a study of inertial effect in the wellbore in pressure...
TRANSCRIPT
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
DOCTOR OF PHILOSOPHY
by
KIYOSHI SHINOHARA
A p r i l 1980
@ COPYRIGHT 1980
by
K I Y O S H I S H I N O H A M
ii
I c e r t i f y t h a t I have read t h i s t h e s i s and t h a t i n my opinion i t i s f u l l y adequate, i n scope and q u a l i t y , as a d i s s e r t a t i o n f o r t h e degree of Doctor of Philosophy.
( P r i n c i p a l Advisor)
I c e r t i f y t h a t I have read t h i s t h e s i s and t h a t i n my opin ion i t is f u l l y adequate, i n scope and q u a l i t y , as a d i s s e r t a t i o n f o r t h e degree of Doctor of Philosophy.
I c e r t i f y t h a t I have read t h i s t h e s i s and t h a t i n my opinion i t is f u l l y adequate, i n scope and q u a l i t y , as a d i s s e r t a t i o n f o r t h e degree of Doctor of Philosophy.
Approved f o r t h e Un ive r s i ty Committee on Graduate S tudies :
Dean of Graduate S tud ie s
iii
Dedicated t o my p a r e n t s , Kisaburo and Maki, t o my son, Daisuke,
and t o my wi fe , Kiyomi
iv
ACKNOWLEDGEMENT
The author wishes t o express h i s s i n c e r e apprec ia t ion t o Professor
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
r e sea rch adv i so r . The author is g r a t e f u l t o P ro fes so r s W. E. Brigham,
S . K. Sanyal, and R. N . Horne, Department of Petroleum Engineering, f o r
s e rv ing on h i s reading committee.
The author i s a l s o indebted t o the Nippon S t e e l Corporat ion, Stanford
Geothermal Program a t Stanford Un ive r s i ty , and t h e Ha l l ibu r ton O i l Well
Serv ices Company f o r f i n a n c i a l suppor t .
I n a d d i t i o n , thanks are due t o D r . M. Sengul, Marathon O i l Company,
f o r a s s i s t a n c e w i t h numerical Laplace inve r s ion . The scrupulous h e l p i n
e d i t i n g and typing of t h e manuscript by M s . E l i zabe th S . Luntze l and t h e
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 g r a t e f u l l y acknowledged.
F i n a l l y , t h e au thor wishes t o express 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 h e r cons t an t suppor t , encouragement, and understanding
dur ing t h i s work.
V
ABSTRACT
I n a "slug t e s t ," a f i n i t e amount of l i q u i d i s removed suddenly from
a s t a t i c w e l l , causing a p e r t u r b a t i o n which can be used t o o b t a i n informa-
t i o n on t h e r e s e r v o i r . I d e a l l y , t h e wel lbore l i q u i d would s t a r t t o move
up t h e w e l l t o r e p l a c e t h e removed l i q u i d , r each a maximum v e l o c i t y , then
begin t o d e c e l e r a t e as t h e l i q u i d level approaches 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 u s e f u l method as a short- t ime w e l l test . The
s o l u t i o n f o r t h i s problem i s o l d , and t h e problem has been inves t iga t ed by
many people. However, t h e s o l u t i o n s a v a i l a b l e thus f a r do n o t r i go rous ly
inc lude t h e i n e r t i a l e f f e c t of movement of t h e l i q u i d i n t h e wel lbore , and
do n o t exp la in p re s su re 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 t u a t i o n s i n t h e
wel lbore completely. I n e r t i a de l ays t h e s tart of movement of t h e wel lbore
l i q u i d , and momentum can cause i t t o eject above t h e f i n a l s t a t i c level .
O s c i l l a t i o n s i n p re s su re and l i q u i d l e v e l may r e s u l t .
Actua l ly , t h i s kind of problem can be involved i n t h e s t a r t of l i q -
u id production f o r a l l w e l l s , and i n t h e shut- in of h igh- ra te water in j ec-
t i o n w e l l s . It i s a l s o important t o d r i l l s t e m t e s t i n g of l i q u i d producing
format ions.
I n t h i s s tudy , a new and complete s o l u t i o n f o r t h i s problem i s ob-
ta ined and t h e e f f e c 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 s o prepared and used t o s tudy t h i s c lass of
problem. The a v a i l a b l e f i e l d d a t a were i n t e r p r e t e d and d iscussed . A new
s o l u t i o n f o r f low per iod d a t a a n a l y s i s f o r a c losed chamber test w a s pre-
pared as an extens ion of t h e gene ra l s lug test s o l u t i o n .
v i
TABLE OF CONTENTS
ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vi
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 . SLUG TEST ANALYSIS . . . . . . . . . . . . . . . . . . . . . . 5
2- 1 Mathematical Formulation . . . . . . . . . . . . . . . . . 5
2-2 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 . . . . . . 18 2-2-2.2 Early Time Solutions . . . . . . . . . . . 20
2- 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 . . . . . . . . . . . . . . 24
2-2-3.3 Albrecht-Honig Method . . . . . . . . . . 25
2-2-3.4 Comparison of Results . . . . . . . . . . 25
2-2-4 Results and Discussion . . . . . . . . . . . . . . 32
2- 2-4.1 Effect of a on Solutions . . . . . . . . . 32
2-2-4.2 Effect of CD on Solutions . . . . . . . . 55
2- 2-4.3 Effect of s on Solutions . . . . . . . . . 59
vii
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 7
2-3-1 Example 1 (Typical DST Flow Period Data) . . . . . 1 0 7
2-3-2 Example 2 (Understanding Field Data) . . . . . . . 112 2-3-3 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 . . . . . . . . . . . . . . . . . 133
4 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 138
5 . NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7 . APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A . Investigation of the Analytical Laplace Transform Inversion of Eq . 40 . . . . . . . . . . . . . . . . . . 148
B . Separation of Real and Imaginary Parts of Eq . 48 and Eq.49 . . . . . . . . . . . . . . . . . . . . . . . . . 153
B-1 For Early Times . . . . . . . . . . . . . . . . . . . 154 B-2 For Late Times . . . . . . . . . . . . . . . . . . . 155
C . Field Data Example . . . . . . . . . . . . . . . . . . . . 157
C-1 Well. Reservoir Data. and Measured Pressures . . . . 1 5 7
C-2 Pressure Drop Caused by Friction . . . . . . . . . . 166
viii
D . Deriva t ion of F i n i t e Dif ference So lu t ions . . . . . . . . . 169
D-1 Closed Chamber T e s t . . . . . . . . . . . . . . . . . 169
D-2 S l u g T e s t . . . . . . . . . . . . . . . . . . . . . . 172
D-3 Buildup Test . . . . . . . . . . . . . . . . . . . . . 173
E . Computer Program . . . . . . . . . . . . . . . . . . . . . 174
E-1 S t e h f e s t Method . . . . . . . . . . . . . . . . . . . 175
E-2 Albrecht-Honig Method . . . . . . . . . . . . . . . . 179
E-3 Computer Program t o Ca lcu la t e t h e P res su re Di s t r ibu- t i o n s I n s i d e The Reservoir . . . . . . . . . . . . . 183
E- 4 Computer Program t o Simulate Closed Chamber Test. Slug T e s t . and Buildup Test . . . . . . . . . . . . 189
ix
LIST OF TABLES
1. DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FOR c = 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 FOR cD = 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 . . . . . . . . . 54
8. DIMENSIONLESS WELLBORE PRESSURE VERSUS tD/CD FOR CDeZs = 10 . . 82 9. DIMENSIONLESS WELLBORE PRESSURE VERSUS tD/CD FOR CDe2s = 10 . . 84
4
5
10. CORRESPONDENCE BETWEEN THE SYMBOLS USED IN PETROLEUM ENGINEERING AND THOSE IN GROUND WATER HYDROLOGY . . . . . . . . . . . . . . 1 2 6
11. ADJUSTED PROPERTIES OF WELL-AQUIFER SYSTEMS REPORTED BY VAN DERKAMP16. . . . . . . . . . . . . . . . . . . . . . . . . I 2 7
C-1 WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 1 . . . . . 157
C-2 WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 2 . . . . . 1 5 9
C-3 WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 3 . . . . . 160 C-4 MEASURED LIQUID LEVEL (TAKEN FROM GRAPHS IN VAN DER KAMPI6) . . . 165
X
LIST OF FIGURES
. . . . . . . . . . . . . . . . . . 1 TYPICAL DRILL STEM TEST DATA 1 2
2 . TYPICAL DRILL STEM TEST DATA 2 . . . . . . . . . . . . . . . . . 2
. . . . . . . . . . . . . . . . . 3 SCHEMATIC DIAGRAM OF A SLUG TEST 6
4 . DIMENSIONLESS LIQUID LEVEL IN THE WELLBORE VS DIMENSIONLESS TIME FOR cD = 103 AND s = o BY NUMERICAL INTEGRATION OF EQ . 4 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 . DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR cD = 103 AND s = o . . . . . . . . . . . . . . . . . . . . . . 34
6 . DIMENSIONLESS LIQUID LEVEL IN THE WELLBORE VS DIMENSIONLESS
D TIME FOR C = l o 3 AND s = 0 . . . . . . . . . . . . . . . . . 35
7 . DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR cD = 1 0 3 AND s = o IN CARTESIAN COORDINATES . . . . . . . . .
TIME FOR CD = 103 AND s = 0 IN CARTESIAN COORDINATES . . . . .
S T O ~ G E C O N S T ~ T - F O R TO . . . . . . . . . . . . . . . . . .
36
8 . DIMENSIONLESS L I Q U I D LEVEL IN THE WELLBORE VS DIMENSIONLESS 37
9 . LOG a AND LOG a VS LOGARITHM OF DIMENSIONLESS WELLBORE 4 3
10 . . . . F O R s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 44
11 . . . . F O R s = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 45
12 . . . . F O R s = 2 0 . . . . . . . . . . . . . . . . . . . . . . . . . 46
1 3 . . . . F O R s = 5 0 . . . . . . . . . . . . . . . . . . . . . . . . . 47
14 . . . . 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 . . . . . . . . . . . . . . . . . 50
17 . EXPONENT OF DIMENSIONLESS WELLBORE STORAGE CONSTANT VS SKIN FACTOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
18 . LOG al AND LOG a2 VS SKIN FACTOR FOR CD = 1 . . . . . . . . . . 52
xi
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 . 58 Dl n
22. LOG t VS SKIN FACTOR FOR CD = 10’ . . . . . . . . . . . . . . 60 D, I
23. EFFECT OF SKIN FACTOR ON DIMENSIONLESS WELLBORE PRESSURE FOR c =io3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 D
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 oo ) . . . . . . . . . . . . . . . . . . 65
26. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR A LARGE DIMENSIONLESS WELLBORE STORAGE CONSTANT (CD = . . . 66
3 27. DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR CD = 10 AND 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 . . . . . . . 70
PRESSURE DISTRIBUTION INSIDE THE RESERVOIR FOR CD = 10
cD 3 7 29.
30. 3 AND s = a = 0 WITH DIMENSIONLESS TIME . . . . . . . . . . . . . . . 7 1
31. EFFECT OF a ON PRESSURE DISTRIBUTION INSIDE THE RESERVOIR AT t = 102, FOR cD = 103 AND s = o . . . . . . . . . . . . . . . . 73 D
3 FOR CD = 10 . . .AT tD -= 3x10 , A N D s = O . . . . . . . . . . . . . 74 32.
33. ... AT t = 10 , FOR C = 10 AND s = 0 . . . . . . . . . . . . . . 75 34. 76
35. ... AT t = 10 FOR CD = 10 AND s = 0 . . . . . . . . . . . . . . 77 36. 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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
38. DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10 . . . . . 86 39. DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe = 10 . . . . . 87 2s 2
x i i
4 0 . DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR Cge2s = 10 3 . . . . . . 41. DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe 2s = 10 4 . . . . . . 4 2 . DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe 2s = 10 5 . . . . - .
4 4 . DIMENSIONLESS WELLBORE PRESSURE vs tD/cD FOR CDe2s = 10 1 0 . . . . . 4 5 . DIMENSIONLESS WELLBORE PRESSURE vs tD/cD FOR CDe2s = 10 1 5 . . . . . 93
46 . DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10 20
4 7 . DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe 2s = 10 30
88
89
90
91
92
4 3 . DIMENSIONLESS WELLBORE PRESSURE vs tD/cD FOR cDe2s = 10 8 . . . . . .
. . . . . 94
95 . . . . . 48 .
4 9 .
5 0 .
51 .
52.
5 3 .
5 4 .
5 5 .
5 6 .
5 7 .
5 8 .
DIMEN3IONLESS LIQUID LEVEL IN THE WELLBORE VS tD/CD FOR S C e = 1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
. . . . . . . . . . . . . . . . . . . . 97
. . . . . . . . . . . . . . . . . . . . 98
D 2
2s 3
. ..vs t /c FOR CDe2s = i o
... VS tD/CD FOR C e = 10
D D
D ... vs tD/cD FOR CDe2s = io 4 . . . . . . . . . . . . . . . . . . . . 99
5 ... vs t /c FOR CDe2s = i o 2s 8 ... VS tD/CD FOR C e = 10
2s 1 0 ... VS t /C FOR CDe = 10
15 ... VS t /C FOR C e2' = 1 0
20 ... VS tD/CD FOR C e2' = 10
2s 30 ... VS t /C FOR CDe = 10
. . . . . . . . . . . . . . . . . . . . 1 0 0
. . . . . . . . . . . . . . . . . . . . 1 0 1
. . . . . . . . . . . . . . . . . . . . 102
. . . . . . . . . . . . . . . . . . . . 1 0 3
. . . . . . . . . . . . . . . . . . . . 1 0 4
. . . . . . . . . . . . . . . . . . . . 105
D D
D
D D
D D D
D
D D FIELD DATA IN EXAMPLE 1 * t t - * . - t . . . 1 0 9
5 9 . RESULT OF MATCHING IN THE LOG-LOG SCALE IN EXAMPLE 1 . . . . . . . . 110
6 0 . FIRST INTERPRETATION OF THE DATA IN EXAMPLE 2 . . . . . . . . . . . 1 1 3
6 1 . SLUG TEST SOLUTION IN DIMENSIONLESS FORM IN EXAMPLE 2 . . . . . . . 1 1 4
6 2 . COMPARISON OF ACTUAL DATA AND CALCULATED RESULTS IN EXAMPLE 2 . . . 1 1 5
6 3 . COMPARISON OF ACTUAL DATA AND CALCULATED RESULTS IN EXAMPLE 3 . . . 1 1 7
6 4 . PRESSURE DISTRIBUTION INSIDE THE RESERVOIR BASED ON SLUG TEST ANALYSIS IN EXAMPLE 3 . . . . . . . . . . . . . . . . . . . . . . 1 1 9
x i i i
65. PRESSURE DISTRIBUTION INSIDE THE RESERVOIR BASED ON BUILDUP TEST ANALYSISINEXAMPLE3 . . . . . . . . . . . . . . . . . . . . . 1 2 0
66. 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 FOR CLOSED CHAMBER TESTS WHEN cD = 103 AND s = o . . . . . . . . 134
FOR 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 4 8
x iv
1. INTRODUCTION
Often i n d r i l l stem tes ts ( h e r e i n a f t e r abbrevia ted as "DST"), flow
per iod d a t a are charac te r i zed by a p ressure t r a c e which i n c r e a s e s wi th
i n c r e a s i n g t i m e , showing t h e accumulation of l i q u i d i n t h e d r i l l s t r i n g .
I n some cases t h e pressure- time t r a c e i s l i n e a r a t t h e beginning of t h e
f low per iod, and then becomes concave t o t h e t i m e a x i s , showing an i n i t i a l
apparent cons tan t f l o w r a t e and then a decreas ing f lowra te . I n o t h e r cases
t h e pressure- time t r a c e i s concave t o t h e t i m e a x i s from t h e beginning of
t h e f low per iod, showing a decreas ing rate throughout t h e f low per iod
(see F igs . 1 and 2 ) .
I n cases where t h e formation p r e s s u r e is too low t o l i f t a column of
t h e r e s e r v o i r l i q u i d t o t h e s u r f a c e , t h e w e l l may a c t u a l l y s t o p f lowing
before t h e DST tester va lve i s c losed. This r e s u l t s because t h e 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 equal t o t h e i n i t i a l formation p ressure .
I n rare c a s e s wi th h igh p r o d u c t i v i t y format ions , t h e l i q u i d l e v e l i n t h e
wel lbore may i n i t i a l l y o s c i l l a t e around 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 event , t h e i n i t i a l p o r t i o n of a DST f low per iod may b e viewed
as a test i n which a column of l i q u i d whose head is equal t o t h e i n i t i a l
formation p ressure has been removed ins tan taneous ly .
cushion, t h e concept becomes more complicated, b u t i s s t i l l v a l i d .
consider t h a t a head of l i q u i d equ iva len t t o t h e d i f f e r e n c e i n p ressure
between i n i t i a l formation p r e s s u r e and t h e p ressure above t h e 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 "slug
test" by F e r r i s and Knowlesl i n 1954. The word "slug" r e f e r s t o an
-1-
-2 -
FIG.
/,------
,/ ;I 1: TYPICAL DRILL STEM TEST DATA 1
FIG. 2: TYPICAL DRILL STEM TEST DATA 2
-3-
i n i t i a l volume o f l i q u i d removed from t h e wel lbore t o i n i t i a t e flow.
Cooper e t a1. ,3 i n 1967, r epor t ed t h e r e s u l t s of a f i e l d test i n a s t a t i c
water w e l l from which a f l o a t w a s suddenly removed, g iv ing t h e appearance
of t h e ins tan taneous removal of a q u a n t i t y of water equal t o t h a t d i sp l aced
by t h e f l o a t . Actua l ly , t h e Cooper e t a l . d a t a i n t e r p r e t a t i o n method w a s
based on s t u d i e 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 involv ing t h e
cool ing of a ba tch of ho t water w a s r epor t ed by Beck, J aege r , and New-
s t ead 2 i n 1956.
4 Maier presented an approximate a n a l y s i s of t h e equ iva len t DST problem
van Pool len and Weber,5 i n 1970, and Kohlhaas,‘ i n 1972, appl ied i n 1970.
t h e Cooper e t a l . 3 s o l u t i o n t o DST f low per iod d a t a a n a l y s 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 tudy by Jaeger i n 1956,7 which included
a s u r f a c e r e s i s t a n c e s imi lar t o t h e s k i n e f fec t , ” ’ most r e c e n t works do
no t i nc lude wel lbore damage e f f e c t s . A s o l u t i o n inc lud ing 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 t h e u s e of
t h e s o l u t i o n s w a s n o t demonstrated. Papadopulous e t a l . presented ex-
tended r e s u l t s f o r t h e zero s k i n e f f e c t case i n 1973.
Although
12
The most complete d i scuss ion 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 presented i n 1975 by Ramey e t a l . I3 Three new s l u g tes t type-
curves were developed f o r t h e a n a l y s i s of f low per iod d a t a . This s tudy ,
which w a s reviewed i n t h e Earlougher14 monograph i n 1977, d id inc lude t h e
wel lbore s k i n e f f e c t .
The s o l u t i o n s mentioned thus f a r d i d no t cons ider t h e i n e r t i a of l i q -
u id moving i n t h e wellbore. In deep, h igh p roduc t iv i ty w e l l s , t h e i n e r t i a
of t h e l i q u i d i n t h e wel lbore cannot be neglec ted . Sometimes t h i s e f f e c t
causes o s c i l l a t i o n s of t h e p re s su re and t h e l i q u i d l e v e l i n t h e wel lbore
about s t a t i c p o s i t i o n s .
-4 -
The i n e r t i a l e f f e c t of l i q u i d i n t h e wel lbore was f i r s t i n v e s t i g a t e d by
Bredehoeft e t a l . I5 i n 1966, us ing an analog computer.
s o l u t i o n was n o t given.
b i l i t y assuming an exponen t i a l ly damped f l u c t u a t i o n w a s presented by van
d e r Karnpl' i n 1976; however, he d i d n o t provide a gene ra l s o l u t i o n and d i d
n o t i nc lude a s k i n e f f e c t . Thus f a r , no gene ra l s o l u t i o n inc lud ing 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 wel lbore wi th no o s c i l l a t i o n h a s been presented , t o our
knowledge.
However, a gene ra l
An approximate method t o determine t h e permea-
Another r e l a t e d w e l l t e s t i s t h e c losed chamber test , l7 which has
become popular f o r p o l l u t i o n p r o t e c t i o n e s p e c i a l l y f o r o f f s h o r e w e l l s .
It i s a d e v i a t i o n from conventional 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 a n a l y s i s of f low per iod d a t a from c losed chamber tests has never been
o f f e red , t o our knowledge, and bottomhole p re s su re d a t a f o r t h e f low period
have been d iscarded o r have n o t even been r epor t ed i n many cases.
The purposes of t h i s s tudy are t o d e r i v e a complete s o l u t i o n f o r t h e
s l u g test , inc luding 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 wel lbore
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 of parameters on t h e gen-
eral s o l u t i o n ; t o advance t h e understanding of s l u g tes t d a t a ; and t o s tudy
t h e a n a l y s i s of f low per iod d a t a from c losed chamber tests as an extens ion
of gene ra l s l u g test s o l u t i o n s .
2 . SLUG TEST ANALYSIS
The s l u g tes t problem w a s s t a t e d by Ramey e t al .13 as t h a t posed by a
formation whose i n i t i a l p re s su re , pi, is a t most less than t h e p re s su re
t 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 tes t packer i s set j u s t above t h e formation wi th t h e
d r i l l s t r i n g empty except f o r t h e a p p r o p r i a t e cushion l i q u i d , a l l t h e
elements f o r a s l u g tes t would b e p re sen t . A t zero t i m e , t h e tester
valve is opened t o expose t h e formation suddenly t o t h e cushion p re s su re ,
(po + Patm
formation begins t o produce, t h e produced l i q u i d i s s t o r e d wi th in t h e
d r i l l s t r i n g , and t h e l i q u i d level begins t o rise. The i n i t i a l production
r a t e w i l l be h igh and w i l l g r adua l ly d e c l i n e as accumulating l i q u i d i n t h e
d r i l l s t r i n g causes an inc reas ing back p re s su re . Usually t h e l i q u i d pro-
duc t ion ceases by i t s e l f , and i n some cases t h e l i q u i d l eve l i n t h e w e l l -
bore o s c i l l a t e s be fo re i t reaches equi l ibr ium.
) , i n t h e d r i l l s t r i n g above t h e valve ( s e e F ig . 3 ) . A s t h e
The mathematical formula t ion of t h e s l u g test problem is explained i n
Sec t ion 2-1. The s o l u t i o n f o r t h i s problem i s obta ined and i n v e s t i g a t e d
i n Sec t ion 2-2. F i e l d d a t a examples are d iscussed i n Sec t ion 2-3.
2-1 Mathematical Formulation
I n o rde r t o o b t a i n a s o l u t i o n which inc ludes 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 wel lbore , a momentum balance i s r equ i r ed . The l i n e a r
momentum balance i n t h e wel lbore s tates t h a t t h e ra te of change of momen-
tum is equal t o t h e n e t f o r c e on t h e l i q u i d :
-5-
-6-
X -
L
hf_
I i i I
I t
i I
SURFACE
- DRILL STRING
- yP
,TESTER VALVE
~ PAC K E R
- - -FORMATION
FIG. 3: SCHEMATIC DIAGRAM OF A SLUG TEST
-7-
L
2 2 - rr 2 p - rr 2 pf(L+x) g P d 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 parentheses i n t h e lef t- hand s i d e i s t h e product
of t he mass of l i q u i d i n t h e wellbore 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 wellbore. The f i r s t term i n t h e right-hand s i d e is t h e
force caused by t h e wel lbore bottomhole pressure .
fo rce caused by t h e atmospheric pressure .
force , and t h e f o u r t h term is t h e f r i c t i o n fo rce .
The second term is t h e
The t h i r d term i s t h e g r a v i t y
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 wellbore, r i s t h e i n s i d e f P
rad ius of t h e d r i l l s t r i n g , t h e tub ing , o r t h e casing p ipe i n which t h e
produced l i q u i d e n t e r s , L is t h e l i q u i d l eng th whose head i s equ iva len t
Pa tm t o t h e i n i t i a l formation p re s su re , pi, minus atmospheric p re s su re ,
x is t h e l i q u i d l eve l i n t h e wellbore measured from t h e s t a b l e p o i n t , p
i s t h e wel lbore bottomhole p re s su re (we w i l l c a l l t h i s t h e "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^
fac to r .
W
Rearranging Eq. 1:
dx dpf - 2 d x
d t ( g ) 2 f (LSX) - - = Pf(L+X) 2 + Pf dt d t pw P a t m - pf(L+x)g
For a cons tant compress ib i l i t y l i q u i d , t h e fol lowing r e l a t i o n holds i n
t he wellbore approximately:
-8-
Then, Eq. 2 becomes as fol lows:
dx C f
d t P
Since t h e head sf t h e l i q u i d l e n g t h , L, i s equ iva len t t o t h e i n i t i a l f o r-
mation p ressure , pi, minus t h e atmospheric p ressure ' Patm:
PW'Pi 'w"a t m - L g = -
pf pf (5)
So, Eq. 4 becomes:
f dx a t d t
d t P
PW'Pi
pf - --
For t h e formation, i f we adopt t h e fo l lowing assumptions, t h e w e l l -
known d i f f u s i v i t y equation'' der ived from t h e c o n t i n u i t y equat ion and
Darcy's l a w can be used.
1 ) Horizonta l , i s o t r o p i c , homogeneous, i so the rmal , r a d i a l , i n f i n i t e
porous medium of cons tan t th ickness , h , p o r o s i t y , 4 , and permeab i l i ty , k ,
2) Constant l i q u i d v i s c o s i t y , 1-1, and s m a l l cons tan t t o t a l system
c o m p r e s s i b i l i t y , c t '
3 ) No f l u i d f low a c r o s s t h e h o r i z o n t a l boundaries and n e g l i g i b l e
g r a v i t y e f f e c t .
4 ) The square of t h e p ressure g r a d i e n t wi th r e s p e c t t o r a d i a l d i s t a n c e
is n e g l i g i b l e .
-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,. '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 level
in the wellbore, and p
(we will call this "dimensionless wellbore pressure").
D D W
D D
is the dimensionless wellbore bottomhole pressure WD
-10-
Using these dimensionless variables, the following equations are de-
rived :
From Eq. 7: n
aPD aPD +- e- - -- aLPD 1
2 rD ar, at, arD
From Eq. 6: .-I
I dPwD dxD (Lfx) -- f (Lfx) dxD dxD Pf cfg - - -- 4r dtD dtD 2 dtD dtD [($,’+ P
In 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-filled
reservoirs.
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
-11-
the order of the (SJ term is the same as that of the pressure drop term 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 represents
wctrw
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 XD a * -
'wD 2 -k XD dtD
In 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:
-12-
The i n i t i a l l i q u i d l e v e l i n t h e wel lbore i n dimensionless form i s a l-
w a y s- l , because:
Pfgx (t=O>
(Pa tm+P f Lg - [ P f g { L+x ( t = 0 'i +P a tm 1 x ( t =O) = D D
- x (t=O) x (t=O)
- -
- - - 1
This i s t h e second i n i t i a l condi t ion.
The i n i t i a l v e l o c i t y of t h e l i q u i d l e v e l i n t h e wel lbore can be con-
s ide red zero f o r t h e usua l s l u g tests; however, t o make t h e s o l u t i o n
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 of l i q u i d l e v e l i n t h e
wel lbore i s given as some value . This cond i t ion i s t h e t h i r d i n i t i a l
cond i t ion :
= x ' ( t =O) D D tD'O
Next w e w i l l consider t h e boundary cond i t ions . Since w e assumed
t h a t t h e r e s e r v o i r i s i n f i n i t e :
R i m pD(rD, tD) = 0
r-fo3 D
This i s t h e f i r s t boundary cond i t ion .
-13-
From a material balance on t h e wellbore:
Using t h e dimensionless v a r i a b l e s de f ined i n Eqs. 8 t o 11, t h e fol lowing I I
equat ion is obta ined:
dxD 1 - = - -
r =1 dtD cD D
C
bore s t o r a g e cons tan t def ined as fol lows:
is t h e dimensionless wel lbore s t o r a g e c o n s t a n t Y 2 l and C i s t h e w e l l - D I
~
C 2 27rr h@ct
c = W
2 7rr
c = P pfg
10 A s t h e t h i r d boundary cond i t ion , w e can u s e t h e fo l lowing equat ion
obta ined from a p r e s s u r e balance a t t h e sandface based on t h e d e f i n i t i o n
of t h e s t e a d y- s t a t e s k i n f a c t o r . 899
I
A s a summary of t h i s s e c t i o n , t h e fo l lowing equat ions are obta ined
as t h e r e s u l t of mathematical formulat ion of t h e s l u g test problem.
I
i
-14-
L
L
Initial Conditions:
Boundary Conditions:
2-2. Solutions
x (t -0) = - 1 D D-
D -=A(%) dx
D r =1 D dtD
PWD =['D-'(%)] rD=l
Since Eq. 13 and E q s . 18 through 27 are linear equations, solutions
may 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
-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 the D D Laplace transform variable.
Substituting the initial condition Eq. 19, Eq. 28 becomes:
44 This 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 is 0 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
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 E q s . 32 and 3 3 into Eq. 3 4 :
- 'wD = B { Ko(fi) + s f i K1(a) }
Applying the Laplace transformation with respect to time t o Eq. 18:
- - c s b 2 2- *wD a { R XD - Rx (t -0) - x;)(tD=O) 1 + XD = - D D-
- D' x is the Laplace transform of x
From E q s . 35 and 36: D
Applying the Laplace transformation with respect to time to Eq. 2 4 :
r =1 D
- Rx, - x (t =O) = - - D D
-17-
From E q s . 33 and 38:
Substituti
C D { G D D D -x (t = O ) l
f i e K1 (a) B =
Eq. 39 into Eq. 37:
From E q s . 35, 39 , and 40:
CD{sfiK1(fi) + KO(fi)}{-XD(tD=O) + a 2 !Lx'(t =o)) D D -
n e - 'wD (a2!L2+CDs!L+l) &K1 (a) + CDRKO (,a)
From E q s . 32, 39, and 40:
(39)
(40 )
( 4 1 )
Equations 4 0 , 41, and 42 are the solutions for x DY PwD$ and P D in
the 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.
