Gas reservoir engineering ( U. Abdulhamid )
March 13, 2017 | Author: Ali Abukhzam | Category: N/A
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A. A. Urayet, Ph.D. Associate Professor /Al-Fateh University
· NATU,RAL-G.As
--,. .
RESERVOIR ENGINEERING
Prepared By: Dr. Urayet, Abdulhamid
• Part of the Technical Training Program organized by: The Petroleum Research Center, PRC Tripoli, 2004
AAU/C/CONSULATANT/TRAINING/SIAlA/GASOUTLINES.doc
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This course offers a comprehensive review of all aspects related to Natural-Gas Reservoir Engineering from the Producing Formation to the surface:
..•
i
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- The course has been :divided into 8 main chaplers as follows:
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o Definitions and Classifications oGas Laws & Gas Deviation Factor o Natural-Gas Physical Properties o Gas Volumetrics and Gas Material Balance Calculations o Inflow Performance & Deliverability Testing o Transient Pressure Analysis o Vertical Flow Performance o Gas Field Development This course does not consider advanced aspects related to Condensate-Bearing Gas reservoirs such as vapor-liquid equilibrium, phase behavior, etc. The analytical solutions are presented without derivations except when necessary to understand the main assumptions involved and/or the sources of uncertainty in the results. Practical application to actual field data are discussed and practical analysis procedures are illustrated through Example Problems
AAU/C/CONSULATANTITRAINING/SIALA/GASOUTLINES.doc
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Natural-Gas Reservoir Enqineerinq
CONTENTS 1. 1. 1 1.2 1.3 1.4 1. 5
Chapter 1 : DEFINITIONS AND CLASSIFICATION Definitions and Terminology Dry Gas Reservoirs Wet Gas Reservoirs Condensate-bearing Gas Reservoirs Impurities :i "!
2. 2.1 2.2 2.2.1 2.2.2
2.2.3
2.2.4
3. 3. 1 3.2
......
~ :
-
.
Chapter 2 : GAS DEVIATION FACTOR Definitions and Terminology Gas Deviation Factor Deviation factors for single-component system Deviation factors for pure natural gases • Standing I Katz correlation • Brill and Beggs z-factor correlation Gas deviation factor I Effect of impurities • The additive z-factor method • The modified Pc and Tc method (Wichert & Aziz) Gas deviation factor I Practical _aspects • Treatment of heavier hydrocarbon components • Unavailability of the gas chemical composition • Effect of water vapor Chapter 3: NATURAL-GAS PHYSICAL PROPERTIES Gas Formation Volume Factor Isothermal Compressibility • Trube correlation • McCain, W.D.Jr. numerical correlation Viscosity of Natural Gases • Carr, et. al. Graphical correlation • Lee, Gonzalez, and Eakin numerical correlation Water Content of Natural Gases Hydrate Formation Conditions
3.3
3.4 3.5
4. 4. 1 4.2 4. 2. 1
4.2.2
4.3
;~
Chapter 4 : GAS VOLUMETRICS Introduction The Volumetric Method Calculation of the gas-initially-in-place • The lsopach method • The HPV method • The Grid method Calculation of the recovery factor • Recovery factor for volumetric reservoirs • Recovery factor for water drive reservoirs Material Balance Calculations
AAU/C/CONSULATANT/TRAINING/SIALA/GASOUTLINES.doc
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Cont'd 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5
,.
5.
•
9.
Mathematical formulation Analysis of free-gas reservoirs Volumetric reservoirs Water-drive gas reservoirs Treatment of condensates in the gas MBE • · Gas equivalent of hydrocarbon condensates • Gas equivalent of condensing water vapor ~
5.1 5.1.1 5.1.2
5.2 5.2.1 5.2.2 5.3
Chapter 4.: GAS VOLUMETRICS
Chapter 5 : INFLOW PERFORMANCE AND DELIVERABILITY f l ..;. ' TESTING ~ Flow-after-Flow Test Flow-after-flow procedure Flow-after-flow test analysis The Empirical method The Modified method The Exact (Pseudo-gas Potential) method Isochronal Testing Isochronal test procedure Isochronal test analysis Accounting for the Condensates in the Analysis
6. 6.1 6.2
Chapter 6 : TRANSIENT PRESSURE ANALYSIS Introduction
6.3 6.4
The Pw Analysis Method The Pseudo-Gas Potential Method
7. 7.1
Chapter 7 : VERTICAL FLOW PERFORMANCE General Flow Equation in Pipes
7.2
The Trial and Error T - z Method Static bottom-hole pressure calculations Flowing bottom-hole pressure calculations Sukkar & Cornell Method Practical Aspects Gas flow in the annulus Temperature distribution in the wellbore Effect of liquid production
The
7.2.1
7.2.2 7.3 7.4
8. 8. 1 8. 2 8. 3 8.4 8.5 8. 6
P;
1 '
Analysis Method
Chapter 8 : IMPORTANT RESERVOIR ASPECTS OF GAS FIELD DEVELOPMENT Introduction Recovery in Gas Reservoirs Reservoir Deliverability Number of Wells and Well Spacing Deliverability of the Gas Production System Predicting Future Reservoir Performance
AAU/C/CONSULATANT/TRAINING/SlALA/GASOUTLINES.doc
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INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Chapter 1 DEFINITIONS AND CLASSIFICATION NATURAL GAS is defined as a vaporous miXture of hydrocarbons, usually containing varying amounts of impurities .
19
To be more specific, a NATURAL GAS consists principally of the more volatile members of the Paraffin series (Cn H20+2). The "more volatile" members signify hydrocarbons containing one to four carbon atoms per molecule, such as Methane (CH 4 ), Ethane (C2 H 6 ), Propane (C3Hg), and Butane (C4H10>· In addition to the hydrocarbon components, different types of impurities, such as Carbon Dioxide (C0 2 ), Hydrogen Sulfide (H 2 S), and Nitrogen (Nz), are commonly present in natural gases. The chemical composition of the natural gas is pre-determined by the geologic environment of the source rock, · deposition, maturation, migration and trapping. Natural free gas accumulations occure in different environments such as a single gas phase (i.e. no liquid hydrocarbons) with interstitial water in a volumetric reservoir, or as a single gas phase underlain by a thin column of condensing hydrocarbons, or as a gas cap overlying an oil reservoir, or finally as a single gas phase underlain by a water aquifer. Natural gas occurs in different environments including conventional gas reservoirs, gas in tight sands, gas in tight shales, gas occluded in coal, and gas in geopressured reservoirs. However, in this course we only consider Conventional Gas Reservoirs. Conventional Natural gas reservoirs are, commonly, classified into three main categories, as follows: 1. Dry natural gas reservoirs 2. Wet natural gas reservoirs 3. Condensate-bearing gas reservoirs In general, Drv gas reservoirs are characterized by the negligible amount of hydrocarbon liquids condensing at the separator conditions, Wet gas reservoirs are characterized by more liquids condensing from the gas at separator conditions, and possibly in the production string, whereas, Condensate-bearing
AAU/GAS/GASl.doc
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I Natural Gas Reservoir Engineering
Dr. A. A. Urayet
gas reservoirs are characterized by liquids condensing, not only in the separator and production string, but also in the formation, (most noticeably, in the area surrounding the well bore), as the pressure in the reservoir declines below the dew point pressure due to production. In practical field applications, the above classification has been based, in general, on the amount of the separator Liquid Hydrocarbons which are associated with the production of gas from the natural gas reservoir. ·
1.1
DEFINITIONS AND TERMINOLOGY
It is important to introduce, here, some basic definitions which are related to gas engineering in general, and which will be used throughout these notes. More definitions related to each specific topic will be introduced in following chapters.
Liquid Recovery , LR LR = volume of hydrocarbon liquids (in STB) accumulating under separator conditions per one million scf (MM scf) of gas produced to the surface . LR is normally used to classify the gas reservoir into dry or wet. It is important to note that this definition should be used with caution when applied to the case of condensate-bearing gas reservoirs, since LR, in this case, represents the total amount of hydrocarbon liquids accumulating in the separator, which includes the liquids condensing from the gas under separator conditions, in addition to, liquids which have, already, condensed in the formation, and are produced once the critical saturation of liquid has been exceeded in the formation. Even though some engineers use the term Gas Oil Ratio as the inverse of the Liquid Recovery, however, it is advisable to restrict the use of the GOR term to oilreservoir calculations and terminology, and the term LR to the gas reservoir terminoloqv. ·
AAU/GAS/GAS 1.doc
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INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Mole Fraction . x; The chemical analysis of a gas mixture is normally expressed in terms of a mole fraction, x;, of each component, i, where,
number of moles of the component (i)
Xi=~~~~~~~~~~~~~~~~~
Eq (1.1)
total number of moles of the gas Typical examples of the chemical analysis of different natural gases will be given later in Tables (1.1, 1.2, 1.4, and 1.5). It is important to note that in these examples, the sum of the mole fractions of all components heavier than the normal Heptane, are grouped together under the term "Heptanes plus, C;''. Some laboratories report the gas analysis to c;0 , or to C~. especially in case of gas containing appreciable amounts of heavier hydrocarbon components.
Apparent Molecular Weight. M Eq (1.2) The Molecular weights of the different hydrocarbon and non-hydrocarbon ·components, M;, normally present in natural gases and crude oils are given in Appendix-A, Table-1.1. The number of pound-moles of a gas mixture, n, can be calculated as follows,
e I
n = mass of the gas mixture (lb) Apparent molecular weight
m M
m L,xi.Mi
Eq (1.3)
Gas Specific Gravity, y g
Yg =
Mgas
M
Mair
28.966
AAU/GAS/GAS I .doc
Eq (1.4)
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Natural Gas Reservoir Engineering
Dr. A. A. Urayet ~
t '
Example -1. 1 The chemical composition of the free gas produced from the Braebum field in Canada is given in columns (1 and 2) of the attached Table. Calculate the apparent molecular weight , M, and .the gas specific gravity, r g.
Solution • • •
Use Table 1.1/Appendix-A to read the molecular weights,of the different gas components, (column 3) Multiply the mole fraction of each component by its molecular weight as shown in column 4.: Using Eq (1.2), calculate the molecular weight of the gas as follows: M = LXi.Mi = 16.7794
•
Using Eq ( 1.4) calculate the specific gravity of the gas as follows :
rg
=
Mgas Mair
=
M
28.966
= o.5793 ·..
Braeburn Gas field, Canada Components
molecular weight, M (3)
(4)=(2)X(3)/100
Methane Ethane Propane (lso. +N) Butane (lso. +N) Pentane
96.39 0.75 0.11 0.03 0.10
16.043 30.070 44.097 58.124 72.151
15.4638 0.2253 0.0485 0.0174 0.0722
Carbon Dioxide Hydrogen Sulfide Nitrogen
0.97 1.04 0.61
44.010 34.076 28.013
0.4269 . 0.3544 0.1709
(1)
I
-
X;.M;
composition x;( mole%) (2)
100.00 .
AAU/GAS/GAS I .doc
16.7794
4
- - .~"'
/
INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Recombined Reservoir-Gas Specific Gravity The heavier components of the gas produced from a free gas reservoir would condense into liquid at the surface conditions. In order to obtain the composition of the reservoir gases and define the gas properties under · reservoir conditions, it is necessary to monitor the flow rates, the compositions, and the related properties of the gas and condensate streams. Once the necessary separator data is obtained, accurate determinations of the gas composition in the reservoir can be obtained experimentally or analytically. For quick estimation of the reservoir gas gravity, the following equation can be used:
Ygs
+
4584 Yes
Yg =
Res
Eq (1.5)
1+132,800 Yes MoRes where,
rg rgs rcs Res
M0
= specific gravity of the reservoir gas = specific gravity of the separator gas = specific gravity of the stock tank condensate =gas to condensate ratio, ( scf separator gas I one STB condensate) = Molecular weight of the tank oil
If the value of M 0 is not known, it can be estimated using the following equation:
IM
0
0
6084 AP! -5.9
AAU/GAS/GAS I .doc
Eq (1.6)
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Natural Gas Reservoir Engineering
Dr. A. A. Urayet
Example -1. 2 Calculate the specific gravity of gas in the reservoir, given the following data : separator pressure =680 psia specific gravity of stock tank condensate 0.72 specific gravity of the separator gas =0.66 Liquid Ratio 48 STB condensate I MM scf
= =
Solution
1) calculate
AP!=
141.5
-131.5
0.72
Yes 2) R cs
= 1,000,000 = 1,000,000 = 20 833 LR
3) Calculate
rg =
=0
M0
Ygs
3J
= 141.5 -131.5 =65.03
+
48 6084
AP! - 5.9
4584 Yes R
cs 1+132 800 Yes
'
AAU/GAS/GAS I .doc
MORCS
-
' 6084
65.03-'- 5.9
=102.892
0.66 + 4584(0.72) ____ 20_..::,_83_3__ - 0.7835 1 + 132,800(0.72) (102.892)(20,833)
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I Natural Gas Reservoir Engineering 1.2
Dr. A. A. Urayet
DRY GAS RESERVOIRS
Although small amounts of heavier hydrocarbon components can be present, most DRY natural gases consist predominantly of Methane (CH 4 }, the mole percentage. of which may .be as .high as 98% ·of the hydrocarbon fractions. Furthermore, Methane (CH 4 ) and Ethane (C2 H6) would constitute as high as 99% of the total hydrocarbon mole composition. Considering this Universally accepted classification, there are NO Dry Gas Reservoirs in Libya. From a practical engineering point , the DRY gas is usually defined as a gas
which is associated with less than TEN barrels of separator liquids per Million scf of gas produced (i.e. LR< 10 STB I MM sen. The very small amount of hydrocarbon liquids will usually have an API gravity of 70+, corresponding to liquid specific gravity less than 0.7.
TABLE 1.1 Typical Composition of natural DRY gas reservoirs COMPONENTS
COMPOSITION ( mole % ) Examole-A *
Examole - B **
Methane Ethane Propane (lso. +N) Butane (lso. +N) Pentane Hexane Heptanes plus
96.55 2.15 0.45 0.18 0.07 0.25 0.04
96.39 0.75 0.11 0.03 0.10
Carbon Dioxide Hydrogen Sulfide Nitrogen
0.31
-----
0.97 1.04 0.61
100.00
100.00
96.85 99.01
98.98 99.75
C1 % (C1 + C2) %
---
---
* Example -A: West Cameron Pool, Louisiana Gulf Coast, USA ** Example - B : Belloy Formation , Braeburn Field , CANADA
AAU/GAS/GAS I .doc
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INatural Gas Reservoir Engineering
Dr. A. A. Urayet r~ \
1.3
WET GAS RESERVOIRS
Some natural gas reservoirs yield condensed liquids at ratios of 10 - 20 barrels of separator liquids per Million scf of gas produced. Such gases are termed WET. They contain greater quantities of the less volatile {heavy) hydrocarbons (C3+) than do DRY gases. According to this Liquid Recovery criteria, only a small number of Libyan gas reservoirs can be correctly termed WET. Two examples of naturally occurring WET gas reservoirs in LIBYA are presented in TABLE 1.2. It is noted that the Liquid Recovery in both reservoirs is 10- 20 STB I MM scf.
