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Analyzing Injectivity of Polymer Solutions With the Hall Plot
R.S. Buell, SPE, Chevron U.S.A.; H. Kazeml, SPE, Marathon Oil Co.;
and F.H. Po.ttmann, SPE Colorado School of Mines '
Summa..,. The Hall plot was originally used to analyze
water-injection wells. This paper demonstrates that the Hall plot
can also be used to analyze injection of polymer solutions. In
particular, it is possible to determine the in-situ and residual
resistance factors ?f a polymer solution fro~ the Hall plot. The
analysis methods developed are used to examine two field injection
tests and one hypothet-Ical example. The analytical results are
verified with a reservoir simulator.
Introduction Polymer floods, micellarlpolymer floods, and
injectivity- or productivity-profIle-modification treatments are
the most common applications of polymer solutions. The
interpretation of injection pressures and rates associated with
polymer solution injection is important to the efficient
application of the solutions. The Hall plot l-3 is a useful tool
for evaluating performance of injection wells.
The Hall plot was originally developed for single-phase,
steady-state, radial flow of Newtonian liquids. Since the advent of
poly-mer and micellar solutions for EOR, it has also been applied
to the injection of these solutions. Moffitt and Menzie4 used the
Hall plot to evaluate injection of polymer solutions but did not
verify the validity of the Hall plot for this application. This
paper verifies the validity of the Hall plot for evaluating polymer
solution injection.
Because of the complex nature of polymer solution flow through
porous media, exact analytical solutions are generally not
possi-ble. However, some relatively simple approximate analytical
so-lutions can be developed. To verify the analytical solutions for
polymer solution injection, a two-phase, radial, numerical
reser-voir simulator was developed. 2 The simulator is designed to
con-sider the more important phenomena and effects that occur when
polyacrylamide or polysaccharide polymer solutions are injected
into porous media. The simulator has the following characteristics:
slightly compressible flow, two-phase flow, non-Newtonian
rheol-ogy, adsorption/retention with permeability reduction,
concentra-tion effects, skin, and wellbore storage. It was used to
history match two field injectivity data sets.
Development of the Hall Plot The Hall I plot was originally
proposed to analyze the performance of waterflood injection wells.
Hall simply used Darcy's law for single-phase, steady-state,
Newtonian flow of a well centered in a circular reservoir:
q ........................ (1) 14I.2Bwl'w[ln(relr w) +s]
Hall integrated both sides with respect to time to obtain
Wi
Separating the integral of Eq. 2, Hall then rearranged to
obtain
141.2Bwl'w[ln(r elr w) +s] Wi+i'Pedt . ........... (3)
kkrwh
The relation between surface and bottomhole pressures for
steady-state vertical flow is given by
pwf=ptrtlpf+pgD . ............................... (4) Copyright
1990 Society of Petroleum Engineers
SPE Reservoir Engineering, February 1990
Hall substituted Eq. 4 into Eq. 3 to arrive at
141.2Bwl'w[ln(r elr w) +s] --------Wi +J(Pe + tlprpgD)dt.
kkrwh
.................................... (5)
Hall simply dropped the second term on the right side ofEq. 5
and plotted the integral of wellhead pressures with respect to time
vs. cumulative injection, which came to be known as the "Hall
plot." By plotting in this format, Hall observed that if an
injection well was stimulated, the slope decreased, and if a well
was damaged, the slope increased. While Hall's conclusions
regarding changes in slope are valid, the second term on the right
side of Eq. 5 is often not negligible in comparison with the other
terms and there-fore usually cannot be dropped.
In industry applications, the Hall integrals i'Pifdt and i'Pwfdt
fre-quently are used. The slopes calculated from these integrals
should not be used for quantitative calculations unless a
correction proce-dure is applied. Fig. 1 is a Hall plot based on
the data for Well A, where the integral fPwfdt has been plotted vs.
cumulative in-jection. Several changes in slope can be seen on the
plot, but there has been no change in transmissibility or skin. The
changes in slope are caused by changes in rate, which occur because
the integral fPedt has been neglected. Fig. 2 is a Hall plot based
on data for Well C. The three most common forms of the Hall
integral have been plotted for the same data. For each integration
method, the slopes of the curves are quite different.
