

and in contact with fluid in the other half containing 

no tracer elements. Dispersion of tracer elements 

across the interface will occur as a function of time 

and of the coefficient of molecular diffusivity. It can 

be shown that in a semi-infinite region the standard 

deviation of the diffusion across the interface is

(18a)

In the case of a porous medium it is necessary to 

take the "lithology" into account, and

(18b)

The lithologic factor, /, accounts for the effect of the 

solid boundaries of the medium which inhibit the diffusion of the tracer elements. For media of nearconstant grain size its value is always close to 0.71 

(R. P. Rhodes, oral communication, 1959). From 

table 1, Dm =0.64X10~5 cm2/sec, and L=length of porous 

block=119 cm. We substitute these values in equation 18b

and by equation 16b

<r*=0.7o-=2X10- 2X

(18c)

(18d)

By equation 12c

where d=grain diameter=0.048 cm, and making the 

numerical substitutions,

H=7.5 X10V

By equation 176

(19a) 

(19b)
TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-27

The curve of cr* plotted against H* calculated by 

equations 18d and 19b is given on figure 29 (dashed 

line). On the same figure are shown the theoretical 

curve previously developed (solid linej and the results 

of experimental measurements (dash-dot line). It is 

immediately apparent that for /7*<0.7 Off<0.005) 

molecular diffusion becomes more important than 

amplification by the medium, and to the extent that 

these data may be extrapolated to natural aquifers 

it is a matter of some significance. The Reynolds 

number for an aquifer which has a grain diameter on 

the order of 0.5 mm and in which ground-water velocity is on the order of a few feet per day is about 0.005. 

It is not uncommon for flow through natural aquifers 

to occur at R<0.005; hence, it would appear that in 

such aquifers, unless aquifer nonuniformity had an 

overriding effect, the principal mechanism producing 

transverse dispersion is that of molecular diffusion. 

Conversely, for /7*>7 (Jff>0.05) the mechanism of 

molecular diffusion without amplification by the 

medium has a small effect on dispersion, and cannot 

account for the experimental results.

The second idealized mechanism listed is that of 

unrestricted molecular diffusion with amplification by 

the medium, and with zero circuit}7 . By examination 

of figure 29, it would appear that this mechanism is 

of special importance in the range H*=0.7 to H*=7 

(R 0.005 to 5=0.05). This is an intermediate range 

where the assumptions leading to the construction of 

both the solid-line curve and the dashed-line curve 

break down. Hence, it is concluded that in this range 

a theoretical curve should be drawn that is higher than 

both the curves just mentioned. The labor of performing a mathematical analysis of this case appears 

to be formidable and, for the present purpose, unnecessary. A curve of "reasonable fit" (dotted line) 

is drawn by inspection on figure 29. The experimental 

data lie almost wholly outside the range covered by 

this case and thus are not affected.

The third mechanism postulated is that of zero 

molecular diffusion and nonzero circuity. By definition, when circuity is nonzero, any two originally contiguous fluid paths will tend to separate farther and 

farther apart as flow proceeds downgradient, independently of the molecular diffusion of the tracer elements. 

(It is well to note, however, that in practice fluid paths 

always are identified by the motion of tracer elements.) 

Because of the requirements of continuity, if any two 

contiguous paths separate, other paths must enter to 

fill the intervening space. The process is the sum of a 

large number of independent random events, and, by 

the central limit theorem, it should be expected to 

produce a normal distribution of tracer elements across

an originally planar interface (such as in the experimental arrangement). In addition to being independent of molecular diffusion, the fluid paths are fixed 

(within the range of validity of Darcy's law) and do 

not change with change in velocity. Circuity, then, 

is a characteristic of the medium and is a constant.

It is a matter of observation (H.E. Skibitzke, oral 

communication, 1959) that flow through grossly nonuniform media is circuitous, within the meaning of 

circuitous as previously defined. Fluid paths are 

deflected away from the direction of bulk-flow gradient 

in response to spatial changes in media permeability, 

and paths (identified by dye streams) are seen to be 

separated, twisted, and inverted. Whether or not 

dispersion resulting from aquifer nonuniformity would 

be normally distributed probably depends on the 

magnitude and shape of the nonuniformity. Probably, 

certain kinds of nonuniformity could yield a normally 

distributed dispersion. The significant point to be 

noted is dispersion in grossly nonuniform media may be 

orders of magnitude greater than the dispersion either 

measured or calculated for uniform media.

A question may now be raised that appears to lead 

to an explanation of the twofold difference in magnitude 

between prediction and experiment indicated on figure 

29. Can any randomly packed granular medium be 

exactly uniform? The two words "randomness" and 

"uniformity" are mutually exclusive except in a statistical sense. Any randomly packed "uniform" medium 

will consist of a multitude of small regions, the size of 

each being, say, on the order of a few grain diameters, 

and each having a permeability slightly different from 

all the others. The medium is called uniform when 

their sum-total (that is, their statistically expected) 

effect on flow is the same in all parallel sections of the 

medium. The medium is also isotropic when their 

sum-total effect is the same in every direction. Therefore, if circuity is a nonzero function of medium nonuniformity, and if, by the argument just given, all 

randomly packed media are at least slightly nonuniform, it follows that the circuity of the experimental 

porous medium was in fact nonzero. A nonzero 

circuity could account for the nearly constant difference 

in magnitude between prediction and experiment, but 

could not account for the observed decrease in dispersion with increase in fluid velocity. The writer concludes that the observed dispersion may best be 

explained by assuming that a velocity-dependent 

dispersion resulting from tracer-element diffusion amplified by the medium is superposed on a constant 

dispersion which results from circuity of the medium.