TABLE 1.2 Typical Composition of natural WET gas reservoirs COMPONENTS
Methane Ethane Propane {lso. +N) Butane (lso. +N) Pentane Hexane Heptanes plus Carbon Dioxide Hydrogen Sulfide
COMPOSITION ( mole % ) Examole-C * 89.132 3.433 1.064 0.676 0.349 0.320 0.990 3.303 0.733
Examole - D ** 85.83 5.16 2.32 1.38 0.57 0.29 0.35 3.85 0.25
100.00
100.00
Other fluid 12rogerties Liquid Recovery (STB/MM scf) Seoarator conditions Gas specific qravitv Liquid qravitv (API)
11
16
449 psia /91° F 0.6764 51.3
NA 0.688 NA
* Example - C : NC 41-G , offshore Trip.oli , Libya ** Example - D : Sahl Field I Shegega reservoir , Libya
AAU/GAS/GAS I .doc
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INatural Gas Reservoir Engineering 1.4
Dr. A. A. Urayet
CONDENSATE-BEARING GAS RESERVOIRS
Condensate-Bearing gases differ in composition from the DRY and WET gases in that they contain appreciable amounts of the heavier hydrocarbons such as Butanes (C4),
P~ntanes (C5), Hexane (C6 ) ,and Heptanes plus (Cj).
Condensate-Bearing gas reservoirs will produce more than TWENTY barrels of separator liquids per Million scf of produced gas at normal field separator conditions. Rich Condensate-Bearing gases have been identified with producing ratios as high as (400) barrels of liquid condensate per Million scf of produced gas. The average API of the produced liquids ranges normally from an API gravity of the light crudes ( i.e. 45 ) to a maximum of 60 . The International terminology for the different classes of NATURAL gases is presented in Table 1.3. Note that the Condensate-Bearing gases are classified into three categories which are termed RICH, MODERATE, and LEAN. According to this classification, many of the Libyan Gas Reservoirs can be correctly termed Condensate-Bearing Gas reservoirs. The chemical compositions for three free gas reservoirs in Libya, representing the three classes of Condensate-bearing gases, are given in Table 1.4 for illustration.
TABLE-1.3 Practical Field Classification of Natural Gases CLASSIFICATION
Dry Wet Condensate - Lean Condensate - Moderate · Condensate - Rich
LIQUID RECOVERY ( STB I MM set) 100
GOR ( scf I STB) > 100,000 50,000 - 100,000 20,000 - 50,000 10,000 - 20,000 < 10,000
NOTE: In the normal field practice in Libya, no distinction is made between the "Moderate" and the "Rich" categories. Ttie term "Rich" is used to represent any free gas reservoir with LR > 50 STB I MM scf.
AAU/GAS/GAS I .doc
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Appendix-A I Table 1.1
Physical Constants of Hydrocarbons
1
Crltlcol conatonta
No.
...l••...•: ·-lk
Compound
:i-
•
1•
>
IL
1 2 3
4 5
16.0.C3 30.070 .C.C.097 58.12.C 58.12.C 72.151 72.151 72.151 16.178 16.178 86•178 16.178 86.178 100.205 100.205 100.205
Moth0tto Ethono Propotio -autono
l•oltvt-
6
I
-Pontono 7 l•opontono I Noo...,.tono 9 -Hoiruo - · 2-Mothrlpontono . 3-Mothrlpontono Noohoir13 2,3-0lmothrlbvtono 14 -Hoptono 15 2-Mothrlhoxono 16 3-Mothrlhoaono 17 3-E~rlpontono 11 2,2-01-thrlpontono 19 2,4-0lmothrlpontono 20 3,3-01-thylpontono 21 Trlptono 22 n-Octono 23 Dliaobutrl 2.C faooctofto 25 n-Nonono 2' n-Docono 27 Cyclopontono 28 Mothylcyclopontono 29 Cycloho•30 Mothylcydoftoxono 31 Ethylono· 32 Propono 33 1-Bvtono :M Clo-2-Butono 35 Trono-2-But.fto . . . . . .u t - . -1-Ponteno 38 1.2.:..allf04139 1,3-BvtoclloRO"
100.205 100.205 100.205
100.205
100.205 114.232 114.232 114.232 128.259 142.286 70.135 e.c.162 14.162 98.189 28.05.c 42.081 56.108 -56.108 56.108 56.108
.co ·-.cl · Acotrlono Q
43 .c.c 45 '6
BonaTolEthrlbonaono
-x,,1one
111-Xylono p-Xylono .cs Styrene 49 laopropylbonaono 50 Motftyl Alcohol 51 Ethyl Alcohol 52 Cor.bon Monoxide 53 · CorbOfl Oloaldo 5.C Hydrogen Sulfide 55 Sulfur Oloaldo 56 Ammonlo 57 Air S4 Hrdro9en 59 Oargen 60 Nitrogen 61 Chlorine 62 Wotor 63 Helium 6.C Ht'droaen Chloride
41
NH1
HzOi Hz Ci
Na Clz HaO Ho HCI
-258.69 -127•.CS -.C3.67 31.10 10.90. 96.92 . 12.12 .ct.10 155.72 140.c 1.C5.19 121.52 136.36 209.17 19.C.09 197..32
200.25 17.C.S.C 176.19 186.91 177.58 258.22 228.39 210.63
303•.C7
345. .CS '. 120.65 161.25 177.29 213.68 . -15.c.62 -53.90 20.75 38.69 33.58 19.59 85.93 : 70~135. 51.53 .S.C.o92 2.C.06 S.C.092 . 93.30 68.119 26.038 -1.190 176.17 78.114 231.13 . 92.1"1 ' 277.16 106.168 29L97 106.168 282 •.Cl 106.168 281.05 106.168 293.29 10.C.152 .306.:M . 120.195 148.1(2) 32.G.42 172.92(22) '6.069 28.010 -313.6(2) .C.C.010 -109.3(2) -76.6(2.C) :M.076 1.c.0(7) 6.C.059 -28.2(2.Cl 17.031 -317.6(2) 28.96.C -.C23.0(2.C) 2.016 -297 . .C(2) 31.999 -320 . .C(2) 28.013 -29.3(24) 70.906 212.0 18.015 .C.003 36 ..C61 -121(16)
(5000) (800) 190• 51.6 72.2 15.570 20.4.C 35.9 .c.956 6.767 6.098 9.856 7•.CO.C 1.620 ·2.211 2.130 2.012 3.492 3.292 2.773 3.374 0.537 1.101 1.708 0.179 0.0597 9.91.C 4.503 3.26.C 1.609 226 •.C 63.05 45.s.c 49.80 63 •.CO 19.115 (20.) (60.) 16.672 3.22.c 1.032 0.371 0.26.C 0.326 o.:M2 (0.2.C) 0.188 .C.63(22) 2.3(7)
0
c·i
1! 1~ :u:
......
IL
-296.46d -297.89d -305.8.Cd -217.95 '..;255.29 -201.51 -255.83 2.11 -139,58 -2«.63 -1.C7.72 -199.38 -131.05 -180.89 -181 •.CS -190.86.
-182.63
-210.01 -12.12 -70.18 -132.07 -161.27 -6.C.28 -21.36 -136.91 -224.« 43.77 -195.87 -272 •.C50 -301 •.c5d -301.63d -218.06 -157.96 . -220.61 -265.39 -213.16
-164.02
-230.7.C -114.• .Cl.96 -138.9.C -138.91 -13.30 -5.C.12 55.86 -23.10 -140.• 82 -1"3.82(22) -173 •.C(22) -3.C0.6(2)
.
:! ~
a'•:
0
•
;;
a
t :
667.8 707.8 616.3 560.7 529.1 .CS.8.6 490• .C '6.C.0
436.9 436.6 453.1 4"6.8 .CS3.5 '396.8 396.5 408.1 '19.3 402.2 396.9 .C27.2 .C28 •.c • 360.6 360.6 372 . .C 332. 30.C. ·653.8 S.CS.9 591. 503.5 729.8 669. $83. 610. 595. 580 • 590. (653.)
628.
(558 •.C) 890..C 710•.C 595,9 523.5 5.CU 513.6 509.2
sao.
39".0(6) 88.(7) 212.(7)
-117 .2(7) -103.9(7) -107.9(2)
154.(7) 0.9.C92(12)
-.C3.C.8(2.C) -361.8(2.C) -3.C6.0(2.Cl -U9.8(2A) 32.0
"65 •.C 117.C.2(21) 925.3(21) 501 .(17) 1071.(17) 1306.(17) 11.C5.(2.Cl 1636.(17) 5.C7.(21 188.1(17) 736.9(2.C) .C9J.0(2.C) 1118 . .C(2.Cl 3208.(17)
925.(7)
-173.6(16)
1198.(17>
f
>
-116.63 90.09 206.01 305.65 27.C.98 385.7 369.10 321.13 453.7 .C35.83 4"8.3 420.13 .C.C0.29 512.8 .C95.00 503.78 513•.CS .cn.23 475,95· 505.85 496 •.C.C 56.C.22 530.« 519.46 610.68 652.1 '61.S .C99.35 536.7 570.27 .CS.58 196.9 295.6 32.C.37 311.86 292.55 376.93 (339.)
306.
. "'12.) 95.31 552.22 605."55 . 651.2.C 615.0 651.02 6.C9.6 106.0 '76 ..C "62.97(21) .C69.58(21) -220.(17) 87.9(23) 212.7(17) 315.5(17) 270.3(2.Cl -221.3(2) -399.8(17) -181.1 (17) -232 . .C(2.Cl 291.(17) 705.6(17) 12.C.5(17)
0.0991
0.0788 0.0737 0.0702 0.072.C 0.0675 0.0679 0.067.C 0.0688 0.0681 0.0681 0.0667 0.0665 0.0691 0.0673 0.06'6 0.0665 0.0665 0.0668 0.0662
0.0636
0.0690 0.0676 0.0656 0.068" 0.0679 0;059 0.0607 0.0586 0.0600 0.0737 0.0689 0;0685 0.0668 0.0680 0.0682 0.0697 (0.0649) 0.065" (0.0650) . 0.0695 0.0531 0.05.C9 0.056• 0.0557 0.0567 0.0572 O.OS.Cl 0.0570 0.0589(21) o.o580C2U o.os32C17> o.03.C2C23) 0.0459(2.C) o.o306C2"> o.068H17> o.0517C3> o.5167(2.Cl 0.0382(2.C) 0.051.ccm o.028H17> o.o5oocm o.0208C17>
INatural Gas Reservoir Engineering
Dr. A. A. Urayet .~
{ "
TABLE 1.4 Typical Composition of CONDENSATE-bearing gas reservoirs COMPOSITION ( rnole % )
COMPONENTS
Methane Ethane Propane (lso.+N) Butane (lso. +N) Pentane Hexane Heptanes plus Carbon Dioxide Hydrogen Sulfide Nitrogen
-
Examole- E*
Examole - F**
Examole-G***
78.90 4.36 1.37 0.78 0.53 0.38 1.71 11.16
66.64 7.63 4.47 3.50 2.46 2.06 7.47 3.03
0.72
80.77 2.07 0.80 0.72 0.45 0.13 3.26 9.50 1.91 0.39
100.00
100.00
100.00
54 0.77 52 MODERATE
125 0.84
--
----
2.74
Other fluid 12ro12erties LR{STB/MM scf) Gas specific aravitv Liquid aravitv (API ) Classification
28.5 0.78 NA LEAN
* Example - E: Hateiba Field, Zmam/Waha/Gargaf, Libya ** Example - F: 137 S/K Field, offshore Tripoli, Libya ***Example - G : NC 41-A ,offshore Tripoli , Libya
AAU/GAS/GAS 1.doc
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37 RICH
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I Natural Gas Reservoir Engineering
Dr. A. A. Urayet
1.5 IMPURITIES As mentioned earlier, natural gases contain impurities in varying amounts. This constitutes serious problems not only in the design of the field equipment and . . transportation facilities, but also in gas safes contracts. in gas flaring (especially · near metropolitan areas), in evaluating the heat content of fuel gas ,(i.e. Btu problem), in artificial lift design (i.e. corrosion problems), in the design of tertiary recovery projects (Miscibility problems), and of course in the gas treatment required to reduce the impurities to minimum tolerable levels which do not violate environmental regulations. . Impurities associated. with natural gases may include one or more of the following items : 1. 2. 3. 4. 5.
Carbon Dioxide Hydrogen Sulfide Sulfur Compounds Oxygen Nitrogen
6. Water Vapor 7. Mercury 8. Dust 9. Helium 10. Free liquids
New treatment and process technology allows the reduction of impurities to tolerable limits, however, the costs of special treatment may constitute a heavy load on the economics of the project, especially in the offshore gas development projects. Almost all the Libyan natural gas reservoirs are contaminated by Carbon Dioxide, Nitrogen, and Hydrogen sulfide in varying amounts. Other types of impurities are rare occurrences, and (usually) with negligible concentrations.
•
Typical Libyan gases containing impurities are illustrated in TABLE 1.5 . As can be seen from the table, the Carbon Dioxide ( C02 ), and Hydrogen Sulfide ( H 2 S) can be considered the major (and consequently the most troublesome) impurity constituents in the Libyan gas production and treatment practices. It is very hard to set up a general guideline to "tolerable" concentrations of each of the above impurities, since this will depend on the specific use of the gas, the market place safety regulations, the Btu rating, the pressure base of the gas measurements, etc. However, in general, the industrial use of natural gas in most industrialized countries requires ttw concentration of water vapor to be less than (6) pounds per million standard cubic feet of gas, and the hydrogen sulfide content to be less than 1000 grains per million scf of gas .
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TABLE 1.5 Examples of some Libyan Gas Fields containing high percentages of impurities
COMPONENTS
COMPOSITION ( mole % ) Examole- H*
Examole - I**
Methane Ethane Propane (lso. +N) Butane (lso. +N) Pentane Hexane Heptanes plus
72.130 4.290 2.150 1.237 0.587 0.498 2.293
70.15 4.66 2.11 1.11 0.54 0.35 1.20
13.9 1.5 1.1 0.8 0.3 0.2 5.5
Carbon Dioxide Hydrogen Sulfide Nitrogen
12.204 1.702 2.909
13.58 0.12 6.18
65.8 0.6 10.3
100.00
100.00
-
100.000
•
Examole-J***
* Example - H : ** Example - I : *** Example - J :
AAU/GAS/GAS 1.doc
NC 41 - C2 , Metlaoui formation , offshore Tripoli , Libya NC 41 - C1 , Reinche formation , offshore Tripoli , Libya NC 41 - D ,offshore Tripoli , Libya
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Water vapor. Carbon dioxide. and Hydrogen sulfide are normally present in varying amounts in all natural gas reservoirs. All three impurities can cause serious problems in the production of the wells. in the operation of the surface facilities. and in the transportation and marketing of the natural gas. This special treatment normally constitutes a heavy load on the initial investment · required for the field development; as well as, on the operating costs. In brief, . the main production problems caused by these three types of impurities can be summarized as follows :
Effect of Water Vapor Every natural gas reservoir has a certain initial liquid-water saturation which is determined by the geologic environment. In addition, the gas phase will always contain a certain amount of water vapors. Even if only the single gas phase is produced, the reduction of pressure and temperature in the well bore, in the flow lines, and in the surface facilities will result in the condensation of the water vapors to form a free water phase . The presence of liquid water in communication with Carbon dioxide and/or hydrogen sulfide will result in severe corrosion of the equipment due to the formation of acids. In addition, the condensation of water in the flow lines will reduce the area available for the gas flow, which increases the pressure drop in the line, and consequently, the Horse Power requirements. In addition to that, a serious problem will appear when the temperature of the natural gas falls below a certain temperature limit, which allows the liquid water molecules and the gas molecules to combine and form a solid phase known as " Hydrates". The conditions necessary for the formation of Hydrates will be discussed in more detail in Chapter-3.