Injection data must be plotted in the form of Eq. 2 to make
valid quantitative calculations; i.e., cumulative injection should
be plot-ted vs. f(Pwf-Pe)dt. The slope of the Hall plot from Eq. 2
is then given by
1412Bwl'w[ln(r elr w) +s] ..................... (6)
kkrwh
Eq. 6 will not be appropriate when multiple fluid banks with
sig-nificantly different properties exist in the reservoir.
Advantage. and DI.advantage. The Hall plot is a steady-state
analysis method, whereas falloff tests, injection tests, and
type-curve analysis are transient methods. Tran-sient pressure
analysis methods determine the reservoir properties at essentially
one point in time. The Hall plot is a continuous mon-itoring
method; i.e., reservoir properties are measured over a period of
weeks and months. The Hall plot, therefore, can help identify
changes in injection characteristics that occur over an extended
period.
Hall's method has several advantages. Integrating the pressure
data with the Hall integral [f(Pwf-Pe)dt] has a smoothing effect on
the data. Data acquisition for the Hall plot is inexpensive
be-cause only the recording of cumulative injection and surface
pres-sures is required. Surface pressures must be converted to
bottomhole pressures (BHP's), correcting for hydrostatic head and
friction loss-
41
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M~ 350 bbl/d - 100 'r Iwfd! / I I VV, -~-~- ' I I/V I I r ! 1~
_~/f[';~_~ ~- =1=== I ~ I _--+- I I
----!----- I I I I I ' -,--,--+~, I --.----,,--,-+-,-,---~ ,
,
0 ~OOO 4000 6000 8000 10000 ie-ODD HODO 16000 CUMULATIVE
INJECTION (bbl)
Fig. 1-Comparlson of Hall Integration methods, Well A, P. =
1,000 psi.
es. Injection and falloff tests usually require running gauges
on wire-line to depth, which is an additional expense.
The greatest disadvantage of the Hall plot is that the skin, s,
and transmissibility, khl/Jo' are combined in the slope. It is
possible to determine one of these if the other is known, but the
determination of both skin and transmissibility is not possible
with the Hall plot. To use the Hall plot effectively, running
falloff or injection tests periodically is still necessary to
determine the individual values of transmissibility and skin.
Quantitative Analysis Newtonian Fluids. In a mature waterflood,
the transmissibility usually will not change significantly with
time; therefore, any change in the slope of the Hall plot will be a
result of skin effects. Assum-ing no change in transmissibility,
the new skin can be calculated as follows for water injection:
kkrwh s2 =sI - (mHI -mH2), .................... (7)
141.2Bw/Jow
where subscript 1 denotes the old slope and skin and subscript 2
the new slope and skin. This relationship can be useful in
recog-nizing formation damage or fracturing.
When a waterflood begins, two-phase flow will exist in the
near-wellbore region. As the water moves away from the wellbore,
water and oil banks form if the oil saturation is large enough. A
simpli-fied method to analyze this situation is to apply Darcy's
law in a series manner. Because the oil displacement is governed by
the Buckley-Leverett5 equation, the saturations and relative
permea-bilities are not constant within each fluid bank; however,
for sim-plicity, they can be assumed to be constant within each
bank. The accuracy of the results is not significantly compromised
with this assumption. The slope of the Hall plot for a water and
oil bank is given by
(water bank) (oil bank)
.................................... (8)
The interface between the oil and water banks is rbl. The
inter-face of the oil and water banks can be estimated with Eq. 9,
which results from the Buckley-Leverett equation in radial
coordinates. 6
r1I = 5.615Wi (a/w ) +ra . ....................... (9) q,7rh asw
F
The quantity (a/wlaSw)F is the derivative of the fractional-flow
curve at the flood front. The water saturation and the derivative
of the fractional-flow curve at the flood front are determined
with
42
.... & I'w,d' / Go ~ .~ y ~----Og I.~ ,/ .~ // ~ f[pw'p.~,
a. o /' .... g ....12 ~ ---- ~r-Co a:o /
------
CI:il w'" 1#
-------
I PHdt .,.0 ~ zg -:il ~ V :18 C:il X" ~ ~
... 10DO 2000 3000 '000 5000 6000 7000 8000 CUMULATIVE INJECTION
(Barrels)
Fig. 2-Comparlson of Hall Integration methods, Well C.