-18-
2-2-2.1 T r i a l f o r A n a l y t i c a l Invers ion
Appendix A shows t h e mathematical t rea tment of t h i s t r i a l . A s
shown i n Appendix A, i t wzs n o t p o s s i b l e t o f i n d t h e po les of t h e func-
t i o n a n a l y t i c a l l y except a t t h e o r i g i n .
po les (except a t t h e o r i g i n ) e x i s t o r no t ( i . e . , whether t h e v a l u e s of B
which s a t i s f y E q s . A-19 and A-20 e x i s t o r n o t ) , w e assumed t h a t t h e sum
of r e s i d u e s -ike is zero. Then, from Eq. A-21:
I n o rder t o check whether t h e
-
where:
2 A,(u) = (a2u4-C su +1) J1(u) - C UJ (u) D D O
2 A2(u) = (a2u4-c D su +I) Y 1 (u) - C ~ U Y ~ ( U )
L L A,(u) = {a u xD(tD=O) - C sx ( t -0) - aL$(tD=O)} uJl(u) - C D D x ( t D =0) Jo(u) D D D-
2 2 2 A4(u) = { a u xD(tD=O) - C sx ( t =O) - a ~ ' ( t -0)) UY,(U) - C x ( t =o) y0(u) D D D D D- D D D
J and J are t h e Bessel func t ions of t h e f i r s t k ind, o rder zero and
u n i t y . Y and Y are t h e Bessel func t ions of t h e second kind, o rder zero and 0 1
u n i t y . I n o rder t o check whether t h e assumption t h a t t h e r e i s no po le 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 a s i n t e g r a t e d numerically us ing t h e Rom-
berg method.23 Figure 4 shows an example of t h e r e s u l t s . The s o l u t i o n be-
2 5 2 haves reasocably f o r a v a l u e s less than 10 ; however, t h e s o l u t i o n f o r a
6 v a l u e s g r e a t e r than 10 does no t make sense f o r t h e example case when
-19-
-20-
3 - CD = 10
Eq. A-21, p lays an important r o l e f o r a2 v a l u e s g r e a t e r than lo6 f o r
t h i s example case .
w a s wrong.
i n t h e wel lbore o s c i l l a t e s f o r some cond i t ions .
t h e left- hand s i d e of t h e Laplace p lane t o a l low converging o s c i l l a t i o n t o
happen.
and s = 0. This means that t h e sum of t h e r e s i d u e s , >Re, i n -
Then, t h e assumption t h a t o t h e r r e s i d u e s are n e g l i g i b l e
This a l s o can be guessed from t h e f a c t t h a t t h e l i q u i d level
The po le must e x i s t on
I n conclus ion, i t w a s n o t p o s s i b l e 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 u s e numerical Laplace transform invers ion t i o n .
methods t o o b t a i n t h e e n t i r e s o l u t i o n . This procedure w i l l be d iscussed i n
t h e fo l lowing s e c t i o n s . The e a 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 s
can be obta ined a n a l y t i c a l l y , and w i l l be d iscussed i n t h e remaining p a r t
of t h i s s e c t i o n . The same d i s c u s s i o n i s a p p l i c a b l e t o Eqs. 4 1 and 42, be+
cause t h e i r denominators are t h e same as t h a t of Eq. 40, and i t seems t h a t
t h e r e i s no p o l e c a n c e l l a t i o n between denominator and numerator.
2-2-2.2 Ear ly T ime S o l u t i o n s
Rearranging Eq. 40:
1 2 2 KO q m c1 R + C*SR + 1 + c,a
R XD(tD=o) + { cr2x;(tD=0) - R x = D
(48) S i m i l a r l y , from Eq. 41:
-21-
For early times, we can say that R + m .
modified Bessel functions: 24
Then, from the characteristics of
Substituting Eq. 50 into E q s . 48 and 49:
1 - (51) + CI)SR + 1 + c,a R R
x = D
x (t =O) xD(tD=O) x'(t =O) - + a s R + a D D
R +
!L2 2 3 a 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> = - 2 a
These early time solutions will be used to check the solutions obtained by
the numerical Laplace transform inversion methods in the next section.
-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- e I
ments:
1 K (a) 2- a 1 (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 R D - 1 R x = D
2 + a xA(tD=O) as + 0
+ O a s R + O
Then :
2 XD(tD) = a x'(t =O) 6(tD)
PWDCtD) = 0
D D
-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 e w e are now looking a t t h e l a t e
t i m e zone, 6 ( t D ) = 0 .
so :
x ( t ) = O D D
These r e s u l t s f o r l a t e times agree wi th an understanding of t h e phys ica l
phenomena.
2-2-3 Numerical Laplace Transform Invers ion
A s shown i n t h e previous s e c t i o n , w e cannot obt:ain t h e real space
s o l u t i o n a n a l y t i c a l l y except f o r t h e approximate so1ut:ions a t e a r l y times
and l a t e t i m e s . Therefore , t h r e e numerical Laplace t ransform invers ion
methods were used i n t h i s study. The r e s u l t s obtained by t h e s e methods
were compared mutual ly , as w e l l as t o t h e r e s u l t obta ined from 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 .
f o r c losed chamber t e s t a n a l y s i s and i s explained i n Sect ion 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 d i f f e r e n c e computer program i s presented i n Appendix E ,
2-2-3.1 S t e h f e s t Method
A simple and a c c u r a t e numerical Laplace transform invers ion method
This method i s based on t h e fo l lowing w a s presented by S t e h f e ~ t ~ ~ i n 1970.
a lgor i thm der ived assuming a s p e c i a l p r o b a b i l i t y densi ty28 wi th which t h e
expec ta t ion of a func t ion becomes similar t o t h e Laplace t ransform of t h e
func t ion .
-24-
N N N 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 also
causes 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 method
is based on the following algorithm which is an approximation of the Fouridr
cosine transform inversion of the function assuming f(t) is a real function.
?(a)} + c N Re{ r (a+y)}cos (?)I 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-
-25-
2-2-3.3 Albrecht-Honig Method
The t h i r d method w a s presented by Albrecht and Honig3' i n 1977.
This method is a v a r i a t i o n of t h e Ve i l lon method, and i s based on t h e follow-
i n g a lgor i thm.
-a t
N
k- 1 + 1 R e { i( a + y) } (-l)k
+ e T R(N) f a r 0 < t < T/2 (69)
The d e f i n i t i o n of v a r i a b l e s and c o n s t a n 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 f u n c t i o n shou3ld
be reasonably smooth.
method i s presented i n Appendix E.
M w a s taken as 16. The computer program f o r t h i s
2-2-3.4 Comparison of R e s u l t s
Since t h e s e t h r e e numerical Laplace transform i n v e r s i o n methods
have l i m i t a t i o n s on t h e f u n c t i o n t o which they can b e app l ied ( t h e f u n c t i o n
should b e smooth, o r should n o t have d i s c o n t i n u i t i e s o r r a p i d o s c i l l a t i o d ) ,
i t is necessary t o test t h e r e s u l t s obta ined by t h e s e numerical Laplace tzans-
form invers ion methods t o a s s u r e r e l i a b i l i t y . In o r d e r t o i n v e s t i g a t e t h e
r e l i a b i l i t y of t h e s o l u t i o n s , t h e case of C
a t y p i c a l case.
by numerical Laplace t ransform invers ion methods and 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 .
shown i n Fig . 5.
test s o l u t i o n s which neg lec t i n e r t i a l e f f e c t (a = 0) and t h e p resen t 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
Solu t ions f o r t h i s case f o r v a r i o u s a va lues were obta ined
The dimensionless wel lbore p r e s s u r e s f o r t h i s example c a s e are
One of t h e main d i f f e r e n c e s between t h e previous s l u g
The wel lbore p ressures drop i n s t a n t l y (p = 1)
-26-
f o r t h e i n e r t i a - l e s s cases, b u t pass through a minimum (maximum i n p ) on
Fig. 5. Furthermore, t h e r e is an "overshooting" f o r each va lue of a
shown on Fig. 5.
This behavior causes computation d i f f i c u l t i e s .
D 2
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 i n v e s t i g a t e t h e overshoot ing p o r t i o n of t h e s o l u t i o n a t e a r l y
2 3 times, s o l u t i o n s f o r t h e a = 10
f e r e n c e and S t e h f e s t 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 t h e modified Bessel f u n c t i o n s in
order t o u s e t h e computer program developed by t h e Marathon O i l Company f o r
t h e Ve i l lon a lgor i thm, and t h e s e approximations were no t r easonab le f o r t h i s
t i m e range. Thus t h e Ve i l lon method was n o t used f o r t h i s test. Table 1
shows t h e r e s u l t s .
Honig method a g r e e u n t i l t = 0.9; however, a f t e r t h i s t i m e , t h e r e s u l t from
t h e Albrecht-Honig method starts o s c i l l a t i n g , and a f t e r t = 20, they a g r e e
again .
t h e f i n i t e d i f f e r e n c e method. From t h e s e r e s u l t s w e i n f e r t h a t t h e Albrecht-
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 of t h e f u n c t i o n t o app ly
t o t h e s u b j e c t problem, and t h e r e s u l t by t h e S t e h f e s t method i s more re-
l i a b l e than t h a t from t h e Albrecht-Honig method.
f o r t h e S t e h f e s t method wi th N = 16 happens f o r o t h e r combinations of CD, s,
and a va lues , b u t only a t t h e s p e c i a l t i m e of t
no t happen f o r t h e S t e h f e s t method wi th N = 10.
r e s u l t i s n o t known; however, when N i n c r e a s e s , t h e number of odd r e s u l t s i n-
creases. We suspect t h a t t h i s might be caused by roundoff e r r o r increased
by t h e l a r g e N value . 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 h e S tehfes tmethodwi th N = 1 6 i s c l o s e r t o t h e r e s u l t from t h e
case were c a l c u l a t e d by t h e f i n i t e d i f -
The r e s u l t by t h e S t e h f e s t method and by t h e Albrecht-
D
D = 0.9 - 20 agrees wi th t h e r e s u l t from The r e s u l t by S t e h f e s t f o r t D
The s t r a n g e p o i n t a t t = 2 D
= 2.
The reason f o r t h i s odd
This phenomena does D
-27-
TABLE 1: DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FOR 3 3 CD = 10 , s = 0, and a2 = 10 BY VARIOUS METHODS
STEHFEST METHOD ALBRECHT-HONIG N = 10 N = 16 METHOD D t
0.10000D 00 0.20000D 00 0.30000D 00 0.40000D 00 0.50000D 00 0.60000D 00 0.70000D 00 0.80000D 00 0.90000D 00 0.10000D 01 0.20000D 01 0.30000D 01 0.40000D 01 0.50000D 01 0.60000D 01 0.70000D 01 0.80000D 01 0.90000D 01 0.10000D 02 0.20000D 02 0.30000D 02 0.40000D 02 0.50000D 02 0.60000D 02 0.70000D 02 0.80000D 02 0.90000D 02 0.10000D 03 0.20000D 03 0.30000D 03 0.40000D 03 0.50000D 03 0.60000D 03 0.70000D 03 0.80000D 03 0.90000D 03 0.10000D 04 0.20000D 04 0.30000D 04 0.40000D 04 0.50000D 04 0.60000D 04 0.70000D 04 0.80000D 04 0.90000D 04
0.0214529976 0.0578522690 0.1019990041 0.1510830739 0.2033681582 0.2575993243 0.3126960014 0.3682835860 0.4239998059 0.4782305760 0.9210326406 ( 1.1209044423 1.1501383129 1.1153425412 1.0730757852 1.0409903521 1.0204467957 1.0084321135 1.0018058894 0.9925003643 0.9868769238 0.9817682235 0.9773025744 0.9732328125 0.9694145433 0.9657707501 0.9622576376 0.9588482118 0.9281217579 0.9009117264 0.8758621810 0.8524251531 0.8302951976 0.8092746401 0.7892235773 0.7700368590 0.7516319677 0.5998270822 0.4882504455 0.4029095646 0.3361799338 0.2832016559 0.2406405455 0.2061099426 0.1778534067
0.0214507363 0.0578439567 0.1019870617 0.1510840462 0.2033970438 0.2577032417 0.3131997766 0.3681005270 0.4211825268 0.4866216143 -1.6019925079) 1.1239439997 1.1592637356 1.1193314975 1.0679626756 1.0308154293 1.0106315690 1.0021360619 0.9996989531 0.9933859569 0.9863506528 0.9815167752 0.9772050251 0.9731945947 0.9694000645 0.9657663945 0.9622578679 0.9588506142 0.9281251965 0.9009144810 0.8758645594 0.8524275719 0.8302974954 0.8092769553 0.7892259629 0.7700393313 0.7516344375 0.5998351744 0.4882674117 0.4029255755 0.3361788777 0.2831698941 0.2405712959 0.2060032650 0.1777147562
0.0214505620 0.0578434726 0.1019861843 0.1510828088 0.2033259769 0.2576497229 0.3130012126 0.3686016063 0.4238777015 0.4753216573 1.4896993142 1.0351876301 1.2738960311 1.2260014287 0.9760606407 1.1037051060 1.0754371487 0.9435804441 1.0521378927 0.9926207621 0.9862765264 0.9818258892 0.9777967584 0.9736910945 0.9696838148 0.9659079126 0.9623413326 0.9589228953 0.9282446282 0.8999731574 0.8753262051 0.8525508066 0.8304213432 0.8094012954 0.7893507636 0.7701646121 0.7517603409 0.5999662337 0.4884032500 0.4030656960 0.3363229901 0.2833177809 0.2407225771 0.2061577871 0.1778724918
FINITE DIFFERENCE SOLUTION
0.0296930354 0.0700351029 0.1168018516 0.1677057573 0.2212252448 0.2762451511 0.3319048262 0.3875206592 0.4425412130 ' 0.4965186962 0.9267022871 1.1258199853 1.1618849632 1.1259700598 1.0774909751 1.0415048465 1.0218452864 , 1.0138078186 1.0116362626
-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 the overshooting p o r t i o n of t h e s o l u t i o n
than i s t h e r e s u l t by t h e S t e h f e s t method wi th N = 10. The ex i s tence
of t h e overshooting p o r t i o n of t h e s o l u t i o n a t e a r l y t i m e s w a s assured
by t h e Ve i l lon method f o r smaller a va lues . Table 2 p resen t s t h e re-
s u l t s . Although no t shown, t h e s e r e s u l t s ag ree wi th t h e e a r l y t i m e
a n a l y t i c a l s o l u t i o n expressed by Eq. 56.
Table 3 shows a comparison of t h e
case ( see a l s o Fig . 5 ) .
The r e s u l t by
7 f o r t h e a2 = 10
methods agree q u i t e w e l l .
r e s u l t s by t h e s e four methods
The r e s u l t s from t h e s e four
t h e S t e h f e s t method wi th 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 agrees b e s t wi th o t h e r
r e s u l t s .
Table 4 shows t h e comparison of t h e r e s u l t s by a l l f o u r methods
8 f o r a2 = 10 , f o r which t h e s o l u t i o n has an o s c i l l a t i o n .
t h e Albrecht-Honig method and by t h e Ve i l lon method agree very w e l l .
This is probably because t h e a lgor i thms f o r these two methods are similar.
However, t h e r e s u l t s show a h igher ampli tude of t h e o s c i l l a t i o n than do
t h e r e s u l t s by t h e S t e h f e s t method and 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 .
This might be caused by t h e h igher s e n s i t i v i t y t o t h e o s c i l l a t i o n of t h e
func t ion of t h e s e methods, which appeared a t t h e overshoot ing p o r t i o n of
t h e s o l u t i o n f o r t h e a2 = 10
r e s u l t by t h e S t e h f e s 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 than t h e r e s u l t by t h e S t e h f e s t method wi th
N = 10.
The r e s u l t s by
3 case i n t h e Albrecht-Honig method. The
From t h e s e comparisons of t h e r e s u l t s obta ined by va r ious methods
f o r t h e example case , w e conclude t h a t s o l u t i o n by t h e S t e h f e s t method
wi th N = 16 gives t h e most r e l i a b l e s o l u t i o n f o r p resen t problems. The 4
-29-
TABLE 2: DIMENSIONLESS WELLBORE PRESSURE VERSUS DIMENSIONLESS TIME FOR
AND s = 0 BY THE VEILLON METHOD 2 3 VALUES WHEN CD = 10 SMALL a
-4 a2 = 10
tD
5.00000000D-06 6.00000000D-06 7.00000000D-06 8.00000000D-06 9.00000000D-06 1.00000000D-05 2.00000000D-05 3~00000000D-05 4~00000000D-05 5.00000000D-05 6.00000000D-05 7~00000000D-05 8.00000000D-05 9~00000000D-05 1~00000000D-04 2~00000000D-04 3~00000000D-04 4~00000000D-04 5~00000000D-04 6~00000000D-04 7~00000000D-04 8~00000000D-04 9~00000000D-04 1~00000000D-03 2*00000000D-03 3~00000000D-03 4*00000000D-03 5*00000000D-03 6*00000000D-03 7 - 00000000D-03 8*00000000D-03 9*00000000D-03
8.20475719D-02 1.07018134D-01 1.33722237D-01 1.61899765D-01 1.91326994D-01 2.218070051)-01 5.52322567D-01 8.61512986D-01 1.09554509M-00 1.23809928M-00 1.29629459D+00 1.28982849IHOO 1.24256281Dt-00 1.17674162IHOO 1.10970029IHOO 1.01220135IHOO 1.00412114IHOO 1.00377037IHOO l.O0246443D+OO 1.00186191D+00 1.00148557IHOO 1.00121204IHOO 1.00100964IHOO 1.00085534IHOO 1.00026381IHOO 1.00010884IHOO l.O0003922D+OO 9-99999204D-01 9.99972619D-01 9.99953199D-01 9.99938042D-01 9.99925627D-01
tD
5 - 00000000D-08 6 00000000D-08 7*00000000D-08 8 - 00000000D-08 9~00000000D-08 1~00000000D-07 2- 00000000D-07 3- 00000000D-07 4*00000000D-07 5*00000000D-07 6 * 00000000D-07 7~00000000D-07 8*00000000D-07 9~00000000D-07 1~00000000D-06 2*00000000D-06 3~OOOOOOOOD-06 4~00000000D-06 5 00000000D-06 6~00000000D-06 7 * 00000000D-06 8*00000000D-06 9~00000000D-06 1~00000000D-05 2 00000000D-05 3~00000000D-05 4~00000000D-05 5 00000000D-05 6~00000000D-05 7~00000000D-05 8*00000000D-05 9*00000000D-05
PWD
8.20475740D-02 1.07018138D-01 1.33722244D-01 1.61899776D-01 1.91327010D-01 2.21807028D-01 5.52322800D-01 8061513818D-01 1*095547OlDtOO 1*23810272D+00 1.29629985IHOO 1-28983565D+00 1- 242571751~00 1*17675205IHOO 1~10971184IHOO 1*01221536~+00 1*00413878IHOO 1*00379068IHOO 1.00248714IHOO 1~00188678IHOO 1*00151244D+OO 1.00124077D+00 1*0010401OIHOO 1.00088745IHOO 1 00030922IHOO 1*00016446D+OO 1~00010345D+00 1*00007101D+OO 1-00005128IHOO 1- 00003816IHOO 1.00002887Dt-00 1*00002196D+OO
-30-
z 0 GI Gl ti
9 m
r l - N m 0 0
i3 a 5 a rl 0 0
m rl 0 m a co m h 0
rn m c\i N h m 0
m 0 m rl O U rlrl e b rlrl
U 0 a U a
U m co a m e rl 0 N
a m 4 m U m 0 m N
m 0 h co m \c) m m N
m ;d a 0 m
0 W rl 0 rl
a m b N c o d h l m
m 0 4 a m co N m a
m 4 a m N U rl a W
m h m m rl a m N a
U U co m
h- U U r U
m * m rl h 0 co U
rl U m m m m co U 0
m rl 0 co h \c) 03 m N . . . . . . . . . . . . . . . . . . E! 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
. . . . . . . . . . . . . . . . . . ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
n LI
a 0 a m a h m N N 0
0
h m m a m m m 0 m 0
0
a h h m h h U 0 to 0
0
N 0 N r-i 43 m m 0 rl rl
0
m U a rl co m a rl U rl
0
U 0 co N m h 0 hl h rl
0
h N N U a3 co 0 N 0 N
0
m rl 0 b a rl m rl m N
0
co co N a co m N 0 a N
0
m W 0 e 0 a rl co co N
0
U h a m co co h N m a 0
0 m m m m 03 N rl a a 0
N U N m N U co h U U 0
m 0 m h a 0 m W U m 0
N m m m 0 03 N co m N
0
a rl m co h h m N N 0
0
m m m m m m m m m U e u e U e U U U 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I n 0 0 0 0 rl
n 0 0 0 0 N
n 0 0 0 0 m
n 0 0 0 0 U
n 0 0 0 0 m
n 0 0 0 0 a
n 0 0 0 0 h
n 0 0 0 0 co
n 0 0 0 0 m
n 0 0 0 0 rl
n 0 0 0 0 N
n 0 0 0 0 m
n 0 0 0 0 .t
n 0 0 0 0 in
n 0 0 0 0 a
n 0 0 0 0 h
n 0 0 0 0 m
n 0 0 0 0 m - ._ - . . . . . . . . . . . . . . . . . .
l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m a 0 X H
!2
W
n 0 m H W c E w Fr X w H m
Q d
II
z
0 d
II
z
a U
-31-
. . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
I l l
~ c a ~ ~ n n n ~ n n ~ n n n ~ n a a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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
-32-
S t e h f e s t method wi th 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
following s e c t i o n s , al though t h e s o l u t i o n s were checked by o t h e r 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 should decrease as t h e
s o l u t i o n has more o s c i l l a t i o n s . Bessel func t ions are c a l c u l a t e d using the
polynomial approximations, 32 as shown i n the computer programs i n Appendix
E .
double p r e c i s i o n . Sometimes r e s u l t s show a s l i g h t o s c i l l a t i o n around t h e
c o r r e c t answers i f t h e program i s run i n s i n g l e p r e c i s i o n . This e n t i r e
d i scuss ion is a p p l i c a b l e no t only t o t h e func t ion p ( s e e Eq. 41), b u t
a l t o t o t h e func t ion % ( s e e Eq. 3 9 ) and pD ( s e e Eq. 4 2 ) .
The numerical Laplace t ransform invers ion program should be run i n
WD
2-2-4 Resu l t s and Discuss ion
The s o l u t i o n s were obta ined by t h e S t e h f e s t method wi th N = 1 6 f o r
a range of parameters. These s o l u t i o n s w e r e checked by t h e r e s u l t s ob-
ta ined by o t h e r numerical Laplace t ransform i n v e r s i o n methods, and by 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 .
Based on t h e s e s o l u t i o n s , t h e e f f e c t of parameter va lues on t h e solu-
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 h e s o l u t i o n are i n v e s t i g a t e d i n t h i s
s e c t i o n .
2-2- 4.1 E f f e c t of a on So lu t ions
The dimensionless number a w a s def ined i n Eqs. 1 6 and 1 7 :
o r :
-33-
The dimensionless number a r e p r e s 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 wel lbore on 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 inc lud ing t h e ine 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 wel lbore , t h e case when C = 10
as a t y p i c a l example, and t h e s o l u t i o n s f o r va r ious a values w i l l be
obta ined f o r t h i s case. Figure 5 shows t h e dimensionless wel lbore pres-
pwD, versus t h e dimensionless t i m e , t and Figure 6 shows t h e d i - s u r e ,
mensionless l i q u i d level i n t h e wel lbore , %, ver sus t both i n semilog
coord ina te s . F igures 7 and 8 show t h e same r e s u l t s on Cartesian coordi-
n a t e s . The t abu la t ed r e s u l t s obta ined by t h e S t e h f e s t method are given
i n Tables 5 and 6 .
I n o rde r t o i n v e s t i g a t e t h e
3 and s = 0 w i l l be used D
D’
D’
13 The s o l u t i o n f o r a = 0 ag rees wi th t h e previous s l u g test s o l u t i o n ,
which n e g l e 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 wel lbore . Consid-
e r i n g t h e range of dimensionless times i n which f i e l d d a t a usua l ly l i e ,
it can be 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 i d i n t h e wel lbore i s
n e g l i g i b l e when a2 is less than about 10 On t h e o t h e r
hand, when a2 is g r e a t e r than about l o 4 , f i e l d d a t a should n o t fo l low t h e
a v a i l a b l e s l u g test s o l u t i o n .
a2 - > 10
eluding 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 wel lbore presented h e r e i n
should be used t o ana lyze f i e l d d a t a . This is t r u e even though no o s c i l l a-
t i o n i n p re s su re o r l i q u i d level i s ev iden t . I n f a c t , i t is n o t rare f o r
a t o t ake va lues g r e a t e r than 10 f o r h igh i n i t i a l formation p res su re
w e l l s o r h igh permeabi l i ty r e s e r v o i r s .
f o r t h i s example case, t h e l i q u i d level i n t h e wel lbore shows an o s c i l l a t i o n .
4 f o r t h i s example.
This can b e seen by comparing t h e a = 0 and
4 cases on F igs . 5 and 6. I n t hese s i t u a t i o n s , t h e s o l u t i o n s in-
2 4
2 8 When a is g r e a t e r than about 10
-34 -
i
-35-
> c4 0 F w . H VI
E
U
-36-
n U
cn cn w
II
-37-
-7- “-l----
z H
0
II
m
rn z w E n
z H
i
-38-
m h 0 0 0 0 0 0 0 0
0
m 0 N 0 0 0 0 0 0 0
0
0 h m 0 0 0 0 0 0 0
0
m \o m 0 0 0 0 0 0 0
0
U co h 0 0 0 0 0 0 0
0
U N 0 rl 0 0 0 0 0 0
0
m co N rl 0 0 0 0 0 0
0
0 \o m rl 0 0 0 0 0 0
0
N m co rl 0 0 0 0 0 0
0
m m rl N 0 0 0 0 0 0
0
rl m co m 0 0 0 0 0 0
0
N U m 0 rl 0 0 0 0 0
0
h 03 co m rl 0 0 0 0 0
0
rl 0 co rl N 0 0 0 0 0
0
m 0 N co N 0 0 0 0 0
0
N m 0 m m 0 0 0 0 0
0
b h rl N U 0 0 0 0 0
0
h U U m U 0 0 0 0 0
0
0 m \o co m 0 0 0 0 0
0
n 0 m m h m r-l 0 0 0 0
0 I
N rl m m h m rl 0 0 0
0 I
0 \o m rl co m 0 0 0 0
0
m \o m m rl m 0 0 0 0
0
-0-l h m m m \o 0 0 0 0
0
m N co 0 0 03 0 0 0 0
0
m N \o m m m 0 0 0 0
0
m rl h rl rl rl rl 0 0 0
0
rl \o m U b N rl 0 0 0
0
m m \o m m 0 m 0 0 0
0
\o U rl U m rl m 0 0 0
0
03 0 rl
II
b N
b 0 rl
I1
b N
\o 0 rl
II
8 N
m 0 rl
II
b N
U 0 rl
II
b N
0
II
b
co N h O 0 0 0 0 0 0
0
U m 0 N 0 0 0 0 0 0
0
0 0 h m 0 0 0 0 0 0
0
rl m \I) m 0 0 0 0 0 0
0
N U co h 0 0 0 0 0 0
0
U U N 0 rl 0 0 0 0 0
0
m m 03 N rl 0 0 0 0 0
0
m m m m rl 0 0 0 0 0
0
N N m co rl 0 0 0 0 0
0
m m m rl N 0 0 0 0 0
0
U rl m 03 m 0 0 0 0 0
0
0 N U m 0 rl 0 0 0 0
0
m h 00 03 m rl 0 0 0 0
0
rl rl 0 03 rl N 0 0 0 0
0
0 m 0 N co N 0 0 0 0
0
0 N m 0 m m 0 0 0 0
0
U \o b rl N U 0 0 0 0
0
\ o m \ o m U U U \ o m a 3 Urn 0 0 0 0 0 0 0 0
0 0 . .
N U rl m rl co m 0 0 0
0
rl m U co m m \o 0 0 0
0
U co rl \o 0 0 co 0 0 0
0
\o N m m m m m 0 0 0
0
rl m rl m rl rl rl rl 0 0
0
N m m 0 U b N rl 0 0
0
0 m h U N cn 0 m 0 0
0
h m U 0 m U rl m 0 0
0
rl \o m m 0 N 0 0 0 0
m co m m u3 m 0 0 0 0
0 m 0 m \o m 0 0 0 0
N m rl U co b 0 0 0 0
h m m U N 0 rl 0 0 0
m U U m co N rl 0 0 0
m 0 co m m m rl 0 0 0
co co 0 N m co rl 0 0 0
rl m rl m m rl N 0 0 0
0 N m 0 m co m 0 0 0
co co m co m m 0 rl 0 0
N m m m h 03 m rl 0 0
co N rl \o co h rl N 0 0
h U h h h rl co N 0 0
0 m co N rl 0 m m 0 0
N \o m m rl rl N U 0 0
U m \o h \o m m U 0 0
h U 0 0 U m co m 0 0
0 U m 0 0 h b m rl 0
U 00 b U m 0-l \o h m 0
h 0 N rl m 03 4- 0 m 0
UI m m m 0 rl m m a 0
h m \o 0 m U to h h 0
h rl N m \o h rl N m 0
\o \o 0 m N U h rl N rl
b rl m 0 b U 0 \o h N
\o rl N m m N co U N U
N m 00 N h 0 0 0 0 0
m co rl m N co
U m m h 0 m
\o 0 U 0 0 m
m m co 0 m m
0 rl N \o m N
\o U h co U m
rl rl 0 co N U
rl m 0 03 m m
0 m 0 N N 0
03 m m 0 U U
m 03 rl m U 0
rl N m U Lo h
rl \o N N b m
\o rl N \o m .t
m co co \D m rl
\o m h h U m
rl \o \o 0 N N
m co h h U m
\o m m h h h
m co N N h m
N m U 0 co
co m m co N
0 m m m
U co h N
m m m rl m
b N m m m
n w n n n n n n n n n n n n ~ n n n n n ~ ~ n n n n n n n n n n o o o o o o o o o o o o o o o o o o o o c ~ o o o o o o o o o c, O O O O O O O O O O O O o O O O O O O O c ~ o o o o o o o o o
o o o o o o o o o o o o o o o o o o o o c ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c ~ o o o o o o o o o r l N m U m \ o h c o m ~ N m u m \ o b c o m r l N c r ) U m \ o h c o m r l N m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o o o o o o o o o o o o o o o o o o o ~ c l o o o o o o o o o
n m n
tl W I4 f4
2
n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n ~ ~ ~
d d d d d d d d d d o o o o o o o o o o o o o o o o o o o o o o o o
ooooooooooooooooooooooooooooooo 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 U m a h c o m r l N m U m a h c o m r l N m U m a h c o m r l N m U m \ o h ~ c n r l . . . . . . . . . . . . . . . . . . . . . . . .
