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SOCIETYOF PETROLEUMENGINEERSOF AIME
6200
NcvthCentralExpressway
Dallas,Texas 75206
~ R
SPE 4885
TRIS IS A PREPRINT
--- SUBJECTTO CORRECTION
Determination of Laminar, Turbulent, and
Transitional Foam Flow Losses in Pipes
By
R.il.Blauer,B.J, Mitchell,and C.A. Kohlhaas,MembersSPE-AIME,
. .-
~o~~do Schoolof Mines
@Copyright 1974
Ameriwan Institute of Mining Metallurgical. and Petroleum Engineers. inc.
Thispaperwas preparedfor the llthAnnualCaliforniaRegior:aleetingof the Societyof
PetroleumErxgineersf AIME, to be held in SanFrancisco,Calif.,April
4-5 1974.
Permissionto
copy is restrictedto an abstractof not morethan 300 words.
Illustrationsmay not be copied.
. .
The abstractshouldcontainconspicuousacknowledgmentof whereand by whom the paper is present
Publicationelsewhereaftermlblicationin the JOURNALOF PETROLEUMTECHNOLOGYor the SOCIETYOF
PETROLEUMENGINEERSJOURNALis usuallygrantedupon requestto the Editcwof the appropriate
journalprovidedasreementto givepropercreditis made.
Discussionof thispaperis invited.
Three copiesof any discussionshouldbe sentto the
Societyof PetroleumEkgineersoffice,
Such discussionmaybe presentedat the abovemeetingan
with tinepaper,may be ccnsi.deredarpublicationin one of the two SPE magazines.
ABSTRACT
,,
foam in oilwell treatments were the
In arriving at a mekhod for pre-
calculation of friction loss, wellhead
dieting friction losses iillaminar,
pressure, and the resulting density and
carrying capacities during treatment.
transitional, and turbulent. flow re-
gimes for flowing foam, it was found
Lockhart and Martinelli,l devel-
Reynolds numbers and Fanning friction
factors could be calculated with ef-
oped the most popular emperical corre-
lation for predicting friction losses
fective foam viscosity, actual foam
density, average velocity, and true
of flowing two-phase fluids wi~hin
horizontal pipes. This method was
pipe diameter.
Further, it was found
the relationship between Reynolds
sufficiently accurate only for pressure
losses within pipes smaller than 2-
number and Fanning friction factor for
inches in diameter.2
Other emperical
foam was identical to that of single-
3 o5 based on mechanical
orrelations I
phase fluids.
The test data were taken
energy or momentum losses proved in-
under controlled flow in capillary tubes
sufficient also.
and l-1/4-inch and 2-3/8-inch oilfield
seamless tubing.
Friction losses of
foam within 2-7/8-inch tubing and
Bertuzzi, Tek, and Poettmanns6
4-1/2-inch and 5-1/2-inch casing dur-
energy dissipation function which has
ing fracture treatments have been
extensive oilfield use fox predicting
pressure losses within horizontal pipes
accurately predicted using the proposed
for non-uniformly mixed, two-phase
method.
.
fluids was found not reliable for foam.
INTRODUCTION
Failures of these emperical two-
Foam has been used for several
phase correlations for the prediction
of preswme losses for foam flow may be
years as a wellborec lean-out and drill-
lodged in the theoretical developments
ing fluid and to a very limited extent
by Einst,ein7
and HatschekB* .
They
as a fluid loss, diverting, and frac-
turingfluid.
Applications have been
point o~t foam should be treated as a
limited because the behavior of the
single-phase fluid with viscosities
foam could not be confidently pre-
significantly greater than either phase
Mitchelll used these theories to ex-
dicted.
Major problems when using
perimentally find viscosities of foam.
References and illustrations at end.of.
paper.
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DETERMINATION OF IJUKINAR, TURBULENT, AND TRANSITIONAL
-
~nn~ n~nv.~
nnrnm rnn7 r
nccwe
TM DT17i~CS
SPE 4885
L
z u
-L ~, J-lWwv L- ~ ~. K . I. J =* -U
Hel
also showed foam approximately
bubbles are no longer spheres and re-
exhibits Bingham plastic behavior and
form to parallelepipeds during flow.
presented plastic viscosities and yield
strength data.
