A STUDY ON FLOW THROUGH poms MEDAUM  

THESIS FOR THE DEGREE 0E PH,.D. ”
MICHIGAN STATE UNIVERSITY —
HON~HSIEH SU- .
1968

THESiS

 

LIBRARY
Michigan State

1m L‘QIVCISitY

 

This is to certify that the

thesis entitled

"A STUDY ON FLOW THROUGH POROUS MEDIUM"

presented by

Hon—Hsieh Su

has been accepted towards fulfillment
of the requirements for

Ph D degree in Civil Engineering

 

of "
rt"

2 ,. "’i
k; /(. Zone-k,

Major professor

 

Date March 13, 1968

 

0-169

 

 

 

 

 

AESTRA 'I'

A STUDY ON FLOW THROU’IIH POROUS MEDIUM

by Hon—Hsieh Sn

The present investigation is a probabilistic study of the porous
medium and the mechanism of fluid. flow in the pore system. If the
porous medium is such that its pores have different orientation with
respect to certain direction and the hydraulic gradient is the sole
cause for the fluid particle to move in the pores, then the tortuous
paths of the pores will result in a dispersion of fluid flow in a porous
medium.

A canal network model of a porous ‘rziedium was assumed in this
study. An orientation factor was used to represent the preferred
orientation of the pore canals. For [given canal distributions, it was
possible to calculate the degree of dispersion at a given time. 1' theo-
retical analysis of dispersion was made based on the assumption that
a fluid particle would travel in a pore canal network system by follow-
ing the probability distribution function for the choice of direction. A
functional relationship was derived for the dispersion that includes the
longitudinal distance of fluid Iiarticle movement, the length of the unit.
canal, and the orientation fat tor of the porous medium. It was also
shown that the orientation factor of the porous ‘E'nedium could be re-
lated to the ratio of permeability coefficients measured in two perpen-
dicular directions in the porous medium.

A laboratory experiment was set up to investigate the dispersion
phenomenon. The laboratory analysis showed that the dispersion

phenomenon in a porous medium is a macrosc0pically measurable

 

Hon—Hsieh Su

quantity. The factors affecting; the quantity of dispersion were found
to be the length of the pore canals, the distance of travel by the fluid
particles, and the orientation of the pore canals. It was also found
that the packing characteristics such as porosity, packing uniformity
which were not considered in the present theoretical analysis also
affect the dispersion. The experimental results showed that the
functional relationships as derived theoretically appeared to be
qualitatively correct.

Therefore, it is in principle possible to predict the dispersion
from a knowledge of the characteristics of the porous medium. From
the results of this study, it can be stated that the assumption of the
canal network model and a corresponding probabilistic calculation
appear to help in explaining the dispersion phenomenon of the fluid

flow through porous medium.

   

A. STUD“? or: FLOW TURC on F‘CRCUS l\-’IEDIU II

”ion-335mb Su

T317513

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR on ~ 1080

Department of Civil Engineering

1968

 

(75 5..
/&/§:/é~ y

'. CKNOWLE DGEMENT

The writer ishes to express his indebtedness to all of his
committee members for their valuable assistance and guidance
during his c0urse work and the development of this thesis. Sincere
appreciation is due to Doctor H. Wu who had given his constant
and valuable suggestions and instructions during the preparation of
this thesis and to Doctor K. J. l-‘rnold for his critical review of
this thesis. The writer also v ishes to ex press his appreciation to
Doctors C. E. Cutts, R. K. Wen and O. L. Andersland for their

review of the manuscript.

ii

   

TABLE OF CONTENTS

ACKNOWLEDGEMENT

TABLE OF CONTENTS .

LIST OF FIGURES

LIST OF TA BLES

LIST OF APPENDICES .

CHAPTER 1. INTRODUCTION.

CHAPTER 2. LITERATURE REVIEW

CHAPTER 3. THEORETICAL ANALYSIS

1.

Z.

8.

Dispersion Phenomenon

Canal Network Model

Pore Geometry

probability Distribution Function in Fluid Flow

Longitudinal DiSplacement of Fluid Particle
after N Consecutive Steps

The Mechanism of Fluid Flow in a Porous Medium ,

The Longitudinal Dispersion of Fluid Flow in a
Porous Medium

The Permeability Coefficient and Orientation Factor.

CHAPTER 4. EXPERIMENTAL PROGRAMS

1.

Sample Preparation .
Dispersion Measurements
Permeability Measurements.

Presentation of the Experimental Data .

iii

vii

, viii

10
10
12

14

16

18

28
32
36
36
36

38

 

 

 

CHAPTER 5. ANALYSIS OF EXPERIMENTAL RESULTS .

1. The Break Through Curves
2. DiSpersion as a Function of Distance
3. The Characteristics of Pore Geometry
4. Dispersion Coefficients
CHAPTER 6. DISCUSSIONS AND CONCLUSIONS .

LITERATURE CITED

iv

45
45
45
46
49
64

69

 

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

3-2.

3-3.

3-4.

U»)
I
U1

5~2a.

LIST OF FIGURES

The schematic sketch of a canal network system
in a porous medium and a random path chosen by
a fluid particle

The schemati: sketch of grain particle arrange—
ments in a porous medium for two extreme cases

The distribution of pore canals as represented

by the fraction of surface area over a hemisphere .

The probability distribution curves of the choice
of directions at a junction in a porous medium

The schematic diagram for the relationship
between the fixed values of 20, f, K; and the

random variables in paths of a porous medium .

Relationsliip between orientation factor n and
C5 in Eq. (73) - - ' '

Numerical values of <cosl-l>, «335?», «059»
and (Sech) for various n values.

Relationsuip between the Orientation factor n
and the permeability ratio in an anisotrOpic porous
medium calculated from Eq. (76)

Schematic sketch of the experimental set-up for
the Ineasurements of disr)ersion of fluids through
porous media,

Sections of permeability apparatus-
Typical breakthrough curve with respect to time

Typical. break thr0ugh curve with respect to
distance ,

The relationship of 7:1? and the percentage of fluid
particles that has reached the longitudinal dis-—
tance of 2711‘ Sample No. S11

The relationship of (Zv‘fré and the percentage of
particles that has reached the longitudinal dis—
tance of er. Sample NO. S11

Relation between standard deviatiOn and the
longitudinal distance, S .

ll

ll

15

-22

'3]

34

41

.42

56

56

57

58

59

 

 

‘0‘
r.” i

 

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure IV-Z.

Figure

Figure

5-3b.

5-3c.

5-5.

IV—1.

IV—3.

IV-4.

Relation between standard deviation and the
longitudinal distance, P1.

Relation between standard deviation and the
longitudinal distance, P2.

Relation between standard deviation and the
longitudinal distance, P3.

Relation between standard deviation and the
longitudinal distance, P4.

Relation between standard deviation and the
longitudinal distance, N

Relation between porosity and the orientation
factor n obtained from Figure 3—8

Schematic sketch of the possible patterns of
flow paths in a porous medium consisting of
plate shaped particles

Permeability and porosity for sample P1.
Permeability and porosity for sample P2.

Permeability and porosity for sample P3.

Permeability and porosity for sample P4.

Figure V-l through Figure V—ll. Break thrOugh curves

vi

.59

.60

. 60

.61

.6]

.62

.63
.86
.86
.87
.87

113

through
132

   

 

 

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

LIST OF TABLES

4-1. Samples for dispersion measurements. . 43
4-2. Samples for permeability measurements . . 44
5-1 The average standard deviations . . 53
5-2. Orientations of the porous media as calculated
from the permeability data . . 54
5-3. Length of pore canal obtained from experimental
data..................55
111-]. Test of applicability of Saffman's equations . . 81
III-2. Comparison of the numerical results of standard
deviation for various theories. . 82
IV-l. Permeability test data for sample P1 . 83
IV-Z. Permeability test data for sample P2 . 83
IV-3. Permeability test data for sample P3 . .84
IV-4. Permeability test data for sample P4 . 85
IV-S. Permeability test data for sample N . 85
V-1 through V- 11. Dispersion measurements . . . 88
through
112

vfi

   

APPENDIX I.

APPENDIX 11.

APPENDIX III.

APPENDIX IV.

APPENDIX V.

LIST OF APPENDICES

Derivations of the Relationships .

An Estimate of Error in Assuming an Average
Cosine of the Angle of Orientation for a Given
Path in a Pore Canal Network.

The Numerical Comparison of Present Theory
to those of Saffman's and De .Iosselin de Jong's

The Presentation of Permeability Measure-
ments .

The Presentation of Experimental Data on
Dispersion ..

viii

.71

.74

.79

.83

.88

   

CHAPTER 1

IN TRODUC TION

Many studies on the flour... of fluids through porous materials
have been undertaken. Darcy's law is valid for most cases when an
average rate of flow through a porous medium is considered. Many
investigations have been made to study the coefficient of permeability
in Darcy's eq uation. Most of these attempts were based on the so-
called capillary model. This is the sii‘nplest model of a porous
medium and consists of a bundle of parallel capillaries with uniform
cross sections. This model plus Poiseuille's viscous flow equation
leadsdirectly to Darcy's equation. Unfortunately, the perrzieabilities
derived from this type of model are not entirely satisfactory when
used to describe the phenomenon of flow through porous materials.

It is apparent that difficulties with the capillary model arise
because the capillaries are all parallel and that they have identical
cross sectional area and length. The model is far from reality and
therefore it is unreasonable to expect a consistent functional relation-
ship among several measurable prOperties.

The shortcomings of the capillary model Rec1 some investiga—
tors to an entirely different approach based on the statistical treat—
ment of the porous medium. Dispersion phenomenon was used to
study the movement of fluids in the porous medium and investigations
have been reported in the past ten years. Most of these are theoret-
ical studies.

The present investigation is a probabilistic analysis of fluid

flow through porous medium. A porous medium in this investigation

 

at “
{‘0’ L

 

2
is assumed to consist of a network of pore canals linked together. A
fluid particle is assumed to follow a probability function in choosing
a flow path while travelling from one position to another in the porous
medium. Different paths have different lengths between two positions
and also require different travel times. If, during continuous flow, a
fluid is replaced abruptly by another miscible fluid so that they occupy
two distinct phases at the beginning, the difference in flow paths
causes a mixing of the two fluids. This mixing is the phenomenon of
dispersion. A theoretical analysis of dispersion based on these
assumptions was made and a functional relationship was derived for
the dispersion in porous media.

A laboratory experiment was set up to investigate the disper~
sion phenomenon. The experimental results showed that the disper-
sion is a function of the medium prOperties. The characteristics of
dispersion as observed in the experiments are in general agreement
with the theoretical predictions. The theoretical relationship between
the standard deviation of dispersion and the distance was found to be
correct

It is hoped that a study of this nature would lead to a better
understanding of the basic aspects of the fluid flow through porous

materials .

 

af“

 

..

 

CHAPTER 2
LITERATURE REVIEW

The well knov n Darcy's law (1856), an empirical expression
of the flow of fluids through porous materials based on measurements

of the flow of water through sands and sandstones, may be written as

=kAP ...............l
v M (l

where v is the velocity of flow, Ap is the pressure head difference
between two points in the porous medium, and Az is the distance
between them. The coefficient of permeability k as defined in the
above equation is the rate of flow of fluids across a unit cross section—
al area of the porous medium under a unit pressure gradient.
Refinements to Darcy's equation were made by many investi-
gators such as Blake (2.), Kozeny (9). and Carman (3) in attempts to
generalize the equation for flow through porous materials. While
keeping the fundamental form of Eq. (1) unchanged, these studies tried
to relate the permeability coefficient to the physical and geometrical
characteristics of the porous medium. A widely known equation is

the Kozeny — Carman equation

1 e3 (2)
R: m A . . . . . . . . ..
RI/‘SCZJ (1'6)£

 

where fl is the viscosity of the fluid in the porous medium, 6 is

the porosity, S0 is the specific surface area, k1 is a constant ex-

pressed as

 

of "
Q’.’ i

 

4

in which Le is the actual path of the fluid flow and L1 is the distance
between the two sections of porous medium under consideration. k
is a constant representing the shape effect of the pores in the medium.

In the above treatment, the porous medium is represented by a
bundle of parallel capillaries, and the laws of viscous flow are applied
to flow in the capillaries. Relationships between the various macro-
scqaically measurable quantities are deduced from this model. Ex-
perimental evidence shows that the theory applies with considerable
accuracy to porous media composed of nearly spherical particles of
relatively large size. However, it is unsuccessful in describing the
flow characteristics of clays, which are composed of fine, plate-
shaped particles.

Two groups of factors have been brought out to explain the
failure of the Kozeny-Carman equation. The first is the forces at the
liquid-solid interface. This includes the assumption of high viscosity
close to the particle surface or the presence of immobile films of
adsorbed fluids at the particle surface, Terzaghi (19) and Zunker
(20). Other investigations were reported by Bastwo and Bowden (1),
Elton (7), and Michaels and Lin (12). The second is the change in the
packing characteristics of the material. This includes the orientation
of the particles and pore size distribution. Michell (l3) and Lambe
(10) studied the effect of the particle orientation and Olsen (14) studied
the effect of changing pore size distributions. The experimental
evidence shows that changes in the packing characteristics affect
the nature of fluid flow in fine-grained soils and the relationship
between the property of the porous medium and the fluid flow remains

indefinite.

 

 

and“
Qt.” t

 

 

5
Other investigators tried the statistical approach. In 1950,
Childs and Collis—George (4) proposed a theory wherein flow was
determined by the pore radii and by the probability of continuity of
pores of different radii. Spherical particles were assumed. Marshall
(11) in 1958 and Quirk (15) in 1959 proposed alternatives to Childs and
Collis-George's method. Scheidegger (17), (18) in 1954 derived the
differential equations of motion of fluid through a porous medium
from probability theory. An average ensemble represents all parts
of the porous medium under the so-called ”hypothesis of disorder".
The geometrical conditions for the motion of a particle prevailing at
a Spot in a porous medium are assumed to be entirely uncorrelated
with those at any other spot of that material. The movement of fluid
particles in such a medium is then considered as steps with respect
to time or distance. Then, by virtue of the Central Limit Theorem,
the probability of a specific particle being at x at time t is a
Gaussian distribution
—3/Z (x—Tj

p(x,t)=(4TTDt) exp(—m_). . . . . . . . (4)
where 3? is the average distance and D is called the factor of disper-
sion. However, Scheidegger's differential equation for fluid motion
in a porous medium is too complicated for practical application.
Experimental results by Day (5) confirm the existence of the disper-
sion phenomenon in porous materials but are not extensive enough to
verify Scheidegger's theory. Scheidegger’s statistical treatment does
not define the microsc0pical mechanism of the fluid movement in the
pores. Therefore, his result contains a numerical constant describ-

ing the granular properties of the porous medium which can only be

 

 

at “
Q" L

 

 

6
determined by experiment. According to Scheidegger, the dispersion
constant should have equal magnitude in both longitudinal and trans—
versal dispersion. However, Day’s experimental result indicates
that there is a marked difference of about 6 to 8 times in longitudinal
and transversal dispersion.

De Josselin de long (6) in 1958 also derived an expression to
describe the fluid movement in a porous medium from probability
considerations. The pore system of a packed material is represented
by a network of unit canals continuous throughout the medium. The
probability of a fluid particle moving in this network to travel a
certain distance within a certain time was calculated. Molecular
diffusion is not considered. The computation gives explicit values

for the coefficients of the longitudinal and transversal dispersions as

follows:
0‘ - u (320 (m 3 - logs)" (s)
X ’ T:— T . . . . . . . .
G" : L (320 (A + 3 ~10gr))1/‘2 . . . . . . . . (6)
z T I:— T

In these eXpressions G‘X and G'Z are the standard deviations
in transversal and longitudinal dispersion respectively; u is the
residence time for elementary canal in principal flow direction; Z0
is the average distance travelled along the longitudinal direction;

L is the length of the unit canal; log r is Euler's constant and is
equal to O. 577 approximately; and 7\ is a function of distance 20.
De Josselin de Jong also performed one experiment and measured

the dispersion in sand. The measured longitudinal dispersion co-

 

 

 

 

7
efficient increases with the distance and the relationship between the
standard deviation and the square root of distance is linear as indica-
ted by Eq. (6) However, De Josselin de Jong's computations are
rather complicated and apply only to a porous medium with com—
pletely random pore orientation.

Saffman (16) in 1959 also applied statistics to the dispersion
in a porous medium. Saffman considered three conditions: (a)
molecular diffusion is very large compared to velocity of flow, (b)
molecular diffusion is very small compared to velocity of flow, (c)
intermediate case in which the fluid is comhletely mixed across each
channel but not along the channel. Case (c) is analogous to De
Josselin de Jong's model. Saffman obtained solutions for all three
cases. He arrived at the following expressions for the standard de-
viation of the longitudinal dispersion of a fluid particle

in a porous medium after time T

/

21,2
G‘=(TVL.S)..............(7)

2

where the quantity 82 is represented by the following equations. For
idealized fluid particles where to 3 t1 :00 , or fluid particles very

close to idealized conditions

1 54VT2
Z::§‘(10gl.)..........(8)

1 0-1
TlOg—C-TZ"°""""(9)

 

 

 

 

 

 

 

Vt /L
0/
_/ O yZ
n log L )
3 VT
8 = ,1 log 27 ,, (10)
0 2L.
3\'tO/L
if . ’ ,y >> 1.
F.(log'riz)z

Therefore, the standard deviation increases linearly with the
10garithm of distance. In the above equations, V is the average
velocity of the fluid flow, T is the duration of time, L is the length
of the unit pore canal, “5 is the average number of displacements,

t is the estimate of the time for appreciable diffusion along the pore

o
and t1 is the estimate of time for appreciable diffusion across the
pore. 1

Thus two essentially different approaches have been employed
to study the flow of fluids through a porous medium; namely, inves-
tigation of the factors affecting the permeability coefficients in the
classical Darcy's law based on the capillary model and the statistical
approach based 0n the canal network model. The capillary model has
proved itself to be an adequate model for permeability of certain
types of porous materials but it does not apply to dispersion. The
statistical approach has been applied to fluid flow through porous
materials only in the last ten years and is not yet completely develop-
ed. However, statistics have been used to describe diffusion prob-

lems successfully and the recent application of statistics as described

in the preceding paragraphs gives a consistent explanation of the

 

 

 

9
dispersion phenomenon. Therefore it appears that the statistical
approach is a tool that deserves more attention in the study of the

flow through porous materials.

