WATER POLLUTION CONTROL RESEARCH SERIES • 1606 DLL 09/70
DISPERSION IN HETEROGENEOUS
NONUNIFORM ANISOTROPIC POROUS MEDIA
ENVIRONMENTAL PROTECTION AGENCY * WATER QUALITY
OFFICE
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes the
results and progress in the control and abatement of pollu-
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Reports should be directed to the Head, Project Reports
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Office, Environmental Protection Agency, Washington, D,C. 20242.
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DISPERSION IN HETEROGENEOUS NONUNIFORM ANISOTROPIC POROUS MEDIA
Robert A. Greenkorn
School of Chemical Engineering
Purdue University
Lafayette, Indiana
for the
WATER QUALITY OFFICE
ENVIRONMENTAL PROTECTION AGENCY
Grant No. 16060 DLL (Formerly WP-10K&8-03)
September 1970
For salo by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $1
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EPA Review Notice
This report has been reviewed by the Water Quality Office,
EPA, and approved for publication. Approval does not signi-
fy that the contents necessarily reflect the views and poli-
cies of the Environmental Protection Agency, nor does mention
of trade names or commercial products constitute endorsement
or recommendation for use.
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ABSTRACT
The objective of this project is to study the theory and measurement of
dispersion during miscible flow in heterogeneous nonuniform anisotropic
porous media. An understanding of the dispersion phenomenon is necessary
to predict flow of miscible fluids in non-homogeneous porous media. A
significant literature review of dispersion in non-homogeneous porous
media was accomplished. The continuum and statistical theories of flow
in porous media were studied. A series of experiments in models of uncon-
solidated porous media were run to study the effects of heterogeneity,
nonuniformity, and anisotropy on dispersion.
The non-homogeneities are defined in relation to permeability of the
media and dispersion coefficient* are correlated with respect to perme-
ability. Tracer output curves from models were interpreted for disper-
sion. Dispersion changes significantly with permeability. When models
of different permeability are connected, the dispersive effect is not
reciprocal. Dispersion changes significantly with nonuniformity. Like-
wise there is a significant effect when models of different nonuniform-
ity are interconnected (second-order heterogeneity). Anisotropy was
included in two kinds of models (linear and radial) by packing alternate
layers of beads causing flow through these layers at different angles.
The usual dispersion model is not adequate for "block box" prediction
of the tracer output curve for data on anisotropic shperes.
The results have immediate practical applicates, especially in tracing
of contaminents in surface and subsurface flow such as movement of pes-
ticides, fertilizers, acid mine drainage, feed-lot waste, etc. through
the soil.
This report was submitted in fulfillment of project 16060DLL under the
partial sponsorship of the Federal Water Pollution Control Administration.
Key words - Heterogeneous, nonuniform, anisotropic, porous media, dis-
persion, permeability.
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CONTENTS PAGE
Conclusions vj
Recommendations v^£
Introduction ^
Theory 10
Dispersion During Flow in Linear Heterogeneous Porous Media 18
Dispersion During Flow in Nonuniform, Heterogeneous Porous 32
Media
An Exploratory Study of Permeability and Dispersion 51
Phenomena in Anisotropic Porous Media
Permeability and Dispersion During Flow in Linear Hetero- 61
geneous Anisotropic Porous Media
Appendices
1. Dispersion in Heterogeneous Nonuniform Anisotropic 78
Porous Media
2. On Dispersion in Laminar Flow Through Porous Media 81
3. A Statistical Model of a Porous Medium with Non- 82
uniform Pores
ii
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FIGURES PAGE
1 DISPERSION AND FLOW 2
2 OBSERVED PERMEABILITY FOR UNIFORM MEDIA 4
3 OBSERVED PERNEABILITY FOR NONUNIFORM MEDIA 5
4 DESCRIPTION OF MODEL 19
5 DIAGRAM OF EQUIPMENT 21
6 k vs d5Q 24
7 DISPERSION COEFFICIENT VERSUS VELOCITY 28
8 D/V vs R^ 30
9 PLEXIGLAS MODEL — SINGLE TUBE 33
10 EQUIPMENT SET-UP 36
11 SIZE DISTRIBUTION OF BEADS USED IN PACKING MODELS 37
12 DISPERSION COEFFICIENT VERSUS VELOCITY (SIX NON- 43
UNIFORM MODELS)
13 DISPERSION COEFFICIENTS VERSUS VELOCITY (FIRST 44,45,46
AND SECOND ORDER HETEROGENEITY)
14 D/V vs Rk (HOMOGENEOUS) 47
15 D/V vs Rj^ (HETEROGENEOUS) 48
16 D vs J FOR VARIOUS S 50
17 MODEL SKETCH (DOUBLE FIGURE) 52
18 SKETCH OF APPARATUS 54
19 -TF vs angle 57
20 1//~D vs angle 60
21 LENTH APPARATUS CORED AT 45° INTERVALS 62
22 DESCRIPTION OF MODELS 63
23 EQUIPMENT DIAGRAM 65
ill
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PAGE
24 BREAKTHROUGH CURVE 66
25 AVERAGE PERMEABILITY VERSUS FLOW RATE 67
26 /T vs ANGLE 69
27 FLOW PATTERN IN MODEL 72
28 /~D vs ANGLE 74
iv
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TABLES
— PAGE
1 Model Properties 23
2 Velocity and Dispersion Coefficients 26
3 Porosities and Slopes of Bead Size Distribution 38
4 Permeability of Models ^Q
5 Permeability of Composite Models 4^
6 Permeability of Anisotropic Model cc
7 Dispersion Coefficients 55
8 Dispersion Coefficients -JQ
9 Ratio of Longitudinal to Transverse Dispersion Coefficients 79
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CONCLUSIONS
1. There Is a significant effect on dispersion due to difference in
average particle size or average permeability.
2. The exponent of the relation D = a v is dependent upon heterogeneity.
3. It appears there is a difference in overall dispersion through a
heterogeneous system depending on the arrangement of heterogeneities.
(That is, permeabilities cannot be "lumped" into one average).
4. When determing average permeability of glass spheres based on bead
diameter, a valve of d between d-n and d-- should be used.
5. There is a significant effect of nonunifonnity on the dispersion co-
efficient even though the distributions have the same average perme-
ability.
6. The parameter "a" in the expression D = a v is dependent on the
nonuniformity of the system (i.e. the slope of the log permeability
distribution.)
7. Permeability in heterogeneous anisotropic porous media is dependent
on the direction of flow.
8. Dispersion in anisotropic porous media is dependent on the direc-
tion of flow.
9. Both permeability and dispersion in 2 d porous media exhibit tensorial
properties in that 1/v^ and 1//D versus angle plot as ellipses.
10. Both permeability and dispersion in 1-d anisotropic porous media
exhibit tensorial properties in that »^k and Sto versus angle plot as
ellipses.
11. We were unable to prove experimentally whether the permeability and
dispersion tensors are symmetric.
12. At low velocity, the major directions of the permeability tensor and
dispersion tensor are the same but as velocity increases the dispersion
becomes a circle (i.e. principal values the same.) If one extra-
polates the comparison to high flow rate, the major directions of
the two tensors become orthoganol. The low velocity behavior (principal
directions parallel) was observed in 1-d anisotropic models and the
relation between relative valves of dispersion and velocity extra-
polated to high flow velocities. The high velocity behavior (principle
directions orthoganal) was observed in an exploratory 2-d study.
13. The "diffusion form" of the dispersion equation cannot be used for
predicting concentration breakout in heterogeneous anisotropic porous
media.
vi
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RECOMMENDATIONS
1. Data in consolidated material media should be obtained.
2. The data should be used to develop a correlation - probably of the
form
DL X + 2y
where X and y are determined from nonunif ormity data, and X and y
are made dependent on angle and velocity.
3. More realistic models, such as the telegraph equation, where the
effects of the media are distributed, should be studied both ana-
lytically and numerically.
A. A use able scheme for predicting flow in real media using the above
should be developed.
vii
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NOMENCLATURE
Symbol Explanation
a A proportionality constant
A Cross-sectional area of the porous medium
b_, b., .. Coefficients in regression equation
C Concentration of tracer
C_ Inlet tracer concentration
C Dimensionless concentration, C/CQ
C1 Proportionality constant
d Parameter reflecting directional effect due to method
of packing the porous medium
d _ Effective particle diameter used in permeability studies
dsn Average particle diameter
^ftA» difi Particle diameter of the 84% and 16% cumulative fractions
D Dispersion coefficient
g Gravitational constant
h1, h_ Fluid heights in manometer tube
k Permeability
K Constant in Darcy's Law, dependent on properties of
fluid and porous medium
L Length of porous medium
n Constant used to designate value of exponent
n Unit vector normal to flow
p , p Fluid pressures at top and bottom of porous medium
L O
P Combination of static pressure and force of gravity
Pe Peclet number, vL/D
Q Volumetric flow rate
viii
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R Reynolds number based on d-Q
p _ Reynolds number based on dcn
d[50] J 50
Re Reynolds number based on 6 and u
p Reynolds number based on permeability
S Slope of log-normal line describing the distribution
of particle diameters
t Time
v Seepage velocity, Darcy Law velocity
v Frontal velocity, u/
V Volume of fluid displaced
V Value of one pore volume of a porous medium
V' Volume of empty plexiglass tube
z Vertical distance
a Constant dependent on the pore system, coordinate axes
$ Proportionality constant, coordinate axes
y Coordinate axes
T Proportionality constant
6 A diameter (average bead, pore, etc.) associated with
the porous medium
n D/vz
9 Number of pore volumes of a fluid displaced, V/VQ
X Major elliptical axes
X Minor elliptical axes
p Fluid viscosity
V Kinematic viscosity of fluid
£ vt/z
p Fluid density
ix
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Porosity of porous medium
Proportionality constant
Proportionality constant
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INTRODUCTION
This report presents the results of a study of dispersion during
flow in nonhomogeneous porous media. The theories of dispersion were
reviewed, a statistical theory derived, and an experimental program
of measuring dispersion completed. Since most real media are non-
homogeneous, that is they are heterogeneous, nommiform, and anisotropic
it is necessary to understand the effects of these non-homogeneities
when attempting to predict dispersion in such systems.
We start with definitions of dispersion and the non-homogeneities,
heterogeneity, nonuniformity and anisotropy. After a brief literature
review we derive the mathematical model we will use to interpret the
tracer experiments, a brief discussion of certain statistical terms,
the experimental techniques, and the results and interpretation of
these results for a series of four experiments. The conclusion and
recommendations presented earlier are from this interpretation.
Definitions. Dispersion, as considered here, is the macroscopic
mixing caused by uneven cocurrent laminar flow in fixed beds of real
media. The behavior of a traced fluid in one dimension is indicated
schematically in Figure la with some boundary conditions. We imagine
fluid flowing according to Darcy's Law
v - -
vx n dx
and further imagine we can mark a line of particles, the X's in the
figure. As a result of velocity variation in path selection, this line
spreads as flow progresses. If we plot the concentration of marked
particles C (normalized by the input concentration C ) versus through-
put measured in multiples of total pore volume, we characteristically
obtain a breakthrough curve as shown in Figure Ib. We usually apply
a diffusion model to dispersion with such velocities such that there
is no significant diffusion effect, so
where D is called the dispersion coefficient. The approximate solution
of this equation, with boundary conditions of Figure 1 is
r i L~v^t
£--i[l-erf{ - Ml (3)
Co 2 2/~Dt
where L is the length of the media. At a pore volume throughput equal
to 1 for the idealized situation of Figure 1, the slope of this break-
out curve gives the D in equation (2) which is given by
-------
>,
X
<
>i
t < 0 C«
t > 0
1.0
C. 0.5
C0
r\
C =
b
$&
«
OoOSt
6*-£*t
X
X
X
K
-*
xz
0
Co
•
t>0 C = C(X,t;0)
a
/
f
V/V0
b
Figure 1 - Dispersion and Flow
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i Lv
1 x
D » -r-
4ir 3 C/C
3 V/V
0 V/V
o
Thus the slope of this curve is a rough measure of the dispersion taking
place in flow through the medium.
We base our definitions of heterogeneity, nonuniformity and
anisotropy on the probability density distribution of permeability of
random macroscopic elemental volumes selected from the medium, the
permeability being expressed by the one-dimensional form of Darcy's
Law, Equation (1) .
By a uniform medium we mean one in which probability density
function for permeability is either a Dirac 6-function or a linear
combination of n-functions that satisfy the relation
N
f(k) - ±S1 ^6 (5)
where
N
Examples of this sort of behavior are shown in Figure 2.
By nonuniform we mean a medium in which the probability density
function cannot be constructed with a finite number of weighted 6-
functions as shown in Figure 3.
