Clays and Clay Minerals, Vol. 41, No. 3, 373-379, 1993.
POROSITY-SIZE RELATIONSHIP OF DRILLING MUD FLOCS:
FRACTAL STRUCTURE
HENING HUANG
Atlantic Oceanographic and Meteorological Laboratory
National Oceanic and Atmospheric Administration
4301 Rickenbacker Causeway, Miami, Florida 33149
Abstract--The porosities of flocs formed from a used drilling mud were determined by measuring sizes
and settling speeds of individual flocs. These flocs were produced in a Couette-type flocculator under a
variety of combinations of fluid shear and solid concentrations. In the calculation of floc porosities, a
floc settling model was employed that can consider the effects of creeping flow through a floc on its settling
speed. Results show that floc structure can be well described as a fractal with a fractal dimension of 1.531.64 for the floc size range tested. The effects of flocculation conditions, such as fluid shear and solid
concentration, on floc porosity and structure were examined. It was found that floc porosity and fractal
dimension were not influenced by solid concentration, but they increased as fluid shear decreased. Empirical expressions for the porosity of drilling mud flocs are obtained from both the floc settling model
and Stokes' law. For solid volume fraction in flocs, the relative difference between these two expressions
could be as much as 38%. However, the fractal dimensions estimated based on the two settling models
are nearly the same.
K e y Words--Drilling mud, Flocs, Fractal, Porosity.
INTRODUCTION
from computer simulations (Meakin, 1988). Computer
models
have been developed by many investigators,
The flocculation of clays or fine-grained sediments
such
as
Void
(1963), Sutherland (1967), Goodarz-Nia
in water bodies is of importance in many fields, such
(1977),
Lagvankar
and Gemmell (1968), Meakin (1984),
as wastewater treatment, water purification, and parand Mountain et al. (1986). These studies have shown
ticulate waste (e.g., used drilling muds) disposal in the
ocean. The focculated particles known as flocs are typ- that fractal dimension depends on the condition offloc
formation and that D may range from less than 1.7 to
ically characterized by their tenuous and loose porous
3.0 (Rogak and Flagan, 1990).
structure. The physical properties offlocs, such as denOn the other hand, a n u m b e r of experimental studies
sity, settling speed, permeability, and strength, are obon
the floc porosity or density-size relationship have
viously influenced by the floc structure.
been
performed by many investigators for flocs formed
In recent years, many studies have shown that floc
from
a variety of materials, such as Lagvankar and
structure can be described in terms of the concept of
fractal geometry, i.e., floc porosity (or effective density) Gemmell (1968) for Fe2(S04)3 flocs; Matsumoto and
and floc size are consistent with power-law relation- Mori (1975) for bentonite and alum flocs; Magara et
al. (1976), T a m b o and Watanabe (1979), and Li and
ships:
Ganczarczyk (1987) for activated sludge flocs; Glasgow
1 - e oc d ~ - 3
(1)
and Hsu (1984) for kaolin-polymer flocs; Gibbs (1985b)
for clay flocs; T a m b o and Watanabe (1979) for aluAp: = ( p f - Pw) ~ d~-3
(2)
m i n u m flocs; Weitz and Oliveria (1984) for gold colloid
where e is the floc porosity; 1 - e is the solid volume flocs; Klimpel and Hogg (1986) for quartz flocs; Gibbs
fraction in a floc; d: is the floc size (diameter); Ap: is (1985a) and Burban et al. (1990) for river sediment
the floc effective density; p:is the floc density; pw is the
flocs; and Alldrege and Gotschalk (1988), Kajihara
density of water in the floc; D is the fractal dimension (1971), Hawley (1982), and McCave (1975) for marine
for a self-similar structure.
aggregates. These studies have shown that the power
In Eqs. 1 and 2 the n u m b e r 3 represents the Euclid- law for floc porosity (or effective density) is at least
ean dimension for the three dimensional space. Since valid over a limited range of floc sizes.
a fractal dimension is typically less than the Euclidean
I n the experimental studies mentioned above, three
dimension, Eqs. 1 and 2 indicate that floc porosity methods were used in the determination of floc density
increases and effective density decreases as floc size (or porosity). One of these is the equivalent density
increases.