-39-
h 0 rl
II
N 8
m m co m h
rl m m a N co h m r-4 rl
0
U e m co a N m co m rl
0
m 0 e U N m 2
h U 0 m 0 co N a 0 N
0
U m 0 0 co U N co rl U 0
N h a h
N 0 rl h
rl a N U m rl N 0 rl co 0
co m N co h m h h a co 0
h rl U In
U h m U m m h rl rl m 0
m 0 a a h m m h m a3
0
m h h
m m h N rl co U m rl U 0
0 N m h a 0 N m U m 0
rl m In
a 03 N m N N m N U N
0
U 0 m m rl m
m h m m a 0 m a h rl
0
0 CO u) e
h m m rl m 03 m rl m 0
0
UI 0 rl a m
03 a m a m
m 0, U d w m r l r v m c o UfCJOh m r l r l m
w h m r 3 m a c o w a m 0 0 0 0 0 0 0 0
0000
$ 3 c o a
. . . .
- a 0
2 a 0
m a rl cn
0 co ;9 m m rl io m
h 0 m
;9 m 0 0 N
ni a m m m
2 a m rl h 0 m m rl
0
in U m m 0
0
. . N a m 0 m
. - m h a co
rl m 0
rl 0 m rl 0
0
II 6 0 a
;9 m co 4 a 0 N
h a N 0
0
m 0 rl 0
0
h co N
0
h rl 0
0
N * rl
0
d m 0
h
0
co 0 0 0 0
03 rl U h m m rl co m m 0
0 m N
m r-l U U 0 co co h a h
0
cu U m 0 0 rl co U U co 0
co a h
rl m h
m co rl a co m 03 U co m 0
m a m h m
U h h
m M
h
o, U U m rl 03 N m \o U co 0
m U rl m U m N m N co 0
m .t
h h a rl 0
m Q\ N 0 U m m rl a h
0
h ?-I
0 a rl m m h co 0 m U 0
co m 0 0 m 0 U U 0 e 0
a m h h a 0 0 h m m 0
0 m m 03 m m
.t 0 m 0 a
2 In m N m m m h
0 m 03 W co
c;r U
N m m
._ m m co m
0 rl
hi m 0 co
ui N m N m
& m U
?n m co m N ;D
U co m h
0 co 0 Q\
& a m 0 ?-I
rl a m h
co a a N
U m rl N m 0
0 . .
rl 0
2 rl N m 0 m 0
h 0 U 0 0
rl 0 rl co h
U LA
U 0 m 4 m 0 rl 0
0
II
25 N
rl rl m m 0
03 co m co 0
m m co N
0
a m m 0
rl 0 co 0
4 0 a
0 U N
h a 0
G co 0
c\l 0
rl 0
0
O Q O O 0000 rl
0
rl
0 rl 0 0 0 . . . .
0 0 0 0 0
U 0 rl
II
8 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
r l o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o
E U
~ ~ ~ ~ ~ ~ _ _ _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I O O O O O O O O O O O d O O O O O O O O o o o o o o o o o o o o o o
K z 0 u N N N N N N m m m m m m m m m U U U U u U v U U m m m m m m m ~ m a
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-40-
00 0 rl
II
8 N
h 0 rl
II
8 Fr(
Fi H rn rn W
0 H rn z W
g
E n rn 3 rn ffi W 3
2 5 sw E w d
0 !a
W
3 W GI c3 H
H 4 rn rn w g 13 E E rn
n .. \o
W
3
u3 0 rl
II
8 N
VI 0 rl
II
8 N
U 0 d
II
8 N
0
II
8
n U
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r l d r l 4 0 0 d 0 0 0 0 d 0 d d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
r l r l r l r l 0 0 d 0 0 0 0 0 0 d 4 r l d 0 0 0 o o o o o o o o o o o o o I l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l I l l l l l l l l l l l
o d r l r l d d r l d o r l o o o o o o o o o o o o o o o o o o o o o d o 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r l r l r l r l o o o o o o o o o o o o o o o o o o o o o o o o o o o a o 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I I I I I I I I I I I I I 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o a o 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ' 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
-41-
co 0 rl
II
23 N
b 0 rl
II
23 cv
a 0 rl
II
b N
m 0 rl
II
a N
U 0 rl
II
23 N
0
II
b
b m U r l rlrl o m c o m a m m m m m m m m m . .
U N U 0 0 m m m m m
0 rl m N 0 0 co m m m m m & . . .
m cv b N 0 rl N co m m
m m w co a a m b o\ m
b m o r l m m m c o m b N c o corn a m m m m m . .
m 0 a rl m m 0 m m o\
N m 0 U 03 m U 0 co m
me40 a m 0 ocoo c o e m o r l m a a m c o b c o W U U mace m m w . . .
m m m 0 a rl 0 53 m co
0 0 b N N N m U co b
b o\ rl 0 U rl rl rl m 0
a m a N a m m b N U
r l m u a r l m m b m m U Q N o 3 m N a m ".?
rl m m U rl a m m b rl
0 0 m N N rl m m rl rl
c o m b m m o r l m a N d U u m o o a o O r l N r l 0 N m N m 0 0 0 . . .
W m U N m m m U 0 0
m m u o d m r l N N U N b m r l rlrl 00 0 0 . .
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I I I I I I I I I I I I I I I I I I l l
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
a m N N m a m a b m U U a m b W m m m m . .
m co co a m 0 m m m m
4 0 u m arl m a o r l N a b U N U coa m m . .
03 0 rl m N m U N U m
a c o m m b N O N b m m e O b brl Q U coco . .
U b m o a r l o m m m r l m
b a U . . .
b U 0 rl a rl m 0 m m
N m O U m m con m N m r l a 0 Q U b m N N . .
m a m U N m b m m rl
a U b 0 m m m rl b rl
0 a b m b U rl m U rl
m a m N rlrl U N r l m m N coa 0- m N 0 0 . .
U a co a N N m b rl 0
m N U N co m m N rl 0
N m m b N r l O b m m c\1a m m O M 4 0 00 . .
N 0 b rl a rl m b 0 0
N 0 0 m a a a m 0 0
m rl m m rl m N rl 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
m o N VI m VI m m m m .t N m .t N N h .t m m N m m m N m m N 4 h \c) co m m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O O O O O O O O O O O O O O O O d O O O o o o o o o o o o o o o o o I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
U N b m rl b N 03 a m
m m N U o u N m a a m o b U U r l a a m m . .
m m U 00 0 m 0 co m m
0 N N N m U b b N m
N rl m m co U a 0 0 m
N r l u m m w
b m coco . .
rl rl 0 m co U rl 0 m co
a Prl m m VI U rl m 0 co
m co m m 03 0 rl m co b
m m m a m m m m a b
rl m co hl co m m rl m b
m b rl
m m a m c o m U b 4 0 url N m C O N w o U U . .
m m 00 m 0 b rl a m m
m rl m a rn a rl m co N
N co b m oi a m 0 U N
o w U b O b m m N m 0 4 O b a b O b N r l . .
U O 0 N m m U m rl
Nr l r l 000 . . .
0 m rl Lo - 0 m m 0 rl 0
U m N m b h iJi co 0 0
r l m r l N c o b rlrl .t 4 C u & m m b a 00 00 . .
m N m U m a a m 0 0
0 b b O .-
.
u m r l m o \ m 0- 0 4 eo\ b N rlrl 0 0 0 0 . .
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 1 I I I I I I I I I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I
N N e 4 m m m m m m m m m U U U U U U u U U m m m m m m m m ~ a a a a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o a o n 0 0 0 0 b
n n 00 0 0 00 00 com
n 0 0 0 0 rl
n 0 0 0 0 N
n n 00 00 00 00 m e
n 0 0 0 0 m
n 0 0 0 0 a
n n 00 00 00 00 b c o
n 0 0 0 0 m
n 0 0 0 0 rl
n n n 0 0 0 000 0 0 0 000 N m U
n 0 0 0 0 m
n 0 0 0 0 a
n n 00 00 00 00 b c o
n 0 0 0 0 m
n 0 0 0 0 rl
I? 0 0 0 0 N
n n n 0 0 0 0 0 0 0 0 0 000 m U m
n 0 0 0 0 Q
n n n 0 0 0 0 0 0 0 0 0 0 0 0 b c o m
n 0 0 0 0 rl
ca 0 0 0 0 N
n n 0 0 00 0 0 00 m e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-42-
The a va lues f o r which t h e l i q u i d l e v e l i n t h e wel lbore o s c i l l a t e s depend
on t h e va lue of t h e dimensionless wel lbore s t o r a g e cons tan t , CD, and
the 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 genera l l i m i t s on t h e e f f e c t of a on t h e re-
s u l t s , s o l u t i o n s f o r va r ious C and s va lues were obta ined. Figures 9
through 1 5 show t h e r e s u l t s . c1 is def ined as t h e va lue of a below which
pwD i s e s s e n t i a l l y t h e same as t h e a = 0 ( i n e r t i a- l e s s ) case s o l u t i o n when
pwD 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 neglected
f o r p r a c t i c a l purposes if a < al.
which o s c i l l a t i o n of t h e l i q u i d l e v e l i n t h e wel lbore occurs .
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 could no t happen and j u s t beyond which
o s c i l l a t i o n s i n l i q u i d level could happen. A l i n e a r r e l a t i o n s h i p between
l o g a and l o g a versus l o g C w a s found f o r t h e e n t i r e range of C in- 1 2 D D
D
1
a i s def ined as t h e va lue beyond 2
This num-
corresponds t o t h e c r i t i c a l damping cond i t ion discussed by van der 2
"Critical damping" w a s used h e r e t o refer t o a cond i t ion under
v e s t i g a t e d when s i s g r e a t e r than 5, and f o r 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- log s c a l e f o r d i f f e r e n t s va lues .
shows how t h e a c t u a l curves d e v i a t e from s t r a i g h t l i n e s f o r CD = 1 when s
i s smaller than 5 . Then, w e conclude t h a t t h e e f f e c t of a on t h e s o l u t i o n
can b e es t imated from t h e following:
va lues g r e a t e r than 10 D
Figures 16 and 1 7 show t h e c o e f f i c i e n t of thes 'e
Figure 18
-2 1.25 1.077 - 0.0385 log s cD a = 1.99x10 s 1
0.85 1.077 - 0.0385 log s cD a = s 2
-43-
' 12
IO
-i-----f-- ~ - - - , -
-----I------
FIG. 9: LOGalAND LOG a2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGE
CONSTANT FOR s = 0
-44-
8
2
0
-2
“4 I
FIG. 10: LOG al AND LOG a2 VS TOGARITHM OF DIMENSIONLESS WELLBORE STOIUGE
CONSTANT FOR s = 1
1:
I(
I
f
tJ
J
r t!
c
c . - d
- 4
-0.1412 CD 1.05 sal
FIG. 11: LOG al AND LOG a2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGE
CONSTANT FOR s = 5
FIG. 1 2 : LOG al AND LOG a2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGE
CONSTANT FOR s = 20
LOG c*
F I G . 13: LOG al AND LOG a2 VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGd
CONSTANT FOR s = 50
12
I C
8
E
ti
::4 -I
2
, C
. - 2
-4
F I G . 1 4 : LOG a AND LOG a2 VS LOGARITHM O F DIMENSIONL,ESS WELLBORE STORAG'Lt 1
CONSTANT F O R s = 100
-49-
-50-
W L3 0 a
W n 0 m I1
cu U
I n I n l 0 I
- - 8 901 ONW V 301
cu 0
- I
I
_.
u 0 I4
u 0 Gi
1 .OE
I .Q5
I .04
= 1.03
I
-51-
10 s
FIG. 17: EXPONENT OF DIMENSIONLESS WELLBORE STORAGE CONSTANT VS SKIN FACTOR
-52-
2
C U (3 0 -1
-I
- 2
-3
I I I
- T T
T T
I I I
0 2 4 6
S
FIG. 18: LOG al AND LOG a2 VS SKIN FACTOR FOR CD = 1
1 1
I
L
L
-53-
When s < 5, cD - > lo3:
1.05 CL = 0.141 CD 1
1.05 = 3.98 CD 2
When s < 5, CD < lo3: use Figs . 9 , 10, and 11.
I n van der Kamp's paper,16 t h e parameter which c o n t r o l l e d t h e c r i t i -
cal damping cond i t ion w a s presented as d.
1 2 s 8.5 1 -r '(E)' &n [ 0.79 rf (,I ] C
d = 8T
This parameter d is expressed as follows i n
( s e e the correspondence between t h e symbols
and those used i n ground water hydrology i n
'DRn [ s] 4a
d =
(74)
t h e nomenclature of t h i s s tudy
used i n petroleum engineer ing
Table 10, Sect ion 2-3-4).
(75)
However, t h e value of d a t which c r i t i c a l damping occurs could n o t be ob-
ta ined i n van der Kamp's a n a l y s i s and w a s hypothesized as 0.7.
d a t a presented showed d to be between about 0.4 and 1 . 2 .
i s mainly due t o the u n c e r t a i n t y i n the va lue of t r a n s m i s s i v i t y .
t h e a
t a ined i n t h i s s tudy , d can be c a l c u l a t e d .
g i b l e f o r water w e l l s .
Table 7 shows the r e s u l t s .
when d i s between 0.45 and 0.52
The a c t u a l
This u n c e r t a i n t y
Since
va lue which corresponds t o t h e c r i t i c a l damping cond i t ion w a s ob- 2
Often the s k i n f a c t o r i s negl i-
Then, d may be obta ined us ing Eq. 73 and Fig. 9 .
A s a conclus ion, c r i t i ca l damping should occur
f o r p r a c t i c a l C va lues . D
-54-
TABLE 7 : THE VALUE OF d AT WHICH CRITICAL DAMPING OCCURS
5 10
10
l o 3
l o 4 10
10
10
d
0.47
0.47
0 .46
0.45
0 . 4 8
0 . 5 1
0.52
-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 r WD
than 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 t h a t t h e wel lbore p ressure becomes less than t h e
i n i t i a l cushion head i n t h e wel lbore . This i s caused by t h e i n c r e a s e i n
k i n e t i c energy of t h e l i q u i d by rap id movement of t h e l i q u i d up t h e w e l l -
bore . This overshooting remains about 30% f o r a l l small a values s t u d i e d .
However, t h i s phenomenon probably has no u s e f u l meaning f o r f i e l d d a t a
i n t e r p r e t a t i o n because t h e dimensionless t i m e i s u s u a l l y l a r g e even f o r a
minute of real t i m e , and might n o t be seen i n f i e l d d a t a because of f r i c -
t i o n .
2-2-4.2 E f f e c t of C, on So lu t ions
F igure 1 9 shows t h e dimensionless wel lbore p ressure ,
Y
versus
dimensionless t i m e , tD, and Fig. 20 shows t h e dimensionless l i q u i d l e v e l ,
xD, versus t
age c o n s t a n t , CD, when t h e s k i n f a c t o r , s , i s zero .
t h e corresponding s o l u t i o n s h i f t s toward smaller t D
The i n c r e a s e i n C decreases t h e e f f e c t of a. This phenomenon agrees w i t h
Eqs. 70 and 72. On t h e o t h e r hand, t h e s o l u t i o n f o r a = 0 s h i f t s toward
’wD ’
f o r t h r e e d i f f e r e n t va lues of dimensionless wel lbore 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
inc reas ing t when C i n c r e a s e s . This is because t h e l i q u i d t akes a
longer t i m e t o occupy t h e l a r g e r wel lbore . Then, i t can be 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 h e in-
e r t i a l e f f e c t f o r t h e l i q u i d i n the wel lbore .
D
A t a g lance , t h i s statement
appears s t r a n g 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 = 0 D
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 ious C and s values were obta ined. Fig-
ure 2 1 shows t h e dimensionless t i m e
D
a t which pwD becomes 0 .9 . We ’ tD1’
-56-
I I I I I 1- -
(! P /' \
L.. .
B 2 2 m 2 0 V W
d
-I----- --- _. _- -.__
I I I I / /
I OJ
el
ii
-57-
r- ---- '- --.- I-~-__ _-..
\
\ \
\
z H
0 0
H n
a 4-
.."
3 0
H z s VI 2 0 u W
3 0 .H VI
VI VI W J z 0 H VI
H n
.. 0 N
-58-
I
FIG. 21: LOG t VS LOGARITHM OF DIMENSIONLESS WELLBORE STORAGE CONSTANT Dl
-59-
can say t h e s e l i n e s are approximately p a r a l l e l f o r s - > 5.
s e n t s t h e logar i thm of t ve rsus s when C = 10 . We o b t a i n an exac t
s t r a i g h t l i n e f o r s - > 5.
t a ined :
Figure 22 p re -
3 D Dl
As a r e s u l t , t h e following r e l a t i o n w a s ob-
When s > 5: -
0.83 1.04 cD = 0.17 s
When s < 5 , i t is necessary t o o b t a i n a s o l u t i o n f o r each case.
The a va lue , which i s necessary f o r o s c i l l a t i o n s i n l i q u i d l e v e l ,
i n c r e a s e s wi th i n c r e a s i n g C as seen i n Fig. 20. This phenomenon agrees
with Eqs. 7 1 and 73. This means t h a t when t h e CD
l a t i o n of t h e l i q u i d l e v e l i n t h e wel lbore w i l l n o t happen e a s i l y .
D
va lue i s l a r g e , o s c i l -
2-2-4.3 E f f e c t of s on So lu t ions
F igure 23 shows t h e dimensionless wel lbore p r e s s u r e 9 PwDY ver-
and F ig . 24 shows t h e dimensionless l i q u i d tD , s u s t h e dimensionless t i m e ,
l e v e l , %, versus t
the s k i n f a c t o r i n c r e a s e s , t h e s o l u t i o n f o r a given va lue of a s h i f t s to-
ward smaller t i m e s . An inc reased s k i n f a c t o r decreases t h e e f f e c t of a.
This phenomenon agrees wi th Eqs. 70 and 73. On t h e o t h e r hand, the solu-
t i o n f o r a = 0 s h i f t s toward inc reased t i m e when t h e s k i n f a c t o r i n c r e a s e s .
This agrees w i t h Eq. 73.
l i q u i d t o flow i n t o t h e wel lbore because of t h e s k i n e f f e c t . Then, s i m i -
l a r l y t o t h e e f f e c t of CD, i t can be s a i d t h a t t h e g r e a t e r t h e s k i n fac-
t o r , f o r a wide range of a v a l u e s , t h e less 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 moving i n t h e wel lbore .
f o r t h r e e d i f f e r e n t va lues of s k i n f a c t o r , s . When D
This i s because it i s more d i f f i c u l t f o r t h e
-60-
-.-I----- --! --..------ 7-
s
3 F I G . 22: LOG t VS S K I N FACTOR FOR CD = 10 Dl
-61-
‘\
‘\ \
m 0 I+
II
V n
w
z 0 H
-62-
.I_- -*-- 7 T- --I_-
e Y
I
-63-
Equation 76, r epor t ed i n t h e previous s e c t i o n , shows how f a r t h e s o l u t i o n s
s h i f t , depending on t h e s va lues gene ra l ly .
The overshooting of p a t e a r l y t i m e s f o r small a va lues decreases
This is because t h e l i q u i d cannot f low
WD
wi th inc reas ing s k i n f a c t o r , s.
i n t o t h e wel lbore f r e e l y because of t h e f low r e s i s t a n c e caused by t h e s k i n
f a c t o r , and cannot permit t h e l i q u i d column i n t h e wel lbore t o accelerate
s u f f i c i e n t l y t o induce t h e overshooting.
For a nega t ive 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 oppos i t e effeelt
would be expected. But f o r nega t ive va lues of t h e s k i n f a c t o r , t h e so lu-
t i o n d ive rges i n some cases. This may be caused by t h e d e f i n i t i o n of t h e
s t eady- s ta t e s k i n f a c t o r ( i . e . , a sudden i n c r e a s e of p re s su re a t t h e sand-
f a c e f o r a nega t ive s k i n f a c t o r ) .
u se t h e e f f e c t i v e wel lbore r a d i u s , r ' = r e , i n s t e a d of t h e a c t u a l w e l ~ l -
bore r a d i u s , r as an approximation. Using t h i s e f f e c t i v e wel lbore
r a d i u s , CD and a w i l l be rep laced by C e2' and
t h e va lues of a and a
C e2' i s g r e a t e r than 10 . a and a because of t h i s m u l t i p l i c a t i o n . This means t h a t a nega t ive
s k i n f a c t o r i n c r e a s 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 wel lbore
on s l u g test s o l u t i o n s .
obvious from t h e c h a r a c t e r i s t i c s of t h e curves i n F ig . 18.
I n o rde r t o avoid t h i s problem, w e can
-S W W
W'
r e s p e c t i v e l y . Then, D
i n Eqs. 72 and 73 w i l l be mul t ip l i ed by e O*ls when
3 A nega t ive s k i n f a c t o r g ives smaller va lues olf 1 2
D
1 2
3 I f C e2' i s less than 10 , t h i s tendency is more D
Another i n t e r e s t i n g r e s u l t concerns an apparent cons tant f l owra 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. This phenomenon i n
1 3 , 3 3 DST f low per iod d a t a has been d iscussed by several i n v e s t i g a t o r s .
It w a s suggested t h a t c r i t i c a l f low choking might be t h e cause of t h i s
phenomenon. This explanat ion w a s a l s o given i n t h e Earlougher monograph.
Cr i t i ca l f low choking can happen f o r mul t iphase gas- l iquid flow. C r i t i c a l
choking i s n o t l i k e l y i f on ly l i q u i d f lows, 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
-6 4 7
system. However, t h e same apparent cons tan t f l o w r a t e phenomenon a t e a r l y
times i s o f t e n observed f o r l i q u i d flow. Judging from t h e s o l u t i o n s ob-
t a i n e d i n t h i s s tudy , a l a r g e s k i n f a c t o r may b e one explanat ion of t h i s
phenomenon. F igure 25 shows a l i n e a r r e l a t i o n between t h e dimensionless
wel lbore p ressure , and t h e dimensionless t i m e , tD, a t e a r l y t i m e s f o r PWD , 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)',
W
t h i s s o l u t i o n means a l i n e a r r e l a t i o n s h i p between t h e we l lbore p ressure ,
4 and t h e t i m e , t , f o r t from 0 t o 10 . This l i n e a r i t y would appear
t o b e a cons tan t f l o w r a t e per iod. The reason why t h e l a r g e s k i n f a c t o r
might be a cause 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 cannot
flow i n t o t h e we l lbore r a p i d l y because of t h e l a r g e s k i n f a c t o r ( i .e . ,
t h e p r e s s u r e drop occurs mainly a t t h e sandface) , then t h e back p r e s s u r e
caused by t h e accumulated l i q u i d i n t h e wel lbore i s r e l a t i v e l y n e g l i g i b l e
compared t o t h e p r e s s u r e i n s i d e t h e formation, and t h e p r e s s u r e g r a d i e n t
i n s i d e t h e formation remains almost cons tan t f o r a t i m e .
PW* D
For t h e same reason, w e can expect t h a t t h i s phenomenon happens
f o r a l a r g e wel lbore s t o r a g e case because t h e f l u i d head i n t h e wel lbore
does n o t i n c r e a s e as r a p i d l y i f t h e wel lbore 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, v e r s u s t h e dimensionless t i m e , tD, f o r t h e case when C = lo3'. A
d 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 can b e seen a t e a r l y t i m e s .
However, such a l a r g e we l lbore s t o r a g e cons tan t is i m p r a c t i c a l . For
f e a s i b l e dimensionless we l lbore s t o r a g e c o n s t a n t s , CD of 10
phenomenon cannot b e seen.
bore p ressure ,
10 and CD = lo5, r e s p e c t i v e l y .
the solutions f o r t h e s e cases.
Theoreti-
Figure 26 shows t h e dimensionless wel lbore p ressure ,
D
2 6 - 10 , t h i s
Figures 27 and 28 show t h e dimensionless w e l l -
f o r t h e cases when C = tD , D ve rsus dimensionless t i m e , 'wD ' 3 There is no ev iden t l i n e a r p o r t i o n of
-65-
a h
I 0 Q -
3
-66-
c-- T
0
0
a !A d
m m w
H n
-4 c4 0 Fr
m m W d z
m o 3 4
.. Q c\l
c3 H Fr
D
D
x Q +- s
Q
u
3
cn w z 0
0
0
0
m w IJ 2 0 H m E E n .. 03 CY
2 k
-69-
One more cause of apparent cons tan t f lowra tes is 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 wel lbore . Figure 29 shows t h e dimensionless
f o r t h e case when tD , versus dimensionless t i m e , 'wD ' wellbore p ressure ,
3 CD = 10 , s = 0 , and a2 = l o7 . A 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 can be
3 3 seen f o r t va lues from 6x10 t o 8 . 4 ~ 1 0 . However, t h e s t r a i g h t p o r t i o n D does n o t start from t h e zero t i m e , and i t does no t cont inue f o r a long
t i m e . Therefore, al though t h i s may happen, w e i n f e r t h a t t h i s is no t the
main exp lana t ion f o r t h e apparent cons tan t f l o w r a t e phenomenon a t e a r l y 1
t i m e s .
,
2-2-4.4 Radius of I n v e s t i g a t i o n
The p r e s s u r e 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 b e c a l c u l a t e d by ob ta in ing t h e real space s o l u t i o n of Eq. 42 f o r
va r ious parameter va lues . F igure 30 shows t h e p r e s s u r e d i s t r i b u t i o n in+l
s i d e t h e r e s e r v o i r f o r a simple example case when C = 10 and s = a = 0
at v a r i o u s t i m e s . The p r e s s u r e g r a d i e n t wi th r e s p e c t t o r a d i a l d i s t a n c e
reaches a maximum va lue s h o r t l y a f t e r s t a r t of product ion and then t h e
p r e s s u r e g r a d i e n t dec reases g radua l ly wi th t i m e ( i .e. , t h e f l o w r a t e de-
creases) and t h e i n v e s t i g a t i o n r a d i u s i n c r e a s e s w i t h time.
p r e s s u r e ve rsus r a d i u s i s l i n e a r i n t h e r e s e r v o i r c l o s e t o t h e w e l l ; how-
ever , f a r from t h e w e l l p r e s s u r e versus r a d i u s i s convex t o t h e coardi-
n a t e of r a d i a l d i s t a n c e f o r e a r l y times.
r a d i u s becomes concave t o t h e r a d i a l coord ina te a f t e r t h e f lowra te starts
decreas ing.
3 D
A graph of
The graph of p r e s s u r e ve rsus
I n o r d e r 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 t h e p ressure d i s t r i b u-
3 t i o n i n s i d e t h e r e s e r v o i r , t h e case of C = 10 and s = 0 w a s s e l e c t e d
as a t y p i c a l example, and t h e p ressure 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
I)
-70-
0
0. I
0.2
0.3
a 3 0.4 a-
8.E
0.7
0.E
I I I
/ I /
f
-3 t, X I 0
FIG. 29: DIMENSIONLESS WELLBORE PRESSURE VS DIMENSIONLESS TIME FOR 3 7 C = 10 , s = 0, AND a2 = 10 AT EARLY TIMES D
-71-
0
II
b I I
m
n V ffi 0 Fr
E 0 3 c4 w m W ffi
W E w H
.. 0 m
-72- w a s obta ined f o r v a r i o u s a va lues a t c e r t a i n t i m e s .
show t h e r e s u 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 e r e n c e e x i s t s
between t h e p r e s s u r e d i s t r i b u t i o n f o r a = 0 and a2 = lo5, and t h e pres-
Figures 31 through 35
s u r e remains t h e same as
The p r e s s u r e d i s t r i b u t i o n
8 t h e p r e s s u r e f o r a2 = 10
The p r e s s u r e g r a d i e n t f o r
2 8 t h e i n i t i a l formation p r e s s u r e f o r a = 10 .
5 f o r ci = 0 and ci2 = 10 remain t h e same, and
starts showing some response a t t = 3x10 . a = 0 and a = 10 are decreas ing ( i . e . , t h e
3 D
2 5
f l o w r a t e i s decreas ing and t h e wel lbore p r e s s u r e i s recover ing) ; however,
t h e p r e s s u r e g r a d i e n t f o r a2 = 10 i s s t i l l i n c r e a s i n g a t tD = 10 . The 8 4
2 8 wellbore p r e s s u r e f o r a = 10 becomes g r e a t e r than t h e i n i t i a l formation
p ressure , pi; i .e . , t h e l i q u i d level i n t h e wel lbore o s c i l l a t e s .
t h e o t h e r hand, t h e wel lbore p r e s s u r e f o r a = 0 and a2 = 10
On
become almost 5
4 t h e s a m e as t h e i n i t i a l formation p ressure , pi, a t t = 5x10 . The w e l t - D bore p r e s s u r e f o r a l l t h r e e a v a l u e s are n e a r l y t h e same as t h e i n i t i a l
formation p ressure , pi a t tD = 10 . a va lue , t h e later t h e p r e s s u r e d i s t r i b u t i o n s tar ts changing and t h e longer
5 We conclude t h a t t h e g r e a t e r t h e
i t con t inues changing, and t h e smaller t h e p r e s s u r e g r a d i e n t 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 almost t h e 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 drop was found a t va lues of r of 100 and less. D
F igure 36 shows t h e e f f e c t of t h e s k i n f a c t o r , s , on t h e i n v e s t i-
ga t ion rad ius . When t h e s k i n f a c t o r i s l a r g e , t h e p r e s s u r e drop occurs
mainly a t t h e sandface , and t h e p r e s s u r e g r a d i e n t i n s i d e t h e r e s e r v o i r is
small ( i . e . , t h e f l o w r a t e is s m a l l ) . However, t h e i n v e s t i g a t i o n r a d i i are
almost t h e same f o r a l l va lues of s. Then, w e can s a y approximately t h a t
t h e s k i n f a c t o r does 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 u s g r e a t l y .
