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7/23/2019 SPE-7690 McDonald a.E. Approximate Solution for Flow on Non-newtonian Power Law Fluids Through Porous Media
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SPE7690
APPROXI MATEOLUTI ONSORFLOWONNON-NEWTONI AN
POWERLAWFLUI DSTHROUGHOROUSMEDI A
by Alvis E. McDonald, Member SPE-AIME,
Mobil Research & Development Corp.
Copyright
1979.American
n s t i t u t e o f M in in g , M e t s l lu r g lr x l , s n d P e t r o le w mE ng in e e r s , I n c .
T hl s pa p er m sp rm en t t i a t ~e 19 7 9So t i ew o t Pt i r ol e um Ef l g i nm rs o f A IM Ef i t i Sy o os i um o n Rm ew o lr 3 mu i at i o , l h e l di n De n ve r . Co l o ra d o. F eDr ua w l . 2 . 1 9 79 . T h em a te r l al i s su o i ed t o
c o r re c t i on by t he a u th o r. P er mi s si o n t oc o py i s r t i t ri ~ ed t o an a bs t m~ o f no t mo r e t n a n3 0 0 w o rd s. W ri 1 8 62 0 0N Ce n tr a l Ex Ry . , D a l la a, T x. 7 5 20 6 .
ABSTRACT
1. SUMMARY AND CONCLUSIONS
Individual well modeling with r-z geometry is
often done with a small number of radial
A small number of radial cells (e.g., 6 or 7)
cells (e.g., 6 or 7).
This usually leads to
.1s usually adequate for individual well
small space truncation error for black oil
models using r-z geometry. Our results indl-
systems.
ior power Iaw fluids a finer grid
cate that solution errors are small for such
1s required. An example problem Is readily
models when the fluid is a black oil.
But
solved using 20 radial cells.
Coarser defl-
errors are “ not small for power law fluids.
n l~ lo n
leads to unacceptable truncation error.
The problem discussed here requires 20 radial
cells to reduce truncation errors to accep-
A method is shcwn to Improve finite dlffe-ence
table levels.
Improved accuracy in the space
accuracy by space differencing the time derlv-
dependent approximation can be obtaindd by a
ative. An analytical solution to the differ-
suitable space differencing of dp/dt. With
ence equation is developed, and used to vall-
such im~rnvsd accuracy it becomes tenable to
date approximate numerical solutions.
use a Ieduced number of radial cells to model
power law behavior.
Three figures, five tables illustrate results.
ii
iNTRODUCTION
.—
Recent papersl’2 developed the following
equation describing radial flow of a non-
Newtonian power law fluid in a porous medium:
References at end Gf paper.
1
n
~g
&+~~=nBr
(1)
a r2
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APPROXIMATE SOLUTIONS FOR FLOW OF NON-NEWTONIAN
7690
POWER LAW FLUIDS THROUGH POROUS MEDIA 17
---
where
P=
= azphiz
B = constant involving Injection rate and
formation injectivity
Pt
= sp/at
n = exponent in the shear rate-viscosity
Equations (4) and (5) will be solved by finite
relation
p= pressure in atm (kilo Pascal)
difference methods. Results are to be com-
r = radius in cm (meter)
pared with the analytic solution given in Ref-
t = time in sec (see)
erence 1.
Quantities given in parentheses are to be used
iii.
FiRST METHOD-AN EXACT TiME, DiSCRETE
when working with the international system of
SPACE FORMULATION (METHOD OF LINES)
units. Engineering units are used In tables
A set of ordinary differential equations can
and figures.
be obtained by differencing (4) w.r.t. the
initial and boundary conditions are given by:
space variable x, retaining a continuous time
representation.
