A Study of Inertial Effect in the Wellbore

8/6/2019 A Study of Inertial Effect in the Wellbore

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A STUDY OF INERTIAL EFFECT IN THE WELLBORE

IN PRESSURE TRANSIENT WELL TESTING

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOROF PHILOSOPHY

byKIYOSHI SHINOHARA

A p r i l 1980


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@ COPYRIGHT 1980by

K I Y O S H I S H I NOHAM

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Dedicated t o my parents, Kisaburo and Maki,t o my son, Daisuke,

and t o my wife, Kiyomi

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ACKNOWLEDGEMENT

The au thor w i shes t o exp res s h i s s i n ce r e app rec i a t i on t o Pro fesso r

H. J . Ramey, Jr., Department of Petroleum Engineering, f o r h i s guidance,

understanding, encouragement, and c r i t i c a l review of t h e manuscript as a

resea rch adviso r . The au thor is g r a t e f u l t o P r o fe s so r s W. E. Brigham,S . K. Sanyal, and R. N . Horne, Department of Petrole um Engine ering , f o r

serv ing on h i s reading commit tee .

The author i s a l s o i nd e bt ed t o the Nippon S te el Corporat ion, Stanford

Geothermal Program a t Stanfo rd Un ive r s i t y , and t he Ha l l i bu r ton O i l WellServices Company for f inancia l suppor t .

In add i t i on , t hanks are due to D r . M. Sengul, Marathon O i l Company,

fo r as s i s ta nc e wi th numer ical Laplace inver s ion . The scrupulous hel p in

ed i t i ng and typ ing of th e manuscrip t by M s . El i zabe th S . Luntzel and the

d r a f t i n g of t h e f i g u r e s by M s . Terri Ramey are grateful ly acknowledged.

F i n a l l y , t h e a u th o r wi sh es t o e x p r es s h i s sincere a p p r e c i a t i o n t o h i s

wife, Kiyomi, f o r he r con sta nt suppor t , encouragement , and understanding

dur ing t h i s work.

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ABSTRACT

I n a " slug t e s t ," a f i n i t e amount of l i q u id i s removed suddenly from

a s t a t i c w e l l , causing a per tur bat ion which can be used t o obt ai n informa-

t i o n on th e r e s e rv o i r . I d ea l ly , t h e we l lbo re l iq u id would s t a r t t o move

up the w e l l t o rep la ce the removed l iq u i d , reach a maximum v e l o c i t y , then

b eg in t o d e c e l e r a t e as t h e l i q u i d level a ppr oa che s t h e i n i t i a l s t a t i c

l i q u i d level. This is a useful method as a short- t ime w e l l tes t . The

so lu t io n fo r th i s p ro b lem i s old, and t he problem has been inves t ig ated by

many people. However, th e so lu ti on s av ai la bl e thus f a r do not r igo rous ly

in c lu d e th e in e r t i a l e f f ec t o f movement of th e l iq u id i n th e we l lbo re , and

do n o t e x p l a in p r e ss u r e o s c i l l a t i o n s a n d l i q u i d level f l u c tu a t i on s i n t h e

we l lbo re completely. I n e r t i a d e lay s th e s tar t of movement of t h e we ll bor e

l i q u i d , and momentum can cause i t t o eject ab o v e th e f in a l s t a t i c level .

O s c i l l a t i o n s i n p r es s u r e and l i q u i d l e v e l may r e s u l t .

A c tu a ll y , t h i s ki nd of problem can be involved i n t h e s t a r t of l i q-

uid p roduct ion fo r a l l w e l l s , and i n th e sh u t- in of high- ra te water i n j ec-

t i o n w e l l s . It i s a l s o i m po rt an t t o d r i l l s t e m t e s t i n g o f l i q u i d prod uc in g

format ions .

I n t h i s s tu dy , a new and complete solut ion for th is problem i s ob-

t a in ed and th e e f f ec t s of important parameters are i n v e s t i g a t e d . A f i n i t e

d i f f e r e n c e s o l u t i o n w a s a l so p repared and used t o s tudy th i s class of

prob lem. The ava i l ab l e f ie l d da ta were in t e rp re t ed and d iscussed. A new

s o l u t i o n f o r f l ow pe r io d d a t a a n a l y s i s f o r a closed chamber test w a s pre-

pared as an extens ion of th e genera l s lug test s o l u t i o n .

v i


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TABLE OF CONTENTS

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . vABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ViTABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . xi1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 12 . SLUG TEST ANALYSIS . . . . . . . . . . . . . . . . . . . . . . 5

2-1 Mathematical Formulation . . . . . . . . . . . . . . . . . 52-2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . 1 4

2-2-1 Solutions in Laplace Space . . . . . . . . . . . . 1 5

2-2-2 Analytical Approach to Laplace Transform Inversion. 17

2-2-2.1 Trial for Analytical Inversion . . . . . . 182-2-2. 2 Early Time Solutions . . . . . . . . . . . 202-2-2. 3 Late Time Solutions . . . . . . . . . . . 22

2-2-3 Numerical Laplace Transform Inversion . . . . . . . 23

2-2-3.1 Stehfest Method . . . . . . . . . . . . . 23

2-2-3. 2 Veillon Method . . . . . . . . . . . . . . 242-2-3. 3 Albrecht-Honig Method . . . . . . . . . . 25

2-2-3.4 Comparison of Results . . . . . . . . . . 252-2-4 Results and Discussion . . . . . . . . . . . . . . 32

2-2-4. 1 Effect of aon Solutions . . . . . . . . . 322-2-4. 2 Effect of CD on Solutions . . . . . . . . 552-2-4.3 Effect of s on Solutions . . . . . . . . . 59

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2-2-4.4 Radius of Investigation . . . . . . . . . 69

2-2-4.5 Batch Injections . . . . . . . . . . . . 7 9

2-2-4.6 Application of Solutions . . . . . . . . 7 9

2-3 Field Data Examples . . . . . . . . . . . . . . . . . . . 1 0 72-3-1 Example 1 (Typical DST Flow Period Data) . . . . . 1 0 7

2-3-2 Example 2 (Understanding Field Data) . . . . . . . 1122-3-3 Example 3 (Comparison of Results of Slug Test

Analysis and Buildup Test Analysis) . . . . . . 116

2-3-4 Example 4 (Oscillation Case) . . . . . . . . . . . 123

3

.ANALYSIS OF FLOW PERIOD DATA IN CLOSED CHAMBER TESTS

. . . . .129

3-1 Mathematical Formulation . . . . . . . . . . . . . . . . 129

3-2 Results and Discussion . . . . . . . . . . . . . . . . . 1334. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 1385. NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . 1416. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 1457 . APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A . Investigation of the Analytical Laplace TransformInversion of Eq . 40 . . . . . . . . . . . . . . . . . . 148

B. Separation of Real and Imaginary Parts of Eq . 48 andEq.49 . . . . . . . . . . . . . . . . . . . . . . . . . 153

B-1 For Early Times . . . . . . . . . . . . . . . . . . . 154B-2 For Late Times . . . . . . . . . . . . . . . . . . . 155

C. Field Data Example . . . . . . . . . . . . . . . . . . . . 157C-1 Well. Reservoir Data. and Measured Pressures . . . . 1 5 7C-2 Pressure Drop Caused by Friction . . . . . . . . . . 166

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D . Der ivat i on of F i n i t e D i f f erence So lu t i ons . . . . . . . . . 169D-1 Closed Chamber T e s t . . . . . . . . . . . . . . . . . 169D-2 S l u g T e s t . . . . . . . . . . . . . . . . . . . . . . 172D-3 Buildup Test . . . . . . . . . . . . . . . . . . . . . 173

E . Computer Program . . . . . . . . . . . . . . . . . . . . . 174

E-1 Stehfest Method . . . . . . . . . . . . . . . . . . . 175E-2 Albrecht-Honig Method . . . . . . . . . . . . . . . . 179

E-3 Computer Program t o Ca lcu late t he P res sur e Dist r ibu-t i on s In s id e The Reservoi r . . . . . . . . . . . . . 183

E- 4 Computer Program t o Si mu la te Closed Chamber Test.

Slug T e s t . and Buildup Test . . . . . . . . . . . . 189

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LIST OF TABLES

1. DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FORc = 103, s = 0, AND a2 = 103 BY VARIOUS METHODS . . . . . . . . 27

SMALL a2 VALUES WHEN cD = 103 AND s = o BY THE VEILLON METHOD . 29

cD = 103, s = 0 , AND 0.2 = io BY VARIOUS METHODS . . . . . . . . 30

D

2 . DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FOR

3. DIMENSIONLESS WELLBORE PRESSUTE VERSUS DIMENSIONLESS TIME FOR

4. DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FORcD = 103, s = 0, AND a2 = 108 BY VARIOUS METHODS. . . . . . . .

31

cD = 103 AND s = o . . . . . . . . . . . . . . . . . . . . . . . 38

TIME FOR c = 103 AND s = o . . . . . . . . . . . . . . . . . . 40

5. DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FOR

6. DIMENSIONLESS LIQUID LEVEL IN THE WELLBORE VERSUS DIMENSIONLESS

D

7. THE VALUE OF d AT WHICH CRITICAL DAMPING OCCURS . . . . . . . . . 548. DIMENSIONLESS WELLBORE PRESSURE VERSUS tD/CD FOR CDeZs = 10 . . 829. DIMENSIONLESS WELLBORE PRESSURE VERSUS tD/CD FOR CDe2s = 10 . . 84

4

5

10. CORRESPONDENCE BETWEEN THE SYMBOLS USED IN PETROLEUM ENGINEERINGAND THOSE IN GROUND WATER HYDROLOGY . . . . . . . . . . . . . . 1 2 6

11. ADJUSTED PROPERTIES OF WELL-AQUIFER SYSTEMS REPORTED BYVAN DERKAMP16. . . . . . . . . . . . . . . . . . . . . . . . . I 2 7

C-1 WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 1 . . . . . 157C-2 WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 2 . . . . . 1 5 9C-3 WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 3 . . . . . 160C-4 MEASURED LIQUID LEVEL (TAKEN FROM GRAPHS IN VAN DER KAMPI6) . . . 165

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LIST OF FIGURES

. . . . . . . . . . . . . . . . . .1 TYPICAL DRILL STEM TEST DATA 1 22. TYPICAL DRILL STEM TEST DATA 2 . . . . . . . . . . . . . . . . . 2

. . . . . . . . . . . . . . . . .SCHEMATIC DIAGRAM OF A SLUG TEST 64. DIMENSIONLESS LIQUID LEVEL IN THE WELLBORE VS DIMENSIONLESS

TIME FOR cD = 10 3 AND s = o BY NUMERICAL INTEGRATION OF EQ.4 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FORcD = 103 AND s = o . . . . . . . . . . . . . . . . . . . . . . 346. DIMENSIONLESS LIQUID LEVEL IN THE WELLBORE VS DIMENSIONLESS

DTIME FOR C = l o 3 AND s = 0 . . . . . . . . . . . . . . . . . 35

7. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FORcD = 1 0 3 AND s = o IN CARTESIAN COORDINATES . . . . . . . . .