method (Lagvankar and Gemmell, 1968; Gibbs, 1985a,
At the present time much of our understanding of
1985b). It is based on the principle that if the density
the structure and properties of fractal flocs has come
o f a floc is equal to the density of the solution in which
Copyright 9 1993, The Clay Minerals Society
373
374
Huang
it is suspended, it will not sink or rise. However, this
is not generally a usable method because the measurements must be made in a matter of seconds; the pore
water o f a floc is quickly replaced by the standard density solutions, thereby changing the floc density (Gibbs,
1985a, 1985b).
Matsumoto and Mori (1975) measured floe densities
by using the Oden balance method and photo-extinction method simultaneously, provided that the relationship between the size and settling speed of flocs
was known. However, this method has not been used
by others.
The third method is as follows. The sizes and settling
speeds of flocs are first measured in a settling tube.
Assuming that the settling of each individual floc satisfies a settling model, Stokes' law for example, the
porosity e of a floc is determined from:
1 -
e-
18~Ws
(3)
g(pp -- p w)d/
where Ws is the measured floc settling speed; # is the
dynamic viscosity of water; g is the gravity acceleration; and 0p is the density of solid particles in the floc.
This method is questionable because Stokes' law is
only valid for an impermeable sphere. Since a floc is
of highly porous structure, the ambient fluid will penetrate the floc; the settling speed of the floc is, therefore,
higher than that of an impermeable particle with the
same size and the same effective density as the floc
(Neale et al., 1973; Matsumoto and Suganuma, 1977;
Masliyah and Polikar, 1980; Ooms et al., 1970).
In this study, the porosities of flocs formed from a
used drilling m u d are determined by measuring sizes
and settling speeds of individual flocs. A floc settling
model that can consider the effects of the creeping flow
through a floc on its settling speed is employed in the
floc porosity calculation. In the following, we first describe the experimental methods, then the floc settling
model. Results for porosities and fractal dimensions
and discussions are then presented.
MATERIALS A N D E X P E R I M E N T A L
METHODS
The material used in the study was a used drilling
m u d from a platform in the Santa Barbara Channel.
Table 1 shows the components of this drilling mud.
Excluding water, about 65% (by weight) of the m u d is
barite and 30% is bentonite, while the rest consists of
small amounts of additives. The median size (diameter) of the disaggregated drilling m u d particles (test
sample) is about 6 #m and 90% of the solid mass is in
< 17 ~m particles.
The drilling m u d flocs used in the settling tests were
produced in a horizontal Couette-type flocculator
(which is 254 m m long, and 50 m m in diameter) at a
variety of combinations of fluid shears (50, 100, 200
s -1) and solid concentrations (10, 50, 100, 200, 400
Clays and Clay Minerals
Table 1. Components of the used Santa Barbara Channel
drilling mud.
Component
Weight
(%, excluding
water)
Barite
Aquagel
Drispac (starch)
Therma-Thin
Barabrine
Caustic
65.14
29.86"
2.5
1.6
0.56
0.35
Generic description
Bentonite
Polyanionic cellulose
a = This measured value includes some formation clays that
enter the mud system during drilling operations.
mg/liter), and at a pH of about 8. The procedure for
flocculation tests was essentially the same as that described by Tsai et al. (1987), so only a brief description
is given here. In doing a flocculation test, the sample
suspension at a known concentration was first disaggregated in a blender and was then put into the flocculator. The flocculator was run at a constant rotational
speed to produce a uniform fluid shear. At certain time
intervals (5, 10, or 20 min, depending on flocculation
conditions), the flocculator was stopped, and samples
were withdrawn from the flocculator for particle size
analysis using a Malvern Particle Sizer 3600E. Then
the flocculator was filled and run again. This procedure
was continued until successive samples showed that
the average median diameter over time approached a
constant. It was then assumed that a steady state of
flocculation had been reached. After this, a small
a m o u n t of suspension containing flocs was taken using
a pipette from the flocculator and immediately introduced into a settling tube for the setting speed measurement.