-73-
\
M-
w H m H
n
z
m H n
m v ) W
rl kl 0 II
w c i u u w k l f f i F r o W F r
-74-
r% 0 Err
n
cc) 0 rl X m
I1
\
J 0
0 2 w m w p:
U . . (A z H
.. hl cc)
-75-
-76-
n
-
7------- -r- -P- ------. -$------- ---I--- Y)
0 .--
e 0 4 x Ul
II
u n
p: U 0 z W
-. .
v)
2 E w
W 5 v) z H
..
-77-
1 !S
J 0 --
I
n x
Fr m 0 rl
II
u n
.. W pr;
W n U
U
-78-
.I
-7 9-
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. 37 D
= 10 3 4 , 10 , and lo5, among which actual field data are D
As a summary of this section, the investigation radius depends on the
dimensionless wellbore storage constant, (+,, and is not affected by the
dimensionless number, a, or the skin factor, s, as much. The main pressure
drops 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 the initial 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 the
drawdown case and is positive for the injection case.
From the definition of
2-2-4.6 Application of Solutions -S
Using 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 a D D’
-80-
I1 cl
V f3
0 I I I I I
0
t3 1;
D x
H
z 0
.. m m
-81-
graph.
w e l l test d a t a a n a l y s i s i n 1973, and by Ramey e t al.IL3 f o r convent ional
s l u g test so lu t ions , which do n o t inc lude t h e inertia:L e f f e c t , as an apprcxi-
mation i n 1975.
This has been done by Earlougher and Kersch31i f o r convent ional
I n our s o l u t i o n s w e have one more parameter bes ides
C e2', and t h e coord ina te is t h e same, tD/C,,. Tables 8 and 9 show t h e D
accuracy of t h i s approximation f o r t h e example cases when t h e product ,
CDe2s = 10 and %e2s = 10 f o r d i f f e r e n t a values . Since t h e d e t a i l e d
d i s c u s s i o n of t h i s approximation w a s g iven i n Ramey e t a l . , l3 i t i s no t
4 5
repeated he re .
range of
t i o n curves f o r t h e s l u g tes t
This approximation appears good enough f o r a p r a c t i c a l
and ae2'. Figures 38 through 47 show t h e r e s u l t i n g solu-
inc lud ing 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 wel lbore wi th r e s p e c t t o t h e wel lbore p r e s s u r e , and F igs . 48 through
57 show t h e same r e s u l t s w i t h r e s p e c t t o t h e l i q u i d :level i n t h e wel lbore .
Although t h e s e f i g u r e s are i n reduced s izes , f u l l - s i z e f i g u r e s are avail-
a b l e . CDe2s and D D'
ae2' However, as can be seen,
We might be a b l e t o o b t a i n t h e matched p o i n t s f o r t / C
t h e o r e t i c a l l y from t h e s e s o l u t i o n curves .
i t is p r a c t i c a l l y imposs ible t o select a unique curve matched t o t h e a c t u a l
f i e l d d a t a , because t h e r e are s o many similar curves caused by t h e new
parameter ae . 2s
There 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 tudy
t o o b t a i n some informat ion f o r r e s e r v o i r s ( t h e main purpose i s t o determine
t h e va lue of pe rmeab i l i ty ) . One way i s t o check t h e p o s s i b i l i t y t h a t t h e
i n e r t i a of t h e l i q u i d i n t h e wel lbore w i l l a f f e c t f i e l d performance. This
procedure should be adopted when t h e e f f e c t of t h e i n e r t i a of t h e l i q u i d
i n t h e wel lbore on t h e f i e l d d a t a i s n o t clear; otherwise we should go
d i r e c t l y t o t h e second usage of t h e s o l u t i o n s which w i l l be expla ined la ter .
-82-
h
U 0 rl
II
m N
01
V d 0 Fr
W
u cn
n
n
n
2 F3 2 2 2 2
d
3
cn
PI
0 w IJ
=z cn cn W GI
H cn 8 E n .. co W GI m 2
0 rl 0 rl
II
m 01
8
e N
h 0 rl
II m U N
01
0
U 0 rl
II
m U
N 01
0
rl m rl
rl
II m
0
II m
rl m ri
4
II
m
0
II
m
rl m rl
rl
II
m
0
II m
n
n V \
u
m a a m rl 0 0 0 0 0
rl rl m co N 0 0 0 0 0
N h 0 m U 0 0 0 0 0
b N m rl a 0 0 0 0 0
rl m m m h 0 0 0 0 0
m a N h m 0 0 0 0 0
N N a m rl rl 0 0 0 0
a m N U m rl 0 0 0 0
a m 0 m m rl 0 0 0 0
m U h m h rl 0 0 0 0
.a- 0 W N r- 0 0 0 0 0
0 h m U 0 a 0 0 0 0
U m m N U co 0 0 0 0
4 N U m 03 0 rl 0 0 0
a rl m U U m rl 0 0 0
m m U a 0 a rl 0 0 0
h 0 m U h co rl 0 0 0
N a U h U rl N 0 0 0
co a m m N U N 0 0 0
m 0 03 a m m m 0 0 0
m I- m U 0 a co 0 0 0
N N a 0 b m rl rl 0 0
m a a U m U m rl 0 0
0 rl a m m 0 m rl 0 0
0 m a rl m a N N 0 0
m m m h rl U a N 0 0
h a) h N 0 N 0 m 0 0
rl rl a 0 U 0 U m 0 0
h U rl N h U U I- 0 0
m a co m 0 0 0 0 0 0
0
h m m h m rl 0 0 0 0 0
0 I
m a m m N 0 0 0 0 0
0
a m rl co m 0 0 0 0 0
0
h m m rl m 0 0 0 0 0
0
m m m m a 0 0 0 0 0
0
U co 0 0 co 0 0 0 0 0
0
m a m m m 0 0 0 0 0
0
m b rl rl rl rl 0 0 0 0
0
rl a U I- N rl 0 0 0 0
0
N 0 a m 0 m 0 0 0 0
0
a 0 m m rl m 0 0 0 0
0
m m N h m b 0 0 0 0
0
U a m 0 h m 0 0 0 0
0
m b N U rl N rl 0 0 0
0
U N a m a U rl 0 0 0
0
m co co m N h rl 0 0 0
0
m 0 N co co m rl 0 0 0
0
U m m I- m N N 0 0 0
0
m co co co ro rl m 0 0 0
0
0 U m a m m co 0 0 0
0
m 03 h 0 h a rl rl 0 0
0
U a m m N rl m rl 0 0
0
m h m m N m N N 0 0
0
* rl N m m 0 a cv 0 0
0
m rr) U m 0 03 m N 0 0
0
a m a m N a m m 0 0
0
rl a I- h a m m I- O 0
0 W
n
a N m m a 03 m 0 0 0
0
h m rl rl 0 a b m rl 0 0
c:
co a m m U m m N 0 0
0
a 0 rl a U rl co m 0 0
0
m rl rl m 0 m rl m 0 0
0
0 0 rl m U N m a 0 0
0
m rl h h m co m h 0 0
0
m rl rl N U 0 m m 0 0
0
m m h m m h 0 rl rl 0
0
m m rl 03 I- co a N rl 0
0
0 co rl m a 0 a 0 m 0
0
co a a m m m m 0 m 0
0
U U h N a b a rl h 0
0
U a 0 co a N m m m 0
0
a h a m U m co m rl rl
0
U co a h 0 N rl rl a rl
0
m co a 0 co m co m a, rl
0
N 0 m m rl U m a 0 N
0
m U 0 m \D rl U 0 N U 0
a a rl U m b h h m m 0
m h m U N a U m m h
0
a U h N 0 m a h m co 0
h a a co co h U 0 rl m 0
m a co I- rl a rl 0 a m 0
N 03 N co co U N N m m 0
03 I- I- N 0 co m 4 rl 0
rl
co N 0 b b a N rl N 0
rl
U m m N U N a rl co m 0
W
h m m a m co N m a b
0
0 U b U m h a a co e
0 U U h 0 N U U co m 0
h 0 N a m cv co a rl 0 a
m 0 rl rl co I- h co N 0
rl
m N m m co m rl m N rl
a m U m 0 m 00 m N 0
Yl
U m a a rl m U rl a rl
rl m N N m e b m N 0
rl
m m m N N a m N N rl
m m l-i m m U m co rl 0
rl
co m rl co m m 0 N h 0
rl m h 03 h 0 a U rl 0
rl
03 m rl rl m a m m m 0
m U co co N m a ri rl 0
rl
m rl m h rl m m m 4 0
0 0 0 m 0 h 0 m 0 0
rl
I- co m N m N m b 0 0
rl a m m b U m N rl 0
rl
m 0 0 m co U m m 0 0
h m m N m rl b m m N N
0 I
W
co m m co N m 0 U 0 0
N 0 N m N N m 0 0 0
rl
co m co 0 m a m 0 0 0
0 m rl a rl 0 co m m ch
0
U 0-3
U m 0 m I- m m m
U 0 rl N m rl 0 m m m 0
U co co 0 a co m co m m 0
N e m 0 a U co h m m 0
U U a N N a m h m m 0
m m m m a rl m a m m 0
0 m a h b m U a m m 0
rl cv m e rl b m N m m 0
a m a h I- 0 m m co m 0
N m I- m h m 0 h co m 0
I- 0 rl co a co N U 03 m 0
U N 43 0 m N a rl co m 0
rl ri h a I- N 0 m b m 0
01) rl b N co co U a h m 0
U a m a a m m m h m 0
h a m a a * m rl h m 0
m 0 I- m m m a co U m 0
c o N a m a a m m 0 0 m h CnN coco m m m m . .
a m m h m m a I- m m
co rl m U U co rl h m m
m U a m 0 N h a m m
N U N I- m 03 N a m m
rl co N m rl N I- N m m
co m a m 0 h h a co m
N m m co a rl 0 U co m
co co N rl rl m m rl co m
N rl I- N U m h 03 I- m
U rl N 0 U rl N a I- m
N U m co rl N I- M I- m
U m 03 h rl h N rl h m
a m N m U a m 03 U m
0 r l r l r l r l r l ~ r l r l r l r l r l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I v
m m m m m m m m m N N N N N N N N N r l r l r l r l r l r l r l r l ~ o 0 0 l l l l l l l l l l l l l l l l l l I 1 1 1 1 1 1 1 1 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r i N m e m a b c o m r l N m U m a h c o m r l N m U m ~ h c o ~ r l N ~
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~
d d d d d d d d d d d d d d d d d d d d d d d d d d d d d ~ ?3
-83-
0 rl 0 rl
II
111
01
23
u N
I- O rl
II
111
01
?5
u N
u 0 rl
II 111
Q)
c3
u N
rl m rl
rl
II
111
0
II 111
rl m rl
rl
II
111
0
It 111
rl m rl
rl
II
111
0
II
111
n
n u \
u
~ _ _ _ _ _ . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
I l l
m m m ~ - m m m o m r l m m r l r l m ~ ~ ~ - \ o m \ o m o m m I - m c o \ o e m r l u c o N m m r l N u m m m \ o c o m m m N r l O m r l ~ u ~ I - m m r l \ o m \ o u N O \ o N m N c o r l I - r l c o N o u I - N r l r l u \ o c o ~ r l I - o r l u m \ o m r l m \ o m I - m u m m m m c o m r l u I - \ o I - o m m c o r l N m \ o N m r l m \ o r l m u m \ o r l O u m c o m c o N u r l c o u I - m c o \ o m u N u r l m m c o N u N c o N \ o c o r l N O N m O m r l N o o \ o \ o m N O r l N m c o N e c o N N m I - I - m u m N O u r l m f i \ o m N O c o \ o u N r l I - \ o c o r l m 0 \ o m 0 \ o m N r l r l o o o o m m c o m c o c o c o \ o m e u m m N N N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . .
N N co h 0 m
co 0 0 m 0 co
co m m 0 I- co
I- co m \o m co
m e co 0 N co
m N N m 0 co
\o m m 0 I- \o
0 m I- m \o m
0 co I- \o 0 m
I- co I- \o \o N
rl co 0 m m N
co m m u 0 N
m rl m \o 0
rl rl 0 m m 0
0 m 4 0 N 0
UJ 0 rl 4 0
m 0 m 0 0
u \o fi 0 0
u N \o \o 0 0
\o U co m 0 0 ~ ~ ~ ~ _ _ _ _ _ . . . . . . . . . . . . . . . . . . . . . . . . .
looooooooooooooooooooooooo
n 0 0 0 0 u
n 0 0 0 0 m
n 0 0 0 0 \o
n 0 0 0 0 I-
n 0 0 0 0 a,
n 0 0 0 0 m
n 0 0 0 0 rl
n 0 0 0 0 N
n 0 0 0 0 m
n 0 0 0 0 u
n 0 0 0 0 m
n 0 0 0 0 UJ
n 0 0 0 0 I-
n 0 0 0 0 co
n 0 0 0 0 m
n 0 0 0 0 rl
n 0 0 0 0 N
n 0 0 0 0 m
n 0 0 0 0 e
n 0 0 0 0 m
n 0 0 0 0 \o
n 0 0 0 0 b
n 0 0 0 0 co
n 0 0 0 0 m
n 0 0 0 0 rl . . . . . . . . . . . . . . . . . . . . . . . . .
~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-84-
m co rl m N 0 0 0 0 0
rl m m rl m 0 0 0 0 0
N m W m h 0 0 0 0 0
h 0 0 03 0 rl 0 0 0 0
0 m m W m rl 0 0 0 0
N co m W W rl 0 0 0 0
4 W N W m rl 0 0 0 0
m rl U W N N 0 0 0 0
U rl h W In N 0 0 0 0
W * m co co N 0 0 0 0
I .
h m m N 0 m 0 0 0 0
* U m 0 m m 0 0 0 0
h 03 0 m 0 m rl 0 0 0
\o U m m W W rl 0 0 0
\o 0 W m m 0 N 0 0 0
N co W N rl U N 0 0 0
U 0 h m m b N 0 0 0
rl 0 0 U co rl m 0 0 0
N U 0 h h m m 0 0 0
h 03 m m m W h 0 0 0
m 0 \o h m 0 N rl 0 0
co rl * N h m W rl 0 0
rl 0 h U 0 N rl N 0 0
h N m 0 m m m N 0 0
m rl m m m h 0 m 0 0
m U m W 0 W m m 0 0
h m 0 W m m 0 * 0 0
m 0 m N
N rl II 0 r l o II
m U
N a,
I1 v)
m 0 m N
m II 0 r l v )
I1
v) U
N a,
I1
v)
m 0 m N
W II 0 r l m II
v) U
N a
* O
I1 v)
w z 0 V
i
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \./ 1 1
W U h N rl 0 0 0 0 0
0
rl W m 0 m 0 0 0 0 0
0
rl m u) rl m 0 0 0 0 0
0
co N h m h 0 0 0 0 0
0
0 0 rl h m 0 0 0 0 0
0
m m U rl N rl 0 0 0 0
0
0 h m W U rl 0 0 0 0
0
m m m N h rl 0 0 0 0
0
m m co co m rl 0 0 0 0
0
N rl co m N N 0 0 0 0
0
h a) m W rl m 0 0 0 0
0
m m 03 m m co 0 0 0 0
0
N m m b W rl rl 0 0 0
0
m 0 co m rl m rl 0 0 0
0
m 0 0 0 h co rl 0 0 0
0
0 rl m U m N N 0 0 0
0
U m rl W 0 W N 0 0 0
0
rl 03 m m co m N 0 0 0
0
co m m b \o m m 0 0 0
0
co co m a) rl U h 0 0 0
0
N N rl cn N h rl rl 0 0
0
03 co m h 0 N W rl 0 0
0
0 0 co m 0 co 0 N 0 0
0
m U W 0 rl m m N 0 0
0
h 0 cn N m N 0 m 0 0
0
m b U * U rl m m 0 0
0
N co N W m 0 0 U 0 0
0
m m \o co m rl m N 0 0
0
co co N N N co rl m 0 0
0
rl e m 0 m m m h 0 0
0
0 N m W m U h 0 rl 0
0
h h 03 m W 0 W m rl 0
0
U m W b co 0 m W rl 0
0
co m N W W U U m rl 0
0
m U h m m 0 U c\l N 0
0
m m h 0 W \n m m N 0
0
m h N W 0 rl m co N 0
0
- U m W h m m 0 m N 0
z
m m W m m m 0 rl m 0
0
N co m m 4 Q co N N rl
0
m m m m a N m U m rl
0
W rl \o 03 m m co m co rl
0
m m m rl 0 h W W rl N
0
co rl U m W N m W U N
0
co N m N m * m W h N
0
m rl W m co h * v, 0 m 0
co 0 U N cy) co U m Vl m 0
m U U U 0 W h h U 03
0
m m U U m rl m m N m 0
h m m h N U W m W m 0
0 m rl rl N h co m m m 0
0 m cn N 0 0 N m 0 0
rl
0 rl W m h W U U rl 0
rl
co U m 0 U b N rl 0 0
0
m m m U N m 0 m 0 0
0
m h 0 rl m U rl m 0 0
0
m m m m rl m m h 0 0
0
rl e m m m b W m 0 0
0
N N m m m co 0 N rl 0
0
co N m 0 m h m U rl 0
0
m m 03 N m N 4 h rl 0
0
m co W h h N b m rl 0
0
* h U W m h m N N 0
0
03 m U rl U N W 0 m 0
C
m N m N 0 co m 0 co 0
0
rl m N U N m N rl rl rl
0
rl h 03 m co U rl N U rl
0
m h m h N h m N h rl
0
rl W h rl 0 m m m 0 N
0
03 m m 0 * co W m m N
0
m m m co U-l m m m W N
0
N rl co 0 m rl U N m N
0
Cr) U W m 03 0
U U m 0
0 v, m 0 co m rl W N h
0
In m m m m m m W N m 0
h rl N m U co m m h m 0
0 N N m co m m m m m 0
rl rl rl N W m m N rl 0
rl
W m m U m co \o h rl 0
rl
m m rl N N co m W rl 03
0
h m m m m rl 0 rl 0 rl
rl
N rl rl m m co h N co 0
rl
co m W U co co 0 h m 0
rl
m m m rl m m m W rl 0
4
U U m U co m N 0 rl 0
rl
m co m U h m W co 0 0
rl
rl m rl m 0 m co h 0 0
4
m U m rl rl m m W 0 0
rl
W m m m co 0 0 W 0 0
rl
m m m N W W m rl 0 0
rl
m W U h m m m 0 0 0
rl
m m U 0 rl U W m m m 0
co U h W N N rl m m m 0
0 U 0 m rl 0 h co m m 0
N N co U U m m 03 m m 0
r- m m m rl 0 0 co m m 0
03 h m W 0 m W h m m 0
0 U U 0 \o m m h m m 0
0 4 W m rl m 03 U m m 0
0 m b m W m m N m m 0
m m co m U m m 0 m m 0
m rl m W U U N co 03 cn 0
* m 0 0 h 03 rl W co m 0
0 N co 0 rl h rl e co m 0
N * h N m co rl cv co m 0
co U m rl I-. m N 0 co m 0
m m m m m m m m m N N N N N N N N N r l r l r l r l r l r l r l r l r l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n n n n n n n n n n n n n n n n n n n n n n n n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r l N m U m W h c o m r l N m * m W h c o m r l N m U m W b c o m .~ . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
n 1- v
0 0 0 0 0 0 0 0 0 r l r l r l r l r l r l r l r l r l N N N N N N N N N N n n n n n n n n n n n n n n n n n n n n n n n n n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 oooooooooooooooooooooooooooo 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r l N m U m W h c o m r l N m U m W h c o m r l N m U m W h c o m r l
-85-
0 co rl h rl m m U 0 0
0
m 0 U m N m h m 0 0
0
co m N m rl U rl m rl 0
0
m m h W W co W 0 N 0
0
0 m N W m N m W N 0
0
0 m m N m N 0 N m 0
0
co co N m m b b b m 0
0
m rl rl h U m m m U 0
0
h m 0 m m m m m U 0
0
co 0 rl rl m h rl m m 0
0
0 co h W h m rl m 4 rl
0
m m rl I- b 0 m co W rl
0
m W b W m rl 0 rl N N
0
m co 0 m m co W co W N
0
0 h N rl rl U U rl rl m 0
m a3 0 W W m m co U m 0
N rl \D m W m 0 rl co m 0
00 m W m m W W h 0 U
0
rl rl rl co W m h co N U 0
N h U rl N m m m co m 0
m 03 h h h W m m 0 rl
0
N 0 m m m m N m 03 0
0 I
h m m U m m m 03 m 0
0 I
U W U h m m m co m 0
0 I
m 0 U h 0 N 0 W rl 0
0
m h N N 03 rl m 0 m 0
0 I
W m rl U d 0 m 0 m 0
0 I
0 co U h m 0 0 N rl 0
0
U N rl co h m 0 0 N 0
0
U m N rl N co 0 U m 0
0
U W h m rl N N h N 0
0
co 0 h 0 0 rl W m rl 0
0
m 0 m N
N 4 . II
r l m 0
II
m U
N a 8
m h N rl N 0 m U 0 0
0
m N m m rl h W m 0 0
0
m 0 h m m h 0 m rl 0
0
U N rl m m rl W 0 N 0
0
m m m m W U N W N 0
0
N co m U m U m rl m 0
0
N h m 0 rl m W h m 0
0
m h co N 0 h U m U 0
0
U m h 0 m h N m U 0
0
co rl W m rl m 0 m m 0
0
m co co m m W 0 m 7 4 rl
0
h 0 m co U rl co 03 W rl
0
m m N rl W N m 0 N N
0
03 m rl U m 0 W 03 W N
0
co m m h m W m rl rl m 0
co 0 rl W U m co m U m 0
W m h m m 0 0 d co m 0
U c\1 rl I- N N W h 0 U 0
03 m U rl m W h co N U 0
N m N h co W m m co m 0
m rl m m co W m m 0 rl
0
U rl rl W m W N m co 0
0 I
rl N h m 0 h h m N 0
0
U co m N 0 h 0 0 N 0
0
co U N m m co 0 U m 0
0
0 0 co co co rl N b N 0
0
rl W U 0 W 0 W m rl 0
0
0
II
m
U m co W N rl m U rl 0
rl
0 m N N U 4 m 03 h m 0
N rl h m 0 rl 0 In m m 0
m W W rl m W m W m m 0
rl U m m h m m m rl m 0
W h N N co rl m m 0 m 0
m 03 rl U m m m m h co 0
N co rl W U N rl m m co 0
m m h 0 rl U m m U 03
0
rl m rl rl m 0 03 U N h
0
m rl U a.7 co m co W N W
0
N m N N m 0 m m U m 0
rl m h W U m U W h U
0
m U W m h m co h rl U 0
W h m co 4 0 co h W m 0
W m h h W U h U N m 0
0 N rl m m m m h oo N
0
0 N h 0 N U U m m N
0
m m rl 0 N co rr) 0 m 0
0
co I- U m 0 N m rl U 0
0
m N m U co 0 W m 0 0
0
h m m co m N 0 03 0 0
0
h h W m m m co W 0 0
0
W m d rl W m co W 0 0
0
W N m U m U 0 W 0 0
0
m 0 m N
II
rn m 0 rl
II
m U
N al
B O
m co m m 0 co U b rl 0
rl
N U m U rl U rl co h m 0
rl m N m co W rl W m m 0
h N m W m m rl m rl m 0
rl m rl 0 m rl rl m 0 m 0
rl m m W N m co h co co 0
N m m U W m rl m h co 0
r-l m W W m N m co m co 0
N N m 0 0 m rl m U 03
0
0 m 03 m W m W U N h
0
0 N 0 co 0 co h W N W 0
m W N U m rl N m U m 0
co W 0 h h 03 m W h U 0
U W 0 co 03 U 03 h rl U 0
0 rl N co h W h h W m 0
m U m m U N h U N m 0
0 rl W U W a3 m h co N
0
m m W rl W m U m m N
0
0 U h m m m m 0 m 0
0
co N W m 0 m m rl U 0
0
m 0 W W m b h m N 0
0
rl h N h m N 0 W rl 0
0
W h N m VI 0 0 N rl 0
0
W 0 rl W m 0 W m 0 0
0
m 0 m W U N 0 03 0 0
0
W U h 03 m U 0
0 0
0
II
m
m 0 m N
W II 0 r l m II
v) U
N a,
8
N m N co 0 rl m co h m 0
co m rl 03 m U 0 0 W m 0
co h h 0 U co m N
2 0
U m m W m m h W N m 0
U 0 h 0 h m 4 rl rl m 0
m U W 0 U h 0 W m 03
0
m m m m N 43 U rl co co 0
m N m 0 m rl m h W co 0
m m W N m m m m m m 0
W m I- N co 0 rl 0 U co 0
0 I- W m U h m rl N h
0
m co N 0 W m W U N W
0
h N W rl h m W m U m 0
m 0 rl CO m m N m h U
0
h m U N 0 W 0 h rl U
0
m N W h U 0 N h W m 0
m W N co m N m U N m 0
03 co m m 03 rl m h co N
0
rl rl 0 m m W N m m N
0
03 U rl 0 4 h W 0 m 0
0
co rl m W W I- m rl U 0
0
0 0 N I- W m co W 0 0
0
rl W h N 0 m 0 W 0 0
0
rl 0 m N W m h m N 0
0
N m m m N m 0 W rl 0
0
U N W U m 0 0 N rl 0
0
N N m m 4 rl W m 0 0
0
co W N h m N 0 a3 0 0
0
0
II
m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . QUI 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-86-
a s”
1 . ;
0.1
I d2 I d I 10
V G D
2s FIG. 38: DIMENSIONLESS WELLBORE PRESSURE vs tD/cD f o r dDe = 10
-87-
1.:
1.1
0. t
0.s
Q ,5
0.L
-3.2 io- - 1
90 1 10 2 io
2 FIG. 39: DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10
3 to
-88-
1.4
0.1
*' CI.:
2s 3 FIG. 40 : DIMENSIONLESS WELLBORE PRESSURE VS t D / C D FOR C D e = 10
FIG. 41: DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10 , 4
-90-
“t 0.8
0.7,- I 0.6 I i- I
0.2 c
FIG. 4 2 : DIMENSIONLESS WELLBORE PRESSURE VS t D / C D FOR C e2’ = 1 0 5 D
-9 1-
8 FIG. 4 3 : DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10
-92-
Q.C
0.4
0.2
0.2
0. I
0
- 0.i
-0.2
FIG. 4 4 : DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe 25 = 10 10
-93-
I ----7----- - - -p_------ T---"
0.71
I I
15 FIG. 4 5 : DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10
-94-
-7- ---.-- r----- i---~--”- -I ” --I--
I
20 FIG. 46 : DIMENSIONLESS WELLBORE PRESSURE VS tD/CD FOR CDe2s = 10
-95-
FIG. 4 7 : DIMENSIONLESS WELLBORE PRESSURE vs tD/cD FOR CDe2s = 10 30
-96-
,
I I I I I I -
I I I I I I * 0 0 0
5 -
- 0 cu * rq ab I
Ci I
0 0 I I 0
I
-97-
u) 0 It
v) -. II
" \ y C) I1
v, i V
cu 6 \ CI II
I I I I I I I * cu 0 cu d- a aD 0 0 0 0 -
I 0 I
d I I
-98-
\
I I I I I I I
6 -
0 I
0 t
0 I I
I
m 0 cu rl
0 - II
-99-
I I I I I I I * cu 0 cu 0 0 0 d- (D oq
0 ci 0 I I 1 I
0 -
'0
N '0 -
-100-
I I I I I I
cu
"/ I t
v)
I I I I I I I '0 - - 0 cu d- (0 aD I I d- cu 0
t 0
t 0
t 0 I
1 0 0
I
-101-
U
I I I I i I I I - (D * cu 0 cu (0 Q)
d 0 6 0 I
0 8
0. I
0 I
00 0 t i
n H b s m
.. m ln
-102-
I I I I I I I L cu c\1
m cu n 0 -
I I I I I I I d- c\I 0 cu * aD a 0 0 o 0
I 6 ci I I
0 r) -
0 d 0 rl
-103-
r I I 1 1 I I
d= tn 0)
N Y.
II
tn
7 \\
I I I I I I I * N 0 cu * (D 0 0
I I ci
I I l d d
O X
-104 -
1 I I I 1 I I
I\ II
rr) d- cu
d- 0 0 - -
II
v) v) d-
0 - II
"'/ cu a#
U
I I I I I I I
0 1 * N 0 cu (D a0
0 1 0 0 I
0 I
ci I I
0 N 0 N rl
0 - II
m N
a,
V n
0 -
cu 0 I
, -
a 3 0 H I4 cn cn w I4 z 0 H cn z; E n .. \D m
2 Fr
-105-
rr) a n
I- ? A o)
A I \Y
U e cu Q)
rr) 0
I I I I I I I I (D e cu 0 cu * (0 cx)
(5 0 d d I
0 I
0 I
0 I
R 0
w E
I
F i r s t , w e apply
d a t a . When t h e r e is a
t h e e n t i r e t i m e range,
-106-
convent ional s l u g test type-curves13 t o t h e f i e l d
curve which matches t h e f i e l d d a t a very w e l l f o r
w e should c a l c u l a t e t h e s k i n f a c t o r and permeabi l i ty
from t h e matched p o i n t s of C e'' and t D / C D . Using t h e s e v a l u e s of s k i n
f a c t o r and pe rmeab i l i ty , t h e ct va lue and c1 v a l u e can b e obta ined. I f the
c1 va lue is smaller than t h e ct va lue , t h e obta ined pe rmeab i l i ty and s k i n
f a c t o r are r e l i a b l e , and t h i s is t h e end of t h e s l u g test d a t a a n a l y s i s
f o r t h i s case.
proceed t o t h e second usage of t h e s o l u t i o n s .
matched curve i n t h e convent ional s l u g t e s t type-curves a t t h e beginning,
w e have t o go t o t h e second usage of t h e s o l u t i o n s a l s o .
D
1
1
I f t h e ct va lue i s g r e a t e r than t h e ctl va lue , w e should
, When we cannot f i n d a w e l l -
One more way t o u t i l i z e t h e s o l u t i o n s presented i n t h i s s tudy is
t o use t h e s o l u t i o n curves i n Fig. 38 through 57 as type- curves. This
procedure may be adopted when 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
wel lbore on t h e s l u g test d a t a is n o t n e g l i g i b l e , and t h e r e f o r e t h e con-
v e n t i o n a l s l u g test d a t a a n a l y s i s cannot b e app l ied .
method t o ana lyze t h e s e d a t a .