 

af"
,,.uo.

 

 

 

CHAPTER 3

THEORETICAL ANALYSIS

1. DispersiOn Phenomenon.

 

Dispersion is a phenomenon observed in a porous medium as a
mixing of two miscible fluids when a fluid flowing in a porous medium
is abruptly replaced at its bottom by another completely miscible
fluid. This phenomenon is also called a ”miscible displacement"and
is different from the molecular diffusion. The microstructure of the
porous medium results in a tortuous path for a fluid particle travel-
ling through the pores. This tortuosity of the flow path causes the
individual fluid particles travelling in the pores to arrive at different
places after a given time interval. The relationship between this
dispersion and the structure of the porous medium is a basic char-

acteristic of the flow through a porous medium.

2. Canal Network Model.

 

In order to study the mechanism of fluid flow in the porous medi-
um, a canal network model is constructed for the pore canals. The
following assumptions are made regarding this model.

(1) The pores between grains of the porous medium can be

represented by a system of unit canals joined together to form

a network as illustrated schematically in Figure 3—1.

(2) The unit canals in the model of the porous medium have

a length and an average cross sectional area that are represen—

tative of the average size of the pores that form the canals.

(3) The grains in the porous medium are assumed to be

rigid.

   

 

11

 

Figure 3-1. The schematic sketch of a canal network system
in a porous medium and a random path chosen by
a fluid particle.

/
/ y:
/
Completely random Completely oriented

Figure 3-2. The schematic sketch of grain particle arrange-
ments in a porous medium for two extreme cases.

 

ad"
we” r

 

 

 

12
(4) The external forces on the fluid in the porous medium are
homogeneous and time independent. The gravitational force is
neglected.
(5) The pressure gradient in each canal is proportional to
the cosine of the angle between the direction of the canal and
the principal direction of the gradient.
(6) Any pa rt in the porous medium is macroscopically iden-
tical with other parts in the same sample. This implies that
a fluid particle travelling in the porous medium finds exactly
the same probability function for displacement at any point in '

the medium.

3. Pore Geometry.

Since a porous medium is constituted by packing , of the grain
particles, the shape, size and direction of the individual canal in the
medium depend upon those of its adjacent grain particles. The
directions of the canals in a porous medium which is made up of
uniform spherical particles are approximately randomly oriented.
For a porous medium consisting of plate-shaped particles, anisotropy
is dependent on its packing characteristics and the canals usually show
SOme preferred orientation. In Figure 3-2, schematic sketches of
these two extreme conditions are shown for plate shaped particles.

A mathematical expression for the distribution of the canal
direction is required. For a randomly oriented porous medium
(Figure 3—3), let the direction of canal be represented by the fraction
of surface area on a hemisphere. Then the probability for a fluid
particle to find a direction defined by dA which is contained between

0, {6 and 9 + d0, {3 1' did may be represented as

 

 

13

r2 sinG d0 dd
p : Z‘Wr1

 

sin0d0d¢.............(ll)

_l
"Z—T’T

 

 

Figure 3-3. The distribution of pore canals as represented by the
fraction of surface area over a heniiSphere.
An anisotronic porous medium is one in which particles show

preferred orientations. It is assumed here that the distribution of

canal direction for the general anisotropic case has the following form

a
pzz—T—f—sinnadgdd............(12)

in which n is a positive number (equal to or greater than unity) that
characterizes the particle orientation. a is a normalization constant
Which can be expressed by the following equation. (See Appendix 1,

section 1, for derivation).

n(n-Z) (n—4) .

_ . . 2 b
a - (n-l)(n-3). . . . (Tr-T") . . . . . . . . (13)

   

 

14
where b I l for n even and b I O for n odd. It is seen that when n is
equal to unity, Eq. (12) represents the case for a randomly oriented
porous medium and when n approaches infinity, Eq. (12) represents
a case where all the canals are lined up at 0 equals 90 degrees, a

completely oriented porous medium.

4. Probability Distribution Function of Fluid Flow.

 

A fluid particle travelling in a porous medium will have to make
a choice of direction whenever it arrives at a junction of canals. This
choice of direction not only depends on the pore canal distribution as
discussed in the preceding section but also depends on the direction
of the individual canals. The discharge of each individual canal is
prOportional to the cosine of O by virtue of assumption (5) in section
2. Therefore, those canals making angles perpendicular to the grad-
ient will have no discharge at all. This means that the probability of
a fluid particle entering such a canal is zero even though there may
exist a large portion of canals in that particular direction. The
probability of a fluid particle to take up the direction defined by an
area dA which is contained between 0, (J and O + d0 , ¢+d¢ is
assumed to be equal to the proportion of the discharge in that direc-
tion to the total discharge (all canals in all directions), The dis-

charge of a canal with angle 0]- is

q] qO c050] . . (4)

Where q0 is the maximum possible discharge of a canal and is equal
to the discharge of that canal with 01 equal to zero.

Therefore, the discharge toward the direction dA can be de—

fined as, by combining Eq. (14) and Eq. (13).

 

 

 

..

 

 

15

aq .n
qodzzfio stcosOdOde . . . . . . . . . (15)

Let Q be the overall average discharge for all directions, then,

211 3‘."
Q IJ dfl aqo sinngcosOdO 1 aqo . . . . . (16)
0 o 27

Let g9“ denote the fraction of the discharge in the directioa

defined by dA which is contained between 0, 121 and o + d0, 11 + def.
g9?) : (10¢ : EL sian cosO d0 def . . . . . (17)
‘Q‘ 21T

Eq. (17) is the probability distribution function for the choice of
direction based on discharge and geometric distributions of the pore
canals. Probability distribution functions for various orientation

factors are computed frOm Eq. (17) and are shown in Figure 3—4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 u I 1 l I
ZTngg = (n+1) sinnO d0 d¢
Z A
‘a
e?
f: \
N \\
3 1 /W%\ \
:1 =1 /
E n / / / K \\\\
/ n13 n= 1 \ \
/ X1311)
0 o 13 2 3 0 so so 7 8 9
O in degrees
Figure 3—4. The probability distribution curves of the choice

of directions at a junction in a porous medium.

 

af“
Q‘fl’ \

 

 

16

5. Longitudinal Displacement of Fluid Particle After N Consecu-
tive Steps.

The process of the fluid particle movement in the porous medi-
um is considered as a series of consecutive steps of displacement in
the unit canal network. Here a step of displacement is defined as the
journey through a unit canal. The distance of this displacement in
the gradient direction is called the longitudinal displacement and is

equalto
ziILcosgj...............(18)

Now, after N consecutive steps the total longitudinal displacement is
N

221.231cosej............(19)
j:

in which Z is a function of the choice of direction at every step of
displacement and therefore is dependent on the probability distribu-
tion of O and (5.

If the probability distribution of O and G is that given by Eq.
(17), then the average displacement of each step in the longitudinal
direction may be evaluated as follows. Let E(z) be the average lon-

gitudinal displacement for each step in a unit canal, then

E(z)=JLCOSOg9¢. . . . . . . . . . . . (20)
8
Substitute Eq. (17) into Eq. (20) and integrate, it is obtained that
- L(n—+l) (n-l) . . Tr b
E(z) — W( ‘7) O I I v I e O (21)

The variance of z is evaluated as (see Appendix 1, section 2)

 

"‘
'1’ i

 

 

l7
2 2L2 L. n+1) (n—l) 1r
Var(z) :G-z : F173 ' ( n+ n n— (T)b)2 (22)

The average displacement of a fluid particle in the longitudinal

direction after N consecutive steps is
E(z):NE(z)...............(23)
and the variance of Z is the sum of the variances of the N steps
2
Var(ZN) = N Var-(z) I NG-Z . . . . . . . . . (24)

From Eq. (21) through Eq. (2.4) it can be seen that the average
displacement and the standard deviation of each step not only depend
on the length of the unit canal but also depend on the canal orientation
factor n.

According to probability calculus, after N repeated trials the
distribution approaches the Gaussian distribution provided N is
sufficiently large. Therefore, the probability for a fluid particle to
arrive at a distance between Z and Z+dZ along the gradient direction

after N steps is

I (24-12(2))2
MZN) =(2rrNo‘zz,)”zeXp( - 2N¢ZZ ). . . . . (25)

where 0'2 is the standard deviation of each step as expressed by

Eq. (22) and Z is the longitudinal displacement travelled by the fluid
particle after N repeated steps as represented by Eq. (19). 13(2) is
the average longitudinal displacement after N steps as represented by

Eq. (21) and Eq. (23).

 

4“ “
V..-
t

 

 

18

6- The Mechanism of Fluid Flow in a Porous Medium.

 

Considering now the mechanism of fluid flow in a porous medi—
um- Let t0 denote the shortest time possible for a fluid particle to
pass through a unit canal of length L. Then by virtue of the preced-

ing assumptions made with regard to the canal network model,

t)- : tosecgj . . . . . . . . . . . . . . . (26)

Where tj is the time required for a fluid particle to pass through a
canal making angle Oj with the gradient direction. If N canals are

taken, then the total time required TN will be

N
T'tZsech............(Z7)
1:

Therefore, the distribution function represented by Eq. (25) in
the preceding section involves an unknown distribution of time which
is dependent on the path taken by an individual particle. Two fluid
particles may pass through the same number of canals arriving at
different longitudinal distances and spending different times. Also,
two fluid particles may pass through different number of canals by
taking different paths and arriving at different longitudinal distances
for the same amount of time. Furthermore, two fluid particles may
pass through different paths in arriving at a given longitudinal dis-
ta nce but the time spent and the number of canals traversed are
different.

If, in a given porous medium, a large number of fluid particles

are introduced and the journey of each individual fluid particle is

to 110wed closely. Then, for a given longitudinal distance 20' the

 

at “
«.9
I

 

19
number of unit canals traversed by individual particles to arrive at
20 may be averaged as N— 0’ and the time required for each indivi-
dua 1 particle to arrive at this longitudinal distance may be averaged

a s .TZO‘ Letting L, the length of the unit canal and to, the minimum

time required for a fluid particle to pass through a unit canal: be

equal to unity, the following two equations may be written from Eqs.

( l 9) and (Z7).

ZOINZO<cosO>ZO. . . . . . . . . . . . (28)
TZOINZO<secC>ZO. . . . . . . . . . . . (29)

Where < FED—>20 is the average longitudinal distance of each step and
is a representative value of the orientation of the given porous medium
which is defined by zO/NZO and (5733020 is the average time re-
quired for one step of journey for the given porous medium.

In Eqs. (28) and (29), N20 is the average number of canals and
T20 is the average time required for a given 20. If an individual
fluid particle travelling an arbitrary path is considered, then the
number of canals travelled N20’ and the time required TZo are ran—
dom variables depending on the path taken. This is illustrated in

Figure 3-5 as case (c). The following two equations may be written

to describe these random variables:

ZO=NZO<COSQ>Z° . . . . . . . . . . . . (30)

Z . . . . . . . . . . l
TZO NZO<secO>ZO ° (3)

 

0“ "
cm"

 

 

 

20
New, if the average number of canals in the above case is con-
sidered as fixed, then for this fixed N20, the average distance tra-
velled by the fluid particles and the average time required to complete

thi 5 fixed number of steps may be expressed as

2_1N <cosO>_ ...........(32)
N Z N

0
TNZEZO<sch>E§I . . . . . . . . . . . . (33)

a nd similarly the individual fluid particles in an arbitrary path may

have distributions of longitudinal distance ZN and TN as follows:

Z‘N NZ < c050 >

O

(34)

Z l

TN Nzo<sec9>fi............(35)

This is the case illustrated in Figure 3-5 as case (b), and it can
be seen that this is exactly the case investigated in the preceding
section and expressed in Eq. (25). The conditional probability distri-
bution for the longitudinal displacement Z-- may now be rewritten in

N
the following form

(2 ) 1 27 ( (Zfi _ 2W2.) (29)
p -- 3 — .7. exp — Z“ T . . . -a
N (ZWNZOTZ) N20 6-2

 

 

The ab0ve expression is the distribution function of longitudinal distan-
Ces travelled after a fixed number of canals N and is obtainable by
knowing exactly the pore canal distribution. However, a measurement
0f Such a distribution in a laboratory is practically impossible.

Another type of distribution may be investigated. If the average

   

21
time TZO in Eq. (2.9) is fixed, then there exists a distribution in the
longitudinal displacement for this fixed time and there also exists a
distribution of the number of canals taken by an individual fluid
particle. This is the case of (a) in Figure 3—5 and may be expressed

in the following equations:

2— : E— I - o o o o o o o o O n o O

T T Q cos©> T (36)
__ _ __ / _

TZO- Nszeca>T . . . . . . .. . . . . (37)

FOr the individual particle in an arbitrary path, the random variables

may be expressed in the forms

..:.. -............38
2T NT<cosQ>T ()

TZN—<sec9“............39
ZQT >T ()

The distribution of longitudinal displacement for fixed time as
expressed in Eq. (38) may be measured in the laboratory although
difficult. It is difficult to measure the quantity of fluid particles at
different positions at the same time.

In summary, for any path chosen in a porous medium, the
prOgress of a fluid particle can be described in the following three
cases (see Figure 3~5).

(a) For a fixed time of T20, the number of canals and the

distance are variables dependent on the paths.

(b) For a fixed number of canals NZ , the time Spent and

o
the distance travelled are variables depending on paths.

   

22

.Esmpog msonoa a mo 33.3 E moaning Someday of was Z AH. .oN mo

 

 

 

 

 

m03m> voxfi on,» asbestos azmcowuflou 2.3 you Emnwmwb oSwEosum 9.5. .mum madmrm
m seen
H . .
IN #2 H
5 pump
21 .Z- . A
In P 2
3v ammo -lr.
W
0»
a2
13 ommo

 

 

 

 

 

   

 

 

23
(C)

For a given distance 20' the time spent and the number
of canals traversed are variables depending on paths.

The relationship between the random variables in the above

three cases may be evaluated as follows:

From Eqs. (32) and (38)

 

: NZ°<°OSO>E. . . . . . . . . . . (40)
N'T <COSO>7f~

Ni
r—ll 2|

and from Eqs. (33) and (39)

 

 

 

 

 

T— N secG -
———.N I 20 < >N. (41)
T20 N‘rf <sec0>rf
Therefore, by combining Eqs.(40) and (41)
._.__ __ __ _ / $__ __
ZN : (Nzolz (<SeCQ>N\ ” N 720 (42)
Z— N:— __ / _ __
T T (c050) Tfisecu) T TN"
BY comparing Eqs. (38) and (34)
z— N" < c050 -
l:T >T..........(43)
ZN NZ() < COSO>N
Therefore,
NZo _ ZN <COSO> T (44)
NT Z71? <cosO> N

Substitution of Eq. (44) into Eq. (42.) will result in

 

 

 

 

24

Z

 

(45)

21% <§§€O>E (cos—@213 (c050)?
EN '

i 3 _.._
<sec9>1f @059)erJ

 

This is the relationship between the longitudinal diSplacement
for a fixed time i“ and the distribution of longitudinal displacement
for a fixed number of canals N. Under the assumption that for each

individual path Z3. and ZN do not differ much, the distribution Z

may be derived using Eq. (45). In Eq. (45) the expression inside
the brackets may be analyzed as follows. The ratio of T20 to EN may
be considered as approximately unity in a porous medium of large
dimension compared to the length of the individual canals. Also
<C059>7f and <COSQ>N for an individual path can also be consider-
ed approximately equal by assuming that fOr a given path the average
Orientation is not changed greatly by the addition of a relatively
Small number of unit canals. However, the values of <cosG>f
times <5ec9> 7f and /<-c—o_56>-1<I times <m)fi are not necessarily
equal or approximately equal to unity depending on the particular
porous medium and the individual path. It is a function of the path
inc luding the number of canals and the canal orientations. In order
to simplify the analysis, these two quantities, <c050>7f<sec0>f and

(MRI <§€E§>fi , may be assumed to be equal to unity. The
probable error involved in this assumption is investigated and pre-
Sented in Appendix II.

Now, if we compare Eqs. (30) and (38)

 

7.0 _ NZO<COSQ>ZO (46)
Z711 N21: (c050)?