TIME FOR CD = 103 AND s = 0 IN CARTESIAN COORDINATES . . . . .

S T O ~ G C O N S T ~ T - F O R TO . . . . . . . . . . . . . . . . . .

368. DIMENSIONLESS L I Q U I D LEVEL IN THE WELLBORE VS DIMENSIONLESS

379. LOG a AND LOG a VS LOGARITHM OF DIMENSIONLESS WELLBORE

4 310. . . . O R s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 4411. . . . O R s = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 4512. . . . O R s = 2 0 . . . . . . . . . . . . . . . . . . . . . . . . . 4613 . . . . O R s = 5 0 . . . . . . . . . . . . . . . . . . . . . . . . . 4714 . . . .FOR s = 100 . . . . . . . . . . . . . . . . . . . . . . . . . 4 8

15.LOG al AND LOG a2 VS SKIN FACTOR

. . . . . . . . . . . . . . . .4 9

16. LOG A AND LOG B VS SKIN FACTOR . . . . . . . . . . . . . . . . . 5017. EXPONENT OF DIMENSIONLESS WELLBORE STORAGE CONSTANT VS SKIN

FACTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118. LOG al AND LOG a2 VS SKIN FACTOR FOR CD = 1 . . . . . . . . . . 52

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19. EFFECT OF DIMENSIONLESS WELLBORE STORAGE CONSTANT ON DIMENSION-LESS WELLBORE PRESSURE FOR S = 0 . . . . . . . . . . . . . . . . 56

20. EFFECT OF DIMENSIONLESS WELLBORE STORAGE CONSTANT ON DIMENSION-LESS LIQUID LEVEL IN THE WELLBORE FOR S = 0

. . . . . . . . . .57

21. LOG t VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGE CONSTANT . 58Dl n22. LOG t VS SKIN FACTOR FOR CD = 10 . . . . . . . . . . . . . . 60D,I23. EFFECT OF SKIN FACTOR ON DIMENSIONLESS WELLBORE PRESSURE FOR

c = i o 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61D

24. EFFECT OF SKIN FACTOR ON DIMENSIONLESS LIQUID LEVEL IN THE WELL-BORE FOR CD = lo3 . . . . . . . . . . . . . . . . . . . . . . . 62

25. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR A

LARGE SKIN FACTOR ( s = l o o ) . . . . . . . . . . . . . . . . . . 6526. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR A

LARGE DIMENSIONLESS WELLBORE STORAGE CONSTANT (CD = . . . 663

27. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR CD = 10AND s = a = 0 AT EARLY TIMES . . . . . . . . . . . . . . . . . . 67

28. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR

= lo5 AND s = a = 0 AT EARLY TIMES . . . . . . . . . . . . . 68...FOR CD = 10 , s = 0, AND a2 = 10 AT EAIU,Y TIMES . . . . . . . 70PRESSURE DISTRIBUTION INSIDE THE RESERVOIR FOR CD = 10

cD3 7

29.

30.3AND

s = a = 0 WITH DIMENSIONLESS TIME . . . . . . . . . . . . . . . 7 1

31. EFFECT OF a ON PRESSURE DISTRIBUTION INSIDE THE RESERVOIR ATt = 102, FOR cD = 103 AND s = o . . . . . . . . . . . . . . . . 73D

3FOR CD = 10...AT tD -= 3x10 , A N D s = O . . . . . . . . . . . . . 7432.

33. ...AT t = 10 , FOR C = 10 AND s = 0 . . . . . . . . . . . . . . 7534. 76

35. ...AT t = 10 FOR CD = 10 AND s = 0 . . . . . . . . . . . . . . 7736. EFFECT OF SKIN FACTOR ON INVESTIGATION RADIUS . . . . . . . . . . 78

3

4 3

D D

D

D

4 3...AT t = 5x10 , FOR CD = 10 AND s = 0 . . . . . . . . . . . . .5 3

37. EFFECT OF DIMENSIONLESS WELLBORE STORAGE CONSTANT ON INVESTIGATION

RADIUS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8038. DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10 . . . . . 8639. DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe = 10 . . . . . 872s 2

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65. PRESSURE DISTRIBUTION INSIDE THE RESERVOIR BASED ON BUILDUP TESTA N A L Y S I S I N E X A M P L E 3 . . . . . . . . . . . . . . . . . . . . . 1 2 066. COMPARISON OF ACTUAL DATA AND CALCULATED BUILDUP RESULTS FOR

EXAMPLE3

. . . . . . . . . . . . . . . . . . . . . . . . . . .1 2 1

6 7 . LIQUID LEVEL I N THE WELLBORE FOR 2-C WELL . . . . . . . . . . . . 124

68 . LIQUID LEVEL I N THE WELLBORE FOR 9-A WELL . . . . . . . . . . . . 125

69. SCHEMATIC DIAGRAM OF A CLOSED CHAMBER TEST . . . . . . . . . . . . 130

7 0 . DIMENSIONLESS WELLBORE PRESSURE pGD VS DIMENSIONLESS TIME FORCLOSED CHAMBER TESTS WHEN cD = 103 AND s = o . . . . . . . . 134FOR CLOSED CHAMBER TESTS WHEN cD = 103, s = 0 , AND a = o . . . .71. DIMENSIONLESS LIQUID LEVEL I N THE WELLBORE VS DIMENSIONLESS TIME 135

A-1 INTEGRATION PATH . . . . . . . . . . . . . . . . . . . . . . . . . 1 48

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1. INTRODUCTION

Often i n d r i l l stem tes ts ( he r e ina f t e r a bbr e v ia te d as "DST"), flowpe r iod da ta are charac te r ized by a pr e ssu r e t r a c e which inc r e a se s wi th

i n c r e a s i n g t i m e , showing t h e a c cu mu la ti on of l i q u i d i n t h e d r i l l s t r i n g .

In some cases the pressure- t im e t r a c e i s l i n e a r a t th e beginning of t he

flow period , and th en becomes concave t o th e t i m e a x i s , s h o w i n g a n i n i t i a l

apparent cons tant f l owra te and then a de c r e a s ing f lowr a te . I n o th e r cases

the pressure- t i me t r a c e i s concave to the t i m e ax is f rom the beginning of

th e f low per iod , showing a decreas ing rate throughout t he f low per iod

(see F igs . 1 and 2 ) .

I n cases where the format ion pressure is t oo low t o l i f t a column of

t h e r e s er v o i r l i q u i d t o t h e s u r f ac e , t h e w e l l may ac tua l ly s top f lowing

be f o r e the DST tester va lve i s c losed . Th i s r e su l t s bec ause the head of

l i q u i d i n t h e d r i l l s t r i n g becomes e q u al t o t h e i n i t i a l f or ma ti on p re s su r e.

I n rare c a se s wi th h igh p r oduc t iv i ty f o r m at ions , t h e l i q u i d l e v e l i n t h e

w e ll b or e may i n i t i a l l y o s c i l l a t e ar ou nd t h e e v e n t u a l l y s t a b l e s t a t i c l e v e l .

I n any e ve nt , t h e i n i t i a l p o rt i o n of a DST flow period may be viewed

as a test i n which a column of l i q u i d whose head is eq ua l t o t h e i n i t i a l

formation pressure has been removed instantaneously.

cushion, the concept becomes more complicated, but i s s t i l l v a l i d .

c ons ide r tha t a head o f l i q u i d e q u i v a l e nt t o t h e d i f f e r e n c e i n p r e s s u re

betwe en i n i t i a l f o r m at ion p r e s su r e and the p r e s su r e a bove the tester va lve

has been removed.

I f t h e r e i s a l i q u i d

We

This kind of test is similar t o a w e l l response test c a l l e d a "slugtest" by F e r r i s a nd Knowlesl i n 1954. The word "slug" r e f e r s t o an

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-2-

FIG.

/,------

,/;I

1: TYPICAL DRILL STEM TEST DATA 1

FIG. 2: TYPICAL DRILL STEM TEST DATA 2


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-3-i n i t i a l volume o f l i q u i d removed f rom the wel lbore t o i n i t i a t e f low.Cooper e t a 1 . , 3 i n 19 67 , r e p o r te d t h e r e s u l t s o f a f i e l d test i n a s t a t i cwater w e l l from which a f l o a t w a s suddenly removed, giving the appearance

of t he ins tanta neous removal of a q u a n t i t y of water e q u a l t o t h a t d i s p l a c e d

by t he f l o a t . Ac tua l l y, t h e Cooper e t a l . da t a i n t e rp re t a t i on method w a s

based on s t ud ie s by J a e g e r 2 y 7 i n 1956. A f i e l d a p p l i c a t i o n i nv o lv i ng t h ecool ing of a batch of hot water w a s rep ort ed by Beck, Ja ege r, and New-s t ead

2i n 1956 .

4Maier presented an approximate an al ys i s of th e equivale n t DST problem

van Pool len and Weber,5 i n 1970, and Kohlhaas, i n 1972, appl iedi n 1970 .the Cooper e t a l . 3 s o l u t i o n t o DST f l ow per iod da t a ana lys i s .most c u r r e n t s t u d i e s r e f e r t o t h e s t u d y by J ae g e r i n 1956,7 which includeda s u r f a c e r e s i s t a n c e similar t o th e sk in ef fec t , mos t rec ent works do

not inclu de wel lbore damage ef fe c t s . A s o l u t i o n i n c l ud i n g t h e s k i n e f f e c t

was presented by Agarwal e t a l . OY1l i n 1970 and 1972, a l though th e use of

t h e s o l u t i o n s w a s no t demonstrated. Papadopulous e t a l . presented ex-

ten de d r e s u l t s f o r t h e z e r o s k i n e f f e c t case i n 1973.

Although

12

The most complete discussion of DST a p p l i c a t i o n s of t h e s l u g test so lu-

t i o n s w a s pre sen ted i n 1975 by Ramey et a l . I 3 Three new slug test type-curves were developed fo r t he ana lys i s o f f l ow per iod da t a . Th i s s t udy ,

which w a s r ev iewed i n t h e Earlougher14 monograph i n 1977, did inc lud e th ew e ll b o re s k i n e f f e c t .