For N nodes with equal incre-
ments D = Xi+l
- xi for each node pair (i,
Plt=fj -PO
(2*1)
i+l), (4) can be difference as follows:
[ 1
j~P2-PI) + ~ (PI-P3) - (3@D)Q =
‘%lr=rw
=Q
r r.re
=0
(2.2)
where
blp~
(6.1)
Q=& ,
[
1
nd where
~ (~-*) (Pi+~-Pi) + (&+~) (Pi-~-Pi) =
h = thickness in cm (meter)
hip;
(i = 2,3,...,N)
(6.2)
k = permeability in Darcy (micrometer)
q
= injection rate in cm3/sec (m3/see)
Equation (6.2) is obtained by using the second-
rr, = drainage radius in cm (meter)
order correct approximations
r; = wellbore radius in cm (meter)
Pi+l
- Pi-1
P; = 2D
+O(D2)
p = viscosity in cp (milli-Pascal-see)
in Equations (1) and (2) apply the logarithmic
transformation
Pi+~ - 2Pi + Pi-l
P? =
+ 0(02)
xagnr
(3)
D2
to obtain
At the end point i=l these expressions are
xx
‘px=bpt~
(4)
not valid.
Instead we use the second-order-
np
correct approxi mations:
p I @o
= Po
I
[
y” $ 8P2 - ~pl
- P3
1
6Dp~
+ 0(02),
Px]Wgn rw=Q
(5)
then use boundary conditions (S) to r@Plac@
pxl~gn re= 0
p; by Q.
A Matrix Representation
where
Equations (6) comprise a set of coupled ordi-
(2 + +)x
nary differential equations.
To simpli fy
b = b(x) _ nBe
later treatment multiply both sides of these
equations by 2D2 and use the substitutions
px= sp/ax
.
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.
.
“7690
A. E. McDonald
177
fi = 2D2bl
I
and where ~ and ; are the vectors
c = -Zo(sn + D Q
CN = (2n - D)pO
Then Equations (6) become:
-7npl +8np2 - np~+cl
(2n + D)Pl - knp2 + (2n - D)P3
= f2p;
Multiplying (7) by F-l we obtain
+t
+
(2n + D) pN-~
P
=A~+b,
- 4npN-~ + (2n - D) PN=
f~qP;-l
I
where
(2n + D)pN.l - knpN + cN
=fp
N; A = F-lM ,
+
In matrix notation these equations are written
b = F-l: ,
as:
The solution to Equation (8) is given In
F~t=M;+2,
(7)
Appendix A.
where F and M are the matrices
f2
.
‘N-1
‘N
d
[
-7n
2n D
8n
-4n
-n
2n-D
.
2n+D
-4n
2n+D
(8)
Dfscusslon
Since the method is exact with respect to t
the only errors arise from dlscretlzing the
space variable x.
Consider the problem given
on Page 5 of Reference 1$ namely:
n
=2
B = 5.228 X 10-4
u = 3.2726 Cp
k a ).1 Darcy
q = 368.o2 cm3/sec (inJection)
h = 914,4 cm
rw = 7.62 cm
r
e = 15000 cm (Reference 1 uses = for re)
Comparison with the exact pressure risel,
Pw - po, at r= rw, is shown in Table 1.
The comparison is excellent for D = .096, and
fair for D= .399.
For larger values of D,
f~nite difference accuracy decays rapidly.
The number of nodes corresponding to each
value of D is shown in parenthesis in the
table.
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APPROXIMATE SOLUTIONS FOR FLOW OF NON-NEWTONIAN
7690
POWER IAW FLUIDS TNROUGH POROUS MEDIA
---
The 7 node cas~ is of special interest, since
this 1s a typic~l number
model coning problems.
Figure 1 compares the 7,
solutions with the exact
entire drainage radius.
of nodes used to
IO, and 20 Rode
solution for the
On this plot the
80 node solution is Indistinguishable from
the exact.
Evidently 20 nodes are satisfac-
tory for the duration of the well test. But
f e w e r nodes lead to erroneous analysis, be-
cause of the magnification of truncation
error with Increasing time.
The reader is
reminded that ~ s ace truncation error Is
+
ncluded, since the so ut~on
i s exac~the
time variable, t.
It 1s tempting to conclude that well modeling
should never be done with fewer than 20 radial
nodes. But first let us repeat the above com-
parisons for a single phase, sl]ghtly-
compressible black oil system.
The problem
is as given for Figure 1, except that vis-
cosity is taken as constant (i.e., n = = in
Equation (l), and B becomes 1.4432
X
10-3.
See Reference 1 for computation of B).
Figure 2 shows no essential differences in
computed solution among the several grid spac-
ings, and all are reasonably accurate.
Iv.