 

 

 

 

25

and from Eqs. (39) and (31)

“T20 _ N'T (see9>:f

 

 

 

-. O C O C O C O O O . C O (47)
TZO NZO<SCCQ7ZO
Therefore, from Eqs. (46) and (47)
T2 I. ZQTZO (c059? <SeC0>ZO
o m”: . . . . . . . . 48
ZT <COSQ)ZO<SCCQ>T ( )

where T20 is the distribution in time required to arrive at a given
longitudinal distance 20 as represented by case (c) in Figure 3-5.
T 20 is a measurable quantity and the laboratory set-up for this
measurement is usually mOre convenient because many time readings
can be taken at one location whereas there is a serious limitation on
the number of locations that can be measured at a given time. The
quantity inside the brackets of Eq. (48) may be considered as approx—
imately equal to unity if it is assumed that for a fluid particle travel-
ling in a given path the average orientation will not change appre-
ciably from one stage to another. This is a reasonable assumption
Considering that the range of the dispersion in the porous medium is
Usually small compa red to the overall size of the'medium.
Approximations other than that obtained by replacing the
bracket of Eq. (4-8) by unity are possible, for example, by substi—
tL112ion of Eqs. (32) and (34) into Eq. (45), Eq. (45) may be rewritten

in the following manner:

2% EN (57550)};1 <cosO> ,1; TZ

- —--—

ZN ZR <SeCQ>T <COSQ>1_\I T

O

 

ZT —
N

 

 

 

 

 

 

26
or /_ _
Z— : Z- (secl9>—1\‘I {c059}? TZO
T N < _ _ _ . . . . . . (49)
secO>T <COSQ)N TN

Therefore, the distribution of longitudinal displacement for
fixed T is now expressed as proportional to the distribution of longi-
tudinal diSplacement for fixed N. Based on the foregoing discussions,
the quantities in the bracket of Eq. (49) may be c0nsidered as approx-
ima tely equal to unity except the ratio of (5550) 1:; to <sec0>fiy
This ratio may be larger or smaller than unity dependent on their
average orientations because they represent two different paths in a
porous medium.

The following additional aSSumptions are necessary with regard
to Eqs. (45), (48) and (49) in order to simplify the relationships
between random variables.

(1) In a given path the average orientation of the path is

approximately the same at different stages. This implies that

if <eos<9>17I is obtained for a fixed N steps, then, provided
the fluid particle continue this path until a fixed time "T, the
new <C059>T will still be approximately equal to .<C059>fi°
Therefore, the quantity

<cosG>fi2 <5ec0>zo

____.____._.______———
<cosQ> Zo<sec0>¢f

in Eq. (48) will be approximately equal to unity.

(2) The ratio TZO/TN is assumed to be approxunately
equal to unity. Since N is the average number of steps re-
quired to arrive at a given longitudinal distance Z0, therefore,

the average time required for particles to complete a longitu-

 

 

 

 

27
dinal distance of ZO should be approximately equal to the
average time required for completion of N steps.
(3) In a given path if an average <cosQ> is obtained, then
there exists an average <5ec0> such that (c050) = l/(secQ)
This assumption will inevitably produce an error whose
magnitude depends on individual paths. An investigation of the

magnitude of error for this assumptiOn is presented in Appendix

II.
(4) The ratio of <sec0>fi to <5ec0>f in Eq. (49) is assumed
to be approximately equal to unity. The possible error involved

in this assumption is investigated and presented in Appendix II.

Based on the above assumptions, we may replace Eq. (45) by

z-
N

zf=:—- (50‘
21?

and Eq. (49) may be replaced by

._ : _. )
zT ZN (51,
Also, Eq. (48) may be replaced by
.7. _.
Z“ = "0 T20 (52)

Eq. (52) is to be used to obtain the distribution of the longitudi-
nal displacement for a fixed T from the measured distribution of time
for a given distance 20. Ens. (‘30) and (51) are to be used to obtain
the theoretical distribution of the longitudinal displacement from the
known distribution of longitudinal displaceirient for a fixed number of

steps N as expressed by Eq. (25a).

28

7. The Longitudinal Dispersion of Fluid Flow in a Porous Medium.

 

During the displacement of one miscible fluid by another, the
break through curve obtained from Eqs. (50) or (51) gives the particle
concentration at the mixing front. The degree of dispersion is mea-
sured by the standard deviation. It is necessary to derive the ex-
pressions for the expected mean and standard deviation for the dis-
tribution of longitudinal displacement at a fixed time fzo.

Define /U{ as the "it'th moment about an arbitrary point and
fi as the "i"th moment about the mean. Then, by definitiOn the

following general equations may be written (see Kendall (8)).

/u.i'=[w(x)ip(x)dx ............(53)

/u~i = S (x—fl{)ip(x)dx . . . . . . . . . . . (54)

We have

/é:/uz+/*1Z.............(55)

and 2 3 4
/"4/ :/*4 +4/‘f/A3+éy"1/*z+ 4/“,1/“1‘7‘i . . . . (56)
where /U~2 is the variance and since for the normal distribution
/‘1 =/*3 = 0
and
f4 = 3/2
therefore, Eq. (56) may be written as

2 ,2 ,4
/U»4 = 37LZ+6/LL1/u.2+/LLI . . . . . . . . . (56a)

   

 

29
Now, from Eq. (50), the expected value for 23. may be evaluated as
- — ‘1 Z
Elzrfl-ZI:I E(ZN‘)............(57)
Z
and from Eq. (55)» E(Z’N) may be written as
Z 2 3
Bug) fight—(ZR) . . . . . . . . . . . . (58)
Therefore, by substitution of Eq. (58) into Eq. (57), it is obtained that

-1

Hz?) = EN (03%;). . . . . . . . .(59)

The variance of Z]: may be obtained from Eq. (55)

- 3 2
Var(Z—,-) -mfi) - EMT) (60)
and from Eq. (50), since
3 _—z 4
EllezzfiHZfi) ............(61)
Also from Eq. (56a)
4 4 _ Z 2 4
E(z§) 23W§+6z§ (TI—q +251— . . . . . . . (62)

By substitution of Eqs. (63), (61) and (59) into Eq. (60), it is obtained
that

_ -2 4
Var(Z-,T) :2leI (G—"N'l’4—Z-NZGE—qz). . . . . . . (63)

For the case of n 3 1, from Eq. (24)

N

(TE—:31: ...............(64)
N18

   

 

 

30

and from EqS. (21) and (Z3)

_ _ 2—

23': - 3NL . . . . . . . . . . . . . . . (65)
therefore

2 - 1 —

(TN .1? fit, . . . . . . . . . .. . . . . (66)

SubstitutiOns of the above three equations into Eqs. (59) and (63)

give the following expressions:

-— L
E Z— : _. _.
VarlZ—) : Er. +i) . . . . (68)
T 3 N 24 a I l l g a u
and
GE. - L L
’ Z“+—.............69
T 3—(N Z4 ) ( )
Eq. (59) and Eq. (63) are the general expressions for the ex—
pected mean and the variance for 271-“. Substitutions of Eq. (21) and

Eq. (22) into Eq. (59) and Eq. (63) give the expressions in terms of

the orientatiOn factor n.

_ Z
Elzflzzfi'l'llm -Cn). . . . . . . . (70)
_ Z 2
63le : 4ZK‘ [.(Wn— - Cn)+ 2L2( (WC—n— _ cn)2 , . (71)

where the term Cn is expressed as

_(n+l)(n-l). .. 1Tb
C_W(T)........(7Z)

 

‘ ‘-
Qnol'i

 

 

31
and b I l for n equals even numbers and b 2 0 for n equals odd numbers.
Let us define a numerical constant Cr'1 to include all the terms that

coutain the orientation factor n in Eq. (70) and Eq. (71) as
Cfi=W-Cn............(73)

In Figure 3—6 the relationship of Cl'1 against n is shown. It is
noted that C1,) has its maximum value at n : 4 and then decreases

rather slowly as n increases.

 

 

 

 

 

 

 

 

 

1.
— g / \
o ‘ \<
“5 >
v . 75
.3
as
>

. SO . l

l 3 5 i 9 ll
Orientation factor n.
Figure 3-6. Relationship between orientation factor n and

ch in Eq. (73).

The above analysis for the expected mean and variance for Z?
are based on the relationship of random variables as represented by
Eq. (50). On the other hand, Eq. (51) may be used instead of Eq. (50).
Since in Eq. (51) the distribution of longitudinal displacement for a
fixed time is considered to be equal to the distribution of the longitu-
dinal distance for a fixed TV, the expected mean and the variance for

2T may be represented by Eqs. (21), (22), (23) and (24).

 

 

ad "
em”
k

 

 

111‘

 

32

8. The Permeability Coefficient and Orientation Factor.

 

In Figure 3—1 a fluid particle enters the porous medium at
point 0 and arrives at point 01 after travelling N unit canals. The
permeability is defined as the velocity of flow for a unit pressure
gradient. If the permeability of the porous medium is measured in
two perpendicular directions as K2 and Kx and if (aim) and <CW>
are used to denote the average value of the flow path through the
porous medium with respect to these two gradient directions, then a
relationship between canal orientation and the permeability coeffi-
cients can be established in the following manner.

If K2 and KX are defined as the permeability coefficients along
Z-direction and X—direction respectively, then the following relation-

ships may be written:

Kz=Zo/TZO=L<c—o§7 MOORE) . . . . . (74)

szxo/TXO:L<Eas—q3> /t0<schL—>. . . . . (75)
Therefore,

K, _ was» «“65”

 

F-<m7 <W7 .........(76)
x
The quantities on the right hand side of Eq. (76) can be calcu-
lated using the equations derived in the preceding sections. Since the
distribution of canal direction for the general anisotrOpic case was
assumed to have the form represented by Eq. (12) and the theoretical
probability function of the choice of canals based 0n discharge and

geometric distribution was derived as Eq. (17) for a gradient in

 

 

 

',

 

33

Z dlreCthu, the quantities (c059) and (secQ) "iay be expressed as

(go—5.9) =5; [g 89¢ c059

  

n+1 2W .,
= 2T sinnO cos“0d13d)21 (77)
o o
(78)

Similarly,
(seco) = $15: Tr(:sin“ 96m

The numerical values of (c059) and (secQ) are calculated
frorn the above two equations for variOus n values and plotted in

Figure 3—7.
For the gradient in X direction, Similar derivations would lead
section 3)

to the results as follows (see appendix I
(79)

((2%..— f" jfisinn +ZQCOSZ¢de¢

("05”) = '3 2n(n-2) .
Odd nimbers of n and is equal to/z for

Where B is equal to 1/T1 for

even numbers of n
.L
n+1) n—l) . . ‘1" TI
( ( f sinngdgdw.

(”CW = 32501-2)
The numerical values of (c056) , <cos¢> , <5ec9> and (sec?)
'5, .— . It

(801

.L
‘2.“ o

are calculated for various n values and are s lO\ n in FI’Y‘JTC 3 7

is Seen that the values for (Eds—55> and (COT?) , (5—6755) and (W)
are Identical for the case of isotrOpical porous rredium The ratios
are also calculated from Eq. (77) usma these values

lor K and K
2 x
810 plotted as shown in FleIe 3 8

 

v0"
\

 

 

 

 

 

cosW)

and <

Values of <cosO> .

34

 

 

 

 

 

 

 

 

 

 

 

 

 

Values of (sece) and <

 

sec (a).

1. 0
(cos?) for X-gradient.
/ /O
o 8 / /
' f
0. 6 d
\ _ I
<5ecG> for Z-gradient.
O 4
m\
/ NO
(c056) fOr Z—gradient.
0.2 \\O\C mp,”
4
(secy) for X-gradient.
O
l 5 10 15 20
Orientation factor n.
Figure 3-7. Numerical values of <cosO> , (sec0> ,

\RfsV) and <s'ecy) for various n values.

 

 

 

 

35

 

 

 

 

 

 

 

 

 

 

 

 

l. 2
Assumed pore canal distribution
function.

a
: —— ' n
p 2T1 Sin OdOdfl
l. i)
.x 1
x; l
\
N
M
“5 .
m 0.7-
u
.3
a
>
'o
u
‘5
g 0.5
.2
a:
U
o. 2 \O\O\
\NO
0 J
1 5 10 15 20 25
Orientation factor n.
Figure 3-8. Relationship between the orientation factor n

and the permeability ratio in an anisotropic

porous medium calculated from Eq. (76).

 

"" i

 

CHAPTER 4

EXPERIMEN TA 1_. PROG RA MS

1. Sample Preparation.

 

Packings of Spherical, plate shaped, and cylindrical particles
were used to make the porous medium for the experiments. Ottawa
sand was used to represent the Spherical particles. The sieve anal-
ysis of the Ottawa sand sho'v-ss that 97. 7 percent of the particles have
a diameter beta-seen 1). 5‘? to 0. 84 mm. Plate shaped particles were
made out of plastic sheets. The plates were cut with a metal cutter
carefully controlled to give uniform particle size. Besides the
Spherical and the plate shaped particles, cylindrically shaped par-
ticles were prepared from nylon filaments. A summary of the

sample properties is given in Table 4—1.

2. DiSpersion Measurements.

 

To measure the dispersion, a percolation apparatus as shown
on Figure 4-1 was designed. The percolation cylinder was made
Of lucite. The cylinder was 6. 35 cm. in diameter and 14 cm. in
length. Pairs of electrodes were inserted at various distances from

the bottom of the cylinder. The locations of the electrodes were as

 

 

follows:
Position of electrodes Distance from base
Position 1 l. 25 cm.
Position 2 5. 05 cm.
Position 3 8. 85 cm.
Position 4 12; 65 cm.

36

 

 

 

 

 

37
The electrodes consisted of ‘5: mm. stainless steel wire and were
connected to an Impedance Bridge of 1000 cycles per second. This
set—up enabled the measurement of the electrical resistivity of the
fluid around the electrodes.

The percolation cylinder was packed with the particles to make
up the porous medium. The uniformity of the packing was c0ntrolled
carefully during the process of packing. A 1. 20 cm. thick layer of
glass heads 6 mm. in diameter was placed underneath the percola—
tion cylinder to serve as a filter. Between the porous medium layer
and the filter a wire screen with 0. 5 mm. opening was inserted.

The system was first saturated with O. 001 Normal NaCl
solution. Then at the bottom of the cylinder the liquid was replaced
with 0. 1 Normal NaCl solution. The liquid at the bottom of the
percolation cylinder was connected to a constant head supply tank of
O. 1 Normal NaCl solution. Thus the liquid flow in the cylinder was
upward and the flow rate was measured volumetrically at the outlet.
The velOCity of the flow in the porous medium was controlled by ad—
justing the head between the liquid surface in the supply tank and the
outlet elevation of the cylinder. Different velOCities were used for
the same sample to obtain a range in the duration of time in order to
obtain any information regarding the time effect on dispersion.

The flow velocities in the dispersion measurements were kept
sufficiently small to produce laminar flow in the pore system. The
actual velocities were all less than 1. 0 cm/min. And since the
largest size of the pore canal can be considered as to be appr0xi-
mately between 0. l and 0. 3 cm. , the Reynold's number can be

calculated as

 

 

ao‘ "
1'0"

 

 

38
VDf’ l l
R = 7‘ = W“ to 7.66—

where V is the velocity, D is the size of the pore, f is the density
of the fluid and /U- is the viscosity of the fluid. A Reynold's number
of unity is usually considered as the boundary between laminar and
turbulent flow.

The time history of the change in NaCl concentration at each
electrode position was determined by measuring the electrical re—

sistance of the system with Impedance Bridge. From calibration

 

test the relationship between the NaCl concentration and the elec—
trical resistance was found to be linear in the IOg-log plot. There-
fore, by measuring the electric resistance against time, the concen-
tration of fluid around the electrode can be obtained by direct

interpolation.

3_. Permeability Measurements.

 

The permeability constant for the porous materials used in
the dispersion measurements were determined by using a perma-
ability apparatus shown in Figure 4-2. The apparatus was made of
lucite and was designed to measure fluid flow in two perpendicular
directions.

The porous materials were packed in the apparatus from the
tOp. Preferred particle orientation was produced in the direction
perpendicular to the direction of packing. After the medium was
saturated with water, flow was introduced in the vertical direction
and the rate of flow was measured at the outlet. The permeability

coefficient was calculated by Darcy's law.

 

r

 

-~"s

Q

 

 

39

The longitudinal and transverse permeabilities K2 and Kx are

defined respectively as the permeability of flow perpendicular and

parallel to the direction of packing. According to the relationship

derived in Chapter 3, the average orientation of the porous medium

can be evaluated from the K2 and Kx data using Figure 3—8.

4.

Presentation of the Experimental Data.

 

(l) Dispersion with respect to time.

Electric resistivity of the fluid in the percolation cylin-
der was measured at three electrode positions. The resis—
tance was used to obtain the NaCl concentration of fluid at the
electrode position by direct interpolation from the calibration
curve. The measurements were taken at various time intervals
after the 0. l N solution was introduced. Results of these
measurements are tabulated in Table V-l through Table V-ll
in the appendix. The time required for the NaCl concentra-
tion to reach 50% of O. 1 Normal NaCl concentration was taken
as the average time, TZO.

(Z) Diapersion with respect to distance.

As derived in Chapter 3, the break through curve with
respect to distance at a given time can be calculated from the
time distribution at a given distance. These break through
curves are shown on Figure V-l through Figure Vcll in
Appendix V. The electrode positions were taken as the average
distance of the break through curves. The standard deviations

with respect to distance are also shown in the figures.

 

a.“ “
.'.’l

 

 

ll

 

40
(3) Permeability coefficients.