Einsteins theory is valid for
foam qua+ities less than 0.52. The
Krug and Mitchells12 modifica-
derivation of his two-phase viscosity
tions in existing single-phase fluid-
is based on an energy balance and the
flow equations account for both the following assumptions;
compressibility of the gas within the
foam and the resulting changes in the 1.
Solid spherical particles
viscosity. The resultant equations
are suspended in a homo-
were applicable to the circulation of
foam within a wellbore; however, these
geneous fluid.
equations were limited to laminar flow
2.
All particles are homoge-
and lacked emperical justification.
neous, weightless, and
have identical volumes
The foregoing theories and empir-
and diameters.
icism and additional field and labora-
tory work have been combined to devise
3.
Spacing of the particles
a method for predicting the type of
is uniform and the parti-
regime in which the foam is flowing
cles do not touch.
and a method for calculating the pres-
sure losses of foam flowing within 4. There is no slip at the
horizontal o: vertical pipes.
surface of the particles.
EFFECTIVE VISCOSITY OF FOAM
Einsteins equation for the vis-
cosity of foam is
Foam in this study is a homogene-
ous mixture of air or nitrogen, fresh
water; and a surface-active agent.
llf
The gas phase exists as microscopic
=pl l.orE2.5rT p).. . . 2
?
gas-bubbles suspended in the water and
surfactant solution.
In practice these
Hatscheks theories explain foam
bubbles may occupy between 10 and 95
viscosity for the bubole-interference
percent of the total foam volume.
and bubble-deformation quality ranges.
The interference between the spherical
Foam quality is the ratio of gas
bubbles in foams with qualities between
volume to the total foam volume.
0.52 and 0.74 requires additional work
be applied to initiate and maintain
flow.
This additional work accounts fo
J . . . . . . . . .
T,P Vf
(1)
h+gh foam apparent viscosity. Hatscheks
viscosity for bubble-interference foam
is
Since the gas is compressible, tempera-
ture and pressure of the foam must be
lJf
specified.
= P1 1.0 + 4.5 rT,p) . . . . (
Figure 1 was developed from the
His next argument was the deforma-
theories of Einstein and Hatschek and
tion of the bubbles within foams which
laboratory measurements by Mitchell.
have,qualities above 0.74 causes the
Three separate theories, each with a
geometric shape of the bubbles to pro-
physical model dependent on quality,
teed from spheres to dodecahedra and
were required to fully determine and finally to parallelepipeds, and this
understand foam viscosity.
bubble configuration is the only one
which can flow in laminae.
The vis-
The spherical gas-bubbles in foams
cosity of the foam caused by the shear
with qualities between 0.00 and 0.52
of the fluid between the parallelepipeds
are uniformly dispersed in the liquid
gas bubbles is
and do not contact other bubbles. Flow
is Newtonian.
At 0.52 quality the
spherical bubbles are packed loosely
in a cubic arrangement and begin to
Pf =
Ml+ . . . . . . . (
interfere by contacting each other
T,P
during flow.
Above 0.74 quality the
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,
)?4885 R. E. BLAUER. B. J.
MITI
Mitchell showed that foam behaves
approximately as a Bingham plastic
fluid in fully developed laminar flow.
His shear stress-shear rate relation-
ship for
foams
with shear rates above
20,000 see-l is linear for any quality.
The relationship for foam flowing with
a shear rate below 20,000 see-l can be
linearized by subtracting the apparent
yield strength.
His shear stress-shear
rate equation for Bingham Plastic Foam
is
(T-ry)=llp@ . . . . . . . . ..(5)
Figure 1 shows Mitchells plastic
viscosity and Figure 2 shows the ap-
parent yield strength. He showed
experimentally determined plastic
viscosities agree well with Einsteins
and Hatscheks theories.
Additional work showed ar.effec-
tive viscosity which combines plastic
viscosity and yield point is signifi-
cantly more reliable than plastic
viscosity alone in the determination
of Fanning friction factors. The use
of effective viscosity of Bingham
plastic fluids is technically precise;
however, it is interesting to note
that most Bingham plastic fluids en-
countered in the oilfield have small
yield strengths compared with their
plastic viscosities. Thus, the ef-
fective viscosity and the Bingham
plastic viscosity are of similar magni-
tude and are commonly interchanged.