The fourth case listed, that of unrestricted molecular 

diffusion with amplification by the medium and with
C-28 FLUID MOVEMENT IN EARTH MATERIALS

nonzero circuity, includes all postulated mechanisms. 

For reasons already given, the effect of medium amplification becomes unimportant at very low values of the 

H number; at high values of the H number the effect of" 

molecular diffusion beyond individual pore spaces 

becomes unimportant. At intermediate values of 

this number the two mechanisms are of comparable 

magnitude. Medium circuity is assumed constant 

at all values up to the limit of validity of Darcy's law. 

The portions of the theoretical curves given on figure 

29 appropriate to each of the various ranges, and for 

the particular medium studied, were previously indicated. The effect of nonzero circuity may be included 

by increasing the magnitude of all theoretical curve 

values by some constant amount, say 0.5 hi this case. 

The fifth case listed, that of restricted molecular 

diffusion with amplification by the medium and with 

zero circuity, is the model analytically developed in 

this report. It is not a comprehensive description of 

transverse dispersion, but it probably is the most 

complex combination of mechanisms that it is practical 

to treat analytically. The further development of 

analytical procedures probably will require the accumulation of systematic experimental data; in particular,

data on the effects of increasing nonuniformity of the 

medium will be required.

REFERENCES

Crank, J., 1956, The mathematics of diffusion: London, Oxford

Univ. Press, 347 p. 

Feller, William, 1950, An introduction to probability theory and

its application: New York, John Wiley & Sons, 419 p. 

Kitagawa, K., 1934, Sur le dispersement et I'e'cart moyen de

l'6coulement des eaux souterraines: Kyoto (Japan) Imper.

Univ., Coll. Science Mem., ser. A, v. 17, p. 37-42. 

McBain, J. W., and Liu, T. H., 1931, Diffusion of electrolytes,

non-electrolytes, and colloidal electrolytes: Am. Chem. Soc.

Jour., v. 53, p. 59-74. 

Miller, K. S., 1953, Partial differential equations in engineering

problems: New York, Prentice-Hall, 254 p. 

Pai, Shih-I, 1956, Viscous flow theory, I-Laminar flow: New

York, Van Nostrand Co., 384 p.

Rifai, M. N. E., Kaufman, W. J., and Todd, D. K., 1956, Dispersion phenomena in laminar flow through porous media:

California Univ., San. Eng. Research Lab., Prog. Kept. 2,

157 p. 

Scheidegger, A. E., 1958, Typical solutions of the differential

equations of statistical theories of flow through porous media:

Am. Geophys. Union Trans., v. 39, no. 5, p. 929-932. 

Taylor, Geoffrey, 1953, Dispersion of soluble matter in solvent

flowing slowly through a tube: Royal Soc. (London) Proc.

A, v. 219, p. 186-203.
APPENDIX

Derivation of equation 9a of text:

n= 1,3,5 . . .

COS

n

exp-( (20a)

Integrating both sides of 20a with respect to y over 

interval w=0 to w=a

r c *,-& Jo 2 n

_ ~~0 ,I 

*

By equation 8

- 

«=1,3,5 .

exp 

\2o/

(20b)

(20c)

=-+- 

 O ' ^.2 ^-1 X * TT n= 1,3,5.

Note: When t= oo, g=4

When f=0, 2=4+4 X)

^ T « = 1,3,5, n2 22

Derivation of equation lOa oi text:

The preceding solution for q(a,t) converges satisfactorily for moderate and large t (low fluid velocity), but 

not for small t (high fluid velocity). It turns out that 

an equation that will converge satisfactorily for small t 

can be developed from the error-function solution of the 

original differential equation. This solution of the 

differential equation as given by Crank (1956, p. 15), 

letting h=a, and Z=*2a is

We desire g(a, t), where

(22)

The error function is itself an integral, and a method 

integrating it with respect to y over the distance a is not 

apparent. However, the integral is easily differentiated with respect to y. We may take advantage of 

this fact by developing the following relationship:

Start with the differential equation for diffusion in 

one dimension

(23)

Integrating with respect to y over the distance a

/o^ N (24a)

Intrgrating left side of above, and interchanging order 

of integration and differentiation of right side, we get

den 5 r«

 ^r\ H djMoJ

Because the fluid region is bounded at ?/=

hence,

(24b)

(24c)

Referring to 22 for any a constant, g is a function of 

time only. Hence, we may differentiate 22 with 

respect to time, substitute it in 24c, and get

dt=7T7 i-'a/ (25a)

We now integrate both sides with respect to time

(25b)

where A=constant of integration

when £=0, %=1=A

Thus,

q(a,t)=l- dt (25c)

C-29