Permeabilities were measured in the laboratory in two
perpendicular directions using the procedure described in
section 3. The results of the permeability determinations are
listed in Table IV-l through Table IV-S, and also shown in
Figure IV—l through Figure IV-4 in Appendix IV. A summary
of these results is shown in Table 4-2. Different perme—
abilities are obtained for all the porous media used except the

Ottawa sand.

 

 

 

 

 

Q‘O‘I

 

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Side View

 

l‘—_5.lcm ——1'
MT 1 1m

 

M

 

 

. E ,__
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Section A -A'

Figure 4~2. Sections of permeability apparatus.

 

 

 

43

 

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N.oe eosxso.m m.emm
m.mm aosxsv.m m.~>m
m.¢a «oaxmm.v a.w>~
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0:5an

 

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scuacaaau

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: : Nam

.. : Mvm

: __ mmm

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: __ Nmm

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Sample
test no.

8

PPM
131312
PP13

PP'
12

PP21

PPZZ

PP21

I
PP22

P133l

PP3Z

PP31

PP'32

PP41

PP42

PP43

I
PP41

PP'
42

N!

Table 4-2.

Material

Otta wa
sand

plastics
II

Nylon
filaments

H

44

Samples for permeability measurements.

Particle
Dimensions

 

. 7mm dia.

.3“

.3X.

. l". 03C1n

! !

Porosity
%

35.

34.

41.

39.

46.

43.

43.~

34.

38.

29.

34.

31.

34. L’

35.

00

00

60

60

20

.60
.20

.20

.20

Perm.

8.

o 0‘ 0‘

10.

10.

ll.

11.

ll.

10.

10.

ll.

12.

ll.

cm/min.

85

52

70

22

20

.05

.24

.42

42

1'7

00

Direction
of HOW

Longit.
& Transv.

Transv.

ll

Transv.

Longit.

 

 

 

 

 

CHAPTER 5

ANALYSIS OF EXPERIIVIENTAI.. RESULTS

1. The Break Through Curves.

 

The procedures for measuring the dispersion of fluid flow in a
porous medium as described in Chapter 4 give a relationship between
salt concentration of the mixing fluid front at the electrode positions
and time. Figure 5—18 shorr s a typical curve for this concentratiOn
versus time. If there is no diSpersion phenomenon taking place and
the molecular diffusion is neglected, then there "would be simply an
abrupt change in concentration from 0. 001 to ' . 1 Normal at the mix-
ing front. This is the horizontal line shown on Figure 5-13.

NOW, if T20 is the average time, then the distance a fluid par-

ticle would have travelled at time TZO can be calculated from Eq. (52)

...._,_._...-

 

1“

in which T20 is the tirr‘ie in a curve represented by Figure 5~la. the
distance thus c-f:lculated gives the break through curves with respect
to distance at a given time :20. Figure 5~lh shows such a curve
calculated from Figure 33-13. The calculated break through curves
with respect to distance are presented on 17ig-;Lire V-l through Figure
V—ll in Appendix V. One si jiiificant prOperty noted in these break

through curves is that the curves are asymmetric.

2. Dispersion as a Function of Distance.

 

Eq. (69) and Eq. (7l) indicate that in a given porous medium,

the standard deviation of a break tlii'OLi,3;li curve is a linear function

45

 

 

 

 

 

 

46
of the Square root of the average distance. Table 5-1 is a summary
of the standard deviations obtained in the Cl‘lSDBI‘SiOl’l measurements.
The values of the standard deviations (Table ‘-‘-l) are obtained from
Figure V-l thrOugh Figure Vsll Ly taking averages of the vertical
distances between 0. 5315 and 0.. 085 Normal NaCl concentrations. This
assumes that the measured distribution may be approximated by a
normal distribution.

Figure 5-33 through Figure 5-3f show. the relationship betvreen
the average‘standard deviatiOn and the average distance in square root
scale. With no exception, the relationship is linear. Eq. (69) shows
an existence of a finite value of the standard deviation at 20 equals to
zero. However, since the value of l. is very small, it is practically

insignificant.

3. The Characteristics of Pore Geometry.

 

Eq. (71) also shows that the standard deviation of a break through
curve is a function of the orientation factOr n. This orientation func-
tion is represented by Eq. (73) as Ch and plotted against n in Figure
3-6. It can be seen from Eq. (71) and Figure 3—6 that the orientation
factor n is probably the least influential factor among the variables in
the right hand side of Eq. (7l).

As derived in Chapter 3 the orientation factor n can be obtained
from the permeability data. Table 5-2 shows the calculations and
results of the orientation factor obtained using Figure 3-8 and the
permeability data. The permeabilities of the porous materials were
measured in two perpendicular directions as described in Chapter 4.

The permeability data thus obtained permit the evaluation of the

 

 

 

.
""k

 

 

 

47
orientation factor of the porous medium. The results are given in
Table 5-2. Figure 5-4 shows the relationship between orientation and
porosity. For the samples tested, the orientation factor ranges from
1. 0 to 2. 0 approximately.

In Figure 5-3 the lepe 0‘. the line represent the ratio (Fla/£0
which is :1 function of both n and 1,. as was derive-l in Eq. (71}. Since
the values of r: for the known porous materials and gaorosities may be
obtained from Figure 5-4, the value of 13 can. be calculated with Eq.

(71). Table 5-3 shows the calculated values of L. It is seen that in

 

general, L. increases pith decrease in porosity and therefore with
increase in orientation factor n. For randomly oriented case (n = 1‘},
the following comparison between the value of la and the dimensions of
the grain particles constituting; the porous media can be made. For
Ottawa sand, the calculated value of L is 0. 13 cm vhich is approxi-
mately twice the average diameter of the particles. For plates, the
calculated 1.. ranges from ". 12 cm to 0. 37 cm which is approximately
0. ‘5 to l. 2 times the longest side of the plates. For cylindrical par—
ticles, the value of 1.. calculated is approximately two—thirds of the
length of the particles. Since for a loose packing of uniform spheri—
cal particles, tue length of the pores should not be eXpected to be
larger than the particle diameter, the calculated L for Ottawa sand
indicates that tlhe agreement between theory and experiirient is rather
poor. This suggests the error introduced in the assumptions of the
theoretical derivations. Althoiigh the calculated values of 1-. for the
porous materials for plates and cylinders fall in more reasonable

range the same magnitude of errors may be involved in these cases

also.

 

 

 

 

 

 

48

However, the above method of calculating the value L is based
on the relationship represented by Eq. (5t). If Eq. (51) instead of
Eq. (50) was used to derive the standard deviation, then the standard
deviation of the dispersion will be represented by Eq. (24) and the
average length of pore canals L calculated will be approximately
four times that calculated from Eq. (71). Therefore, the standard
deviatiou as given by Eq. (71) seems to be in better agreement with
the experimental reSults.

If Eq. (50) is the true representation of the distribution and if
Zfi is normally distributed, then nyz should also be normally dis—
tributed. Similarly Zr‘f should be normally distributed if Eq. (51) is
to be the true representation of the distribution. The measured dis-
tributions for sample 511 are plotted on probability papers as Figure
S-Za and S-Zb. The distribution curves on these figures do not
reveal any significant difference between the two assumptions made
with regard to the distribution.

Different porosities were used in the diSpersion measurements
in order to produce a range in orientation. It is found that porosity
affects both orientation and the length of canal. From Figure 5-4 it
is seen that the range of porosity used produces only a range of
orientation factor n from 1. O to 2.. O. This seems to indicate that the
actual pore canal orientation does not follow closely the orientation
of the grain particles in the porous medium. For the plate shaped
particles a very dense packing should produce a particle orientation
very close to a horiZOntal. The calculated n values from measured
data (Table 5—3) do not indicate much increased n values for a dense

packing. This may be due to the possibility that a dense packing of

 

 

 

 

49
plates may produce clusters of particles in a porous medium and
fluid flow may occur largely through the continuous pores between
these clusters (see Figure 5—3). Tuerefore, the orientatiOn factor
n is no longer controlled by the orientation of the plates. The contin—

uous pores between tne clusters will apparently reduce the value of n

 

due to the existence of nearly vertical pores. This explanation is
also supported by the observation that there is an almost consistent
increase in the calculated value of L for decrease in porosity or

increase in density for plate samples (Table 5-3). Since the sizes

 

of the pores between the clusters are influenced by the size of the
clusters, it may be expected that the value of L for a dense packing
where clusters are likely to exist would be greater than that in a
loose packing where the fluid flow takes place mainly through the
pores '_.eti'eer. plates and Herefore the value of L is centrolled by

the size of the indix idual plates.

4. Dispersion Coafl'icient.

 

Since the break through curve as shown in Figure 5—lb repre-
sents the distribution of the fluid particles that have travelled a lon-
gitudinal distance Zia after a given time -, the magnitude of the
standard deviation of the break through curve is a measure of the
degree of dispersion and may be designated as the 'dispersion co—
efficient'. It is clear from the preceding analysis that this disper-
sion coefficient is a function of the average distance of journey, the
length of the unit canal, and the orientation factor n.

A numerical comparison is made in Appendix III for the

measured dispersion coefficient in this investigation with those

0

 

 

 

 

 

50
which could be predicted from the present theory and the theories of
De Josselin de Jong and Saffman. De Josselin de Jong's theory was
derived based on the assumption that the porous medium consists of
uniform spherical particles and Saffman's theory assumed a randomly
oriented straight pores in a porous medium. Therefore, the equa-
tions from both theories could be applied to the experimental data of
Ottawa sand samples in the present investigation where the orientation
factor n is equal to unity and also c0uld be roughly compared to the
cases of lmsepackings of plate shaped particles.

In order to apply these theories, the length of the unit canal of
pores must be evaluated. Assuming the magnitude of the molecular
diffusion to be negligible, this can be done by substituting different
length of L into the equations representing different theories, (Eqs.
(6). (7), (69)), and compare the results with the measured dispersion
coefficients. For a porous medium consisting of spherical particles,
the length of the_pores is not likely to be greater than the diameter of
the particles even for a loose packing. Therefore, it is reasonable
to assume that L lies between )5 to that of the diameter of the parti-
cles. For the Ottawa sand samples, a best agreement is obtained
when L is taken as the diameter of the particles. (The value of L
should be reduced if molecular diffusion is taken into account).
Calculations of the dispersion coefficients from different theories
are shown in Table III—Z.

From this result, it can be seen that when L is taken as O. 07
cm where the De Josselin de Jong's theory agrees best with the ex-
perimental results, then the Saffman's equations would give approxi—

mately 10% to 30% higher values and the present theory would give

 

 

 

 

 

51
approximately 30% lower values compared to the De Iosselin deJong’s
results . The difference between De Josselin de Jong's and the pre-
sent theory can readily be seen by comparison of Eqs. () and (’9)
The De Josselin ye .Iong's theory would approximately give a higher

dis per sion coefficient by a factor equal to

(A + 3‘4 — lOgi")

 

which is approximately l. 4. The fact that the present theory is con—
sistently underestimating the measured criSpersion coefficient may be
censidered as the result of the errors involved in the assumptions

made in the course of the theoretical derivation (see ppendix ll).

 

 

A difference of similar magnitude is seen between these theo-
riCS when applied tc loose Ixackir. 's of ltwlate shaped particles ("Fable
III—2). A pore 23.1.1! length of .‘. 1 cm is assumed for P1 and P3
Samples and H- 3 cm {01‘ pa and p4 samples. The calculated results
“'53:: compared to the measured ones suggest that for plate shaped
pa 1“ ticles the lengths of the pores are likely to be smaller than the
longest side of the plates constituting the porous medium.

Frauri te preceding analyses, it can be stated that the present
thCtOry consistently underestimates the dispersion coefficient. The
amOunt of error could be appr0§;llliatei)' 30% or even more for the
"Ta Se of n = l. Thereiore, it is probable that for the cases of n
Fs'reater than 1 or for anisotroyic porous iriedia, similar magnitude Of
e3'5‘Ji‘0rs are existinw. It is not possible to accurately predict or calcu-
late the length of the pores for an anisotrOpic porous medium packed
with plate shaped particles. However, by assuming that the error
it1VOlved in the present theory is approximately 30%. it can be 63“-

thelted that the length of the pores should be smaller than the longest

 

 

 

 

 

52

side of the plates and probably comparable to One-half the length of

the IOngest side.

 

 

 

!

.1..|

 

 

53

 

 

Table 5-1. The Average Standard Deviations
Sample Porosity Electrode Average
No. % Position 0‘20 in cm.
51 & s2 34. 25 1 0.270
2 0.452
3 0. 705
1"1 1 37. 92 1 0.310
O. 747
3 1.080
P1 2 43. 31 1 0.400
2 0.809
3 1.190
P13 50.42 1 0.307
2 0.625
3 0.818
F) 44.52 1 0.442
2 1 2 1.120
3 1. 510
1322 52.18 1 0. 350
2 0.843
3 1.160
P 38.92 1 0.418
3 1 2 0.820
3 1.060
P 52
3 44.61 1 0.2
2 2 0. 540
3 0.682
P 0 425
4 44. 25 1 .
1 2 0. 763
3 l. 068
P 468
33.46 1 0-
42- 2 0.925
3 1.338
N & N 49 19 1 0.388
1 2 2 0. 736
3 0.992

 

 

 

54

Table 5-2. Orientations of the porous" media as calculated

from the permeability data.

 

 

 

Poro us Porosity Kz Kx Kz/K Value
Material % cm/min. cm/min. x of n
:5 35.00 8.85 8.85 1.000 1.00
1331 32.00 5.83 8 56 0.681 2.00
36.00 6.85 8.89 0.771 1.70

40.00 7.88 9.20 0.857 1.45

44.00 8.90 9 51 0.936 1.25

47.60 9.80 9 80 1.000 1.00

I>2, 36.00 8.02 10.41 0.770 1.70
40.00 9.00 10.80 0.833 1.50

44.00 9.98 11.30 0.883 1.35

48.00 10.95 11.77 0.930 1.25

P3 34.00 8.03 10.00 0.803 1.55
36.00 9.31 10.65 0.874 1.30

38 00 10.61 11.29 0.940 1.20

40.00 11.87 11.91 0.997 1.05

13.; 30.00 8.74 10.48 0.834 1.50
32.00 9.40 10.74 0.875 1.40

34.00 10.06 11.01 0.914 1.27

36.00 10.74 11.28 0.952 1.20

38.00 11.42 11.55 0.989 1.10

Pq 49.20 11.00 12.75

 

Q‘d

 

 

 

55

 

 

 

Table 5-3. Length Of pore canal obtained from experimental data.

Sample Porosity O‘iZ/fZ—O c' ... L H

No- % T n 11 cm.
5 34.25 0.21 1.00 0.083 0.133
P 11 37.92 0. 35 1.60 0.087 0. 352
P 1 2 43.31 0. 39 1.26 0. 085 0. 447
P 1 3 50.42 0.26 1. 00 0. 83 0.204
P21 44. 52 0. 52 1. 31 0.0855 0. 791
P22 52.18 0.40 1. 00 0.083 0. 482
P31 38.92 0. 35 1.11 0.845 0. 362
P32 44.61 0.20 1.00 0.083 0.120
3P41 ‘ 44.25 0. 35 1.00 0. 083 0. 369
P42 33.46 0. 445 1. 37 0. 086 0. 576
N 49.19 0. 33 1.00 0.083 0. 328
"‘ The values of Ch are obtained from Figure 3-6.

** The values of l. are calculated from Eq. (71) assuming
that the values of n obtained from Figure 5-3 are appli-
cable to Eq. (71), (72) and (73).

 

 

 

 

2.0

in minutes

Time

56

T l I I r l

 

r 1 l

Sample test NO. : P411
Position 2

Theoretical break through curve
if no dispersion takes place.

 
 
 

 

 

 

 

 

 

JD Break through curve observed.
:3 ‘
I I L l L l I 1 l
.001 .02 .04 .06 .08 .10

Distance Z in cm.

Salt concentration measured in Normal

Figure 5-la. Typical break through curve with respect

to time.

 

I l r l 1 r l 1 1
Sample test No. : P
T = 11. 20 min.

     
  
 
   
 
 

411

Break through curve
calculated from Figure 5-1. x

 

 

Theoretical break through
curve if no dispersion
takes place.

-

 

 

l l l

.02

 

L
. 08 . 06 . 04
Salt concentration in Normal

.001

Figure 5-1b. Typical break through curve

to distance.

with respect

 

 

 

 

 

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
~o\\o\
9 - Nku
17.0 = 8.85cm \O‘K‘K‘
T- : 13.331‘11111. RR
8 , i 0
; o
7 %
lE—4 L
N
2c: 6 1
8 i roN
.‘3 ”\km
52" 5 *7 \
:8 :20 : 5°05lcm \‘N
73: 1T 2 7.046616. >\\°
"So
:3
3
3
Z
0
MN
”fix
”‘0
\l 0 “\N-oa
1 N9.
2‘0 : 1.2!5cm. b
T : 2.40mi11.
0
Z 10 20 3O 4O 50 60 70 8O 9O 95 98

Figure S-Za.

Percentage of fluid particles

The relationship of 7.5:: and the percentage of fluid
particles that has reached the longitudinal dis-
tance of Zr’f. Sample No. S] 1.