By a heterogeneous medium, we mean one in which permeability
distribution is at least bimodal. Figure 2a is homogeneous and
uniform and Figure 3a is homogeneous and nonuniform. Figures 2b
and 2c are heterogeneous and uniform, and Figures 3b and 3c are
heterogeneous and nonuniform.
By anisotropy, we mean the permeability varies with direction
in an elemental volume in the medium.
The probability density function for permeability is, in general,
a function of location and orientation. We can describe this
function with five independent variables: the rectangular coordinates
x (i • 1, 2, 3) for location and the angular coordinates 9 and fy for
orientation:
fk2
J * f^.S.iJOdx
P(k (7)
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f (K)
(a)
f (k)
(b)
f (k)
(c)
Figure 2 - Observed Permeability for Uniform Media
4
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f (k)
(a)
f (k)
(b)
f (k)
(c)
Figure 3 - Observed Permeability for Nonuniform Media
5
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If the probability density distribution is independent of 9 and
i|» the medium is isotropic; if the distribution is expressible by a
finite linear combination of
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The article of Chung and Wen2 contains an extensive tabulation of
experimental work, including such items as systems and range of vari-
ables studied. The more recent article of Greenkorn and Kessler3
(This review was performed as part of this project — a summary is
attached as Appendix 1) examines dispersion in heterogeneous, non-
uniform and isotropic porous media with emphasis on statistical and
continuum theories which have been derived for the nonideal case.
Harleman, Melhom, and Rumer1* have studied to some degree the
effect of heterogeneity as reflected by permeability change on
dispersion in porous media. They took data on some ten different
media of uniform size which had either spherical or angular shapes,
and from these data developed a correlation for the dimensionless ratio
of the longitudinal coefficient of dispersion to the kinetic viscosity
as a function of the permeability Reynolds number. This correlation
is used in a later paper by Shamir and Harleman5 in which a specific
type of heterogeneous media, that is, layered media, is studied. In
this latter study they considered dispersion in flow perpendicular and
parallel to layers. They determined the dispersion for the various
layers from the correlation of the earlier paper of Harleman et alk and
then calculated the resulting dispersion at the end of the column. In
addition, they also took experimental data at various points along the
column and showed that there was approximate agreement between the
superposition bf the various dispersion coefficients, the experimental
data and the equivalent curve resulting from looking at the concentration
output from the total length of the model.
The effect of direction on permeability, anisotropy, has been
measured in a few instances and the theory of direction permeability
has been worked out.
Scheidegger6 has tested the theory on existing literature data.
Greenkorn, Johnson, and Shellenberger7 have shown the theory works on
natural sandstones and they correlated various parts of the permeability
tensor with measureable lithological characteristics such as grain size,
grain orientation, and layering. To the authors' knowledge there are
no measurements of dispersion in anisotropic media which have been
interpreted as anisotropies. Shamir and Harleman5 indicate it is
conceivable to consider a layered medium as an anisotropie. However,
they have done no calculations, nor have they attempted to measure
dispersion in such media.
There is a vast amount of theoretical literature associated with
the continuum theory of dispersion and the tensor representation of
the dispersion coefficient3. However, this theory has not been utilized
to interpret data from anisotropic porous media.
*
Various attempts have been made to correlate dispersion with the
characteristics of porous media, but only moderate success has been
achieved, probably because of the inherent difficulties in isolating
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the effects of the various characteristics.1* These characteristics can
be best understood through a brief description of a porous medium. Perhaps
the simplest description is: a porous medium is a solid with holes in
it. But for mathematical purposes it is described as a solid containing
a complex network of pore spaces or voids. Sheiddeger8 classifies pore
spaces into voids, capillaries, and pore spaces. In voids the walls
have negligible effect on the hydrodynamical phenomena in their interior,
while in capillaries the walls do have a significant effect. In pore
spaces, the molecular structure of the fluid is a significant property.
Voids can be classified further: extremely large voids are called
"caverns," while extremely small ones are called "molecular interstices,"
the remainder being simply designated "voids." The pore space in the
media used in this work fall into the void classification.
Pore spaces are also classified as to whether they are inter-
connected or non-interconnected. Fluid flow is possible only through
interconnected pores, and this interconnected portion of the medium is
called the effective pore space. Non-interconnected pores are called
"dead-end pores."
Finally, the porous medium as a whole can be classified as
consolidated or unconsolldated. A consolidated porous medium has a
fixed structure, for example, limestone or sandstone. Unconsolidated
porous media are beds made of sand, glass beads, catalyst pellets,
etc., such as their effective pore space depends upon the packing
procedure used in the construction of the bed. The porous media used
in this work (plexiglas tubes packed with glass beads) are unconsolidated.
Permeability can be defined as a measure of the ease with which a
fluid will flow through a porous medium — the higher the permeability,
the easier will be the flow. It is a statistical average of the fluid
conductivities of all the flow channels in the medium, and normally
considered a point property of the medium where a point is defined as
a region over which Darcy's Law will hold. (A region large enough for
volume averaging, see Whitaker9).
In all applications of Darcy's Law, whether it be in homogeneous,
heterogeneous, or nonuniform porous media, it has been concluded that
there is a critical seepage velocity above which the law is not valid.
This critical velocity is characterized by a Reynold's number defined
by
Re-^f (8)
where £ is the diameter associated with the porous media (average bead
diameter, average pore diameter, etc.). However, investigations have
been made since 1920 to determine the value of this critical Reynolds
number, with the conclusion that it lies somewhere between 0.1 and 75.
Sheidegger8 explains this as due to the Indeterminacy of the diameter 6.
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In porous media constructed of glass beads, the pore size
distribution is related to the distribution of bead diameters. When
a medium is packed with beads of various sizes, the small beads can
rest in the interstices between the large beads. The size of the
average pore space can thus be characterized by dimension smaller
than the median bead size.1
Particle size distribution in naturally occurring, unconsolidated
porous media often follows a log-normal distribution, that is, the
particle size versus cumulative percent plots is merely a straight line
on log probability paper. This is also often the case with commercially
produced mixtures of glass beads. Raimondi, Gardner and Patrick10 found
that three mixtures of glass beads all had roughly the same permeability
when their effective bead diameters crossed at approximately the bead
diameter corresponding to the ten percent cumulative fraction. This
effective diameter, dio» was used for 6 in calculating the Reynolds
numbers for this work.
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THEORY
Dispersion coefficients are most often obtained by measuring the
mixing of a trace element resulting from flow through the porous
media for which the dispersion is to be estimated. Either a step
input in tracer concentration or a rectangular input is utilized and
the resulting breakthrough curve is analyzed with the dispersion model.
When the traced solution displaces the untraced solution filling the
voids of a porous medium, the concentration interface between these
two solutions does not remain as a sharp plane; rather, as flow
progresses, there is a gradual mixing of the traced and untraced fluid
due to local velocity fluctuations. These continuous variations of
velocities both in magnitude and direction throughout the tortuous
path in the void space of the porous media cause mixing of the two
fluids and a mixed region is formed between the traced and untraced
solution. If one then collects the output of solution from such a
porous media, a breakthrough curve such as is sketched in Figure Ib
results, where the pore volume is the volume of the porous medium
multiplied by its porosity. The problem is to describe the output
tracer concentration as a function of time or pore volume. If the
porous structure is assumed random a statistical model can be used to
fit breakthrough curves such as Figure Ib.11 The result for flow
in the y direction as
.2
(9)
This is a solution of the diffusion equation with convection
for an infinite media with a A-function input.
We adopt the diffusion model here but consider a step input and
use the boundary conditions
C(0,t) - C ; t * 0
o
C(X,0) - 0 ; x * 0 (11)
C(»,t) - 0 ; t * 0
The solution of Equation (10) with these boundary conditions is
C/C - \ [erfc {^-£} + exp {-} erfc {±-^-4 (12)
° 2 2/~^ " 2v^T
10
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where erfc is the complementary error function, £ » vt/x and n - D/vs,
v is the Darcy velocity given by Equation (1).
Most often tracer data are interpreted with Equation (12). If n
is less than .0075, the second term may be ignored and the error will
be less than 5%12, where n is the reciprocal of the Peclet number
vL/D and therefore the Peclet number must be greater than 133 in order
to drop the second term of Equation (12).
Equation (12) can be put into the form (with last term dropped)
1 l - V/Vn
C/C - f [1 - erf ( ° } (13)
2/vTviT
0 Lv~~
x
which shows that the theoretical breakthrough curve for a given medium
is determined by the dimensionless Peclet number. The dispersion
coefficient can be determined by taking the derivative of the break-
through curve at the inflection point
(14)
In using the above model we are assuming that all of the mixing is
due to convective transport rather than molecular diffusion. Diffusion
is not important where the Reynolds number Re - Dv p/y is greater
than 10-3. 13 x
Correlation of Dispersion and Permeability. Harleman et, al1* have studied
dispersion-permeability correlations in porous media. They correlate
their data on the basis that permeability can be related to the average
particle diameter by
k - C'dfo (15)
When we adopt the diffusion equation (10) and the solution (13) as
a model for dispersion we have tacitly assumed calculation is directly
proportional to pressure, since the fundamental equation of flow
expresses flux as proportional to the gradient of a potential function.
Since in Darcy's Law, equation (1), the flux is proportional to velocity
we would expect the dispersion coefficient resulting from using equation
(10), to be dependent on velocity. Indeed when dispersion data is
investigated as a function of velocity the data follows an equation of
the form:
11
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D - parameter (velocity) (16)
Scheidegger8 points out when using equation (10), which is justified
on statistical arguments, that there are two possible extremes for the
form of the dispersion coefficient velocity dependence; the first
directly proportional to velocity:
D - av (17)
the second proportional to velocity squared
D - av (18)
Equation (17) represents the case where there is enough time for
complete mixing of invading and original fluid in each pore.
Equation (18) represents the case where mixing is not complete in
each pore.
Relations such as Equations (16) (17) and (18) can be expressed as
tensor equations and the "parameter" a becomes a fourth order tensor
and the velocity dependence is represented by a dyadic product of
velocity.11* Usually in this form a linear velocity dependence is
assumed and the tensor for anisotropic is reduced to a constant type
of relation of a form.15 l6 (Reference 16 reports a theoretical study
of the dispersion tensor referenced as part of this paper — a summary
is attached as Appendix 2).
Dy - X6ljV + 2y (19)
The X and u are related to the properties of the medium. For one-
dimensional flow the dispersion coefficient in the direction of flow
(the longitudinal coefficient) takes on the simplified form
Dll " (X + 2y)v (20)
which is of the form of Equation (17).
Harleman, et al" assume this form (Equation 17 or 20) to obtain
their correlation expression from Equation (15).
(21)
V V
and
12
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(22)
Poreh17 has taken into account the possibility that mixing in
the pores is not complete and he obtains constitutive type relation
for the dispersion values of the form
(01 + ct2 ^-) s + «3 - . (23)
D2 ij £>2
which is of the form of Equation (18) for one dimensional flow.
Thus in general we would expect a combination of mechanisms in
the pores depending on the velocity of flow and the properties of
the media so an equation of the form of Equation (16) should represent
one- dimensional experiments with the parameter a function of the media
properties and the experiment dependent on velocity and media properties,
since variation in pore size will cause different intersititial velocities.
At very slow velocities one would expect n -*• 2 and at relatively high
velocities in a permeable media n •*• 1.
Baring and Greenkorn17 (This paper reports a theoretical study and
was in part performed as part of this project — a summary is attached
as Appendix 3) have shown that a statistical model of a porous media
with non-uniform pores predicts permeability is of the form
k - *2Cg(a,3) 23)
where 4> is porosity is the average pore radius, C is a numerical
constant and g(a,B) is a function of the first and second moments
of the pore radius distribution. We might expect the parameter C1 in
equation (15) to be a function of porosity and the nonuniformity of the
pore size distribution. The average value to be used for comparison of
data is not necessarily dso but the average value for the particles pore
size distribution utilized. For most unconsolidated media, the distribution
is skewed and a value of dm to dzo must be used to obtain permeability.
Haring and Greenkorn show that the form of the dispersion coefficient
in nonuniform media depends on both the values of the pore radius
distribution and the pore length distribution as well as average pore
length and velocities. We might expect the parameter of Equation (16)
to be dependent on these distributions.
D - eg (a, 3) f(a,b) vh(ln a,b,v) (24)
13
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In order to correlate data according to Equations (21) and (22) we
must assume a uniform medium with complete mixing in the pores. With
these assumptions
- j (<&>) = C' (25)
and we can replace <£> in Equation (24) with . Combining with Equation
(23) gives an equation of the form
D - — — TT ^ g(ct>B)^ f(a,b) vh(ln {a,b,a,0,<£> v}) (26)
c'c172 *
For uniform media g(a,@) and t(a,b) are constants and h * 1 and
t/k~
D - B Y± v (27)
or
which is of the form of Equation (22) .