Effect of Hydrogen Sulfide , H2S The hydrogen sulfide is present in all natural gas reservoirs in concentrations ranging from a trace to as high as 25%. The presence of hydrogen sulfide represents one of the main problems to the production engineer, since the hydrogen sulfide will combine with water to produce corrosive acids . Hydrogen sulfide also constitutes a severe health and environmental problem. The flaring of natural gas containing Hydrogen sulfide will result in sulfur dioxide
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which is very toxic. Extensive gas treatment is, especially, required if the flaring center is in the proximity of inhabited areas.
Effect of Carbon Dioxide . C02 The effect of carbon dioxide is not as critical as the hydrogen sulfide, mainly, because it does not constitute an environmental problem. The main production problem associated with the presence of carbon dioxide in natural gases is the formation of corrosive acids in the presence of water. Another problem would be the lowering of the heat content of the natural gas, consequently, requiring special treatment before marketing .
Finally, from Reservoir Engineering point of view, it is important to remember that the presence of impurities (especially at high percentages) greatly affects the physical properties of the natural gas, especially the viscosity and compressibility, (as will be discussed in chapter-2). Neglect of the effect of impurities in any Reservoir or Production Engineering calculations will always result in erroneous evaluation of the fluid properties, leading, of course, to improper reservoir analysis, production and development.
AAU/GAS/GAS I .doc
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Chapter 2
GAS DEVIATION FACTOR The most important gas properties are primarily determined by the Gas Composition, Pressure, and Temperature. Prior to introducing the different techniques of evaluating these properties, it is necessary to know understand the basic gas laws governing ideal single-component gases, and to learn how to account for the deviation of the Hydrocarbon natural gas performance from these laws.
2.1
DEFINITIONS AND TERMINOLOGY
Prior to discussing the different fundamental gas laws, and the techniques used in the evaluation of the different physical properties of natural gases, it is important to introduce the following important definitions:
Critical Pressure . Pc , and . Citica·1 Temperature . Tc The critical pressure, Pc. and the critical temperature, Tc. for a single component gas are defined as the Pressure and the Temperature at the Critical Point (i.e. the point at which the intensive properties of the gas and liquid phases are identical). In simple terms, the critical temperature, Tc. can be defined as the temperature above which the gas CANNOT be liquified, regardless of the pressure exerted on the svstem. Whereas, the critical pressure, Pc. is defined as the pressure at which the gas phase will be in equilibrium with the liquid phase when the system is at the critical temperatrure. The critical values for the more common hydrocarbon components of natural gases and the impurities normally associated with them are given in AppendixA, Table1 .1.
AAU/GAS/GAS2.doc
INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Pseudo-Critical Pressure. Ppc, and,Pseudo-Critical Temperature. Tpc_ The Pseudo-Critical Pressure, Ppc. and the Pseudo-Critical Temperature, for any hydrocarbon mixture are defined and calculated as follows :
Tpc.
Eqs (2.1)
and,
where, n
=number of components x; = mole fraction of the i-th component Pei= critical pressure of the i-th component, and Tc;= critical temperature of the i-th component
Pseudo-Reduced Pressure, P,, and, Pseudo-Reduced Temperature, T, The Pseudo-Reduced Pressure, Pr, and the Pseudo-Reduced Temperature, Tr, for a pure natural gas system are defined and calculated as follows:
and,
T. =_I_ r T
Eqs (2.2)
Pc
•
where, P and T are the Pressure and the Temperature of the gas system respectively.
The Law of Corresponding States The Law of Corresponding States expresses that "All PURE gases have the same characteristics at the same values of reduced pressure and reduced temperature". By characteristics, it is meant the physical properties such as deviation factor, density, and viscosity. This Law has proven to be a very powerful tool in the analysis of the behavior (and the evaluation of the physical properties) of gases in general, and natural gas systems in particular.
AAU/GAS/GAS2.doc
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I Natural Gas Reservoir Engineering
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Different investigators have shown that this law applies very effectively for any gas mixture. as long as the gas components have similar molecular characteristics O.e. if the components are closely related· chemically). Consequently, this law is expected to apply very effectively for pure hydrocarbon gases since they are all of the paraffin group. However, deviations will occur with increasing percentage of impurities (since they have a different molecular and chemical structures). The degree of deviation will depend on the type of impurity component in the natural gas; with the highest deviations occuring in the case of Hydrogen Sulfide (H 2 S), lesser deviations in the case of Nitrogen (N 2 ), and the least deviations would be in the case of Carbon Dioxide (C02).
Extensive Properties These are the physical properties which are dependent on the quantity of gas available, such as Volume (V), and, mass (m), and,
Intensive Properties These are the physical properties that are independent of the quantity of gas, such as density, specific gravity, viscosity, and, gas compressibility. These are the properties which are basically of interest to the reservoir engineer.
Equation of State The "Equation of State. EOS " can be defined as an equation which relates the three fundamental thermodynamic properties of any gas system, namely, Volume (V), Pressure (P), and, Temperature (T). An "Ideal Gas" is defined as a gas which has the following characteristics: 1) the total volume of the gas molecules is considered negligible with respect to the volume occupied by the gas, (i.e. the volume of the container), 2) there are no attractive or repulsive forces between the gas molecules, nor, between the gas molecules and the container, and, 3) all collisions between the molecules are perfectly elastic.
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Natural Gas Reservoir Engineering
Combining the three fundamental gas laws which govern ideal gas behavior, namely, Boyles Law, Charles and Gay Lussac Law, and Avogadro Principle, the Equation of State for an Ideal Gas can be written in the following form: Eq (2.3)
IPV=nRT
I
where, n =the number of pound (lb) moles of gas present in a Volume, V, P = Pressure of the system, psia, T =Temperature of the system, degrees Rankin, and, R
= the Universal Gas Constant, R = 10.72 lb.;ole , (English system) in .0 R
Example 2.1 The volume of a vessel containing 5 lbs of methane is 2 ft behavior, calculate:
3
.
Assuming ideal gas
a - the pressure of the gas system, if the temperature is maintained at 50 °C, b - the gas volume at standard conditions.
Solution m n = number of lb-moles = -
M
a) T° F
= 32 + 2_(T
0
5
C)
=
5 16.043
= 0.31166
lb-moles
= 32 + 2_(50) = 122° F 5
PV= nRT - nRT _ 0.31166(10.72)(122 + 460) _ . - 972 .23 psta Pv 2.0 b) At standard conditions, P
PV
=nRT
or,
V
= nRT = (0.31166)(10.72)(60 + 460) =118.18 p
AAU/GAS/GAS2 .doc
= 14. 7 psia, and T =60 ° F 14.7
4
scf
INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Equation Of State for Natural Gases At the standard pressure of 14.7 psia, and standard temperature of 60 °F, the form of Eq (2.3) can be used with negligible error for any gas system, since the assumptions regarding the definition of Ideal gas are generally met. However, the ideal-gas law is completely inadequate to describe the behaviour of natural gases at the high reservoir. or even separator, pressures and temperatures, since at such conditions the molecules would be nearer to each other and consequently the volume of the molecules and the effect of the attractive forces cannot be considered Negligible. Also, the high gas temperatures would result in higher kinetic energy which would result in more frequent collisions. Numerous forms of the EOS have been developed to describe the natural gas behavior. Some of the best known forms are:
The Beattie-Bridgeman Equation this equation is best suited for single component gases, and consequently its use should be restricted to Dry natural gases composed mainly of Methane. The equation contains five constants which have been determined empirically from laboratory measurements.
The Benedict-Webb-Rubin Equation, (RF-2.1) this equation is considered, currently, as one of the best and most suitable forms of the Equation of State describing the Pure (i.e. less than 2 mole % impurity) Dry, Wet and Lean Condensate-Bearing natural gases (LR30%) of impurities. The equation contains 21 constants which have been determined from experimental data of 264 natural gas and condensate systems. In addition to its accuracy for all normal
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reservoir pressure and temperature ranges, the main advantages of this equation is that its use is a direct procedure (i.e. does not require trial and error procedure), and it can be differentiated easily to obtain the gas compressibility.
2.2 GAS DEVIATION FACTOR In the field of Petroleum Engineering, a much simpler method is used to account for the deviation of natural gas behavior from the Ideal Gas Law. This is achieved by introducing One single factor to correct for all the different sources of deviation; whether the deviation is due to the high pressure and temperature of the system, or to the presence of more than one hydrocarbon component, or due to presence of non-hydrocarbon components (i.e. impurities). This factor which is termed the "Gas Deviation Factor, z", is included into the Equation of State for an ideal gas, Eq (2.3), as follows:
I
Eq (2.4)
PV =znRT
The term "Gas Deviation Factor" will be used through out these notes, even though many technical people in the industry use the term "Gas Compressibility Factor" to signify the same factor, z. Our choice is made mainly to avoid confusion in terms, since the term "Compressibility factor, C", is used in all branches of science and engineering to signify the relative change in volume with respect to the change in pressure and/or temperature.
2.2.1
Deviation Factors for Single Component System
The Deviation Factors for different single hydrocarbon component gases were determined experimentally, and, then plotted and correlated as a function of pressure, P, and Temperature, T. These correlations are found in all Reservoir Engineering books. The characteristic shape of the z-factor correlation is illustrated in Figure (2.1) for Methane. This figure clearly indicate the following important features: •
The z -factor will decrease in value with increasing Pressure until a certain minimum is reached, then the z-factor will start increasing afterwards with any increase in pressure, (i.e. dzldp becomes positive).
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Dr. A. A. Urayet
Natural Gas Reservoir Engineering
1.20
Methane 1.10
1.00
le
0.9
.. ....0u
.!':
0.8
0.7
c 0
·;;
"' ·;;
0.6
-0
....
"'
(.'.)
0.5
0.4
0.3 -40 -22
0.2
-""'-I--\- -
4 -+------1----~---+---f
68 32
140 104
0.1 0
500
1000
1500
6000
5000
7000
8000
Pressure psi a
Pressure pounds per square inch absolute
Fig. 2_.I
AAU/GAS/GAS2.doc
Gas deviation factor for methane
7
9000
10000
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Natural Gas Reservoir Engineering
Dr. A. A. Urayet
•
The minimum value for the z-factor is dependent on the Temperature isotherm value (lower minimum value of z, for lower Temperature, as seen from Fig 2.1 ). Note that the minimum value of z will be lower for heavier gases (such as Ethane).
•
It is noted that beyond a certain Pressure range (usually between 50007000 psia), the deviation factor, z, of the lower-Temperature isotherm will start having higher values than the deviation factor, z, of the highertemperature isotherms.
2.2.2
Deviation Factors for Pure Natural Gases
Even though laboratory measurements are the most accurate method for determining the z-factor for any gas mixture, however, the extensive effort and time, and the sophisticated measuring devices required, have encouraged the trend toward developing semi-empirical techniques in order to evaluate the zfactor of any gas mixture.
Standing I Katz Correlation The determination of the Deviation Factor for a PURE (i.e. negligible percentage of impurities) multi-component natural gas can be achieved, to a high degree of accuracy, by employing the concept of Pseudo-Reduced Pressure and Pseudo-Reduced Temperature. The Deviation Factors were determined experimentally for a large number of natural gases and were correlated as a function of Pr, and Tr, in line with the principle of Corresponding States. The results were presented in the "Standing I Katz z-factor charts" shown in Appendix-A I Figure 2.1. These extended charts cover the normal range of Field Pressures and Temperatures. Linear extrapolation of these curves beyond (Pr= 24) can be applied with minimum error.
Consequently, in order to calculate the z-factor for any pure natural gas system at a certain pressure and temperature, the following procedure can be applied : 1- Read from Appendix-A I Table 1.1 the critical values of pressure and temperature for each component,
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2- Use the chemical analysis of the gas mixture to calculate Ppc, and Tpc for the gas, using Eqs (2.1 ), 3- Calculate Pr, and Tr, of the gas system, using Eqs (2.2), 4- Use the Standing I Katz charts, Appendix-A I Fig (2.1) to read the value of the z-factor corresponding to the calculated Pr and Tr values; The proper procedure for the use of Standing I Katz charts to calculate the Gas Deviation Factor is illustrated in Example Problem 2.2.
It is important to note that the accuracy of the above procedure, (which is developed for pure gases) can still be maintained, even in the case of gases contaminated by SMALL concentrations of C02 and N2 (less than 2% maximum for each), and by smaller concentrations of H2S (less than 1%), provided that these components are included in the calculation of the PseudoCritical Pressure and Temperature. Different investigators have shown that further corrections must be introduced if higher percentages of impurities are present in the gas mixture (Reference 2.3).
Example 2.2 The chemical composition of the gas produced from Braeburn field in Canada is given in the attached Table (columns 1 and 2). Calculate the gas deviation factor, z, for the following conditions:
a - P = 3000 psia , and T = 230 °F b- P
= 1000 psia , and T = 150 °F
Assume that x (iso-C4) = 0, and x(iso-C5 ) = 0
Solution: 1. Read the values of Pc and Tc for each component from Appendix-A I Tablet. 1, and enter into columns 3 and 4.
2. Calculate (x; . Pei) and (x; . Tci) for each component as shown in columns 5 and 6.
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3. Calculate: n
PPc
n
= L:Xi·Pc;
= 677.32
TpC = LXi.Tc.I i=l
,and,
i=l
=350.62
4. Calculate Pr and Tr as follows : Case-a P,
r
Tr
= _!...._ = PPc
Case-b
3000 677.32
= 4.429
Pr
PPc
= _!_ = 230 + 460 = 1.97 TPc
= _!__ =
1000 677.32
= 1..476
Tr=_!_= 150+460 =1.74 TPc 350.62
350.62
Using Standing and Katz charls, Appendix-A I Fig 2. 1, read:
z =0.92
z = 0.94 Calculation of pseudo-critical constants
( 1)
(6)
(2)
(3)
(4)
composition x;( mole%)
Pei ( psia)
(o R)
Methane Ethane Propane (lso.+N) Butane (lso.+N) Pentane
96.39 0.75 0.11 0.03 0.10
667.8 707.8 616.3 550.7 488.6
343.37 550.09 666.01 765.65 845.70
643.69 5.31 0.68 0.17 0.49
330.97 4.13 0.73 0.23 0.85
Carbon Dioxide Hydrogen Sulfide Nitrogen
0.97 1.04 0.61
1071.0 1306.0 493.0
547.90 672.70 227.60
10.39 13.58 3.01
5.32 7.00 1.39
677.32
350.62
components
I
AAU/GAS/GAS2.doc
100.00
10
Tei
(5) X;.
Pei
X;.
Tei
INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Brill and Beggs z-factor Correlation
I Different investigators have tried to represent the Standing/Katz z-factor charts mathematically for computer use. However, most of the available mathematical expressions would require the solution of a system of Non-Linear equations numerically such as the "Hall-Yarborough correlation" and the "Dranchuk, Purvis and Robinson" correlation. One of the simplest and most practical expressions available in literature for the evaluation of the z-factor is the Brill and Beggs Correlation which has the following form:
1-A
D
z=A+--+c.Pr
Eq (2.5)
eB where,
A= 1.39(T,. - 0.92)o.s - 0.36T,. -0.101
6 _ 0.066 . .2 0.32Pr B-(0.62-0.23Tr)Pr +(T _ -0.037)Pr + . ( 9 T. _ 9) r 0.86 10 r · C = (0.132- 0.32 log T,.)