The so lu t i o ns men tioned t hus f a r d id no t cons ide r t h e i n e r t i a o f l i q -

uid moving i n t he wel lbore . In deep, h igh produ ct iv i t y w e l l s , t h e i n e r t i a

o f t he l i qu id i n t he we l lbo re canno t be neg l ec ted . Sometimes t h i s e f f e c t

c a u se s o s c i l l a t i o n s o f t h e p r e s s u r e and t h e l i q u i d l e v e l i n t h e w e ll b or e

about s t a t i c p o s i t i o n s .


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-4-

The i n e r t i a l e f f e c t o f l i q u i d i n t h e w e ll bo re was f i r s t i n ve s ti g at e d by

Bredehoeft e t a l . I 5 i n 1966, usi ng an analog computer.s o l u t i o n was not g iven .

b i l i t y assuming an exponent ia l ly damped f luc tua t io n w a s presented by van

d e r Karnpl' i n 1976; however, he did no t provide a genera l so lu t i on and d idno t i nc lude a s k i n e f f e c t . Thus f a r , no g e n e r al s o l u t i o n i n c l ud i n g t h e

case when t h e w e l l behavior is a f f e c t e d by t h e i n e r t i a l e f f e c t of t h e

l i q u i d i n t h e w e ll b or e w i t h no o s c i l l a t i o n h a s b ee n p r e se n te d , t o o ur

knowledge.

However, a genera l

An approx imate method t o d ete rmi ne t h e permea-

Another re la ted w e l l t e s t i s the closed chamber test, l7 which hasbecome popular f o r p o l l u t i o n p r o t ec t i o n e s p e c i a l l y f o r o f f s ho r e w e l l s .

It i s a dev i a t i o n from conven tiona l d r i l l s t e m t e s t i n g . A s o l u t i o n u s e f u l

f o r ana ly si s of f low period da ta from closed chamber t es t s has never been

off ere d, t o our knowledge, and bot tomhole press ure d at a fo r t he f low period

have been discarded o r have not even been rep ort ed i n many cases.

The purposes of th i s s tudy are t o d e ri v e a comple t e so lu t i on fo r t he

s l u g test, i nc lu di ng t h e i n e r t i a l e f f e c t of t h e l i q u i d i n t h e w el lb or e

and t h e s k i n e f f e c t ; t o i n v e s t i g a t e t h e e f f e c t s o f p a r am et er s on t h e gen-

eral so lu t ion ; t o advance the unders tanding of s l u g tes t da t a ; and t o s t udy

th e ana ly si s of f low period da ta from closed chamber tests as an ex tens ion

o f genera l s l ug tes t s o l u t i o n s .


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2. SLUG TEST ANALYSIS

The slug tes t problem w a s stated by Ramey e t a l .13 as that posed by afo rmat ion whose i n i t i a l p res su re , pi, i s a t most less t h a n t h e p r e s s u ret o l i f t a column of r e s e r v o i r l i q u i d j u s t t o t h e s u r f a c e of t h e e a r t h .

When a d r i l l s t e m t es t packer i s set j u s t above t he fo rmat ion with t he

d r i l l s t r i n g empty e xc e pt f o r t h e a p p r o p r i a t e cu sh io n l i q u i d , a l l t h e

elements for a s l u g tes t would be present . A t zero t i m e , t h e tes ter

valve is opened t o expose th e format ion suddenly t o t h e cushion pressu re ,

(po + Patmformat ion begins t o produce, t h e produced l iqu id i s s t o r e d w i th i n t h e

d r i l l s t r i n g , and t h e l i q u id level beg ins t o rise. The i n i t i a l produc t ion

r a t e w i l l be high and w i l l g r a d u a ll y d e c l i n e as a cc um ul at in g l i q u i d i n t h e

d r i l l s t r i n g c au se s an increas ing back pressure . U su a ll y t h e l i q u i d p ro-

duc t i on ceases by i t s e l f , and i n some cases t h e l i q u i d l e v e l i n t he w e l l -

b or e o s c i l l a t e s b e fo r e i t reaches equi l ib r ium.

) , i n t h e d r i l l s t r i n g above t h e valve ( s ee F ig . 3 ) . A s t h e

The mathematical for mulat i on of t h e s lu g test problem is explained i n

Sect ion 2-1. The so l u t ion f o r t h i s problem i s ob ta ined and i n ves t i g a t ed

i n S e c t io n 2-2. Fie ld data examples are d i scus sed i n Sec t i on 2-3.

2-1 Mathe matical Formu latio n

I n o r d e r t o o b t a i n a s o l u t i o n which i n c l ud e s t h e i n e r t i a l e f f e c t of

t h e l i q u i d i n t h e we l lb o re , a momentum balance i s r equ i red . The l i ne a r

momentum bal anc e i n t h e wel lbo re states t h a t t h e ra te of change of momen-

tum is e qu a l t o t h e n e t f o r c e on t h e l i q u i d :

-5-


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-6 -

X-

L

hf_

I

iiII

iI

SURFACE

- DRILL STRING

- P

,TESTER VALVE

~ PACKER

- -FORMATION

FIG. 3: SCHEMATIC DIAGRAM OFASLUG TEST


23/208

-7-

2 2 - r r2 p - r r2 pf(L+x) gPd t d t P pw p a t m"[ p f m p (L+X) dx ] = TI:

The term i n s i d e t h e p a re n th e se s i n t h e l e ft- h an d s i d e i s the produc t

of the mass o f l i q u i d i n t h e w e ll b or e and t h e v e l o c i t y of t h e l i q u i d

level i n t h e we llbo re . The f i r s t term i n t h e r ig ht-hand s ide is t h e

force caused by th e wellbore bottomhole pressure .

for ce caused by th e atmospheric pressu re .

f o r c e , and th e f ou r th term is t h e f r i c t i o n f o r ce .

The second term is t h eThe third term i s t h e g r a v i ty

p i s t h e d e n s i t y of t h e l i q u i d i n t h e w el lb or e, r i s t h e i n s i d ef P

r a di u s of t h e d r i l l s t r i n g , t h e t u b i ng , o r t h e c as i ng p i p e i n which t h e

produced l i qu id en te rs , L is th e l iq ui d lengt h whose head i s e qu iva le n t

P atmt o t h e i n i t i a l f or ma ti on p r e ss u r e, pi, minus atmospheric pressure ,x i s t h e l i q u i d level i n th e wel lbore measured from the s ta bl e poi nt , p

i s the wel lbore bottomhole pressu re (we w i l l c a l l t h i s t he "wellbore pres-

sure"), g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n , and f i s t h e Moody friction^f a c to r .

W

Rearranging Eq. 1:

dx dpf -2d xd t

( g ) 2 f (LSX) --=Pf(L+X)2 + Pf dt d t pw P a t m - pf(L+x)g

For a c o ns t a nt c o m p r e s s i b il i t y l i q u i d , t h e f o ll ow i ng r e l a t i o n h o l d s i n

the wellbore approximately:


24/208

-8-

Then, Eq. 2 becomes as fo l lows:

dxC

f

d t P

Since the head sf t h e l i q u i d l e n g t h , L, i s e qu iv al en t t o t h e i n i t i a l f o r-mation pres sure , pi, minus the atmospheric pressure

' Patm:

PW'Pi'w"a t m - L g = -pf pf (5)

So, Eq. 4 becomes:

f dxa t d td t P

PW'Pipf- -

For the formation, i f w e adopt the fol lowing assumptions, t he w e l l -

known d i f f u s i v i t y equation' ' deri ved from th e co nt in ui ty equat ion and

Darcy's l a w can be used.

1) Hor izon tal , i so t ro p ic , homogeneous, i so thermal , ra d i a l , i n f i n i t e

porous medium of cons tan t thi ckn ess , h, po ro si ty , 4 , and permeabil i ty , k ,2) Co n s tan t l i q u id v i sco s i ty , 1-1, and s m a l l co n s tan t to t a l sy s t em

co mp ress ib i l i t y , ct '3) No f l u i d f low acros s the hor izon t a l boundar ies and neg l ig ib l e

g r a v i t y e f f e c t .

4 ) The sq ua re of th e p r essu re g r ad ien t w i th r e sp ec t to r ad ia l d i s t an ce

is n e g l i g i b l e .


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-9-

These assumptions appear to be reasonable for slug tests. Then:

p is the pressure inside the reservoir, r is the radial distance, t is the

time, and c is the total system compressibility defined as the sum of the

reservoir liquid compressibility and the formation compressibility.

/

t

In order to arrange the equations in dimensionless form, the following

dimensionless variables are introduced:

P,. '

..

Pfgx% = Pi4P0+Patm) (= -*)Pi-Pw-wD - pi- (Po+Patm)

p is the dimensionless pressure, t is the dimensionless time, r is the

wellbore radius, r is the dimensionless radial distance, x is the di-mensionless liquid level in the wellbore, x(t=O) is the initial liquid levelin the wellbore, and p

(we will call this "dimensionless wellbore pressure").

D D W

D D

is the dimensionless wellbore bottomhole pressureWD


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-10-

Using these dimensionless variables, the following equations are de-

rived:

From Eq. 7: n

aPD aPD+- e

-

---

LPD 12 rD ar, at,arD

From Eq. 6:

.-I

IPwD dxD(Lfx)--f(Lfx) dxD dxD Pf fg- - - -4r dtD dtD 2 dtD dtD[($,+ PIn order to obtain an equation which we can handle analytically, the folci

lowing assumptions are adopted:

I

5 )

6 )

The pressure drop caused by friction in the wellbore is negligible.

The compressibility of the liquid in the wellbore is negligible.

7) The

8) L is

Assumption 5

(ST term is negligible.much greater than x.

can be checked using available field case data. It was found

that this assumption is reasonable, especially when there is cushion liq-

uid in the wellbore. Examples of the pressure drop caused by friction in

the wellbore are shown in Appendix C, and will be explained in Section

2-3. Assumption 6 is reasonable because we are considering liquid-filledreservoirs.

exist in the wellbore before the test starts.

To satisfy assumptions 7 and 8, the cushion liquid should

It can be seen in Eq.14that


27/208

-11-the order of the (SJ term is the same as that of the pressure dropterm caused by friction.