From the above we would expect if we correlate dispersion data
versus permeability or grain size In the form of Equation (21) or
(22) that there would be an exponent on the velocity different than 1
(but between 1 and 2) and that the porosity would be a function of the
nonuniformity of the porous system. More generally we would expect to
correlate
(29)
where B is a function of nonuniformity, n a. function of velocity and
nonuniformity.
A general form of Darcy's Law can be written
v - = Vp (30)
where k.. is the permeability tensor. The fact that the permeability
in an anlsotropic porous medium has a tensorial nature leads to the
following conclusions: (1) The pressure gradient and the seepage
velocity in general are not parallel; (2) If the permeability tensor
is symmetric, there are three orthogonal axes in space which are termed
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the principal axes of the permeability tensor.
It is necessary to relate the components k of the permeability
tensor to the actual measurements of the directional permeability.
There are two separate cases to be considered here, which in the
limit lead to the same result. The first is the case where the flow
velocity is assumed parallel to the outward directed normal, (width/
length of model is small) and the second case where the pressure gradient
is assumed parallel to the outward directed normal n (width/length of
model is large).
Consider cutting out of a porous medium a narrow tube parallel to
the outward normal and measuring its permeability in the usual manner.
In this case because this is a narrow pencil, the seepage velocity v
can be assumed parallel to the normal nu The pressure drop along the
tube, denoted by p , is then given by the following
P« " E * VP " U[n • (k'1' n)v (31)
n •"• — n
Define a "directional permeability" kl such that
Therefore
k' -- -, - (33)
n (n • (If1 • n»
If the principal axes of the permeability tensor are chosen as
coordinates and if the angle of n with the coordinate axes are called
a, 3 i and y» the direction permeability kA, is given by
In two dimensions , therefore, if the square root of the directional
permeability / k ' is plotted against the corresponding directions in
an anisotropic porous medium, one obtains an ellipse. The two axes of
this ellipse are in the direction of the principal axes of permeability
and their length is equal to the square root of the principal values of
the permeability tensor.
In the second case we choose a system in which the pressure drop
is given by the boundary conditions , and we measure the component of
velocity which is parallel to the pressure gradient. The directional
15
-------
permeability is defined by Darcy's law from the given pressure and the
velocity component parallel to it. Denoting the pressure gradient by
PN in the direction of in and the velocity component parallel to n by
v, then by similar derivation6,7 ~
k " k Cos2a + k coa2® + k
n
22 coa + k33 cosY (35)
Therefore, in the two-dimensional case, if the reciprocal of
is plotted against the corresponding directions in an anisotropic n
porous medium, one obtains an ellipse. The axes of this ellipse are
in the direction of the principal axes of permeability and their length
is equal to 1 over the square root of the principal values of the
permeability tensor.
Experimental data, then, are interpreted in one of the two forms,
depending on which type experiment is performed. Equations (34) and
(35) represent two different methods of fitting an ellipsoid to
directional permeability data. The question arises as to what per-
meabilities we actually measure in an anisotropic laboratory model.
Let's consider a model of length L and width W through which a fluid
is flowing lengthwise. For W/L ratios approaching zero, the stream
lines are all forced to be in the direction of the measured permeability,
so Equation (34) would apply for this case. For W/L ratios approaching
infinity, the pressure gradient and the measured permeability directions
are parallel; the stream lines are oriented in some other direction.
For this case, Equation (35) would apply. For intermediate W/L ratios,
the measured permeability would fall between two limiting values kA
and k^j. Scheidegger8 has analyzed the ratios of the two limiting
permeability expressions for their maximum possible deviation, and has
found that the degree of anisotropy has to be appreciable before the
difference becomes significant.
We have made the assumption that the permeability tensor is
symmetric, that is, k^. «= k . However, Whitaker19, in his treatment of
Darcy's law, concludes3 thatj there is no general reason to indicate
the permeability tensor should be symmetric. On the other hand,
Greenkorn and Kessler20 and Guin, Kessler, and Greenkorn2 ' have shown
from statistical model studies that one x^ould expect a symmetrical
permeability tensor. It is certainly conceivable to imagine that in a
porous medium if the interstitial velocity is such that the form drag
dominates flow and further that there is a distinct particle orientation
such that the form drag would be a function of direction that the
permeability tensor might not be symmetric. It is not clear at what
values of seepage velocity and particle size distribution this might
happen.
There has been no attempt to show how one would interpret
experimental data in terms of the dispersion tensor. We assume for
the moment that the dispersion tensor is symmetric and thus if we
select a coordinate system moving with the average velocity of the
16
-------
fluid we would adopt a diffusion model for the dispersion in any porous
medium. If we further assume that there is a flux relationship between
the dispersion coefficient and the concentration gradient such as
(36)
then we would have an analogous case to that of the permeability tensor
as given by Darcy's Law and we would expect the same general argument
might hold. That is, for the two cases, if we plotted the measured
dispersion coefficient as determined from the solution of the above
equation, we would plot the square root of v^D~ or 1//D versus
direction for the anisotropic media and expect to obtain ellipses as
in the case of permeability. This of course assumes that dispersion is
represented by a symmetrical tensor. We have a practical dilemma
here in that we cannot obtain the dispersion coefficient by measuring
the concentration change over a length and relating it to the flux;
rather, we measure the concentration breakout curve and fit a solution
of the diffusion equation to this breakthrough curve to obtain the
parameter in the solution, the dispersion coefficient. In principle,
if we maintained a constant concentration gradient across the medium
(and this is concentration of marked particles resulting from flow) and
measured flux we could obtain the dispersion coefficient in an analogous
fashion to the permeability. Thus we are presented with an interesting
dilemma — that we must measure or obtain a measure of D by solving the
second-order diffusion equation. This complicates matters, since for
a radial geometry it becomes very difficult to solve this equation and
in this work we resort to doing a linear analysis. Thus the measure
of dispersion we utilize is not the true dispersion coefficient but
rather the slope of the breakout curve which we interpret to be a
relative measure of dispersion as a function of direction. We then com-
pare plots of this coefficient versus the permeability. Similar
comments as to the difference between the two limiting cases for
permeability would then apply to the dispersion coefficient.
17
-------
DISPERSION DURING FLOW IN LINEAR HETEROGENEOUS POROUS MEDIA22
Experimental Procedures
In order to study the effects of heterogeneity on dispersion
we packed Lucite cylinders 6" long and 1-1/8" in diameter with glass
beads in four ranges of spherical sizes. Figure 4 is a schematic of
one of the models. Each of the models was packed with a different
size of glass bead and the porosity, permeability, and dispersion of
each model determined as well as the result of combining all four
of these models in various arrangements to give a set of composite
models that were heterogeneous by definition of Equation (7). (Note
that in a realistic situation in order to obtain heterogeneity we
must use bead sizes that do have a size range associated with them,
so we are actually investigating the situation of Figure 3b, but our
emphasis was on the effect of change in average permeability, the
heterogeneity rather than variance of the permeability itself, the
nonuniformity.) We used a statistical design for the permeability
and dispersion measurements to investigate the effects of various
parameters and the effect of replication.
The models were packed by continuously vibrating them on a
shake table with a frequency of vibration of about 60 cycles per
second. The beads were poured into the model slowly, taking about
an hour and a half to pack a 6" model. After filling constraining
screens and spacers were placed at the end of each of the models
and caps screwed in place.
Porosity. The porosity of the models was determined by the liquid
saturation method. Water with a very small of detergent was
introduced at the bottom of the model at a very slow rate, and the
air forced out of the top of the model. The detergent was added
to aid in completely wetting the surface of the packing. A small
amount of food coloring was added as an aid in visualizing the
filling of the model. The volume of water occupied by the pore
space was determined as the difference in the weight of the model
before and after filling, with due regard taken to correct for
water remaining in the inlet and outlet parts of the model. The
porosity then was calculated as the ratio of the pore volume to
the total volume of the model. Once the porosity had been determined,
the model was purged with distilled water to remove soap and
coloring and from that point on the model was never dried again.
(In fact, we kept the models immersed in water when they were not
in use and whenever they were connected or unconnected. This was
done underwater to limit the air entering the models.)
18
-------
I"
0-RING
SPACER
CLOTH
SCREEN
CLOTH
SCREEN
\
SPACER^
-itr
i=i
_> t.
i
1
i
i
i
i
ii
L
0-RING
I4.93cm.
OUTSIDE DIA,
I*'
INSIDE DIA.
2.86 Cm.
,11
i
1
1
1
1
1
h—
L_
I
-J.
1
1
•
j
1
— 1
1
i
i
i
LV-
ii
i
i
Figure 4 - Description of Model
19
-------
Permeability. The permeability of each model was determined with
Equation (1) in the form
where Q/A is the Darcy velocity. The measurements were made in a
temperature-controlled room where the temperature was held between
76-77°F. The permeabilities for the models were determined by
measuring pressure drop for several flow rates for each of the
models. Flow rates were measured by the amount of time for
collection of a given amount of effluent. The pressure drop was
measured using a Pace differential transducer, model KP-15 and a
Pace transducer indicator, model CD-25. The transducers were
calibrated with U-tubes and deadweight tester.
Dispersion. Breakthrough curves were determined for each model by
means of an electrochemical system. Potassium chloride was used as
a chemical tracer since mobilities of the potassium and chloride
ions are almost the same and hence the liquid junction potential is
minimized. A silver-silver chloride electrode in conjunction with a
calomel reference electrode were used to determine the concentration
of the chloride solution. The potential was monitored on the
Sargent SR balancing potentiometric recorder. (The preparation of
silver-silver chloride electrodes is discussed by Schumacher and
t It
Garland. ) In order for the potentials of both the model fluid and
the displacing fluid to be recordable on the 0-125 mv scale, the
model fluid was made .02 molar KCl and the displacing fluid was .4
molar KCl. These gave potentials of approximately 80 mv and 13 mv
respectively.
Permeability and dispersion were measured simultaneously, that
is, the flow rate and pressure drop were monitored while the model
fluid was being displaced. Figure 5 shows a schematic diagram of
the equipment used.
Since there is a density difference between the .02 molar and
.4 molar KCl solutions, the displacing solution was always
introduced into the bottom of the model to minimize possible
gravity fingering.
The actual procedure for the displacement was as follows:
the model was purged with .02 KCl molar solution; the desired flow
rate was obtained; the pressure drop was noted for the permeability
measurement; the value controlling the .02 molar solution was shut
off and the valve controlling the .4 molar solution was opened
until the same pressure drop was observed on the transducer-
indicator; the recorder was turned on immediately. The flow rate
20
-------
Ag-AgCl
Electrode
To
Nit rogen
0.02 M
KC1
Reservoir
Tank
x
0.4 M
KC1
Reservoir
Tank
Filter
"*!
H«
(-•
IT
ro
•-«
T
~ I
t
L_
1S.UI
Porous
Media
[
i
1
3
Calomel
Electrode
Transducer
n.
_r
Transduce
Indicator
To
Recorder
Figure 5 - Diagram of Equipment
-------
was controlled by making sure the pressure drop remained constant
throughout the displacement. After the displacement was completed,
the model was purged with .02 molar KC1 solution and preparation for
another run.
A calibration curve was prepared from potential readings based
upon the two solutions. With the calibration curve the concentrations
of KC1 were obtained from the recorder. A plot of dimensionless
concentration versus pore volume was constructed where
c* » C - .02 C - .02
C .4 - .02 .38 (38)
The dispersion coefficients were obtained from these curves and
Equation (4).
For each separate model, duplicates of three different
velocities were used. After the 24 experiments in the individual
models, the four models were screwed end-to-end in each of four
arrangements and duplicates of three velocities of these composite
models were run.
Results
Table 1 shows the measurements of porosity and permeability
for the single models and the measurements of permeability for
three of the four composite models. The permeability was not
determined for the final composite model (3412) since the results
for the previous three had been so consistent.
The values of porosity are reasonable although we did not make
duplicates of the porosity measurements of these particular models,
since once they were saturated we did not wish to dry them, we did
measure the porosity of similar models several times and the
estimate of error of porosity results from these measurements.
The permeability measurements were actually made during the
dispersion runs and represent a minimum of two separate measurements
on each model and in some cases more. The error in the permeability
measurement was about 3%. For the composite models the measurement
error was much less, because we had better control of the pressure
measurement across the longer model.