\\
D = Anti log( 0.3106 - 0.49 T,. + 0.1824 T,.2 ) The Brill and Beggs correlation is within the accuracy requirements for all practical Reservoir Engineering problems. They are most accurate in the range 1.2 < Tr < 2.4 , and Pr < 13. The proper procedure for the use of the Brill and Beggs correlation to calculate the Gas Deviation Factor is illustrated in Example Problem 2.3.
Example 2.3 Use the data given in Example 2.2 (part-a) to calculate the z-factor using the Brill and Beggs z-factor correlation
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Natural Gas Reservoir Engineering
Solution The pseudo-reduced values, Pr and Tr (calculated in Example 2.2) are:
Pr
= _!__ = PPc
Tr
= 4.429
3000 677.32
= _.!__ = 230 + 460 = 1. 97 350.62
TPc
Consequently, 05
A= l.39(Tr - 0.92) · - 0.36Tr -0.101 A= 1.39(1.97 -0.92) 0 ·5 - 0.36(1.97)- 0.101=0.614126 6
66 B = (0.62 - 0.23Tr )Pr + ( 0.0 - 0.037)P} + OC~;~ Tr -0.86 10 r
9)
3 66 2 - 0.03 7)(4.429) + 0. ~~;:;~? B = (0.62 - 0.23xl .97)(4.429) + ( 0.0 1.97 - 0.86 10 . -
B =1.1707703
C = (0.132- 0.32 IogTr) C = 0.132 - 0.32 log(l .97) = 0.03777081 2
D = Antilog(0.3106-0.49Tr + 0.1824Tr )
D = Anti log(0.3106 - 0.49(1.97) + 0.1824(1.97) 2 ) = 1.1302543 Consequently ,
z =A+
1-A B
D
+c.Pr
e
z
= 0.614126 + l :1°;~9\~~; + 0.03777081( 4.429)1.1 6
z = 0.9358
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Dr. A. A. Urayet
Gas Deviation Factor I Effect of Impurities
The presence of impurities (such as C02, N2, H2 S, etc.) in Natural Gases affects the values of the gas Deviation Factor. The magnitude of change in the value of z depends on the type and mole percent of the impurity. Two approaches have been suggested in the literature for including the effect of the impurities in the calculation of the Deviation Factor. These two approaches can be termed the "Additive z-factor method", and the "modified Pc and Tc method". These two approaches can be summarized as follows:
The Additive z-factor Method In this method the Deviation factor is considered to be equal to the arithmetic sum of the individual z-factors of the different components, each weighted according to its mole percentage in the gas mixture. Thus, the Deviation Factor of a natural gas can be represented by the following equation: Eq (2.6) where,
= sum of the mole fractions of the hydrocarbon components
Xhydr
xco2, Xtt2s ,
and xN2
=mole fractions of the different impurities, and,
zco2, ZH2s,
and ZN2
=deviation factors of the different impurities.
It is important here to note that the Deviation Factor for the hydrocarbon components, Zhydr, should be calculated as if only hydrocarbons are present in the system. In other words the Tpc and Ppc used in Eq 2.5 (or entered to the Standing I Katz charts) to evaluate zhydr should be calculated as follows:
Ppc =I Pei ·Yi where,
Yi=
and
xi(hydr)
Eqs (2.7)
LXi(hydr)
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·-----------------------.-------""' j Natural Gas Rese~oir Engineering
-,,,,,,,,,
__ ----
,,,,_,_,
,,,.
Dr. A. A. Urayet ,•
It has been suggested to use certain eccentric factors to be multiplied by the zfactors of the different impurities to obtain more accurate estimate of the Natural Gas Deviation Factor. However, such accuracy is rarely required for Reservoir Engineering calculations. Different investigators have demonstrated that Eq (2.6) will have an error less than 0.1 % for a Nitrogen mole percent of 10%. Similar demonstrations have been made for C02 and H2 S.
The Modified Pc and Tc method
(~
In this method the calculated value of Ppc and Tpc are modified, and then the new values of Pr and Tr calculated on this basis are entered to the Standing I Katz charts or Brill and Beggs correlation, Eq 2.5, to evaluate the Deviation Factor. It is important to note that the modified values have NO physical significance, and should only be understood as a mathematical correction to the Katz charts. One of the simplest (and yet very accurate ) methods available in literature for the modification of Ppc and Tpc was presented by Wichert and Aziz, (RF-2.6). In this method, the modified pseudo-critical properties can be calculated as follows:
' =T -& Tpc pc------. ---
p
,
' PP .TP
=
Pc
c
TPc
c
Eqs (2.8)
+ B(l - B)&
where, Tpc and Ppc are the critical temperature and pressure for the gas mixture including the impurities, and,
A = Sum of the mole fractions of Hydrogen Sulfide and Carbon Dioxide
B = mole fraction of the Hydrogen Sulfide in the gas mixture= X (H2S)
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Natural Gas Reservoir Engineering
Note : Quick estimate of (8) can be obtained from the graphical form of the correlation given in Appendix-A I Fig (2.2) . The Wichert and Aziz correlation should prove very useful in the analysis of Libyan gas reservoirs. The correlation has an average absolute error of 0.97% (for P < 7026 psia, and T < 300 F) for the range of data used in the development of the correlation which included C02 concentrations as high as 54.4 %, and H2 S concentrations as high as 73.8 %. There are many other correlations available in the literature, however, most of them require a trial and error procedure, and consequently, make them fit for computer use mainly . The proper procedure for the use of the Wichert and Aziz correlation to calculate the Gas·oeviation Factor is illustrated in Example Problem 2.4.
Example 2.4 Use the data given in Example 2.2 (parl-a) to calculate the z-factor using the modified Pc and Tc method, (Wichert and Aziz correlation).
Solution: the modified Pc - Tc method A= X (H2S) + X ( C02) i
e
B
=(1.04+0.97)% =0.0201
=X (H2S) =0.0104
s = 120(A 0 ·9 -Al. 6)+15(Bo.s -B 4 ) =120(0.0201°· 9 -0.02011. 6 )+15(0.0104° 5 I
TPc
-
0.0104
4
) =
4.8633
= TPc - e = 350.62-4.86 = 345.76 I
p
I
Pc
pp .Tp
=
c
TPc
c
+ B(l - B)s
(677.32)(345.76) = 667.84 350.62+0.0104(1- 0.0104)( 4.8633)
and, consequently ,
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Natural Gas Reservoir Engineering
Tr =_I__= 230+460=1. 996 TPc 345.76 Pr = _!_ = 3000 = 4.492 PPc 667.84 Substitution of the modified Pr and Tr in the Brill and Beggs correlation, (as illustrated in Example 2.3) will give a value of ( z 0.9432 ), as compared to the value of (z 0.9358) obtained by using the Brill and Beggs correlation directly (without any modifications for impurities) in Example 2.3. The very small difference in the results clearly proves that Brill and Beggs correlation can be used with very small error for any natural gas with Low concentrations of C02 L or Hz..S.
=
=
2.2.4
Gas Deviation Factor I Practical Aspects
Treatment of the heavier hydrocarbon components
/-
The composition of a Natural Gas is reported, normally, in a form similar to that shown in Tables 1.1-1.5 (Chapter-1 ). It is important to note that the mole percentage of all the heavier components (i.e. C7 +, or C9+ or sometimes C13+) is normally reported as one single value. In such cases, the molecular weight and the specific gravity of this group of components should be reported. These two properties have been correlated in literature with (Pc) and (Tc) as shown in Appendix-A I Fig 2.3. And, consequently, the heavier components can now be included in the normal procedure of calculating (Pr) and (Tr) which would be introduced to the Standing/Katz charts to calculate the z-factor. Another method preferred by many reservoir engineers is to use the physical properties of Octane ( C8H 18 ) for the C7+ fraction, properties of ( C10H22) for the C9+ fraction, etc. This should always give good results for Dry and Wet gases.
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Dr. A. A. Urayet
Natural Gas Reservoir Engineering
Unavailability of the gas chemical composition In many instances the accurate chemical composition of the natural gas might not be available. This normally occurs when field calculations are required at the early life of the field (to make preliminary estimates of the gas in place and recoverable reserves). In this case the specific gas gravity can be used to estimate the values of (Pc) and (Tc) for the gas system from the available charts shown in Appendix-A I Fig (2.4). Sutton R.P. presented the following relationships for the calculation of the pseudocritical properties using the specific gravity of the natural gas:
Ppc =756.8-131.0(yg)-3.6(yg) 2 + {y.J..lzs. +'+.'LG· Xc 6 z.- lt- X1\J2 and,
This simple correlation should be corrected using the correction graphs given in Appendix-A I Fig 2.4 to account for impurities.
The proper procedure for the application of these practical techniques to calculate the Gas Deviation Factor is illustrated in Example Problem 2.5.
*Effect of water vapor Finally, the gas laboratory analysis is usually run after the gas sample has been dried. Consequently, the water vapor content is not reported. However, different investigators have concluded that the Deviation Factors calculated by the previous techniques are little affected by the water vapor content, and consequently no corrections are required for normal Reservoir and Well Productivity calculations.
AAU/GAS/GAS2.doc
17
Dr. A. A. Urayet
Natural Gas Reservoir Engineering
Problem 2. 5. The chemical composition of a Libyan gas is given in columns 1 and 2 of the accompanying table. The following data is also available: reservoir pressure reservoir temperature specific gravity of the reservoir gas molecular weight of C 7 + specific gravity of C1 +
= 2852 psia 227°F =0.688 100.1 = 0.80
=
=
Calculate the gas deviation factor. Use the pseudo-critical constants evaluated from: a) the individual Pci and Tc; , b) the Sutton correlation
Solution: a) using individual Pei and Tei 1) enter with the molecular weight and specific gravity of C7 + to Fig 2.3 (Appendix-A), and read Pc; (C1 +) = 1040 psia, and Te; (C1 +) = 505 °R 2) Read from Table 1. 1 (Appendix A) the values of P ci and Tc; for the individual components, and enter into columns 3and 4 3) calculate x; .Pc; and x;. Te; as shown in columns 5 and 6 n
4) calculate
PPc = L,xi.Pq
=680.7 psia
i=I
5) enter with the values of C0 2 and H 2 S mole percentages to the Wichert and Aziz graph, Fig 2.2 (Appendix-A), and read 5 - 6 6) use Wichert and Aziz correlation, Eqs (2.8), to correct Ppc and Tpc as follows:
AAU/GAS/GAS2.doc
18
Dr. A. A. Urayet
\ Natural Gas Reservoir Engineering
I
TPc = TPc -
8
= 382.3 - 6 = 376.3 I
p' = PPc .TPc _ (680.7)(376.3) = 67 0.0l Pc TPc +B(l-B)& 382.3+0.001(1-0.001)(6) 7) calculate :
Tr=_!__= 227+460=1.826 376.3
TPc and,
e
P. = _.!!__ = 2852 = 4.26 r PPc 670
8) finally, enter with the values of Pr and Tr to the Standing I Katz correlation, Fig 2. 1 (Appendix-A), and read z =0.905
b) using Sutton correlation 1) substitute the value of y g
= 0.688 into the
Sutton correlation, Eqs.(2.9), as
follows:
Ppc = 756.8-131.0(yg )-3.6(yg) 2
= 756.8-131.0(0.688)-3.6(0.688) 2 = 665 and,
Tpc = 169.2 + 349.5(yg)-74.0(yg) 2
= 169.2 + 349.5(0.688)- 74(0.688) 2 = 374.6 2) use the appropriate correction charts in Fig 2.4 (Appendix-A) to read: pseudo-temperature correction for C0 2 = - 3 ° F pseudo-temperature correction for N 2 0 pseudo-temperature correction for H 2 S = 0
=
AAU/GAS/GAS2.doc
19
I
Dr. A. A. Urayet
Natural Gas Reservoir Engineering
= + 14 psi
pseudo-pressure correction for C02 pseudo-pressure correction for N2 pseudo-pressure correction for H2S
=O =O
3) consequently, Tpc = 374.6-3 = 371.6 °R and,
4) claculate :
Ppc = 665+14=679 psia
Tr =_I_= 227 + 460 =I.SS TPc
and,
371.6
P. = _!___ = 2852 = 4.2 r PPc 679
5) finally, enter with the values of Pr and Tr to the Standing I Katz correlation, Fig 2.1 (Appendix-A), and read z =0.908 Calculation of pseudo-critical constants (Example 2.5)
( 1)
components
Methane Ethane Propane n- Butane n- Pentane Hexane Heptanes + Carbon Dioxide Hydrogen Sulfide Nitrogen
I
AAU/GAS/GAS2.doc
(2)
(3)
(4)
composition x;( mole%)
Pei ( psia)
Tei (o R)
85.83 5.16 2.32 1.38 0.57 0.29 0.35 3.85 0.10 0.15
667.8 707.8 616.3 550.7 488.6 436.9 505.0 1071.0 1306.0 493.0
343.37 550.09 666.01 765.65 845.70 913.70 1040.00 547.90 672.70 227.60
100.00
20
( 5) Xi.
Pei
(6) Xi.
Tei
573.17 36.52 14.30 7.60 2.79 1.27 1.77 41.23 1.31 0.74
294.71 28.38 15.45 10.57 4.82 2.65 3.64 21.09 0.67 0.34
680.7
382.32
Dr. A. A. Urayet
Natural Gas Reservoir Engineering
REFERENCES RF-2.1
"Predicting Phase and Thermodynamic Properties of Natural Gases with the Benedict-Webb-Rubin Equation of State", Gas Technology, SPE Reprint series , No 13 , vol.1
RF-2.2
"The Prediction of Volumes, Compressibilities, and Thermal Expansion Coefficients of Hydrocarbon Mixtures", Avasthi and Kennedy SPEJ, June 1968 (95)
RF-2.3
"Volumetric Behavior of Natural Gases Containing Hydrogen Sulfide and Carbon Dioxide'', Macry Georgos and Govier Gas Technology, SPE Reprint series , No 13 , vol. 1
RF-2.4
"A New Equation of State for z-factor Correlations", Yarborough and Hall Gas Technology, SPE Reprint series, No 13, vol. 1
RF-2.5
" How to Solve Equation of State for z-factor'', Yarborough and Hall Gas Technology, SPE Reprint series , No 13 , vol. 1
RF-2.6
"Calculate Z's for Sour Gases'', Wichert E. and Aziz K. Hydr.Proc., May 1972, (119-122)
RF-2.7
"Compressibility Factor for High-Molecular-Weight Reservoir Gases", Sutton , R.P. , paper SPE 14265 ( 1985 SPE Annual) , Las Vegas , Sept. 22-25
RF-2.8
"Compressibility of Natural Gases'', Albert Trube Petroleum Transactions, AIME, vol. 20, 1954,(264)
RF-2.9
"Viscosity of Hydrocarbon Gases under Pressure" by Carr , Kobayashi , and Burrows Petroleum Transactions, AIME, Vol. 201 , 1954, (264)
AAU/GAS/GAS2.doc
21
Dr. A. A. Urayet
Natural Gas Reservoir Engineering
RF-2.10
"The Viscosity of Non-Polar Gases at Normal Pressure" by Stiel L., and Thodos G. AIChE Jour. 1961 , 7 ,(611)
RF-2.11
"Calculation of the Viscosity of Technical Gas Mixtures from the Viscosity of the Individual Gases" by Heming F.,and Zipperer L. Gas and Waserfoch , 1936 , 79 , 49
RF-2.12
"The Viscosity of Natural Gases" by Lee A., Gonzalez M., and Eakin B. Journal of Petroleum Technology, Aug. 1966, (997)
RF-2.13
"The properties of Petroleum Fluids" 2nd edition , Penn Well Books , Tulsa (1989) 120,175,214,318,513,525-28
RF-2.14
"An hnproved Method for the Determination of the Reservoir-Gas Specific Gravity for Retrograde Gases" Gold,D.K., McCain, W.D.Jr., and Jennings ,J.W. JPT (July 1989) 747-52, Trans. AIME, 278
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Pt; 4-=t -4
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
Chapter 3
NATURAL-GAS PHYSICAL PROPERTIES (RELATED TO RESERVOIR ENGINEERING) As has been mentioned in previous chapters, the most important gas characteristics are determined primarily by the Gas Composition, Pressure, and, Temperature. New, easy to use, and improved correlations for the evaluation of different natural-gas properties are being introduced continuously in the literature. The accuracy of any reservoir or production engineering calcu~ations will be primarily affected by the accuracy of calculating these properties:. Even though there are numerous gas properties which are of importance in the industrial use of natural gases, such as thermal conductivity, enthalpy, entropy, gasoline content, etc.; however, in this section, only the physical properties which are related to Reservoir Engineering will be considered; namely, the Gas Formation Volume Factor (Bg ), the Gas Compressibility (Cg), the Gas Viscosity (ug), the Water Content of natural gases, and the Hydrate-formation conditions.