However, foams high yield strength
requires the use of effective viscosity.
The effective viscosity in this
study is developed with the Hagen-
Poiseuille law and the B.ckingham-
Reiner equation.
The Buckingham-Reiner equation
for laminar flow of
fluids within pipes
Q=
n AP D 9
128 Bp LG
Bingham plastic
is
[1-:>
+*(>)Kl
. . . . .
O.**
(6]
The Hagen-Poiseuille law for
Newtonian
laminar
flow within pipes is
n AP Dq gc
Q
. {7;
~28peL .
IELL, C. A. KOHLHAAS
For the Newtonian turbulent flow
relationships to be valid for a Bingha
plastic fluid, the flow rate in equa-
tion 6 must equal the flow rate in
equation 7. A practical solution of
this equality shows the effective vis-
cosity of the Bingham plastic foam is
gc~D
Ye
= ~p + ~vy . . . . . .
.
Figure 3 illustrates the effectiv
viscosity of foam based on equation 8.
EXPERIMENTAL APPARATUS AND PROCEDURE
Osborne Reynolds experiments in
which streams of dye were injected int
water flowing within pipes showed the
frictional pressure losses were propor
tional to average fluid velocity when
the dye remained as a stream. This fl
regime was called laminar or stream-
line flow.
When mixing of the dye and
water occurred the pressure losses wer
approximately proportional to the squa
of average fluid velocity. Due to the
appearance of the dye mixing with wate
the flow regime was called turbulent.
Therefore, the flow regimes of a fluid
within pipes can be determined by
measuring pressure losses and flow rat
and comparing their functional relatio
ships.
To generat~ these functional re-
lationships for foam two horizontal
capillary tubes and two horizontal
strings of tubing were used in flow
tests.
The large diameter difference
of the tubes and tubing assured any
friction loss correlation developed
would have wide application.
The tube
and tubing also
served to
produce full
developedlaminar and turbulent flow.
regimes, respectively.
Pressures at each end of either
the tubes or tubing, flowing tempera-
ture, gas volume, and liquid weight
were measured for each flow test. A
mass balance, the real gas law, and
derived equations were used to calcu-
1 late foam-volumetric flow rate, qualit
foam average velocity, shear rate, sh
stressl and effective viscosity.
The capillary tubes used were ea
24-inches long with internal diameters
of 0.0483-inch and 0.0924-inch.
The
foam for these tests consisted of an
aqueous solution of 1 Adofoarn BF-1
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FOAM FLOW FRICTIO
9 commercial anionj.c surface active
agent, and nitrogen.
Two calibrated
temperature-compensated gauges recorded
flowing pressures. The foam was flowed
into a plastiic separator which was con-
nected to a wet-test meter.
The separa-
tor was weighed to determine the mass
of liquid which was flowed during a
test.
The effect of quality changes on
&he foam properties were kept to a mini-
num by keeping the tubes discharge
pressure above 500 psi.
Calculated
velocities were less than 20 feet per
second. Figure 4 is a diagram of the
capillary tube apparatus.
The large diameter tubes were made
of eight joints of new J-55 tubing. The
l-1/4-inch 2.40-pound tubing was 194.75
feet long with an internal diameter of
1.380-inch.
The 2-3/8-inch, 4.7-pound
tubing was 181.32 feet long with an
internal diameter of 1.995-inch. An
aqueous solution of 1 Dowell F58B,
an anionic surface-actice agent, was
pumped through the tubing strings at
rates as high as 7 BPM.
Nitrogen was
combined with the water at rates to
3000 SCFM.
This produced foam volu-
metric rates to 34.5 BPM.
A water
storage tank was gauged before and
after each test to determine water mass
and volume rates.
Pump displacements
of liquified nitrogen were used to
determine nitrogen gas volumetric rate.
Pressure losses at the extreme ends of
the tubing during the laminar flow tests
were measured by a single water mano-
meter inclined at 85 from vertical.
Turbulent flow pressure losses were
recorded by six differential pressure
transducers which were connected across
each joint of tubing.
Pressures at eacl
end of the tubing strings Were measured
by two calibrated laboratory gauges.
Test pressures to 3000 psi were used.
Figures 5 and 6 show the tubing appar-
atus.