Thus f a r t h e r e is no
As mentioned before , i t is p r a c t i c a l l y impossible t o select t h e
s u i t a b l e parameter va lues as C e2' by matching wi thout knowing t h e s k i n
f a c t o r , s , unfor tuna te ly . Therefore, we have t o assume t h a t t h e s k i n
f a c t o r may b e obta ined by o t h e r methods, such as bui ldup test a n a l y s i s .
2s Then, w e can choose t h e proper se t of s o l u t i o n curves f o r a c e r t a i n C e
va lue from Figs . 38 through 57, o r w e can p repare t h e s o l u t i o n curves f o r
t h i s C e'' va lue us ing t h e computer program i n Appendix E.
2s s o l u t i o n curves t o f i e l d d a t a , t h e matched p o i n t s of cCe
D
D
Applying these D and t D / C D can
-107-
be obta ined and t h e pe rmeab i l i ty can b e c a l c u l a t e d from e i t h e r t h e matched
a va lue o r t h e matched t /C
app l ied t o t h e a n a l y s i s of both t h e wel lbore 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 wel lbore .
value . The methods presented h e r e i n can be D D
2-3 F i e l d Data Examples
I n o r d e r t o i n v e s t i g a t e t h e s l u g test and check t h e v a l i d i t y of t h e
s o l u t i o n ob ta ined i n t h i s s tudy , a v a i l a b l e f i e l d d a t a were i n v e s t i g a t e d
u s i n g t h e s o l u t i o n .
t h i s s e c t i o n .
A few t y p i c a l examples are shown and d i scussed i n
The example 1 i n Sect ion 2-3-1 and t h e example 2 i n Sec t ion
2-3-2 were publ ished be fore as examples of s l u g test d a t a .
has d i f f e r e n t r e s u l t s f o r t h e s l u g test a n a l y s i s and t h e bui ldup test
a n a l y s i s .
2-3-3.
level i n t h e wel lbore .
The example 3
The reason f o r t h i s d i f f e r e n c e i s i n v e s t i g a t e d i n Sect ion
The example 4 i n Sec t ion 2-3-4 shows o s c i l l a t i o n s of t h e l i q u i d
2-3-1 Example 1 (Typical DST Flow Per iod Data)
To u s e t h e s l u g test s o l u t i o n s presented i n Figs . 38 through 4 7 ,
it i s necessary t o know t h e c o r r e c t va lue of t h e cushion l i q u i d head; i n
o t h e r words, t h e p r e s s u r e equ iva len t t o t h e cushion l i q u i d head, p . One
should measure p be fore opening t h e tester valve.
n o t be rep laced by t h e minimum wel lbore p ressure measured because i t i s
shown i n t h i s s tudy t h a t t h e minimum measured wel lbore p r e s s u r e is n o t
always equal t o (p + p ). When t h e cushion l i q u i d head is not known,
a logar i thmic coord ina te f o r p
matching t o avoid t h e e r r o r caused by t h e i n c o r r e c t cushion l i q u i d head.
0
This p ressure should 0
0 a t m
should b e used i n s l u g test type-curve WD
This example shows how f i e l d d a t a can b e m i s i n t e r p r e t e d because of
an i n c o r r e c t v a l u e of p The d a t a f o r t h i s example were published by 0'
-108-
Ramey et a1.,I3 and reviewed i n a previous paper.33
d a t a and measured p r e s s u r e s are i n Table C-1 i n Appendix C.
shows wel lbore p r e s s u r e v e r s u s t i m e .
e a r l y times.
643 p s i g ( 4 . 5 3 ~ 1 0
l a t e d s t r a i g h t l i n e a t t = 0 i n Fig . 58, 560 p s i g ( 3 . 9 6 ~ 1 0 6 p ) , w a s used
as po.
The a c t u a l r e s e r v o i r
F igure 58
There i s a bend i n t h e curve a t
1 3 I n t h e Ramey et a l . paper, t h e p r e s s u r e measured a t t = 0, 6 p ) was used as po. I n a previous paper,33 t h e extrapo- a
a The former obta ined kh = 698 md-ft m2-m) and s = 6.55,
and t h e la t ter obta ined kh = 763 md-ft ( 2 . 3 0 ~ 1 0 -3 m 2 -m) and s = 6.55. On
t h e o t h e r hand, v a l u e s of kh = 377 md-ft (1.13~10-'~ m2-m) and s = 0.80
were obta ined from t h e bui ldup tes t a n a l y s i s . 13
Assuming t h a t t h e i n i t i a l cushion l i q u i d head i s not known, t h e
log- log scale type- curves presented by Ramey et a1.I3 were used i n t h i s
s tudy.
b e s t .
5 It w a s found t h a t t h e C e2' = 10 curve matches t h e f i e l d d a t a D However, it w a s almost imposs ible t o s e l e c t t h e b e s t matched log-
log type- curve wi thout informat ion from t h e buildup test a n a l y s i s , be-
cause of t h e l a c k of d a t a a t l a t e flow t i m e s . It is recommended t h a t one
should measure p
times ( f o r i n s t a n c e , u n t i l pwD = 0.01) i n a s l u g test.
t h e matching r e s u l t .
m -m) and s = 0.79 were obta ined.
be fore t h e test starts and o b t a i n f low d a t a a t l a t e 0
5 Figure 59 shows
-13 From t h e matched p o i n t s , kh = 420 md-ft ( 1 . 2 8 ~ 1 0
2 These r e s u l t s are c l o s e r t o r e s u l t s of
t h e bui ldup test a n a l y s i s than t h e p rev ious ly repor ted r e s u l t s of s l u g tes t
a n a l y s i s .
The f i r s t two d a t a p o i n t s a t e a r l y times i n Fig. 59 appear t o re-
s u l t from i n e r t i a l e f f e c t . The same c h a r a c t e r i s t i c s can be seen i n t h e
e a r l y t i m e d a t a i n Fig. 58. However, t h i s i s no t t r u e . I f t h e two e a r l y
p o i n t s i n Fig. 59 show t h e e f f e c t of i n e r t i a , t h i s means t h a t a i s about
4 1 . 4 ~ 1 0 and t h e pe rmeab i l i ty of t h e r e s e r v o i r i s about s i x t y times g r e a t e r
-109-
FIG. 58: FIELD DATA I N EXAMPLE 1
-110-
d
-111-
than t h a t obta ined from t h e bui ldup test a n a l y s i s . This i s n o t l i k e l y ,
because w e b e l i e v e a t least t h e o rder of magnitude of t h e r e s u l t of t h e
bui ldup test a n a l y s i s .
p o i n t s of t h e f i e l d d a t a w e r e n o t caused by t h e i n e r t i a of t h e l i q u i d
i n t h e wel lbore .
Therefore , w e conclude t h a t t h e f i r s t t h r e e
I n o r d e r t o make c e r t a i n t h a t w e can n e g l e c t t h e i n e r t i a l e f f e c t
t h e l i q u i d i n t h e wel lbore on t h e f i e l d d a t a , t h e dinoensionless number
a w a s c a l c u l a t e d u s i n g Eq. 72: 1
of
3 c1 = 5.81~10 1
Using t h e r e s u l t of s l u g test d a t a a n a l y s i s i n t h i s study:
2 c1 = 2 . 7 3 ~ 1 0
Since t h e f i e l d example a va lue is less than 01
i n e r t i a l e f f e c t is n e g l i g i b l e f o r t h i s s l u g test a n a l y s i s .
i t can b e s a i d t h a t t h e 1’
The apparent cons tan t f l o w r a t e per iod a t early times which appeared
i n t h i s example cannot be expla ined by t h e l a r g e sk in e f f e c t proposed i n
t h i s study.
bore , as suggested by Ramey e t
This might be caused by c r i t i c a l f low somewhere i n t h e w e l l -
and reviewed by Earlougher. 14
F i n a l l y , i n o r d e r t o check whether t h e assumption t h a t t h e f r i c t i o n
l o s s i n t h e wel lbore i s n e g l i g i b l e is reasonable o r n o t , t h e a c t u a l pres-
s u r e drop by f r i c t i o n i n t h e wel lbore w a s c a l c u l a t e d i n Appendix C. The
magnitude of t h e p r e s s u r e drop by f r i c t i o n i s very s m a l l even though t h e
f low is t u r b u l e n t . The n e g l e c t of f r i c t i o n appears t o be reasonable f o r
t h i s example.
-112-
2-3-2 Example 2 (Understanding F i e l d Data)
6 This f i e l d example w a s f i r s t presented by Kohlhaas and w a s used as
Later t h i s example was an example of t h e f low per iod d a t a i n t e r p r e t a t i o n .
i n v e s t i g a t e d f u r t h e r by Ramey e t a1.,13 and by a previous paper. 33
a c t u a l d a t a are shown i n Table C-2 i n Appendix C. F igure 60 shows t h e
i n t e r p r e t a t i o n of t h e s e d a t a i n t h e previous study.33
of t h e pressure- time curve is l i n e a r (dashed l i n e ) . The bend i n t h e d a t a
i s caused by a change i n diameter from t h e d r i l l c o l l a r (2.5 i n . [ 6 . 3 5 ~ 1 0 - ~ m]
I D ) t o t h e d r i l l p i p e (3.8 i n . [ 9 . 6 5 ~ 1 0 - ~ m] I D ) .
t h e pressure- time curve is l i n e a r aga in . However, t h i s i n t e r p r e t a t i o n of
t h e d a t a might no t be c o r r e c t . F igure 6 1 shows t h e s l u g test s o l u t i o n i n
dimensionless form f o r t h i s example based on t h e r e s u l t s of t h e bui ldup
test d a t a a n a l y s i s ,
and c1 = 5 . 9 9 ~ 1 0 . seen because of t h e l a r g e s k i n f a c t o r . F igure 62 shows a comparison be-
tween t h e f i e l d d a t a and t h e c a l c u l a t e d r e s u l t s . The c a l c u l a t e d r e s u l t s
based on t h e a c t u a l p
(except f o r some e a r l y t i m e p o i n t s ) b e t t e r than t h e c a l c u l a t e d r e s u l t s
based on an ad jus ted p = 205 p s i g (1.5lx10 p ) do. This means t h a t t h e 0 a
f i r s t i n t e r p r e t a t i o n of t h e d a t a i s wrong i f we b e l i e v e t h a t t h e r e s u l t s
of t h e bui ldup test d a t a a n a l y s i s are r e l i a b l e .
shoot ing a t e a r l y t i m e s i s no t clear, even though t h i s phenomenon looks
similar t o t h e r e s u l t of wel lbore phase r e d i s t r i b u t i o n on bui ldup tests
repor ted by F a i r .
The
The i n i t i a l p o r t i o n
The second p o r t i o n of
= 1.77x104, k = 56.7 md (5.60x10’14 m2), s = 44.6,
The s t r a i g h t p o r t i o n of t h e l i n e a t e a r l y times can b e
cD 2
6 = 161 p s i g ( 1 . 2 1 ~ 1 0 pa) agree wi th t h e f i e l d d a t a
0
6
The reason f o r t h e over-
35
-113-
FIG. 60: FIRST INTERPRETATION OF THE DATA I N EXAMPLE 2
-114-
m w l-a z s E E m
n z H
z 0 H E-r
0 Cn
5
.. d W
c3 H Fr
-115-
FIG. 62: COMPARISON OF ACTUAL DATA AND CALCULATED RESULTS IN EXAMPLE 2
-116-
2-3-3 Example 3 (Comparison of Resu l t s of Slug T e s t Analys is and Buildup
T e s t Analys is)
The d a t a i n t h i s example were presented i n il previous study. 33 The
r e s e r v o i r d a t a and t h e measured p ressure are shown i n Table C-3 i n Appen-
d i x C.
a b l e .
r e s u l t from t h e s l u g test a n a l y s i s and t h a t from t h e bui ldup tes t a n a l y s i s ,
and t h t such a d i f f e r e n c e i s o f t e n observed. Also, i t is suggested t h a t
t h e r e s u l t from t h e bui ldup test a n a l y s i s i s more r e l i a b l e than t h a t from
t h e s l u g test a n a l y s i s by s e v e r a l i n v e s t i g a t o r s , 6y13y33 because i t is d i f -
f i c u l t t o match t h e d a t a f o r l a r g e v a l u e s of C eZs uniquely i n t h e s l u g
test a n a l y s i s .
( 2 . 4 2 ~ 1 0 - l ~ m2) and s = 24.3. These r e s u l t s a g r e e ?well wi th those from
log- log type-curve matching i n s l u g test a n a l y s i s . This means t h a t t h e
measured p i s r e l i a b l e . On t h e o t h e r hand, fo l lowing t h e bui ldup test
a n a l y s i s , k = 4,321 md ( 4 . 2 6 ~ 1 0 m ) and s = 33.4. The dimensionless
wel lbore s t o r a g e c o n s t a n t ,
r e s u l t s from t h e s l u g test a n a l y s i s are used, ci = 3 . 4 6 ~ 1 0
Since a < a
t i o n i s r e l i a b l e . I f t h e r e s u l t s from t h e bui ldup test are used,
a = 6 . 1 2 ~ 1 0 . Using t h e s o l u t i o n s i n t h i s s tudy based on t h e s e d a t a , t h e
f low per iod d a t a were c a l c u l a t e d .
l a t e d p o i n t s based on t h e s l u g test a n a l y s i s agree wi th t h e f i e l d f low
per iod d a t a ve ry w e l l ; however, t h e c a l c u l a t e d r e s u l t s based on t h e buildup
test a n a l y s i s do n o t a g r e e wi th t h e f i e l d f low per iod d a t a . I n o rder t o
i n v e s t i g a t e t h e reason f o r t h i s d i f f e r e n c e , t h e buil.dup w a s s imulated 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 which is explained i n Appendix D ,
Not on ly t h e f low per iod d a t a , bu t a l s o t h e bui ldup d a t a are a v a i l -
It i s repor ted t h a t t h e r e is a s i g n i f i c a n t d i f f e r e n c e between t h e
D According t o t h e s l u g test a n a l y s i s , k = 2,452 md
0 -2 2
is 8 . 6 4 ~ 1 0 3 . cD , Simila-r t o Example 1, i f t h e
2 4 and a1 = 1.14~10 . t h e r e s u l t ob ta ined by u s i n g t h e convent ional s l u g test s o h - 1’
2
Figure 63 shows t h e r e s u l t . The calcu-
and
-117-
3
I--- ---'-- ' 7 .; ---- 1 ------r',-
0 * e 9 ="
. ( I *
0
0
'y"
(1
G 3
I i 0 ii') cu
-118-
whose computer program is i n Appendix E.
s imula t ion r e s u l t s based on t h e s l u g test a n a l y s i s and t h e bui ldup test
a n a l y s i s , r e s p e c t i v e l y . It appears t h a t t h e depth of i n v e s t i g a t i o n i s
almost t h e same f o r both t h e s l u g test and t h e bui ldup tes t .
shows t h e f i e l d d a t a and t h e c a l c u l a t e d p o i n t s on a Horner bui ldup graph.
F igures 64 and 65 show t h e
F igure 66
Since w e do no t know t h e e x t e r n a l r a d i u s of
t h e r e s e r v o i r i s almost i n f i n i t e l y l a r g e by
s imula t ion . This i s one of t h e reasons why
tween f i e l d d a t a and t h e c a l c u l a t e d va lues .
{ pi - (po + patm) 1 , w e can s h i f t t h e 'wD
t h e r e s e r v o i r , we assumed t h a t
s e t t i n g r = 10 i n t h e
t h e r e i s some d i f f e r e n c e be-
5 De
However, s i n c e p = pi - W
c a l c u l a t e d p o i n t s up and down
i n p a r a l l e l by changing t h e i n i t i a l formation p r e s s u r e , pi.
d i f f e r e n c e has no s i g n i f i c a n c e . Our i n t e r e s t i s i n t h e s l o p e of t h e l i n e .
The s l o p e of t h e l i n e of c a l c u l a t e d p o i n t s based on t h e s l u g test a n a l y s i s
i s about 12.5 p s i / c y c l e ( 8 . 6 2 ~ 1 0 Pa /cyc le ) , as shown as s l o p e 1 i n Fig .
66.
from t h e s l u g test a n a l y s i s and used t o c a l c u l a t e t h e dimensionless numbers.
The s l o p e of t h e l i n e of c a l c u l a t e d p o i n t s based on t h e buildup test analy-
sis i s about 7 p s i / c y c l e ( 4 . 8 3 ~ 1 0 Pa /cyc le ) , as shown as s l o p e 2 i n F ig .
66.
bui ldup test a n a l y s i s based on t h e f i n a l f lowra te .
which has a s l o p e t h e same as t h e s l o p e 1 and passes through some f i e l d
d a t a p o i n t s a t e a r l y t i m e s , and a l i n e whose s l o p e is t h e same as t h e
s l o p e 2 and connects f i e l d d a t a p o i n t s mainly a t l a t e times, as shown i n
Fig. 66. This means t h a t t h e r e s u l t of s l u g test a n a l y s i s corresponds
t o t h e bui ldup test d a t a a t e a r l y times, and t h e r e s u l t of t h e bui ldup
test a n a l y s i s corresponds t o t h e buildup d a t a a t l a t e t i m e s . On t h e o t h e r
hand, judging from t h e r e s u l t of t h e buildup s imula t ion , on ly one semilog
Then t h i s
4
This s l o p e corresponds e x a c t l y t o t h e pe rmeab i l i ty which i s obta ined
4
Again, t h i s s l o p e corresponds t o t h e pe rmeab i l i ty obta ined from t h e
We can f i n d a l i n e
el
p: w X H
w H m H
n
2
-120-
z H
m w
-121-
-122-
s t r a i g h t p o r t i o n should be apparent on t h e Horner graph as long as t h e
assumption t h a t t h e r e s e r v o i r i s homogeneous holds . Therefore , i t appears
l i k e l y t h a t t h i s r e s e r v o i r i s n o t homogeneous, and t h i s has been r e c e n t l y
repor ted i n ano ther s tudy, i n which t h e dashed l i n e wi th t h e s l o p e 3 i n
Fig. 66 was used f o r t h e bui ldup a n a l y s i s .
why t h e r e is a bending of t h e l i n e a t la te t i m e s , as shown i n Fig . 66,
which looks l i k e t h e e f f e c t of a boundary, a l though t h e r e i s l i t t l e poss i-
b i l i t y t h a t t h e boundary is apparent i n shor t- t ime w e l l tests.
he te rogene i ty appears t o b e t h e main reason why t h e d i f f e r e n t r e s u l t s were
obta ined by t h e s l u g test a n a l y s i s and by t h e bui ldup test a n a l y s i s .
summary, t h e s l u g test a n a l y s i s g i v e s some reasonable r e s u l t s , as does
t h e bui ldup test a n a l y s i s , and i f t h e r e is a s i g n i f i c a n t d i f f e r e n c e between
t h e r e s u l t s of t h e two, w e should s e a r c h f o r t h e reason.
crease unders tanding of t h e r e s e r v o i r .
36
This he te rogene i ty e x p l a i n s
This
In
This should in-
The s l o p e s of t h e l i n e s of c a l c u l a t e d r e s u l t s based on bo th test
ana lyses seem t o become l a r g e r a t long bui ldup t imes. The reason f o r t h i s
i s n o t clear, a l though i t is probably caused by t h e p r e s s u r e g r a d i e n t i n-
s i d e t h e r e s e r v o i r a t t h e shut- in t i m e . This gradient, r e f l e c t s t h e de-
c l i n i n g f l o w r a t e s dur ing product ion, and h a s a convex p o r t i o n i n t h e reser-
v o i r . How long t h e semilog s t r a 2 g h t p o r t i o n i n t h e Horner graph appears i n
t h e bui ldup d a t a a f t e r a shor t- t ime product ion i s beyond t h e range of t h i s
s tudy. Buildup behavior i n DST a n a l y s i s deserves f u r t h e r a t t e n t i o n . It
is noteworthy t h a t F e n ~ k e ~ ~ repor ted r e c e n t l y t h a t p r e s s u r e bui ldup d a t a
f o r a w e l l produced f o r a s h o r t t i m e should fo l low t h e s l u g test type-
curves. Thio observat&on means t h a t t h e s l u g test s o l u t i o n should
be proper f o r at least t h e i n i t i a l cleanup bui ldup dat.a, as w e l l as f o r t h e
-123- flow per iod d a t a . Buildup a n a l y s i s a f t e r shor t- t ime product ion should be
s t u d i e d i n t h e near f u t u r e .
S imi la r t o Example 1, t h e p r e s s u r e drop caused by f r i c t i o n i n t h e
wel lbore was c a l c u l a t e d f o r t h i s example i n Appendix C. The order of
magnitude of t h e f r i c t i o n l o s s w a s s t i l l small, even though n o t as small
as t h a t f o r Example 1. The assumption of n e g l i g i b l e f r i c t i o n seems t o
hold f o r t h i s example case a l s o .
2-3-4 Example 4 ( O s c i l l a t i o n Case)
The d a t a f o r t h i s example are from a water w e l l tes t , and were pre-
sented by van d e r Kamp.I6
measured l i q u i d level i n t h e wel lbore f o r a w e l l c a l l e d 2-C and ano ther
w e l l c a l l e d 9-A, a t New Brunswick, Canada, r e s p e c t i v e l y . The l i q u i d l e v e l
read from t h e s e graphs are i n Table C-4 i n Appendix C. The l i q u i d level
i n t h e wel lbore of t h e w e l l 2-C shows an o s c i l l a t i o n , and t h e l i q u i d level
i n t h e wel lbore of t h e w e l l 9-A shows no o s c i l l a t i o n . The p r o p e r t i e s of
t h e wel l- aquifer system were given i n ground water hydrology symbols.
correspondence between t h e symbols used i n ground water hydrology and
t h o s e used i n petroleum engineer ing were repor ted by Ramey et a l .
10 shows t h i s correspondence, inc lud ing t h e new dimensionless number, a ,
proposed i n t h i s s tudy.
as t h e p r o p e r t i e s of t h e wel l- aquifer system as t h e r e s u l t s of t h e pump
test and t h e response test.
s u l t s i n t h e s imula t ion . Table 11 summarizes t h e v a l u e s used. Figure 67
shows a comparison between t h e f i e l d d a t a and t h e c a l c u l a t e d r e s u l t s f o r
t h e w e l l 2-C using t h e s l u g test s o l u t i o n presented i n t h i s s tudy.
c a l c u l a t e d r e s u l t s based on t h e t r a n s m i s s i v i t y , T = 0.0061 m / s , shows
b e t t e r agreement w i t h t h e f i e l d d a t a than does t h e c a l c u l a t e d r e s u l t s based
The s o l i d l i n e s i n F igs . 67 and 68 show t h e
The
Table
I n van d e r Kamp,16 d i f f e r e n t va lues were repor ted
We s e l e c t e d v a l u e s which y i e l d b e t t e r re-
The
2
I ,
i
-124-
n W m-
oo w o t- II
%-- 3 0 z J O - 4 0 0
n W v) Q,
llJ3 ' X
L
-125-
h
I ln
w p? 0
n H
-126-
TABLE 10: CORRESPONDENCE BETWEEN THE SYMBOLS USED I N PETROLEUM ENGINEERING
AND THOSE I N GROUND WATER HYDROLOGY
Petroleum Engineer ing
cD
D t
Ground Water Hydrology
2 r
L s
T - t 2 S-r .
S
-127-
TABLE 11: ADJUSTED PROPERTIES OF WELL-AQUIFER SYSTEMS REPORTED BY
VAN DER KAMP"
SYMBOLS OF GROUND WATER HYDROLOGY
E f f e c t i v e l eng th of f l u i d column, L(m)
Transmiss iv i ty , T (m / s ) 2
WELL 2c WELL 9A
16 11
0.0068 0.0017
C o e f f i c i e n t of e las t ic s t o r a g e , S 1. ~ x I O - ~ 1. 1x1~-4
Radius of w e l l f i l t e r , r (m)
Radius of w e l l cas ing, r (m)
f
C
0.051 0 051
0.051 0.051
SYMBOLS OF PETROLEUM ENGINEERING
Dimensionless wel lbore s t o r a g e cons tan t , 4 . 5 5 ~ 1 0 ~ 3 3.57~10 cD
tD Dimensionless t i m e , 3
t (-sec) t ( -see) 5.94~10 4 1 . 6 8 ~ 1 0
Skin f a c t o r , s (assumption) 0 0
Dimensionless number, a 3 6 . 2 9 ~ 1 0 4 2 . 3 9 ~ 1 0
-128- 2
on t h e t r a n s m i s s i v i t y , T = 0.0068 m /s, which w a s r epor ted by van d e r Kamp.
The t r a n s m i s s i v i t y of 0.0061 m /s
va lue than i s t h e t r a n s m i s s i v i t y of 0.0068 m /s.
same comparison f o r the w e l l 9-A.
well- aquifer p r o p e r t i e s r epor ted by van d e r Kamp agree ve ry w e l l w i th t h e
a c t u a l da ta .
a q u i f e r .
2 is be l i eved t o be c l o s e r t o t h e t r u e
2 Figure 68 p r e s e n t s t h e
The c a l c u l a t e d r e s u l t s based on t h e
2 The t r a n s m i s s i v i t y of 0.0017 m /s is r e l i a b l e f o r t h i s
I n t h e case of shal low water w e l l s producing from h igh pe rmeab i l i ty
format ions , t h e s k i n f a c t o r , s, i s o f t e n n e a r l y zero. Then, knowing t h e
dimensionless wel lbore s t o r a g e cons tan t , C (or t h e c o e f f i c i e n t of e last ic D
s t o r a g e , S) , w e can p repare type- curves f o r v a r i o u s v a l u e s of t h e dimen-
s i o n l e s s number, a, us ing t h e s l u g test s o l u t i o n presented i n t h i s s tudy,
and can o b t a i n t h e pe rmeab i l i ty , k ( o r t h e t r a n s m i s s i v i t y , T ) , from t h e
matched a value .
m i s s i v i t y shows how t h i s type-curve matching method works.
This example i n which w e could determine a b e t t e r t r a n s-
F i n a l l y , t h e s p i k e , which appears b e f o r e t h e ze ro time (as seen i n
Figs . 67 and 68) , can be seen i n a l l test r e s u l t s r epor ted by van d e r Kamp.
This phenomenon might b e caused by t h e way t h e test w a s performed, be-
cause a f l o a t w a s suddenly removed from t h e wel lbore t o g i v e t h e i n i t i a l
c o n d i t i o n of t h e test.
face.
bottom of t h e f l o a t . However, t h e water i n t h e annulus between t h e f l o a t
and t h e cas ing suddenly drops i n t o the f l o a t c a v i t y causing t h e s p i k e and
a new l i q u i d l e v e l above t h e f l o a t bottom a t t h e i n s t a n t of withdrawal.
This procedure causes a s t r a n g e i n i t i a l water sur-
The f i r s t drop probably p r o j e c t s toward t h e l i q u i d level a t t h e
3 . ANALYSIS OF FLOW PERIOD DATA I N CLOSED CHAMBER TESTS
A c losed chamber tes t is a v a r i a t i o n of a conventional d r i l l stem test
I n c losed cham- (DST) and i s used f o r s a f e t y o r f o r p o l l u t i o n p r o t e c t i o n .
b e r tests , l7'I8 t h e w e l l i s s h u t i n a t t h e s u r f a c e o r a t t h e top of t h e
c losed chamber when t h e f l u i d is being produced.
t h e s u r f a c e o r a t t h e top of t h e c losed chamber, al though i n some cases
kept c losed i n t o mainta in t h e p ressure i n t h e wel lbore when t h e tester
va lve i s c losed. These opera t ions are done t o prevent the r e s e r v o i r f l u i d
from flowing ou t a t t h e s u r f a c e , which may cause p o l l u t i o n problems and
perhaps danger, e s p e c i a l l y f o r o f f s h o r e w e l l s .
diagram of a c losed chamber test .
The w e l l may be open a t
Figure 69 shows a schematic
Usually t h e wel lbore p ressure d a t a f o r t h e f low per iod have been d i s-
carded, and a method t o analyze t h e s e d a t a has no t been presented thus f a r
( t o our knowledge).
t h i s problem, and t h e s o l u t i o n i s i n v e s t i g a t e d i n Sect ion 3- 2.
Sec t ion 3-1 p r e s e n t s t h e mathematical formulat ion of
3-1 Mathematical Formulation
Since t h e p r e s s u r e of t h e trapped gas i n t h e c losed chamber, p ch'
resists t h e movement of t h e r e s e r v o i r l i q u i d , p should be used i n s t e a d of
t h e atmospheric p ressure ,
wel lbore ( see Eq . 1). The momentum balance equat ion i n t h e wel lbore f o r
t h e c losed chamber test becomes:
ch
i n t h e momentum balance equat ion i n t h e 'atmy
'ch-'atm 2 d x L - + gx = c - - 2
d t P f Pf
-129-
(77)
X
L
--hf
-130-
SURFACE VALVE
/ SURFACE
/
/ X -
DRILL STRING /
,TESTER VALVE
PACKER
J FORMATION -
FIG. 69: SCHEMATIC DIAGRAM OF A CLOSED CHAMBER TEST
-131-
The d e f i n i t i o n of t h e v a r i a b l e s and c o n s t a n t s are t h e same as i n Eq. 1.
From t h e gas equation:
Li i s t h e l e n g t h of c losed chamber, x i s t h e l i q u i d l e v e l i n t h e wel lbore ,
x (0 ) i s t h e l i q u i d l e v e l i n t h e wel lbore a t zero t i m e , z i s t h e compres-
s i b i l i t y f a c t o r of t h e gas i n t h e c losed chamber, z i s t h e i n i t i a l compres- i
i s t h e i n i t i a l c h , i s i b i l i t y f a c t o r of t h e gas i n t h e c losed chamber, and p
p r e s s u r e of t h e closed chamber.