3.1
GAS FORMATION VOLUME FACTOR
The Formation Volume Factor, Bg , is not an independent gas property, it is merely an expression of the relationship between the volume of a certain amount of gas (say n-lb moles) under reservoir conditions, Vr, and the volume of the same amount under standard conditions, Volume Factor is expressed as follows:
B g
. Vr - 0.0282 zT (ft3) V:S P scf
B = Vs = 35.35_!_(scf) g Vr zT ft3
AAU/GAS/GAS3.doc
, or,
Vs.
Consequently, the Gas Formation
Eq (3.1)
Eq (3.2)
I
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
Example 3.1 The following data is available for a newly discovered natural gas reservoir Gas Initially In Place , GllP = reservoir pressure = = reservoir temperature gas specific gravity
=
17. 53 Billion ft 3320 psia 195 ° F 0.75
3
a - calculate the gas formation volume factor b - calculate the GllP in scf
Solutlon - since the gas composition is not available, the gas specific gravity can be used to calculate the pseudo-critical properties. Enter Fig 2.4 (Appendix-A) with the value of
r g =O. 75, and read,Ppc =665 psia,
and Tpc =410 ° R
'
- calculate
P. =~ = 3320 = 4.99 r ppc 665
T, =_I_= 195+460=1.60
and,
Tpc
I
410
- enter with the calculated values of Pr and Tr to the Standing/Katz charts, Fig 2.1 (Appendix-A), and read z 0.845
=
a) substitute into Eq (3.2) as follows:
B = 35.35~ scf g zT ft3 B = 35.35. g
(332 0) = 212.046 (0.845)(195 + 460)
b) calculate Gl/P (set)
=GllP (ft3 ) x Bg (.scf -) 3 ft
9
= 17.53 x 10 x 212.046 = 3717.2 B set
AAU/GAS/GAS3.doc
2
scf
jt 3
.·
I Natural-Gas Reservoir Engineering 3.2
Dr. A. A. Urayet
ISOTHERMAL COMPRESSIBILITY
The "coefficient of compressibility " which represents the ability of the substance to be compressed is a complex function of the chemical composition and . physical properties of the substance. This coefficient is Highly pressure-sensitive in the case of gases, Less pressure:-sensitive in the case of liquids with gas in solution such as oil, Even Less pressure-sensitive in the case of nonhydrocarbon liquids (such as water), and is, practically, Insensitive to pressure variations in the case of solids. The " Isothermal Compressibility " can be defined as the "relative" change in the volume of a substance due to the change in pressure imposed on the system, when the Temperature is constant. Using this definition, the Isothermal Compressibility, "c", for any substance can be written as follows:
lc=-_!_.dV ·
Eq (3.3)
V dP
I
This equation is combined to the Real-Gas law (Eq 2.4) to derive an expression for the Coefficient of Isothermal Compressibility for natural gases (normally abbreviated as "Gas Compressibility') as follows:
c = _ _!__ dV = _ P _!!:_(znRT) g V dP znRT dP P
c =g
P .nRT.-(-)=-d z P( z(-)+-.-1 1 dz)
znRT
dP P
z
p2
P dP
and,
Eq (3.4) If the Z vs. P plot has already been developed by the engineer for a certain gas reservoir, then Eq 3.4 can be used effectively to calculate the gas compressibility.
AAU/GAS/GAS3.doc
3
I
I Natural-Gas Reservoir Engineering
Dr. A.
A.
Urayet
(
Inspection of Eq 3.4 clearly indicates that : •
the units of the coefficient of isothermal compressibility is 1/psia.
•
for an Ideal gas the value of (z) is constant and is equal to 1; consequently, the coefficient of iso-thermal compressibility is inversely proportional to the pressure, (Le. Cg =1 IP).
•
for natural gases, the gas compressibility could be higher or lower than the "ideal" gas compressibility depending on the pressure of the system; that is, at low pressures, the term dz/dP is negative and consequently, Cg > 1/P, whereas at high pressures, dzldP is positive, and consequently, Cg < 1/P.
Direct substitution of Pr following expression:
=P/Ppc (i.e. P
=Ppc.Pr) into Eq (3.4) would result in the
IC g = 1 - .1( - -1-dz -) Ppc Pr z dPr where
Ppc
Eq (3.5)
I
is the pseudo-critical pressure of the gas.
Calculating the Gas Compressibility There are different numerical approaches to the evaluation of the isothermal gas compressibility. One of the simplest and more accurate involves the concept of Pseudo-Reduced Compressibility, c,, which was defined by Trube (RF-3.1) as follows: or,
Eq (3.6)
Combining Eqs 3.5 and 3.6, the following expression is obtained,
Eq (3.7)
AAU/GAS/GAS3.doc
4
I
I
,,... -
I Natural-Gas Reservoir Engineering 3.3
Dr. A. A. Urayet
VISCOSITY OF NATURAL GASES
The viscosity of a fluid, in general, represents the resistance of that fluid to flow. It is a character of the fluid and is independent of the media in which the fluid is flowing. More specifically, the Dynamic Viscosity of a fluid represents the ratio between the shear stress per unit area divided by the velocity gradient perpendicular to the plane of shear. · In petroleum engineering, the viscosity term appears in all the dynamic equations representing the fluid flow; in the porous media of the reservoir system, into the wellbore, up to the well head, and through the flow lines and pipelines to the gathering systems. In petroleum engineering, and especially in Reservoir Engineering, the Dynamic Viscosity is expressed in units of "centipoise". The centipoise is defined as (1 gm 2 4 mass I (sec)(100 cm ), and is equivalent to (6.72x10 lb mass I (ft)(sec)). The measurement of the dynamic viscosity of a natural gas in the laboratory requires difficult procedures, and consequently graphical and numerical correlations have been developed for the calculation of the viscosity of a certain gas at any given pressure and temperature. There are two main approaches in the literature for the evaluation of natural gas viscosity under reservoir conditions: • •
Graphical techniques that require the knowledge of the Atmospheric Viscosity in addition to the Pseudo-reduced Pressure and Temperature of the gas. Numerical techniques that require the evaluation of the Gas Density under reservoir conditions.
3.3.1
Carr. Kobayashi and Burrows Graphical Correlation
A widely used graphical correlation for evaluating the viscosity of the gas was developed by Carr et.al. The procedure involves two main steps as follows: •
•
Knowing the molecular weight or the specific gravity of the gas, use Carr et. Al correlation (Fig 3.3 Appendix-A) to read the atmospheric viscosity, µa at the required temperature. Calculate Pr and Tr and enter to Carr et. Al. correlation (Figs 3.4 or 3.5 Appendix-A) to read µg I µa, and hence calculate the gas viscosity µg .
AAU/GAS/GAS3.doc
7
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
Experimental investigation has concluded that natural gas viscosities calculated using this correlation will have less than 2% error for all P, values less than 10. The Carr et. al. correlation includes Corrections for Carbon Dioxide, Hydrogen Sulfide, and Nitrogen. It is not advisable to extrapolate the correction graphs beyond the 15% impurity mole percent limit indicated on the graph, ·
3.3.2
Lee , Gonzalez. and Eakin Numerical Correlation
Different investigators have tried to represent the behavior of natural gas viscosity in numerical· forms. Most of the expressions available in the literature represent the viscosity as a function of temperature, mixture density, and molecular weight. The effect of the impurities is incorporated implicitly in the evaluation of the gas Deviation factor which would be used to calculate the gas density. Practically one of the most accurate formulas for the calculation of the gas viscosity is that developed by Lee, Gonzalez, and Eakin (RF-3.6), which has the following form:
µg
= (10-4 )K.exp(Xpy)
Eqs (3.9)
where,
p
= 1.4935(10)-3 PM zT
K
= (9.379 + 0.01607M)Tt. 5
(209.2+19.26M + T) 986 X = 3.448 + .4 + O.OI009M T Y _;__ 2.447 -0.2224X and,
p = gas density, gm/cc P = Pressure, psia
T =Temperature, 0 R M = Molecular weight of the gas mixture
AAU/GAS/GAS3.doc
8
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
The standard deviation factor of the calculated viscosity using the above correlation (as compared to the measured lab data) is only 2.69 % in spite of the wide range of data used in the derivation of the correlation. The data range included 100 < P (psia) < 8000, 100 < T ('F) < 340, and C02 content of 0.9 - 3.2 mole%. 4
.
"O'
It is important to note that this correlation should lbe used ·u r when applied to some Libyan reservoirs which have very high C02 content, (reaching to more than 60% in some cases).
a
Example 3.3 Given the gas composition shown in Table 3.3 (columns 1 and 2), calculate the gas viscosity at T 200 °F and P 3520 psia , using
=
=
a) the Carr et.al graphical correlations b) the Lee, Gonzalez, and Eakin correlation
Solution: a) using the Carr et.al. correlations
-
1-Tabulate the molecular weights of the individual gas components, using Appendix A, Table 1.1, as shown in column (3), 2- Calculate the molecular weight of the natural gas, in column (4),
M = l.,xi.Mi, as shown
3- Enter the Carr et.al. correlation (Fig 3.3/Appendix-A) with the calculated M=20.0527, and read, µa =0.0128 at T 200 °F,
=
4- Calculate y g
=
20.0527 = 0.6923 28.966
~
0.7
5- Use the specific gravity of the gas as well as the individual mole fraction of each of the impurities to read the correction values as follows:
AAU/GAS/GAS3.doc
9
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
correction for C02
= +0.0003
for
Xco2 = 5.42%
correction for H2S
= +0.0000
for
=+0.0002
for
xH s= 0.83% 2 XN2 =2.03%
'
.
correction for N2 6- Calculate µg
= µa+ corrections
=0.0128 + 0.0003 + 0 + 0.0002
=0.0133 cp
(2) composition mole fraction, xi
(3)
(4)
M·l
X·M· l l
Methane Ethane Propane iso- Butane n- Butane
0.8216 0.0528 0.0314 0.0021 0.0093
16.043 30.070 44.097 58.124 58.124
13.1809 1.5877 1.3846 0.1221 0.5406
Carbon Dioxide Hydrogen Sulfide Nitrogen
0.0542 0.0083 0.0203
44.010 34.076 28.013
2.3853 0.2828 0.5687
I
1.0000
( 1) components
20.0527
7) Calculate Ppe and Tpe for the gas mixture, using the individual component Pei and Tei as explained in earlier examples. The results should be Ppc = 670 psia, and Tpe = 385 °R. 8) Calculate
Pr = -
p
Ppc and,
=
3520
670
= 5 .254
2 460 Tr = __!_ = 00 + = L714 Tpc
AAU/GAS/GAS3.doc
385
10
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
9) Enter the Carr et.al. correlation, Fig (3.5) with the values of
Py
and
Tr
and
µ
read ~=1..7
µa
= (l.7)(µa) = (1.7)(0.0133) = 0.02261 cp
10) Calculate µg
b) Lee, Gonzalez. and Eakin correlation 1) Enter with the values of Py and
Tr
to the Standing/Katz correlation, Fig (2.1),
and read z = 0.89 2)Ca/cu/ate:
p = 1.4935(10)-3 PM zT
= 1.4935(10)-3 (3520)(20.527) = 0.18371 p . (0.89)(200 + 460) K = (9.379 + 0.01607M)Tl. (209.2+19.26M + T)
5
5
K = (9.379 + 0.01607(20.527))(660)1. = 130 _18 (209.2+19.26(20.527) + 660) X = 3.448 +
x
= 3.448 +
986 T
.4 + 0.01009M
986.4 + 0.01009(20.527) = 5.1497 200+460
Y = 2.447 - 0.2224X
y = 2.447 - 0.2224(5.1497) = 1.3017
AAU/GAS/GAS3.doc
II
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
3) Finally, calculate the natural gas viscosity at the reservoir conditions,
µg
= (10- 4 )K.exp(Xpy)
µg
= (10- 4 )(132.123).exp(5.199(0.18371)1. 360158 ) = 0.02296 cp
Note that the results of both techniques are practically identical (1.5% difference). The vety small difference can be attributed to the inherent error in reading the graphs.
AAU/GAS/GAS3.doc
12
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
3.4 WATER CONTENT OF NATURAL GASES All natural gases contain varying amounts of water vapor. The amount of water required to saturate a natural gas (normally termed Saturation Content) is a function of pressure and temperature. At any given pressure, the saturation content. "mwv" decreases as the temperature is decreased, leading to the condensation of liquid water which will result in the formation of Hydrates in the gas stream, and will result in severe corrosion to the field equipment and surface facilities in the presence of free water in communication with C02 and/or H2S in the gas stream. " /'
9
._
"\
~
v\-i t
'"-· \ './.
"Sweet natural gases" are defined as natural gases containing NO (or negligible) amounts of carbon dioxide and hydrogen sulfide. A plot of the water vapor content of "Sweet natural gases" as a function of pressure and temperature is given in Appendix-A I Fig 3.6. It is important to note that the value of the water vapor content read from the plot should be corrected for the water salinity and the gas gravity. An illustration of the proper use of this plot is given in the following example.
Example 3.4 The following information is available from a natural gas well : daily production rate gas gravity stream pressure stream temperature
= = = =
7.5 MMscf I day 0.70 1500 psia 110°F
Calculate the total amount of free water condensing in the stream, when the temperature is lowered to 80 °F. Assume sweet gas. Also, assume that there was no free water at the original temperature of 110 °F.