Welge's7 method. As the oil bank is pushed away from the
well-bore, the water-bank term will dominate owing to the
logarithmic nature of Eq. 8.
Non-Newtonian Fluids. The analysis methods for non-Newtonian
fluids are similar to the methods developed in the previous
section, except permeability reduction must be considered. The
apparent viscosity of the non-Newtonian fluids is taken to be a
constant within each fluid bank. Eq. 10 is for an injection
sequence of polymer and then water. The reservoir is assumed to be
initially oil-saturated. Three fluid banks will be created: oil,
polymer, and water.
[ /JowBw[ln(rb2Ir w) +s] /JopBw In(rbI lrb2)
mH=141.2 +--'------hkakrw hkakrp
(water bank) (polymer bank)
+ /JooBo In(relrbl) I ............................... (10) hkkro
j
(oil bank)
Eq. 10 can be rewritten with just one absolute permeability and
one aqueous-phase viscosity after the introduction of resistance
fac-tor, Rj' and residual resistance factor, Rrj' which are defined
below.
water mobility (kkrw)//Jow Rj = = ................ (11)
polymer mobility (kakrp)//Jop absolute permeability before
polymer k
and Rrj= .... (12) absolute permeability after polymer ka
Because residual resistance factor and resistance factor are
useful in the evaluation of polymer performance, Eq. 10 has been
rewrit-ten with these definitions:
(water bank) (polymer bank)
+ /JooBo In(relrbl) I ............................... (13) hkkro
j
(oil bank)
In Eqs. 10 and 13, apparent viscosity is assumed to be constant
through space; i.e., the non-Newtonian rheology is ignored. The
variation of apparent viscosity in space can be taken into account
by applying Darcy's law, with the definition of effective viscosity
used in the same series manner as used to develop Eq. 10. For the
simple case of a power-law fluid bank occupying the whole
reser-
SPE Reservoir Engineering, February 1990
- 0 >:~ II ~ :ag e. .. III r Wat" I-
-
CUMULATIVE INJECTION (barrels)
Fig. 5-Hall plot, rate-contrOlled history match, Well B.
and resistance factor estimates provided by Milton et ai. were
used as a first approximation and were then adjusted to obtain the
best possible history match. All pressure data were recorded at the
sur-face. For history-matching purposes, all surface pressures were
cor-rected to BHP with friction included.
The residual resistance factor was estimated to be 1.05 on the
basis of transient well testing. To obtain the best possible match
with the field data, reducing the polymer solution viscosity and
in-creasing the residual resistance factor was necessary. The best
his-tory match was obtained with the rheology given in Table I and
a residual resistance factor of 1.33.
The rate-controlled (Neumann) boundary condition and match-ing
pressures were used to obtain Figs. 5 and 6. The reservoir was
modeled with single-phase flow because Milton et ai. considered the
reservoir to be at ROS owing to extensive waterflooding. The Hall
plot generated from history matching can now be used to ap-ply the
analytical procedures developed. The Hall plot shown in Fig. 5 has
three distinct sections: water, polymer, and water injec-tion.
Applying Darcy's law in a series manner yields an equation for each
section. For this example, four unknowns will be assumed: kkrw,
Rrf, Rfl, and Rj2. All other parameters are assumed to be known.