 

 

 

 

 

58

 

 

20: 8.85cm mom
13. 38min. N

HI
ll

 

 

 

 

u
4

lo

 

 

 

 

Longitudinal displacement (2'. :f)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l. 5
\Q\fl\ N
\ONN
\cx
.. ‘ \KK
I. 0 20 1 2 cm \n
.. _ \ o
T ” 2.4G31nih. \\0
0. 5
2 5 10 20 3O 4O 50 60 7O 80 9O 95 98

Percentage of fluid particles

Figure S—Zb. The relationship of (2:5))é and the percentage of
particles that has reached the longitudinal dis—
tance of ZT' Sample No. 811

   

 

 

59

UT

 

Sample 5, T. {-7 out: dia. Sphere

O 51 8.: 82, PoroStity = 34.4% 25! 35.0%

./K

 

m cm.

'.
h-q
n

0?,

 

kl!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O
O
l v; 9 16 25
Average longitudinal distance in cm.
Figure 3-3a. Relation between standard deviation and
the longitudinal distance, S.
3
Sample Pl, 0. 3x0? 1x0. 03 cm‘
E] P1 {, Porosity 37. 9%
2 Op”, Porosity 43.3%
E- 0 P13 , Porosity 50. 4%
U V
.S I
”-4 . K
g 1 ¢//
0 /
0 1 4- 9 16 25

Average longitudinal distance in cm.

Figure 5-3’0. Relation between standard 'jeviation and
the longitudinal distance, P1.

 

60

 

 

 

 

 

 

 

 

 

 

 

Sample P2, 0. 3x0. 3x09 03 cm
.Pz1 Porosity 44. 5%
0P2; Porosity 52. 2%
E
U
c /
lf_——¢ /
N .
b 1.0 /O//C
0 o

l 4 9 16 25
Average longitudinal distance in cm.

Relation between standard deviation and
the longitudinal distance, P2.

Figure 5—3c.

 

 

 

3.0
Sample P3, 0.3x0 leOOo cm.
.1321 Porosity 38.9%
2 O 0P2; Porosity 52.2%
E
a.)
.E i/
'E—i /
N
E 10

w/w
54M

4 9 16 25
Average longitudinal distance in cm.

 

 

 

 

 

 

 

Fa

Relation between standard deviation
and longitudinal distance, P3.

Figure 5-3d.

 

q-o'.

 

 

61

 

3.0
Sample P,,_, 0. 3xd. 3x0. 06- cm,
@1341 Porosity 44.3%

2 o .134; Porosity 33.5%

 

 

1.0 /

 

 

 

 

 

 

 

 

0 o
1 4 9 16 25
Average longitudinal distance in cm.

Figure 5-3e. Relation between standard deviation and
longitudinal distance, P4.

 

 

 

 

 

 

 

 

 

 

3. O
Sampllt N, 0. 05 cm. dia. 0. 5 cm cylindrical
- QN1 3: NZ, Por015ity 49.2% & 49.8%
E 2.0
-E
is
g N /
l. 0 f
0 0
1 4 9 16 25
Average longitudinal distance in cm.
Figure 5-3f. Relation between standard deviation and

longitudinal distance, N.

 

 

 

 

Orientation factor n.

N

H
0

0

U1

62

 

 

0 P1
0 P2
In
‘\ I P3
13 P4

 

 

 

 

 

 

 

 

Figure 5—4.

40 45 V 50 55

Porosity, ”/0

RelatiOn between porosity and the orientation
factor n obtained from Figure 3-8.

 

 

 

 

 

(a) Assumed pattern oi flow paths in either loose
or dense packing of plates.

 

(b) Probable pattern of flow paths in a dense packing
through pores between clusters of plates.

Figure 5-‘3. Schematic sketch of the possible patterns of flow
paths in a porous medium consisting; of plate
shaped particles.

 

 

 

 

t

i
..

 

 

CHAPTER 6
DISCUSSIONS AND CONCLUSIONS

Statistics were applied to study the mechanism of fluid flow in a

porous medium. If the medium is such that its pores have different

orienta tions with respect to certain direction and the hydraulic gradi~
cut is the sole cause for the fluid particles to move in the pores, then
the tortuous paths of the pores will immediately result in a dispersion
of flui (3

flow in the porous medium. It was expected that this treat—

ment would give a functiOnal relatiOnship between the characteristics

 

0f the fluid flow and the porcus medium. Two sets of equations were

derive (:1 representing the relationship between the standard deviation
Of 21 br eak through curve, distance, pore dimension and the orienta—
tion Ea ctor of the pore canals.

In order to simplify the theoretical analysis, a canal network
model was used to represent the pore system in a porous medium. A
Serie s of assumptions with regard to the orientation of an individual
por e and the average orientation of a path were made. These assump-
tions introduce an inevitable error into the result of the theoretical
derivations whose magnitude depends on the particular path.

Since anisotropy is a basic property of a porous medium, the
tb‘eQretical consideration was based on an assumption that the direc-
tions of pores will have a distribution of preferred orientation and

that this distribution can be expressed mathematically. Therefore,

the degree of dispersiOn as represented by the standard deviation of

 

the break through curve contains a factor indicative of the orientation

Of the porous medium. It was also shown that this orientation factor

QChild be related to the ratio of permeability coefficients in two perpen—

 

64

m

 

 

 

 

65
dicular directions in the porous medium.
The fol‘owing conclusions can be drawn from the theoretical
cons iderations of this study:
(1) A mathematical relationship representing the break
through curve of a miscible fluid front in a porous medium may
be derived from a theoretical consideration of canal network
model and an assumption of pore direction distribution functions.
(2) The degree of dispersion as represented by the standard
deviation of a break through curve is a function of the longitu-

dinal distance, the length of the pores and the orientation of

 

the porous medium.

(3) According to the functional relationships derived, the
standard deviation is directly proportional to the square root

of the longitudinal distance for a given porous medium.

(4) The standard deviation for different porous media, other
factors being equal, increases with the square root of the length
of the unit canals.

(5) The effect of the orientation of the porous medium when
other factors being equal, is such that the standard deviation
increases with an increase in the orientation factor n, for n
smaller than 4, and decreases with increasing n, for n greater
than 4.

(6) The magnitude of error introduced in the assumptions of
the theoretical considerations depends on the particular path
chosen. When the longitudinal distance of travel by a fluid par-
ticle is considered, this error may amount to 10% to 20%. An

accurate estimate of the error in an actual case is not possible.

 

 

 

 

66

The results of the experimental analysis in the laboratory
showed that the dispersion phenomenOn in a porous medium is a
macrosc0pically measurable quantity. However, the experimental
res ults agree only qualitatively with the basic equations of diSper-
sion derived from a theoretical consideration in this study. The
dispersion as represented by the standard deviation of the break
through curve was found to be a prOperty of the porous medium.
The medium properties that affect the diSpersion were found to be
the length of the pore canal, the distance of travel by the fluid par-

ticles, and the orientation of the pore canals.

 

Comparisons of the experimental results with the theoretical
EQUations showed that the present theory is consistently under-
estimating the degree of dispersion by an amount up to approximately
30 %. This may be an indication of the magnitude of error involved
in the assumptions made in the theoretical analysis. Since the length
Of the pores in a porous medium consisting of Spherical particles can
Only be roughly estimated, and an estimate of the length of pores in
an anisotrOpic pOrous medium consisting of plate shaped particles
Would be much less accurate, the above mentioned difference between
the theoretical and experimental results would prevent a reliable
Calculation of the medium prOperties from the experimental data.
However, a qualitative deduction of the effects of the factors affect-
ing the diSpersion phenomenon in a porous medium is still possible.

The experimental data on the permeability measurements on
two perpendicular directions in anisotrOpic porous media showed
that the orientation factors thus obtained seemed to be much less
than expected. This suggests that for a dense packing the fluid flow

may largely occur in pores between clusters of particles. Therefore,

 

 

 

 

67

the length and orientation of the pores in such a porous medium are
likely to he greatly influenced by the size of the clusters and also the
distribution of the clusters in the porous medium. The evaluation of
medium properties in such a case “Quid be very difficult and the
deviations from theory woald tend to be greater.

The follo‘xxing conclusions may be drawn from the experimen—
tal analysis of this study:

(1) he standard deviation of a break through curve is found

to increase with square root of the longitudinal distance as pre-

dicted by the theoretical analysis.

(2) Generally Speaking, the length of pores as evaluated from

 

the experimental data increases ith the increase in the size
of the grain particles. An actual lemrth of the pores is a func-
tiOn of the packing characteristics and the factors affecting the
length are not clearly known. The formation of clusters of
particles may be one probable explanation.

(3) The standard deviation increases with increase in the
length of the pores as indicated by the theoretical analysis.

(4) The orientation factor of the porous medium is a function
of the packing characteristics. It increases with decrease in
porosity. The experimental results indicate that for a dense
packing, the orientation of pores does not follow closely with
the direction of the grain particles.

(5) For the range of the orientation factors used in this
investigation, the standard deviation increases with increase
in the orientation factor.

(6) Anisotropy is the result of preferred orientation of grain

 

particles or pores. Permeability coefficients in two different

 

 

 

 

68
directions in a porcus medium may be used to study this

anisotropic cha racte ristics.

Based on the above discussions and conclusions, it is possible

to a ssume a statistical model to study the mechanism of dispersion

phenomenon in a porous medium. From this result, it shOuld also

be possible to investigate the characteristics of the porous medium
by Observing the dispersion phenomenon of the fluid flow in the
medium. Although an assumption of the identical unit canals joined

tOgether in a porous medium is an oversimplification, and the as-

sumptions with regard to the orientations of the paths introduce an

 

i~11evitable error, this type of study provides an understanding of the
mechanism of the fluid flow taking place in the porous medium.

The packing characteristics of the porous medium which were
not COnsidcred in the present theoretical analysis such as porosity,

packing uniformity, are found to affect the dispersion. An accurate

e stimate of the size and shape of the pores was not possible in the

present investigation. These are affected also by the packing char-

acteristics. This effect is very complicated and further investigation

is recommended along this line.

 

 

 

 

(1)

(Z)

(3)

(4)

(5)

(6)

(7)
(8)
(9)
(10)
(11)

(12)

(13)

(14)

LITERATURE CITED

Bastow, S. H. and Bowden, F, P.; Physical properties of
surfaces. Proceedings, Royal Society of London. A151,
1935, p220.

 

Blake, F. C.; The resistance of packing to flow. Trans-
actions, American Institute of Chemical Engineers, Vol. 14,

I922, p415.

Carman, P. C.; Permeability of saturated sands, soils and
clays. Journal of Agricultural Science. Vol. 29, 1939, p262.

 

Childs, E. C. and N. Collis-George; The permeability of
porous materials. Proceedings, Royal Society of London.
Vol. 201A, 1950, p392.

 

Day, P. R. ; Dispersion of a mOving salt-water boundary
advancing through saturated sand. Transactions, American
GeOphysical Union, Vol. 37, 1956, p595.

 

De Josselin de Jong, C.; Longitudinal and transverse diffu—
sion in granular deposits. Transactions, American Geo-
physical Union, Vol. 39, No. l, 1958, [$7.

 

Elton, G. A. H.; Electroviscosity, Proceedings, Royal
Society of London. Vol. 194A, 1948, p259.

 

Kendall, M. C.; The Advanced Theory of Statistics, Vol. 1,
J. B. Lippincott Company, 1943.

 

Kozeny, J.; Berin Wien Akad. 1927, 136a, p271.

Lambe, T. W. ; The engineering behavior of compacted
clays. ASCE Proceedings, Journal of Soil Mechanics.
Vol. 84, Paper T655, 1958.

 

 

Marshall, T. J.; A relation between permeability and size
distribution of pores. Journal of Soil Science. Vol. 9, p1.

 

Michaela, A. S. and C. S. Lin; Effect of counterelectro~
osmosis and sodium ion exchange on permeability of

Kaolinite. Ind. and Eng. Chem. Transactions, Vol. 47,
1955, p1249.

 

Mitchell, J. K. ; The importance of structure to the
engineering behavior of clay. MIT Thesis. 1956.

Olsen, H. W. ; Hydraulic flow through saturated clays.
MIT Thesis. 1961.

69

 

 

 

 

 

(15”
(led
(179

(lifl

(19)

(20)

7O

Quirk, J. P.; Permeability of porous media. Nature,
Vol. 183, 1959, p387. 13';- :3}

Saffman, P. G. ; A theory of dispersion in a porous medium.
Journal of Fluid Mechanics. Vol. 6, 1959, p321.

 

Scheidegger, A. E. ; Statistical hydrodynamics in porous
media. Journal of Applied Physics. Vol. 25, 1954, p994.

 

Scheidegger, A. E. ; Statistical approach to miscible dis-
placement in porous media. The Canadian Mining and
Metallurgical Bulletin. 1959, p26.

 

 

Terzaghi, C. ; Determination of permeability of clay.
Engineering News Record, Vol. 95, No. 21, 1925, p832.

 

Zunker, F.; Z. PflErnahr. Dung, A. 25, p1.

 

 

 

 

 

APPENDIX I

DERIVATIONS OF THE RELATIONSHIPS

1 . Derivation of Eq. (13).

Eq. (12) is the probability distribution function of canal direc-
tion at any point in a porous medium. Let P be the toal probability

Of the choice of directions, then

an 1"
P= [ did 3— sinnOdQ = 1
o 0 2”

or

in
J a sinne d6
0

The normalization constant a is evaluated as follows:

iTT
[sinnede _ (n—1) (n-3). . 3. 1(1)
0 n(n—Z) . . . 4.2 2

A”
n (n-1) (n-3) . . 4.2
sin 9d9
0 n(n-2) . . . 3.1

for n odd. Therefore, the normalization constant a may be express-

H
l—d

fOr n even and

 

ed as

n(n-2). . . . 2b
a=(n_1)(n_3) . .(T')' . . . . . . (13)

 

Where b = 1 for n even and b = O for n odd.

2. Derivation of Eq. (22).

 

Since 2 = L. c056, the variance of Z can be expressed as

71

 

 

 

 

72

Var(Lcos€1) = JLZCOSZQ gee! - 5(212.
8

where gel! is represented by Eq. (17) and E(z) is represented by
Eq. (21) in Chapter 3. Now, substituting Eq. (17) into the above

equation, it is obtained that

2 2H .t" 2
Z 1
[L C05 8 g9}? j I 12.11)}; sinne cos°9d9
g 0 0 ’-
2

21..

n+3

 

Therefore

2L2 L(n 1)(n-1) . .
7.73" ' (Glenn) (ilblz-

 

Var(z) = (22)

3- Derivations of Eq. (79) and Eq. (80).

 

For the gradient in X—direction, letting 4’ be the angle of
orientation measured from the X- axis, then the following expres-

sion may be written:

 

cos 5" = sine cos/9’
and
' (n+1) (IL-l) . +
gag 3 B 2n(n-2) sinn lecosfldedfli

Where B = 1/11 for n odd and E z? for n even. In the above
eXPI‘eSSiOU. géfl is defined as the probability function of the choice
Of canals based on the discharge and geometric distribution for
X-gradient.

Therefore, the quantities (cosy) and <17cosy’) may be

Written as

 

 

 

 

 

73

(CO—51;) =f [9’ egg cosyb

 

B(2n(+l_)2)n 1) .f:f1rihn 28cos Zededy.

and

< l/cossb)=f(‘a [j] ggg/(l/cosfib)
BM )(n I) 0"1: yrfusinn GdBdK.
B2n(n-- 2) .

 

(79)

(80)

 

 

APPENDIX II

AN ESTIMATE OF ERROR IN ASSUMING AN AVERAGE

 

COSINE OF THE ANGLE OF ORIENTATION FOR

A GIVEN PATH IN A PORE CANA 1.. NETWORK

 

The basic relationships between the distance Z, time of travel
’I‘, and the number of canals’ N in terms of canal orientation angle

a r e as follows:

 

N

Z= L X, cosej. . . . . . . . . . . . (19)
j:
N 1 N

T = to Z cosBj = t0 Z: sech . . . . . . . (27)

1:1 j=l

Assuming that there exists an average cosine such that

N
N<cosfl> = Z: cosgj . . . . . . . . . . . (82)

1:1

and N
N<sec°7= Zsece- . . . . . . . . . . . (83)

.- l
).

Fer N larger than unity, the above two equations do not lead to a
Common value of the orientation angle, that is, (c059? is not equal
to 1/<sec0> . The factors affecting the average values are, for a
given path, the number of unit canals traversed, and the magnitude
and combination of the individual canal orientation angles. For N
becoming larger, the average cosine value may be evaluated by
using the probability distribution function derived in Chapter 3 as

fOllows:

74

 

«v'

 

 

 

75

2W 1"

<c056> =J fcose gag . . . . . . . . . . . . (84)
0 0
2W gr

<sec9> zfo j; sece gag . . . . . . . . . . . . (85)

whe re 8812/ is represented by Eq. (17) in the following form:

n+1
geal—rn simecosededu. . . . . . . . . . , , ()7)

An evaluation of <cose) value from the above equations

1‘88 ults in the following magnitudes of errors for various n values:

n = 1, 4, 7, 1o, 13, 16 0°

.1.
Valueof 1.33 1.45 1.49 1.51 1.52 1.53 2
<cosG)(sec8>

Assuming that (c058) is evaluated from Eq. (75) then Eq.

(19) may be written as
Z=NL<cosG>.............(86)

and Eq. (27) may be written by taking into consideration the error

inVolved as
T=ANtO<C——51—S-g>.............(87)
Where A is a correction factor. For n = l, A equals 1/1.33 =0, 75
and for n =16, A would equal to 1/1. 525 = 0.655.
As stated previously the error involved would actually depend
on the number of canals, paths and combination of the canal orienta—

tions. Therefore, in Figure 3-5, path 1 and path 2 in case (b) will

not have the same magnitude of error with the same number of

 

 

 

 

76

cana 15 N. Similarly, considering path 1 in case (a) and case (b),

the percent error will not be the same because the number of canals

and the combinations of the angles are different.