The settling tube made from plexiglass is 1 m high
and 10 x 10 cm wide in cross-section. Three windows
(one for photography, the other two for flashing) are
located at a distance of 35 cm below the water surface
in the tube; at this distance, flocs will have reached
their terminal settling speeds. A horizontal-axis Nikon
SMZ-2T microscope-camera system is mounted against
the photography window. The magnification of this
microscope-camera system can be 16- or 20-fold. A
flash for exposing film is placed either on the opposite
side of the tube from the camera or beside the camera.
A lantern always accompanies the flash, lighting up the
flocs in the tube for observation and focusing from the
microscope. Polaroid 667 Coaterless black-and-white,
professional instant pack films were used for photography.
In performing measurements, the suspension containing flocs was poured gently onto the water surface
in the settling tube and the flocs were allowed to settle.
As one (usually more than one) floc arrived in the field
of view of the camera, two consecutive flashes were
produced with the time between them varying from 2
Vol. 41, No. 3, 1993
Floc fractal structure
to 5 s, depending on the settling speed of the floc. This
resulted in two positions of a floe being recorded on
one photograph. This is called the double-exposure
photographic method. For each test, eight to 14 such
double-flash, single-frame photographs were taken, and
about 10 to 20 flocs were shown on these photographs.
The settling speed of a floc was determined by the
distance between its two successive positions on the
photograph and the time interval between the two
flashes. The size of a floc was measured by averaging
its long and short axis from the photograph. The reproducibility of settling speed measurements as well
as flocculation experiments were tested by performing
duplicate tests at different times with a span from a
few weeks to a few months. Consistent results were
obtained from these duplicate tests. The results of settling speeds as well as flocculation experiments are described in detail elsewhere (Huang, 1992).
The problem of the creeping flow relative to a floc
was initially studied by Brinkman (1947a, 1947b) and
later by Sutherland and T a n (1970), Ooms et al. (1970),
Neale et al. (1973), Epstein and Neale (1974), and Adler ( 1981). It should be pointed out that the term "floc"
was originally used by Brinkman (1947a, 1947b) in his
analysis, and actually referred to an isotropic porous
sphere.
In Brinkman's analysis, Darcy's law (which applies
to a low porosity medium) was extended to describe
the flow through a floc which is of high porosity. The
resulting equation is:
#
(4)
where k is the permeability of the floc; p is the pressure;
V is the velocity of the fow.
Outside of the floc, the governing equation for the
flow is
Vp = u a v .
(5)
By solving Brinkman's flow equation (Eq. 4) in a floc
and Stokes' flow equation (Eq. 5) outside of the floc
and coupling them on the floc surface, Brinkman obtained ~2, the ratio of the resistance experienced by a
floc to an equivalent solid sphere that has the same
diameter and the same density as those of the floc:
~2 -
2fl2[1 - (tanh/3//3)]
2/32 + 311 - (tanh/3//3)]
(6)
where/3 is the normalized floc radius given by:
a~
/3 = '2N/k"
k=}-2
3+
1
e
e
where dp is the diameter of the solid particles in the
floc and 1 - e is the solid volume fraction of a floc as
in Eq. 1. Assuming that a floc consists only of two
parts, solid particles and water, then according to the
mass balance in the floc, the floc porosity (e) can be
related to its density by the following expression:
l - e -
py-~
(9)
Pp - Pw
The settling speed of a floc can be expressed as:
[~
ws =
g P f ~ Pwd ]'/2
3~2Co
Pw
~J
(7)
The permeability can be estimated by the following
expreSsion (Brinkman, 1947a):
(10)
where Co is the drag coefficient.
Then the porosity of a floc can be obtained from:
3pwflCo
2
1 - e = 4g(pp - p~)dy w s.
A FLOC-SETTLING SPEED MODEL
vp = - - ~ v + ~AV
375
(11)
For a Reynolds n u m b e r (Re = wsds/~', where u is the
kinematic viscosity of water) less than unity, the drag
coefficient is expressed as:
CD-
24
Re
R e ~ 1.0.