SECOND METHOD
- A DISCRETE TIME,
DISCRETE SPACE FORMULATION
Discretizatlon in the time domain
done with the first-order-correct
tion
1s often
approxima-
(9)
where d Is a suitable small time increment,
and the remainder term R iS - ~ Ptt + 0(d2)
At times higher-order approximations are war-
ranted. But we will see presently that space
truncation errors are dominant for the present
problem, and the above representation for pt
is quite satisfactory.
Time truncation error is controlled by re-
quiring d to be smell enough that R IS neg-
ligible. In general one requires
Ip(t + d) -
p(t) < s , for suitable e. An
estimate for the magnitude of the remainder
term R is then given by
+’
2I’ t+d:: t P t-d l
v
1
~Lk@d)P(t) I + P(t) - p(t-d) I
___
I.e. ,
R2
so that
Itself
~
d,
the error term
n magnitude.
does not exceed pt
n many cases p is a
monotonic function with p(t +d) - p(t) ~
P(t) - P(t - d), and the error term Is severa
orders of magnitude smaller than pt.
In practice d is not held constant, but allow
to vary In such away that Ip(t+d)
This p~r~ ~~’
s as close as possible to c.
time step size to grow, while forcing the
pressure-change toierance to be met.
Now return to Equation (7) and replace the
t by differences of the form (9). As a
P\
notational convenience pl(t+d) will be simply
written as pl,
and pi(tl will be represented
by Pi. The result is
where
[1
P-:2
.
‘N
Multiplying the last result by F
“ d there
follows:
~ - ~ = F-ld(M$&) .
Thus we must solve the system
(1 - F-’dM); = ~+ F-’d&
If we let
nd
vi=~
Yi=y
the system to be solved Is written explicitly
as
G;=~,
(lo
where
‘F8::-4v@vN-yN-’l
i
L_L____
2vN+yN 1-4VN
J
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.
7fMl A. E. McDonald
1
r
‘1
- 2D(3n + D)Q/fl
1
‘2
.
‘N- 1
PN+(2n
- D)pO/fN
Discussion
For the problems solved earlier by the first
method, compare with the solution to (10).
Errors due to time dlscretization are com-
pared with the corresponding “ continuous
time” solutlon from Section Ill. Time step
size is chosen to restrict
where c ~ 0.1, 1, or 10 atmospheres (1.47,
14.7, and 147 psi).
Results are tabulated for 7 nodes in Table 2.
The error at 5 hours with e = 1.47 psi is only
0.15 psi, or about 0.017%. The error with
e = 14.7 is also small, namely 1.26, or about
0.14%. Comparing time discretization error
of 1.26 with space discretization error of
-66.99 (an 8% error), we see that the spatial
error IS dominant. The 147 psi change cri-
terion is sufficiently accurate for modeling
purposes. Similar conclusions hold for the
10 and 20 node cases, as shown in Tables 3
and 4.
v. THiRD METHOD
- AN iMPROVED CONTINUOUS TiME
DiSCRETE SPA CE FORMULATION
With uniform grid spacing D= xl+, - xi for
each i, define
Pi +
-
PI-1
~xP~ = 2D
Then, as is well known, Equation (4) can be
written as
Space-Differencing The Time Derivative
The following grouping of terms Is convenient:
The term inside the second brarket is just
blpit.
Thus
[
“’ xx’ i - ’ xdw’:=‘:p~‘0(
=bp
Ii”
(12)
Taylor’s series expansion for blp~ yields
dxx(bip;)
=~fpflxx+~P,JXXXX’0(’4)=
Then (12) can be written as:
Now neg ect terms involving D4, and also the
i
xxx
term ~pi
Note that this still leaves
a semnd order error term.
The result is
n
After using Equations (llJ, multiplying by
2D2, and applying the substitution
~2
*
91 =
Tbv
there follows:
(2n+D)pl-,- 4npl + (2n-D)pi+l ‘9i-lP~-l +
to
(13)
09\Pf ‘9i+.lPiW
(1 =2, 3,
.0,
N),and gN+l E O)
Proceeding In the same way one can show that
the second-order-correct relation which
n~glects ~ p~ is given by:
12
‘7np, + 8np2 - np3 - 2D(3n+D)Q =
1991Pf
- 8g2p; + g3p; s
(14)
0(D4) = blp; .
where Q= p; (see Equat on (5)).