The permeability data are plotted in Figure 6 for the
individual models (1, 2, 3, 4) according to Equation (15).
k - 2.52 x 10~4 d (39)
22
-------
TABLE 1 - MODEL PROPERTIES
Model Bead Size Range
micron
(cm)
1 590 - 840
2 250 - 350
3 105 - 149
4 44-62
1234
4321
2143
3412
* **
Porosity Permeability
(%) (darcy)
34.0 120 + 2
34.0 35.8 +
34.3 5.04 +
36.5 .487 +
1.82 +
1.84 +
1.85 +
— — ~_
1.2
.06
.012
***
.01
.01
.01
ff
Measurement error + 1% (Estimated from repeated measurements)
2 duplicates at each of 3 velocities
Estimated from individual model permeability - 1.75 d
23
-------
10
-7
10
r8
10
-6
10
-I
I
10-2
Figure 6 - k vs d
24
50
10'
-------
The data show that a straight-line relationship is a reasonable fit
on log-log plot, and the slope of 2 also is a reasonable number.
The value of the intercept for this relationship was lower than that
predicted by Ramondi et al. for spheres, and it is lower than the
data given by Harleman et al. also for spheres. The value of d
should be the one at the correct average since the distribution of
bead sizes may be skewed. The bead sizes and permeabilities
reported here are less than those given by the previous authors.
(We may be seeing an effect due to the nonuniformity of the pack.)
The dispersion coefficients were calculated from breakthrough
curves using Equation (4). Table 2 gives the dispersion
coefficients calculated in this manner. For the first three models,
which are the larger bead sizes, there is a distinct tail in the
breakout curve. The Peclet number in all runs was greater than 133,
and the Reynolds number was larger than 10~^, so the effects of
asymmetry and diffusion should not be involved. Actually, the
value of the dispersion coefficient is not changed if one attempts
to fit the curve more exactly utilizing Equation (12). (The
asymmetry may be due to entrance-exit effects though the models
should be large enough considering the bead size. It may be a
result of the manner in which the models were packed.)
The data for the individual models 1, 2, 3, 4 are plotted in
Figure 7 as a function of velocity, and the equations representing
these figures are:
Model 1: D - 1.0 x 10~3 v1'45 (40)
Model 2: D = .53 x 10~3 v1*33 (41)
Model 3: D = . 39 x 10~3 v1*25 (42)
Model 4: D » .24 x 10~3 v1'15 (43)
There is a significant difference in dispersion coefficient due to
the bead size or permeability. Also, there is a significant
dependence on velocity above the first power. The trend shows the
smaller the beads (or lower the permeability and probable higher
intersitial velocity) the smaller the exponent and the intercept.
These data are similar to the data of Harleman et_aJL.^ except they
are for somewhat smaller particle sizes and indicate that the
nonuniformity is having an effect.
The dispersion coefficients for the composite models are
plotted as a function of velocity in Figure 7 and the Equations
representing these figures are:
Arrangement 1234: D • 1.65 x 10~3 v1*05 (44)
Arrangement 4321: D - 1.50 x 10~3 v1*07 (45)
25
-------
TABLE 2 - VELOCITY AND DISPERSION COEFFICIENTS
Model
1
2
3
4
1234
4321
V
(cm/sec)
.0420
.0420
.0580
.0575
.0750
.0750
.0421
.0421
.0580
.0580
.0750
.0755
.0420
.0417
.0583
.0583
.0750
.0750
.0426
.0420
.0604
.0602
.0765
.0762
.0416
.0420
.0588
.0586
.0750
.0750
.0416
.0418
.0585
.0585
.0752
.0752
D x 10
(cm /sec)
3.74
4.40
5.57
5.85
10.3
9.08
1.55
1.80
3.03
2.56
3.50
3.74
,36
,15
,65
,02
,29
3.04
.754
.672
1.21
.866
1.54
1.78
4.12
3.97
7.61
5.71
7.62
7.30
4.16
3.85
5.46
5.53
8.24
6.81
26
-------
TABLE 2 (cont.)
Model v D x 103
(cm/sec) / 2, N
(cm /sec)
2143 .0416 3.61
.0416 3.11
.0585 5.83
.0585 4.28
.0750 6.50
.0750 6.59
3412 .0416 3.10
.0416 3.34
.0586 4.79
.0580 4.73
.0752 6.33
.0750 6.13
27
-------
10.0
5.0
D x ios
cm2/sec
1.0
0.5
O.I
1.0
3412
/ /
//* /
/ /
//
COMPOSITE MODELS
INDIVIDUAL MODELS
5.0
10.0
v cm/mln.
Figure 7 - Dispersion Coefficient Versus Velocity
28
-------
Arrangement 2143: D » 1.17 x 10"3 v1*14 (46)
Arrangement 3412: D - 1.10 x 10~3 v1'15 (47)
There is not nearly the spread of dispersion coefficients with the
composites as there is with the individual models. As can be seen,
these data all fall close to that for individual model //I, the
larger bead size, which indicates that the dispersion seems to be
dominated by the higher-permeability material.
It is interesting to note that there is a significant difference
in the results from the different arrangements. The way in which
the material disperses in one permeability material has an effect
on the way it disperses into the next layer. Unfortunately this
seems to indicate that reciprocity does not hold in heterogeneous
media, and one must preserve the location of a given level of
permeability, ergo, a given dispersive media, when trying to
calculate mixing during flow in such media.
The individual and composite dispersion coefficients are
correlated as a function of permeability Reynolds number as given
by Equation(29) in Figure (8). Equations for the data are:
Model 1: - - 187 E^'2 (48)
Model 2: - = 215 R^'2 (49)
Model 3: - - 536 R^'2 (50)
Model 4: ~ = 1215 R^'2 (51)
Arrangement 1234: - - 3110 R?'2 (52)
v 1C
Arrangement 4321: ^= 2945 Rj^2 (53)
Arrangement 2143: - - 2550 Rj^2 (54)
Arrangement 3412: - » 2365 R?;*2 (55)
V K
4
Harleman et al. report
2 - 54 2 (56)
Although the power of the exponent is the same the values of the
intercept are higher than Equation (56) and are definitely a function
of the bead size. It may be that the nonuniformity of the bead
29
-------
10.0
5.0
1.0
0.5
0.1
0.05
x COMPOSITE DATA
o INDIVIDUAL DATA
0.01
10
-4
5 X
10'
-3
5 x I0"3 10
R,
Figure 8 - D/v vs R,
30
-------
size causes the intercept to change. On the other hand the
heterogeneous models (the composites) also take an additive jump
in intercept, although the change of intercept of Equations (52) -
(55) is less drastic than the individual models Equations (48) - (51)
The similar exponent is confusing at this point since the velocity
dependence of the models change.
31
-------
DISPERSION DURING FLOW IN NONUNIFORM, HETEROGENEOUS POROUS MEDIA25
Six different porous medium models were constructed of glass beads
to obtain six possible combinations of two permeabilities and three
slopes (variation in permeability). The models were constructed as
shown in Figure 9. Each model was run at three fluid velocities and to
check the effect of packing each medium was run in both possible
directions. Thus a2X3X3X2 factorial design of experiments
was set up without replication.
The assumption was made that the direction effect due to packing
would be negligible. Then all possible combinations of the six models
taken two at a time (fifteen combinations) were run at the same three
velocities and the permeability and dispersion measured. Each combination
was also run at both possible directions to investigate any effect due
to order of models.
The distribution of bead diameters can be quantitatively described
by the slope, S, of the log-normal line
S - log d8A - log d16 - log ^ (57)
16
where d_, and d.., are the bead diameters of the 84% and 16% cumulative
fractions, respectively. The nonuniformity was built into the models
in this study by preparing three sets each of two different size-range
permeability beads where within each size range the distribution of
bead diameters, that is* the slope as given by equation above changed
but where the three lines crossed at d... This provided media which
had roughly the same permeability but a different distribution of
bead sizes (permeabilities) within a given bead-size range; the
possibility of comparing this second-order heterogeneity; and finally
the possibility of comparing the two different permeability levels
(heterogeneity).
Eight replicates, randomly chosen from the 126 total experiments,
were run as a measure of experimental error.
Size Distribution of Beads. To determine the size distribution of
glass beads in a given mix, samples were analyzed with a photomicroscope.
Bead diameters were measured off the photograph produced, and the
actual bead diameters were then calculated from the value of the amount
of magnification used. The number of beads appearing on a photomicrograph
varied according to the mean bead diameter and the magnification used;
however, 150-200 diameters were measured for each mix before the dis-
tribution was plotted on log-probability paper.
32
-------
• II
If
r*"
QJL.
12
T—f
^DOUBLE i" 0-RING
~~~*
WIRE SCREEN
TUBE OUTSIDE DIA. -if
TUBE INSIDE DIA.
-WIRE SCREEN
DOUBLE V 0-RING
Figure 9 - Plexiglas Model - Single Tube
33
-------
The mixes of beads were made up having size distributions inter-
secting at 10% cumulative fraction by a trial and error mixing of the
available distributions from photomicrographs made of each new trail
mix. The final distributions resulted from mixing various amounts of
14 different commercially available sized glass beads.
Packing Procedure. The glass beads were packed in the plexiglas
tubes to give cylindrical porous media 12 inches long and approximately
1 1/4 inches in diameter. The tubes were constructed such that they
could be joined end to end in order to run combined combinations of
first and second order heterogeneities.
To pack a tube, beads were continuously added while the tube is
vibrated on a shake table at 50-70 cycles per second. The rate of
addition of the beads was held uniform throughout the packing process
and it was controlled so that each of the six tubes took 45-60 minutes
to fill. The vibrating was stopped with the flow of beads. Wire
screens were then mounted on the models to hold the beads in place
and caps were screwed on.
Porosity. The porosity was determined by saturating the porous
media with distilled water, wetting the media before and after saturation.
This difference in weight minus the weight of water accumulating in the
fittings of the tube ends was converted to volume units and gave values
of one pore volume of the porous media. Porosity was calculated as the
ratio of the pore volume to the total volume.
Distilled water was used for these measurements and contained a
small amount of food coloring and a very small amount of liquid
detergent. The food coloring served to help visualize the saturating
and help detect any air pockets, while the detergent acted as a
wetting agent to help completely wet the surface of the media.
With the tube in a vertical position water was introduced in the
bottom, thus forcing the air out the top. The rate of water addition
was varied to keep the air-water interface in the medium as flat as
possible, which took about fifteen minutes. Before making permeability
and dispersion measurements, each model was purged with regular distilled
water to remove the food coloring and the detergent from the medium.
Permeability. Permeability was calculated from Darcy's Law using
data obtained by forcing fluid through the media with a Ruska proportioning
pump. Flow rates used were 224, 128, and 64 cc/hr. (Approximately 1.5,
.8, and .4 cm/min.) Pressure drops were measured with a Pace differential
pressure transducer and a Pace transducer indicator. Permeability was
calculated from equation (37).
Dispersion. Equation (4) was used to calculate the dispersion
coefficient for each run. The electrochemical system described by
Koonce and Blackwell23 was used to determine the breakthrough curves
from which dispersion could be calculated. The system consisted of a
34
-------
silver-silver chloride electrode in conjunction with a Calomel reference
electrode and potassium chloride was used as the chemical tracer. The
potential between the electrodes was a measurement of the chloride
concentration in the immediate vicinity of the silver-silver chloride
electrode. Figure 10 is a schematic of the experimental setup.
Throughout each experiment the electric potential was monitored
on a Sargeant SR balancing potentiometric recorder, which is 0-125
millivolt range. The fluid in the model (.02 molar KCL) gave a
potential reading of about 80 millivolts, while the displacing fluid
(.4 molar KC1) gave a potential of about 12 millivolts. Due to the
slight density difference between the two fluids , the displacing fluid
was introduced into the bottom of each model to eliminate possible
gravity fingering.
As a check on the electro-chemical system, a colorlmetric analysis
of the K in the displaced fluid was conducted on approximately 15% of the
runs. For this test, a Renco automatic fraction collector was used to
continuously collect 5 ml samples of the effluent stream. Two milliliter
samples of this 5 ml sample was analyzed for potassium concentration by
adding 25 ml of distilled water, .5 ml 1-molar sodium hydroxide at
approximately .01 gm, 2, 2', 4, 4', 6, 6' hexanitrodiphenylamine. A
Klett-Summerson photoelectric colorimeter with a 1 mm cell was calibrated
so the readings could be easily converted to KC1 concentration in the
effluent stream of the model.
Dispersion and permeability measurements were made simultaneously
for each run by recording the electropotential and reading the pressure
drop from the transducer indicator. For each run, the model was first
purged with .02 molar KC1 supplied by a gravity feed tank. The pro-
portioning pump was then set for the desired flow rate of .4 molar KC1
and the pump andrecorder turned on. After completion of the displacement,
the model was again purged with .02 molar KC1 to prepare it for the next
run.