(12)
For a Reynolds n u m b e r larger than unity, there are
well over 30 equations in the literature relating the drag
coefficient to the Reynolds n u m b e r (Haider and Levenspiel, 1989). A n expression developed by Concha
and Almendra (1979) may be used and is written as:
(
9"06~ 2
Co = 0.28 1 + R~/2 ]
R e <- 103.
(13)
Experiment results for settling speeds of porous
spheres made of steel wool (Re ~ 0.6, Matsumoto and
Suganuma, 1977) and made of a semi-rigid plastic foam
slab ( R e ranges from 0.2 to 120; Masliyah and Polikar,
1980) are in excellent agreement with Eq. 10. Therefore, Eq. 11 may be more reasonable than Eq. 3 for
determining floc porosity and is used in this study.
RESULTS
During the experiments the settling speeds and sizes
of a total of 216 flocs were measured. The sizes of these
flocs were in the range from 30 to 300 #m, and speeds
were in the range from 95 to 559 u m s 1, corresponding
to the Reynolds n u m b e r ranging from 0.003 to 0.165.
In the calculation of floc porosities, we assume that
= 0 . 0 1 cm 2s 1 , 0 p = 2 . 6 5 g c m 3, a n d p w = 1.02466
g cm 3 (which corresponds to the sea water from Santa
Barbara Channel at a salinity of 33.642 ppt and a temperature of 16.0~
dp is assumed to be 6 ~m, the
median size of the disaggregated drilling m u d particles.
With these parameters, floc porosities were c a l c u -
376
Huang
Concentration
z
__Q 0.100
I-o
[E
Lt.
LU
=
9
9
e
(mgtL)
Clays and Clay Minerals
Concentration (rng/U,
9 10
Concentration (mg/L
9 10
10
50
9
9
100
9
9
e
50
100
e 200
200
4OO
9
50
100
200
400
.J
o
o
0.010
(a)
(c)
(b)
I
0.002
10
IO0
FLOC DIAMETER (prn)
I
100
100010
I
100
100010
1000
FLOC DIAMETER (,urn)
FLOC DIAMETER (ym)
Figure 1. Solid volume fraction in flocs as a function of floc diameter, for fluid shears: a) 50 s ~, b) 100 s -~ , and c) 200 s -~ ,
respectively. Solid lines are from power regressions.
lated from Eq. 11 using the settling speed data. Results
are plotted in Figure 1a through Figure 1c in the way
o f the solid v o l u m e fraction in flocs as a function o f
floc diameter for fluid shears o f 50, 100, and 200 s ',
respectively. We can see from these figures that solid
concentration at which the flocs are formed has no
distinguishable effects on floc porosity. Although scattering exists, the data suggest that for the same size,
the flocs formed at different solid concentrations but
at the same fluid shear may have the same porosity.
By overlapping Figures l a - l c , we find that flocs
formed at higher fluid shears are more dense than flocs
formed at lower fluid shears. This is in agreement with
the concept o f "Mechanical syneresis" described by
Yusa (1977) as the shrinkage and densification of loose
and bulky flocs due to mechanical forces applying locally unevenly and fluctuating over the surface offlocs.
This is also consistent with experiment results o f Klimpel and Hogg (1985) for quartz flocs.
Another observation o f Figure 1 is that the relationship between 1 - e and dF may be fitted by a power
law:
1.53 for fluid shears o f 50, 100, and 200 s ', respectively.
The uncertainties for the coefficients A and m (or the
fractal dimension D) are estimated. A value for A or
m determined from the least-squares method may be
assumed to be the mean or the o p t i m u m value; its
uncertainty or " e r r o r " can be in terms of the standard
deviation az,A for A or a m for m. at,~ and a,~ are calculated using a method described by Bevington (1969).
We obtain that atr~ = 0.1686, 0.1779, 0.1524, and a m
= 0.0367, 0.0394, 0.0356 for fluid shears o f 50, 100,
and 200 s- ', respectively. The uncertainty for D should
be the same as that for m.
It is found that both A and m vary linearly with fluid
shear; the relationships o f A and m with fluid shear are
obtained by linear regressions. Then an empirical expression for the porosity o f drilling m u d flocs is written
as:
1 - e = (0.0825G
+
4.864)df {7"429•1764G+1"32)(16)
where G is in s-~ and dI is in/~m.