The bracket in Eq. (45) may be written as follows:

<5ec8>fi <cosG)I§- (c056)? T20

,—-_,......(88)
<5ec9>$ <c058>fi (c059) N IN

The discussions in Chapter 3 showed that the ratios

(c058) f
(c059) N
and

a:

Zo

 

l

H

N

can be considered as approximately equal to unity.

Let <sec8>fi = Al/<c_o_s-6>fi
and (seed)? 2 AZ/<C086>N

then. the quantity in the bracket as expressed by Eq. (88) can be

represented by Alf/"“2 and, from Eq. (45)

H
ZIN

z— _.............(89)
ZN

If A1 is assumed to have the magnitude of error equivalent to
that for canal orientation n = l, and A2 is assumed to have the

magnitude of error for n =2 16, then

Al/AZ - 0. “IS/0.655 : 1.145

 

 

 

«of
\

 

 

 

77

The :- efore, the error involved in this case will be approximately

15 p e rcent. The above investigation with regard to the assumption

of E: q . (88) to be equal to unity can also be analyzed in the follow-

ing manner. Since '<c059>-I;-I is for the average of the total paths,

and <cose>fi is for an arbitrary path, it is obvious that ifécost‘N

is 121 rger than <cosO> N” then (sec6)-fi will be smaller than

<se c8)fi, and by assuming that <5ec0>fi is approximately equal

to < sectVT, the product,

(secs) N (c' 058) N

 

(seed) T {c059} N-

will not deviate from unity appreciably.

In Eq. (49), the bracket may be written here as follows:

(SCCG) ‘N <COSQ>T TZO

 

<9<9 T"""""(90)
sec>Tcos>fifi

B"=1Sed on the preceding discussions the ratios, <cosfl>T/(cose) “1:1

and TZO/TN will be approximately equal to unity. However, the

45—556? 1;
<sec9> T
represents the ratio of the average secant of all paths over an arbi—
trary path and the error in assuming it to be unity will produce an
errOr of magnitude dependent on the average orientation of the
individual paths. Since the range of the dispersion in a porous
nmedium is usually small compared to the distance of the journey

of individual particles, the assumption of this ratio to be equal to

 

 

 

ad
a",

 

 

78

unity for the purpose of this analysis may not affect the result

Sign i ficantly.

 

 

 

 

 

APPENDIX III

THE NUMERICAL COMPARISON OF PRESENT THEORY

TO THOSE OF SAFFMAN'S AND DE JOSSELIN DE JONG'S

It is shown in Appendix I that a certain amount of error is

introduced by the assumptions made in the derivation of the equa-

tions in Chapter 3.

The c0nsequent error can be estimated by a comparison of

the present theory with the theories of Saffman and De Josselin de

Jong,

The theory presented by Saffman may be summarized as

77
(5; = (TVI.S“)Z

 

 

 

 

 

(7)
where the quantity S2 is represented by the following equations:
3 _ 1 54VT 2
S‘jfi-(logldl..........(7a)
4VT ”1-.
. o/
If y Y >> 1
n‘1ogn‘
l 3Vt 1
522T10gO--17.........(7b)
L
, VtO/L
1E << 1
l . Y
fi/‘(Iog 3.129.);
1 27VT
Z : """_‘ .1 .. o . . . . . . . . 7C
'5 6 log 2L. ( l
. 3Vt /1-
1f 0 >> 1.

79

 

   

 

1"

 

80

De Josselin de Jong's equation may be written as

Z :

1. 320 3 V1
-- ..._._ At— -1 0 . . . . . . .
3 ( L ( 4 Carl) (6)

In order to apply Saffman's equation it is necessary that

certain tests have to be made to determine the applicability of the

equations. By taking the velocity range used in the dispersiOn

measurements, a molecular diffusivity of k = 1.5 x lO—ScmZ/sec,

and an orientation factor of unity which corresponds an average path

05 ((2059) = 0. 654, the results of the tests are tabulated in Table

III- 1

 

- It is seen that both Eq. (7a) and Eq. (7c) apply to the flow

condition in the present investigation. From Eq. (7), Eq. (7a) and

EQ- (7c), standard deviations are calculated and compared with
thOSe obtained from De Josselin de Jong's equation and the present

1:heOry in Table III-2. The average values of the corresponding

Statlclard deviations obtained in the diapersion measurements are

also shown in Table 111-2.

 

 

 

 

 

ad
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APPENDIX IV

THE PRESENTATION OF PERMEABILITY MEASUREMENTS

Table IV-l. Permeability test data for sample P1.

Sample Weight

 

 

 

Test of Direct.

No. Sample Porosity Head Discharge K of Flow
(gms) (%) (cm) (cm3/min) (cm/min)

P11 130.91 34.50 0.28 5.25 6.45 Transv.

0.60 11.40
0.93 17.80

 

P12 127.35 36.10 0.35 7.50 6.95 Transv.
0.70 14.10
1.02 23.40

P13 106.16 46.80 0.46 12.90 9.60 Transv.
0.76 21.00
1.08 30.50

Pll' 130.91 34.50 0.50 12.80 8.75 Longu.
0.85 22.00
1.20 31.00

p12' 116.36 41.70 0.50 15.00 9.34 Lougit

0.95 26.00
1.22 33.00

Table IV—2. Permeability test data for sample P2.

Sample Weight

 

 

 

Test of Direct.

N0. Sample Porosity Head Discharge K of Flow
(gms) (%) (cm) (cm3/mia (dmfmin)

P21 122.70 39.00 0.25 6. 50 8. 75 Transv.

0.40 10. 50
0.70 18.00

P22 107.75 46.40 0. 37 ll. 50 10.52 Transv.
0. 63 18. 20
l. 08 32. 50

83

 

 

C“,

 

 

84

Table IV~2. (continued)

Sample Weight

 

 

 

Test of Direct.
No. Sample Porosity Head Discharge K of Flow
(gms) (’70) (cm) (cm3/min) (cm/min)
P21 ' 120.05 39.20 0.25 7. 50 10.70 Longit.
0.71 21.00
1.03 31.50
P22 ' 113. 79 43.10 0. 20 6.50 11.22 lsongit.
0.61 20.60
0.93 31.00
532:3' 112.83 43.50 0.36 11.75 11.28 Longih
0.68 22.50
1.00 33.00

Table IV—3. Permeability test data for sample P3.

Sample Weight

 

 

 

Test of Direct.
7:13; Sample Porosity Head Discharge K of Flow
(gms) (‘79) (cm) (cm3/lmin) (cm/min)
P33 1 132.80 34.00 0.20 3.50 8.04 Transv.
0.50 11.60
0.80 20.00
1.20 28.00

P32 123.40 38.60 0.17 5.40 11.00 Transv.
0.53 17.00
0.73 27.00

193,1' 140.08 29.60 0.45 10.90 8.65 'Longit
0.75 19.50
1.15 29.10

P321 131.52 34.20 0.22 6.30 10.10 Longit.
0.61 18.50
0.98 29.50

 

1-01

 

 

 

 

85

Ta ble IV-4. Permeability test data for sample P4.

Sample Weight

 

 

 

Te st of Direct.

No- Sample Porosity Head Discharge K of Flow
(wag (%) (cm)(cm3hnm)(anhnm)

P41 ' 140.62 29.60 0.45 13.40 10.41 I-.ongit.

0.67 20.40
1.09 33.50

 

P42' 129.61 35.20 0.29 9.25 11.17 "
0.61 20.00
0.94 30.90

p41 140.50 31.00 0.30 8.00 9.05 Transv.
0.50 13.50
0.90 24.00

PZ1Z_ 131.80 34.40 0.25 8.00 10.24 Transv.
0.60 18.50
1.00 30.50

F243 129.50 35.60 0.30 9.50 10.24 Transv.
0.55 17.00
1.10 33.50

Table IV-5. Permeability test data for sample N.

Samp1e Weight

 

 

 

Test of
EL~_ Sample Porosity Head Disqharge K of Flow
(gms) (070) (cm) (cm' /rnin) (cm/min)

P“, 101.70 49.20 0.18 6.80 12.75 Transv.
0. 43 16. 00
0. 59 21. 80
0. 79 29. 00

N11 101.70 49.20 0.40 12.25 11.00 Longit.
0. 72 22. 80

1.04 32.80

 

Q“

 

 

Permeability constant in cm/mln.

0

Permeability constant in cm/min.

H
H

b-d
O

\O

13

12

ll

10

86

 

   
 

 

 

 

    

 

 

 

l 1 l 1
Sample P1

— -l

l" —l
. Longitudinal

"' u—l
0 Transversal
1 1 1 1

30 34 38 42 46 50

Porosity in %
Figure IV—l. Permeability and porosity for sample P1,
r 1 1 T
Sample P2

#- II‘
0 Longitudinal
i- 0 Transversal
1 1 1 1

34 40 44 46 50

Porosity in %
Figure IV-Z. Permeability and porosity for sample P2,

 

 

 

 

87

 

 

 

 

 

 

 

, 13 r 1 1 I I
-S
\2812 _ Sample P3 _
E
s:
.'—‘11 1- '4
afi—D
c:
an
‘3
gIO
O ’- .0
L:
i?
-.: 9 - ~
:5 .1..ongitudinal
CK
Q,
E 8 b OTransversal "‘
a:
De
7 l l l l l
28 32 36 40 44
Porosity in ”/0
Figure IV—3. Permeability and porosity for sample P3.
, a
.E 1.: 1 I l I 1
\E s 1
E 121— amp e P4 _
o
.E
g, 11— ..
s:
cc
.3.)
(D
g 10— .1
o
>.
+3
: 9 l- A
'2 0 Longitudinal
o
181 8b OTransversal
a)
0..
7 1 _L l 1 l

 

 

28 32 36 40 44
Porosity in ‘70

Figure IV—4. Permeability and porosity for sample P4.

 

 

 

 

APPENDIX V

THE PRESENTATION OF EXPERIMENTAL

 

DA TA ON DISPERSION

 

Table V-la. Dispersion measurements, 511' Vm = 0.714 cm/min.

 

 

 

 

 

 

 

Position 1 Position 2 Position 3
Conceal Concenh Concent
Time in in Time in in Time in in
min- sec. Normal min. sec. Normal min. sec. Normal
0 O .001 0 0 .001 O 0 .001
15 .0012 5 0 .0021 10 0 .0014
30 .0014 15 .0024 15 .0016
45 - 30 .0026 30 .002
1 0 .002 45 .0029 45 .0027
15 .0025 6 0 .0038 11 0 .0036
30 .0038 15 .0057 15 .0052
45 .0082 45 .0078 30 .0075
Z 0 .0168 7 0 .0137 45 .0091
15 .0375 15 .0223 12 O .012
30 .058 30 .0325 15 .017
45 .076 45 .0465 30 .0228
3 0 .085 8 0 .057 45 .0305
15 .092 15 .069 13 O .0395
30 .095 30 .080 15 .047
45 .098 45 .090 30 .057
9 0 .096 45 .0625
15 .098 14 0 .071
15 .0755
30 .081
45 .088
15 O .094
15 .096

Table V-lb. Dispersion measurements, 512' V = 0. 57 cm/min.

m

 

 

 

\fosition 1 Position 2 Position 3

T‘ Concent. Concefi. Concent.
{me in in Time in in Time in in

m; sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001

30 .0012 5 30 .0016 12 0 .0012
1 0 .0014 6 0 .0019 30 .0015
88

 

 

 

89

Table v- 1b. (continued)

 

30 .0019 15 .0021 13 15 .0021
45 .0025 30 .0022 30 .0027

2 0 .0038 45 .0024 45 .0033
15 .008 7 O .0026 14 0 .0049

30 .0129 30 .0036 15 .0065
45 .0255 45 .0053 30 .0074

3 0 .043 8 15 .007 45 .010
15 .060 30 .010 15 0 .013
30 .071 45 .0152 15 .0168

45 .082 9 0 .0225 30 .0223

4 O .092 15 .0295 45 .027
15 .094 30 .039 16 0 .0345

30 .097 45 .051 15 .041

10 0 .0615 16 30 .048

15 .071 45 .056

30 .080 17 0 .062

45 .089 15 .068

ll 0 .095 45 .075

15 .098 18 15 .087

30 .092

45 .094

19 0 .098

Table V-lc. Dispersion measurements, 813, V = 1.43 cm/min.

 

 

 

 

m
\Eosition 1 Position 2 Position 3
. ConcenE Concent ConcenE
THnae u; in Thnein in innm n: in
HELD; 2:32. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
15 .0012 2 30 .0014 5 45 .0117
30 .0025 45 .0021 6 0 .020
45 .0065 3 0 .0040 15 .032
1 0 .0255 15 .0112 30 .045
15 .053 30 .0205 45 .060
30 .073 45 .0375 7 0 .0735
45 .085 4 0 .059 15 .082
‘2 0 .090 15 .075 30 .089
15 .095 30 .086 8 0 .0925
45 .092

5 O .096

 

 

«a

 

 

90

 

 

 

 

 

 

Table V-ld. Dispersion measurements, 514’ Vm = 1.065 cm/min.
Position 1 Position 2 Position 3
Torment. Concefit. Concent.
Time in in Time in in Time in in
min- sec. NOrmal min. sec. Normal min. sec. Normal
0 0 . 001 O 0 . 001 0 0 . 001
15 .0011 3 30 .0014 7 0 .0016
30 .0012 45 .0017 15 .0022
45 .0018 4 0 .0026 30 .0034
l 0 .0036 15 .0044 45 .0062
15 . 0085 30 . 0090 8 O .0090
30 .0238 45 .0127 15 .0165
45 .055 5 0 .0275 30 .026
Z 0 .077 15 .0355 45 .0372
15 .092 30 .053 9 0 .051
30 .097 45 .0705 15 .065
45 .098 6 O . 088 30 .079
15 . 097 45 .085
10 0 . 090
15 .095
30 .097
45 . 098

Table V-le. Dispersion measurements, 821, Vm = 0.432 cm/min.

 

 

 

 

 

 

 

dosition 1 Position 2 Position 3
. Conceni. Concent. _ Concent.
Tlfhe in in Time in in Time in in
31& sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
30 .0013 5 0 .001 15 45 .0011
1 0 .0022 6 0 .0011 15 30 .0013
30 .0037 7 30 .0013 16 30 .0016
2' 0 .0051 8 30 .0018 17 30 .0028
30 .013 9 30 .0036 18 15 .0047
‘3 0 .030 10 0 .0063 19 0 .0105
30 .0535 30 .0115 30 .0145
4= 0 .073 11 0 .0195 20 0 .025
30 .078 30 .036 30 .038
5 0 .083 12 0 .055 21 0 .051
6’ 0 .089 30 .067 30 .062
'7 0 .091 13 0 .076 22 0 .0725
30 .080 30 .083
15 0 .092

 

 

 

 

 

91

Table V-lf. Dispersion measurements, S Vm = 0. 342 cm/min.

 

 

 

 

 

 

22'
Position 1 Position 2 Position 3
‘ Concenf. Concenf. Concent.

Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 O .001 0 0 .001
30 .0014 8 0 .001 17 30 .001

1 0 .002 9 0 .0011 18 30 .0011
30 .0029 10 0 .0014 19 O .0013
2 0 .0037 ll 0 .0018 20 0 .0015
30 .0063 30 .0025 21 0 .0018
3 0 .0147 12 O .0036 22 0 .0026
30 .0295 30 .0063 23 0 .0056
4: 0 .053 13 0 .0118 30 .0096
30 .073 30 .0167 24 0 .Ol02
51 0 .084 14 0 .0282 30 .0197
30 .088 30 .044 25 0 .029

62 0 .092 15 O .060 30 .040
30 .077 26 0 .052

16 0 .088 30 .063

27 0 .071

30 .079

28 0 .088

30 .095

Table V-lg. Dispersion measurements, 523, Vm = O. 787 cm/min.

 

 

 

 

 

 

Position 1 Position 2 Position 3
Concent. Concent. ConcerRT
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
15 .0013 2 45 .001 8 0 .001
30 .002 3 45 .0012 9 O .0016
45 .0036 4 30 .0017 15 . 002
1 0 . 0047 5 0 . 0029 30 . 0025
15 .0093 30 .0091 45 .0035
30 .023 6 0 .0242 10 0 . 0051
45 .051 15 .037 15 .008
2- 0 .0725 30 .058 30 .0106
15 .090 45 .073 45 .0173
7 0 . 081 11 O . 0295
15 . 085 15 . 040
45 . 089 30 . 055
45 . 067
12 0 . 073
15 . 081
30 . 087
45 . 091

13 45 . O93

 

 

 

92

 

 

   

 

 

 

 

Table V-Ih. Dispersion measurements, 524, Vm = 0.625 cm/min.
Position 1 Position 2 Position 3
Wncenfi Concent. Coucent.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 O . 001 0 O . 001 0 0 .001
30 .0014 4 45 .0011 9 45 .0011
45 .0021 5 30 .0014 10 30 .0012
l 0 .0031 6 O .002 11 30 .0017
15 .0039 30 .0037 12 0 .0025
30 .0063 7 0 .0083 30 .0043
45 .015 15 .0165 45 .007
2 0 .0235 30 .0225 13 0 .0103
15 .046 8 0 .043 15 .0125
30 .062 15 .053 30 .0175
45 .078 30 .065 45 .0285
3 30 . 092 45 .069 14 0 . 035
4 0 .093 9 0 .073 15 .047
10 0 . 091 30 .0555
45 . 062
15 O . 066
30 . 075
16 0 . 085
30 .091
5 l7 0 . 0925
I
1' Table V-Za. Dispersion measurements, P111 , Vm = . 575 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. Cm.
Time in in Time in in Time in in
min, sec. Normal min. sec. Normal min. sec. Normal
\ __ __ _ __________ __
0 0 .001 0 0 .001 0 0 .001
30 .0018 3 30 .0011 9 15 .0021
45 .0025 4 0 .0015 45 .0028
1 0 . 0037 30 . 0022 10 15 .0042
15 .0065 5 O .0036 45 .0062
30 .0122 30 .006 11 15 .0088
45 .0185 6 O .0103 45 .012
Z O .031 30 .012 12 30 .015
15 .0485 7 0 .0183 13 0 .019
30 .064 30 .0285 30 . 024
45 .078 8 0 .0375 14 0 .0305
3 0 .086 30 .049 30 .0365
15 .092 9 0 .060 15 0 .0455
30 . 070 30 . 052
10 O . 080 16 0 . 062
30 . 088 30 . 070
ll 0 .091 17 O .075
30 . 095 30 . 083
18 0 . 089
30 .095

 

 

1's

 

 

93

Table V—Zb. Dispersion measurements, P112 , Vm = . 306 cm/min.

