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Design of Muds for Carrying Capacity
R. E. Walker SPE AIME LamarU.
T. M. Maye s
SPE-AIME Milchem Inc.
ntrodu tion
Hole cleaning san important function
of
a drilling liquid.
The ability to predict the degree of cleaning possible for a
given mud and flow rate is a definite advantage in plan
ning and completing a successful drilling operation. The
prediction is normally made by calculating the transport
velocity the difference in the annular velocity and the
slip velocity of the particles by assuming the slip veloc
ity is equal to the terminal settling velocity
of
the particle
a stationary liquid. The settling velocity
s
easily calcu
lated if the annular flow s turbulent or the particle settles
in the turbulent regime. Under these conditions the slip
velocity depends on the density difference between the
mud and the particle and on the particle shape and size.
The slip velocity s not a function of the liquid viscosity.
However these settling velocities may be high up to 74
ft/min for a shale sphere Vs in. in diameter and high
annular velocities are needed to clean the hole.
There are many cases where high annular velocities are
unavailable and/or undesirable. Annular velocities may
be low because of pump limitations or an enlarged hole
o r may be low where risers are used. Also it may be
necessary to restrict annular velocity
t
minimize the
equivalent circulating density or to maintain laminarflow
opposite drill collars. If turbulent-regime slip velocities
are too high forthe annular velocities the viscous proper
ties of the liquid must be increased until the particle falls
in a transition or laminar regime where slip velocities are
influenced by viscous forces. Unfortunately none of the
methods proposed in the literature for predicting terminal
settling velocities in the transition or laminar regimes are
suitable for field application. Most of the theoretical and
experimental work
s
with s p h e r e s 1 ~ 8 1 0 ~ 1 3 ~ 1 4 or with liq-
uids that are almost Newtonian. In work using non
Newtonian liquids the rheological measurements are not
sufficient for defining properties in the experimental
range
6
or
the prediction equations use rheological mod
els with constants defined in a shear-rate range different
than that of the experimental work.
4
One study 9 uses
viscosities not associated with that of the liquid surround
ing the particle while another stud
y
provides a solution
so complex that it
s impractical for field use.
An assumption that the terminal settling velocity will
be the same as the slip velocity
s
questionable because of
the complex motion of the particle in the annulus. The
moving liquid has a velocity profile near parabolic in
form that s affected by hole geometry liquid flow
properties and pipe rotation.
The particles tilt with the
velocity profile which in turn affects their settling veloc
ity. Drillpipe rotation introduces a centrifugal force caus
ing radial migration
5
and some particles are also
trapped near the
p i p e 1 8 ~ 1 9
If
the mud
s
non-Newtonian
the viscosity
of
the liquid around the particle
s
depen
dent on the settling velocity and the flowing- velocity
profile. An additional complication s the various shapes
a cutting or caving may have.
Improvement in methods for predicting successful
hole cleaning s dependent on a better understanding
of
how viscous forces retard particle settling. Practical ap-
The ability to predict the degree ofhole cleaning possible with a given mu n flow rate is an
advantage successfully planning and completing a drilling operation. A simple reasonably
accurate mathematical prediction technique
s
developed that can be used in the field.
JULY 1975 893
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plications
of
methods for improving hole cleaning require
a simple approach. The objectives
of
this
paper
are. to
relate the size, density,
and
slip velocities
of
cuttings
and
cavings to the shearstress
vs
the shearrate relationship
of
the l iquid and to establish a simple mathematica l pro
cedure applicable to field use.
ttling
elocity
The
settling velocity of any particle depends on a number
of
factors such as the density and flow properties
of
the
liquid and the volume, density, and shape of the particle.
Nonnally
the relations are correlated
by
plotting a drag
coefficient vs a Reynolds number. The drag coefficient is
defined as twice the net vertical force volume times
density difference) divided by the product of the particle
velocity squared, projected area of the particle, and the
liquid density. The Reynolds number contains a liquid
viscosity
t rm
convenient for Newtonian liquids,
bu t
unsuitable for non-Newtonian liquids.
Work
with Newtonian liquids shows that the drag
coefficient vs the Reynolds-number relation can be di
vided into three regimes: turbulent, laminar, and transi
tion.