EXPERIMENTAL CORRELATIONS
Mitchell showed that average foam
flowing velocity and quality can be ac-
curately determined using a mass bal-
ance and the real gas law.
The methods
used to calculate flowing properties in
tubing were only slightly different
from the capillary tube. Equations for
the capillary tube are presented in the
following discussion.
Since the flow tube is in a state
of dynamic equilibrium, the mass enter-
ing the tube as gas or liquid must leav
the tube,
DETERMINATION OF LAMINAR, TURBULENT, AND TRANSITIONAL
LOSSES IN PIPES
SPE 48
M~n=Mout . . . . . . . . . . (9
in which
M~n = Mil + M
+ M
lg gg
+M;lo . (1
M
out
= lC ld
+ lg + Wt + g
gd
. (1
The mass of liquid in the separator
is
s Wlc. , . . .
* (
lC = gc
Applying the real gas law and Dalto
law of partial pressures, the mass of
gas and water vapor flowing through th
wet-test meter is
M
Wt
=M
w
+ lm
(Pm -
Pv)
Vmm
+
v 1 m
RT
9 9
Z1 R T1
The mass of water vapor and
placed from the separator is
(
gas dis
(Pa -
Pv) v~c fiq
M
Vs
gd= Z9RT9
+
v lC 1
. (
lRTl
Total measured flowing mass of liq
and gas is
lt =
gt
thus ,
L+
Pv Vmnl
1 gc zl R T1
,-
(Pa
- Pv) Vmm
RT
9 9
(Pa -
Pv) Vlc Mg
R Tg
.*
9
(
(
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FM *002
m. a.
Pun
UQJS m. u .
A.LL&b-uu.
b.
n.
-U
A.L.IA-u
The volume of gas entering into
Average shear rate and shear rate
solution is,negligible.
Equations 15
at the wall of the tube for a Bingham
and 16 are valid for large diameter
plastic fluid is
tubing if the mass and volume are re-
placed by mass rate and volumetric rate.
@a=@w= * (Zz
Volumes of saturated gas flowing
at the tube pressure and temperature
are determined by the perfect gas law.
Shear stress at the wall is
The average pressure of the flow tube
can be used only if the differential
pressure is small in comparison. Data
.~-
T
4L
. (
points with .high differential pressure
w
Y
and low average pressure were deleted
for the final analysis.
The Reynolds number and Fanning
friction factor are
The volume of saturated gas in the
tube is
vf D of
R=~
v
=V
Mgt R t
Sg
gt
vlv=-
Mg (Pt-pv) Zg
f
AP D
= L ~f Vf2
.*9**
*
tpv
+~
(Pt-Pv) (17)
Plots of the shear stress-shear
rate relationships are given in Figure
7 through 9.
Foams of constant qualit
The volume of liquid in the tube
show two distinct flow regimes.
Below
is
critical shear rates for each pipe
diameter and foam quality the slope of
. lt
the shear stress-shear rate function
It pl
Iv
(18
) 1.0, indicating larninarBingham plasti
flow.
Above this critical shear rate
the slope is approximately 2.0 and the
foam is in turbulent flow. Calculated
True foam quality is
slopes of the turbulent shear stress-
shear rate relationship are between 1
Vs
and 2.01.
T,P= V +V;
(19)
Sg
l?hecritical shear rate for foam
corresponds with a critical Reynolds
number between 2000 and 2500.
Figure
True foam density is
10 is a conventional single-phase flu
Moody diagram with the foam data suPe
lt + C3t
imposed.
The algebraic standard devia
Pf=v
,.
(20) tion of friction factors calculated w
gt Vlt
this data compared with the Colebrook
method normally used for single-phase
fluids is 0.05 for the laminar flow
Average foam-flowing velocity in
regime and 0.188 for the turbulent fl
the flow tube is
regime.
vfv .***.***
APPLICATION OF METHOD FOR PREDICTION
(21) OF FRICTIONAL LOSSES IN PIPES
EXAMPLE CALCULATIONS
The effective foam viscosity, tru
e
density and pipe diameter, and average
1.
Laminar Flow
foam velocity is used to calculate
shear stress at the wall of the pipel
Determine. the pressure-loss
average shear rate, Reynolds
number ?
gradient of 0.80 quality foam
and Fanning friction factor.
flowing within a l.,00-inch
diameter pipe at 1.0 BPM.