Then :
[Li-x(o) 12 - A + & - - - _ _ _ pi-pw
PfL pfL (Li-x)z i p c h , i 'a tm 1 d2
d t 2 L x - (79)
In o r d e r t o o b t a i n a dimensionless form of t h i s equat ion, w e w i l l u s e t h e
D, tD, and p WD dimensionless v a r i a b l e s x
t h e dimensionless number a def ined i n Eq. 1 6 . I n a d d i t i o n t o t h e s e dimensibn-
def ined i n Eqs. 11, 9 , and 1 2 , and
less v a r i a b l e s and number groups, t h e following dimensionless groups are
introduced: - P a t m -
pD atm Pi-(P 0 +Patm)
p,h,i P i - (P,+Patm)
B =
-132-
, r e p r e s e n t s t h e dimensionless atmospheric a t m
PD The dimensionless va lue ,
p ressure .
i n i t i a l p ressure of t h e c losed chamber and t h e volume of t h e closed chamber
The dimensionless groups 6 and L r e p r e s e n t t h e e f f e c t of t h e Di
on t h e closed chamber tes t s o l u t i o n , r e s p e c t i v e l y . Then, Eq. 79 becomes:
2 [LD -x,(O>lz i a t m
'wD - [ 'LD -XD)Zi P - _ - 2 XD a *- 2 -I- XD
dtD i
i s atmospheric, which i s 'ch, i ' I f t h e i n i t i a l c losed chamber p ressure ,
o f t e n t r u e , Eq. 83 becomes:
'D
Since Eqs. 83 and 84 are non l inea r because of t h e las t term, t h e s o l u t i o n
w a s obta ined us ing 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 which i s explained i n Ap-
pendix D. Appendix E shows t h e computer program f o r t h i s f i n i t e d i f f e r - I
ence s o l u t i o n . In o rder t o see t h e dimensionless wel lbore p ressure wi th in
t h e same range, t h e fo l lowing new dimensionless v a r i a b l e , p* i s i n t r o - ~
duced :
WD '
The v a r i a b l e p*
t h e sum o f t h e l i q u i d cushionfiead, po, and t h e i n i t i a l c losed chamber pres-
s u r e,
t i o n wi th p def ined previously:
r e p r e s e n t s t h e dimensionless wel lbore p ressure based on WD
as an equ iva len t cushion head, which has t h e fo l lowing rela- 'ch, i '
WD
-133-
- 'wD ':D l-@+pD
- a t m
The r e s u l t s , t o be d iscussed i n t h e next s e c t i o n , are presented i n terms
of t h e dimensionless wel lbore p ressure , PED
3-2 R e s u l t s and Discussion
I n o rder t o i n v e s t i g a t e t h e c losed chamber test s o l u t i o n and t h e
parameters which a f f e c t t h e s o l u t i o n , t h e case f o r C = 10 and s = 0 was
aga in s e l e c t e d as a t y p i c a l example.
3 D
The s o l u t i o n s f o r some parameter
v a l u e s were obta ined f o r t h i s problem using 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
explained i n Appendix D. To determine t y p i c a l o rder of magnitudes of
parameter va lues , l e t u s consider t h e fo l lowing example case . Suppose t h e
i n i t i a l formation p ressure , pi, i s 3,000 p s i a (2 .07~10
cushion head, po, i s 2,000 p s i a ( 1 . 3 8 ~ 1 0
'a t m 9
chamber,
7 Pa) ; t h e l i q u i d
7 Pa); t h e atmospheric p ressure ,
i s 1 4 . 7 p s i a
'ch, i'
(l.O1x1O5 Pa) ; t h e i n i t i a l p ressure of t h e c losed
5 is 1 4 . 7 p s i a ( 1 . 0 1 ~ 1 0 Pa) which i s l i k e l y ; t h e l eng th
u a t m example, w e
combination
B = 0 . 1 and
and (3 = 0.1
of t h e c losed chamber, Li, i s 2,000 f t (610 m); and t h e l i q u i d d e n s i t y ,
2 3 i s 0.8 ( 8 . 0 ~ 1 0 kg/m ) ( t o wa te r ) . Then t h e dimensionless number
P, i s 0.015, f3 i s 0.015, and LD is 0.70. Therefore , as a t y p i c a l
Of'
I
s e l e c t 0.01 as p Datm i
of dimensionless numbers, w e w i l l consider t h e case when
and f3, and 1 as LD . I n a d d i t i o n t o t h i s
L = 1, as t h e high i n i t i a l c losed chamber p ressure case,
and L = 0.01 as t h e h igh i n i t i a l closed chamber p r e s s u r e i
D
D, and small c losed chamber volume case .
Figure 70 p r e s e n t s t h e dimensionless
d imensionless t i m e , tD, and F ig . 7 1 shows
wel lbore p ressure , p iD, ve r sus
t h e dimensionless l i q u i d l e v e l
t- /
Y
.. 0 b
1 ; I I r I I / I
H IJ
m m w ;I
i5
E
H VI 2 w n
rl r-
..
u H Fr
-136-
i n t h e wel lbore , x v e r s u s t f o r t h e s e parameter va lues . It was assumed
t h a t t h e gas c o m p r e s s i b i l i t y f a c t o r i s 1. A s we can see, t h e r e are l a r g e r
d i f f e r e n c e s between t h e x s o l u t i o n s than between t h e p* s o l u t i o n s f o r D WD
t h e s e example cases .
pGD, is based on t h e sum of t h e l i q u i d cushion head, p
c losed chamber p ressure ,
p ressure , p w e w i l l see t h e same order of d i f f e r e n c e i n t h e s o l u t i o n s wD’
as t h a t i n t h e x s o l u t i o n s . Then, t h e dimensionless wel lbore p ressure ,
p:D, i s a u s e f u l dimensionless v a r i a b l e t o handle t h e closed chamber test
s o l u t i o n ; from now on, w e w i l l d i s c u s s t h e p* s o l u t i o n s .
D’ D
This i s because t h e dimensionless wel lbore p ressure ,
and t h e i n i t i a l 0’
I f we u s e t h e dimensionless wel lbore
D
WD
For t h e t y p i c a l c a s e (B = 0.01, L = 1) and f o r pract ical purposes, Di
t h e r e i s no d i f f e r e n c e between t h e s l u g test s o l u t i o n and t h e c losed cham-
ber test s o l u t i o n i n terms of t h e p* s o l u t i o n s , a s seen i n Fig . 70. We WD
can say t h a t t h e s l u g test s o l u t i o n can be app l ied t o analyze t h e f low
per iod d a t a of t h e c losed chamber t e s t f o r t h i s t y p i c a l case us ing t h e
d imensionless wel lbore p ressure , pED. The o t h e r two c a s e s are u n l i k e l y
i n f i e l d t e s t i n g ; however, t h e s e two cases were s tud ied i n o rder t o in-
v e s t i g a t e t h e e f f e c t of t h e dimensionless numbers B and L
t ions .
on t h e solu- Di
The f i r s t example, B = 0 .1 and L = 1, r e p r e s e n t s t h e case when t h e
p ressure of t h e c losed chamber i s inc reased be fore t h e test starts . The Di
second example, P = 0.1 and L = 0.01, r e p r e s e n t s t h e case when t h e
c losed chamber i s p ressur ized be fore t h e test starts and t h e volume of
t h e c losed chamber i s ve ry small.
Di
The s o l u t i o n s f o r t h e s e cases are d i f -
f e r e n t from t h e s l u g tes t s o l u t i o n , e s p e c i a l l y a t l a t e t i m e s . Then,
s t r i c t l y speaking, t h e s l u g test s o l u t i o n cannot be used t o analyze t h e
f low per iod d a t a from a c losed chamber tes t f o r t h e s e cases. I f t h e s l u g
-137-
t e s t s o l u t i o n were used, t h e r e would be some e r r o r i n t h e r e s u l t s .
ever , i t should be remembered t h a t t h e s e two examples are no t l i k e l y i n
a c t u a l t e s t i n g .
How-
I f pi - (po + patm ) i s smaller than t h i s example, t h e
dimensionless numbers B and L become
dimensionless number B on t h e s o l u t i o n Di
e f f e c t of t h e dimensionless number L Di
i n i t i a l p ressure of t h e c losed chamber
l a r g e r . Then, t h e e f f e c t of t h e
i n c r e a s e s . On t h e o t h e r hand, t h e
decreases . However, as long as t h e
is atmospheric, i t is u n l i k e l y
t h a t t h e dimensionless number B w i l l become g r e a t e r than 0 .1 i n a c losed
chamber t e s t .
ence between t h e s l u g test s o l u t i o n and t h e c losed chamber test s o l u t i o n
i s no t l a r g e f o r p r a c t i c a l purposes, as long as t h e volume of t h e c losed
chamber i s no t too s m a l l , as shown i n Fig . 70. Therefore, our d i s c u s s i o n
on t h e e f f e c t of t h e dimensionless numbers f3 and L on t h e s o l u t i o n holds
genera l ly .
I f t h e dimensionless number B i s less than 0.1, t h e d i f f e r -
Di
A s a summary of t h i s s e c t i o n , t h e s l u g test s o l u t i o n can be app l ied t o
analyze t h e f low per iod d a t a from a c losed chamber tes t except f o r t h e case
when t h e i n i t i a l p ressure of t h e c losed chamber i s ve ry high compared t o
t h e atmospheric p ressure , o r t h e volume of t h e c losed chamber i s ve ry small
For t h e s e s p e c i a l cases, s o l u t i o n s should be obta ined us ing t h e f i n i t e d i f -
fe rence s o l u t i o n i n Appendix D t o analyze t h e f low per iod d a t a
c losed chamber tes t . Phase change is not considered i n t h i s a n a l y s i s .
from t h e
4 . CONCLUSIONS
The e f f e c t of t h e i n e r t i a of t h e l i q u i d i n t h e wel lbore on t h e s l u g
test s o l u t i o n was i n v e s t i g a t e d .
responds t o Froude's number, w a s def ined t o determine how t h e i n e r t i a of
t h e l i q u i d i n t h e wel lbore a f f e c t s a s l u g tes t s o l u t i o n . The e f f e c t s of
wel lbore s t o r a g e and t h e s k i n f a c t o r on t h e s l u g test s o l u t i o n were d i s-
cussed, and t h e i n v e s t i g a t i o n r a d i u s of a s l u g test w a s s tud ied .
f i e l d d a t a from s l u g tests were analyzed.
d a t a a n a l y s i s i n c losed chamber tests was s tud ied as an extens ion of t h e
gene ra l s l u g test s o l u t i o n .
A new dimensionless number, a, which cor-
Some
The s o l u t i o n f o r t h e f low per iod
A s a r e s u l t of t h i s s tudy , t h e fol lowing conclus ions appear warrantedi.
1. So lu t ions f o r t h e s l u g test fne luding 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 wel lbore depend on a dimensionless number, a, def ined i n
t h i s s tudy.
2 . Approximately, 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 wel lbore is
n e g l i g i b l e f o r p r a c t i c a l purposes i f a i s less than a
70 and 72 .
def ined by E q s . 1,
3 . Approximately, t h e o s c i l l a t i o n of t h e l i q u i d level i n t h e wel lbore
may happen when a i s g r e a t e r than a def ined by Eqs. 7 1 and 73. The
van d e r Kamp parameter d i s approximately 0.5 f o r c r i t i c a l damping.
2 '
4 . Increased wel lbore s t o r a g e i n c r e a s e s t h e range of a wherein 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 wel lbore is n e g l i g i b l e , and decreases
t h e tendency f o r t h e l i q u i d level i n t h e wel lbore t o o s c i l l a t e .
5. Increased s k i n f a c t o r (wellbore damage) i n c r e a s e s t h e range of a
wherein 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 wel lbore is n e g l i g i b l e , -138-
-139-
and decreases t h e tendency f o r t h e l i q u i d l e v e l i n t h e wel lbore t o o s c i l -
la te .
we l lbore more s i g n i f i c a n t , and i n c r e a s e s t h e tendency of t h e l i q u i d l e v e l
i n t h e wel lbore t o o s c i l l a t e . P o s i t i v e s k i n f a c t o r dec reases t h e over-
shoot ing of p a t e a r l y times f o r small a va lues .
Negative s k i n f a c t o r makes 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 the
WD
6. How t h e wel lbore s t o r a g e and s k i n f a c t o r f o r c e t h e s l u g test s o l u t i o n
t o s h i f t on t h e t i m e axis can be es t imated by Eq. 7 6 .
7. General ly speaking, t h e h igher t h e i n i t i a l formation p ressure and
permeabi l i ty , and t h e lower t h e l i q u i d v i s c o s i t y , t h e d r i l l s t r i n g r a d i u s
and t h e s k i n f a c t o r are, t h e more t h e i n e r t i a of t h e l i q u i d i n t h e w e l l -
bore a f f e c t s t h e s l u g test s o l u t i o n .
8. A l a r g e s k i n f a c t o r may be one reason f o r apparent l i n e a r i t y be-
tween t h e wel lbore p ressure and t i m e o f t e n observed i n DST flow per iod d a t a
a t t h e beginning of a f low per iod.
9. The i n v e s t i g a t i o n r a d i u s f o r a s l u g tes t depends on t h e dimension-
The l a r g e r t h e dimensionless wel lbore less wel lbore s t o r a g e cons tan t , CD.
s t o r a g e cons tan t i s , t h e deeper t h e i n v e s t i g a t i o n r a d i u s i s . The s k i n
f a c t o r , s, and t h e d imensionless number a do 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 u s as much. The depth of i n v e s t i g a t i o n f o r t h e bui ldup test and t h e
s l u g test are almost t h e same i f t h e shutdown i s done a t l a t e t i m e . The
depth of i n v e s t i g a t i o n is on t h e o rder of hundreds of wel lbore r a d i i , i n
many cases.
10.
duc t ion , but a l s o t o ba tch i n j e c t i o n .
11.
The s o l u t i o n presented i n t h i s s tudy i s a p p l i c a b l e not only t o pro-
There are two ways t o u t i l i z e t h e s o l u t i o n obtained i n t h i s s tudy
In i n t e r p r e t a t i o n of f i e l d d a t a . One way i s t o u s e t h e s o l u t i o n as a type-
-140-
curve when t h e s k i n f a c t o r , s, i s known by o t h e r means, such as buildup
t e s t a n a l y s i s .
t h e matched t /C value .
cases f o r which t h e convent ional i n e r t i a - l e s s s l u g test type- curves can-
n o t be app l ied . The o t h e r way t o u s e t h e s o l u t i o n is t o check whether t h e
r e s u l t obta ined by t h e convent ional s l u g test a n a l y s i s i s r e l i a b l e o r n o t .
A f t e r t h e convent ional s l u g test a n a l y s i s , a
a i s less than a
reasonable.
Permeabi l i ty can b e obta ined from t h e matched a v a l u e o r
This type-curve can be used f o r l a r g e a va lue D D
should b e c a l c u l a t e d . I f 1
the r e s u l t of convent ional s l u g test a n a l y s i s should b e 1’
12. It is important t o measure t h e cushion l i q u i d head a c c u r a t e l y be-
f o r e opening t h e tester v a l v e and t o o b t a i n enough d a t a a t l a t e times i n
t h e s l u g test.
i n log- log scale should b e used t o match t h e s l u g test d a t a .
When t h e cushion l i q u i d head i s no t known, t h e s o l u t i o n
13. Several i n v e s t i g a t o r s have concluded that s l u g test r e s u l t s are
no t as r e l i a b l e as bui ldup test r e s u l t s .
on d i f f i c u l t y i n type-curve matching. The r e s u l t s of t h i s s tudy i n d i c a t e
s l u g test r e s u l t s should be r e l i a b l e and a g r e e wi th bui ldup test r e s u l t s .
This obse rva t ion appears based
14. Fur the r work is necessary t o understand t h e buildup d a t a a f t e r a
shor t- t ime production.
s h o r t product ion test may be analyzed as a s l u g test seems t o b e a ve ry
important idea .
The Fenske3’ obse rva t ion t h a t bui ldup fol lowing a
15. The d e v i a t i o n of c losed chamber s o l u t i o n f r o m t h e s l u g test solutkon
depends upon t h e i n i t i a l p r e s s u r e of t h e c losed chamber and t h e volume of
t h e c losed chamber.
i s n e a r l y atmospheric, and t h e volume of t h e c losed chamber is no t too
small, t h e s l u g test s o l u t i o n can be app l ied t o analyze t h e f low per iod
d a t a of t h e c losed chamber test.
A s long as t h e i n i t i a l p r e s s u r e of t h e c losed chamber
5. NOMENCLATURE
a = positive number greater than the real part of the singularities of function; a constant defined in Eq. D-3, in Appendix D
A = arbitrary constant; coefficient of the equation which relate Cg and 1 a
B = arbitrary constant; coefficient of the equation which relates C and D 2 a
4 2 C = wellbore storage, L T /M
= dimensionless wellbore storage constant 2
= compressibility of theliquid in the wellbore, LT /M
= total system compressibility, LT /M
cD
f
t
C
2 C
d = parameter which controls the critical damping condition
D = diameter of the pipe, L
D E = exponent of C
f = Moody friction factor
g = gravitational acceleration, L/T 2
h = thickness of the formation, L
= modified Bessel function of the first kind, order zero IO
= Bessel function of the first kind, order zero
= Bessel function of the first kind, order unity
JO
2 k = permeability, L
= modified Bessel function of the second( kind, order zero
= modified Bessel function of the second kind, order unity
= variable of Laplace transform
KO
K1
R
L = liquid length whose head is equivalent to the initial formation pressure minus atmospheric pressure, L; liquid length in the well- bore in Appendix C y L
-141-
Li
LD i M
N
NRe
P
Patm
'ch
P i
PO
P W
'wD
'WD -
P:D
Pwf
q
r
r
r
r
C
D
D e
-142-
l e n g t h of c losed chamber, L
dimensionless l e n g t h of c losed chamber
number of nodes
number of terms
Reynold's number
p ressure a t t h e p o i n t i n t h e r e s e r v o i r , M/LT
atmospheric p ressure , M/LT
p ressure of c losed chamber, M/LT
2
2
2
L i n i t i a l p r e s s u r e of c losed chamber, M/LT
dimensionless p ressure
D Laplace transform of p
dimensionless atmospheric p ressure
d imensionless p ressure a t t h e node i a t t h e t i m e s t e p n
i n i t i a l formation p ressure , M/LT
p ressure equ iva len t t o t h e cushion l i q u i d head, M/LT
2 wel lbore p ressure , M/LT
2
2
dimensionless wel lbore p ressure
Laplace t ransform of p
dimensionless wel lbore p ressure based on t h e i n i t i a l c losed cham- b e r p ressure
flowing wel lbore p ressure , M/LT
product ion f lowra te , L /T
WD
2
3
r a d i a l d i s t a n c e from t h e a x i s of t h e w e l l , L
r a d i u s of cas ing , L
dimensionless r a d i a l d i s t a n c e from t h e a x i s of t h e w e l l
dimensionless e x t e r n a l r a d i u s
r Di
ArD i r e
r P
r
r
r '
S
W
W
Re
Re I S
S
t
A t
t D
t Dl
t
T
P
U
X
x ( t =O) =
-143-
dimensionless r a d i a l d i s t a n c e a t node i
increment of dimensionless r a d i a l d i s t a n c e a t node i
e x t e r n a l r a d i u s , L
r a d i u s of p ipe , L
r a d i u s o f sandface, L
wel lbore r a d i u s , L
e f f e c t i v e wel lbore r a d i u s , L
r e s i d u e
real p a r t of
s k i n f a c t o r
c o e f f i c i e n t of e l a s t i c s t o r a g e
t i m e , T
shut- in t i m e , T
dimensionless t i m e
increment of dimensionless t i m e
dimensionless t i m e a t which p becomes 0 .9 f o r a = 0
product ion time, T
upper t i m e l i m i t f o r numerical Laplace transform invers ion methods ; t r a n s m i s s i v i t y , L2 / T
v a r i a b l e i n Appendix A; v e l o c i t y of l i q u i d column i n t h e wel lbore i n Appendix C, L / T
l i q u i d l e v e l i n t h e wel lbore , L
i n i t i a l l i q u i d level i n t h e wel lbore , L
dimensionless l i q u i d l e v e l i n t h e wel lbore
= i n i t i a l dimensionless l i q u i d l e v e l i n t h e wel lbore
= i n i t i a l v e l o c i t y of t h e dimensionless l i q u i d l e v e l i n t h e
0
WD
wel lbore
D Laplace transform of x
-144-
Y = Bessel func t ion of t h e second kind, o rder zero
Y = Bessel f u n c t i o n of t h e second kind, o rder u n i t y
z = compress ib i l i ty f a c t o r of t h e gas i n t h e c losed chamber; complex
0
1
v a r i a b l e i n Appendix A
z = i n i t i a l compress ib i l i ty f a c t o r of t h e gas i n t h e c losed chamber i
GREEK NOMENCLATURE
a = dimensionless number def ined i n Eq. 1 6
a = va lue of a below which p becomes t h e same as t h e a = 0 case solu- 1 WD t i o n when p becomes 0 .9 .
WD
a = va lue of a beyond which t h e l i q u i d level i n t h e wel lbore o s c i l l a t e s 2
f3 = dimensionless number def ined i n Eq. 80; complex v a r i a b l e i n Appendix A
y = E u l e r ' s cons tan t
X
Pf = d e n s i t y of t h e l i q u i d i n t h e wel lbore , M/L
@ = p o r o s i t y
= v a r i a b l e i n Appendix A
3
6.
1.
2.
3.
4 .
5.
6.
7.
8.
9.
10.
11.
12.
13.
REFERENCES 1 "
F e r r i s , J . G . , and Knowels, D.B.: "The Slug Test f o r Estimating Trans- m i s s i b i l i t y , " U.S. Geol. Survey Ground Water Note 26 (1954), 1-7.
Beck, A . , J aeger , J . C . , and Newstead, G.: "The Measurement of t h e Thermal Conduc t iv i t i e s of Rocks by Observations i n Boreholes," Aust. J. Phys. (1956), 9, 286.
Cooper, H.H., Jr . , Bredehoeft , J . D . , and Papadopulos, 1,s.: "Response of a Finite-Diameter Well t o an Ins tantaneous Charge of Water," Water; Resources Research (1967) , 3, No. 1, 263-269.
Maier, L.F.: "DST I n t e r p r e t a t i o n Ca lcu la t ions for. Water Reservoirs ," Ha l l ibur ton Serv ices Limited Report, Calgary, Alber ta , Jan. 2, 1970.
van Pool len , H.K., and Weber, J . D . : "Data Analysis f o r High I n f l u x Wells," Paper SPE 3017, presented a t t h e 45th Annual F a l l Meeting, SPE of AIME, Houston, Oct. 4-7, 1970.
Kohlhaas, C.A.: "A Method f o r Analyzing Pressures; Measured During D r i l l s t e m- T e s t Flow Per iods ," J. P e t . Tech. (Oct. 1972), 1278.
Jaeger , J . C . : "Conduction of Heat i n an I n f i n i t e Region Bounded Inte$b- a l l y by a C i r c u l a r Cylinder of a P e r f e c t Conductor," Aust. J. Phys. (1956), - 9, No. 2, 167.
'
van Everdingen, A.F.: "The Skin Ef fec t and Its In f luence on t h e Prodwc- t ive Capacity of a Well," Trans. , AIME (1953), -- 198, 171-176.
Hurst , W.: "Establishment of t h e Skin E f f e c t and I ts Impediment t o Fluid-Flow I n t o a Well Bore," P e t . Eng. (Oct. 1953), 25, B-6.
Agarwal, R.G., Al-Hussainy, R., and Ramey, H . J . , Jr.: "An Inves t iga- t i o n of Wellbore Storage and Skin E f f e c t i n Unsteady Liquid Flow: I. Ana ly t i ca l Treatment," SOC. P e t . Eng. J . (Sept. :L970), 279.
Ramey, H . J . , J r . , and Agarwal, R.G. : "Annulus Unztoading Rates as In- fluenced by Wellbore Storage and Skin Ef fec t , ' ' - Soc. P e t . Eng. J . (Oct. 1972), 453.
Papadopulos, I . S . , Bredehoeft, J . D . , and Cooper, H.H. , Jr.: "On t h e Analysis of 'Slug Test' Data," Water Resources Research (Aug. 1973), - 9, No. 4 , 1087.
Ramey, H . J . , Jr., Agarwal, R.G. and Mart in , I. : "Analysis of 'Slug T e s t ' o r DST Flow Period Data," J. Can. P e t . Tech. - (July-Sept. 19751, 1-11.
-145-
-146-
1 4 . Earlougher, R.C., Jr.: "Advances in Well Test Analysis," Monograph 5 , SPE, Dallas, Texas ( 1 9 7 7 ) , 96- 101.
1 5 . Bredehoeft, J.D., Cooper, H.H., Jr., and Papadopulos, I . S . : "Inertial and Storage Effects in Well-Aquifer Systems: An Analog Investigation," Water Resources Research ( 1 9 6 6 ) , 2, No. 4 , 697-7017.
1 6 . van der Kamp, G.: "Determining Aquifer Transmissivity by Means of Well Response Tests: The Underdamped Case," Water Resolurces Research (Feb. 1 9 7 6 ) , - 1 2 , NO. 1, 71- 77.
1 7 . Alexander, L.G.: "Theory and Practice of the Closed-Chamber Drillstem Test Method," J. Pet. Tech. (Dec. 1 9 7 7 ) , 1539- 1544.
1 8 . Marshal, G.R.: "Practical Aspect of Closed-Chamber Testing," 3 . Can. Pet. Tech. (July-Sept. 1 9 7 8 ) , 82- 85.
1 9 . Matthews, C.S., and Russell, D.G.: "Pressure Buildup and Flow Tests in Wells," Monograph 1, SPE, Dallas, Texas ( 1 9 6 7 ) , 4- 7.
2 0 . Binder, R.C.: Fluid Mechanics, Prentice-Hall ( 1 9 4 9 1 , 82- 83.
2 1 . van Everdingen, A.F., and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems," Trans., AIME ( 1 9 4 9 ) . 1 8 6 , 305- 324.
2 2 . Karman, V.T., and Biot, M.A.: Mathematical Methods in Engineering, McGraw-Hill ( 1 9 4 0 ) , 61- 63,
2 3 , David, P.J., and Rabinowitz, P.: Numerical Integration, Blaisdell Publishing Co. , Waltham, Mass. ( 1 9 6 7 ) , 1 6 6 .
24 . Watson, G.W.: A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press ( 1 9 4 4 ) , 202 .
2 5 . Churchill, R.V. : Operational Mathematics, McGraw-Hill ( 1 9 7 2 1 , 4 5 9 .
26 . Carslaw, H.S., and Jaeger, J.C.: Operational Methods in Applied Mathe- matics, Oxford Univ. Press ( 1 9 6 3 ) , 248- 249.
27 . Stehfest, H.: "Numerical Inversion of Laplace Transforms," Communica- tion of the ACM (Jan. 1 9 7 0 ) , - 1 3 , No. 1, 47- 49.
28 . Gaver, D.P.: "Observing Stochastic Process and Approximate Transform Inversion," Oper. Res. ( 1 9 6 6 ) , - 1 4 , No. 3 , 445-459.
2 9 , Veillon, F.: "Numerical Inversion of Laplace Transform," Communication of the ACM (Oct. 1 9 7 4 ) , - 1 7 , No, 10, 587- 589.
3 0 . Dubner, H., and Abate, J. : "Numerical Inversion of Laplace Transform and the Finite Fourier Transform," J. ACM (Jan, 1 9 6 8 ) , - 1 5 , No. 1, 115- 1 2 3 .
-147-
31. Albrecht, P., and Honig, G.: "Numerical Inversion der Laplace Trans- formierten," Angewandte Informstik (1977), 8, 336-345.
32. Abramowitz, M., and Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1J.S. Dept. of Commerce, A.M.S. 55 (1964), 378-379.
33. Shinohara, K., and Ramey, H.J., Jr.: "Analysis of 'Slug Test' DST F l d h Period Data with Critical Flow," Paper SPE 7981, presented at the 49th Annual California Regional Meeting, SPE of AIME, Yentura, Apr. 18-20, 1979.
34. Earlougher, R.C., Jr., and Kersh, K.M.: "Analysis of Short-Time Trans- ient Test Data by Type-Curve Matching," J. Pet. Tech. - (July 1974), 793-800.
35. Fair, W.B., Jr.: "Pressure Buildup Analysis with Wellbore Phase Redis- tribution," Paper SPE 8206, presented at the 54th Annual Fall Meeting;, SPE of AIME, Las Vegas, Sept. 23-26, 1979.
36. Stright, D.H., Jr.: "DST Analysis with Pressure Dependent Rock and Fluid Properties," Paper SPE 8352, presented at t:he 54th Annual Fall Meeting, SPE of AIME, Las Vegas, Sept. 23-26, 1979.
37. Fenske, P.R.: "Type Curves for Recovery of a Discharging Well with Storage," J. Hydrol. (1977), - 33, 341-348.
38. Carslaw, H.S., and Jaeger, J.C.: 2. g., 71.
39. Churchill, R.V.: 9. G., 177-178.
40. Churchill, R.V.: 2. - cit., 170.
41. Carslaw, H.S., and Jaeger, J.C.: 2. &., 76.
42. Carslaw, H.S., and Jaeger, J.C.: 2. - cit., 173.
43. Frick, T.C.: Petroleum Production Handbook, Vol. ~ 11, McGraw-Hill Book Co., New York (1962), 31-34.
44. Wylie, C.R.: Advanced Engineering Mathematics, McGraw-Hill (1975), 402-403.
45. Wylie, C.R.: - op. &., 315.
7. APPENDICES
A. INVESTIGATION-OF THE ANALYTICAL LAPLACE TRANSFORM INVER- SION OF EO. 40
Applying the Mellin Inversion Formula38 to Eq. 40:
Since the integrand has a branch because of we have t o choose an
integration path which does not contain the origin in evaluating Eq. A-1,
as shown in Fig. A-1.
B
0
FIG. A-1: INTEGRATION PATH
-148-
-149-
From t h e Chauchy i n t e g r a l theorem: 40
Re are t h e r e s i d u e s of t h i s func t ion .
From Carslaw and Jaeger,41 i f R + 00:
Then :
(A-3
We can eva lua te t h e i n t e g r a t i o n
X = u2eiT i n t o Eq. A-1:
by rep lac ing X by u 2 e ia . S u b s t i t u t i a g
26 Since:
(A-6
-150-
(A-7)
Substituting Eqs. 44 through 47 into Eq. A-8:
2 'u tD du OD -A3 (u)+iA4 (u)
A1 (u) -id2 (u) e
du
(< -9)
2 -iT* by setting X = u e .