In
the turbulent regime, the only resistance slowing
the fall of the particle is causedby the
momentum
forces
of
the liquid; viscosity plays no part. Thus, if a particle is
falling
the turbulent regime, increasing the
mud
viscos
ity will not slow the settling rate until the viscosity is
increasedsufficiently to force a change from the turbulent
to the transition or the laminar regime. In the laminar
regime, the entire resistance slowing the fall is caused by
the viscous forces of the liquid; the momentum forces are
negligible.
The
drag coefficient in this regime varies
inversely with the Reynolds number.
Between
the two
regimes is the transition regime, where both viscous
and
momentum
forces retard the falling particle.
If
the flow in anannulus is turbulent, a particle slips in
turbulence. If
the flow is laminar, a particle may slip in a
turbulent, transition,
or
laminar regime depending
on
its
geometry and on the viscous proptrt ies of the liquid.
Work with Newtonian l i q u i d s ~ indi ca te s tha t the
laminar-transition-regime change for a particle occurs at
a Reynolds number between 0.1 and
0.3 and
the change
from
the transition to turbulen t
regime
occurs
at
a
Reynolds n\lmber of 100.
The
approach used in this paper to develop a useful
method forpredicting slip velocities is to assume a simple
set
of conditions and develop predictive equations. The
results are compared with laboratory experiments and
with the results
of
the work
of
others.
The problem is simpl if ied by assuming a disk shape
that is the simplest shape consistent witha cutting form. It
also assumes that the qisk falls flat ,side down, which
represents the condition for the highest terminal settling
velocity. Drag coefficients are established
fo r
this disk
shape and orientation, and the relations with Reynolds
numbers for each flow regime are assumed from Newto
nian l iquid data. An assumption is then made about the
viscosity term in the Reynolds number to develop the
settling-rate equation as a function of the liquid shear
stress and the shear rate.
E,quation evelopment
Terminal sett ling velocity equations are developed by
combining the drag coefficient and the Reynolds number
894
1
= 2
3 4
4
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+
25
895
Fig. l-Comparison
of calculated with measured settling
velocities.
-
25
b ------1
1-
0
4 0 L 6 0 ~ ~ 8 O ~ I ~ O ~ ~ 1 2 ~ o ~ l ~ o
OBSERVED SETTLING VELOCITY,
Ft
/min
CONTROL
LIMITS
..
. . : : .
•
.
.
-
20
of
20
- 10 - - - - - - - - - - - - - - - - - - - -
- 15 •
f/
::>
z
i
+ 15
+ 10
Discussion of
Experimental Data
The method used
to
evaluate the accuracy of the equa
tions considered the difference between measured and
calculated settling velocities. The measured particle ve
locity was used to calculate the Reynolds number.
There are
87
sets
of
useful experimental data, with
10
additional sets negated because
of
unstable fall. The data
include a Reynolds-number range from
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\
\
TABLE
2A-EXPER-IMENTAlRESULTS
DIFFERENCE B ETWEEN OBSERVED
AND
CALCULATED SLIP VELOCITIES
Liquid 2
l iquid
3
Terminal Velocity
Terminal Velocity
(ft/min)
Reynolds
(ft/min) Reynolds
Observed Difference* Orientation
Number .Observed Difference*
Ori-entation Number
113.
2.6 Flat 203.
68.
4.3
Flat 50. 88. 10.1 Flat
100.
27.9
4.1 -Flat 8.1
48.5
3.6
Flat , 50.
Unstable 1.1 29.2 5.5
-Flat
23.9
5.3
1.2 Edge
0.4 16.4
2.8 Flat
12.6
_101.
(4.8)
Flat 71.
48.2
0.0 Flat
17.8
-67.
6.9 Flat 37.9
. Unstable
2.6 38.4
4.3
Flat
20.0
12.8
6.0
Edge
0.8 21.7 3.7
Flat 8.9
19.3
(11.9)
Stable
3.2 35.9
(5.7)
45° 10.6
8.9
(2.7)
Edge- 0.6 Unstable
5.9
2.8
1.5)
Edge
0.1
Unstable
2.5
.22.2
3:9
Edge 5.1
39.7 1.6 Flat 39.2
8.5 1.6 Edge
0.8
24.7
4.2
Flat
20.0
4.1
(9.8)
Flat
0.5 25.4
(3.5)
Edge 12.6
1.9 (3.3)
Edge
0.1
16.3
0.7
Flat
6.6
0.44
(3.4)
Stable
.