DISCUSSION
1 -
e = Ads"
(14)
where A and m are the experimentally determined coefficients.
Using the least-squares method, we obtain that A =
9.168, 12.852, 21.461, m = 1.3594, 1.3886, 1.4735
with the corresponding correlation coefficient r =
0.9748, 0.9751, 0.9791 for fluid shears o f 50, 100, and
200 s -t, respectively. The power law relations with
these A and m values are shown in Figure 1 as solid
lines.
Comparing Eq. 14 with Eq. 1, we find:
D = 3 - m.
(15)
Therefore, floc structure can be well described in
terms o f the concept o f fractal geometry. The fractal
dimensions determined from Eq. 15 are 1.64, 1.61 and
We can compare the empirical expression for porosity and the fractal dimensions obtained above by
using the floc settling model with those obtained by
using Stokes' law. An empirical expression for the settling speeds o f the same drilling m u d flocs has been
found to be (Huang, 1992):
Ws = ad~
(17)
where ws is in u m / s and dF i s in #m; a = 12.851,
15.839, 23.265, and b = 0.59, 0.58, 0.52 for fluid
shears o f 50, 100, and 200 s ', respectively. With the
assumption that the settling o f individual flocs satisfies Stokes' law, the same empirical equation as Eq.
14 is obtained from Eq. 3 with .4 = 14.865, 18.312,
26.912, and m = 1.41, 1.42, 1.48 for fluid shears o f
50, 100, and 200 s -~, respectively. Th e corresponding
Vol. 41, No. 3, 1993
Floc fractal structure
50
45
40
LU 35
L)
z
HJ
,,u_ 25
a
__ 2o
U.I
n-
. . . . . . . _~.~. ~ . . ~ .~.- . . . . . . . . . . . . . . . . . .
lO
I
I
I
I
I
I
I
50
100
150
200
250
300
350
400
FLOC DIAMETER (,urn)
Figure 2. Relative differences for solid volume fraction in
floes defined by Eq. 19. Solid line for G = 50 s ~, dashed line
for G = 100 s -~, and dash-dotted line for G = 200 s -~.
fractal d i m e n s i o n s are 1.59, 1.58, a n d 1.52. N o t i c e
t h a t t h e v a l u e s o f m a n d D are n o t significantly d i f f e r e n t f r o m t h o s e e s t i m a t e d b a s e d o n t h e floc s e t t l i n g
model.
T h e n , an e m p i r i c a l e x p r e s s i o n for floc p o r o s i t y est i m a t e d b y Stokes' law is w r i t t e n as:
1 - e = ( 0 . 0 8 1 1 G + 10.57)dr (4-857•1764a+1.38) (18)
w h e r e G is in s 1 a n d dI is in # m .
T o c o m p a r e t h e t w o e m p i r i c a l e x p r e s s i o n s for solid
v o l u m e f r a c t i o n in floes, w e define t h e r e l a t i v e differe n c e R D b e t w e e n (1 - e)zq. 16 a n d (1 - e)Eq. 18 as t h e
following:
RD =
(1
-
e)Eq.
18
--
(l
-
e)Eq.
16
(1 - - e)Eq. 16
(19)
377
Figure 2 s h o w s t h e relative difference as a f u n c t i o n
o f floc d i a m e t e r for fluid shears o f 50, 100, a n d 200
s -1. W e can see f r o m this figure t h a t the difference
ranges f r o m 17% to 38% for the floc size range a n d
fluid s h e a r range tested. T h e differences are greater at
a fluid s h e a r o f 50 s -1. T h i s is c o n s i s t e n t w i t h t h e fact
t h a t floes f o r m e d at l o w e r fluid shears are l o o s e r so
t h a t t h e i r settling b e h a v i o r is m o r e d e v i a t e d f r o m
S t o k e s ' law t h a n floes f o r m e d at h i g h e r fluid shears.