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L
APPROXIMATE SOLUTIONS FOR FLOW OF NON-NEWTONIAN
7690
POWER LAW FLUIDS THROUGH POROUS MEDIA
180
To simplify the following discussion, replace
Equation (14) by (14)-(13), where in (13) i
1s set to 2.
The result is:
-($jn+D)p, + 12np2
- (3@Jp3
- 2P(3n+D)Q=
.
(15)
Matrix Representation
Equations (13) dnd (15) can be written in the
form
+t
Hp
= K~ +Z ,
(16)
where
[
18g,
-lt g2
9,
1092 93
.
H=
1
(17)
r-(9n+Dj 12n -(3n-D)
1
Zn+i)
-4n 2n%D
*
K=
.
.
2n+D -4n
2n-D
1
2n+D -4n
J
+
e=
[
‘2D(3n+D)Q
o
.
.
o
(2n-D)po
To solve as before, write (16) as
+t
p =A;+&
where
A= H-lK
~= H-l;
18
Then proceed as in Section iii to solve ordi-
nary differential system (18).
Discussion
We again treat the problemof Section iii,
For the improved method of this seution, tab-
ulated results are shown in Table 5.
cmn-
pared to Table ~, these results show Improved
accuracy for D - .399. Plotted results are
shown in Figure 3.
Results are acceptable tor
all but the 7 node (D = 1.264) case.
The method of this section appears more
complicated than that of Section iii. This
is true for simple problems such as the one
presented here, where it is feasible to cal-
culate an exact solution to the different
equations. in general, however, the problem
is too complicated to solve analytically.
Usually it is too difficult to solve as a
system of ordinary differential equations.
instead one discretlzes the time domain, us-
ing finite differences. For such cases the
present formulation adds very little effort
to the overall calculation, while considerably
reducing space truncation error.
To use the
method we require a substitution such as the
one used in Equation (12) to write
Similar techniques have been employed by3
other writers. Kantorovich
and Krylov
proposed such a method for harmonic equations.
Laumbach4 approached the matter differently
but nevertheless concluded by space-differenc-
ing the right hand side, pt. Higher order
methods are treated by Swartz%md Ciment and
Leventha16$7 .
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.
“7690
A. E. McDonald
181
NOMENCLATURE
Upper Case Gothic
A auxiliary matrix, near Equation (8).
Redefined near Equation (18).
B constant involving inJection rate and form-
ation injectivity.
D node spacing (equal increments),
x coord inate.
F auxiliary matrix, near Equation (7).
G
auxiliary matrix, near Equation (10).
H
auxiliary matrix, near Equation (16).
K
M
Q
R
v
auxiliary matrix, near Equation (16).
auxiliary matrix, near Equations (7) and
(lo).
number of nodes in x direction.
shorthand for p(t). Used near Equaf:ion(lO)
vector containing pressures Pi on
dlscre
tized grid.
qp/2rkh .
truncation error, or remainder term for
time difference at Equation (9).
modal matrix, Appendix A.
Lower Case Gothic
bi
I
nBe(2+ xi
vector containing the bi.
auxiliary vector, near Equation (7).
d
‘i
9i
h
k
n
P
q
r.
spacing in time domain,
used
for time
difference.
auxiliary vector, near Equation (16).
2D2bi .
element of matrix H, near Equation (17).
thickness in cm (meter).
permeability in Darcy (micrometer*).
exponent in shear rate-viscosity relation.
pressure In atm, kilo Pascal, or
psi(Tables and Figures).
vector containing pressures pi on dls-
cretized grid.
inJection rate in cm3/sec (m3/see).
radius in cm, meter, or teet (figures),
t time in see, hours (Tables and Figures).
~ auxil~ary vector, Appendix A.
v~ element of matrix G, near Equation (10).
;’ auxiliary vector, Appendix A,
x in r, a transformed coordinate .
yi element of matrix G, near Equation (10).
Greek
d difference operator, as follows:
f’i+l-f’i-l
( f ? i=~
P{+l-2Pi +Pi-1
LSxxpi=
~2
s error control parameter.
A matrix of eigenvalues, Appendix A.