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ceed t hese l imit s, six are i n t he t ur bulent r egime; in .each
c as e, the d iffere nce is less t ha n 20 p erc en t
of
the mea
sured velocity. The four remaining cases-occurred with
disks wh ose thickness was equ al to or o n e ~ h a l f of th e
diameter; in each i nstance, t he par ti cl e was i n t ransit ion
fall.
Terminal velocities
of
thfee non disk shapes were
measured
in L iq uid s 4 and 5. All were
made
from
lA-in.-thick plastic. One
s h a p e ~ a s
a s qua re w it h I- in .
s id es ; a no th er w as a t ri an gl e m ad e b y c ut tin g t he s qu are
along a diagonal; and the third was a
x
Vz-in.rec
tangle. The dif ferences i n vel ocit ies wer e in
the
_sa me
ra nge as those obtained for disks when the effective
d ia me te rs w er e c al cu la te d by the hydraulic-diameter
equation:
3
d
== . . _
10)
c
In summary
the e qu at io ns p re di ct ed t he v el oc it y
within ±10 ft/min for 88 percent of the test sets and closer
t ha n e it he r 10 ft/min or 20 p erc en t -of the m ea su re d
velocities
over
95
percent
of the time.
The
over-all error
could b e r ed uc ed b y m od if yi ng th e e qu at io n fo r t he tur
bulent r egime, as t he dif ference i n this r egime averages
4.2
ft/min, and
by
alt er ing t he equat ion i n t he ~ a m i n a r
re gi me , w he re the a ve ra ge -e rr or is ft/min.
The
corr elat ion is
deemed
s uf fi ci en tl y a cc ur at e f or f ie ld
application.
Disk
o ri en ta ti on a pp ea rs to b e i nf lu en ce d
by
many
factors. Within
the
limits of
this
study, f lat fall always
o cc ur re d a bo ve a R ey no ld s n um be r of 13 and edge fall
occurred predominantly below H ow ev er , w he n t he
disk thickness equaled its diameter,
flat
fall occurted at
a R ey no ld s n um be r
of
2 and s tab le fall occu rred at a
Reynolds
number
of 1
When
unstabl e fall occur red, i t
was in t h e R e y n o ~ d s
number
range
of
2. 5 to 10. The
largest errors in the predicted-to-observed velocities in
t he l aminar and t ransit ion r egimes occur red wit h disks
having thickness-to-diameter ratios
of
0.5,and 1.0. Thi s
w as p ro ba bl y b ec au se t he e qu at io ns do n ot a cc ou nt fo r
the viscous drag
on
the peripheral area
of
the disk.
Equation Evaluation
Experiments that simulate continuous
t r a n ~ p o r t
i n a well
bor e and provi de rheologi c al i nfo rmat ion are l imit ed.
Field dat a _containing r equi si te i nf or mati on are almost
nonexistent. - ,
Williams and Bruce
18
measured the ,transport rate for
disks in an annulus with Newtonian fluids where pa.rticle
sli p was t ur bulent . The t er mi nal s e t t U n g v e l o c i ~ y ,eqia
tion, Eq. 6, is equivalent to the slip-velocity equation
developed in their work. .
Sifferman et al measured, the transport velocity
of
simulated cavings carried up an annulus with gel muds. A
comparison between their c a l c u l ~ t e d slip veiocities; es- ,
tablished by subtracting the t ~ a n s p o r t velocity from the
bulk-liquid velocity, and the terminal settling velocities
predi ct ed wit h Eqs. 6 and 8 are l isted
in
Table 3. Evalua
tion was limited to four sets of data; one set
~ h e r e
par ti cl es wer e i n t ur bulent f al l, one set i n t ransit ion fall,
a nd tw o sets i n
laminar
fall. Agreement was excellent for
t he t ur bulent and t ransit ion r egimes, but not as .good for
the laminar regime. The laminar conditions were calcu
lated for two differentparticle orientations because a shift
TABLE 2B-EXPERIMENTAL RESULTS DIFFERENCE BETWEEN OBSERVED AND CALCULATED
SLIP
VELOCITIES
L iq uid 4
L iq uid 5
Particle Shape
Terminal Velocity
Terminal Velocity
Diameter
Thickness Specific ft/min)
Reynolds
ft/min.)