Solution 1) Enter the vapor content chart (Fig 3. 6) at T read mwv
=59 lb water I MM scf of gas
2) make the necessary corrections as follows :
AAU/GAS/GAS3 .doc
13
=110 °F, and P =1500 psia , and
I Natural-Gas Reservoir Engineering
Dr. A. A. Urayet
- use the gas gravity correction chart to read CG given gas gravity of 0. 70 . - the corrected mwv 3) similarly, at T
=0. 99, for T =110 ° F,
=mwv x CG =59 x 0.99 =58.4
and the
· lb I MM scf
=80 °F, the corrected mwv will be equatto 25
lb/MM scf
4) consequently, the amount of water condensing from the natural gas will be: mwv (at T= 110°F) -mwv(at T = 80°F) = 58.4- 25 = 33.4 lbl MM scf 5) finally , the free water accumulating daily in the gas stream will be equal to : mw
=(33.4 lb!MM scf)( 7.5 MM scf!day) =250.5 lb of free water
3.5 HYDRATE-FORMATION CONDITIONS The presence of liquid water in the same stream with natural gas results in the formation of Hydrates which are solids resembling ice in appearance. Hydrates form under pressure at temperatures as high as 85 °F (29.4 °C). The presence of liquid water is necessary for the formation of Hydrates. This is one reason why the gas stream must be de-hydrated to certain minimum limits. The Hydrates (which should not be confused with free liquid water condensing from the gas) are materials that have fixed chemical composition. but exist c.~ '6 without chemical bonds (clathrates). Their structure consists of several water . :z. molecules associated with each hydrocarbon molecule. Consequently, the 1 ~; ' presence of liquid water in sufficient quantities will always result in the formation "" 11 ,J (1 0 of Hydrates. H1
0
e
· -?- · · ·
l{. ~(j' A.IC'
.,~ 0..
40
10
20
30
~
Per~ent
60
50
70
80
H2 S
Fig.J..'1 Pseudocritical temperature adjustment factor,
E3 ,
0
R. (After Wichert and Aziz.)
384 10.0 I . _ ~ )...__
r-!--1 -t-~-r+-W----
~"I S:>,. I I
1''- 'I
~
I
~ '
1.0
Properties of Natural • ! I • I le so I I I c: I I I• I I I ii 0
Cl
1500 1000 SaNrallon Preasura, psla
0
0.6
~
2000
Fig. 9.10-Effect of COz solubility (in terms of saturation pres:· sure) on water viscosity.
0.4
9.4.6 Water/Brine Viscosity. Fig. 9.9 presents the viscosities of pure water and NaCl brines as functions of temperature and salinity. . The following equations (except for the pressure correction Ao) are presented by Kestin et al.,29 who report an accuracy of ± 0.5% in the range 70 < T < 300°F, 0 < p < 5,000 psia, and 0 < C,,.. < 300,000 ppm (0< c,,.
'/ I / I I II I
I
I
I
I
1
lh
I
//
I I ~I~/ '/I /, JI ~lo~,, I II 1/ I I I~ ~~~1 //; /~~/// ~ J llh ~//I;~~ /ffi""' ~ ~~~v ,....... I -
J
J /
3.0
1
/
'/
/
0
2.5
2.0
1.5
/,_,,, ....
O~,
1.0 0.1
0.2
0.3
~~t:::::::
0.5
1.0
2.0
3.0
5.0
1
,,,
10
20
Pseudoreduced pressure, P,
Fig.
3:.,S Viscosity ratio versus pseudoreduced pressure. (After Carr et al.)
30
~;1 ~;1 Natural Gas Reservoir Engineering
HW# Due by:
..m1 e'4 Dr. A. A. Urayet
IGas Properties
I~ov3, 2002
Problem-1 The Q reservoir is a wet gas reservoir with an average LR=12 STB/MMscf. The following· information is available: Component Methane Ethane Propane N - Butane lso- Butane N- Pentane lso - Pentane Hexane Heptanes plus Carbon Dioxide Hydrogen Sulfide Nitrogen
X; 85.00 2.50 0.92 0.60
g:~~
0.08 0.15 0.80 4.70 2.91 1.82
Calculate the gas physical properties (i.e. Ppc, Tpc, MW,
y g• z, Bg, Cg, and µg) at the initial ~.
reservoir conditions ofPi=2200 psia and Ti=l75°F
100.00 Assume M(C7+)=110 and rg(c1 +> =0.800
Problem-2 The U reservoir is a wet gas reservoir with an average LR=14 STB/MMscf. The following information is available:
Yg
= 0.695
X(H2S) X(Ni) X(COi) Pi Ti
=2.20% = 1.20 % = 5.75 % = 2550 psia = 198 °F
Calculate the gas physical properties z, Cg, and µg .
IMPORTANT NOTES: ~ Use Numerical equations rather than graphical correlations whenever possible. ~ In evaluating any gas physical property use all different techniques studied in class or which you might have read in literature.
AAU/C/UNIV_F0203/NATGAS/PROBLEMS&FIGS/HW1 .doc
.\/o.
Natural Gas Reservoir Engineering
Dr. A. A. Urayet
REFERENCES
RF-1
"LinearAquiferBehavior" Nabor G. W. and Barham R.H. .JPT, May 1964 ,(561-563)
RF-2
" Calculation of Water Influx for Bottom Water Drive Reservoirs" by : D. R. Allard and S. M. Chen SPE Reservoir Engineering (May 1988),369-379
RF-3
"A Mathematical Model for Water Movement about Bottom Water Drive Reservoirs" by : K. H. Coats SPE Jour. (March 1968) 44-52; Trans. AIME,225
RF45
"The Material Balance as an Equation of a Straight Line Part -11, Field Cases " D. Halvena and A.S. Odeh JPT, July 1964, (815-822)
AAU/GASNOLillvfETRICS.doc
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I Natural Gas Resel'Voir Engineering
Dr. A. A. Urayet
·chapter 4 GAS VOLUMETRICS
4.1
INTRODUCTION
Natural gas accumulations can be classified into three main categories : •
• •
Non-Associated (Free) Gas existing with the interstitial water in the pore space. The free gas could be in dynamic communication with an underlying or surrounding water aquifer (i.e. ''water drive reservoir"), otherwise, it is termed "Volumetric". Associated Gas existing as a free "gas cap" overlying crude oil. Dissolved Gas existing "in solution" with the crude oil in the reservoir.
In this section only the Non-associated (i.e. free) gas volumetrics will be considered, since the other two categories are normally treated in the oil reservoir volumetrics. ·
Gas Volumetrics The term "Gas Volumetrics" is used, here, to signify different techniques related to the calculation of the volume of gas initially in place, GllP or Gi, and to the estimation of the produce-able reserves. Two main techniques are commonly employed in Gas Volumetrics, which are:
1 - The Volumetric Method •
•
This method is used to calculate the GllP, and is applicable as soon as the first discovery is made. Consequently, it is .the primary tool in the techno/economic evaluation of the gas property and the field development project. The basic data required in this method consists of the petrophysical properties obtained from well logs and core analysis, the geologic maps, the initial reservoir conditions in terms of pressure and temperature, and, finally, the gas PVT data.
AAU/GASNOLUMETRICS.doc
I Natural Gas Reservoir Engineering •
Dr. A. A. Urayet
The credibility of the GllP estimated by the Volumetric method will be improving as more wells are drilled and more accurate reservoir description and reservoir geologic and petrophysical maps become available.
2 - The Material Balance Method •
•
•
This method is used to validate the GllP estimates obtained by the Volumetric Method, as well as, to determine the ultimate reserves, and, the reservoir pressure performance as a function of production. The basic data required includes the historical production and reservoir average pressure data, in addition to the gas PVT data. Consequently, this method can be used once the production has started from the reservoir. However, The results obtained by Material Balance can be misleading at the very early stage of production, (i.e. produced gas < 3% of GllP). The credibility of the results will continue improving with continuing production from the reservoir. Normally, once more than 10% of the GllP have been produced, the results obtained by this method will have a high degree of credibility.
Prior to presenting the different techniques employed in Gas Volumetrics, it is important to introduce the following basic terms and definitions:
Gas Initially In Place, GllP or G It is the total volume of gas (in scf) existing initially in the reservoir.
Cumulative Gas Produced, GR It is the total volume of gas (measured in scf) produced from the start of production to the time when the calculations are performed.
Proven Gas Reserves The Proven Gas Reserves are the estimated gas volumes (in scf) which geologic data and engineering calculations demonstrate with high degree of certainty to be recoverable from the reservoir, under the existing economic conditions and the available engineering technology.
AAU/GASNOLUMETRICS.doc
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Natural Gas Reservoir Engineering
Dr. A. A. Urayet
Unit Recovery It is the technically proven reserves (in scf) which can be produced from one unit volume of the pay (normally one acre-ft).
Cumulative Water Production, WQ It is the total volume of water (in STB) produced with the gas since the start of field production to the time when the calculations are performed.
Cumulative Water Influx, W! It is the total volume of water (in reservoir barrels) which has encroached from the water aquifer to the formation section occupied initially by the free gas, since the start of production to the time when the calculations are performed.
Net pay thickness. h0 It is the thickness (at the well bore) of the reservoir portion occupied initially by the free gas, after excluding the zones which have negligible contribution to gas production (such as shale streaks).
lsopach Map Once the net pay thickness has been defined for each well and placed on the well location map, points of equal thickness are connected, and, the resulting contour map is normally termed "Jsopach map". The choice of the contour interval is normally dependent on the highest net pay value, and on the smoothness of the structure. A typical lsopach map is shown in Fig 4.1.
AAU/GASNOLUMETRICS.doc
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I Natural Gas Reservoir Engineering
Dr. A. A. Urayet
Fig 4.1_ A typical lsopach map
AAU/GASNOLUMETRICS.doc
4
INatural Gas Reservoir Engineering Net Pay Volume
I
Dr. A. A. Urayet
vf
It is the volume of that section of the formation which is occupied, initially, by gas after excluding the zones which are not contributing to the gas production. In order to calculate the Net Pay Volume, Vt, it is necessary first to construct the lsopach map, then the total area enclosed by each contour is measured (using a planimeter, a graph paper, or, the proper computer software). Finally, the Net Pay Volume, Vt. is calculated using the "Simpson Rule" as follows:
- hr Vf -)Lao +2a1 +4a2 +2a3 +4a4 + ..... +4an-2 + 2 an-I +an ] + tn-an
2
Eq (4.1) where, 3 Vt =net pay volume, ft h = contour interval, ft ai = area of the contour- i, ft tn = highest thickness above the n-th contour, ft It is important to note that the application of the Simpson Rule necessitates that the number of contours be EVEN. Another (less accurate) method which can be used in case of odd or even number of contours is the "Trapezoidal Rule" which has the following form:
Well Porosity,¢ It is the average porosity of the net pay of the well. If the porosity is read from the well logs for each foot of the net pay, then,
I
Eq (4.3)
AAU/GASNOLUMETRICS.doc
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INatural Gas Resavoir Engineering
Dr. A. A. Urayet
where,
/
•
X-Y.
0
x-6&-~
c.
/
,.. ,,,,
0
A:\l',C.\S. VOLUivIETRICS.doc
,,,
'
•
./
' ~t:.. For P=14.7 psia. qg=AOFP
>
= 21,774(2852 2 -14.7 2 )0.4062 =13.96 MMscf ID
For P=1000 psia,
qg=21,774(2852 2 -1000 2 ) 0·4062 =13.23 MMscf ID
·:>-
AAU/ GAS/GAS5b.doc
25
r··
Figure (Ex 5.1-A) Empirical Method l.E+o7
I I
~
I
I
fi
slope=2.4618 n=l/2.462=0.4062
~ j I
I
/
J
•
actual data
-best fit
•
I
I
I J
I
I/
AOFP=13.23 MMscf/D 'F
l.E+05
100
10
qg (corrected) Mmscf/day
AAU/GAS /Ex5 I
Ex5 I
26
e
e Inflow Performance Example Problem-5.1
3000.i:;~~=i==i=+:i==r:r=r=J:::c=CJ=r::C""T""9~r-1-r-r-ri-11-i--r.,-r-11-r-i--r--r--,..._,.
2500
1
I I II I I IIII III II I
I~
___
I IIIIIIII IIIIII I I II
~
.. 1'.
2000 I
I
I
I
I
I
I
I
111•11-~ 111111 llitt 111111111111 '
, '" 1ooor=~=t=t:t==t=t=i=±=t:±=±=l=J=J=-f--t-t-~=f=F=J=~r=f=l==f==f=~~~=J=j=tj=±~j
~
500+--+---+--+--+--+--+--+--+-+~--if--l---l--+--1--1---+--+--+--+--l--+--I--+-+~
0+-....__.__,__._-+__.~.__..__.......,.-+-....__.__,__._-+__,,
0
2
4
6
__.__..__.......,.-+-...&.-.....i.....j,.,.--l..-!---1.......1~'"--.i...-.+--...i.-...r...
8 flow rate, MMscf/D
10
12
14
16
INatural Gas Reservoir Engineering b.2
Dr. A. A. Urayet
The Modified Method
(P2 -P.2 ) )>
wf
Plot
vs. q g on linear paper, and fit to a straight line as shown in
qg Fig (Ex 5. 1.-B).
)>
Read from the plot (or calculate using the Least Square Fit):
a= -0.11303 )>
I
b=
4.76x10-8
Consequently, the.inflow performance can be represented by:
( p2
P.2 ) -
wf
qg
= -0.11303 + 4.76x10-8 qg
or, -2
2
(P - Pwf) =
)>
-0.11303 qg + 4.76x10
-8 2
qg
For P=14. 7 psia,
consequently, qg =AOFP= 14.313 MM scf!D
)>
ForP=1000 psia,
2
2
.
8 2
(2852 · -1000 ) ~ -0.11303 qg + 4.76xl0- qg consequently, q g = 13.487 MM scf/D
AAU/ GAS/GAS5b.doc
27
Figure (Ex 5.1-B) Modified Method 0.7
0.6
o.s
e
/
slopt ~ = b = 4 76(10)-8 ntercep =a =-1 I, 11 ;jU;j
0.4
~
bl)
~ .......
..,,_
0.3
~
-I
~
0.2
/
~ .......
0.1
I/
0.0
-0.1
V•
!/
~
M
../
/
v
/
/
/
v
• actual data --best fit
/
I•,
/
/
-0.2 0
2
4
6
8
10
12
14
16
qg (corrected) Mmscf/day
AAU/GAS /Ex5 l
Ex5 1
28
e
e Inflow Performance Empirical vs. Modified
3500-.--.--.-.....-....-....--.--.--.--.--.---.------.---....-------..--..--.---.---.---,,....-,....-.--....-..--..--.,.-.,---.--.--.--.-.....-....-....,.....,.....,.....,........
3000
I I I I I I I I I· I I I I I I I I I I I ·1 I I I I I I I ·1 I I I I I ··. I I I I I I I
2500
I I I I I I I I I ·1 I I I I I I I
--
-........_
....... f""'llllli
......
l
1'."'-
a. 1500
1000
~
"~'X'-
--------I---+-....._.__
· I I I I I I I I'"\\l\.I'- I !\111111111~ 'I v I
llt{
~
500
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 .::111,1~·111·1~b
l I .• ~ ~ I I I I I
I
I
0
I
I
I
I
I 2
I
I
I
I
I 4
I
I
I
I
I 6
I
I
I
I
I
I
I
8 flow rate, MMscf/D
I
I
I 10
I
I
I
I
I 12
I
I
I
'I 14
II
I
I
I
I 16
Natural Gas Reservoir Engineering
b.3
Dr. A. A. Urayet
The Exact Method
0 / t'l'(/})-'!'(Pwf) ,"- ,-. o vs.
qg
qg on l"mear paper, an dfitt 1 o
· htl"me as sh own m · dg c:· a s tra1g
(Ex 5.1-C).