Rfl is the resistance factor of the polymer bank while poly-mer
solution is being injected. Rj2 is the resistance factor of the
polymer bank after polymer injection has stopped and water
injec-tion has begun. Rfl and Rj2 usually will not be equal because
of shear thinning of the polymer solution and because
adsorption/reten-tion will reduce polymer concentrations as the
polymer slug prop-agates through the reservoir. The slope of the
Hall plot for the first water-injection period is given by Eq. 6.
The slope of the Hall plot for the polymer-injection period is
given by
(polymer bank) (water bank)
................................... (15) An oil bank is assumed
not to form in Eq. 15 because of the exten-sive waterflooding. The
slope of the Hall plot for water injection following polymer
injection is given by
[ R,frwBw[ln(rb2Ir w) +s] Rj2JlwBw In(rbl /rb2)
mH3 = 141.2 + hkkrw hkkrw
(water bank) (polymer bank)
+ JlwBWh~::lrbl) J ............................. (16) (water
bank)
Eq. 8 is used to calculate a permeability to water of 90.9 md
with a skin of7.2. The permeability to water used by the reservoir
simu-lator is 91.0 md. With one unknown eliminated, there are now
three 44
i
2~. J:LLt-l~!wj-~ --0 10 20 30 40 50 60 70 80 90 100
TIME (days)
Fig. 6-BHP vs. time, rate-controlled history match, Well B.
unknowns and only two equations. To solve for all the unknowns,
another equation is necessary. The residual resistance factor can
be estimated by taking the ratio of the Hall plot slopes for water
injection before and after polymer injection. The contributions
from the banks farther away from the wellbore become smaller and
smaller as injection continues when Eq. 16 is used. As injection
proceeds, the Hall plot slope after polymer injection, mH3, will
ap-proach the value given by
14I.2RrfJlwBw[ln(r e1r w) +s] mH3= ................. (17)
hkkrw (water bank)
The latest straight-line portion of water injection following
poly-mer is used to estimate mH3, so the influence of the other
banks will be at a minimum. Eqs. 6 and 17 can be combi:1ed to solve
for Rrf. The residual resistance factor can then be estimated
with
Rrf=mH3 /mH!' .................................. (18) The
residual resistance factor is computed to be 1.42. The in-
situ residual resistance factor calculated in the simulator is
1.33. Eqs. 15 and 16 can be used to find the two remaining
unknowns, Rfl and Rj2. Because this problem was simulated with
one-phase flow, all displacement processes are miscible.
Piston-like displace-ment, therefore, is assumed to occur. The
location of the interface between banks can be calculated by
volumetric calculations account-ing for polymer adsorption. Using
Eq. 15 to calculate the resistance factor at the end of the
polymer-injection period results in a resistance factor of 2.22 for
the polymer bank. It can be seen that when numerical values are
substituted into Eq. 15, the water bank is less important than the
polymer bank, supporting the conclusion that the bank in contact
with the wellbore will dominate. The water bank away from the
wellbore can be assumed to be negligible, and the single-fluid-bank
assumption can be used. When this single-fluid-bank assumption is
used, the resistance factor is calculated to be 2.06. The
single-fluid-bank assumption will underestimate the resistance
factor because, in this case, the polymer bank is assumed to extend
to the drainage radius when it actually extends only some fraction
of the drainage radius. The simulator provides pressures for each
cell. The average resistance factor for the poly-mer bank is
calculated with the simulator results to be 2. 17.
The slope of the water injection following polymer is used to
solve Eq. 16. Substituting numerical values into Eq. 16 results in
a cal-culated resistance factor of 4.19 for the polymer bank away
from the wellbore. The average resistance factor of the polymer
bank is calculated with the simulator results to be 3.51. The
resistance factor calculated for the polymer bank away from the
wellbore can be significantly in error. The reason for the larger
errors is that the pressure drop caused by the polymer bank is
small away from the wellbore. A small error in the determination of
the slope re-sults in an increased error in the calculated
resistance factor. Ta-ble 2 compares the approximate analytical
methods with the simulator results.
