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



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

.



.

.

“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.




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



.

.

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)).



L


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 .



.

“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


K

M

Q

R

v


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




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.



.

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

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345 50

345.55 345.92 347.58

348.28

680.30 682.26

684.45 684.77

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583.64 583.48

584.87

592.64 6 9.09

743.44

745.36

747.67

747.99

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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



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Documents

SPE-7690 McDonald a.E. Approximate Solution for Flow on Non-newtonian Power Law Fluids Through Porous Media