Applying these assumptions to Eq. 14, the following result is ob-

tained:

The new group$ ( k 2) is a dimensionless number and representswctrw

the effect of inertia of the liquid in the wellbore.

dimensionless number, a, is equivalent to Froude's number,20 which repre-sents the ratio between inertia force and gravity force.

defined as:

In fact, this

This number is

or:

Then, Eq . 15 becomes:

2

- -2 XDa * - 'wD-k XDdtDIn order to determine the initial conditions, we assume that the

initial reservoir pressure, pi, is the same at any point in the reser-voir. This condition is expressed as:


28/208

-12-

The i n i t i a l l i q u id l e ve l i n t he we l lbo re i n dimens ion les s form i s a l-

w a y s- l , because:

Pfgx t=O>(P a tm+P fLg - Pf g{L+x ( =0 'i+P a m 1x ( t =O) =D D

- x (t=O)x t=O)- -

- -- 1This i s t he s econd i n i t i a l condi t i on .

The i n i t i a l ve l o c i t y of t he l i qu id l ev e l i n t he we llbo re can be con-

s ide red ze ro fo r t he u sua l s l ug tests; however, t o make th e so lu ti on

genera l , i t i s assumed t h a t t h e i n i t i a l v e l o c i t y o f l i q u i d l e v e l i n t h e

wellbore i s given as some valu e. This con dit ion i s t he t h i rd i n i t i a l

condi t ion :

= x ' ( t = O)D D

tD'O

Next w e w i l l consider th e boundary cond i t ion s. Since w e assumed

t h a t t h e r e s e r v oi r i s i n f i n i t e :

R i m p D ( r D , tD ) = 0r-fo3

D

This i s t he f i r s t boundary cond i t i on .


29/208

-13-

From a material balance on the wel lbore:

Using the d imens ionless va r ia b le s de f ined i n Eqs. 8 t o 11, the fo l lowing II

equat ion is obtained:

dxD 1- = - -r =1t D c D D

C

bore s torage cons tan t def ined as fol lows:

is the d imens ionless wel lbore s torage c o n s t a n t Y 2 land C is t h e w e l l -D

I

~

C

227rr h@ctc =W

27rrc = Ppfg

10A s the th i r d boundary condi t ion , w e can use the fo l lowing equat ion

obtained from a pres su re ba l ance a t the sandface based on the def in i t ion

of the s teady- s t a t e s k i n f a c t o r .899

I

As a summary of t h i s sec t i on , th e fol lowing equat ions are obtained

as th e re su l t of mathemat ica l formulat ion of th e s lug test problem.

I


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i

-14-

L

Initial Conditions:

Boundary Conditions:

2-2. Solutions

x (t -0) = - 1D D-

D=A(%)dx

Dr =1D

dtD

PWD =['D-'(%)] rD=l

Since Eq. 13 and Eq s . 18 through 27 are linear equations, solutionsmay be obtained using the Laplace transformation.

place space are given in Section 2-2-1.

inversions of these solutions are considered in Section 2-2-2; however,

The solutions in La-

The analytical Laplace transform


31/208

-15-

the complete real space solutions could not be obtained. Numerical Laplace

transform inversion methods considered in Section 2-2-3 were applied. The

characteristics of the solutions are investigated in Section 2-2-4.

2-2-1. Solutions in Laplace Space

Applying the Laplace transformation with respect to time to Eq .

13:

-p is the Laplace transform of p with respect to time, and R is theD D

Laplace transform variable.

Substituting the initial condition Eq. 19, Eq. 28 becomes:

44This is a modified Bessel equation of order zero, and the solution is:

-PD = AIo(rD&) + BKO(rDfi)

I is the modified Bessel function of the first kind of order zero, K is0 0

the modified Bessel function of the second kind of order zero, and A and

B are arbitrary constants.

To satisfy the boundary condition Eq. 2 2 :

A = O


32/208

so:

Then:

-16-

-PD = ~ ~ ~ ( r ~ f i )

K

Applying the Laplace transformation with respect to time to Eq . 27:

is the modified Bessel function of the second kind of order unity.1

Substituting Eqs. 32 and 3 3 into Eq. 34 :

-'wD = B { Ko(fi) + s f i K1(a) }

Applying the Laplace transformation with respect to time t o Eq. 18:

- - c s b2 2- *wDa { R XD - Rx (t -0) - x;)(tD=O) 1 + XD = -D D--

D'x is the Laplace transform of xFrom E q s . 35 and 36:

D

Applying the Laplace transformation with respect to time to Eq. 24 :

r =1D

-Rx, - x (t =O ) = --

D D


33/208

-17-

From E q s . 33 and 38:

Substituti

C D { GD D Dx (t = O ) lf i e K1(a)=

Eq. 39 into Eq . 37:

From E q s . 35, 39, and 40:

CD{sfiK1(fi) + KO(fi)}{-XD(tD=O) + a2!Lx'(t =o) )D Dn e-'wD (a2!L2+CDs!L+l)K1 (a)+ CDRKO(,a)

From E q s . 32, 39, and 4 0:

( 3 9 )

( 40)

( 4 1 )

Equations 4 0 , 41, and 42 are the solutions for x Y PwD$ and PD inthe Laplace space, respectively. We seek the corresponding real space

solutions. The procedure will be discussed in the following sections.

2-2-2 Analytical Approach to Laplace Transform Inversion

The complete analytical inversions of Eq. 40 through 42 could not

be obtained. The reason is explained in this section and in Appendix A.

As an approximation, the analytical solutions for early times and late

times are obtained.


34/208

-18-

2-2-2.1 Tria l f o r A n a l y t i c a l I n v e r s i on

Appendix A shows th e mathematical t reatme nt of t h i s t r ia l . A s

shown i n Appendix A, i t wzs no t pos s ib l e t o f i nd t h e po l es of t h e func-t i o n a n a l y t i c a l l y e x c e p t a t t h e o r i g i n .

poles (except a t t h e o r i g in ) e x i s t o r n ot ( i . e . , whether th e value s of Bwhich s a t i s fy E q s . A-19 and A-20 exis t o r n o t ) , w e assumed that the sum

of r e s idues -ike is zer o. Then, from Eq. A-21:

In orde r t o check whether th e

-

where:

2A,(u) = (a2u4-C su +1)J1(u) - C UJ (u)D D O

2A2(u) = (a2u4-c Dsu +I) Y1(u) - C ~ U Y ~ ( U )

L LA,(u) = { a u xD(tD=O) - C sx ( t -0) - aL$(tD=O)} uJl(u) - CD Dx ( tD=0) Jo(u)D D D-

2 2 2A4(u) = { a u xD(tD=O) - C sx ( t =O ) - a ~ ' ( t0)) UY,(U) - C x ( t =o) y0(u)D D D D D- D D D

J and J are t h e Bessel f u n c ti o n s of t h e f i r s t ki n d, o r d e r z er o and

un i ty . Y and Y are t h e Bessel func t ion s of t h e second kind, ord er zero and0 1

uni t y . In order t o check whether th e assumpt ion th at t her e i s no pole be-

0 1

s i d e s t h e o r i g i p is c o r r e c t , Eq. 43 w as int eg rat ed numerical ly us ing th e Rom-berg method.23 Figure 4 shows an example of th e r e s u l t s . The solut ion be-

2 5 2haves reasocab ly f o r a v a l u e s less than 10 ; however , t he so lu t io n fo r a

6v a l u e s g r e a t e r t h a n 10 does not make sense for the example case when


35/208

-19-


36/208

-20-

3 -CD = 10Eq. A-21 , p lays an impor t an t ro l e fo r a2 va lues g rea t e r t han lo6 f o rt h i s example case.

w a s wrong.

i n t he we l lbo re o s c i l l a t e s f o r some cond i t i ons .

t he l e f t-hand s i de of th e Laplace p lane t o a l l ow converg ing os c i l l a t i o n t o

happen.

and s = 0. This means that th e sum of th e residu es, >Re, i n-

Then, t he assumption th at o the r res i due s are n e g l i g i b l e

Th i s a l so can be guessed f rom the f ac t t ha t t he l i qu id level

The pole must e x is t on

In conclus ion , it w a s n o t p o s s i b le t o o b t a i n a complete real space solu-

An a l t e r n a t i v e s t e p is t o us e numerical Laplace t ransform in versi ont i o n .

methods t o obt a in the en t i r e so lu t io n . This procedure w i l l be discussed i nth e fo l lowing sect io ns . The ea r l y t i m e and l a t e t i m e approximate s o l u t i o p scan be obta ined analy t ica l ly , and w i l l be d i scussed i n th e remaining pa r t

o f t h i s s ec t i on . The same d i scus s ion i s a p p l i c a b l e t o Eqs. 4 1 and 42, be+

cause their denominators are t h e same as that of Eq. 40, and i t seems t h a t

t h e r e i s no pole c an ce ll at io n between denominator and numerator.

2-2-2.2 Early Time So lu t i ons

Rearranging Eq. 40:

1

2 2 KOq m1 R + C*SR + 1 + c ,aRX D ( t D = o ) + { cr2x;(tD=0) -R=D

(48)Simi lar ly , f rom Eq. 41:


37/208

-21-For early times, we can say that R + m .modified Bessel functions: 24

Then, from the characteristics of

Substituting E q . 50 into E q s . 48 and 49:

1 (51)+ CI)SR + 1 + c,aRx =D

x (t =O) xD(tD=O) x'(t =O )- + a s R + aD DR+ !L22 3a R

(53)s R + a[ 2 D D R a2R2 + CD& + 1 + c D a-pwD = CD a x'(t =O ) -Applying the inverse Laplace transf~rmation~~directly:

Pw(tD> = - 2a

These early time solutions will be used to check the solutions obtained by

the numerical Laplace transform inversion methods in the next section.


38/208

-22-2-2-2.3 Late Time Solutions

Similarly as for early times, we can say that R + 0 for late ti4 s.From the characteristics of modified Bessel functions,26 for small argu- eIments:

1K (a) -a1 (58)

where y is Euler's number.Substituting Eqs. 57 and 58 into Eqs.48and 49:

x (t =O) x (t =O )- + [ a 2 x p D = 0 ) - D RD - 1=D

2+a xA(tD=O) as + 0

+ O a s R + O

Then:

2XD(tD) = a x'(t =O) 6(tD)

PWDCtD) = 0

D D


39/208

-23-

6 ( t D ) i s t h e Dirac d e l t a f ~ n c t i o n . ~ ~ S i n c ew e are now looking a t t h e l a t et i m e zone, 6 ( t D ) = 0 .

so:

x ( t ) = OD D

Th ese r e su l t s f o r l a t e times agree with an understanding of th e physica l

phenomena.