SPE Reservoir Engineering, February 1990
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Well C. Well C was used to evaluate the injectivity of micellar
and polymer solutions. The daily injection data consisted of mice
1-lar solution injection followed by polymer solution injection.
The polymer solution was then displaced with water. The reservoir
data, fluid properties, daily injection history, and polymer
parameters are given in Refs. 2 and 8.
The injection pressures were controlled to prevent reservoir
frac-turing or fracture parting. A falloff test was run before
micellar solution injection began. The test indicated a water
mobility of 27 md/cp [27 md/mPa' s] and a skin of -1.14. The skin
and permea-bility calculated from the falloff test were used in the
simulator for history matching. This reservoir had been
waterflooded extensive-ly before the injection testing. The
reservoir was estimated to be at ROS; therefore, the history match
was done with only single-phase flow.
The history match was conducted in the same manner as for Well
B. The best history match was obtained by adjusting rheology and
resistance factors. All other parameters were taken from available
data and assumed to be correct. The Carreau model was used to
approximate the rheology of the polymer and micellar solutions.
History matching was done with both the rate- and
pressure-controlled boundary conditions.
The Neumann (rate-controlled) boundary condition was used for
Figs. 7 and 8. Table 3 gives the rheology of the polymer solution
used to obtain the best match. The amount of permeability
reduc-tion was much larger than for Well B. The residual resistance
fac-tor used in the best match was 11. 1. The rate-controlled
boundary condition was used for cases with various levels of
adsorption/reten-tion. As with Well B, the results were found to be
relatively insen-sitive to the amount of adsorption/retention.
The Hall plot for Well C was analyzed in the same manner as that
for Well B. There were no initial water-injection data for this
well; however, the slope before polymer and micellar solution
in-jection can be calculated because the skin and transmissibility
are known. The Hall plot slope for water injection is calculated
with Eq. 6 to be 1.68 (psi-D)/STB [0.0387 (kPa' d)/stock-tank m3]
from the falloff testing data before polymer injection. The slope
for the late water-injection period is 16.50 (psi-D)/STB [0.380
(kPa'd)/stock-tank m3]. Eq. 18 can now be used to estimate a
residual resistance factor of 9.80, which is reasonably close to
the simulator value of 11. 10. Table 4 compares the analytical
solutions with the simulator results.
Eqs. 15 and 16 can now be used to calculate the average
resistance factor of the polymer/micellar solution banks. At the
end of poly-mer/micellar solution injection, the average resistance
factor is cal-culated with Eq. 15 to be 29.53. The average
resistance factor can also be approximated by assuming a single
bank that extends to the drainage radius, which results in an
average resistance factor of20.95. The average resistance factor
calculated with the simula-tor is 29.40.
After water injection, the polymer/micellar bank is between 48
and 95 ft [15 and 29 m]. The simulator results were used to
calcu-late an average resistance factor of 14.7. A resistance
factor of 16.6 was calculated with Eq. 16.
The apparent viscosity for Well C is given as a function of
radial distance in Ref. 8. The relative change in the apparent
viscosity within the polymer bank is small. Small changes in
apparent vis-
TABLE 2-COMPARISON OF ANALYTICAL METHOD WITH SIMULATOR RESULTS,
WELL B
Analytical Methods, Analytical Methods, Parameter Multiple Banks
Single Bank kkrw 90.9 90.9 R" 1.42 1.42 R 11 2.22 2.06 Rf2 4.13
~ 0 0 '" .~ V :g /' ." 0 , 0 ., 0 0; .... /v Q. :g Simulatot
Match :!~ / .. Q. Field Cat ... 0 :I g .s ~. If' :=; Mic_Uar SO'UL
:::l g I/) 0
-I V "'. N 7-polymer t---.--- ---- -Wat&r-1---
Simulator 91.0
1.33 2.17 3.51
v
f---~
00 1000 ZOOO 3000 4000 5000 6000 7000 BODO
CUMMULATIVE INJECTION (barrels,
Fig. 7-Hall plot, rate-controlled history match, Well C.