2 -u tD du
Similarly, we can obtain the integration
OD ~A3(~)A2(~)-A4(~)A1(~)}+iIA3(u)A1(u)+A4(u)A2(u) 1 e A1(uI2 + A2(u) 2
(A-10)
From Eqs. A-4, A-9, and A-10:
-151-
Next we w i l l e v a l u a t e t h e r e s i d u e s of t h e func t ion .
r e s i d u e a t t h e branch p o i n t . For small z :
F i r s t , consider t h e
26
- 1 K1(z) = - Z
(A- 12)
(A- 13)
Then t h e r e s i d u e a t h = 0 becomes:
2 htD cD Rn h+C,(Rn 2-Y) lxD(tD=O)+a ‘x’ ( t =o) 1
2 2 LD A Rn x +C,X (Rn 2-y) (a X +CDSA+l) - -
2
e D D x [ { (a A+CDS) - 2 R i m X+O 2
(A- 14)
Since R i m h Rn X = 0, t h e r e s i d u e a t X = 0 = 0. A+O
42 I n o rder t o f i n d t h e o t h e r po les , set h = - B 2 , as done by Carslaw and Jaeger .
The dennminator of t h e in tegrand i n Eq. A-1 becomes:
(A- 15)
26 Since :
- J o ( z ) 5 iYo(z) 0 -
K (+iz) = - - J (2) + iY1(z) 1 - “1 2 1 -
Equation A-15 becomes:
- i .rrB 2 4 2 2 4 2 + - 2 [{(a B -CDsB +l ) J1(B) - CDBJo(B)} i { ( a B -CDsB 3-1) Y1(B) - C,BYO(B)ll
(A-18)
(A-16)
(A-17)
-152-
Then t h e po les have t o s a t i s f y t h e fol lowing equat ions:
2 4 2 (a B --CDSB +1) Y p ) - CDBY0(B) = 10 (A-20)
However, i t was no t pos s ib l e t o f i n d t h e va lues of (3 >which s a t i s f y t he se two
equat ions a n a l y t i c a l l y .
A s a r e s u l t :
(A-21)
The sum of t h e r e s idues of t h e func t ion remains unknown.
APPENDIX B: SEPARATION OF REAL AND IMAGINARY PARTS OF EQ. 48 AND EQ. 49
KO (4) K10' as an expl ic i t f u n c t i o n of R i n a Since we cannot o b t a i n
simple form, t h e e a r l y t i m e and t h e l a te t i m e approximations f o r t h e modi-
f i e d Bessel func t ion are adopted. Then, f o r t h e e a r l y t i m e s we should
handle Eqs. 51 and 53, and f o r t h e l a t e times w e should handle Eqs. 59 and
61.
I n o rder t o o b t a i n t h e real p a r t and t h e imaginary p a r t of t h e func t ions
s e p a r a t e l y , set :
kr T Since w e consider R = a + i - f o r k = 1, 2 , ...,-, and a i s p o s i t i v e and
g r e a t e r than t h e real p a r t s of t h e s i n g u l a r i t i e s of t h e f u n c t i o n ,
Tr o < e < - - - - 2
Then :
2 2 2 R = (a -b ) + i ( 2 a b )
-153-
03-21
03-31
-154-
B-1: FOR E A K Y TIMES
S u b s t i t u t i n g Eqs. from B-1 t o B-5 i n t o Eq. 51:
a 2 a b
r r x ( t =0) + i - x ( t s=O) x D = x D D ( t = O ) { - 2 - i +} + [ a x ' ( t =o> - -
r
- 2 D D 2 D D r D D
1 . -
a 2 { ( a 2 2 -b )+ i (2ab) ) + CDs(a+ib) + CD { 4 7 r+a + E } + 1
Define t h e fol lowing v a r i a b l e s :
A Z a 2 2 2 ( a - b ) + C s a + C D D
2 B r 2 a a b + C s b + C D D
E : a 2 x ' ( t =o) - - a x ( t -0) 2 D D- r D D
F E x ( t =0) 2 D D r
Then:
2 El-iF + -- D } Ai- iB
- x = { - E + a xA(tD=O) - i F
(B-11)
(B-12)
-155-
Similarly, from Eqs. 53, and B-1 through B-11:
- s(a+ib) + + i e 'WD A + iB = CD(E+iF) x
C D [ (AE+BF) [ sa + E } - (AF-BE) [ sb +,/y } ] - -
D2
cD [ (AE+BF) ( sb + } + (AF-BE) [ sa + e } ] D 2
+ i
B-2: FOR LATE TIMES
Substituting Eqs. B-1 through B-11 into Eq. 59:
2 - x = ( a x ' ( t =0) - E - iF D D D
cb-13)
E + iF (B-14) + 2 a {(a 2 2 -b )+i(2ab))+C s(a+ib)+l-CD(a+ib){~ 1 Rn r+y- Rn 2 + i - 8 1
D 2
Define the following variables:
(B-15) 2 2 2 * 1 G E a (a -b ) + C sa + 1 - C a (-Rn r + y - Rn 2) + CDb*- D D 2 2
2 1 8 H E 2a ab + C sb - C b ( Rn r + y - Rn 2) - C a - D D D 2 (B-17)
(B-17)
-156-
Then, Eq. B-14 becomes:
2 E+iF + - D D D } G+iH - x = { a x ' ( t =O)- E - iF
'E+'' } + i [ - F + -- C;F-HE I 2 = ( a x ' ( t =O)- E -I- - I2 2 I D D
Similarly, from Eqs. 61, B-1 through B-11, and B-15 to B-17:
1 0 2 s(a+ib)-(a+ib){- Rn r+i + y - Rn 2) - -
= CD(E+iF) x 'WD G + iH
(B-18)
I; C (E+iF) (G-iH) sa-,(, 1 Rn r+y- Rn 2) + b 2 e +i{sb-b(- 1 Rn r+y - Rn 2)-a, y } - - D 2
I2 (B-19)
Define the following variables:
Then :
cD{ (GE+HF) J-(GF-HE)K) - -
2 I
cD{ (GE+HF)K+(GF-HE) J) + i
I2
(B-20)
(B-21)
(B-22)
APPENDIX C: FIELD DATA EXAMPLE
C-1: WELL. RESERVOIR DATA. AND MEASURED PRESSURES
Four f i e l d examples are considered i n Sec t ion 2. The fol lowing sum-
marized t h e f i e l d d a t a necessary f o r t h e s e examples.
TABLE C-1: WELL, RESERVOIR DATA AND MEASURED PRESSURE I N EXAMPLE 1
Well and Reservoir Data
Formation th i cknes s , h 1 7 feet: (5.2 m)
Po ros i t y , @
Viscos i ty , 1-1
Compress ib i l i ty , c t
F lu id d e n s i t y ,
Formation temperature
Dimensionless wel lbore s t o r a g e cons t an t ,
T e s t i n t e r v a l
T e s t recovery
p f
cD
Dr i l l - p ipe r a d i u s , r Hole- size r a d i u s , r
I n i t i a l p r e s su re , p
P W
i
16 percent
1.0 cp ( M O - ~ Pa ' s )
8 ~ 1 0 - ~ l / p s i ( 1 . 1 6 ~ 1 0 -9 1/P@) 0.8458 ( 8 . 4 5 8 ~ 1 0 ~ kg/m3)
168°F (349°K)
2 . 4 8 ~ 1 0
7782-7796 f e e t (2372-2376 m) 180 f t mud (55 m) 4060 f t . gassy o i l (1237 m) 880 f t s a l t water (268 m) I
2.25 inches ( 5 . 7 2 ~ 1 0 - ~ m)
3.94 inches ( 1 . 0 0 ~ 1 0 - ~ m)
3475 ps ig (2 .396~10 Pa g)
4
7
CONTINUED
-157-
-158-
n
-159-
TABLE C-2: WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 2
Well and Reservoir Data
Formation thickness, h Porosity, $
Viscosity, 1-1
Compressibility, c
Fluid density, p f
Drill pipe radius, r
Hole-size radius, r
Initial pressure, pi
t
P W
Slug Test Data
t (min)(60 s)
0.0 3.29 7.26 11.52 15.83 19.70 23.52 27.38 33.20 37.99 41.04 45.79 50.77 54.74 59.68 64.03 68.49 73.42 78.07 81.94 85.14 87.65 90.80 93.07 96.80 100.33 103.96 107.40 110.98 114.90 117.37
pw 3 (psig)(6.89~10 Pa g)
161 181 215 251 272 298 315 322 345 370 388 407 423 443 471 486 500 525 551 569 580 585 593 605 624 632 64 5 655 677 694 7 01
9 ft (2.74 m) 0.15 0.39 CP (3.9~10-~ Pa s)
1 4 ~ 1 0 ~ ~ l/psi (2.03x10-’ 1/Pa)
1.9 in (4.83~10-~ m)
4 3/8 in (1.11~10-~ m) 7 2240 psig (1.555~10 Pa)
0.65 (6.5~10~ kg/m 3 )
PWD
1.0000 0.9904 0.9740 0.9567 0.9466 0.9341 0.9259 0.9226 0.9115 0.8995 0.8937 0.8817 0.8740 0.8644 0.8509 0.8437 0.8369 0.8249 0.8124 0.8038 0.7985 0.7961 0.7922 0.7864 0.7773 0.7734 0.7720 0.7624 0.7518 0.7436 0.7403
-160-
TABLE C-3: WELL, RESERVOIR DATA AND MEASURED PRESSURE IN EXAMPLE 3
Well and Reservoir Data
Formation thickness, h Porosity, @
Viscosity,
Compressibility, c
Wellbore oil density, pf Format ion temperature Test interval
uO
t
Drill pipe ID
Hold radius, r
Re c over y W
Drill collar
Oil formation volume factor
First flow period
Initial formation pressure, p i
Slug Test Data
t (min)(60 s)
0 2 4 6 8
10 12 14 16 18 20 22 24 26 28 30
70 ft (2.13~10 m) * 4% BV 156.3 cp (1.56x10-' Pa s ) 6 ~ 1 0 - ~ psi -1 (8. ~ O X ~ O - ~ ~ Pa-4)
2 3 0.925 gm/cc (9.25~10 kg/m )
87°F (304'K) i
4,374 ft to 4,444 ft (1333 m to 1355 m) 3.34 in (8.48xlO-* m) 1
4.375 in (1.11x10-~ m> 41.8 bbl (6.65 m ) (3,853 ft oil) (1174 m)
1 630 ft (192 m) ~
2.25 in ID (5.72~10~~ m)
0.999 15 min (900s) 1865 psig (1.286~10 Pa g)
3
7
pwf ( P S W
3 (6.89~10 Pa g)
371.4 384.0 404.6 422.4 441.3 459.1 478.0 496.3 514.6 534.1 552.5 571.9 590.8 607.5 625 2 641.8
1 0 992 0.978 0.966 0.953 0.941 0.929 0.916 0.904 3.891 0.879 0.866 0.853 0.842 0.830 0.819
CONTINUED
-161-
TABLE C-3, CONTINUED
t
(min)(60 s)
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 7 2 74 76 78 80 8 2 84 86 88 90 92 94 96 98
100 102 104 106 108 110 1 1 2 114 116 118 120 1 2 2 1 2 4 126
pw€ (PSii3)
3 ( 6 . 8 9 ~ 1 0 Pa g )
659.0 675.1 691.1 707.7 721.5 737.5 753.0 767.9 782.2 795.4 810.9 825.2 838.0 852.7 865.9 880.2 892.8 906.0 918.6 930.6 943.2 955.3 967.3 979.3 992.5
1002.3 1014.3 1025.8 1036.6 1049.2 1058.4 1069.9 1079.6 1089.9 1101.4 1110.0 1120.3 1129.5 1139.2 1150.7 1158.7 1167.9 1177.0 1186.8 1197.1 1204.5 1213.1 1222.3
PWD
0.807 0.797 0.786 0.775 0.766 0.755 0.745 0.735 0.725 0.716 0.706 0.696 0.688 0.678 0.669 0.659 0.651 0.642 0.634 0.626 0.617 0.609 0.601 0.593 0.584 0.578 0.570 0.562 0.555 0.546 0,540 0.532 0.526 0.519 0.511 0.505 0.499 0.492 0.486 0.478 0.473 0.467 0.461 0.454 0.447 0.442 0.436 0.430
CONTINUED
-162-
TABLE C- 3, CONTINUED
t (min)(60 s)
128 130 132 134 136 138 140 14 2 144 14 6 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 2 00 202 204 206 2 08 210 212 214 2 16 2 18 220
3 ( 6 . 8 9 ~ 1 0 Pa g)
1230.3 1240.1 1247.5 1256.1 1263.5 1272.1 1280.7 1287.6 1295.6 1303.1 1310.5 1319.1 1325.4 1332.9 1339.8 1347.2 1355.2 1360.4 1367.8 1374.1 1380.4 1387.9 1393.6 1399.3 1405.7 1411.4 1418.8 1423.4 1429.7 1435.4 1441.2 1448.1 1452.1 1457.2 1463.0 1468.1 1474.4 1478.4 1484.2 1488.7 1493.9 1499.6 1503.6 1508.8 1513.4 1518.5 1522.5
'WD
0.425 0.418 0.413
0.403 0.397 0.391 0.387 0.381 0.376 0.371 0.365 0.361 0.356 0.352 0.347 0.341 0.338 0.333 0.329 0.324 0.319 ' 0.316 0.312 0.308 0.304 0.299 , 0.296 0.291 0.288 0.284 0.279 0.276 0.273 0.269 0.266 0.262 0.259 0.255 ' 0.252 0.248 0.245 0.242 0.238 0.235 0.232 0.229
CONTINU
0.408 1
~
-163-
TABLE C-3, CONTINUED
t (min) (60 s)
222 224 226 228 230 232 234 236 238 240 242 2 44 246 248 250 252 254 256 2 58 260 262
Buildup Test D a t a
A t (min) (60 s)
0 1 2 3 4 5 6 7 8 9
10 1 2 1 4 16 18 20
t + A t P
A t
570.4 285.7 190.8 143.4 114.9
95.90 82.34 72 .18 64.27 57.94 48.45 41.67 36.59 32.63 29.47
n
(6 .89~10' Pa g)
1527.1 1531.7 1536.3 1540.9 1545.5 1548.9 1552.9 1556.9 1560.9 1566.7 1569.5 1573.5 1577.6 1581.6 1586.1 1589.6 1592.4 1596.5 1600.5 1603.9 1607.3
4 t = 569.4 min (3 .416~10 s) P
PWS (Psi31
( 6 . 8 9 ~ 1 0 Pa g )
1607.3 1816.5 1849.7 1852.6 1854.9 1856.0 1856.6 1857.2 1857.7 1858 3 1858.9 1859.5 1859.5 1859.5 1860.0 1860.0
A t (min) (60 s)
22 24 26 28 30 35 40 45 50 55 60 65 70 75 80 85
t +At: P-
A t 26.88 24.73 22.90 21.31 19.98 17.27 15.24 13.65 12.39 11.351 10.49
9.76 9.13; 8.59 8.19 7.70
--
'WD 0.226 0.223 0.220 0.217 0.214 0.212 0.209 0.206 0.204 0.200 0.198 0.195 0.192 0.190 0.187 0.184 0.183 0.180 0.177 0.175 0.173
%s ( P S W
3 ( 6 . 8 9 ~ 1 0 Pa g)
1860.6 1861.2 1861.2 1861.2 1861 2 1861.2 1862 3 1862.3 1862.3 1863.5 1863.5 1864.1 1864.1 1864.1 1864.1 1864.1
CONTINUED
-164-
TABLE C-3, CONTINUED
A t pws t + A t ( P S W P 3
LIL
(min) (60 SI A t ( 6 . 8 9 ~ 1 0 Pa g )
90 95
100 105 110 115 120 125 130 135 140 1 4 5 150
7.33 1864.1 6.99 1864.6 6.69 1865.2 6 .42 1865.2 6.18 1865.2 5 .95 1865.2 5.75 1865.2 5.56 1865.2 5.38 1865.2 5.22 1865.2 5.07 1865.2 4.93 1865.2 4.80 1865.2
155 160 180 2 00 220 240 260 280 300 320 340 355
t + A t P-
A t --
4.6'7 4.56 4 .16 3.8.5 3.59 3.3'1 3.19 3.0:3 2.90 2 .78 2.6(3 2.60
(6 .89~10 ' Pa g )
1865.2 1865.2 1865.2 1865.2 1865.2 1865.2 1865.2 1865.2 1865.2 1865.2 1865.2 1865.2
-165-
TABLE c-4: MEASURED LIQUID LEVEL (TAKEN FROM GRAPHS IN VAN DER KAEIP'~)
L i q u i d L e v e l , x ( cm)
T i m e , t (SI
0
1
2
3
4
5
6
7
8
9
10
11
1 2
1 3
14
15
16
1 7
18
19
20
21
2-C Well
-3.87
-2.65
-0.76
0.18
0.60
0.50
0.17
-0.13
-0.23
-0.24
-0.20
-0.11
-0.02
0
9-A W e l K
-4.60
-3.61
-2.72 1 -2.10
1
-1.66 I -1.32
-1.08
-0.85 I -0.79 ~
-0.66 i I
-0.57
-0b49 -0.39 ~
-0.30
-0.23
-0.20
-0.13
-0.11 ' -0.09
-0.06
-0.02
i
0
-166-
C-2: PRESSURE DROP CAUSED BY FRICTION
C-2.1 Example 1
We w i l l cons ider t h e p re s su re drop by f r i c t i o n a t t h e end of t h e
apparent cons t an t f l owra te per iod a t e a r l y t i m e s ( i . e . , t = 40 min, i n t h i s
example), which i s c l o s e t o t h e maximum value .
t h e a c t u a l f l owra te , q, f o r t h e apparent cons tant f l owra te per iod .
From Fig. 58 we can o b t a i n
2 dPwf q = _ _ E . -
wr Pf*g d t
- - 1732-712 (0.8458)(62.4)(32.2) 70
( f t 2 ) ( f t 3 /lbm) ( s ec 2 / f t ) (psi /min)
-3 3 = 1126.1 bbl/day ( 2 . 0 7 2 ~ 1 0 m /s)
Then t h e v e l o c i t y of t h e l i q u i d i n t h e wel lbore , u , becomes:
L 2 n r
P
u =
2 - - 1126'1 = 10,196 (bbl/day) ( l / f t )
= 0.663 f t / s (0.202 m/s)
-167- Thus :
PEuD Reynolds Number = - u
2 . 2 5 ~ 2 (0.8458) (62.4) (0.663) [ .,
(1) ( 2 . 0 9 ~ 1 0 - ~ ) (32.2)
= 19,500
Therefore t h e flow i s t u r b u l e n t .
From t h e Moody diagram43 (assuming E / D = 0.01), t h e f r i c t i o n f a c t o r ,
f , i s about 0.04. The l i q u i d l e n g t h i n t h e wel lbore , L a t t = 40 min
( 2 . 4 ~ 1 0 s) becomes (consider ing t h e cushion l i q u i d ) : 3
(560) (144) (0.8458) (62.4:) L = (0.663) (40x60) +
= 3120 f t (951 m)
Then t h e p ressure drop caused by f r i c t i o n i n t h e well-bore becomes:
2 L PU A p = f 9 - .- D 2
(3120) ,, (0.8458) (62.4) (0.663)L 2 = (0.04) zx2.25
1 2
2 2 ( f t ) ( l / f t ) (Rbm/ft3:, ( f t /sec )
3 = 0.83 p s i ( 5 . 7 2 ~ 1 0 Pa)
-168-
C-2.2: Example 3
S imi la r t o Example 1, w e w i l l cons ide r t h e p r e s s u r e drop caused by
f r i c t i o n a t t h e end of t h e apparent cons tan t f l o w r a t e per iod ( i . e . , t = 30
min, i n t h i s example).
Then : q = 363.4 bb l jday ( 6 . 6 8 7 ~ 1 0 - ~ m3/s:i
u = 0.388 ft/s (0.118 m / s )
Reynolds Number = 59.3
Thus t..e f low is laminar and t h e f r i c t i o n f a c t o r , f , becomes:
f = - - 64 - 1.08 NRe
S i m i l a r t o Example 1:
L = 1625 f t (495 m)
Then:
4 Ap = 5.9 p s i ( 4 . 0 7 ~ 1 0 Pa)
APPENDIX D: DERIVATION OF FINITE DIFFERENCE SOLUTIONS -
D-1: Closed Chamber Test
From Eq. 14:
where : i-1 M-1 -
r = (rD > Di e
a =- ' inr De M-1 (D-3 >
r is the dimensionless radial distance at node i, r Di De
is the dimensionless
external radius of the reservoir, and M is the number of nodes.
From Eq. 83:
n+l n n-1 -2x D D +x 2 XD a . (AtD>
1 n+l n 2 D + - (x +XD)
0-4 > -169-
-170-
From Eq. 27:
n+l - n+l s - - - ( P - P 1 pWD pD1 a D2 Dl
From Eq. 2 4 :
n+l n pn+l- n+l
D D - - -- 1 D2 pD1 x -x
a AtD cD
Then t h e system of equa t ions becomes as fo l lows :
When s # 0:
c (1)
-1
1
0
0
0
1
1
1
0
0
0
S - a
E(2)
. . .
E (M-1)
1
0
0
0
1
.
0
. 1
Em)+
n+l D X
n+l 'WD
P"+l Dl
n+l
pD2
n+l
'M-1 P
n+l P
DM
(D-'6)
/
A"
5 n K
At, D
0
-171-
Where :
n A =
E ( i ) = -
- %
2
f o r a + 0 b(1) =
I 1 f o r 01 = 0
2 (AtD)
f o r a # 0 c(1) = [
f o r a = 0
2 2 (At,) n n-1 -
2 "wD x - x
2a (2 - 2a2 J D D
(At,)
- B 2 a
LD -xD(o) PD
LD -x atm -- i
n P D i
f o r a = 0
(D-TO) I
-172-
When s = 0:
- - CDa CDa
1
0 1 E ( 2 )
. .
0 1 E (M-1)
1
n+l D X
n+l P Dl
pD, n+l
L
n+l
'M- I.. P
Pn+l DM
0
1
1
E (M)+1
\
An
L
(D- 12)
E ( i ) , b ( l ) , c ( l ) , and A n are t h e same as def ined i n E q s . D-8 through D - 1 1 .
D-2: Slug Test
From Eq . 18:
n+l-2xnhn-l
(D-13) 2 XD D D 1 n+l n 1 n.+l n + (XD +xD) = - - 2 ('wD a .
(At,)
This equat ion should b e used i n s t e a d of Eq. D- 4 . Then, Eq, D-11 should be
changed as fol lows:
An = [
0
2 (At,)
2a f o r a # 0 - n n-1
D D 2 'wD x - x
(D- 14)
f o r a = 0
The o t h e r equat ions are t h e same as those i n t h e c losed chamber test .
-173-
D-3: Buildup T e s t
A f t e r t h e shut- in , w e w i l l consider t h pressur 'e change only i n s i d e
t h e r e s e r v o i r , Then w e can u s e Eq. D-1 f o r t h i s case . Assuming t h e follow-
ing equat ions t o be t h e boundary cond i t ions :
- 'D1 - 'D2
P = P 'M-1 DM
t h e system of equat ions becomes as follows:
I
E (1)+1
1
0
. 1
0
. . E (M-1
. . I 1
' pn+l
Dl
pn+l
D2
pn+l
D3 . . n+l
P DM-l
n+l P
DM
(D- 15)
[ E (1)+2] Pn Dl
[ E (M-1)+2] pn DM-
(D-16)
APPENDIX E: COMPUTER PROGRAM
The fou r computer programs are shown
gram of t h e S t e h f e s t method t o o b t a i n t h e
he re . The f i r s t one i s t h e pro-
Laplace t ransform inve r s ion f o r
t h e s u b j e c t func t ion . The second is t h e program of t h e Albrecht-Honig
method f o r t h e same purpose. The t h i r d i s t h e program t o o b t a i n t h e pres-
s u r e 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 . The las t i s t h e program t o simu-
l a t e a c losed chanber t e s t , s l u g t e s t , and buildup t e s t . The program of Y T i
t h e Ve i l lon method i s no t shown here . One can f i n d t h e l o g i c i n t h e o r i -
28 g i n a l paper.
-174-
-175-
E-1 STEHFEST METHOD
-176-
DPW( I ) = O . A( I ) = O ,
100 CONTINUE DO 150 K=NI,NTIME 1FCK.EQ.MIl GO TO 101 ALF=lO.**(K-l) GO TO 102
101 ALF=O. 102 CONTINUE
151 FORHAT(lH1,D30.10) WRITE(6.151) ALF
TT= 0 . DO DT= 1. D-2 DO 200 I=l,M TT=TT+DT
r
IF(TT.GE.9.99D-5) DTZ1.D-4 ~ IF(TT.GE.9.99D-4) DT=l.D-3
IF(TT.GE.9.99D-3) DTz1.D-2 IF(TT.GE.9.99D-2) DT=l.D-l IF(TT.GE.9.99D-1) DT=l.DO IF(TT.GE.9.99DO) DT=l.Dl IF(TT.GE.9.99Dl) DT=l.D2 IF(TT.GE.9.99D2) DT=l.D3 IF(TT.GE.9.99D3) DT=l.D4 IF(TT.GE.9.99D4) DT=l.D5 IF(TT.GE.9.99DS) DT=l.D6 IF(TT.GE.9.99DG) DT=l.D7 IF(TT.GE.9.99D7) DT=l.D8 IF(TT.GE.9.99D8) DT=l.D9 IF(TT.GE.9.99D9) DT=l.DlO IF(TT.GE.9.99D10) DT=l.Dll IF(TT.GE.9.99Dll) DT=l.D12 IF(TT.GE.9.99D12) DT=l.D13 IF(TT.GE.9.99D13) DT=l.D14 IF(TT.GE.9.99D14) DT=l.DlS TD ( I )=TT DO 300 J=l,N cc= 2. CO=DLOG(CC)/TD(I) SS=CO*DFLOAT(J) RSS=DSQRT(SS) FA=ALF*SS+SKIN*CD Al=BKO(RSS)/BKl(RSS) Fl=DXO/SS+(ALF*DXDO-DXO~SS)/(ALF~SS*SS+CD~SKIN~~SS+l.
$ +CD*RSS*A 1 ) F2=CD~(ALF"DXDO-DXO/SS)*(SKIN*SS+RSS*Al)/(ALF*:jS~SS
$ +CD*SKIN*SS+l.+CD*RSS*Al) DFl=V(J)*Fl DF2=V(J)*F2 SF=SF+DF 1 SF2=SF2+DF2
3 0 0 CONTINUE DX( I )=SF*CO SP= 0. DPW(I)=SFZ*CO SF2=0.