2-2-3 Numerical Laplace Transform In ve rs io n

A s shown i n t he p rev ious s ec t ion , w e cannot obt:ain t h e real spaceso lu t io n an a ly t i ca l ly ex cep t f o r th e ap p ro x ima te so1ut:ions a t e a r l y timesand l a t e t i m e s . Therefore, th re e numerical Laplace transform inver sion

methods were u sed i n th i s s tud y. The r e su l t s obtained by th es e methodswere compared mutually, as w e l l as t o th e r e su l t o b ta in ed f ro m a f i n i t e

d i f f e r e n c e s o l u t i on .

for closed chamber t e s t an a ly s i s and i s explained i n Section 3 and i n App$p-d i x D.

The f i n i t e d i f f e r e n c e s o l u t i o n was developed mainly

The f i n i t e di f f er en ce computer program i s prese nted i n Appendix E,

2-2-3.1 St eh fe st Method

A simple and ac cu ra te numerical Lapla ce transform inv ers ion method

This method i s based on the followingwas presented by S t e h f e ~ t ~ ~i n 1970.algorithm derived assuming a s p e c i a l p r o b a b i l i t y densi ty28 with which theexpecta t ion of a function becomes similar t o t h e Laplace t ransform of t hefunct ion .


40/208

-24-

NN min (i T)

L

-f is the Laplace transform of the function f(t).cause a more accurate result theoretically; on the other hand, large N alsocauses a larger roundoff error. In this study, 16 was used for N. This

value was selected from a comparison of the results obtained by this meth4~

with those from other methods.

salient points, or rapid oscillations (this depends on the ratio between the

wavelength of the function and the peak of the probability density).

computer program for this method is presented in Appendix E.

A large value of N will

The function should not have discontinuitiis,

The

2-2-3.2 Veillon Method

Another method was presented by Veillon2 in 1972. This methodis based on the following algorithm which is an approximation of the Fouridrcosine transform inversion of the function assuming f(t) is a real function.

?(a)} + cN Re{ r (a+y)}cos (k=l

T

a is positive and greater than the real part of the singularities of f(t>.The function should be reasonably smooth. The program for this method was

developed by Dr. M. Sengul of the Marathon Oil Company. In order to use

this program, it was necessary to obtain the real part and the imaginary

part of the function separately.

dix B. N was taken as 16.

The procedure used is explained in Appen-


41/208

-25-2-2-3.3 Albrecht-Honig Method

The third method w a s presented by Albrecht and Honig3' i n 1977.This method is a va r i a t io n of th e Vei l lon method, and i s based on the fol low-

ing a lgor i thm.

-a t

N

k-1+ 1 R e { i( a +y)} (-l)k+eT R(N) f a r 0 < t < T/2 (69)

The de f in i t i on of va r i ab l es and cons t an t s are t h e same as those used i n the

previous two methods, and R(N) is t h e c o r r e c t i o n term. The function shou3ldbe reasonably smooth.

method i s presented i n Appendix E.

M w a s t aken as 16. The computer program f o r t h i s

2-2-3.4 Comparison of Re su lt s

Since the se th re e numerical Laplace t ransform inv ers ion methods

have l im i t a t i o ns on t he func t i on t o which t hey can be app l i ed ( t he func t i on

should be smooth , o r should not have d i scont inui t i es or rap id o s c i l l a t i o d ) ,i t is necessary to test th e re su l t s ob ta ined by th ese numer ical Laplace t zan s -form inver s ion methods t o a s su re r e l i ab i l i t y . In o r d er t o i n v e s ti g a t e t h er e l i a b i l i t y of t h e s o lu t i o n s, t h e case of C

a t y p i c a l case.

by numer ical Laplace t ransform invers ion methods and th e f i n i t e d i f f ere nce

s o l u t i o n .

shown i n F i g. 5 .

test so l u t i ons which neglect i n e r t i a l e f f ec t (a = 0) and th e pres ent problem

can be seen on Fig. 5.D

3= 10 and s = 0 w a s s e l e c t e d as

D

S o lu ti o ns f o r t h i s case f o r v a r i o us a values were obtained

The dimens ionless wel lbore pressures for th i s example case are

One of th e main di ff er en ce s between t he previous s l ug

The wel lbo re p res su res d rop i n s t an t l y (p = 1)


42/208

-26-f o r t h e i n e r t i a- l e s s cases, but pass through a minimum (maximum i n p ) onFig. 5. Furthermore, there is an "overshooting" f o r e a c h va lue o f ashown on Fig. 5.

This be ha v io r c a use s c om puta t ion d i f f i c u l t i e s .

D2

8

F i n a l l y , t h e a2 = 10 case shows a d i s t i n c t o s c i l l a t i o n .

To inve s t i ga t e the over shoot ing por t ion o f th e so lu t i on a t e a r l y

2 3times, s o l u t i o n s f o r t h e a = 10ference and Stehfest and Albrecht-Honig methods. It w a s necessary t o apply

e a r l y t i m e o r l a t e t i m e approximations f o r th e modif ied Bessel f u n c t i o n s inord er t o us e t h e computer program developed by t h e Marathon O i l Company f o r

th e Vei l lon a lgor i thm, and t hese approximat ions were n o t r ea s o na b l e f o r t h i s

t i m e range. Thus th e Vei ll on method was n o t used f o r t h i s test. Table 1

shows the resul ts .

Honig method a gr ee un t i l t = 0.9; however, a f t e r th i s t i m e , t h e r e s u l t from

the Albrecht-Honig method starts o s c i l l a t i n g , a n d a f t e r t = 20, t he y a g r e e

aga in .

the f i n i t e d i f f e r e nc e method. From the se r e su l t s w e i n f e r t h a t t h e A lb re ch t-

Honig method i s t o o s e n s i t i v e t o t h e o s c i l l a t i o n o f t h e f u n c t io n t o ap p ly

t o th e subjec t problem, and t he r es u l t by th e S te hfe s t method i s more re-

l i a b l e tha n t ha t from th e Albr ec ht-Honig method.

for the S tehfes t method wi th N = 16 happens f o r ot he r combinations of CD, s,and a va lue s , bu t on ly a t t h e s p e c i a l t i m e of t

not happen fo r t he St eh fe st method with N = 10 .

r e s u l t is n o t known; however, when N inc r eases , th e number of odd re su l ts i n-

creases. We suspec t t ha t t h i s might be caused by roundof f e r ro r increased

by the l a r geN va lue . This phenomenon is t h e s u b j e c t of cont inuing s tudy.

The r e s u l t b y t he S te h f e s tm e thodwi th N = 1 6 i s c l o s e r t o t h e r e su l t f r om the

case were c a l cu l a t ed by t h e f i n i t e d i f -

The re s u l t by th e Ste hfe st method and by th e Albrecht-

D

D

= 0.9 - 20 a g r e es wi th th e r e s u l t fromThe r e su l t by S te h f e s t f o r tD

The s t ran ge point a t t = 2D

= 2.The r e as o n f o r t h i s odd

This phenomena doesD


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44/208

-28-f i n i t e d i f f e r e n c e s o l u t i o n i n t h e o ve rs ho ot in g p o r t io n of t h e s o l u t i o n

than i s the r e su l t by the Steh fes t method wi th N = 10. The ex ist enc e

of the overshoot ing por t ion of th e so lu t ion a t e a r l y t i m e s w a s assured

by the Ve ill on method fo r smaller a values . Table 2 presen t s t he re-

s u l t s . Although not shown, th ese res u l t s agree wi th the ear l y t i m e

analy t ica l so lu t ion expressed by Eq. 56.

Table 3 shows a comparison of the

c a s e ( s e e a l s o F ig . 5 ) .

The resul t by

7f o r t h e a2 = 10methods agree qui te w e l l .

re su l t s by t hese four methods

The resu l t s f rom these four

th e Ste hfe st method with N = 16

and N = 10 are q u i t e c l o s e , b u t t h e N = 16 va lue ag rees bes t w i th o the r

r e s u l t s .

Table 4 shows th e comparison of th e re s u l t s by a l l four methods

8f o r a2 = 10 , fo r which t he so lu t i on has an o sc i l l a t i on .the Albrecht-Honig method and by t he Ve il lo n method ag re e very w e l l .

This is probably because the algori thms for these two methods are similar.

However, the results show a higher ampl itude of the os ci l l a t io n than do

t h e r e s u l t s by t h e S t eh fes t method and by t he f i n i t e d i f f e rence so lu t i on .

This might be caused by th e h igher se ns i t iv i t y t o t h e o s c i l l a t i o n of t h e

fu nct ion of th es e methods, which appeared a t the overshoot ing por t ion of

t h e s o l u t i o n f o r t h e a2 = 10re su l t by th e Steh fes t method wi th N = 1 6 i s c l o s e r t o t h e r e s u l t by t h e

f i n i t e d i f f e r e n c e s o l u t i o n t h a n t h e r e s u l t by the Stehfest method with

N = 10.

The re su l t s by

3case i n t he Alb rech t-Honig method. The

From th ese comparisons of th e r e su l t s o btained by vari ous methods

for the example case , w e conclude th at so l u t i on by t he Ste hfe s t method

wi th N = 16 gives the mos t re l i ab le so l u t i on fo r present problems. The4


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-29-TABLE 2: DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FOR

AND s = 0 BY THE VEILLON METHOD2 3VALUES WHEN CD = 10SMALL a

-4a2 = 10tD

5.00000000D-066.00000000D-067.00000000D-068.00000000D-069.00000000D-061.00000000D-052.00000000D-053~00000000D-054~00000000D-055.00000000D-056.00000000D-057~00000000D-058.00000000D-059~00000000D-051~00000000D-042~00000000D-043~00000000D-044~00000000D-045~00000000D-046~00000000D-047~00000000D-048~00000000D-049~00000000D-041~00000000D-032*00000000D-033~00000000D-034*00000000D-035*00000000D-036*00000000D-037 00000000D-038*00000000D-039*00000000D-03

8.20475719D-021.07018134D-011.33722237D-011.61899765D-011.91326994D-012.218070051)-015.52322567D-018.61512986D-011.09554509M-001.23809928M-001.29629459D+001.28982849IHOO1.24256281Dt-001.17674162IHOO1.10970029IHOO1.01220135IHOO1.00412114IHOO1.00377037IHOOl.O0246443D+OO1.00186191D+001.00148557IHOO1.00121204IHOO1.00100964IHOO1.00085534IHOO1.00026381IHOO1.00010884IHOOl.O0003922D+OO9-99999204D-019.99972619D-019.99953199D-019.99938042D-019.99925627D-01

tD5 00000000D-08

6 00000000D-087*00000000D-088 00000000D-089~00000000D-081~00000000D-072-00000000D-073-00000000D-074*00000000D-075*00000000D-076 00000000D-077~00000000D-078*00000000D-079~00000000D-071~00000000D-062*00000000D-063~OOOOOOOOD-064~00000000D-065 00000000D-066~00000000D-067 00000000D-068*00000000D-069~00000000D-061~00000000D-052 00000000D-053~00000000D-054~00000000D-055 00000000D-056~00000000D-057~00000000D-058*00000000D-059*00000000D-05