)
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DETERMINATION OF LAMINAR, TURBULENT, AND TRANSITIONAL
6
FOAM FLOW FRICTION LOSSES IN PIPES
SPE 4885
.
1. a
Average foam velocity,
f
= 17.16 fps
b.
From equation 8, effec-
tive viscosity, ue =
29 CPS
c. Foam density may be ap-
proximated by neglecting
gas in solution and the
vapor pressure of water.
Approximate foam density,
Pf
= Pl(l-r)
= 1.67 PP9
d. From equation 24, Reynolds
=917 NR < 2000,
umber, NR
therefore, flow is laminar.
e. From equation 6, laminar
pressure-loss gradient,
AP
= 0.590 psi/ft.
AL
2. Turbulent Flow
Determine pressure-loss gradient
for the laminar flow example with
a flow rate of 5.0 BPM.
a.
Average foam velocity,
f
= 85.8 fJX3
b.
Effective foam viscosity,
Pe
= 10.9 Cps
c. Approximate foam density,
Pf = 1.67 ppg
d.
Reynolds number, NR =
1.21 x 104
NR < 2000
Therefore flow
is turbulent
Other wells have been fractured
through 4-1/2-inch and 5-1/2-inch casin
and the calculated wellhead pressures
have been within 2 of the actual pres
sures during treatment.
CONCLUSIONS
1. Foam behaves as a single-phase
Bingham plastic fluid.
2. Effective viscosity of foam
must be used for calculating
pressure losses for flowing
foam.
3. Friction losses for foam may
be determined as for a single-
phase fluid using conventional
Reynolds number and Moody
diagram.
4. Friction losses in oilfield
pipes can be accurately pre-
dicted for any combination of
foam flow rate and pipe size.
NOMENCLATURE
D
=
inside diameter of tube, in
f
= Fanning friction factor
9C
= gravitational constant, 32.2
L=
tube length, ft
M= mass at atmospheric conditions
Lbm
M =
mass at flowing conditions, Lb
E= molecular weight
R
= Reynolds number
e. From Figure 12 Fanning
friction factor, f = .0075
P = pressure, psia
f.
From equation 25, turbulent
AP =
pressure drop between ends of
pressure-loss gradient,
tube, psid
Q=
foam quality
P
= 3.57 psi/ft
AL
FIELD AiPLICATION I R= universal gas constantl
Wellhead pressure during a fracture
10.732
(PSIA)(FT5)
treatment using foam flowing at 14 BPM
(Lb-mole) (OR)
in 2-7/8-inch tubing at 7000 feet was
predicted using the proposed method.
T
= temperature, R
~he total calculated-wellhead pressure
was 5250 psi and measured pressure was
5350 psi.
t
= time, sec
v=
volume, ft3
v
=
velocity, ft/sec
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-. - . ---
M,uAaw xi*\, . u . L,AAALAAUALU* L. n.
nw44.uA-L2
t
w weight, Lbf
GREEK SYMBOLS
z
=
compressibility factor r
T,P
= foam quality at specified
temperature and pressure
SUBSCRIPTS
4
shear rate, sec
-1
=
a
=
average tube
pf =
true foam density, Lbm/ft3
e = effective
T
= shear stress, Lbf/ft2
f = foam
Y
= yield strength, Lbf/ft2
9
= gaseous phase
B
=
viscosity, CP
gd
= gas displaced from separator
ACKNOWLEDGEMENTS
99
= gas-phase in gaseous state
The authors wish to thank Dowell
gl
= gas-phase absorbed into the
and NOWSCO for their assistance in the
liquid phase tubing tests.
gm = saturated gas measured by
REFERENCES
wet-test meter
1.
Lockhart, R. W. and Martinelli, R. C
gt = gas flowing in tube Proposed Correlations of Data for
Isothermal Two-Phase, Two Component
in = entering tube Flow in Pipes, Chem. Eng. Procir.,
1949
v.
45
p.
39
1 = liquid
2. Baker, O.: Experiences with Two-
lC =. liquid in the separator Phase Pipelines, paper presented
before the joint meeting of Cana-
ld =
liquid-phase vapor displaced
dian Natural Gas Processing Assn.
from separator and.the Natural Gasoline Assn. of
America, Calgary, Alberta, Sept. 15
lg =
liquid-phase in g..:~us state
1960.