~ Read from the plot (or calculate using the Least Square Fit):
a'
=-7.99049
'b'
=3.009601(10f6
~ Consequently, the inflow performance can be represented by:
l/l(R )- l/f(R 1) l w = -7.99049 + 3.009601(10)- 6 .q g qg
or,
'!'(/} )- l/l(Pwf) = -1.99049.qg + 3.009601(10)- 6 .q:
~
For P=14.7 psia,
'!'(2852)- '!'(14.7) = -1.99049.qg + 3.009601(10)- 6 .qi 5.28(10) 8
= -7.99049.q
g+ 3.009601(10)-6.q: '
consequently, qg =AOFP= 14.6392 MM scf/D
~
For P=1000 psia,
'!'(2852)- '1'0 000) = -7 .99049. q g + 1009601ci 0)- 6 . qi 8 5.28(10) -0.725(10) 8 = -7.99049.q g + 3.009601(10)- 6 .qi '
Consequently, qg = 13. 701 MM scf/D
AAU/ GAS/GAS5b.doc
29
Figure (Ex 5.1-,:C) Exact Method 35.0 '/
30.0
1y
I
25.0
-
ell
C"'
~
I
15.0
cf .._
s I
c
c. .._
s
10.0
~
5.0
v
0.0
-5.0
I/.
I
20.0
........
I
I
I v
I
I/
./ I
• actual data --best fit
.
I
-6 µ:: - - I.; 1~up ut
\'= 3.00 6
.
v
-10.0 0
2
4
6
8
10
12
14
16
q" (corrected) Mmscf/day "'
AAU/GAS /Ex5 1
Ex5-1
30
INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Example 5.2 (Analysis of a typical Isochronal Test) . An ·1so-chronal test was run in a dry gas reservoir, and the following data was collected from the test,
we// status Initial S.I. Flow# 1 S.I. Flow#2 S.I. Flow#3 S.I. Flow#4 Extended Flow Final S.I.
time, hours
(MM scflday)
Pwf, psia
48 12 15 12 17 12 18 12 72 100
--
1952 1761 1952 1680 1952 1420 1952 1100 700 1952
2.62
-3.29
-4.95
-6.25 6.25
---
Calculate the AOFP using the Empirical method.
AAU/ GAS/GAS5b.doc
30
I
.INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Solution
The Empirical Method
>
2 Plot ( P - P;,
1)
vs. qg on a log-Jog plot, and fit to a straight line as shown in
Fig (Ex 5.2).
>
Read from the plot (or calculate using Least Square Fit): n =0.6719
>
Draw a line parallel to the best fit (i.e. with slope=0.6710) passing through the extended flow data point
>
Calculate the value of C for the extended flow rate:
C = extended flow rate = 6,250,000 · = 260 p2 _ P.2 )n (l 2 _ 700 2)0.6719 · 952 ( wf(ext)
>
Consequently, the inflow performance relationship for the stabilized flow can be represented by:
>
For P=14.7 psia. qg=AOFP =
260(19522 -14.7 2 ) 0·6719 = 6.864 MMscf ID
AAU/ GAS/GAS5b.doc
31
I
Figure (Ex 5.2) Empirical Method 10000000.0
.l / ,• I, /
........
1'
~
,J
,,, v
,/
,. ~/
,
,'/ ,,
•
--best fit
,'
1000000.0
-
,· 1' /
1' ,,,
,,,
,, , ·' ,
v
7
;:» iUflC
actual data
= 1 ...Cl o ..
n=1/ .884= IJ.67' 9
,·1' ,,, ,
AOFP=6.864 MMscf/D
, 100000.0 10
qg (corrected) MMscf
AAU/GAS /Ex5 2
Ex5_2
33
INatural Gas Reservoir Engineering
Dr. A. A. Urayet
Chapter 6 TRANSIENT PRESSURE ANALYSIS IN GAS WELLS 6.1
INTRODUCTION
There are three main differences in the formulation of the flow equations describing the oil and gas flow in porous media. These differences can be summarized as follows:
Fluid Properties In the derivation of the Diffusivity Equation for liquids it was safe to assume that liquid properties (such as Bo, Co, and Uo) are constant. This assumption cannot be applied in the case of gas flow, because gas properties are highly sensitive to pressure changes with time and location in the reservoir.
Skin Factor Due to the relatively small liquid velocity toward the well bore, it was safe to neglect the effect of turbulance and thus use Darcy equation to represent steady-state flow of into the wellbore. However, due to the large velocities by which the gas flows toward the wellbore, the original Forchheimer equation should be used to represent the gas flow. This equation takes into consideration the effect of gas Slippage and Turbulence as follows:
I-
dP = uV + j3pV2
dr
I
Eq (6.1)
k
where fJ is a formation constant determined by the petrophysical characteristics of the formation (mainly type of rock, porosity, and permeability). The last term in the above Equation (normally known as the Non-Darcy effect) accounts for the slippage and turbulence. In order to visualize the importance of the Non-darcy effect, consider a gas well producing at a constant rate of (5 MM scf I day) . Also assume that p = 0.001
AAU/GAS//GASPTA.doc
1
Natural Gas Reservoir Engineering
= 0.15, and k = 100 md. characteristics would correspond to /3 =3). Then,
gm/cc, u
= .02 cp, h
= 25 ft,
r/J
Dr. A. A. Urayet
(such petrophysical -.
at a distance 10 ft. away from the well bore
V
=_!I_ =3183. 1 ft/day =1.12 cm/sec 2wh
and consequently ,
f3pV 2
= 3(0.001)(1.12) 2 = 0.003763
uV (.02)(1.12)/(0.1) =0.224 k dP - - = 0.224 + 0.003763 =0.227763 dr It is clear that the Non-Darcy effect, in this case, has negligible contribution (i.e. 3000 psia, i.e. the gas flow can be approximated by liquid flow, and consequently, the analytical solution for the Diffusivity equation will be:
f1-Pwf =162.6
quBg
kh
(logt+log
kP
,./, 'r
U·Y,
2
-3.23+0.87S')
Eq 6.9
lW
where, q =flow rate, scf I day,
Bg = 0.005035. T.z bbl I set, p
p=
_lj'--·+_P_w_if_
2 A plot of Pwf vs. Log t for Draw-down, or, ~vs vs. Log
(t + ~t) I ~t for build-up
will result in a straight line with a slope,
I
Eq 6.10
•
and, Eq 6.11
where,
cti = (1- Swi )cgi + swiCw +cf Mihr = f1 - lhr for Draw-Down test analysis Mihr = lihr - Pwfo for Build-up test analysis I
S =S+Dq
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Dr. A. A. Urayet
Natural Gas Reservoir Engineering
P;
The same procedure, described earlier for the method, for the calculation of the mechanical skin factor, S, and the Non-Darcy flow coefficient, D, can be used in this case .
•
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Dr. A. A. Urayet
6.4 THE PSEUDO-GAS POTENTIAL METHOD
In order to linearize the original diffusivity equation, Eq 6.2, Alhussainy and Ramey introduced the concept of the Pseudo-gas Potential, m(p }, where:
I
m(p) = 2
J.!_dP
Eq 6.12
I
14.7uz
Substituting the m(p) transform into the original Diffusivity Equation, the following form is obtained: 2 o m(p) 1 O m(p) ..-: e'- J.rcr>
\;_
~\il
::r-
2. Sukkar and Cornell method Each of these methods will be applied for the calculation of the static and flowing bottom hole pressure. Other available techniques in the 1ft8ratUre will be discussed in the Final Remarks.
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Dr A. A. Urayet
I
7.2 TRIAL AND ERROR SOLUTIONS FOR VERTICAL FLOW CALCULATIONS (f - METHOD)
z
~
7.2.1
STATIC BOTIOM HOLE PRESSURE
For a static fluid column, qg =O, and consequently, Eq 7.14 can be simplified into the following form:
12J-dp= z 20.01s1srg f dL 1P
1
Eq 7.15
T
A practical method for the calculation of the static bottom-hole pressure is based on substituting the average values of z and T (between the well head and the bottom of the well) into Eq (7.15). Then direct integration and rearrangement of the equation will result in the following simplified form:
-
0.0! 875(yg ).L. cos B Pwh .e
I- Pws -
Eq 7.16
T.z
where,
z
•
= deviation factor calculated at T
Twb
= T.h+T.b w w 2
'
~vh +~vs and P = --""'-'--'--2
=the gas temperature at the bottom of the well
Since Pws is not known, and since the right hand side of Eq (7.16) requires the -
knowledge of Pws to calculate z, consequently, the solution would require a Trial and Error procedure. Normally, two or three trials are sufficient to reach to the engineering accuracy required in such calculations. In order to minimize the number of iterations required to reach the desired accuracy, it is recommended to use the following equation to calculate the initial guess,
AAU/GAS/GASFLOW.doc
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I Natural Gas Reservoir Engineering
I Pws = Pwh (1+2.5xl 0- 5 D)
Dr A. A. Urayet
I
Eq 7.17
-5
where the factor (2.5x10 ) represents the methane gradient under standard conditions . Finally, it is important to note that once the above calculations of the static bottom hole pressure have been carried out, a "gas head factor, Fg" can be calculated for the specific well conditions (i.e. specific geothermal gradient of the well, specific gas composition, etc.) as follows:
Eq 7.18-A
~
This factor can, now, be used for future calculations of the static or well head pressures of the well using the following relationship:
j \ lpws =Pwh(l+Fg.D)
Eq7.18-B
An illustration of the proper use of the Trial and Error Technique for the calculation of the static bottom hole pressure is given in Example Problem 7.1.
7.2.2
'
.'
FLOWINMG BOTTOM HOLE PRESSURE
~
.
•
A practical method for the calculation of the flowing bottom-hole pressure is based on substituting the average values of z and T (between the well head and the bottom of the well) into the General Vertical Flow Equation (7 .14 ). Then direct integration and rearrangement of the equation will result in the following form: -2 s, 25(y g )T.z.f.q .(e -1).L
s.D 5
Eq 7.19 where,
s
_ 2ygL.cosB 53.34.T.z
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I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
=gas flow rate , MMscf/D
.
q D
= inside diameter of pipe , inches , and
f
= Moody
Friction Factor at the arithmetic average temperature and
pressure Since
Pwf is not known, and since the right hand side of Eq (7.19) requires the
knowledge of
-
Pwfto calculate z, consequently, the solution would require a
Trial and Error procedure. Two or three trials are normally sufficient to reach to the engineering accuracy required in such calculations. The above equation can be applied from the well head to the sandface in one step (as the equation is written), or, it can be applied in multi-step calculation which is preferable if the flowing temperature diagram is available. Finally it is important to note that this method (unlike its application to static pressure calculation) is not as accurate as other techniques such as the Sukkar and Cornell method or the Cullender and Smith method. Also, the friction factor needs to be re-evaluated at each trial which makes it tedious to apply for fast calculations. Consequently, it is very rarely used by reservoir and production engineers.
AAU/GAS/GASFLOW.doc
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·.:.;
I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
7.3 SUKKAR AND CORNELL METHOD
This method allows direct calculation of the bottom hole pressure without trial and error. It also gives more accurate results as compared to the average Taverage z method. In this solution, it is assumed that the Kinetic energy term can be neglected (normally< 0.1%), and that the irreversible energy losses can be expressed by friction loss correlations. An average temperature is used, and Equation 7 .14 is re-written in terms of pseudo-reduced pressure as follows:
Prf2 z Ip
0.01875ygL
_ _...::,_r_ d]J.r= - - - - = - -
Prt 1
Bz2
Eq 7.20
T
+--
p2;
where,
P,.1
=
Pi I Ppc
Pr2 =P2I Ppc L = vertical distance between points 1 and 2 , ft , and T =average temperature between points 1 and 2, 0 R B = 667 fq
2
2
T
D5P2 pc
•
In order to simplify the calculations, Sukkar and Cornell re-wrote Eq 7.20 and evaluated the pseudo reduced integrals as follows: If using the ORIGINAL Sukkar and Cornell table of integrals (Table 7.1) then Equation 7 .20 will have the form:
Eq 7.21-A
AAU/GAS/GASFLOW.doc
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I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
where, P,.(b) represents the reduced pressure corresponding to the bottom-hole pressure, P,.(wh) represents the reduced pressure corresponding to the wellhead pressure, and the lower limit (12) a base number for the evaluation of the integral. If using the EXTENDED Sukkar and Cornell table of integrals (Tables 7.2) then Equation 7 .20 will have the form:
Pr(b)
f
0.2
z/ P r
l + Bz
2
dp=
p;
Pr(wh)
f
0.2
zip
0.01875ygL d p +---2 l + Bz T r
Eq 7.21-B
p;
Note the change of the base number to (0.2) and the change of sign on the right hand side of the equation from"-" to"+". For the static pressure calculations the value of B is set equal to zero. Whereas for the flowing pressure calculations it is required first to calculate the friction factor, f, in order to be able to calculate B . It is recommended to use Equations 7.9 (Blasius smooth pipe correlation) and Eq 7 .11 (Jain rough pipe correlation) for the Turbulent flow case, and Eq 7 .6 in the Laminar flow case.
•
An illustration of the proper use of the Sukkar and Cornell method for the calculation of the static and flowing bottom-hole pressure is given in Example Problems 7.1 and 7.2 .
AAU/GAS/GASFLOW.doc
12
I Natural Gas Reservoir Engineering
7.4
Dr A. A Urayet
I
CONCLUDING REMARKS
Gas Flow in the Annulus It is important to note that the equations presented for the static and flowing pressure calculations in the tubing can also be used for gas flow in the annulus. In such case the following modifications are required: • use Deq in the calculation of the Reynolds Number, NRe, where
Deq =De +Di, and De is the inside diameter of the casing, and Di is the •
outside diameter of the tubing. for the calculation of the" f3 " factor, use the following substitution : Eq 7.22
Temperature Distribution in the Well bore The geothermal gradient is different in the various sedimentary basins, usually ranging between 1.3 to 1.9 °F increase for each one hundred feet of depth.
•
In the Sirte basin the average geothermal gradient is 1.6 °F / 100 ft of depth, and consequently, the formation temperature at any depth in the Sirte basin can be approximated by the following correlation:
IT(° F) = 80 + l.6(_Q_) 100
Eq 7.23
where, D is the vertical depth of the formation from the surface. When the fluid flows in the borehole, such as in drilling and production operations, the temperature effect will be different on different formations due to the difference in the thermal conductivities of the formations and the different well hole completions at each depth. Consequently, when the well is closed, temperature equalization will be reached after a very long time (months usually). This is why temperature irregularities in the temperature surveys can sometimes confuse the production engineer, and would be mistaken for an anomaly in the well bore, such as a hole in the casing, or fluid migration behind the pipe, etc.
AAU/GAS/GASFLOW.doc
13
I Natural Gas ReseNoir Engineering
Dr A. A. Urayet
I
~
l~ \ ~./ .
Two main features about the temperature diagram in a gas well are: • rt;ithe gas temperature at the well head will always be higher in case of the (,f~f v flowing well than in case of shut-in. , • unlike the oil temperature diagram, the gas temperature at any depth will be higher than the temperature obtained by linear interpolation between the well head and formation temperatures Consequently, the average gas temperature (to be used in the calculations) is not actually the temperature at a point one half the total depth, nor is it the average between the well head and the formation temperatures. And, consequently, if the flowing temperature diagram is available, more accurate results can be obtained by applying a multi-step calculation (of 1000-2000 feet each step). If a temperature diagram is not available then a geometric rather than linear average should be used. Practical gas field experience shows that the use of the such an average would normally minimize the error in the pressure calculations to a tolerable range of less than 10 psi.