PWD8.20475740D-021.07018138D-011.33722244D-011.61899776D-011.91327010D-012.21807028D-015.52322800D-018061513818D-011*095547OlDtOO1*23810272D+001.29629985IHOO1-28983565D+001- 42571751~001*17675205IHOO1~10971184IHOO1*01221536~+001*00413878IHOO1*00379068IHOO1.00248714IHOO1~00188678IHOO1*00151244D+OO1.00124077D+001*0010401OIHOO1.00088745IHOO1 00030922IHOO1*00016446D+OO1~00010345D+001*00007101D+OO1-00005128IHOO1- 0003816IHOO1.00002887Dt-001*00002196D+OO


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ma0XH!2

W

n0mHWcEwFrXwHm

QdIIz

0dIIz

aU

-31-

. . . . . . . . . . . . . . . . . .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0I l l

~ c a ~ ~ n n n ~ n n ~ n n n ~ n a a0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

. . - . - _ -. . . . . . . . . . . . . . . . . .1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


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-32-

St eh fe st method with N = 16 w a s used t o o b t a i n t h e s o l u t i o n s c i t e d i n t h e

fo l l owing s ec t i ons , a l t hough t he so lu t i ons were checked by other methods

c o n s i s t e n t l y . The r e l i a b i l i t y of t h e s o l u t i o n s ho ul d d e cr e as e as t h e

so lu t i on has more o s c i l l a t i on s . Bessel func t i ons are ca l cu l a t ed u s ing the

polynomial approximations, 32 as shown i n t he computer programs i n AppendixE.

double pr eci s io n. Sometimes r es u l t s show a s l i g h t o s c i l l a t i o n a round t h e

cor re ct answers i f the program i s r un i n s i n g l e p r e c is i o n. T hi s e n t i r e

d i s cus s ion is app l i cab l e no t on ly t o the funct ion p (see Eq. 41), b u ta l t o t o t h e f u n c t i o n % (see Eq. 3 9 ) and pD (see Eq. 4 2 ) .

The numerical Laplace t r ansfor m inver sion program should be run i n

WD

2-2-4 Resul ts and Discussion

The s o l u t i o n s were obtained by the Stehfest method with N = 1 6 f o r

a range of parameters . These so l u t i ons w e r e checked by the resu l t s ob-

t a i ned by other numerical Laplace transfo rm inve rs ion methods, and by t he

f i n i t e d i f f e re n c e so l u t i o n .

Based on these so l u t i ons , the e f f ec t of parameter values on the so lu-

t i o n and t h e c h a r a c t e r i s t i c s of t he so lu t i on are i n v e st i g at e d i n t h i s

s e c t i o n .

2-2-4. 1 Effec t o f a on Solu t ions

The dimensionless number a w a s def ined i n Eqs . 1 6 and 1 7 :

o r :


49/208

-33-The dimensionless number a r e p r es e n t s t h e e f f e c t of i n e r t i a of

t h e l i q u i d i n t h e we l lb o re o n t h e s o l u t i o n .

c h a r a c t e r i s t i c s of the s o l u t i o n i n c l u d i ng t h e i ne r t i a l e f f e c t of t h e

l i q u i d i n t h e we ll bo re , t h e case when C = 10

as a typ ica l example , and the so lu t ion s f o r var ious a values w i l l beo b ta i ne d f o r t h i s case. Figure 5 shows th e dimensionless wel lbore pres-

pwD, versus the d imens ionless t i m e , t and Figure 6 shows the d i -s u r e ,mens ionless l iqu id level i n t h e w e ll b or e , %, ver sus t both i n semilogcoord ina t es . F igu res 7 and 8 show the same r e s u l t s on Cartesian coordi-

n a t e s . The tabula t ed r es u l t s ob ta ined by the S tehf es t method are given

i n T ab le s 5 and 6.

I n o r de r t o i n v e s t i g a t e t h e

3and s = 0 w i l l be used

D

DD

13The s o l u t i o n f o r a = 0 ag rees wi th t he p rev ious s l u g test s o l u t i o n ,

which n e g le c t s t h e i n e r t i a l e f f e c t of t h e l i q u i d i n t h e w e ll b or e . Consid-

e r ing t he r ange of dimensionless times i n whic h f i e l d d a t a u s u a ll y l i e ,

it can b e s a i d t h a t t h e i n e r t i a l e f f e c t of t h e l i q u id i n t h e wel lbore i s

neg l ig ib l e when a2 is less than about 10 On th e ot he rhand, when a2 is grea t e r t han abou t l o 4 , f i e l d da t a should no t fo l l ow thea v a i l a b l e s l u g test s o l u t i o n .

a2 - 10eluding t h e i n e r t i a l e f f e c t of t h e l i q u i d i n t h e w e l lb o re p r e s en t ed h e r e i n

shou ld be used t o ana lyze f i e l d da t a . Thi s is t r ue even though no osc i l la-

t i o n i n pr e ss u re o r l i q u i d level is e v id e nt . I n f a c t , i t is n o t rare f o r

a t o t a k e v a lu e s g r e a t e r t h a n 10 f o r hi g h i n i t i a l f o r ma ti on p r es s u r e

w e l l s o r high pe rmeab i li t y r e se rvo i r s .

fo r t h i s example case, t h e l i q u i d level i n t he we l lbore shows an o sc i l l a t i on .

4

fo r t h i s example.

This can be seen by comparing the a = 0 and

4cases on Figs . 5 and 6. I n t h e s e s i t u a t i o n s , t h e s o l ut i o n s in-

2 4

2 8When a is grea t e r t han abou t 10


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- 3 4 -


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-35-

>

c40Fw .

HVIEU


52/208

-36-

nU

cncnw


53/208

II

-37--7--l--zH

0IIm

rnzwEn

zH

i


54/208


55/208


56/208


57/208


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-42-

The a va lues f o r which t he l i q u id l ev e l i n t he we llbo re o s c i l l a t e s depend

on th e value of the d imens ionless wel lbore s to rag e cons tan t , C D , andt h e s k i n f a c t o r , s .

I n o r d e r t o o b t a i n g e ne r a l l i m i t s on t he e f f ec t o f a on the re-

s u l t s , s o l u t i on s f o r va r io us C and s va lues were obta ined . Figures 9

through 15 show the re su l t s . c1 is def ined as the value of a below which

pwD i s e s s e n t i a l l y t he same as t h e a = 0 ( i n e r t i a- l es s ) case so lu t ion whenpwD i s smaller than 0.9. This means t h a t t h e e f f e c t of a can be neglectedf o r p r a c t i c a l p u r p o s e s ifa < al.which os c i l l a t i o n of t he l i q u id l ev e l i n t he we l lbo re occu r s .

b e r a

Kamp .

which o s c i l l a t i o n s i n l i q u i d level coul d not happen and j u s t beyond which

o s c i l la t i o n s i n l i q u i d level could happen. A l i nea r r e l a t i onsh ip be tween

lo g a and log a ver sus l og C w a s found fo r the e n t i r e range of C in-1 2 D D

D

1

a i s def ined as the value beyond2

This num-

corresponds to the c r i t i c a l damping condi t ion d i scussed by van der2

" C r i t i c a l damping" w a s used here t o refer t o a condi t ion under

vest igated when s i s g r e a t e r t h a n 5, and for C

when s i s smaller than 5.

s t r a i g h t l i n e s i n t h e log- lo g s c a l e f o r d i f f e r e n t s va lues .

shows how the a c t ua l curves devia te f rom s t ra ig ht l i ne s fo r CD = 1when si s smaller than 5. Then, w e conclude t h a t t he e f f e c t o f a on the so l u t i on

can be e st imated from the fol lowing:

va lues g rea t e r t han 10D

Figures 1 6 and 1 7 show the co ef f i c i en t of thes 'eFigure 18

-2 1.25 1.077 - 0.0385 l o g scDa = 1 . 99x10 s1

0.85 1.077 - 0.0385 l o g scDa = s2


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-43-

' 12

IO

-i-----f--- - - , ------I------

FIG. 9: LOGalAND LOG a2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGECONSTANT FOR s = 0


60/208

-44-

8

2

0

-2

4I

FIG. 10: LOG alAND LOG a2VS TOGARITHM OF DIMENSIONLESS WELLBORE STOIUGECONSTANT FORs = 1


61/208

1:

I(

I

f

tJ

J

rt!

c

c. - d

-4

-0.1412 CD1.05sal

FIG. 11: LO G alAND LO G a2VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGECONSTANT FORs = 5


62/208

FIG. 1 2 : LOG al AND LOG a2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGECONSTANT FOR s = 20


63/208

LOG c*

F I G . 13: LOG al AND LOG a 2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGdCONSTANT FOR s = 50


64/208

12

IC

8

E

ti::4-I

2

, C

. - 2

-4

F I G . 1 4 : LO G a AND LOG a 2 VS LOGARITHM OF DIMENSIONL,ESS WELLBORE STORAG'Lt1

CONSTANT FOR s = 100


65/208


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-50-

W L30a

W n0m1cuUI n I n l

0 I-

8 901 ONW V 301

cu0

-II

_.

u0I4

u0Gi


67/208

1 .OE

I.Q5

I .04

= 1.03

I

-51-

10s

FIG. 17: EXPONENT OF DIMENSIONLESS WELLBORE STORAGE CONSTANTVS SKIN FACTOR


68/208

-52-

2

C

U(30-1

-I

- 2

-3

I I I

-T T

T T

I I I

0 2 4 6

S

FIG. 18: LOG alAND LOG a2VS SKIN FACTORFOR CD = 1


69/208

1 1

-53-

When s < 5, cD - lo3:1.05CL = 0.141 CD

1

1.05= 3.98 CD2

When s < 5, CD < l o 3 : use Figs . 9 , 10, and 11.I n van der Kamp's paper,16 t he parameter which con t ro l l e d t he c r i t i -

cal damping condition wa s presented as d.

1 2 s 8.51-r '(E)' &n [ 0.79 rf (,I ]Cd = 8TThis parameter d is expressed as fo l lows i n

(see the correspondence between the symbols

and those used i n ground water hydrology i n

'DRn [s]4a=

(74)

th e nomenclature of t h i s s t u d y

used i n petroleum engineering

Table 10, Sect ion 2-3-4).