Reynolds
in.) , in.)
_ Q ~ ~ t _
Observed Difference* Orientation
Number
Observed Difference*
Orientation
Number
Disk
1
2.83 128
18.0
Flat
348.
117.
6.5
Flat
.833.
Disk 1
2.83 90.
11.9
Flat
211. 78.
0.3 Flat 4 8 7 ~
Disk
1
l/
s
2.83 56. 1.2 Flat
115.
54.
1.5)
Flat
277.
Disk 1
1/16
2.83
38..8
3.4
FIClt
71.
Disk 1
1/32
2.83
24.0
3.0
Flat
44.0
Disk
1/
2
2.69 123. 8.7
Flat
162.
111.
3.4)
Flat
390.
Disk
2.83
83.
3.5)
Flat
97.
98.
13.6 Flat
303.
Disk
s
2.83
49.1 1.4
Flat
50.
Disk
1/16
2.83 29.8 2.9
Flat
27.3
41.0
1.5) Flat
94.
Disk V4
V4 2.68
51.
6.3)
45°
28.6
85.
1.7
Flat
117.
Disk
V4 Vs
2.68
33.8
1.2 Flat
17.2
47.2
4 .5) Flat
61.
Disk V4
1/16
2.68
23.8 5.5 Flat
10.9
Disk
1
1.38
52.
0.1 Flat
95.
61.
10.8
Flat
282.
Disk
1
V4 1.38 32.5
1.7
Flat
60.
39.
3.4
Flat
178.
Disk
1/
2
1.38 36.3
3.1)
Stable
33.2
60.
5.5
Flat
137.
Disk V
1/
4 1.38
24.6
1.2
Flat
22.5
38.2 1.2
Flat
87.
Disk
V4
1/
4 1.38
14.9
2.1)
45°
6.8
26.6
0.3) Flat
30.5
Disk 1
1/32 8.77 51.
6.1)
Flat
104.
70.
13.0
Flat
360.
Disk V
1/32 8.77 52.
2.0
Flat
53.
Disk
1/
4 1/32 8.77
45.3
8.9
Flat
23.2
Square**
1.04
0.248
1.42
36.8
3.3 Flat 70.
38.3
1
3
Flat
183.
Triangular***
0.66
0.247
1.42 32.5
4.4 Flat
39.2
39.9
0.8
Flat
120.
Rectangular
t
0.82
0.236
1.42
33.2 3.6 Flat
49.8
33.9
3.5) Flat
127.
Average velocity
2.7
2.55
Standard deviation of velocity
5.4
5.4
Standard error
of
the mean 1.1
1.3
*Observed min
us
calculated: ) minus.
**Square plate, I-in. sides, l .04-in. effective diameter.
***45°
isosceles triangle with
I-in.
sides, O.66-in. effectivediameter.
t
x V2 in rectangle, O.82-in. effective diameter.
JULY 1975
897
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in orientation would be expected at a low Reynolds
number.
An
assumption
of
e dg e fall, w hi ch i s li ke ly to
occur under laminar conditions, more closely approxi
mates experimental results .
Additionalinformation
is
needed before drawing defi
nite conclusions but, based
on
the available,data, termi
nal settling and slip velocities can be considered the same
and the predicted velocities are consistent with experi
ments of others.
Application
Slip Velocities
The developed equations can be more easily applied
in
the field by transposing to the following units:
velocity - ft/min .
density
- lb /gal
particle dimensions - in.
shear stress lb
f
/100 sq ft
The specific gravity
of
the solids
is
a ss ume d to be 2 .5 o r
20.8 Ib/gal.
The first step in estimating the slip velocity
is
to calcu
late the shear stress developed by the particle:
T
c
7.9
Yh
c
20. 8 - c . 11)
The
particle thickness can be estimated from samples
over
the shaker, junk basket, viscous plugs,
or
offset
wells). .