The KC1 concentration was read off the continuously recorded
electrode potential at certain time intervals, and the dimensionless
concentration calculated from equation (38) where C is the KC1
concentration. The time was read in units of pore volume, and the plot
of pore volume versus concentration is the breakthrough curve needed to
calculate the dispersion.
Results
Size distribution of the beads. The final size distribution of the
bead used in packing the size porous media are shown in Figure 11. The
parameter used in correlating the size distribution with the dispersion
coefficient was the slope of these log-probability plots. The slopes of
lined in Figure 11 are listed in Table 3.
35
-------
0.02 M KC1
Gravity
Feed Tank
0
Filter
Calomel
Reference
Electrode
6
Recorder
O
Ag-Ag Cl
Electrode
Porous
Media
Outlet for
Flushing with
0.02 M KC1
To Automatic
Fraction
Collector
d-
Transducer
Indicator
Transducer
Filter
Proportionating Pump
Containing 0.4 M KCl
Figure 10 - Equipment Set-Up
36
-------
E
E
tr
UJ
UJ
o
o
<
UJ
CD
1.0
0.5
0.2
O.I
0.05
0.02
0.01
MODEL 6
X
MODEL 5
MODEL 4
MODEL I
i i i
5 10 20
50
80 90 95 98
CUMULATIVE AMOUNT SMALLER THAN
STATED SIZE (%)
Figure 11 - Size Distribution of Beads Used in Packing Models
37
-------
Table 3. Porosities and Slopes of the Bead Size Distributions for the Models.
oo
Model
1
2
3
4
5
6
Slope
0.
0.
0.
0.
0.
0.
184
280
389
182
282
379
Porosity
0.
0.
0.
0.
0.
0.
350
348
333
341
330
321
Small
Small
Small
Large
Large
Large
Description
beads,
beads ,
beads ,
beads ,
beads,
beads ,
small
spread
medium spread
wide
small
spread
spread
medium spread
wide
spread
-------
Porosity. The results of the porosity measurements are shown in
Table 3. Although they do vary from model to model, the values are
within the range expected for this kind of pack. Three porosity
measurements were made on each model, and the maximum variation of
porosity was 1%.
Since the vibrating time in packing each model was the same, the
results indicate that porosity is decreasing as the bead diameter
distribution widens or as the slope of the plots in Figure 11 increases.
When two models were joined to form a composite model, the porosity
used in calculations was the arithmetic average of the two individual
model porosities.
Permeability. The permeabilities were calculated from Equation
(37) with the proper constants. For the nonuniform but homogeneous
medium (single tubes) the permeabilities measured appear in Table 4.
The permeabilities seem to decrease slightly with a decrease in velocity,
and this is probably due to a small amount of air being present in the
medium. Actually this change is within the error of repeated measure-
ment, so the average permeabilities were used in the correlations.
Remondi's et al_10 observations of the permeability-dependence upon d
would predict that models 1, 2, and 3 should have the same permeability,
and likewise models 4, 5, and 6 should have the same permeability.
The data in Table 4 show that this is not quite the case here, but it
appears that the diameter should be greater than d.Q, and probably
about don'
For the homogeneous nonuniform models, the permeability values
were correlated with Equation (39). The equations for these are
Slope 1: k - 2.4 x 10~4 dj^67 (58)
Slope 2: k - 2.9 x 10~4 d*'64 (59)
Slope 3: k - 2.2 x 10~4 d*J51 (60)
In these equations we used d - as the average diameter rather than
d5Q as did Pleshek and GreenTcorn22 and Harle-nan et al**.
The permeability values for the heterogeneous porous media, that is,
porous media that have first and second order heterogeneities, appear
in Table 5. The predicted values of permeability are also listed in
Table 5 and the average variation between measured and predicted
values is about 2% of the predicted permeability.
Dispersion. The dispersion coefficient for each run was calculated
from the breakout curve and Equation (4) in the manner described
earlier. All the curves were roughly symmetrical. Theoretically, the
39
-------
Table 4. Permeabilities of the Homogeneous Porous Media.
Velocity
( cm/min)
1.42
0.81
0.40
1.42
0.81
0.41
1.49
0.85
0.43
1.46
0.83
0.41
1.50
0.86
0.43
1.55
0.88
0.44
Model
1 -
1 -
1 -
2 -
2 -
2 -
3 -
3 -
3 -
4 -
4 -
4 -
5 -
c _
5 -
6 -
6 -
6 -
up
up
up
up
up
up
up
up
up
up
up
up
up
up
up
up
up
up
Permeability
(Darcy)
1.98
1.96
1.94
2.78
2.81
2.73
4.70
4.65
4.40
27.6
26.3
23.1
37.2
34.6
27.0
47.0
42.1
33.5
Model
1 -
1 -
1 -
2 -
2 -
2 -
3 -
3 -
3 -
4 -
4 -
4 -
5 -
c _
5 -
6 -
6 -
6 -
down
down
down
down
down
down
down
down
down
down
down
down
down
down
down
down
down
down
Permeability
(Darcy)
2.00
1.98
1.93
2.89
2.85
2.77
4.76
4.68
4.54
25.4
23.3
20.2
36.8
33.4
28.5
44.6
40.2
34.7
Note: "up" indicates that, during the run, the model was in
the same position (with respect to gravity) as when it was
packed. "Down" indicates the model was in the opposite
direction during the run.
40
-------
Table 5. Permeabilities of the Heterogeneous Porous Media.
Measured
Predicted
Model
Combination
1 -
1 -
1 -
1 -
1 -
1 -
1 -
1 -
1 -
1 -
1 -
1 -
1 -
T _
1 -
2 _
2 -
2 -
2 -
2 -
2 -
2 -
2 -
2 _
2 -
2 -
2 -
2
2
2
3
3
3
4
A
4
5
5
5
6
6
6
3
3
3
4
4
4
5
5
5
6
6
6
Velocity
(cm/min)
1.42
0.81
0.41
1.45
0.83
0.41
1.43
0.82
0.41
1.46
0.83
0.42
1.48
0.84
0.42
1.45
0.83
0.41
1.44
0.82
0.41
1.46
0.84
0.42
1.48
0.85
0.42
Permeability
(Darcy)
2.38
2.35
2.29
2.82
2.76
2.75
3.77
3.81
3.82
3.63
3.68
3.64
3.77
3.72
3.58
3.62
3.60
3.52
5.32
5.19
4.97
5.41
5.39
5.09
5,57
5.62
5.42
Permeability
(Darcy)
2.34
2.32
2.27
2.80
2.77
2.70
3.71
3.65
3.56
3.78
3.72
3.62
3.82
3.76
3.67
3.55
3.53
3.41
5.13
5.10
4.89
5.29
5.24
5.00
5.35
5.31
5.10
41
-------
inflection point of the breakthrough curve should occur at one pore
volume and a. reduced concentration of .5, and the slope for Equation
(4) should be calculated at this point. Experimentally, the inflection
point was displaced to the right on all breakthrough curves, although it
did occur at about PV = .5. This shifting of the breakthrough curve does
not have any effect on the calculated dispersion coefficient and is
probably a result of making errors in the pore volume calculations or
in measurement of porosity. As stated earlier, not all the air can be
removed from the porous media, and thus each run may carry a little bit
of air, increasing the value of one pore volume and shifting the
breakthrough curve to the right. The presence of dead-end pores as
discussed by Coates and Smith26 would cause a shift in the curve to
the left. We would not expect a large volume of dead-end pores in an
unconsolidated media. In articles by Rafai et al11 and Harleman", this
shifting of the breakthrough curve was not discussed.
Another possible cause of the curve shifting could be the lag time
in the electrochemical measurement system. Koonce and Blackwell did
not note any significant lag time. The potassium analysis, which was
made separate from the electrochemical analysis, agreed with the
electrochemical analyses and so it is doubtful that there is a lag
time in the electrochemical measurement system.
The dispersion coefficient versus velocity is plotted on a log-log
scale in Figure 12. For the heterogeneous porous media, that is,
first and second order heterogeneities, the plots of dispersion
coefficients versus velocity are given in Figure 13.
Plots of D/V versus R. on a log-log scale for the homogeneous
nonuniform porous media are shown in Figure 14. Plots for the first and
second-order heterogeneous porous media of D/v versus R, are shown in
Figure 15. K
The equations for these lines in Figure 12 are
£ - 297 R^'01 for dw - 35y (61)
J - 1630 Rj^48 for d10 - 170y (62)
The equations for the lines in Figure 13 are:
QQ
D/V - 534 R£yy , d1Q - 36y (63)
D/V - 105 RH9 , dlD " 36U (64)
R d!0 - 170u
D/v - 985 R'17 , d - 170y (65)
42
-------
0.30
0.10
0.06
0.03
o
0.01
0.06
0.003
MODEL 3
MODEL 2
0.2
0.4 0.6 0.8 1.0
v (cm/mln)
MODEL 6
MODEL 5
MODEL 4
MODEL I
2.0
4.0
Figure 12 - Dispersion Coefficient Versus Velocity (Six
Non-Uniform Models)
43
-------
0.40
0.20
E
o
o.iok
0.08
0.06
0.04
0.02
0.01
0.008
0.006
0.004
0.2
0.4 0,6 0.8 1.0
v (cm/min)
2.0
Figure 13a*- Dispersion Coefficients Versus Velocity (First
and Second Order Heterogeneity)
44
4.0
-------
0.40r
M
E
o
0.20
0.10
0.08
0.06
0.04
0.02
0,01
0.008
0,006
0.004
Q2
0.4 0.6 0.8 1.0
v (cm /mm)
2.0
Figure 13b- Dispersion Coefficients Versus Velocity (First
and Second Order Heterogeneity)
45
4,0
-------
0.40
0.20
0.10
0.08
0.06
0.04
E
o
0.02
0.01
0.008
0.006
0.004
0.2
0.4 0.6 0.8 1.0
v (cm /min)
2.0
Figure 13c- Dispersion Coefficients Versus Velocity (First
and Second Order Heterogeneity)
46
4.0
-------
0.40
0.20
0.10
0.08
0.06
0.04
0.02
0.01
0.008
0.006
0.0041-1
d[lo] = 0.036 mm
"4
d[lo] =0.170 mm
"4
IO" 2X|0" 4X|0
"4
IO"8 2 x IO"3
Figure 14 - D/v vs R, (Homogeneous)
47
-------
0.80
0.60
0.40
BOTH PARTS HAVE
d[lo]« 0.036 mm '
0,20
0.10
0.08
0.06
0.04
0.02
0.01-
0.008
10
-4
2X|0
-4
4X10
R,.
-4
BOTH
d[lo] = 0.036mm
AND
d[lo] = 0.170 mm
BOTH PARTS
HAVE
d[lo]= 0.170
10'
2X10'
Figure 15 - D/v vs R, (Heterogeneous)
48
-------
The plots above show that not only does the dispersion coefficient
vary with the level of permeability, but it does vary with the distribution
of permeabilities as well as with the heterogeneity since the scatter is
the outside experimental area. The magnitude of this difference
suggests the existence of another factor influencing D and that we
expect to be the distribution of beads sizes of permeabilities is
given by the slope of the bead size distribution. To check if the
slope was significant a statistical analysis of the data was performed.
Statistical analysis. The correlation developed tested was of the
form
D - f(S, k, v, d) (66)
An analysis of variance was performed on all the homogeneous nonuniform
media data to determine which main factors and which interactions were
significant. Then a regression analysis was performed to arrive at an
expression for the correlation. Although a linear model of the type
D - bo + ^l S + B2 k + B3 v + b^d (67)
was investigated, greater success was achieved with a logarithmic model.
This is not surprising, since the correlations compared earlier are of
logarithmic form. An analysis of the variables showed that the following
five variables affect the dispersion: log k, log S, log V, d, log k x
log V.
The fact that d remained in the equation indicates that there is a
significant effect due to the packing technique used in constructing the
porous media. The regression equation derived from the analysis for the
up direction is
(.7 + .54 Ivgk)
D - .18 -Z £= (68)
k.07
For the down direction the coefficient .18 becomes .2. Thus although
the effect due to d is significant, it is very small. A plot of
equation (68) and the data is shown in Figure 16. Thus, the slope
(that is) nonuniformity is definitely affecting the value of dispersion
and the scatter of Figures 14 and 15 is due to nonuniformity effects.
It is also interesting to note that log permeability and log velocity
gives a covariant type of effect. Again, we would expect something like
this from the statistical model results of Haring and Greenkorn18. As
with Pleshek's data, the reciprocity does not seem to hold.