(75)

However, t h e va lu e of d a t which c r i t i c a l damping occurs could not be ob-

taine d i n van der Kamp's an al ys is and w a s hypothesized as 0.7.

da ta pre sent ed showed d tobe between about 0.4 and 1 . 2 .

i s mainly due to the uncer t a in ty i n t he va lue o f t r ansmi s s iv i t y .

t he a

t a i ned i n t h i s s t udy , d can be ca l cu l a t ed .

g i b l e f o r water w e l l s .

Table 7 shows the re su l t s .

when d i s between 0.45 and 0.52

The actual

Th is unc e r t a in ty

Since

value which corresponds t o th e c r i t i c a l damping condi t ion w a s ob-2

Of t en t he sk in f ac to r i s negl i-

Then, d may be obtained using Eq. 73 and Fig. 9 .

A s a conclusion, cr i t ical damping should occur

f o r p r a c t i c a l C values .D


70/208

-54-

TABLE 7 : THE VALUE OF d AT WHICH CRITICAL DAMPING OCCURS

51010

l o 3l o 410

1010

d0.47

0.47

0 .4 6

0.45

0 .4 8

0 . 5 1

0.52


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-55-

For small a va lues , t h e r e i s a t i m e range wherein p i s g r e a t e rWDthan 1, as can be seen i n Fig. 5. This happens f o r small a va lues a t

e a r l y t i m e s . This means th a t the wel lbore pre ssur e becomes less than the

i n i t i a l cu sh io n h ea d i n t h e w e ll b or e . T hi s i s caused by the inc reas e i n

k i ne t i c energy of the l i qu id by r ap id movement of the l i qu id up th e w e l l -

bor e. This oversho oting remains about 30% f o r a l l small a va lues s t ud i ed .

However, t h i s phenomenon pro bab ly ha s no use f u l meaning f o r f i e l d da t a

in t e rp re t a t i on because t he d imens ion l es s t i m e i s usua l l y l a r ge even fo r a

minute of real t i m e , and might no t be seen i n f i e ld da ta because of f r i c -

t i o n .

2-2-4. 2 Effec t o f C on Solut ionsF igu re 1 9 shows the dimensionless wel lbore pressure,

Y

versus

dimensionless t i m e , tD , and Fig. 20 shows the d imens ionless l iqu id level ,xD, ver sus tage cons t an t , C D , when the sk in fac tor , s , i s zero .t he co rrespond ing so lu t i on s h i f t s t oward smaller t

D

The incre ase i n C decreases t he e f f ec t o f a. This phenomenon agrees w i t hEqs. 70 and 72. On t he o the r hand, t he so lu t i on fo r a = 0 s h i f t s toward

wD

fo r t h ree d i f f e re n t va lues of d imensionl ess we l lbo re s t o r -D

When C i n c r e a s e s ,D

f o r t h e same a v a l u e , ,I

D

i nc reas ing t when C i nc reases . Th i s is because t he l i qu id t akes a

longer t i m e t o occupy the la rg er wel lbore . Then, i t c a n b e s a i d t h a t t h e

D D

g r e a t e r C i s , f o r a wide range of a, t h e less important w i l l be t he i n-

e r t i a l e f f e c t f o r t h e l i q u id i n the wel lbore .

D

A t a g l a n c e , t h i s statement

appear s st ra ng e. However, i t i s t r u e .

In o r d e r t o e v a l u a t e t h e s h i f t toward i n c r e a s i n g t f o r a = 0D

s o l u t i o n s , t h e s o l u t i o n s f o r va r i ou s C and s values were obtained. Fig-

ure 2 1 shows the dimensionless t i m e

D

a t which pwD becomes 0 .9 . We D1


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-56-I I I I I 1--

(P/'\L...

B22m20VW

d


73/208

-I-------- _._--.__

I I II / /

IOJel

ii

-57-

r----- '--.- I-~-__-..

\

\\

\

zH00

Hn

a4-

.."

30

HzsVI20uW30.HVI

VIVIWJz0HVI

Hn

..0N


74/208

-58-

I

FIG. 21: LOG t VS LOGARITHMOF DIMENSIONLESS WELLBORE STORAGE CONSTANTDl


75/208


76/208

-60-

-

.-

I--

-

---

-

! --..-----

-

-

s

3F I G . 22: LOG t VS S K I N FACTOR FOR CD = 10Dl


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-61-

\\\

m0I+

IIV

n

w

z0H


78/208

-62-

.I_- -*-- T---I_-

e YI


79/208

-63 -Equation 76 , repor ted i n th e prev ious sec t i on , shows how fa r t he so l u t io ns

sh i f t , depending on the s v a lu es g en e ra l ly .

The overshooting of p a t e a r l y t i m e s f o r small a values decreases

This is because the l iqu id cannot f low

WDw i t h i n c re a s i ng s k i n f a c t o r , s.

i n to th e we l lbo re f r e e l y b ecau se o f th e f low r e s i s t an ce caused by th e s k in

fa c t or , and cannot permi t t he l i qu id column i n th e wel lbore to accelerate

su f f i c i e n t ly to in du ce th e o v e rsh oo t in g.

For a n e g at i v e s k i n f a c t o r o r a f r a c t u r e d w e l l , t h e o p po s i t e effeel twould be expected . But fo r negat ive va lues of t he sk i n fa c t o r , t he so lu-

t i o n d iv e rg es i n some cases. This may be caused by th e de f i ni t i on of t he

s t ead y- s t a t e s k i n f a c t o r ( i . e . , a sudden inc rea se of p ress ure a t t h e sand-f a c e f o r a n e g at i v e s k i n f a c t o r ) .

u s e t h e e f f e c t i v e w e ll b or e r a d i u s , r ' = r e , i n s t e a d o f t h e a c t u a l w e l ~ l -b o re r ad iu s , r as an approximat ion . Using t h i s e f fe c t iv e wel lbore

r a d i u s , CD and a w i l l be replaced by C e2' andth e v a lu es o f a and a

C e2' i s g rea te r th an 1 0 .a and a because of t h i s mul t ip l i ca t i on . Th is means th a t a negat ives k i n f a c t o r i n c re a s e s t h e i n e r t i a l e f f e c t o f t h e l i q u i d i n t h e w el lb or e

on s lug test s o l u t i o n s .

o bv io us from th e ch a ra c te r i s t i c s of th e curv es i n Fig . 18.

In o rd e r t o av o id th i s p ro blem, w e can

-SW W

W'respec t ive ly . Then ,

D

i n Eqs . 72 and 73 w i l l be mul t ip l ied by eO * l s when3

A n eg a t iv e sk in f ac to r g iv es smaller values olf1 2

D

1 2

3I f C e2' i s less than 10 , t h i s t en dency i s more

D

Another i n t e r es t i ng re su l t concerns an apparen t constan t f lowra te

phenomenon a t e a r l y t i m e s i n many d r i l l s t e m tests . Th is phenomenon i n

1 3 , 3 3DST f low per iod data has been discussed by several i n v e s t i g a t o r s .

It w a s suggested tha t c r i t i c a l f low choking might be the cause of th is

phenomenon. Thi s ex pl an at io n w a s a ls o given i n t h e Earlougher monograph.

Cri t ica l flow choking can happen fo r mult ipha se gas- liquid flow. C r i t i c a l

choking i s n o t l i k e l y i f o n l y l i q u i d f l ow s, o r i f t h e r e i s no o r i f i c e i n t h e

14


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syste m. However, t h e same appa rent con sta nt fl ow ra te phenomenon a t e a r l y

times i s of te n observed fo r l iqu id f low. Judging f rom the so lu t ion s ob-

t a in e d i n t h i s s tu dy , a l a rg e sk in fac to r may be one explanat ion of t h i s

phenomenon. Figure 25 shows a l i ne ar r e l a t io n between t he d imens ionless

wel lbore pressure , and the dimensionless t i m e , tD , a t e a r l y t i m e s f o rPWD,a l a r g e s k i n f a c t o r case (s = 100). Since p = pi - PwD P i - (Po + Patm)',Wth i s so lu t ion means a l i ne ar r e la t ion shi p between t he wel lbore pressure ,

4and the t i m e , t , f o r t from 0 t o 10 . This l i ne ar i t y would appear

t o be a cons tan t f lowra te per iod . The reason why the lar ge sk in fa c t or

might be a cau se of t h i s phenomenon is t h a t t h e r e s e r v o i r l i q u i d c an no t

f low in to t he we l lbo re r ap id ly because of t he l a rg e sk in f a c to r ( i . e . ,the pressure drop occurs mainly a t the sandface) , then the back pressure

caused by t he accumulated l iq ui d i n the wel lbore i s r e l a t i v e l y n e g l i g i bl e

compared t o th e pressur e in s id e the format ion , and the pr essure gra d ien t

in s id e th e formation remains a lmos t cons ta n t f or a t i m e .

PW* D

For the same reason, w e can exp ect t h a t t h i s phenomenon happens

f o r a l a rge we l lbo re s t o ra ge case because the f lu id head i n the wel lbore

does not increase as r a p i d ly i f t h e we l lb or e s t o r a g e i s l a r g e .

c a l l y , t h i s is t r u e .

pwD, versus the d imens ionless t i m e , tD , f o r t h e case when C = lo3'. Ad e f i n i t e l y s t r a i g h t p o r t i o n of t h e s o l u t i o n c an b e s e e n a t e a r l y t i m e s .

However, such a l a r ge we l lbo re s t o r age cons t an t is imprac t i ca l . Fo r

fea s ib l e d imension l es s we l lbo re s t o rage cons t an t s , CD of 10phenomenon cannot be seen.

bore pressure ,

10 and CD = l o 5 , r e spec t i ve ly .the solutions f o r t h e s e cases.

Theoreti-

Figure 26 shows the dimensionless wellbore pressure,

D

2 6- 10 , t h i sFigures 27 and 28 show the dimensionless w e l l-

f o r t h e cases when C =tD , Dversus dimensionless t i m e ,'wD '3

There is no ev iden t l i ne a r po r t i on of


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a h

I

0Q -

3


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0

a!A d

mmw

Hn

-4c40Fr

mmWdz

m o3 4

..Qc\l

c3HFr


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D

D

xQ+-s

Q

u

3

cnwz0


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One more cause of apparent cons tan t f lowrates is t h e i n e r t i a l ef -

f e c t of t h e l i q u i d i n t h e w el l bo r e. Figure 29 shows the dimensionless

f o r t h e case whentD ,versus dimensionless t i m e ,'wD 'wel lbore pressure ,3CD = 10 , s = 0 , and a2 = l o 7 . A s t r a i g h t p o rt i o n of t h e s o l u t i o n c an b e3 3

s e e n f o r t val ues from 6x10 t o 8 . 4 ~ 1 0 . However, the s t r a i gh t por t ionD

does not start from the zero t i m e , and i t does not cont inue for a long

t i m e . Therefore , although t h i s may happen, w e i n f e r t h a t t h i s is not the

main explanat ion for the apparent constant f lowrate phenomenon a t e a r l y 1

t i m e s .