The second step is to estimate whether the annular flow
is
laminar
or
turbulent.l
6
If
it is laminar, skip to Step
3; if
it is turbulent, calculate the slip velocity for the turbulent
regime:
c
=
16.62 •••••••••• • •••••• •• 12)
V;;;
rhe
third step
is
to determine the turbulent-transition
rt
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MAXIMUM SHEAR RATE
FOR
TRANSITION
REGIME: SLIP
FOR
PARlICL 1
1 0 0 - r - - - - - - - - - - - - r - : 1 / ~ - ; - - ,
A-RT-IC-L E-S-l . . . ..----T-:?----
899
6 •
1000
8 100 2
SHEAR
RATE SEC-I
ig 2-Stress-rate
relations for two field muds
1
1
- - -P-OI -NT -S-- . . - - . .. - - ~ - t - - 7 - - - S H - e : A - R - S - T R - E - S S - D - E - V t - L O - P - E
- : - = ~ - - - -B Y RIVER GFtAVEL
I P ~ T ~ SHEAR STRESS Df :VELOPED
I . I BY l iS
SAND
I I
I I
I
I
I
I
I
0::
« 8
w
::I:
CI
6
o
Q
.Q
I
2
f./
Cf
W
0::
Cf
omenclature
e
= projected cross-sectional area of
particle, ft
2
= particle diameter, in.
d
eq
= equivalent particle diameter, ft
d
p
:=;;:
particle diameter, ft
d
w
:=: inside diameter of glass pipe, ft
w
wall-effect factor defined in Eq. 9
T
dimension factor, Ib
f
/100 ft
2
g
~
acceleration of gravity, 32.17 ft/sec
2
g
= gravitational constant,
32.17
Ibm
-ft/ lb
f
-sec
2
h = particle thickness, in.
h
p
=
particle thickness, ft
e
= perimeter around projected area of
particle, ft
ed drag coefficient defined by Eq. 1
Re
Reynolds number
V
e
=
particle velocity relative to the liquid,
ft/min
V
p
= particle velocity relative to the liquid,
ft/sec
y shear rate, sec-
i
velocity in
95
percent of the cases.
2. Data for comparison is l imited, but the avai lable
information indicates that the terminal settling velocities
calculated by
the proposed equations are numerically
similar to slip velocities calculated from transport-rate
measurements .
3. Limited data indicate that the equations developed
for disks can be used to predict slip velocities and hole
cleaning for other shapes commonly encountered when
drilling.
4. These equations permit comparing hole-cleaning
capabilities of muds with different flow properties and
provide a method to adjust flow properties to clean an
annulus.
h
= I ~ O 2 15
The effective particle thickness is 0.108 in. This gives
a shear stress
of
8.6
Ib/l00
ft
2
in 10-lb/gal mud. The
effective diameterof the largerparticles is estimatedat
14
in.
To clean the hole, the larger particles must have the
same or a lower slip velocity than the l/s-in. sand grains.
The shear rate corresponding to a slip velocity of
17
ft/min for the
14
-in.-diameterparticles is, from Eq. 14, 33
sec-i. The mud must be changed to give a shear stress
equal to or greater than
8.6Ib/l00
ft
2
at a shear rate of
33
sec-
i
Point B in Fig. 2 .
The hole was displaced with a prehydrated gel mud
weighted
in
brine to 10 lb/gal. This mud, which cleaned
the hole and allowed problem-free drilling to continue,
closely matches the required shear stress at
33
sec-
i
, as
shown in Fig. 2.
In
retrospect, the 14-in. gravel has a slip velocity of 45
ft /min in 10-lb/gal brine and 30 ft/min in the guar-gum
mud. In each mud the particles would be carried up the
gauge hole to accumulate in the enlarged hole and would
then fall back when the circulation stopped.
onclusions
1.
A simple set of equations were developed that pre
dict the terminal settling velocity of disks in turbulent,
transition,
or
laminar fall for a wide range of test condi
tions. The equations predict the terminal settling velocity
within ± 10 ft/min in 88 percent of the cases and within
either
±
10 ft/min or 20 percent of measured settling
JULY, 1975
sand grains exert a stress of7 .5Ib/100 ft
2
in the 10-lb/gal
mud. A spherical shape is used for the sand grains, and
the effective thickness to use in Eq. 11 is two-thirds the
sphere diameter. Slip velocity estimates for spheres are
valid in the laminar and transition regimes becauseEqs. 4
and 5 are valid for both spheres and disks, but Eq. 3
turbulentregime is not valid for spheres. The flow in the
annulus was laminar. From the shear stress vs shear rate
plot shown in Fig. 2, the shear rate corresponding to a
stress of
7.5
Ib/100 ft
2
is 91 sec-
i
Point A . This shear
rate is less than the limit for the transition regime Line
C , so the transition equation, Eq. 14, is used to calculate
a slip velocity
of
17 ft/min. A mud velocity
of
27 ft/min
might be considered necessary to clean the hole 17
ft/min to overcome the slip and 10 ft/min to move the
particles through the enlarged hole in a reasonable time .