M viscosity in cp(milli-Pascal-see)
Subscripts
e external boundary.
w wellbore.
i ,N
node numbers.
Superscripts
t used with P-PZ = sp/at
x used with p -px = sp/ax
l’i+l-pi-l
used with & -dxpi = ~
xx used tiith p -pxx = a2p/ax2
Pi+l
‘2Pi+Pi-l
used with d
- dxxpi=
D2
.,
REFERENCES ‘
1.
Odeh, A. S.,
and Yang, H. T., “ Flow of Nona
Newtonian Power Law Fluids through Porous
Media,” SPE 7150, Society of ?etroleum
Engineers of AiME Fall Meeting, October 2-
4, 1978, Houston, Texas.
2.
ikoku, C. U., and Ramey, H. J., Jr.,
“ An investigation of Wellbore Storage and
Skin Effects during the Transient Flow of
Non-Newtonian Power-Law Fluids in Porous
MedIa,l’ SPE 7449, Society of Petroleum
Engineers of AiME Fall Meet ng, October 2-
4, 1978, Houston, Texas.
3. Kantorovich, L. V., and Krylov, V. i.,
Approximate Methods of Hiqher Analysis,
interscience Publishers, inc., (1958),
pp. 185-186.
I
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APPROXIMATE SOLUTIONS FOR FLOW OF NON-NEWTONIAN
7mn
mum IAW HIIIIM T .IRO1lGH
POROUS MEDI A
SPE 182
Wav
““- . . . . . . - - . . . . . . . . . .
- -
4. Laumbach, D. D.,
“ A High Accuracy, FInite-
Equation (A-1) is the required solution to
Difference Technique for Treating the
Equation [8) ~r (18).
Convection-Diffusion Equation,” SPEJ, V. 15,
No. 6, December 1975, pp. 517-531.
Reduction to Iractlcal Form
5. Swartz, B. K., “ The Construction of Finite
Difference Analogs of Some Finite Element
Soiution (A-1) is not computationally useful
Schemes,”
Mathematical Aspects of Finite
if one must evaluate e~t by power series.
Elements In Partial Diff
erential Equations
Rather, we transform to a space where the
(C. de Boor, Editor), Academic Press, New matrix corresponding to A is diagonal, after
York, 1974, pp. 279-312.
6. Cimant, M., and Leventhal, S. H., “ Higher
which the computations become trivial.
Order Compact implicit Schemes for the
Let V be a non-singular matrix and A a dia-
Wave Equat ion,
II Math. of Comp., V. 299
gonal matrix such that
No. 132, October 1975, pp. 985-994.
7: Ciment, M., and Leventhal, S. H., “ A Note
AV=VA.
(A-2)
on the Operator Compact implicit Method Then
for the Wave Equation,” Math. of Comp.,
V. 32, No. 141, January 1978, pp. 143-147.
V-’AV =A,
APPENDIX A
and A is the required diagonal matrix cor-
responding to A.
Once V and A have been de-
Analytic Solution of Ordinary Differential
termined,the probiem simplifies.
Let vectors
=
~andtibe defined by
A
To solve Equation (8) or (18) multiply both (A-3)
F=vt .
sides by the integrating factor e-AtA, and
group terms as follows:
Equation (8) becomes
‘At (@.- A;) = @-Atl,
e
dt
i.e.,
+ (VtiJ =AV~+V; ,
or equivalently}
&= “ -lAV:+; ,
‘i-
dt
$ e
‘At;) = e-At~ ,
i.e.,
The rul es of ordi nary di fferenti at i on apply,
I
even though A Is a matrix, p$ovide$ that
proper relat~ons
to vectors
p
and b are mai n-
The p;;cedure of the preceding section now
ta ined. We now integrate the last result
leads us to the analog of (A-1)
above:
+
u = eAt~~O + A-l~ - A-l: ,
(A-4)
t
I
t
+s~ds = -A-le+s+ t
where
e-A~lo =
blo ,
+
‘o
Sk”’;. .
0 -1
assuming that A
exists. Continuing the pro-
Eq~atfon A-1+ ,together Wth the f ; rSt Of
cess, and
using the convention that
(A-3), prescribes the final solut~or’.
i. e.,
‘t;o+A
“ b -A-’ .