49
-------
c
*E
E
o
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01 -
S = 0.384
S* 0.183
J—I—I—I—L_I—l I i i i i i i . i
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
(0.70 + 0.54 LOG K)
v
mmmmm
0.07
Figure 16 - EL vs T- for Various S
50
-------
AN EXPLORATORY STUDY OF PERMEABILITY AND DISPERSION
PHENOMENA IN ANISOTROPIC POROUS MEDIA"
Experimental Procedures. The model used is an octagonal "box"
made of plexiglas and packed with alternating layers of glass beads
as shown in Figure 17. The medium was made anisotropic by using glass
beads of two different sizes, 590-840 y and 105-149 \i. The anisotropy
is built into the model by packing the two sizes of beads into six
alternate vertical layers. A 1/2 inch hole was drilled into each
lateral face of the mcdel to permit fluid flow through the eight
corresponding directions. The model had a removable plexiglas lid
to allow easy packing of the beads. After packing, a rubber gasket
was inserted, between the lid and the level of beads in the model, and
the gasket was subjected through a hole in the center of the lid to a
constant water pressure of 8 lb./in.2 gauge in order to insure a tight
bead pack and to minimize channeling of the fluid.
The model was packed by slowly pouring beads into their separate
compartments while the model was constantly being vibrated by a
mechanical oscillator. As each section was filled, the aluminum dividers
were slowly removed, but a metal screen remained in place in the beads.
The last packing of the beads were carefully added until the model
was completely packed. Vibration was stopped immediately to prevent
the possibility of the small beads sifting into the larger beads through
the openings in the separation screens.
All tubing used in the experiments was 1/4 inch polyflow tubing
with appropriate fittings. Other equipment consisted of a polyethylene
bottle used as a constant head tank; a 0-30 lb/in.2 pressure gauge to
measure the pressure on the diaphragm; a simple manometer; a flow
conductivity cell; and a 0.25 mv Brown-mull balancing recorder.
Porosity. The porosity of the model was determined by the displace-
ment method. The model was weighed dry, filled with water and weighed
again. The volume of the void space was determined from the weight of
entrapped water. The liquid used in all experiments was distilled water
which had been filtered and deaerated. In the case of the porosity
measurements a small amount of detergent was added to the water to give
better wetting properties and a few drops of red food coloring were
added to the water to aid in visual detection of entrapped air pockets.
The model was dried by forcing hot air through the eight different
directions, care being taken that the air rate was very low so that
there would be no shifting of the bed. The porosity was determined as
.348 and the error is estimated as -f 1.1%.
Permeability. The permeability measurements were performed in a
manner similar to that used by Darcy in his original experiment. The
51
-------
SMALL BEADS
LARGE BEADS
Figure 17 - Model Sketch (Double Figure)
52
-------
model was placed in a vertical position with flow of distilled water
provided by a variable level constant head tank. (See Figure 18 for
experimental setup). After passing through the model, the water was
collected in a graduated cylinder for a known period of time to
determine the volumetric flow rate and the pressure drop was read from
the simple manometer with one leg connected to each side of the model.
The permeabilities were calculated according to Darcy's Law, Equation
(37), expressed in terms of the measurable quantities. In this case,
A was taken to be the area of the opening covered by the screen in
each face of the model. This is the area over which the fluid pressure
is distributed at the time the fluid is entering the porous medium.
The laboratory was maintained at a constant temperature of 77°, so the
viscosity and density of the fluids did not change during the experiment.
The independent variables, therefore, in Equation (37) are Q and Ah.
These are the quantities measured in each direction and in each experiment.
Dispersion. The dispersion experiments were conducted with the same
equipment as was used for measuring permeability. A breakout curve was
determined for each experiment at a given flow rate in a given direction,
by using either a .005 molar or .02 molar solution of KC1 as a displacing
fluid. If the weak KC1 solution was used as a tracer for the first
experiment, the stronger solution was used in the second experiment, so
that the two solutions were used alternatively as tracers throughout
the series of experiments. The concentration was measured with a con-
ductivity cell which was recorded on a strip chart showing the change in
concentration of the effluent. The breakthrough curves for each experiment
were determined from the recorder charts. In order to avoid any possible
gravity effects, the experiments were always run such that the dense solution
flowed from bottom to top of the model.
Sequence of Experiments. The porosity measurements were conducted
first, and the permeability and tracer breakthrough experiments were run
simultaneously. The flow rate and pressure drop needed for the permeability
calculation were recorded during the time In which the KC1 solutions were
displacing one another in the model. It was assumed throughout these
experiments that the density and viscosity properties of the KC1 solutions
were essentially the same as of water.
A total of 48 experiments was run. Three separate flow rates, 10,
15, and 22 cc/min. were run for each of the eight directions and each
run was made in duplicate to provide a basis for an error estimate.
This design of experiments led itself to a straightforward analysis of
variance which determine the effects of flow rate and direction on the
permeability and dispersion coefficients.
Results and Discussion
Permeability. Measurements are shown in Table 6. A simple analysis
of the variance was performed on these data, with results indicating
53
-------
constant
head tank
en
I
Manometer
r,
Conductivity Cell
L.
Effluent
Receptacle
Chart
Recorder
Not used for permeability
measurements
Figure 18 - Sketch of Apparatus
-------
Table 6. Permeabilities
Ul
Direction 1
Direction 2
Direction 3
Direction 4
Direction 5
Direction 6
Direction 7
Direction 8
Flow rate
(cc/min)
9.8
9.8
9.33
9.84
9.5
9.43
10.5
9.33
9.2
9.5
10.3
10.41
9.8
10.1
10.4
9.6
k (darcies)
1442
1640
907
898
178
172
945
1030
1540
1626
1000
1058
266
196
1011
1110
Flow rate
(cc/min)
14.1
15.3
15.33
15.1
14.3
14.6
16.2
14.67
14.5
14.2
14.85
15.8
15.0
15.5
13.6
15.73
k (darcies)
1400
1651
920
941
170
169
1016
914
1530
1533
955
1097
214
202
1140
1008
Flow rate
(cc/min)
20.75
21.0
20.33
19.2
20.33
20.9
20.0
20.3
21.8
20.8
20.5
21.0
20.1
20.5
20.3
19.5
k (darcies)
1440
1700
851
942
175
174
1014
1005
1527
1509
1002
1052
196
196
1050
1078
-------
a highly significant effect due to direction but no evidence of an
effect on permeability due to flow rate or due to an interaction
between the flow rate and direction.
The magnitudes of the permeabilities contained in the octagonal
model are considerably higher than those values obtained for similar
sized beads in linear models. This is due to the assumption of the
area in the calculation of the permeability estimates. However, the
results are reproducable and relative values have meaning.
It should be noted in Table 6 that the permeabilities in opposite
flow directions — for example, directions 4 and 8 — are not identical.
A possible explanation for this lack of symmetry is that the fluid sees
different possible paths for flow as it enters the model. If the fluid
enters in direction 4 (see Figure H) it encounters initially a layer
of small beads, whereas the fluid entering in direction 8 encounters a
layer of large beads which offers less resistance to flow. These paths
of greater and lesser resistances are presumed to be the reason for lack
of agreement between the permeabilities measured in opposite directions
although in general the difference between opposite direction permeabilities
is not large.
The permeability data obtained from the octagonal model was tested
according to the theory of directional permeability as shown earlier.
For the particular geometry of the model used here, one should use the
second case where the velocity is not parallel to the pressure gradient;
thus,
k" = k,. cos2a + k00 cos2B (69)
n 11 ii
If the reciprocal of the square root of each permeability is plotted
against its corresponding direction, an ellipse should be obtained.
In Figure 19 this curve has been plotted and it resembles an ellipse,
although it is not symmetric for the reasons discussed above. The
results indicated, as expected, that the flow rate vector is not in the
same direction as the pressure gradient.
Dispersion. Two separate methods of calculating C/C were used in
order to obtain similar curves for those experiments in wftich the weak
KC1 solution was used as the tracer and in the experiments where the
stronger KC1 solution was used as the tracer. The breakthrough curves
corresponding to the directions of highest permeability are the steepest
and those corresponding to the directions of lowest permeability are
the flattest. The dimensionless concentrations approached 1 after about
1-1/2 to 2 pore volumes for experiments in all directions except the
directions 1 and 5 (those with the highest permeability). In those
directions the breakthrough curves do not approach 1 until 3 or 4 pore
volumes of fluid were discharged. For directions 1 and 5 the curve rises
56
-------
225
315
270
Figure 19 —» vs angle
V l«.
57
-------
rapidly at first and then levels off and rises very slowly. An
explanation for this may be that tracer fluid travels rapidly through
the layer of large beads until the curve reaches a value of .6 or .7.
Then the fluid in the layers on either side of the large bead layer
begins to be displaced. It is the displaced fluid from these adjacent
layers which causes the curve to rise so slowly for the remainder of
the experiment.
In order to compare the breakthrough curves with the permeability,
we decided to calculate the dispersion coefficient in the normal fashion
for a linear porous medium using Equation (4). Although these data would
not correctly describe the exact mixing in this type of model, they
are an approximation and also we should be able to compare the relative
values in such a way that we can determine whether or not there is a
directional dependence on the dispersion and further, if this directional
dependence has any relation to permeability. Thus calculating the
dispersion by Equation (4) is shown in Table 7 as a function of flow
rate and direction. An analysis of variance of these data shows the
effects of both direction and flow rate are highly significant and that
the interaction between flow rate and direction is slightly significant.
Since dispersion is a tensor it is assumed that it appears in the
flux equation in a form analogous to the appearance of the permeability
tensor K in Darcy's law. Therefore with the boundary conditions for
the octagonal model we plotted the reciprocal of the square root of the
dispersion coefficient versus direction and as a function of velocity.
Figure (20) is a plot of this data. A comparison of Figures (19) and
(20) shows that the principal directions of these two ellipses correlate
but are oriented 90° apart from each other. This indicates the dispersion
coefficient increases in the direction of decreasing permeability. It
is interesting to note that the two ellipses are geometrically similar;
the ratio of the major permeability axis to the minor axis is 1.3 and
the ratio for the dispersion ellipse is 1.38. This fact indicates the
anisotropy of the model effects permeabilities and dispersion coefficients
to a similar extent. The fact that the major directions are orthogonal
is disturbing since we would assume that the dispersion would increase
directly with permeability according to the correlations of Harleman et a I1*
58
-------
Table 7. Dispersion Coefficients
VO
Direction 1
Direction 2
Direction 3
Direction 4
Direction 5
Direction 6
Direction 7
Direction 8
Flow rate
(cc/min)
9.8
9.8
9.33
9.84
9.5
9.43
10.5
9.33
9.2
9.5
10.3
10.41
9.8
10.1
10.4
9.6
D (x 103)
(cm^/sec)
9.78
7.12
39.4
25.5
90.1
57.2
66.1
34.3
7.07
8.22
51.7
38.7
130
95.0
46.1
46.7
Flow rate
(cc/min)
14.1
15.3
15.33
15.1
14.3
14.6
16.2
14.67
14.5
14.2
14.85
15.8
15.0
15.5
13.6
15.73
D (x 103)
cm^/sec)
14.4
19.4
74.9
76.4
107
241
89.0
80.0
7.22
8.21
60.0
78.0
158
202
66.4
81.4
Flow rate
(cc/min)
20.75
21.0
20.33
19.2
20.33
20.9
20.0
20.3
21.8
20.8
20.5
21.0
20.1
20.5
20.3
19.5
D (x 103)
(cm^/sec)
36.8
21.1
98.2
70.0
185
283
113
118
19.9
20.9
97.9
129
245
251
83.3
107
-------
225
270
315
45'
Figure 20 - 1//D vs angle
60
-------
PERMEABILITY AND DISPERSION DURING FLOW IN
LINEAR HETEROGENEOUS ANISOTROPIC POROUS MEDIA*•
Experimental Procedures
27
Figure 17 depicts the porous medium used by Lenth and Greenkorn
The medium is made up of alternating 1-inch thick layers of large (590-
840 y) and small (105-149 y) glass beads. The porous media models that
would result from coring the medium in Figure 17 at 45* intervals re-
presents the porous media modelled in this work. The result of this
envisioned coring operation is shown in Figure 21. It can be see that,
depending on flow directions through each medium, model 1 represents
two angles (90° and 270°) of flow through the bed. Model 3 represents
two angles (0, 180°). Because of the symmetry of Lenth's model, model
2 includes four angles (45, 135, 225, 315°). Figure 22 shows the de-
tails of the Lucite tube for making the models and the details of the
packing arrangement.