,

2-2-4.4 Radius of I nve s t i gat ion

The p r e s s u re d i s t r i b u t i o n i n s i d e t h e r e s e r v o i r f o r s l u g tests

can be ca lcu la ted by obta in ing the real space solu t i on of Eq. 42 f o r

var ious parameter values . Figure 30 shows the pressure d i s t r ibu t ion in+ls i d e t h e r e s e r v oi r f o r a simple example case when C = 10 and s = a = 0at var ious t i m e s . The pressure gr ad ien t wi th respect t o r a d i a l d i s t a n c e

reaches a maximum value shortly after s t a r t of product i on and then th e

p res su re g rad i en t dec reases g radua l l y wi th t i m e ( i . e . , t he f l owra t e de-creases) and t he i nv es t i ga t i on r ad ius i nc reases wi th time.

pres su re ve r sus r ad ius i s l i n e a r i n t h e re s e r v o ir c l o s e t o t h e w e l l ; how-

ever , f a r f rom the w e l l pres su re versus r ad ius i s convex to the coardi-

n a t e of r a d i a l d i s t a nc e f o r e a r l y times.

rad iu s becomes concave t o th e ra d i a l coord inate a f t e r the f lowrate starts

decreas ing .

3D

A graph of

The graph of p res sur e versus

I n o r d er t o i n v e s t i g a t e t h e e f f e c t of a on the pressure d i s t r ibu-

3t i o n i n s i d e t h e re s e r v o ir , t h e case of C = 10 and s = 0 w a s s e l e c t e d

as a t yp i ca l example , and t he p res sure d i s t r i bu t i on i n s i de t he r e se r vo i r

I)


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0

0.I

0.2

0.3

a3 0.4a-

8.E

0.7

0.E

I I I

/I/

f

-3t, X I 0

FIG. 29: DIMENSIONLESS WELLBORE PRESSUREVS DIMENSIONLESS TIME FOR

3 7C = 10 , s = 0, AND a 2 = 10 AT EARLY TIMESD


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0IIbI Im

nVffi0FrE03c4wmWffiWEw

H

..0m


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w a s o b tain ed f o r v a r io u s a values a t c e r t a i n t i m e s .

show the resu l t s . A t e a r l y t i m e s , tD= lo2, a s l i g h t d i f f er e n ce e x i s t sbe tween th e p r e ssu re d i s t r ib u t io n fo r a = 0 and a2 = lo5, and the pres-

Figures 31 through 35

su re remains th e same as

Th e p r e ssu re d i s t r ib u t io n

8t h e p r e s s u r e f o r a2 = 10The pressure g rad ien t fo r

2 8t h e i n i t i a l f or ma ti on p r es s u re f o r a = 10 .5

f o r ci = 0 and ci2 = 10 remain the same, andstarts showing some response a t t = 3x10 .a = 0 and a = 10 are decreasing ( i . e . , t h e

3

D

2 5

f lo wra te i s decreas ing and t he wel lbore p re ssure i s recovering); however,

t h e p r e s s u re g r a d i e n t f o r a2 = 10 i s s t i l l i n c r eas in g a t tD = 10 . The8 42 8

wel lb o re p r e ssu re fo r a = 10 becomes g r e a te r th an th e i n i t i a l f ormat ionpressure , pi; i . e . , t h e l i q u i d level i n t h e w el lb or e o s c i l l a t e s .th e o th e r hand , th e we l lbo re p r e ssu re f o r a = 0 and a2 = 10

On

become almost5

4t h e s a m e as t h e i n i t i a l f or ma ti on p re s s ur e , pi , a t t = 5x10 . The w e l t -

D

b o re p r e s s u r e f o r a l l t h r e e a v a lu es are n e a r l y t h e same as t he i n i t i a l

formation pressure, pi a t tD = 10 .a v a lu e , t h e la ter t h e p r es s u re d i s t r i b u t i o n s ta r t s changing and the longer

5We co n c lu d e th a t t h e g r ea te r th e

i t conti nues changing, and th e smaller t h e p r e s s u r e g r a d i e nt w i l l be . Bow-ever, t h e i n v e s t i g a t i o n r a d i u s i t s e l f i s a lmost the same f o r a l l a valwps.A s i g n i f i c a n t p r e s s u r e dr op was found a t values o f r of 100 an d less.

D

Figure 36 shows th e e f f e c t o f t h e sk in f ac t o r , s , o n th e in v es t i-

gatio n radiu s. When th e sk in fa ct or i s l a r g e , t h e p r e ssu re d ro p o ccur s

mainly a t t h e san dface, and th e p r e ssu re g r ad ien t in s id e th e r e se rv o i r i s

small ( i . e . , t h e f l o w r a t e is s m a l l ) . However, the inv es t iga t io n r ad i i arealmost the same f o r a l l values of s. Then, we can say approximately that

t h e s k i n f a c t o r doe s n o t a f f e c t t h e i n v e s t i g a t i o n r a d i us g r e a t l y .


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\

M-

w

HmH

nz

mHn

m v )W

rlkl0 IIw c iu uwk l f f iF r oW F r


90/208

-74-

r%0Err

ncc)0rlXmI1

\

J0

02wmwp:U.(AzH

..hlcc)


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n

-

7-------r- -P- ------. -$----------I--- Y)0.-- e04xUlIIu

p:U0zW

-. .

v)2EwW5v)zH

..


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1!S

J

0--

I

nx

Frm0rlIIu

..Wpr;

WnU

U


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.I


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cD Figure 37 shows how the dimensionless wellbore storage constant,affects the investigation radius for the a = s = 0 case.similar figures for different a and s values.these figures are the same.

age constant, the deeper the investigation radius.

vestigation radius as the deepest point where p

the magnitude of the investigation radii for the dimensionless wellbore

storage constant C

likely to fall, are about 100, 300, and 800, respectively.

We can prepare

The! characteristics of

The greater the dimensionless wellbore stor-

If we define the in-

is 0.05, from Fig. 37D

= 103 4, 10 , and l o 5 , among which actual field data areD

As a summary of this section, the investigation radius depends on thedimensionless wellbore storage constant, (+,, and is not affected by thedimensionless number, a, or the skin factor, s, as much. The main pressuredrops often lie in the region less than 100 r distant from the well pro*duced.

W

2-2-4.5 Batch Injections

The solution presented in this study is not only for drawdown cases,

but also applies for sudden batch injection cases.

the dimensionless liquid level in the wellbore, 5 , the fact that theinitial dimensionless liquid level in the wellbore, %(O), is always -1,as shown in Section 2-1, is also true for injection tests. However, the

actual initial liquid level in the wellbore, x(O), is negative for thedrawdown case and is positive for the injection case.

From the definition of

2-2-4.6 Application of Solutions

-SUsing the effective wellbore radius, r = r e , and a new CO-

W W

ordinate t /C we can shift all of the solutions to the same domain on aD D


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I1clV f30I

IIII

0

t31

Dx

H

z0

..mm


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-81-graph.

w e l l tes t da ta an al ys is i n 1973, and by Ramey e t al.IL3 fo r conven t iona ls l u g tes t solut ions , which do not incl ude th e iner t ia:L e f f e c t , as an apprcxi-mation i n 1975.

This has been done by Earlougher and Kersch31i fo r conven t iona l

I n our s o l u t i o n s w e have onemore parameter bes ides

C e2', and the coord inate is t h e same, tD/C,,. Tables 8 and 9 show theD

accuracy of t h i s approximation fo r th e example cases when the product,

CDe2s = 10 and %e2s = 1 0 f o r d i f f e r e n t a va lues . S ince t he de t a i l eddisc uss io n of t h i s approximat ion w a s giv en i n Ramey e t a l . , l3 i t i s no t

4 5

repeated here .

range of

t i o n c u rv es f o r t h e s l u g tes t

This approximation appears good enough for a p r a c t i c a l

and ae2'. Figures 38 through 47 show the r es u l t i ng so lu-i n cl u di n g t h e i n e r t i a l e f f e c t of t h e l i q u i d

i n t he we l lbo re wi th r e spec t t o t he wel lbore p r e s s u r e , and Figs . 48 through57 show the same r e s u l t s w i th r e s p e c t t o t h e l i q u i d :level i n t h e w e l lb o re .Although th ese f ig ure s are i n r educed s izes , f u l l- s i z e f i g u r e s are avail-

a b l e . CDe2s andD D'

ae2' However, as can be seen,We might be ab l e t o obta in the matched poi n t s f or t / C

t he o re t i c a l l y f rom these so lu t i on cu rves .

i t is p r a c t i c a l l y i m p o s s i b l e t o select a unique curve matched t o th e a c t u a lf i e l d d a t a , be ca us e t h e r e are s o many similar curv es caused by t he new

parameter a e .2sThere are two ways t o u t i l i z e t h e s o l u t i o n s piroposed i n t h i s s t ud y

to o bta in some informat ion fo r re ser voi rs ( t he main purpose i s t o de te rmine

th e va lue of per mea bil i ty ). One way i s t o c h e c k t h e p o s s i b i l i t y t h a t t h ei n e r t i a of t h e l i q u i d i n t h e w el lb or e w i l l a f fe c t f i e l d pe rformance. This

procedure should be adopted when the e f f e c t of th e i n e r t i a of t h e l i q u idi n t h e w e ll b or e on t h e f i e l d d a t a i s no t clear; otherwise we should go

d i rec t l y t o t he s econd usage of the so lu t ions which w i l l be expla ined la te r .


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a

1 . ;

0.1

Id2 I d I 10V G D

2sFIG.38: DIMENSIONLESS WELLBORE PRESSURE vs tD/cD f o r dDe = 10


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1.1

0.

0.sQ,5

0.L

-3.2io- - 190 1 10 2io

2FIG. 39: DIMENSIONLESS WELLBORE PRESSURE VS t D / CD FOR CDe2s = 10

3to


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1.4

0.1

*' CI.:

2 s 3FIG. 40 : DIMENSIONLESS WELLBORE PRESSURE VS t D / C D FOR CDe = 10

8/6/2019 A Study of Inertial Eff

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A Study of Inertial Effect in the Wellbore