=e
(A- 1)
As
in scal ar arithmetic
e
‘t is defined by
the infinite series eAt - i+At * A2 t2 +
3
Z-
A3$-+
. . . . where i is the identity matrix
.
with dimensions N x N.
f Note that we have used the commutative prop-
erty Ae
-At - e-At A
Cal cul at i on of V and A
The only difficulty in solving (A-2) lies in
determining V and A.
The reader undoubtedly
recognized Equation (A-2) as describing the
e$gensystem of A, where the diagonal of A
contains the elgenvalues of A, while the so-
called modal matrix V has the eigenvectors of
A for its columns. We have used the EISPAK
routines, as distributed by Argonne National
Laboratory, to compute V and A.
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Table 1
Comparison of Analytical with Exact Tlma-Olscreta
Spaca Dlffarence Solutions to Equation 1
The,
Flnlta Olffarenca Solutions
HOurS
Analytlc Pres ure? D = .096 D ~ .399 D = .843
0 = l.264—
Hours
Rise. DSI
(80 nodes) (20 nodas)
( 10 nodes)
(7 nodes) 0.1
.1
345.50
345.66
347.78
355.43
376.88 I
I
583.64
583.66 587.88
605.16
65 .50 2
2
679.60
679.60
684.77
708.19
761.15 3
3
742.22
742.22 747.99
771.00
817.48 4
4
789.83
789.83
196.09
8i8.77 859.12 5
Table 2
Olscrete Tlma Solutions for 7-Nods Grid (D = 1.264)
O I s cretized-T Ime Sol urlom
Exact Sol ut Ion
c = 147 Psi
c * 14.7 9s1
C = 1.47 DSI
~sina Eauation (7)
370.18
373.99
376.50 376.88
643.56
647.88
651.07
651.50
754.38
757.65
760.72
761.15
813.19
815.14
817.19 817.46
856.20
857.60 858.93
859.12
893.42
894.42
895.53
895.68
5
828.69
828.69 835.37 859.09 895.68
Tabla 3
Discrete Time SoluTlons for 10 Noda Grid (D = .843)
2
3
4
5
D I s cretl zad-T [me Sol ut Ions
Exact So I ut i on
s = 147 psi
c * 14.7 oai
e
* 1.47 3s1
Uslna EauaTlon (7)
350.58 353.08 355.13 355.43
600.06 602.34 604.84
605.16
702.90
705.26
707.83
708.19
766.53
768.45
770.70
771.00
814.60
816.53
818.50 818.77
855.11
856.80
858.83 859.09
Time,
Hours
9.1
1
2
3
4
.5
Tabla 4
Table 5
0 I screte Time Sol utlons for 20 Noda Gr Id (D = .399)
(hmpar i son of Anal y+ Ical with Exact Time-improved Discrete
Space SC I ut Ions for Several Grid Spacings
O I scretl zad-Time So I u tlons
Exact Sol ution
c = 147 Dsl
c = 14.7 DSl
9
= 1.47 gsl
Ualna Eouatlcm (7)
Finite Di fference Solutlons
Ana l yt I c P ressure
342.74
345.07
347.45
0 = .096 0 * .399
0 * .843
0 = 1.264
347.78
Hours Rise psl
.—
(80 nodes) (20 nodes) (10 nodes)
(7 odes)
583.08
585.24
587.56 587.68
.I
345 50
345.55 345.92 347.58
348.28
680.30 682.26
684.45 684.77
I
583.64 583.48
584.87
592.64 6 9.09
743.44
745.36
747.67
747.99
2
679.60
679.40 J581.30 69 I.1O
715.78
791.26
793.46
795.78
796.09
3
742.22
742.00 744.24
752.20
768.01
830.71 832.65
835.05 835.37 4
789.83 789.60
792.12
800.09 809.82
5
828.69 828.45 831.21
840,81
847.69
7/23/2019 SPE-7690 McDonald a.E. Approximate Solution for Flow on Non-newtonian Power Law Fluids Through Porous Media
http://slidepdf.com/reader/full/spe-7690-mcdonald-ae-approximate-solution-for-flow-on-non-newtonian-power 10/11
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http://slidepdf.com/reader/full/spe-7690-mcdonald-ae-approximate-solution-for-flow-on-non-newtonian-power 11/11
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