In order to maintain heterogeneity of the beds the two sizes of
beads were kept segregated by screens. Once the dividers were in place,
the models were packed by adding beads slowly while the model was con-
tinuously vibrated on a Centron vibrating table. Beads were packed
through the ends of the model where possible, or by small access holes
on the side of the models.
Porosity. The liquid-saturation method was used to determine the
porosity of the models. The models were weighed and then filled with
distilled water which contained small amounts of food coloring and
detergent (two to three drops per liter of water). The difference
between the wet and dry weights, with corrections for fittings and
entrance sections, allowed the void volume to be determined. When the
tubes were filled initially with water almost no air was observed to
be trapped on the inside tube wall. However, when repeated measurements
were attempted by draining, drying, and refilling, air bubbles were
visible on the tube walls. Therefore, the models were filled initially.
and maintained full of water. Before further measurements were made
the food coloring and detergent were removed by flushing with several
volumes of distilled water.
Permeability. The permeability measurements were made by forcing
fluid, (.4 M KC1) through the porous media with a Ruska proportioning
pump. Various constant flow rates were available from the positive dis-
placement pump. The pressure drop was measured with a water manometer.
Permeability was determined at three fluid flow rates (130, 263,
453 cc/hr). The corresponding frontal velocities were about .9, 1.8,
and 3.1 on/min. Room temperature was held constant at 76°F during all
runs; thus the fluid properties remained constant. The permeability was
determined with Equation (37). Replicates were run at all flow rates
in order to determine experimental error.
61
-------
90'
135
180'
r* • <
590-840 microns
PTTI
105-149 microns
Model 2
45°
Model 3
ISO6 —
Model 3
— 0°
Model 2
225
Model 2
315°
Model 1
270°
figure 21 - Lenth Apparatus Cored at 45° Intervals
62
-------
0-RING-*,
ON
U>
14.93
iOO Mesh
Wire Screentt.
1/8" 0-Ring
II
2.86 = INSIDE DIA.
OUTSIDE DIA.
400 MESH
WIRE SCREEN
!•
i
Model 1
Model 2
Model 3
-
IT
590-840
MICRONS
105-149
MICRONS
Figure 22 - Description of Models
-------
Dispersion. Potassium chloride was used as a chemical tracer to
determine the breakthrough curves for each porous medium. A silver-
silver chloride electrode was used in conjunction with a Calomel reference
electrode2 was used to monitor chloride concentrations at the outlet
of each model. The potential developed by the electrode system was
monitored with a Sargeant SR balancing potentiometric recorder. The
recorder range was 0-100 mv.
A .4 M KCl solution was used to displace a .02/nolar Ktl solution. The
two solutions gave potential readings of about 3 and 74 mv respectively.
When a run was completed, the model was backflushed with .02 M KCl sol-
ution in preparation for the next run. The models were positioned
vertically and the .4 M solution was always introduced at the bottom
of the tube. Figure 23 is a diagram of the equipment.
The data (time, millivolts, calibration equation, physical proper-
ties) were punched on computer cards. A CDC 6500 computer was used to
determine the dispersion coefficients. The computer is programmed to
first calculate dimensionless concentration with a space for C- as initial
KCl concentration, C as the tracer solution concentrations, and C is
the KCl concentration of fluid leaving the model. About 16 data points
were used for each dispersion run.
The dispersion coefficient was then determined by iteration with
equation (12). The standard deviation between the data and the values
were predicted by Equation (12) and calculated. Minimization of this
quantity by least sequence was the basis for curve-fitting the data.
The experimental and theoretical breakthrough curves were plotted
automatically (See Figure 24 for an example).
Equation (13), the truncated form of Equation (12) was also used
to calculate dispersion coefficients. In most runs the Peclet number
was less than 133; therefore an error of greater than 5% was expected
in dispersion coefficients calculated by Equation (13).
Results and Discussion
Porosity. The porosities of the three models are 1 - .376, 2 - .372
and 3 - .388. The values vary only slightly from model to model, and
the error in the porosity is estimated less than + .01.
Permeability. The permeability values determined as described by
Pleshek and Greenkorn22. The permeability of models 1 and 3 were esti-
mated as 10 and 62.5 Darcys respectively. Model 2 should fall between
these two values. The permeabilities are summarized as a function of
flow rate in Figure 25. The values given here are in good agreement
with the predicted values, although they were slightly low. (This slight
difference was due to the fact that the values of Pleshek contained silk-
cloth containers at the end of the models and thus his values are just
slightly low.)
64
-------
Gravity
Feed Tank
(0.02 M KC1)
To Automatic
Fraction Collector
Proportionating
Pump
(0.4 M KC1)
Calomel Reference Electrode ]
AG-AGC1 Electrode
Porous
Medium
Filter
P
Recorder
Manometer
Outlet for
Flushing with
0.02 M KC1
Figure 23 - Equipment Diagram
-------
o
o
\
o
0.75-
0.50-
0.25-
0
0
0.5 1.0 1.5
PORE VOLUME
2.0
MODEL = 2 DOWN
FRONT. VEL. * 0.909
STD. DEV. = 0.0107
D <= 0.0905
2.5
Figure 24 - Breakthrough Curve
-------
90r o
80
70
MODEL 3
0
o
60
CO
LLJ
o
o:
2
50
40-
30
20
10-
o
MODEL 2
-o-
-o-
MODEL
.—o-
RELATIVE FLOW DIRECTION
— -UP DOWN
i i
2.0 3.0 4.0 5.0 6.0 7.0 8.0
Q (cc/min)
Figure 25 - Average Permeability Versus Flow Rate
67
-------
An analysis of variance was performed to determine the effect of
flow rate and flow direction on each tube. The mean squares for flow
rate and direction for each tube were compared with the best estimate
of the experimental error. It was found that the effect of flow rate
was not significant for any of the models. However, reversing the dir-
ection of flow was significant in model 1 and highly significant in
model 3. Flow direction reversal was not significant in model 2. The
explanation of the symmetry of permeability in model 2 and the asymmetry
in models 1 and 3 is not apparent. Evaluation of the packing procedure
gives no indication that this result should be expected. Lenth and
Greenkorn27 found that in general permeabilities in opposite flow dir-
ections were not the same.
The data collected for permeability in this experiment are most
closely represented by the first case of the tensor theory of permeability
and thus one would expect the plot of the square root of permeability
as a function of angle to give an ellipse. The results of such a plot
for the data for these models are shown in Figure 26. The ellipse is
a least squares fit determined by the method utilized by Greenkorn et all
Lenth and Greenkorn27 choice of area term used in Darcy's Law
resulted in much larger values of permeability than were found in this
study. However, the directions and relative magnitudes of the major
and minor axes of the permeability found in this work agree with the
results of Lenth and Greenkorn.27
Dispersion. The dispersion coefficient was determined from Equation
(12) by a least-squares curve fitting technique using the computer.
The raw data were fed directly into the computer program and the best
fit in dispersion coefficient was determined in this form.
The results of the dispersion coefficient calculations, standard
deviations, and Peclet numbers are shown in Table 8. There was a slight
shift in the breakout curves such that the value for C/C to .5 did
not always occur at one pore volume. In these cases a sSight shift of
the breakout curve was made to correct for a probable error in the
porosity measurements.
The asymmetry that was observed in some of the curves In this study
was considerably more than Equation (12) could predict. For the late
portions of the breakthrough curve the C/C measured was always below
the theoretical value. This tail-off has been observed by others in
studies of both homogeneous and heterogeneous porous media. As mentioned
previously this tail-off has been attributed to dead-end pores in the
media. The magnitude of the tail-off observed here is more than could
be realistically attributed to dead-end pores. The tail-off in general
gets more exaggerated as the fluid velocity increased. The least of
this effect was seen in the volumes of the displaced fluid are left
behind. In time, this trapped fluid with both diffuse out into the
main stream and be displaced by a slow convective mixing which occurs
in these semi-dead regions. The results was the tail-off or exaggerated
68
-------
0'
o
315
270
225
180
Figure 26 - /k vs angle
69
-------
Table 8. Dispersion Coefficients for Model with Screen Dividers
2
D (on /min)
Model-Position
1-down
1-down
1-down
1-up
1-up
1-up
1-up
2-up
2-up
2-up
2-down
2-down
2-down
2-down
3-down
3-down
3-down
3-up
3-up
3-up
3-up
3-up
Velocity
(cm/min)
3.128
1.814
0.899
3.128
1.814
1.814
0.899
3.161
1.834
0.911
3.161
3.161
1.834
0.909
3.031
1.758
0.871
3.031
3.031
1.758
1.758
0.871
Computer
Least-Squares-Fit
0.9445
0.2412
0.0500
0.9174
0.3258
0.2810
0.0743
0.8251
0.4243
0.2152
1.1032
1.6597
0.7336
0.0905
1.3428
0.8336
0.6446
2.3489
1.8228
0.8519
0.8434
0.3939
By
Slope Method
0.450
0.194
0.038
0.645
0.347
0.281
0.111
0.822
0.396
0.170
0.838
1.77
0.865
0.108
0.957
0.912
0.423
1.50
2.08
1.07
1.07
0.294
(Standard Dev.)
from
Dispersion Eq.
0.1169
0.0878
0.0729
0.0921
0.0853
0.0752
0.0520
0.1069
0.0798
0.0667
0.0921
0.0633
0.0728
0.0107
0.0375
0.0588
0.1194
0.0959
0.0625
0.0826
0.0470
0.0707
Peclet No.
49.44
112.30
268.58
50.90
83.15
96.40
180.75
57.21
64.53
63.24
42.78
28.44
37.32
149.91
33.70
31.49
20.18
19.26
24.83
30.82
31.13
33.02
-------
asymmetry observed in this work.
The flow in the model 2 is envisioned to look as in Figure 27 with
the flow taking the path of least resistance. For an elemental volume
of fluid, this path consists of the longest possible time spend in the
high permeability sections and the shortest time spend in the low perme-
ability sections. A stream line directly along the wall represents
the highest resistance path available for the flow in the tube that is,
the longest possible path in the low permeabilities and the shortest
path through the high permeabilities. This gives rise to "semi-dead"
regions near the wall in each section of the medium especially in model
2. The areas outside of the depicted flow path in Figure 27 show these
slow displacement regions. Again, the result is the asymmetric concen-
tration distributions observed.
The majority of the flow model 3 will take place through the high
permeability section with a ratio of frontal velocities (V1 to V») equal
to the ratio of permeabilities of two sections, about 25 to 1. Assuming
no cross-flow between sections, when the tracer front has traversed
the length of the tube in the high permeability section, it will have
gone about l/25th of the model 1 (90°), with slightly more tail-off
appearing in model 2 (45°) and more yet in model 3 (0°).
The breakthrough curves obtained in this work and in Lenth and
Greenkorn27 appear to be inherent in this approach to anisotropic
porous media, for example, packed beds layered at various angles. The
diffusion model with one parameter, D, simply will not predict the ob-
served asymmetry. The two bead sizes used here have permeabilities
of 120 and 5 Darcys (k]L : k2 - 25). As the degree of heterogeneity
decreases, the degree of asymmetry decreases and at some point nearing
a homogeneous bed (k^ : k- - 1). The application of the diffusion model
to dispersion would describe the observed concentration distribution
satisfactorily.
In addition, in model 1 the five circular dividers between the
bead sections probably caused a slight restriction at each divider,
thus causing an orifice effect in the data as shown in Figure 27.
The results of the computer calculations of dispersion coefficients
were tested against the correlations described previously.
The correlation expressed by Equation (29) was tested. The equations
for the data are: c , ,,
1) D/v - 6.7 x 103 R^"1* (70)
2) DA> - 2.5 x 106 Rk3'5 (71)
3) DA) - 1.0 x 103 \lt26 (72)
Model 3 is in reasonably good agreement with previous workers for the
same reasons discussed above. Models 2 and 3 have greater intercepts
71
-------
HIGH k LOW k HIGH k
T
T-7
a. MODEL I
HIGH k LOW k HIGH k
b. MODEL 2
HIGH k
LOW k
c. MODEL 3
Figure 27 - Flow Pattern in Model
72
-------
and exponents.
We realize that the correlations used for comparison above were
not necessarily intended for extension to porous media of the type
used in this study. The comparison was meant to indicate the results
to be expected when these correlations are applied to heterogeneous,
aniso tropic porous media. It is acknowledged that the observed dis-
persion would decrease to values predictable by the correlations of
previous workers as the degree of heterogeneity decreases to approach
a homogeneous system.
The correlations developed in this study may be used to predict
dispersion in heterogeneous anisotropic porous media with the realization
that only one ratio of section to section permeability was investigated
(25 to 1).
In testing the tensorial properties of dispersion Lenth and Greenkorn
found for the octagonal model that 1//D plotted against direction yielded
an ellipse. This form (1/vlc vs. direction) which Lenth and Greenkorn
used for permeability. The axes for dispersion and permeability were
found to be 90° apart.