T h i s figure suggests that, using Stokes' law in the calc u l a t i o n o f floc p o r o s i t y m a y o v e r e s t i m a t e solid volu m e f r a c t i o n in floes o r effective d e n s i t y o f floes by as
m u c h as 38%.
A c o m p a r i s o n is also m a d e o f t h e fractal d i m e n s i o n s
o f drilling m u d floes w i t h t h o s e o f o t h e r k i n d s o f floes
f r o m s o m e p r e v i o u s studies (Table 2). T h e s e p r e v i o u s
s t u d i e s were all m a d e using j a r testers (or b l a d e - t y p e
flocculators) instead o f the C o u e t t e - t y p e floceulator used
in this study. Jar tester basically is a t a n k w i t h s o m e
sort o f a g i t a t o r o r blade. F l o w s in this t y p e o f a p p a r a t u s
are t u r b u l e n t b u t far f r o m i s o t r o p i e w i t h v e r y high
shears p r o d u c e d n e a r the a g i t a t o r a n d generally low
s h e a r s elsewhere. T h e i n t e n s i t y o f agitation can be
e q u i v a l e n t to a n average fluid shear. N o t i c e t h a t fluid
shears in the studies o f M a g a r a et al. (1976) a n d T a m b o
and Watanabe (1979)are unknown.
In the p r e v i o u s studies listed in T a b l e 2, the fractal
d i m e n s i o n s (D = 1.7-2.1) for q u a r t z floes w e r e g i v e n
by the i n v e s t i g a t o r s ( K l i m p e l a n d Hogg, 1986); the
fractal d i m e n s i o n s for o t h e r floes are n o t directly g i v e n
b y the i n v e s t i g a t o r s b u t d e r i v e d f r o m the settling s p e e d
f o r m u l a (Gibbs, 1985 a, 1985 b), m a k i n g use o f S t o k e s '
law, o r o b t a i n e d f r o m t h e effective d e n s i t y - s i z e relat i o n s h i p s in w h i c h the effective d e n s i t i e s w e r e calculated using solid s p h e r e settling m o d e l s ( T a m b o a n d
W a t a n a b e , 1979; M a g a r a et al., 1976). T h e fractal dim e n s i o n s e s t i m a t e d b a s e d o n S t o k e s ' law o r o t h e r solid
s p h e r e settling m o d e l s m a y be g o o d a p p r o x i m a t i o n s
Table 2. Comparison of fractal dimensions of floes.
Material
D
dp 0~m)
dr 0am)
G (s ])
Drilling mud
Drilling mud
Drilling mud
River sediments~
Illite
Kaolinite
Kaolinite
Montmorillonite
Activated sludge
Activated sludge
Pure quartz
Clay-iron
Clay-magnesium
Color-aluminium
Color-iron
Color-magnesium
1.64"(1.59 b)
1.61 a( 1.58 b)
1.53~(1.52 b)
1.78 b
1.65 b
1.59 b
1.46b-1.97 b
1.35 b
1.47b-1.53 b
1.4 b
1.7b--2.1b
1.92 b
1.91 b
1.67b-1.77 b
1.58b-1.69 b
1.48 b
6c
6r
6r
-0.5-2
0.5-2
-0.5-2
--2--10
------
30-300
30-300
30-300
20-100
20-120
20-130
150-3500
20-120
100-600
200-2000
50-500
200-2000
300-2000
700-5000
500-5000
600-4000
50
100
200
8
6
6
-6
--356--2917
------
Reference
this study
this study
this study
Gibbs (1985a)
Gibbs (I 985b)
Gibbs (1985b)
Tambo and Watanabe (1979)
Gibbs (1985b)
Magara et al. (1976)
Tambo and Watanabe (1979)
Klimpel and Hogg (1985)
Tambo and Watanabe (1979)
Tambo and Watanabe (1979)
Tambo and Watanabe (1979)
Tambo and Watanabe (1979)
Tambo and Watanabe (1979)
a = fractal dimension is estimated based on the floc settling model; b = fractal dimension is estimated based on Stokes' law
or other solid sphere settling models; ~ = median diameter; d = including some flocs collected from Chesapeake Bay.