40 50 60 TIME (days,
Fig. 8-BHP vs. time, rate-controlled history match, Well C.
TABLE 3-HISTORY MATCH, APPARENT VISCOSITY AS A FUNCTION OF
INTERSTITIAL VELOCITY, WELL C
Concentration Interstitial Velocity (ft/O) (ppm) 0.01 0.10 1.00
10.00 100.0 1,000.0
--
0.0 1.00 1.00 1.00 1.00 1.00 1.00 2,445.0 3.00 2.85 2.01 1.55
1.39 1.33 2,800.0 5.00 4.72 3.07 2.19 1.87 1.76 3,100.0 6.99 6.59
4.19 2.91 2.45 2.29 3,430.0 8.49 7.83 4.30 2.83 2.44 2.34 5,000.0
19.98 18.70 12.55 10.51 10.10 10.02
'Savins 13 shear-rate/velocity relation used.
SPE Reservoir Engineering, February 1990 45
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TABLE 4-COMPARISON OF ANALYTICAL METHOD WITH SIMULATOR RESULTS,
WELL C
Analytical Methods, Analytical Methods, Parameter Multiple Banks
Single Bank Rtf 9.80 9.80 R 11 29.53 20.95 R'2 16.63
Simulator 11.10 29.40 14.73
cosity within the polymer bank also occurred with Well B.
Accord-ing to the history matching of Wells B and C, the
non-Newtonian flow effect is relatively small. Apparent viscosity
stays relatively constant within the polymer bank.
Conclusions Using a numerical reservoir simulator, 2 we have
demonstrated that quantitative analysis can be performed on the
Hall plot when non-Newtonian solutions are injected. The best
method for analyzing the Hall plot would be to use a reservoir
simulator like that devel-oped for this study. For the practicing
engineer, however, a simu-lator may not be available. Therefore,
two approximate analysis methods for the Hall plot have been
developed to estimate permea-bilities, resistance factors, and
residual resistance factors.
1. The Hall plot can be used to estimate the performance
charac-teristics of injected polymer and micellar/polymer
solutions.
2. The multibank analysis method will yield more accurate
an-swers than the single-bank method. When the fluid bank in
contact with the wellbore has moved out a substantial distance, the
single-fluid-bank analysis method can be used with acceptable
accuracy.
3. The non-Newtonian rheology effect is small because the change
in the in-situ apparent viscosity of the polymer solutions through
space is relatively small.
4. The amount of permeability reduction has a significant effect
on the Hall plot and simulator results.
5. The transient flow period has little effect on the Hall plot
be-cause, in most field situations, the transient period rarely
lasts more than a few days. Because most Hall plot data are
recorded daily, it is usually not possible to observe the transient
flow period on the Hall plot.
Nomenclature B = FVF, dimensionless D = true vertical hole
depth, ft [m] fw = fractional flow of water, dimensionless g =
gravity constant, 32.2 ft/sec2 [9.81 m/s2] h = formation thickness,
ft [m] k = absolute permeability, md
ka = absolute permeability after polymer, md kro = relative
permeability to oil, dimensionless krp = relative permeability to
polymer, dimensionless krw = relative permeability to water,
dimensionless mH = Hall plot slope, (psia-D)/STB [(kPa
'd)/stock-tank m3]
n = Carreau and power-law-fluid slope parameter,
dimensionless
Pe = pressure at external drainage radius, psia [kPa] Ptf =
surface tubing injection pressure, psia [kPa]
Pwf = bottomhole injection pressure, psia [kPa] t::..pf =
pressure loss caused by friction, psi [kPa]
q = rate, BID [m3/d] rbl = Bank I radius, ft [m] rb2 = Bank 2
radius, ft [m] re = external drainage radius, ft [m] r w = wellbore
radius, ft [m] Rf = resistance factor, dimensionless
Rrf = residual resistance factor, dimensionless s = skin,
dimensionless S = saturation, dimensionless
46
t = time, days ilt = change in time, days Wi = cumulative
injection, bbl
"y = shear rate, seconds-\ A = Carreau rheological parameter,
seconds /./, = viscosity, cp [mPa's]
/'/'e = effective viscosity, cp [mPa's] /'/'0 = viscosity at
zero shear, cp [mPa's]
/./,00 = viscosity at infinite shear, cp [mPa 's] p = fluid
density, Ibm/ft3 [kg/m3]
cf> = porosity, dimensionless
Subscripts F = flood front o = oil P = polymer w = water wf =
injection pressure at r w
Acknowledgments We thank the Colorado School of Mines Petroleum
Engineering Dept., Chevron U.S.A., and Marathon Oil Co. for their
support in presenting and preparing this paper.