A 26lh-in.-diameter hole will have a bulk velocity
of
27
ft/min, which is a reasonable hole size.
A viscous plug was run in the well , removing ~ i n
gravel. Because of the odd shapes of the gravel, the
effective thickness was difficult to determine. The effec
tive thickness was est imated by measuring the shear
stress developed by the particles. This was done by tim
ing the terminal settling velocity of a handful
of
the gravel
dropped into water contained in a verticaI2-in.-diameter
glass pipe
10
ft high. Since the flow regime is turbulent,
Eq. 12 can be used to solve for part icle shear stress; the
resul t can be substituted into Eq. 11 to calculate the
thickness . Or, the effect ive part icle thickness can be
calculated from Eqr 15, obtained by combining Eqs.
11
and 12, using the density of water.
8/18/2019 Walker and Mayes - SPE-4975-PA
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(B-1)
(B-3)
which is Eq.
8.
APPENDIXB
Eq.
13
is developed starting with Eg. A-2, which is
solved for shear rate and is squared to give
T
, . . (A-5)
(B-2)
100 (12)2/3
gc
F
T
100)2 ge d
p
VPLge
FT
100
Eq.
B- 2
is solved for shear rate and the units are changed
to those listed under Application to give
y =
8.62 N
2/3
de
V;;;
IfN
=
100, Eq. B-3 yields
=
186 , (B-4)
de vP;-
which is Eq. 13.
Then, the shear-stress term is replaced with the right side
of Eq. 7 and the velocity term
is.
replaced with the right
side
ofEq.
8 to give
(
NRe J2
=
dpPL
and the shear stress
is
replaced by the right side of Eq. 7
to give
N
Re
= d
p
V
p
PL
;/ge (A-3)
V 100F
Th
p
Ps -
PL
g/ge
Next, the right side
ofEq.
A-3 is used to replaceN
Re
and
the right side
of
Eq. 1 is used to replace
Ne d
in Eq.
A-I
to give'
[
24
v
p
2
PL J =
2g
h
p
ps - PL
100
dpv
p
PL y 2 A-4
100h
p
Ps - PL g
ge
FT
Solving Eq. A-5 for velocity gives
V
p
= ~ 3 / 4 d
p
h
p
Ps -
PL
g
24
v p
L
FTge/
lOO
1
=
(N
Re
2
A-I)
N
ed
3
24 3
Next, substitute for
N
Re andNe d . The viscosity term in the
Reynolds number is replaced by shear stress divided by
shear rate to give
N
Re
=
d
p
VPPLY , (A-2)
T
p
APPENDIX A
Eq. 8 is obtained by combining Eqs.
1
2, 4, and 7 as
follows. First, Eq.
4,
the relation between the drag coef
ficient and theReynolds humber, is inverted and cubed to
give
= shear rate corresponding
toN
Re
= 100, sec-
1
IL =
viscosity, lbm/ft-sec
Pc = liquid density, I
b
m
gal
PL = liquid density, Ib
m
ft
3
Ps = particle density, Ib
m
ft3 .
T
e
=
shear stress, lb
f
/100 ft
2
T
p
=
shear stress,
Ib
mft-sec
2
Acknowledgments
The authors wish to express their appreciation to
Mil
chern, Inc., for permission to publish parts
of
this article
and to the Lamar
U.
Research Committee for their
support.
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Original m n uscriptreceived in Society of Petroleum Engineers office July 31 1974.
Revised manuscript received y 13, 1975. Paper SPE 4975 w s first presented atthe
SPE-AIME
49th
Annual Fall Meeting, held in Houston, Oct . 6-9 , 1974. ©Copyright
1975
American Instituteof Mining, Metallurgical, and Petroleum Engineers, Inc.
This paperwill be included in the 1975 Transactions volume.
900
JOURNAL OF PETROLEUM TECHNOLOGY