378
Huang
for fractal dimensions estimated based on the floc settling model as we have shown for drilling m u d flocs.
We can see from Table 2 that the fractal dimension
ranges from 1.35 to 2.1 and seems dependent on materials from which flocs were formed. Since no experiments were conducted using the same material in both
Couette-type flocculator and jar tester, it is unknown
whether or not the floes formed in a laminar flow have
the same fractal structure as those formed in a turbulent
flow.
The fractal dimension discussed here is a measure
o f the irregularity in floc structure. The smaller the
fractal dimension, the higher the degree o f the irregularity; or the larger the fractal dimension, the more
uniform the floc structure. The up-bound o f fractal
dimension for floc structure is 3, which corresponds to
a uniform structure so that its porosity or density is
not a function of size. Therefore, using fractal dimensions makes it possible to quantitatively describe and
compare the irregularity in structure for floes made
from a variety o f materials and produced under different conditions.
Fractal geometry provides a framework of studying
floc formation process or floc dynamics. According to
the fractal theory, a fractal is constructed from building
units in a self-similar manner, which is mathematically
expressed by a power law relation (of mass and size
for example). A question then is: what are the building
units o f flocs. We notice from Table 2 that the sizes o f
primary solid particles are very small, in a range of less
than 10 #m. The power law relation (e.g., Eq. 16) is
not valid at the size range o f primary solid particles.
This indicates that primary solid particles are not
building units for fractal floes. The possible building
units o f flocs are known as microflocs which are formed
from primary solid particles at the initial stage o f flocculation. Through r a n d o m collision due to f u i d shear,
these microflocs combine to form larger flocs; these
larger flocs may combine to form even larger floes. This
formation scheme o f fractal floes is consistent with a
multistage growth model for floes originally proposed
by Sutherland (1966) and a model called the ordered
structure o f aggregates originally proposed by Krone
(1963).
Floc structure is affected by collision (or transport)
mechanisms by which f o c s are formed. That is, flocs
formed by Brown motion, fluid shear, and differential
settling may have different fractal dimensions. This is
because floc formation schemes for these collision
mechanisms are different. A formation scheme o f flocs
formed by fluid shear has been described above. T h e
growth o f floes formed by differential settling may result from sweeping and catching smaller particles (including primary solid particles and microflocs) by larger particles or flocs. In this study, we find that the fractal
dimension increases with decreasing fluid shear. This
may be accounted for by considering the fact that, at
higher fluid shears, fluid shear is the dominant collision
Clays and Clay Minerals
mechanism for f o c formation; at lower fluid shears,
both fluid shear and differential settling could be important. The variation o f fractal dimensions may be
an indication o f the variation o f the relative importance o f collision mechanisms involved in floc formation. Further studies are needed to i m p r o v e our
knowledge on the effects o f collision mechanisms on
the fractal structure o f floes.
CONCLUSIONS
1) The porosities o f a total o f 216 flocs formed from
a used drilling m u d were determined by measuring floc
sizes and floc settling speeds and by employing a ftoc
settling model in the calculation offloc porosities. Floc
porosity increases (density decreases) as floc size increases. O f the two main parameters offloc formation,
fluid shear has significant effects on floc porosity, while
solid concentration does not. For the same size, the
flocs formed at higher fluid shears are more compact,
or more dense than the flocs formed at lower fluid
shears.
2) The structure o f drilling m u d flocs can be well
described as a fractal with a fractal dimension o f 1.531.64 for the floc size range tested. The fractal dimension
is not influenced by solid concentration, but increases
with decreasing fluid shear.
3) Empirical expressions for the porosity o f drilling
m u d flocs are obtained from both the floc settling model and Stokes' law. For solid v o l u m e fraction in flocs,
the relative difference between these two expressions
is as much as 38%. However, the fractal dimensions
estimated based on the two settling models are nearly
the same. As a first approximation, Stokes' law could
be used to estimate the fractal dimension o f flocs.
ACKNOWLEDGMENTS
This research was supported by the U n i t e d States
Minerals Management Service and the United States
Environmental Protection Agency.
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(Received 28 January 1993; accepted 14 May 1993; Ms.
2318)