 

 

 

Position 1 Position 2 Position 3
’Concenn CoEFEfiE ’COncefiE
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal

0 0 .001 0 0 .001 0 0 .001
30 .0015 5 45 .0011 15 15 .0011
1 0 .0022 7 30 .0015 17 45 .0019
30 .0028 8 30 .0024 18 45 .0026
2 0 .0044 9 0 .003 19 30 .0035
30 .010 10 0 .0053 20 30 .005
3 0 .0136 ll 0 .0088 21 30 .0073
30 .0265 30 .012 22 30 .0106
4 0 .046 12 30 .015 24 0 .0137
30 .063 13 0 .0185 30 .0152
5 0 .079 30 .0237 25 0 .0178
30 .088 14 0 .0295 30 .0205
30 .037 26 0 .0245

15 0 .045 30 .027
30 .054 27 0 .0305

16 0 .062 30 .035
30 .070 28 0 .0395

17 0 .078 30 .044
30 .085 29 O .0495

18 0 .091 30 .055

30 .093 30 O .058

19 0 .095 30 .065

31 0 .071

30 .076

32 0 .080

30 .084

33 0 .089

30 .092

34 0 .096

Table V-Zc. Dispersion measurements, P113, Vm = 0. 44 cm/min.

 

Position 1 Position 2 Position 3

ConcenL LOncenL ConeEfiE

Thnein in Thnein in, Thnein in
min. sec. NOrmal min. sec. Normal min. sec. Normal

0 0 .001 0 0 .001 0 0 .001

30 .0018 3 45 .001 10 45 .0012

l 0 .002 4 45 .0014 ll 45 .0016

30 .0061 5 30 .002 12 45 .0025

2 15 .0213 6 30 .0042 13 30 .0038

30 .0315 7 O .0062 14 30 .0062

3 0 .056 8 0 .0129 16 30 .0155

30 .076 30 .0137 17 30 .019

 

 

 

 

 

94

 

 

 

 

 

Table V-2c. (continued)
4 0 .089 9 0 .0188 18 0 .023
30 . 096 30 . 025 30 . 0275
5 0 .097 10 0 .033 19 0 .0315
30 . 043 30 . 0385
ll 0 . 053 20 0 . 046
30 . 064 30 . 053
12 0 . 073 21 0 . 061
30 . 084 30 . 0675
13 0 . 088 22 0 .076
14 0 . 090 30 . 081
23 O . 089
30 .094
Table V-3a. Dispersion measurements, P121, “,m = 1. 016 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. Concefi'tT "
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec Normal
0 0 . 001 0 0 . 001 0 0 . 001
15 .0023 2 0 .0011 6 0 .0046
30 .006 45 .0024 30 .010
45 .0163 3 15 .0062 7 0 .0168
1 0 .0362 30 .0083 30 .025
15 .0545 45 .0102 8 15 .042
30 .0675 4 0 .0187 30 .047
45 . 080 15 . 026 45 .054
2 30 .095 30 .033 9 0 .062
45 . 042 15 . 068
5 0 . 050 30 . 0745
15 . 058 10 0 . 082
30 . 0655 15 .089
45 . 074 30 .090
6 15 . 085 11 O . 095
45 . 093
7 15 . 095
Table V-3b. Dispersion measurements, P122, Vm = . 793 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. COncenf.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 001 0 0 . 001 0 0 . 001
15 .0012 3 15 .0012 7 0 .0016
30 .0016 45 .0032 9 0 .0115
45 .0036 4 15 .0044 30 .0163

 

 

 

 

 

95
'Table V-3b. (confinued)

l 0 . 0088 45 . 0087 45 . 020
15 .0182 5 0 .012 10 0 .0225
30 .031 15 .0158 15 .0263
45 . 047 30 . 020 45 . 0365

2 O .062 5 45 .0255 ll 0 .041
15 .075 6 0 .0315 15 .0465

30 .085 15 .040 30 .052

45 . 0915 30 . 048 45 . 059

3 O .0965 45 .055 12 0 .064
7 15 . 072 15 . 068
30 . 078 30 . 0725

45 . 081 45 . 077

8 0 . 086 13 15 . 087

15 . 090 45 . 089

30 . 095 14 15 . 091

45 . 097 15 30 .094

Table V—3c. Dispersion measurements, P123, Vm = 0. 581 cm/min.

 

 

Position 1 Position 2 Position 3
Concent. Concent. Concent.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal

0 0 . 001 0 0 . 001 0 0 . 001
30 .0016 3 45 .0011 10 30 .0024
45 .0025 4 45 .0017 ll 15 . 0043
l 0 .0044 5 30 .0033 12 15 .0097
15 .0086 6 0 .0056 45 .0135

30 .013 30 .008 13 30 .020

45 .0232 7 0 .014 14 O .026

2 0 .034 30 .0217 45 .037
15 .045 45 .0262 15 0 .0415

30 .058 8 O .032 30 .050

45 .0685 15 .0365 45 .055

3 0 .075 30 .0405 16 0 . 059

15 .083 45 .0465 15 . 063

30 .089 9 0 .0525 30 .068
4 0 .095 15 .059 45 .0725

30 .097 30 .065 17 0 .077

45 . 070 15 . 081

10 0 . 075 30 . 084

15 , 079 45 . 087

45 . 089 18 0 . 089

11 0 . 092 15 . 091

30 . 093 30 . 093

12 0 . 0955 19 30 . 095

13 0 .097

 

 

 

96

 

 

 

 

Table V-3d. Dispersion measurements, P124, Vm = . 377 cm/min.
PositiOn 1 Position 2 Position 3
Concent Concent Concefiff
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 001 0 0 . 001 0 0 .001
30 .0015 4 45 .001 14 15 .0013
l O .0024 7 30 .0017 15 45 .002
30 .0037 8 0 .0021 17 15 .0039
2 0 .0097 30 .0029 18 15 .0065
30 .0172 9 0 .004 19 15 .0095
45 .024 30 .0057 20 0 .0131
3 0 .0315 10 0 .0067 30 .0162
15 .0385 30 .0093 21 0 .020
30 .046 ll 0 .013 30 .0245
45 .0525 30 .0167 22 0 .0285
4 0 .058 12 0 .023 30 .034
15 .063 30 .0305 23 O .0395
30 . 067 13 0 .038 30 .0445
5 0 .075 30 .046 24 O .0505
30 . 079 14 0 . 0545 45 . 060
6 0 .084 30 .065 25 O .065
15 0 . 070 30 .070
30 . 078 26 0 . 075
16 0 . 084 30 . 081
30 . 088 27 0 . 0845
17 0 . 091 30 . 087
18 0 . 096 28 0 . 091
30 . 094
29 0 . 095
Table V-4a. Dispersion measurements, P131, Vm = . 476 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. Concefiifi
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 O . 001 0 0 . 001 0 0 . 001 ‘
30 .0024 4 45 .0012 12 15 .0016
1 0 .004 5 45 .0014 13 15 .0025
15 . 006 6 30 . 0022 45 . 0035
30 .0066 7 0 .0032 14 15 . 0046
45 .0123 30 .0056 15 0 .006
2 0 . 0215 8 0 . 0072 30 .0078
15 .036 30 .0122 16 0 .0113
30 .049 9 0 .020 30 .0152
45 .062 30 .031 17 0 .018
3 0 . 076 45 . 0305 30 . 0275

15 .085 10 0 .0435 18 0 .035

 

0“

 

 

Table V-4a. (Continued)
15 .051 15 .040
30 .059 30 .0445
45 .065 45 .0495
ll 0 .072 19 0 .055
15 .078 15 .061
30 .084 30 .065
45 .088 45 .070
12 0 .0905 20 0 .073
15 .077
30 .080
21 0 .083
30 .089
22 0 .094
Table V—4b. DiSpersion measurements, P132, Vm = . 695 cm/min.

 

Position 1 Position 2 Position 3

 

 

 

 

 

 

 

 

Concent. Concent. Concent.
'Tinne in in Tfirne in in Tfirne in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 O () .001 0 0 .001
30 .0026 3 0 .001 8 15 .0017
l 0 .0068 4 0 .0026 9 15 .0034
30 .0265 5 0 .005 45 .0054
45 .047 6 15 .0235 10 30 .0081
2 0 .070 30 .033 11 0 .0125
15 .085 7 0 .0555 30 .019
15 .062 12 15 .0335
30 .0755 30 .041
45 .932 45 .046
8 0 .087 13 0 .050
30 .090 15 .056
9 O .0925 30 .062
10 0 .095 45 .070
14 0 .075
30 .086
45 .090
15 15 .095

   

 

98

 

 

 

Table V-4c. Dispersion measurements, P133, Vm = .602 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. COncetTf'.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 001 0 0 . 001 0 0 . 001
30 .0025 4 0 .0012 10 15 .0024
1 0 .0052 30 .0015 ll 15 .0046
30 .014 5 0 .0021 12 15 .0105
2 O .0435 30 .0033 13 0 .014
15 .061 6 0 .0061 30 .020
30 .076 30 .008 14 0 .0285
45 .086 7 O .0152 30 .0395
15 . 0205 45 . 046
30 . 026 15 O . 053
45 . 032 15 .061
8 0 . 041 30 . 069
15 . 0485 16 O . 074
30 . 055
45 . 062
9 0 . 0675
15 . 073
30 . 075
45 . 079
10 0 . 080
45 . 088
12 . 094

 

 

Table V-Sa. Dispersion measurements, p211’ Vm = . 815 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. Concent.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal.
0 0 . 001 0 0 . 001 0 0 . 001
15 .0023 2 15 .0025 7 45 .0174
30 . 0065 45 . 0046 8 30 . 022
45 .0146 3 15 .0093 9 0 .0272
1 0 .0205 4 0 .0139 30 .034
15 .033 30 .0215 45 .0365
30 . 049 45 . 0242 10 0 . 040
45 . 061 5 0 . 0282 15 . 044
2 0 .075 15 .033 30 . 047
30 .084 30 .038 45 .0505
3 30 .095 45 .0425 11 0 . 054
6 0 . 0475 15 .0575
15 . 0535 30 . 060
30 . 058 45 . 063

45 . 0625 12 0 . O66

   

 

 

 

 

 

99
Table V-Sa. (confinued)
7 0 .0675 30 .0725
30 .077 13 0 .0775
8 15 .083 30 .082
9 15 .090 14 30 .086
16 0 .092
Table V-Sb. Dispersion measurements, P212, Vm = . 421 cm/min.
Position 1 Position 2 POSition 3
oncen. Concenh TCEncent
Time in in Time in in Time in in
min. sec. Normal min. Sec. Normal min. sec. Normal
0 0 .001 0 O .001 0 0 .001
30 .0018 3 45 .0015 8 15 .0017
l 0 .0064 5 15 .0039 9 15 .0023
30 .0103 45 .0056 10 15 .0033
2 0 .023 6 0 .0065 ll 15 .005
15 .031 30 .007 12 15 .0075
30 .041 7 0 .009 13 15 .0082
45 .051 30 .0116 14 15 .0106
3 0 .06 8 O .0145 15 30 .015
15 .07 30 .0183 16 30 .0188
30 .078 9 O .0222 17 30 .024
4 0 .088 30 .027 18 0 .0275
30 .0955 10 0 .032 30 .030
10 30 .0375 19 0 .0345
ll 0 .0435 30 .038
30 .050 20 0 .043
12 0 .0565 30 .0475
30 .063 21 0 .0535
13 0 .069 30 .058
30 .076 22 0 .064
14 0 .0795 30 .0695
30 .085 23 0 .075
15 0 .088 30 .080
16 0 .094 24 0 .083
30 .088
25 0 .090

30 .0925

   

 

100

 

 

 

Table V—Sc. Dispersion measurements, P213, Vm = . 569 cm/min.
Position 1 Position 2 Position 3
Concent. CEncent. Concent.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 001 0 0 . 001 O 0 . 001
30 .0029 3 15 .002 7 15 .0031
45 .0062 4 15 .0055 8 15 .0052
l 0 .012 5 0 .0085 9 15 .0085
15 .0156 30 .0121 10 15 .0098
30 .025 6 0 .016 11 15 .0138
45 .0355 30 .0208 12 15 .019‘2'
2 0 .0475 7 0 .0264 13 15 .0262
15 .060 30 .033 14 0 .0325
30 .068 8 0 .040 30 .0385
45 .075 30 .048 15 0 .044
3 0 .081 9 0 .056 30 .0505
30 .085 30 .063 16 0 .057
10 O . 071 30 . 0645
30 . 080 17 0 .0715
ll 0 .084 30 .078
30 . 087 18 0 . 0845
12 0 . 089 30 . 090
30 . 090 19 0 . 095
13 0 . 091
Table V-6a. Dispersion measurements, P221, Vm = . 535 cm/min.
POSition 1 Position 2 Position 3
Concent. Concent. Conce—thT
Time in in Time in in Time in in
m. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
30 .0047 2 45 .0013 9 15 .0062
45 .0091 3 15 .0018 10 15 .007
l 0 .012 45 .0026 45 .0082
15 .0215 4 15 .0037 ll 15 .0093
30 .0325 5 0 .0063 45 .0108
45 .0445 30 .0074 12 30 .0136
Z 0 .058 6 0 .0103 13 0 .0162
15 .070 30 .0144 30 .020
30 .079 45 .0168 14 0 .0248
3 0 .090 7 0 . 020 30 .0292
15 . 023 15 0 . 0355
30 . 027 30 . 0435
45 . 030 16 0 .050
8 O . 034 30 . 057
15 . 0385 17 0 .0635
30 . 044 30 . 0675

45 .050 18 0 .070

 

 

 

 

 

101

Table V-6a. (continued)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 0 . 0545
30 . 060
10 0 . 069
30 . 078
ll 0 . 085
30 . 090
Table V-6b. Dispersion measurements, P272, Vm» = . 295 cm/min.
Position 1 Position 2 Position 3
Toncent. Concent. Concent'.
Time in in Time in in Time in in
min. sec. Normal min. sec. NOrmal min. sec. Normal
0 0 . 001 O 0 . 001 0 0 . 001
30 .0015 4 15 .001 16 45 .003
l 0 .0032 6 30 .0015 18 30 .0042
2 0 .0091 8 30 .0031 19 30 .0052
2 30 . 0205 9 30 . 0044 23 0 . 0069
3 0 .036 11 0 .0066 24 0 .0089
30 .057 12 0 .0096 25 0 .0121
4 0 .072 30 .0122 26 0 .0167
30 .087 13 0 .0155 27 0 .0237
7 0 .095 30 .0192 30 .0295
14 O . 0245 28 0 . 035
30 . 0305 30 . 0415
15 0 . 036 29 0 . 0475
30 . 044 30 . 0535
16 0 . 053 30 0 . 0595
30 . 061 30 . 067
17 0 . 0675 31 0 . 0725
30 . 074 30 . 077
18 15 . 083 32 0 . 078
20 0 . 087 33 O . 081
34 0 . 085
35 0 . 088
Table V-6c. Dispersion measurements, P223, Vm = . 366 cm/min.
Position 1 Position 2 Position 3
Concent. Concetft. ConcenT
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 001 0 0. . 001
3 15 .0011 11 45 .0029
4 0 . 0016 12 45 . 0035
5 0 . 0021 13 45 .0048
6 0 . 0032 15 45 . 0077

 

 

 

 

Table V-6c. (confinued)

102

 

30 .0042 17 30 .0085
7 30 .0047 18 0 .010
8 0 .007 30 .012
30 .009 19 0 .0147
9 0 .012 19 30 .0183
30 .0158 20 0 .0225
10 0 .021 30 .0275
30 .0275 21 0 .033
11 0 .035 30 .0395
30 .044 72 0 .046
12 0 .050 30 .053
30 .060 23 0 .060
13 0 .070 30 .065
30 .075 24 0 .071
14 0 .080 30 .077
15 0 .086 25 0 .081
30 .085
Table V—7a. Dispersion measurements, P311, V = . 0. 95 cm/min.
Position 1 Position 2 Position 3
Concent COncenE’ Loncent
'Thnein in Tunein in Thnein in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .0018 0 0 .001 0 0 .001
30 .0033 2 15 .0015 5 45 .0021
45 .0069 3 15 .0045 6 45 .0062
l 0 .015 4 30 .0273 7 15 .0113
15 .0345 45 .035 45 .014
30 .050 5 15 .052 8 15 .0218
45 .066 30 .064 30 .027
2 0 .077 6 0 .079 9 0 .040
30 .090 30 .090 30 .0515
3 0 .095 7 0 .093 45 .059
10 0 .066
15 .0745
30 .079
45 .085
ll 0 .090
30 .095