References 1. Hall, H.N.: "How To Analyze Waterflood Injection
Well Perform-
ance," World Oil (Oct. 1963) 128-30. 2. Buell, R.S.: "Analyzing
Injectivity of Non-Newtonian Fluids: An Ap-
plication of the Hall Plot," MS thesis, Colorado School of
Mines, Gold-en, CO (1986).
3. DeMarco, M.: "Simplified Method PintJoints Injection Well
Problems," World Oil (1968) 95-100.
4. Moffitt, P.D. and Menzie, D.E.: "Well Injection Tests of
Non-Newtonian Fluids, " paper SPE 7177 presented at the 1978 SPE
Rocky Mountain Regional Meeting, Cody, WY, May 17-19.
5. Buckley, S.E. and Leverett, M.C.: "Mechanism of Fluid
Displace-ment in Sands," Trans., AIME (1942) 146, 107-16.
6. Collins, R.E.: Flow of Fluids Through Porous Marerials,
Petroleum Publishing Co., Tulsa, OK (1961) 149.
7. Welge, H.J.: "Simplified Method for Computing Oil Recoveries
by Gas or Water Drive," Trans., AIME (1952) 91, 195-98.
8. Buell, R.S., Kazemi, H., and Poettmann, F .H.: "Analyzing
Injectivi-ty of Polymer Solutions with the Hall Plot, " paper SPE
16963 presented at the 1987 SPE Annual Technical Conference and
Exhibition, Dallas, Sept. 27-30.
9. Blair, P.M. and Weinaug, C.P.: "Solution of Two-Phase Flow
Prob-lems Using Implicit Difference Equations," SPEl (Dec. 1969)
417-24; Trans., AIME, 246.
10. Carreau, J.P.: "Rheological Equations from Molecular Network
The-ories," PhD dissertation, U. of Wisconsin, Madison, WI
(1968).
11. Vogel, P. and Pusch, G.: "Some Aspects of the Injectivity of
Non-Newtonian Fluids in Porous Media," Enhanced Oil Recovery, F.
John Fayers (ed.), Elsevier Science Publishers, New York City
(1981) 179-96.
12. Milton, H.W. Jr., Argabright, P.A., and Gogarty, W.B.: "EOR
Pros-pect Evaluation Using Field-Manufactured Polymer," paper SPE
11720 presented at the 1983 SPE California Regional Meeting,
Ventura. March 23-25.
13. Savins, J.G.: "Non-Newtonian Flow Through Porous Media,"
Ind. & Eng. Chern. (Oct. 1969) 61, No. 10, 18-47.
51 Metric Conversion Factors bbl x 1.589 873 E-Ol
ft x 3.048* E-Ol psi x 6.894 757 E+OO
'Conversion factor is exact. SPERE
Original SPE manuscript received for review Sept. 27. 1987.
Paper accepted for publica-tion Nov. 2, 1989. Revised manuscript
received June 22, 1989. Paper (SPE 16963) first presented at the
1987 SPE Annual Technical Conference and Exhibition held in Dallas,
Sept. 27-30.
SPE Reservoir Engineering, February 1990