2 0 0 CONTINUE
. . 1 5 0 CONTINUE
-177-
S T O P END FUNCTION BKOCX)
IF(X.GT.2.) GO T O 700 T=X/2. Al=-DLOG(T)*BIO(X)-.577215G6+.4227842*T**2+.23tl6975G*T**4 A2=.0348859*T**6+.00262698*T**8+.0001075*T**10i~.0000074*T**12 BKO=A 1+A2 GO T O 701
T=2./X
IMPLICIT REAL*8 (A-H.0-Z)
7 0 0 IF(X.GT.1.74D2) G O TO 7 0 2
A3=1.25331414-.07832358*T+.021895G8*T**2-.010G~!446*T**3 A4=.00587872*T**4-.0025154*T**5+.00053208*T**6 BKO=(A3+A4)/(DSQRT(X)*DEXF(X)) G O T O 701
7 0 2 BKO= 1. D-70 701 RETURN
END FUNCTION BKl(X)
IF(X.GT.2.) G O TO 7 1 0 T=X/2. Al=X*DLOG(T)*BIl(X~+l.+.l5443144*T**2-.G7278579*T**~ A2~-.~8156897*T**6-.01919402~T**8-.00110404*T**10-.00004686*T**12 BK 1 = ( A 1 + A 2 )/X GO T O 711
T=2./X
IMPLICIT REAL*8 (A-H,O-Z)
7 1 0 IF(X.GT.1.74D2) GO T O 712
A3=1.25331414+.23498619*T-.0365562*T**2+.01504268*T**3 A4=-.00780353*T*~4+.00325G14*T**5-.00068245*T**6 BKl=(A3+A4)/(DSQRT(X)*DEXP(X)) G O TO 711
712 BKl=l.D-70 711 RETURN
END FUNCTION BIO(X)
IF(X.GT.3.75) G O TO 720 IIIPLICIT REAL*8 (A-HP0-Z)
T=X/3.75 A1=1.+3.5156229*T**2+3.0899424*T**4+1.2067492*T~*6 A2=.2659732*T**8+.0360768*T**10+.0045813*T**12 BIO=AI+A2 G O TO 721
T=3.75/X 7 2 0 IF(X.GT.1.74D2) GO TO 7 2 2
A3~.39894228+.01328592*T+.00225319*T**2-.00.1575G5*T**3+.00916281*T*~4 A4=-.02057706~T**5+.02G35537*T**G-.OlG~7633*T**7+.00392377*T**8 BIO=(A3+A4)/(DSQRT(X)*DEXP(-X)) GO T O 721
7 2 2 BIO=l .D70 721 RETURN
END FUNCTION BIl(X)
IF(X.GT.3.75) G O TO 730 T-Xl3.75 A1=.5+.87890594*T**2+.514988G9*T**~+.1508493*T*~6+.02658733*T**8 A2=.00301532*T**10+.00032411*T**12 BI l=(Al+AZ)*X
IMPLICIT REAL*8 (A-Hp0-Z)
-178-
GO TO 731
T=3.7 5 / X A 3 ~ . 3 9 8 9 4 2 2 8 - . 0 3 9 8 8 0 2 4 * T - . 0 0 3 6 2 0 1 8 * T * * 2 + . 0 0 1 6 3 8 0 1 * T * * 3 - . 0 1 0 3 1 5 5 5 * T * * ~ A~~.02282967*T**5-.0289531~*T**~~.01787654*T**7-.00420059*T**8 B I l = ( A 3 + A 4 ) / ( D S Q R T ( X ) + D E X P o ) G O TO 731
7 3 2 BIl=l.D70 731 R E T U R N
730 IF(X.GT.1.74DZ) G O TO 732
END CDATA $STOP / f
-179-
E-2 ALBRECHT-HONIG METHOD
C LAPLACE INVERSION BY ALBRECHT-HONIG C ALf SQUARE OF DIMENSIONLESS NUMBER C ARI=I DETERMINE EXTREME VALUE C C CON VARIABLE TO DETERMINE MAXIMUM DE:SCRETIZATION ERROR C CD DIMENSIONLESS WELLBORE STORAGE CONSTANT C DX 0 INITIAL VALUE OF DIMENSIONLESS LIQUID LEVEL C DXDO INITIAL VALUE OF VELOCITY OF DIMENSIONLESS LIQUID LEVEL
F EXTERNAL FUNCTION C MlNl NUMBER OF FUNCTION VALUE TO BE COPIPUTED IN (T\,TN) C NN NUMBER OF EXTREME VALUE C NS 1 NUMBER OF FUNCTION VALUE REQUIRED FOR CALCULATING INVERSION C NS 2 NUMBE.R OF FUNCTION VALUE REQUIRED FOR CALCULATING CORRECTION TEl C SKIN SKIN FACTOR C Tl LOWER INTERVAL BOUNDARY OF TIME C TN UPPER INTERNAL BOUNDARY OF TIME
ZOTHERW I S E 0 PT I ON A L
IMPLICIT REAL18 (A-H,O-Z) REAL*8 H(2,1000),TT(l000),E(lOOO) COPlllON M M ( 2, 1 0 0 0 1
COPlPLEX*16 F,S EXTERNAL F T \ = O . TN=lOO. NlM1=9 NS1=20 NS2=20 CON=5 NN=8 ART= 1
INTEGERI4 MM,ART,RICHT
CALL L A P I N ( F ~ T l , T N ~ N l M l ~ H I T T 1 N S l r N S l , N S 2 ~ C O N ~ N N ~ M M ~ A R T ~ E ~ I E R ~ WRITE(6,lOOO) IER
1000 FORHAT(Il0)
2 0 0 0 FORMAT(D20.5,F15.10) W R I T E ( 6 , 2 0 0 0 ) ( T T ( J ) , H ( 2 r J ) , J ~ ~ J = l ~ N l M l )
STOP END SUBROUTINE L A P I N ( F ~ T l ~ T N ~ N l M l , H ~ T T I N S l r N S l ~ N S 2 ~ C O N ~ N N ~ M M ~ A R T ~ E ~ I E R ~
C C C C
MINIMUM MAXIMUM ORRECTIVE TERM PASSES (NUMERICAL INVERSION EINEP LAPLACE TRANSFORMATION FCS)
IMPLICIT REAL*8(A-H,O-Z) REAL*8 E ( N S 1 ) , H ( 2 , N l M l ) , K O R . T T o , F A K T
COMPLEX*l6 F,S EXTERNAL F IER=O IF(TH. LE.Tl) IER= 1 IF(NlMl.LT.1) IER=IER+lO IF(NSl.LT.1. OR. NS2.LT.l) IER=IER+100 IFCNN. LT. 1 ) IER= I ER+ 10 00 IF(IER.GT.0) GO TO 250 IF( NN.GT .NS 1 ) NN=NSl IF(NN.GT.NS2) NN=NS2 PI=4.DO*DATAN(I.D0) N l=N lMl+ 1 DELTA=(TN-Tl)/DFLOAT(Nl) I= 1 K2= 1
INTEGERIQ NH(Z,NlMl),ART,RICHT
-180-
NSUM= NS 1 C C CALCULATION OF THE VALUE OF T IN (Tl-TN) C
2 0 DO 100 Kl=l,NlMl N=NN K=I*K 1 + 1 T=DFLOAT(Kl*I)*DELTA+Tl*I V = C 0 N / T
SURE=O.DO PITE=PI/T €INS= 1. DO r
RAL=-O.SDO*DRERL(F(DCMPLX(V,O.DO)))
C C CALCULATION OF THE APPROXIMATE VALUE ECL) TO BE USED C AS AN INVERSE L=1,2,.....NSUM TO BE CALCULATED BY C THE METHOD OF DURBAN C
DO 30 L=l,NSUM W=DFLOAT(L-l)kPITE
EINS=-EINS SURE=SURE+DREAL(F(DCMPLX(V,W)))*EINS
3 0 E(L)=DEXP(V*T)/T*(RAL+SURE) C
C C NN SEEKING FOR THE EXTREME VALUES C
IFCART.NE.1) GO TO 5 0
RIGHT=- 1 IF(E(NSUM).GT.E(NSUM-l)) RICHT=l NSUMPl=NSUH+l Jl=NSUMPl J2=NSlIM
42 RIGHT=-RICHT J 1=J 1- 1 E( J1 )=E( 52)
Jl=Jl-l E(J1 )=E( 1) NZNSUHP 1-J 1 G O TO 5 0
5 2 ~ 5 2 - 1 IF(E(J2)-E(J2-1)) 48,44,47
GO TO 42
GO TO 42
44 IF(J2.GT.2) GO TO 46
46 IFCNSUHPI-J1.EQ.N) GO TO 5 0
47 IF(RICHT.EQ.-l) GO TO 44
48 IF(RICHT.EQ.1) GO TO 44
C C INTERPOLATION AND MID POINT DETERMINATIOM C
5 0 1F(N-3) 52.54,56 5 2 H(K2sKI)=E(NSUH)
GO TO 5 9
GO TO 59
If(N.EQ.4) GO TO 5 8 J l=NSUM-N+3 J 2ZNSUM-2
5 4 H(K2,Kl)=(E(NSUM-2)+E(NSUM))/4.DO+E(NSUM-l)/2.DO
56 SUM=(E(NSUM)+E(NSUM-N+1))+0.25DO+(E(NSUM-l)+E(~tSUH-N+2))*0.75DO
-181-
DO 57 J=J1,J2 57 SUM=SUM+E(J) 58 H(K2,Kl)=SUM/DFLOAT(N-2) 59 MM(KZtKl)=N
1 0 0 CONTINUE IF(I.EQ.3) GO TO 150 1 = 3 K 2 = 2 NSUM= NS 2 G O T O 20
C C CALCULATION OF THE CORRECTED TERM AND CORRECTION TERM OF C OF THE 'MINIMAX'-WEPTE C
150 FAKT=-DEXP(-2.DO*CON) DO 200 K=lrNlMl H(Z,K)=H(l,K)+FAKT*H(2,K) TT(K)=DFLOAT(K)*DELTA+Tl
200 CONTINUE 250 RETURN
END FUNCTION F(S)
C D = 10. **3 ALF=lO.**3 SKIN=O. DXO=- 1. DXDO=O. RS=CDSQRT(S) Q=(ALF*S*S+I.)*RS/(SKIN*RS+BKO(RS)/BKI(RS)) F = C D * ( - D X O + A L F * S * D X D O ) / O RETURN END FUNCTION BKOCX) IMPLICIT REAL*8 (A-H,O-Z) COMPLEX*l6 BKO,X,T,BIO,Al.A2,A3,A4 IF(CDABS(X).GT.Z.) G O TO 700 T=X/2. Al=-CDLOG(T)*BIO(X)-.577215G6+.4227842*T**2+.23OG975G*T*~4 A2=.0348859*T**6+.OU262698*T**8+.0001075*T**10+.000007~*T**12 BKO=Al+A2 G O T O 701
T=2./X
COMPLEX*l6 FsS,BKO.BKl,QpRS
700 IF(X.GT.1.74D2) G O T O 7 0 2
A3~1.25331414-.07832358*T+.021895G8*T**~-.01062446*T**3 A9=.00587872*T**4-.0025154*T**5+.000!53208*T**6 BKO=(A3+A4)/(CDSQRT(X)*CDEXP(X)) G O TO 701
7 0 2 BKO=l.D-70 701 RETURN
END FUNCTION BKl(X) IMPLICIT REAL*8 (A-H,O-Z) COMPLEX*l6 BKl,X,T,BIl,Al,A,A3,A4 IF(CDABS(XI.GT.2.) GO TO 710 T=X/ 2. A1=X*CDLOG(T)*BIl(X)+l.+.l~443144*T**2-.~7278579*T**4 A2~-.18156897*T**6-.01919402*T**8-.00110404*T**10-.00004686*T~*12 BKl=(Al+A2)/X GO T O 711
-182-
7 1 0 IF(X.GT.1.74D2) GO TO 7 1 2 T=2./X A3~1.25331414+.23498619*T-.03G~562*T**2+.01504268*T**3 A4=-.00780353*T**4+.00325614*T**5-.00068245*T**6 B t : l = ( A 3 + A 4 ) / ( C D S Q R T ( X ) x C D E X P ( X ) ) GO TO 711
7 1 2 BKlZ1.D-70 711 RETURN
END FUNCTION BIO(X) IMPLICIT REAL*8 (A-He0-Z) COMPLEX*16 B I O ,X, T, A 1 o A2, A3, A4 IF(CDABS(X).GT.3.75*) GO TO 720 T=X/3.75 A1=1.+3.5156229*T**2+3.0899424*T**4+1.2067492*T**6 A2=.2659732*T**8+.03G0768*T**10+.0045813*T**12 B10=A 1+A2 GO TO 721
T=3.75/X A3~.39894228+.01328592*T+.00225319*T**2-.00157565*T**3+.00916281*T**4 A4~-.02057706*T**5+.02635537*T**6-.01647633*T**7+.00392377*T**8 BIO=(A3+A4)/(CDSQRT(X)*CDEXP(-X)) GO TO 721
7 2 2 BIO=l.D70 721 RETURN
7 2 0 IF(X.GT.1.74D2) G O TO 7 2 2
E N D FUNCTION BIl(X) IMPLICIT REAL*8 (A-H.0-Z)
IF(CDABS(XI.GT.3.75) GO TO 730 T=X/3.75 A1=.5+.87890594*T**2+.51~98869*T**4+.1508493*T**6+.02658733*T**S A2=.00301532*T**lO+.0003241l*T**12 BI l=(Al+A2)*X GO TO 731
T= 3.75/X
COMPLEX*lG B I l , X ~ T , A l , A 2 ~ A 3 , A 4
730 IF(X.GT.1.74DZ) G O TO 7 3 2
A3~.39894228-.03988024*T-.00362018*T**2+.00163801*T**3-.01031555*T**~ A4=.02282967*T**5-.02895312*T**6+.01787654*T**7-.00420059*T**8 B I l = ( A 3 + A 4 ) / ( C D S Q R T ( X ) x C D E X P O ) GO TO 731
7 3 2 BIl=l.D70 731 RETURN
END $DATA $STOP //
E-3
C c C C C C c C C c C c C c C
-183-
COMPUTER PROGRAM TO CALCULATE THE PRESSURE DISTR1:BUTIONS INSIDE THE RES ERVO I R
-184-
RAD16=7.D3 RAD 17= 1. D4 RAD18=3.D4 H = 3 0 N= 16 NI= 1 N TIME= 10 DLOGTW= .69311+7 1805599453DO G(l)=l.DO NH=N/2 DO 5 1 = 2 ~ N
5 G(I)=G(I-l)*I H(1)=2./G(NH-l) DO I O I = Z D N H F I = I IF(I.EQ.NH) GO TO 8 H(I)=FI**NH*G(2~I)/(G(NH-I)*G(I)*G(I-1)) G O TO 10
10 CONTINUE 8 H(I)=FI**NH*G(2*1)/(G(I)*G(I-l))
SN=2*(NH-NH/2*21-1 DO 50 I=l,N V(I)=O. K1=(1+1)/2 K 2 = I IF(K2.GT.NH) K 2 = N H D O 40 K=Kl,K2 IF(Z*K-I.EQ.O) GO T O 37 IF(1.EQ.K) G O TO 38 V(I)=V(I)+H(K)/(G(I-K)*G(2*K-I)) G O T O 40
37 V(I)=V(I)+H(K)/(G(I-K)) G O TO 40
38 V(I)=V(I)+H(K)/G(2*K-I) 9 0 CONTINUE
V ( I 1 =SN*V ( I ) SN=-SN
50 CONTINUE DO 1 0 0 I = 1 ~ 2 0 0 TD(I)=O.DO DX(I)=O.DO DPW(I)=O.DO A(I)=O. DO DPl(I)=O.DO DPZ(I)=O.DO DP3(1)=O.DO DP4(I)=O.DO DP5(I)=O.DO DPG(I)=O.DO DP7(I)=O.DO DP8(I)=O.DO DP9(I)=O.DO DPlO(I)=O.DO DPll(Il=O.DO DP12(1)=O.DO DPl3(I)=O,DO DPlQ(I)=O.DO DP15(I)=O.DO DP16(I)=O.DO DP 17 ( I 1 = 0. D 0
-185-
DP18(I)=O.DO 100 CONTINUE
DO 150 K=NI,NTIME IF(K.EP.NI) GO TO 101 ALF= 10. **(K- 1) GO TO 102
101 ALF=O. 102 CONTINUE
WRITE(6,151) ALF 151 FORMAT(1H1,D30.10)
TT=O. DO DT=l.D3
TT=TT+DT DO 200 I=l,M
IF(TT.GE.9.99D-2) DTz1.D-1 IF(TT.GE.9.99D-1) DT=l.DO IF(TT.GE.9.99DO) DT=l.Dl IF(TT.GE.9.99Dl) DT=l.D2 IF(TT.GE.9.99D2) DT=l.D3 IF(TT.GE.9.99D3) DT=l.D4 IF(TT.GE.9.99D4) DT=l.D5 IP(TT.GE.9.99D5) DT=l.D6 '
IF(TT.GE.9.99D6) DT=l.D7 IF(TT.GE.9.99D7) DT=l.D8 TD(I)=TT DO 300 J=l,N cc=2. CO=DLOG(CC)/TD(I) SS=CO*DFLOAT(J) RSS=DSQRT(SS) RSS 1 = R A D 1 "RSS RSS2=RAD2*RSS RSS3=RAD3*RSS RSSQ=RAD4*RSS RSS5=RAD5*RSS RSS6=RADG*RSS RSS7=RAD7*RSS RSS8=RAD8*RSS RSS9=RAD9*RSS RSSlO=RADlO*RSS RSS11=RADIl*RSS RSS12=RAD12*RSS RSS13=RAD13+RSS RSSlQ=RAD14*RSS RSSIS=RADlS*RSS RSS16=RADlG*RSS RSS17=RAD17*RSS RSS 18=RAD 18*RSS FA=ALF*SS*SS+CD*SKIN*SS+l. AI=BKO(RSS)/BKl(RSS) A21=BKl(RSS)/BKO(RSSl) R22=BKl(RSS)/BKO(RSS2) A23=BKl(RSS)/BRO(RSS3) A24=BKl(RSS)/BKO(RSS4) A25=BKI(RSS)/BKO(RSS5) A2G=BKl(RSS)/BKO(RSSG) A27=BK 1 ( RSS )/BKO ( RSS7 1 A28=EKl(RSS)/BKO(RSS8) A29=BKl(RSS)/BKO(RSS9) AElO=BKI (RSS)/BKO(RSS 10)
-186-
A211=BKl(RSS)/BKO(RSSll) A212=BK\(RSS)/BKOtRSSl~) A213=BKl(RSS)/BKO(RSSl3) A31=Al*A21 A32=Al*A22 A33=Al*A23 A34=Al*A24 A35=Al*A25 A36=Al*A2G 'A37=A 1*A27 A38=Al*A28 A39=AlXA29 A310=A\*A210 A311=Al*A211 A312=Al*A212 A313=Al*A213 Fl=DXO/SS+(ALF*DXDO-DXO/SS)/(FA+CD~RSS.*Al) F2=CD*(ALF*DXDO-DXO/SS)*(SKIN*SS+P.SS*Al)/(FA+(~D*RSS*Al) F3=(ALF*DXDO-DXO/SS)*CD/(FA*A21/RSS+CD*A31) F4=(ALF*DXDO-DXO/SS)*CD/(FA*A22/RSS+CD*A32) F5=(ALF*DXDO-DXO/SS)*CD/(FA*A23/RSS+CD*A33) F6=(ALF*DXDO-DXO/SS)*CD/(FA*A24/RSS+CD*A34) F7=(ALF*DXDO-DXO/SS)*CD/(FA*A25/RSS+CD*A35) F8=(ALF*DXDO-DXO/SS)*CD/(FA*A2G/RSS+CD*A3G) F9=(ALF*DXDO-DXO/SS)*CD/(FA*A27/RSS+CD*A37) FlO=(ALF*DXDO-DXO/SS)*CD/(FA*A28/RSS+CD*A38) F11=(ALFxDXDO-DXO/SS)*CD/(FA*A29/RSS+CD*A39) F12=(ALF*DXDO-DXO/SS)*CD/(FA*A210/RSS+CD*A3101 F13=(ALF*DXDO-DXO/SS~*CD/(FA*A211/RSS+CD*A3111 F14=(ALFsDXDO-DXO/SS)*CD/(FA*A212/RSS+CD*A3121 F15=(RLF*DXDO-DXO/SS)*CD/CFA*A213/RSS+CD*A3131 DFl=V(J)*Fl DF2=V(J)*F2 DF3=V( J 1 *F3 DFQ=V(J)*F& DFS=V(J)*F5 DFG=V(J)*FG DF7=V(J)*F7 DF8=V(J)*F8 DF9=V(J)*F9 DFlO=V(J)*FlO DFll=V(J)*Fll DF12=V(J)*F12 DF13=V(J)*F13 DF14=V(J)*Fl4 DF15=V(J)*F15 SFI=SFl+DFl SF2=SF2+DF2 SF3=SF3+DF3 SFQ=SF4+DF4 SF5=SF5+DF5 SFG=SFG+DFG SF7=SF7+DF7 SF8=SF8+DF8 SP9=SF9+DF9 SF 1 O=SF 1 O+DF 10 SFI 1=SF1 I+DFll SF 12=SF 12+DF 12 SF13=SF13+DF13 SF14=SF14+DF14
-187-
SF15=SF15+DF15 300 CONTINUE
DX(I)=SFI*CO S F 1=0 . DO DPW(I)=SFZ*CO SF2=0. DO DPl(I)=SF3*CO SF3=O.DO DP2(1)=SF4*CO SF4=0. DO DPJ(I)=SFS*CO SF5=0. D O DPQ(I)=SF6*CO SF6=0. DO DP5(I)=SF7*CO SF7=0. DO DP6(I)=SF8*CO SF8=O.DO DP7(I)=SF9*CO SF9=0. D O DP8(I)=SFlO*CO SFlO=O. DO DP9(I)=SFll*CO SF1 1=0. DO DPlO(I)=SF12*CO SF 12= 0. D 0 DPll(I)=SF13*CO SF 13=0. DO DP12(I)=SF14*CO SFttl=O.DO DP13(1)=SF15*CO S F 15=0. DO
DO 1000 I=l,M
1001 FORMAT(DZO.5)
1002 FORMAT(5F20.10)
1003 FORMAT(5F20.10)
1004 FOR11AT(5F20.10)
1005 FORHAT(5F20.10) 1000 CONTINUE 150 CONTINUE
STOP END FUNCTION BKO(X) IMPLICIT REALa8 (A-H,O-Z) JF(X.GT.2.) GO TO 700 T=X/2. A l = - D L O G ( T ) * B I O ( X ) - . 5 7 7 2 1 5 6 6 + . 4 2 2 7 8 4 2 " T + * 2 + . 2 3 0 6 9 7 5 6 * T * * 4 A2=.0348859*T**6+.00262698*T**8+.0001075*T**10+.0000074*T**12 B K O = A 1+A2 GO TO 701
T=2./X A3~1.25331414-.07832358*T+.02189568*T**2-.01062446*T**3 A4=.00587872*T**4-.002515~*T**5+.00053208*T**6
2 0 0 CONTINUE
WRITE(6s1001) TDCI)
WRITE(6,1002) DX(I),DPW(I),DPl(I),DP2(1),DP3(1)
WRITE(6,1003) DP4(I),DPS(I),DP6(I),DP7(I),DP8(1)
WRITE(6,1004) D P 9 ~ I ~ , D P l O ~ i ~ , D P 1 1 ~ I ~ , D P I 2 O , D P 1 3 ( I ~ , D P l 3 ~ 1 ~
WRITE(6,1005) D P I 4 ( I ) ~ D P 1 5 ( I ) , D P 1 6 ( I ) , D P 1 7 ~ I ) , D P l 8 ~ 1 ~
700 IF(X.GT.1.74D2) G O TO 7 0 2
-188-
BKO=(A3+A4)/(DSQRT(X)*DEXP(X)) GO TO 701
7 0 2 BKOz1.D-70 701 RETURN
END FUNCTION B K l ( X ) IMPLICIT R E A L * 8 (A-H,O-Z) IF(X.GT.2.) G O TO 7 1 0 T = X / 2 . Al=X*DLOG(T)*BIl(X)+l.+.l5443144*T**2-.67278579*T**4 A2~-.18156897*T**6-.01919402*T**8-.00110404*T**10-.00004686*T**12 BKl=(Al+AZ)/X GO TO 711
T=2./X A3~1.25331414+.23498619*T-.0365562*T**2+.01504268*T*~3 A4=-.00780353*T**4+.00325614*T**5-.00068245*T**6
G O TO 711
710 IF(X.GT.l.74D2) GO TO 7 1 2
BKl=(A3+A4)/(DSQRT(X)*DEXP(X))
7 1 2 BK1=1.D-70 711 RETURN
END FUNCTION B I O ( X )
IF(X.GT.3.75) G O TO 7 2 0 T=X/3.75 Al=1.+3.5156229+T**2+3.0899424*T**4+1.2067492*T**6 A2=.2659732*T**8+.036076S*T**10+.0045813*T**12 BIO=A 1+A2 GO TO 721
T=3.75/X
IMPLICIT REAL*8 (A-H,O-2)
7 2 0 IF(X.GT.1.74D2) G O TO 7 2 2
A3~.39894228+.01328592*T+.00225319*T**2-.00157565*T**3+.00916281*T*~~ A4=-.02057706*T**5+.02635537*T**6-.01647633*T**7+.00392377*T*~8 BIO=(A3+A4)/(DSQRT(X)*DEXP(-X)) GO TO 721
7 2 2 BIO=l.D70 721 RETURN
END FUNCTION B I 1(X) IMPLICIT REAL*8 (A-H,O-Z) IF(X.GT.3.75) G O TO 730 T=X/3.75 A1=.5+.87890594*T**2+.514988G9*T**4~.1508493*T**6+.02658733*T**8 A2=.00301532*T**10+.00032411*T**12 B I I=(Al+A2)*X G O TO 731
T=3.75/X
GO TO 731 7 3 2 B11=1.D70
A3~.39894228-.03988024*T-.00362018*T**2+.00163801*T**3-.01031555*T**4 A4=.02282967*T**5-.02895312+T*+6+.01787654*T**7-.0042~059*T**8 BIl=(A3+A4)/(DSQRT(X)xDEXP(-XI)
731 RETURN 1
END
7 3 0 IF(X.GT.1.74D2) GO TO 7 3 2
COLLECT 372.1
$DATA $STOP //
-189-
E-4 COMPUTER PROGRAM TO SIMULATE CLOSED CHAMBER TEST, BUILDUP TEST
SLUG TEST, AND
C C C C C C C C C C C C C C C C C C C C
C C C
C C C
c
CALCULATION OF PRESSURE OF CLOSED CHAMBER TEST NOMENCLATURE P( 1 ) LIQUID LEVEL P(2) WELLBORE PRESSURE P(1) PRESSURE AT NODE I R RADIAL DISTANCE RE RADIUS OF OIJETR BOUNDARY DRL INCREMENT OF RADIAL DISTANCE I N LOG SCALE SAC11 ELEMENT OF TRIDIAGONAL MATRIX SB(1) ELEMENT OF TRIDIAGONAL MATRIX SC( 1) ELEMENT OF TRIDIAGONAL MATRIX DK( 1) ELEMENT OF RIGHT SIDE OF EQUATION A(1) COEFFICIE-NT OF THOMAS BCI) COEFFICIENT OF THOMAS CD DIMENSIONLESS WELLBORE STORAGE SKIN SKIN FACTOR M NUMBER OF NODE N NUMBER OF TIME INCREMENT CL DIMENSIONLESS LENGTH OF CLOSED CHAMBER DRO INCREMENT OF RADIUS AT FIRST NODE IMPLICIT REAL*8 (A-H,O-Z) DIMENSION P ~ 5 1 0 ~ , S A ~ 5 1 0 ~ , S B ~ 5 1 0 ~ , S C ~ 5 1 0 ~ , A ~ 5 1 0 ~ , B ~ 5 1 0 ~ ~ D K ~ 5 1 0 ~
INITIAL SET
DO 100 I=1,510 P( I ) = O . SA(I)=O. SB( I ) = G . SC( I) = 0 . A(I ) = O . B(I)=O. DK( I 1 = 0 .
1 0 0 CONTINUE CD= 1. D3 SKIN=O. DO N=100
TT=O, DO DXO=-I. ALF= 1. D6 BETAz1.D-2 DL= 1. DATM-1.D-2 RE= 1. D5 DTO= 1 . D2 DT=l.D2
n=202
M l=M- 1 M2=M-2 M3=M-3 n 4 = 11- 4 DRL=DLOG(RE)/M3 DD=RE**(l./DFLOAT(M3))
CALCULATION OF P
DO 1000 I=l,N IF (I.GT.11 GO TO 2 0 0 TT=TT+DTO
-190-
C INITIAL VALUES C
IF(ALF.EQ.0.) G O TO 208 DK(l)=(l.-DT**2/(2.*ALF))*DXO-(DT**2)*(BETA-DATM)/ALF GO TO 209
DK(I)=-BETA+DATR
IF(SKIN.EQ.0.) G O TO 206 DK( 2)=SKIN*CD*DXO/DT GO TO 207 .
208 CONTINUE
209 CONTINUE
206 DK(Z)=DXO 207 CONTINUE
DO 201 J=3,M DK(J)=O.
201 CONTINUE GO T O 300
C C CALCULATION OF DK C
200 CONTINUE IF(TT.EQ.10000.) DT=1000. TT=TT+DT IF(ALF.EQ.0.) G O TO 210 DK(1)=(2.-DT**2/(2.*ALF))*P(l)-PPl-(DT**2)*P(2)/(2.*ALF)
GO TO 211
DK(l)=-BETA*(DL-DXO)/(DL-P(l))+DATM
IF(SKIN.EQ.0.) G O TO 203 DK(2)=SKIN*CD*P(l)/DT DK(3)=0.
AJ=DFLOAT( J )
,$ - ( D T * * 2 ) * ( ( D L - D X O ) * B E T A / o ) - D A T M ) / A L F
210 CONTINUE
211 CONTINUE
DO 202 J=2,M2
DK(J+2)=-DRLx((DD**(AJ-l.+O.5))**2-(DD**(AJ-~.-O.5))**2) 8 *P(J+2)/(2.*DT)
202 CONTINUE
203 CONTINUE GO TO 204
DK( 2)=P( 1 )
AJ=DFLOAT(J) DK(J+l)=-DRL*((DD**(AJ-l.+O.5))**2-(DD**(AJ-l.-O.5))**2)
DO 205 J=2,M2
t *P(J+1)/(2.*DT) 205 CONTINUE 204 CONTINUE
C C CALCULATION OF SA,SB,SC C
300 CONTINUE IF(ALF.EQ.0.) G O TO 306 SB(l)=l.+DT**2/(2.*ALF) SC(l)=DT**2/(2.*ALF) GO TO 307
306 CONTINUE SB(1)=1 SC( 1 1 = 1 .
307 CONTINUE IF(SKIN.EQ.0.) G O TO 302
-191-
303 C C C
400
404
405
40 1
SA(2)=SKIN*CD/DT SB( 2)=- 1. SC( 2) = 1. SA(3)=1. DR 1 =DRL SB(3)=-(SKIN/DRl+l.) SC(B)=SKIN/DRl
AJ=DFLOAT(J) SA(J+Z)=l.
DO 301 J=2,M3
SB(J+2)=-(2.+DRL*((DD**(AJ-l.+O.5))**2 $ -(DD**(AJ-1.-0.5))**2)/(2.*DTl) SC(J+2)=1.
301 CONTINUE SA(M) = 2. AM=DFLOAT(M2)
?
SB(M)=-(Z.+DRL*((DD**(Atl-1.+0.5))**2 $ -(DD**(AH-1.-0.5))**2)/(2.*DT)) GO TO 303
302 CONTINUE DRl=DRL SA( 2)= 1. SB(2)=-DT/(CDsDR1) SC(2)=DT/(CD*DRl) DO 304 J=2,M3 AJ=DFLOAT( J 1 SA(J+l)=l. SB(J+l)=-(2.+DRL*(CDD**(AJ-l.+O.5))**2
$ -(DD**(AJ-l.-0.5))**2)/(2.*DT)) SC(J+lI=l.
, SA(M-l)=2. 304 CONTINUE
AM=DFLOAT(M2) SB(M-l)=-(2.+DRL*((DD**(AM-l.+O.5))**2
$ -(DD**(AM-l.-0.5))**2)/(2.*DT)) CONTINUE
CALCULATION OF A.B
A(P)=-SC(l)/SB(l) B(Z)=DK(I)/SB(l) IF(SKIN.EQ.0.) G O TO 401 DO 400 J=2,M1 A(J+l)=-SC(J)/(SA(J)*A(J)+SBJ)) B(J+l)=(DK(J)-SA(J)*B(J))/(SA(J)*A(J)+SB(J)) JJ=J+l IF(DABS(B(JJ)).LT.lO.D-70 .AND. J.GT.4) G O - T O 404 CO t j T I N U E GO TO 402 CONTINUE DO 405 K=JJ,Ml A(K+l)=-SC(K)/(SA(K)*A(K)+SBK) B(K+l )=O. CONTINUE G O TO 402 DO 403 J=2,M2 A(J+l)=-SC(J)/(SA(J)*A(J)+SBJ)) B(J+l)=(DK(J)-SA(J)*B(J))/(SA(J)*A(J)+SB(J)) JJ=J+l IF(DABS(B(JJ)).LT.lO.D-70 .AND. J.GT.3) G O T O 406
-192-
4 0 3 CONTINUE
406 CONTINUE GO TO 402
DO 407 K=JJ,M2 A(K+l)=-SC(K)/(SA(K)*A(K)+SB(K)) B(K+I)=O.
4 0 7 CONTINUE 4 0 2 CONTINUE
C C CALCULATION OF P C
IF(I.EQ.1) G O TO 501 PPl=P( 1 ) GO TO 502 f
501 PPl=DXO 5 0 2 CONTINUE
IF(SKIN.EQ.0.) G O T O 503
C C C
J 1=11-J P(J1)=A(Jl+l)*P(Jl+l)+B(Jl+l)
G O T O 50q 500 CONTINUE
503 CONTINUE
J l=M-J P(J1)=A(Jl+I)*P(Jl+l~+B(Jl+l)
505 CONTINUE 5 0 4 CONTINUE
OUTPUT
P(2)=P(Z)/(l.-BETA+DATM) WRITE(6t600) TT,P(l),P(2),P(M-l)
600 FORMAT(D20.5,3F20.10) P(2)=(1.-BETA+DATM)*P(2)
C 1000 CONTINUE
STOP E N D
$DATA $STOP //