 

 

 

 

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table V-7b. Dispersion measurements, P312, Vm = O. 78 cm/min.
Position 1 Position 2 Position 3
‘Cbncent Cfincent _—‘ Cbncent
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 002 0 0 . 001 0 ' 0 . 001
30 .0028 2 15 .001 8 15 .0029
45 . 0042 3 45 .0022 45 .0043
1 0 . 0096 4 15 . 004 9 15 . 007
15 .0172 45 .0082 45 .011
30 .031 5 0 .0093 10 30 .0145
45 . 047 30 . 0178 11 0 .0205
2 0 .063 6 0 .030 15 .0247
30 .080 15 .0375 30 .0285
3 0 . 091 30 . 047 45 . 0335
30 .096 45 .055 12 0 .038
7 0 . 062 15 . 0435
15 . 070 30 . 049
30 . 077 45 . 0535
45 . 084 13 0 . 060
8 0 . 088 15 . 0655
30 . 095 30 . 071
14 0 . 0775
15 0 . 087
16 0 . 095
17 0 .097
Table V-7c. DiSpersion measurements, P313, Vm = . 473 cm/min.
Positon 1 Position 2 Position 3
' Concent. Concent. Concent.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 0017 0 0 .001 0 0 .001
30 .0027 3 45 .0011 10 45 .0014
45 .0031 4 45 .0013 11 45 .0017
1 0 .0038 5 45 .0019 12 45 .0023
15 .0053 6 30 .0029 14 30 .0053
30 .0088 7 0 . 0047 16 0 .0088
45 .0127 30 .008 30 .0112
2 0 .0195 8 0 .0098 17 0 .014
15 .0305 15 .0117 30 .018
30 .0415 30 .0142 18 0 .022
45 .0555 45 .0178 30 .0275
3 0 .066 9 0 .021 19 O .0335
15 .076 15 .0253 30 .041
30 . 082 30 . 0297 45 . 045
4 0 .093 45 .036 20 0 .049

30 .095 10 0 .041 30 ,0575

 

 

 

104

 

 

Table V—7c. (continued)
30 . 0525 21 0 . 064
ll 0 . 060 30 .070
30 .068 22 0 . 077
12 0 . 078 30 . 083
30 . 087 23 0 . 087
13 0 . 0925 30 . 094
30 . 095
Table V-Sa. Dispersion measurements, P3”, Vm = .396 cm/min.
PositiOn 1 Position 2 Position 3
“—T Concent. Concent. Concent.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .0018 0 0 .001 0 0 .001
30 .0037 3 15 .0024 10 45 .0011
1 0 .0041 5 45 .0029 14 15 .0012
30 .0063 8 30 .0054 16 0 .002
2 0 .0081 9 O .0068 17 0 .0037
30 .026 10 0 .0188 18 30 .0075
3 0 .055 30 .0255 19 0 .0113
30 .080 ll 0 .737 30 .016
4 0 . 095 30 . 0495 20 0 . 024
12 0 . 064 20 30 . 031
30 . 0755 21 0 . 040
13 0 . 081 30 .049
22 0 . 060
30 . 068
23 0 . 077
30 .0855
24 0 . 094
Table V-8b. Dispersion measurements, P322, Vm = . 255 cm/min.
Position 1 Position 2 Positon 3
Concent. Concent. cam
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .0034 O 0 .001 0 0 .001
30 .0038 6 45 .0021 19 45 .001
1 O .0042 9 0 .0024 23 30 .0012
30 .0049 11 0 .003 25 0 .0014
2 30 .0068 13 0 .0045 27 0 .0021
3 0 .0129 14 30 .0058 28 0 .0029
15 .0195 15 0 .0074 29 0 .0046
30 .029 30 .0103 31 0 .0081
45 .0405 16 0 .0135 32 0 .0129

 

 

 

 

Table V-8b. (confinued)

 

 

4 0 .0525 30 .0178 30 .0162
15 .065 17 0 .0245 33 0 .020
30 .076 30 .033 30 .025
45 .085 18 0 .040 34 0 .030
5 0 .093 30 .0495 30 .036
19 0 .058 35 0 .0425
30 .069 30 .0495
20 0 .079 36 0 .056
30 .086 30 .064
21 0 .090 37 0 .071
30 .095 30 .076
38 0 .082
39 0 .0905
40 0 .095
Table V-8c. Dispersion measurements, P323, Vm : .261 cm/min.
Posuon 1 Posifinn 2 POsifion 3
Concenfl Concent Concent
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .0037 0 0 .001 0 0 .001
30 .0042 7 30 .0021 21 45 .0011
l 0 .0045 10 0 .0025 25 0 .0015
30 .0055 12 0 .0033 26 0 .0019
2 30 .0082 14 30 .0065 28 0 .0039
3 0 .0167 15 0 .0087 29 0 .0062
30 .037 30 .012 30 0 .0077
45 .047 16 0 .0161 30 .009
4 0 .0575 16 30 .022 31 0 .0112
30 .078 17 0 .028 30 .014
5 0 .091 30 .035 32 0 .0177
18 0 .0435 30 .022
30 .052 33 0 .026
19 0 .061 30 .032
30 .071 34 0 .038'
20 0 .081 35 0 .0505
30 .086 36 0 .063
21 0 .095 30 .0715
37 0 .078
30 .084
38 0 .088
30 .091
39 0 .093

 

 

 

 

106

 

 

 

Table V—9a. Dispersion measurements, P411, Vm = cm/min.
POSition 1 Position 2 Position 3
Concent. Concent COncenT.
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 0029 0 0 . 001 0 0 . 001
15 .0036 4 15 .0011 11 15 .0012
30 .0044 6 30 .0021 13 15 .0018
l 0 .0143 7 30 .0041 15 30 .0034
30 .0287 8 O .0062 16 15 .0052
45 .0042 30 .0094 17 30 .0083
2 0 .0056 9 0 .0107 18 30 .0125
15 .064 30 .0161 19 0 .0163
30 .070 10 0 .0243 30 .021
3 0 .082 30 .036 20 0 .028
30 . 090 ll 0 . 046 30 . 034
4 0 .093 30 .057 21 0 .042
30 . 095 12 0 . 066 30 .047
5 0 .097 30 .072 22 0 .054
13 0 . 078 30 .060
30 . 081 23 0 . 067
14 0 . 085 30 . 072
15 0 .091 24 0 . 077
16 O . 095 25 0 . 085
26 0 .089
27 0 .096
Table V-9b. Dispersion measurements, P412, Vm : cm/min.
Position 1 Position 2 Position 3
Toncent. Concent. COnceTE
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 . 0012 0 0 . 001 0 0 . 001
l 0 .002 9 0 .001 18 15 .0011
2 0 .0049 12 0 .0018 22 15 .0015
3 0 .0087 13 0 .0028 24 30 .0021
30 .0142 14 0 .0046 26 30 .0031
4 0 .0223 15 0 .0061 27 30 .0042
30 .0345 16 O .010 28 30 .0066
5 0 .0465 30 .013 30 30 .0086
30 .058 17 0 .0171 31 0 .0103
6 0 .069 30 .0215 30 .0123
30 .077 18 0 .0272 32 0 .0152
7 0 .085 30 .033 30 .0185
30 .090 19 0 .040 33 0 .0222
8 30 . 095 30 . 0465 30 . 0263

 

 

 

 

107

Table V-9b. (continued)

 

 

 

 

 

 

 

 

20 0 .053 34 0 .031
30 .0595 30 .0355
21 0 .065 35 0 .0405
30 .070 30 .0475
22 0 .075 36 0 .0515
23 0 .079 30 .0565
24 0 .086 37 0 .060
25 0 .090 30 .063
38 0 .0675
30 .070
39 0 .072
40 0 .077
41 0 .080
42 0 .083
43 0 .087
44 0 .089
45 0 .091
Table V-lOa. DiSpersion measurements, P421, Vm = cm/min.
Position 1 Position 2 Position 3
7Concefit ‘Concenh ‘Concent
ifinmin in Thnehi in 'Hnmin in
min. sec. Normal min. sec. Normal min. sec. Normal
0 O .001 0 0 .001 0 0 .001
45 .0087 3 15 .001 13 45 .0037
1 15 .0175 5 30 .0019 15 45 .0075
30 .0245 6 30 .0032 17 0 .0113
45 .0325 8 O .007 18 0 .0125
2 O .0435 9 O .0125 19 0 .0165
15 .053 10 0 .0165 20 0 .0223
30 .062 30 .0207 21 0 .030
45 .069 ll 0 .0265 30 .034
3 0 .073 30 .0323 22 0 .040
30 .086 12 0 .040 30 .044
4 0 .094 30 .047 23 0 .050
13 0 .057 30 .055
30 .064 24 0 .060
14 0 .0705 30 .066
30 .077 25 0 .071
15 0 .0825 25 30 .076
30 .0875 26 0 .0795
16 O .090 27 0 .088
30 .093 28 0 .094

17 30 .096

 

 

«a

 

 

 

108

Table V-10b. Dispersion measurements, P422, Vm = cm/min.

 

 

 

Position 1 PCsition 2 Position 3
COncent. Concent. ConceTitT
Time in in Time in in Time in in
min. sec. Normal min. sec. NOrmal min. sec. Normal
0 0 .002 0 0 .001 0 0 .001
30 .0028 3 15 .0012 9 15 .0019
1 0 .0085 4 15 .0018 10 15 .0029
30 .0175 5 30 .004 ll 15 .0045
2 0 .034 6 30 .0081 12 15 .0069
15 .044 7 30 .0157 13 15 .0103
30 .053 8 30 .023 14 30 .0138
3 0 .0675 9 0 .030 15 30 .0195
30 .079 30 .0385 16 0 .0235
4 0 .085 10 0 .0465 30 .0273
5 0 .095 30 .054 17 0 .0325
ll 0 .062 30 .0375
30 . 069 18 0 . 044
12 0 . 076 30 . 0495
30 . 081 19 0 .055
13 0 .086 30 . 0605
14 0 . 096 20 0 .067
30 . 072
21 30 .080
22 0 . 085
23 0 . 091
24 0 .096
Table V-l la. Dispersion measurements, N11, Vm = 0. 40 cm/min.
Position 1 Position 2 Position 3
Concent. Concent. WEEKL—
Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
30 .0045 6 30 .0013 14 15 .0011
1 0 .0057 7 30 .002 15 15 .0012
30 .0085 8 30 .004 16 30 .0016
45 .0146 9 O .0055 17 30 .0025
2 0 .0235 10 O .0079 18 30 .0033
15 .0355 11 0 .017 19 0 .004
30 .0445 30 .0245 20 0 . 0048
45 .0505 45 .029 30 .0056
3 0 .054 12 O .034 21 0 .0071
30 .063 15 .039 30 .0093
4 0 .068 12 30 .044 22 0 .0118
5 0 .077 45 .049 30 .0158
6 0 .081 13 0 .0535 45 .018
7 0 .083 30 .061 23 0 .0195

 

 

109

Table V-lla. (continued)

14 0 . 067 30 .0235
30 . 070 24 0 .0285

15 0 . 0725 30 . 035
16 0 .074 25 0 . 042
30 .049

26 0 .056

30 .066

27 0 . 067

28 0 .077

Table V-llb. Dispersion measurements, N12, V = .472 cm/min.

 

 

m
Position 1 Position 2 Position 3

Concent. Concent. Cencent.

Time in in Time in in Time in in
min. sec. Normal min. sec. Normal min. sec. Normal

0 0 . 001 0 0 .001 0 0 .001
30 .003 4 15 .0011 11 15 .0011

‘1 O .0049 5 30 .0016 13 15 .0018

15 .0082 8 0 .0089 14 30 ‘ .003

30 .0113 30 .013 16 0 .0048

45 .0195 9 0 .0187 30 .006

2 0 .034 30 .0267 17 0 .0076
15 .043 10 0 .035 30 .0097

30 .0565 30 .046 18 0 .0121

3 0 .074 11 0 .054 30 .0151

30 .082 45 .070 19 0 .019

4 0 .087 12 0 .074 30 .0235
30 .090 30 .080 20 30 .0375

5 0 .093 13 0 .085 21 0 .041
6 0 .095 14 0 .088 30 .0485

15 0 . 090 22 0 . 055

45 . 091 30 .062

23 0 . 067

30 .071

24 0 .076

25 0 . 0815

26 0 .0865

27 0 .0895

   

 

 

110

Table V-llc. Dispersion measurements, N13, V = .641 cm/min.

 

 

n1
Position 1 PositiOn 2 Position 3
ConcenL ConcenL COncéfiE
Time in in Time in in Time in in
nfin. sec. Nornml thin. sec. Nornml nfin. sec. Nornml
0 0 .001 0 0 .001 0 0 .001
15 .0021 2 45 .001 8 45 .0012
30 .0031 4 15 .0016 10 15 .0026
45 .0042 5 0 .0037 45 .0035
l 0 .0076 30 .0064 ll 30 .0057
15 .0145 6 0 .009 12 30 .0092
30 .028 30 .0157 13 30 .0182
45 .045 7 0 .025 14 0 .025
2 0 .062 30 .0375 30 .034
15 .073 8 0 .052 15 0 .043
30 .081 30 .064 30 .0535
3 0 .090 9 0 .078 16 0 .0625
30 .084 30 .071
10 0 .090 17 0 .077
11 0 .093 18 O .085
19 0 .0895
20 0 .0905
Table V—lld. DiSpersionIneasurennents, N2], an I .565 cnuflnin.
Position 1 Position 2 Position 3
COncenL Concenfi Concent
'Thnein in 'Thnein in 'Thnein in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
15 .0025 3 45 .0012 10 45 .0023
30 .0028 4 15 .0015 ll 30 .0035
45 .003 5 15 .0021 12 30 .0071
1 0 .0033 45 .0028 13 30 .0107
30 .0042 6 45 .0055 45 .013
2 O .0053 7 30 .0135 14 0 .0152
30 .0163 45 .0178 15 .018
45 .0265 8 O .0235 30 .0215
3 0 .041 15 .0295 45 .0253
15 .057 30 .0365 15 0 .0295
30 .065 45 .044 15 .034
4 0 .083 9 0 .050 30 .0395
30 .090 15 .057 45 .045
5 0 .093 30 .065 16 0 .051
30 .095 45 .070 15 .056
10 0 .077 30 .061
30 .083 45 .067
11 0 .088 17 O .0715
12 0 .092 15 .074

   

 

'Tatfle V-lld. (confinued)

 

 

 

 

 

30 .077
18 0 .079
19 0 .0855
20 0 .089
21 0 .093
Table V-lle. Dispersion measurements, N22, Vm = .321 cm/min.
Position 1 Position 2 Position 3
TConcenL TCOncefiE ConcenL
Trinne in in 'Tirne in in 'Tinne in in
min. sec. Normal min. sec. Normal min. sec. normal
0 0 .0019 0 0 .001 0 0 .001
30 .0027 6 45 .0011 13 15 .001
1 0 .0032 9 0 .0017 18 30 .0017
30 .0035 10 0 .0022 19 30 .0022
2 0 .0039 30 .003 20 30 .0031
30 .0047 11 0 .0043 21 0 .0036
3 30 .0081 12 0 .0068 22 30 .0052
4 0 .0175 30 .0098 23 0 .0064
15 .026 13 0 .014 30 .008
30 .035 30 .020 24 0 .010
45 .0455 14 0 .0263 30 .0127
5 0 .0575 15 .031 25 0 .016
5 15 .066 14 30 .0365 25 30 .020
30 .072 45 .0415 26 0 .0248
45 .075 15 0 .047 30 .030
6 0 .077 30 .058 27 0 .037
30 .080 16 0 .067 30 .0445
30 .075 28 0 .053
17 0 .083 30 .060
30 .089 29 0 .065
18 0 .092 30 0 .075
31 0 .083
32 0 .090
33 0 .092

 

 

   

112

 

 

 

 

 

 

Table V—llf. Dispersion measurements, N23, Vm : .49 cm/min.
Posifion 1 Posifion.2 Posifion.3
COnceHL COncenL ConcenC
Thnein in Thnein in Thnein in
min. sec. Normal min. sec. Normal min. sec. Normal
0 0 .001 0 0 .001 0 0 .001
30 .0026 4 15 .0011 ll 15 .0011
1 0 .0029 5 0 .0013 13 0 .002
30 .0036 6 0 .0018 14 0 .0032
2 0 .0055 7 0 .0029 45 .0048
30 .0073 30' .0043 16 0 .0075
3 0 .019 8 30 .0084 30 .010
15 .0315 9 0 .0137 17 0 .013
30 .044 30 .0243 30 .0176
45 .057 10 15 .039 18 0 .0225
4 0 .065 45 .0505 30 .030
30 .074 ll 0 .059 19 0 .037
5 30 .084 30 .069 30 .046
12 0 .076 20 0 .054
45 .088 30 .064
21 0 .070
30 .076
22 0 .082

23 0 .085

 

 

113

 

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