In this study the square root of the measured dispersion coefficient
is plotted against the packing angle for all flow rates studied and
is shown in Figure 28. Ellipses shown were calculated by least-squares
fit. Since the dispersion increases with velocity, the ellipses of
course get larger as the velocity increases. The major and minor axes
of the dispersion coincide with the axes of permeability, Figure 26.
27
The ratio of major to minor axes (^iA2) ^or c^e ellipses in Figure
28 are 4.83, 2.10, 1.40 as velocity change from 2.17 cc/min to 4.38
to 7.55. The ratio is roughly inversely proportional to flow rate.
The equation relating X A- to flow rate is
where T is a constant of about 10. These results indicate that the
anisotropy would not appear in these models at a flow rate of about
10 cc/min., and that /D versus packing yield angle should yield a circle.
It should be noted that 10 cc/min. would give a Reynolds number of
.65, which should still be in the Darcy flow regime. It is interesting .
to note that as we increase the flow rate more, the ratio of major to minor
axes reverses, and we do get the major and minor axes not in the same
direction as the permeability ellipse such as reported by Lenth and
Greenkorn27. Please note, this statement is based on an extrapolation
of the data with Equation (73).
73
-------
135
7.55 cc/min
4.38 cc/min
2.l7cc/min
315°
180
270
225°
Figure 28 - f'D vs angle
74
-------
REFERENCES
1. Perkins, T. K., and Johnston, 0. C., "A Review of Diffusion and
Dispersion in Porous Media," Pet. Trans. AIME 228, SPEJ 70 (1963).
2. Chung, S. F., and Wen, C. Y., "Longitudinal Dispersion of Liquid
Flowing Through Fixed and Fluidized Beds," AIChE J 14. 857 (1968).
3. Greenkorn, R. A., and Kessler, D. P., "Dispersion in Heterogeneous
Nonuniform Anisotropic Porous Media," Ind. Eng. Chem. 61 #9, 14 (1969).
4. Harleman, D. R. F., Melhorn, P. F., and Rumer, Jr., R. R., "Dispersion-
Permeability Correlation in Porous Media," J. Hydraul. Div.; Proc.
Am. Soc. Civil Eng. 89 HY2, 67 (1963).
5. Shamir, V. Y., and Harleman, D. R. F., "Dispersion in Liquid Porous
Media," J. Hydraul Div. : Proc. Am. Soc. Civ. Eng. HY5. 237 (1967).
6. Schiedegger, A. E., "Directional Permeability of Porous Media to
Homogeneous Fluids," Geofis Pura Appl. 28. 75 (1954).
7. Greenkorn,R. A., Johnson, C. R., and Shalleriberger, L. K., "Direc-
tional Permeability of Heterogeneous Anisotropic Porous Media,"
Pet. Trans. AIME 231, SPEJ 124 (1964).
8. Schiedegger, A. E., "Physics of Flow Through Porous Media," Mac-
Mil Ian New York (1957).
9. Whitaker, S., "Diffusion and Dispersion in Porous Media," AIChE J.
,13, 420 (1967).
10. Ramomdi, P., Gardner, G. H. F., and Petrick, C. B., "Effect of Pore
Structure and Molecular Diffusion on the Mixing of Miscible Liquids
Flowing in Porous Media," Reprint 43 AIChE-SPE Joint Symposium.
San Francisco (December, 1959).
11. Rifai, M. N. E., Kaufman, W. J., and Todd, D. K., "Dispersion
Phenomena in Laminar Flow through Porous Media," Report #3, Sanitary
Engineering Research Laboratory, Univ. of California, Berkeley,
(July, 1956).
12. Ogata, A., and Banks, R., "A Solution to the Differential Equation.
of Longitudinal Dispersion in Porous Media," U. S. Geologie Survey
Prof. Paper411-A, U. S. Govt. Post Office, Washington (1961).
13. Harleman, D. R. F., and Rumer, R. R., "Longitudinal and Latual
Dispersion in an Isotropic Porous Medium," J. Fid. Mech. 16 #3,
385 (1963).
14. DeJosselin DeJong, G., and Bossen, M. S.t "Discussion of paper
75
-------
by Jacob Bear, on the Tensor Form of Dispersion in Porous Media,"
J. Geophys. Res. J56_ (10), 3623 (1961).
15. Bachmat, Y., and Bear, J., "The General Equations of Hydrodynamic
Dispersion in Homogeneous, Isotropic, Porous Mediums," J. Geophys.
Res. 6£ (12), 2561 (1964).
16. Patel, R. D., and Greenkorn, R. A., "On Dispersion in Laminar Flow
Through Porous Media," AIChE J. _16, 332 (1970).
17. Poreh, M., "The Dispersivity Tensor in Isotropic and Anisotropic Porous
Mediums," J. Geophys. Res. 70 (16), 3609 (1965).
18. Raring, R. E., and Greenkorn, R. A., "A Statistical Model of a Porous
Medium With Nonuniform Pores," AIChE J. 16 477, (1970).
19. Whitaker, S., "The Equations of Motion in Porous Media," Chem. Eng.
Sci. 21, 291 (1966).
20. Greenkorn, R. A., and Kessler, D. P., "A Statistical Model for Flow
in Nonuniform Anisotropic Porous Media," Proc. ASCE-EMD Specialty
Conference on Probablistic Concepts and Methods, (November, 1969).
21. Guin, J. A., Kessler, D. P., and Greenkorn, R. A., "On the Permeability
Tensor for Anisotropic Nonuniform Porous Media," Submitted to Chem.
Eng. Sci. May, (1970).
22. Pleshek, R. C., and Greenkorn, R. A., "Dispersion During Flow in
Linear Heterogeneous Porous Media," To be submitted to Water Resources
Res. (1970).
23. Koonce, K. T., and Blackwell, R. J., "Single-Probe Method for
Measuring Cl-Concentration in a Flowing Fluid," Private Commonwealth
(1967).
24. Shoemaker, D. P., and Garland, C. W., "Experiments in Physical
Chemistry," McGraw-Hill (1962).
25. Niemann, E. H., Greenkorn, R. A., and Eckert, R. E,, "Dispersion
During Flow in Nonuniform, Heterogeneous Porous Media," To be sub-
mitted to Water Resources Res. (1970).
26. Coats, K. H., and Smith, B. D., "Dead-End Pore Volume and Dispersion
in Porous Media," Pet Trans AIME 231, SPEJ 73 (1964).
27. Lenth, E. G., and Greenkorn, R. A., "An Exploratory Study of Perme-
ability and Dispersion Phenomena in Anisotropic Porous Media," To
be submitted to Water Resources Res. (1970).
28. Goad, T. L., and Greenkorn, R. A., "Permeability and Dispersion
During Flow in Linear Heterogeneous Anisotropic Porous Media," To
be submitted to Water Resources Res. (1970).
76
-------
ACKNOWLEDGMENT
The support for this project was provided by the Federal Water
Pollution Control Administration. However a large amount of supplementary
work was directly or indirectly supported by the Petroleum Research
Fund of the American Chemical Society, the Purdue Research Foundation,
the National Science Foundation, and the Esso Production Research
Company. Special thanks is due to these organizations for their help.
77
-------
APPENDIX 1 - Summary of
Dispersion in Heterogeneous
3
Nonuniform Anisotropic Porous Media
This is a state-of-the-art review of dispersion in real porous
media to examine present knowledge and to indicate research
directions.
We limited the discussion to macroscopic mixing caused by uneven
cocurrent laminar flow exemplified by flow of petroleum in reservoirs,
movement of fresh water and waste fluids in aquifers, and flow of
fluids in fixed beds. Countercurrent gas-liquid, liquid-liquid, and
fluidized systems are not considered. Further, the discussion is
directed at dispersion in nonideal media.
We do not pretend to be exhaustive in this review - rather, we
attempt to cover selected papers which illustrate the major directions
of research. The interested reader can find more material in the
bibliography than can be given fair treatment within the scope of
this paper.
We start with definitions of dispersion and of the three
nonidealities, heterogeneity, nonuniformity, and anisotropy. After
a brief general introduction, we consider statistical models of
dispersion, continuum models of dispersion, and data which apply to
systems containing nonidealities.
In general, the most effort for real media seems to be in the
continuum theories. There is relatively less on the statistical
models reported, and meager amounts of data are interpreted in
nonideal media.
If one accepts the form of Equation 31, the effect of
heterogeneity can be incorporated by defining D., as a function of
position. Any useful predicted results will probably have to be
made using numerical techniques for solving Equation 31.
The effect of nonuniformity is incorporated in the form of
dispersion tensor and is summarized in terms of the ratio of the
longitudinal to transverse dispersion coefficients for one-
dimensional flow in Table 9.
These models and the data can be made to agree, except in the
case of Whitaker's results which seem to show that for the
assumptions made to get to a comparable result we must assume the
medium to be uniform. Otherwise, the form of the ratio from
Whitaker is a complex function of several tensors.
78
-------
Table 9. Ratio of Longitudinal to Transverse Dispersion Coefficients
Effect of Konuniformity
Experimental data
Statistical model
Haring and Greenkorn
DL 4 . r27(a + b + 2)j vT,
"
Continuum models
Nikolaevskii
\ ^1
DT" A2
Bear, Scheidegger, de Josselin de Jong
Bachmat and Bear
DL A + 2y
Poreh
|2 2
"l + (0t2 + tt3^ 2
^-— j^
Ot. T Ct_ n
Do
Whitaker
79
-------
To attack anisotropy, we need more experimental data, extension
of the statistical model for this case, and further investigation of
the forms of the dispersion tensor. Empirically, it is appealing to
adopt Bachmat and Bear's model of the tensor and find A and y as
functions of angle.
There are three major questions to be answered after we get a
meaningful amount of data and interpretation for nonideal system.
(1) What is the "correct" model for flow in porous media in
the sense of giving sufficient accuracy? Do we need anything in
addition to curve fitting using the normal distribution (error
function) of Equation 2 and empirical corrections to it?
(2) Can one characterize a porous medium with a reasonable
number of parameters - i.e., at what level do heterogeneity,
nonuniformity, and anisotropy become important?
(3) What is the limiting step in our analysis of the problem:
the mathematical model, the computing techniques, or the experimental
data?
80
-------
APPENDIX 2 - Summary of
On Dispersion in Laminar Flow Through Porous Media
16
Dispersion of a tracer in fluid flow through a porous medium
has been extensively studied. The present communication seeks to
relate the recent work of Whitaker" to the. traditional treatment
of the dispersion equation. Some anomalies in Whitaker's results
are revealed and discussed.
1. Whitaker's analysis of dispersion in porous media results
in an equation identical to the usual dispersion equation.
2. For anisotropic media, the dispersion tensor is related
to the fluid velocity through a tensor of third order as well as
one of fourth order. For isotropic materials, only the fourth-
order tensor is to be retained. However, Whitaker's analysis
indicates (contrary to previous work) that this fourth-order tensor
has only one distinct component for such isotropic media.
3. Whitaker's analysis leads to predicting that the longi-
tudinal dispersion coefficient is three times the value of the
transverse dispersion coefficient for uniform flow in isotropic
materials.
81
-------
APPENDIX 3 - Summary of
A Statistical Model of a Porous Medium
18
with Nonuniform Pores
A random network model of a porous medium with nonuniform pores
has been constructed. Nonuniformity is achieved by assigning two-
parameter distributions to pore radius and pore length. Statistical
derivations result in expressions for bulk model properties which
are consistent with known empirical behavior of porous media such as
capillary pressure, hydraulic permeability, and longitudinal and
transverse dispersion. A series of experiments is suggested whereby
the parameters of porous media structure may be determined from
observed macroscopic behavior by using the expressions developed in
this paper.
A statistical model of a porous medium with nonuniform pores
has been constructed which matches experimental capillary pressure,
permeability, and dispersion data. Several interesting properties
are predicted from such a model. As one might expect, saturation
is a function of capillary pressure and the parameters of flow
radius distribution. The permeability-porosity ratio is a function
of the average radius squared and the pore radius distribution.
Thus, the permeability-porosity ratio causes dissipation due to
entrance-exit effects. The dispersion coefficient is dependent on
the parameters of both the pore radius and length distribution.
Therefore, one would expect dispersion to change as a function of
nonuniformity of the medium, even though the average resistance
and tortuosity are constant. Furthermore, given a pore length
distribution, the tortuosity is constant. Finally, for a nonuniform
medium, a particle following the most probable path is transported
through the medium at a velocity different from the Darcy velocity
by a factor which is a function of the flow radius distribution.
82
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|