11 = , liquid-phase in liquid state 3* Dukler, A. E., et al: Frictional
Pressure Drop in Two-Phase Flow:
lm =
liquid-phase vapor measured by
B-An Approach Through Similarity
wet-test meter
Analysis, AICHE J., 1964, v. 10,
p. 44-51.
lt = liquid at flowing conditions
4.
Hughmark, G. A. :
Hold-Up In Gas-
m= wet-test meter
Liquid Flow, Chem. Eng. Progr.?
1962, V. 58, p. 62-65.
out = existing tube
5. Eaton, Ben A., et al:
The Pre-
P = Bingham plastic diction of Flow Patterns, Liquid
Hold-Up and Pressure Losses Occur-
Sg = saturated gas
ring During Continuous Two-Phase
Flow in Horizontal Pipelines, g.
t = tube Petrol. Technol. , June 1967, p. 315-
328.
v
=
vapor
6.
Bertuzzi, A. F., Tek, M. R., and
Vs = saturated gas displaced from Poettmann, F. H.:
Simultaneous
separator
Flow of Liquid and Gas Through
Horizontal Pipe, Trans. AIME,
w=
wall 1956, V.
207 p. 17-24.
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DETERMINATION OF IAMINAR, TURBULENT, AND TRANSITIONAL
8
7.
Einstein, Albert:
Eine Neve
Bestimmung Der Molekuldimensionenl;
Annalen Der Physik, 1906, v. 19?
SFR 4, p. 289.
8.
Hatschek, Emil:
Die Viskositat
Der Di.spersoide. 1. Suspensoide~
Kolloid 2., 1910, v. 7, p. 301-
304.
9.
:
Die Viskositat Der
Dispersoide.
11 Die Emulsionen
Uno Emulsoide,
Kolloid Z.,
1910
V
8 p.
34-39.
IL
o
.--- ---
}
f
5
0
L
0.0
TN PIPlw
SPE 4885
10.
11.
12.
Mitchell, B. J.: Viscosity of Foam
Ph.D. thesis, Univ. Okla., 1969.
.
Test Data Fill
Theory Gap & Using Foam as a
Drilling Fluid,
Oil and Gas J.,
Sept. 6, 1971, p. 96-100.
Krug, J., Mitchell, .B. J.: Charts
Help Find Volume, Pressure Needed
for Foam Drilling, Oil and Gas J.,
Feb. 7, 1972, p. 61-64.
. -
~ e... ..- -
-~
R //
i
-r --~ .
.-
.
I
.
---- ..._j____--.-- - -- -
I
I
I
DISPERSION
llNTERFEf{E~CE
I
I
1/
1
-. -
i /
I
--1
I
I
I
, 3
lDEFOW\TION-
1
I
I
I
L
>
0.4
0.6
OB
I.0
QUALITY
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Fig.2
- Yield stressoffoam.
1oo-
90
In 80
- .- -
8
: 70
G
o
v
U)
60
-. .
s
z 50
u
g
40 ~
LLJ
>
g 30
.
w
u
20 :
-. -.-
10
- . -.
0
--i___-. .,..
0.6
0.2
0.4
QU
--. ..
.
--
-. .-
,
.-.
1
:
~H~AR
RAf~
s~~-1
1
.--..-.. .
.-+
c).6
79
ALITY
Fig. 3 -
Effective viscosity of foan.
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Fig. 4
- Capillary tube apparatus.
100bbl
AQUEOUS
SOLUTION
NITROGEN
TANK
O-3QG0 ~CF
CHECK VALVE
--.
m
CEMENT SLURRY
CHECK
PUMP, O-78PM
5000psi PRESSURE
TRANSDUCER
@
::::Epsi
TO RECORDER
OILF~~L~_Iu.fj_ ~
ADJUSTABLE
.. ---
- ,- ~-
CHOKE
5000 psi,500psi-
TO RECORDER
DIFFERENTIAL
PRESSURE
TRANSDUCERS
Fig. 5 -
Turbulent flow regime tubing apparatus.
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11/12
~
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6 -
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Fig. 10 -
Floody diagram for foam.
Recommended