Effect of Liquid Production Different correlations are available in the literature for two phase flow, such as Hagedorn and Brown, Duns and Ros, Orkiszewski, Beggs and Brill, and Govier and Aziz. However, different authors have concluded that for gas wells producing less than ~ of condensing water and hydrocarbons per MM scf of gas, then the met~of Sukkar-Cornell can still give very reliable results, if the following adjustments are made:
®ffia,
•
•
Use a TWO-phase z-factor instead of the gas z-factor in all previous equations to account for the condensing liquid hydrocarbons in the tubing. The Elfrik et.al. correlation, shown in Fig (7.3) can be used for this purpose.
•
Use the MIXTURE specific gr.avity, ym, instead of the gas specific gravity, '1:)
~{tr . ,l
Yg , where:
~J
fj
Eq 7.24
AAU/GAS/GASFLOW.doc
14
I Natural Gas Reservoir Engineering
GLR
rL •
Dr A. A. Urayet
I
=Gas Liquid Ratio (scf I STB) =Liquid specific gravity (water=1)
Adjust the gas flow rate in all previous equations to include the produced liquids as follows:
Iqg(corrected) = qg + qe.{3003(1.03- re)}+ qw(7390)
Eq 7.25
where,
qe qw
re
=hydrocarbon condensate flow rate ( STB/D), and , = condensing water vapor flow rate ( STBW /D) , and , =hydrocarbon condensate specific gravity= 141.5 / (131.5+API)
Finally, it is important to note that more accurate, but more sophisticated, techniques for calculating the pressure traverse in the wellbore would include the Temperature as a variable inside the original integral (Eq 7.14), instead of assuming an average temperature. One of the best known methods of this type is the Cullender and Smith numerical method which is normally used in the commercial softwares because it offers higher accuracy especially in case of gas wells producing large quantities of formation or aquifer water. Unlike the Sukkar-Cornell method, this method require a tedious trial and error procedure in addition to the numerical evaluation of the integrals. In order to achieve higher accuracy, this method also requires the calculations to be repeated at small depth increments (normally every 1000 ft), and requires a good temperature survey.
AAU/GAS/GASFLOW.doc
15
I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
Example Problem 7. 1
The following information is available about a certain gas well: 6500 ft formation depth specific gravity r g 0. 70 I
static well-head pressure, Pwh well head temperature bottom hole temperature
2500 psia 136 oF 194 oF
a) Calculate the static bottom hole pressure using the trial and error procedure b) Estimate the future well head pressure when the static pressure at the bottom of the well declines to 2600 psia. c) Calculate the static bottom hole pressure using the Sukkar and Cornell method
S}-
AAU/GAS/GASFLOW.doc
C-=-( -
- -
16
I Natural Gas ReseNoir Engineering
Dr A. A. Urayet
I
Solution a) Calculation of the static bottom-hole pressure using the T:Z method Use the specific gravity of the gas to calculate the pseudo-critical constants as follows:
756.8-131.0(rg )- 3.6(yg ) 2
Ppc =
= 756.8-131.0(0.70)- 3.6(0.7)2 = 663.3 Tpc = =
169.2 + 349.5(rg)-74.0(rg)
2
169.2 + 349.5(0.7)- 74.0(0.7) 2 = 377.6
Trial-1 1) Apply Eq (7. 17) to estimate the first guess for Pws as follows: Pws
= Pwh(l + 2.5xl0- 5 D) = 2500(1+2.5x10- 5 x6500) = 2906 -
2) Calculate P 3) Calculate 4) Calculate
= P.ws + P. 1h = 2906 + 2500 = 2703 11
2
2
psia
psia
2703 = 4.075 663.3 - 194+136 0 165+460 T= = 165 F, and T,. = = 1.655 2 377.6
Pr
-
(at P) =
5) Use Standing/Katz charts or Brill and Beggs correlation to estimate the deviation factor, z = 0.84 6) Use Eq (7 .16) to calculate Pws, 0.01875(rg ).L.cos B 1n
ws =
n
1
wh·
e
AAU/GAS/GASFLOW.doc
T.z
0.01875(0.7)(6500)
= 2500.e
(165+460)(0.84)
= 2941
psia
17
I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
=
There is a difference of 2941 - 2906 35 psi between the assumed and calculated values. Consequently, it is advisable to run another trial.
Tria/-2 Repeat steps 2-6 (using the new calculated value of Pws as a second guess) and calculate ,
2
p = Pws + Pwh = 2941 + 2500 = 2720.5 psia
2
)
2
P) =
2720 5 · = 4.1015 663.3
3)
P.
4
T = 165+460=1. 655
)
r
(at
377.6
r
5) z
=0.842 0.01875(yg ).L.cosB
6)
Pws = Pwh .e
T.z
0.01875(0.7)(6500) = 2500.e (l 65 +46 0)(0. 842 ) = 2940 psia
The difference between the assumed and calculated values is 1 psi only, so there is no need for more trials.
b) Simplified method to estimate future static pressures. 1) Using Eq 7 .18-A, Calculate:
F = g
Pvvs -Pvvh Pwh·D
= 2940- 2500 = 27.08x10-6 /ft 2500(6500)
2} Using Eq 7 .18-B, calculate:
Pws =Pwh(l+Fg.D) or
Pwh
6 :. 2600=Pwh(l+27.08xl0- (6500))
= 2211 psia
AAU/GAS/GASFLOW .doc
18
I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
c) Calculation of the static bottom-hole pressure using Sukkar and Cornell Method 1) Calculate:
Tr =
165 + 460 = 1.655 377.6
an
21_Calculate Pr (at the well head, Pwh) =
2500 = 3.769 663.3
3) Enter Table 7.1 with B=O, Tr=1.655, and P,.(wh)= 3.769. The value of the reduced integral is calculated by interpolation to be
=..:l-:t3269 J. 13 "'1'\
4) Substitute into Eq 7.21-A as follows;
Pr(b) z / p
J
12
r2
dp =
l + Bz
Pr(wh) z Ip
J
12
2
z/
J
12
l + Bz
dp -
4'
''
0.01875y L g
T
2
Pr Pr(b)
r2
~t>
Pr /./3t'J'f-
/.13°'f
«='·~!3:S!\?..
d =l.rl-3169.- 0.01875(0.7)(6500) =l~-0. 1365 =~ Pr JJ (460+ 165)
5) Re-enter Table 7.1 with the value of 0.99619, and
Tr =1.655
and read(by
linear interpolation) the corresponding value of Pr= 4.42945 6) Calculate Pws
= Pr.Ppc = (4.42945)(663.3) = 2938
psia
It is important to note the similar results obtained by the Sukkar and Cornell method and by T-Z method in the evaluation of static pressures.
~o
O' tV
"'.)
AAU/GAS/GASFLOW.doc
19
I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
Example Problem 7. 2
The following information (same as Example Problem 7.1) is available about a certain gas well : formation depth specific gravity,
rg
tubing inside diameter, d; well head temperature bottom hole temperature
•
6500 ft 0.70 1.995 inches 136 oF 194 oF
The well was put on production and the following data was available: flowing well-head pressure , Pwh 1200 psia gas production rate, qg 12.5 MM scf ID Assuming a smooth pipe, calculate the flowing bottom hole pressure using Sukkar and Cornell method
•
AAU/GAS/GASFLOW.doc
20
I Natural Gas Reservoir Engineering
Dr A. A. Urayet
I
Solution 1) The values of the pseudo-critical pressure and temperature (calculated in Example Problem 7. 1) are:
Ppc
= 663.3
psia and Tpc
= 377.6
°R
2) calculate the pseudo-reduced temperature at the average temperature of the well bore,
Tr=_.!_= 460+(194+136)/2 =1. 655 Tpc 377.6 3) calculate the pseudo-reduced pressure at the flowing well head pressure,
=
P. r(wh)
Pwh ppc
= 1200 = 1.S09 l 663.3
4) The calculation of the friction factor requires the prior calculation of the Reynolds Number. Analytically, this number should be evaluated at the average flowing viscosity which is not known. Consequently, it is always sufficient to use the viscosity at the known pressure (well head pressure in this case). - enter with
T=
165 °F, and
r g = 0. 7 into
the Carr et.al. atmospheric
viscosity correlation, (Fig 3.3) and read the gas atmospheric viscosity , µa= 0.0118 cp. - enter with Tr= 1.655, and Pr= 1.809 into the Carr et al correlation, (Fig 3.4 or 3.5) and read_!!_
= 1.17
µa -calculate µgf
=1.17(µa)=l.17(0.0118)=0.01381
cp
- calculate Reynolds Number ,
Re= 20,140 qgrg = 20,140 Dµgf
0 2 -5)(0. 7)
= 6.396x10 6
(1.995)(0.01381)
- finally, substituting in the Drew, Koo, and McAdams correlation (Eq 6.8), calculate the Moody friction factor as follows:
AAU/GAS/GASFLOW.doc
21
I Natural Gas Reservoir Engineering f
= 0.0056 + 0.5NR_~
:i 51 calculate
p-- 667 f.qiT
2
5
2 D .Ppc
6) Enter Table 7.2 with
32
_
-
Dr A. A. Urayet
I
6 032 = 0.0056 + 0.5(6.396xl0 )= 0.00892
667(0.00892)(12.5) 2 (165 + 460) 2 _ 2
5
-
26 · 12
(1.995) (663.3)
B =26.12, Tr =1.655, and P,.(wh) = 1.809. The value of
the reduced integral is calculated by interpolation to be
=0.06345
7) Substitute into Eq 7.21-B as follows;
Pr(b)
J
z Ip r2
0.2 l +
Pr(b)
f 0.2
dp =
Bz
p;
Pr(wh)
J
0.2
z/p r2
dp +
0.01875y L
l + Bz
p;
g
T
z I Pr dp = 0.06345 + O.OlS 75 (0. 7)( 65 00) = 0.06345 + 0.1365 = 0.19995 2 (460+ 165) 1 Bz
+2 Pr
8) for Tr= 1.655 and the value of the integral obtained in step(?), calculate by interpolation from Tables (7.2) the value of P,.(b) =3.42472 9) finally, calculate the flowing bottom hole pressure,
J;vf = Ppc·P,.(wf) = 663.3(3.42472) = 2272psia
AAU/GAS/GASFLOW.doc
22
I Natural Gas Reservoir Engineering
Dr A. A. Urayet /
Field Problem
The following information is available about a certain gas well : formation depth specific gravity'
rg
tubing inside diameter, di tubing roughness, e well head temperature bottom hole temperature
e.
6000 0.73
ft inches . inches oF oF
1.995 0.0006 132 185
The well was put on production and the following data was available: psia MMscf/D
flowing well-head pressure , Pwh 950 gas production rate , q0 8. 5 liquid recovery 24 condensate AP/ 56
a) Using Sukkar and Cornell method calculate the flowing pressure in the tubing at every 2000 ft. depth interval. b) Prepare a plot of the pressure results vs. depth
/ J
I l
(~J¥ J
'
!
~
j J
J
"'·•.
AAU/GAS/GASFLOW.doc
''
'
!
23
Pipe diameter, in feet-d
0.05 0.04 0.0 3 ' 0.0
1
2
I
2~ ...
3
'" "
"
2
3
4 5 6
I I
II
~
I\.
'
Rivqted Steel'
.... ~
' ... , ' "' " ~ " " '"'\.."' I"' . ." ""'""' ~ "N' ~ ' ~ f" ~ ' ~ "~ " " ' '' I\..
~
"'\..
...
....
["-.
Concrete
l"\..
I
I
I
Wood Stave
r....
.....
0.06
-
0.05
-
0.04 0.035
'
...
Cast Iron
"['I
~
-
... !\
-
:'I~
~~
~
2025 1- 0.07
8 10
I\"
-
0.006 ""'"""' ...., I"-. 0.0085"-"""0.004 II... _, 0.002
8 1
l'I
['\
0.0 ~ 1 "'\.."'\.. 0.00
~.po3
4 56
Galvanized Iron
"I\.
I\
I"
~o·o.~"'
"
"'I'\ f'
Commercial Steel or Wrought Iron 0.000,1 ,__ 0.000,08
0.000,06 0.000,05 0.000,04
~'
""
'Q~~l"~f'
·"
0.000.02
...
"""Tu~~
Drawn
0.000,01 0.000,008 0.000,006
'Qa "'- v. ~Oo.
2
3456810
20
'
~-...
0 .014
E 0 u
"' a.
~
0
u. I
0 .012
0. 01
~ 0. 009 '
"'
::J .0 ~ ::J
....;
'
0 ['I·~
tv'
c
0.016
f"
"'"\:!\.. ~
30 40 50 60 BO 100
Pipe diameter, in inches-d
0.018
"
,vo9'~"'\.. ' "
.. , "
I\.~~
~
·"
f'.. 3'-
1
..,
E (.)
"' ~- t?sl.s-. , __ (!>~Qa~' 'I".
""" ~~I
'
'Zoo'
,,.,,.. ·~ '~"~ ·oa QCb O· 0
"" '"
"'\..
0.02
"\"·
'~
.......
01
::J
-
'f\..
' l'...""f\"oos.'I.~ ~~",
I"-.
0.000,03
0.000,005
"
~
.9a.
.s::.
""''I\. ,, "" '' ~.
0.025
.... G>-
IX I" "'0.00 1 ... ~ ... ~ ""\'.~oo_ I\" ~ ['I\."'\.. 0.000,8 ~['I ... I'\ ~ ... I\. 0.000,6 I". I"" I'... ~ 0.000.5 ....... ... "'\.."'\.. I'\..°" 0.000,4 ,... ... ._ Asphalted Cast fron"" f\.. ~ ~ ~ 0.000.3 I" " ~~ I'\ u>--= 0.000.2
""
0.03
"""- 0.008
"
200 300
Figure 1-. i Relative ro~ghness of Pipe materials and friction factors for complete turbulence
.....
11...()
Lines of pseu·do reduced pressure
0.6
,..0
.t>
.
::::
~
:;. Q
Cl.
...
QC
(.,;
TABLE C.2(h) Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation P,
1.1
e
e
0.20 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50
:o.oo
10.SO 11.00 11.50 12.00 12.50 13.00 13.50 14.00
0.0000 0.0033 0.0171 0.0454 0.0861 0.1283 0.1703 0.2120 0.2536 0.2950 0.3362 0.3773 0.4183 0.4591 0.4999 0.5405 0.5810 0.6214 0.6617 0.7018 0.7419 0.7818 0.8217 0.8614 0.9011 0.9407 0.9803 1.0197 1.0591
14.50 1.0985 15.00 1.1377 15.50 1.1770 16.00 1.2162 16.50 . l.2553 17.00 1.2944 17.50 1.3334 18.00 1.3725 18.50 l.4114 19.00 l.4504 19.50 l.4893 20.00 1.5281 20.50 1.5670 21.00 l.6058 21.50 l.6446 22.00 1.6833 22.50 1.7220 23.00 l.7607 ~3.50 1.7994 24.00 1.8381 24.50 l.8767 25.00 l.9153 25.50 1.9539 26.00 1.9924 26.50 2.0310 27.00 2.0695 27.50 2.1080 28.00 2.1465 28.50 2.1850 29.00 2.2234 29.50 2.2619 1 2.3003 " 00
1.2 0.0000 0.0032 0.0158 0.0387 0.0720 0.1119 0.1538 0.1960 0.2382 0.2800 0.3216 0.3630 0.4040 0.4449 0.4856 0.5261 0.5665 0.6066 0.6466 0.6865 0.7262 0.7657 0.8051 0.8444 0.8836 0.9227 0.9617 1.0006 1.0394
1.0781 1.1167 1.1552 1.1937 1.2321 l.2705 1.3087 1.3470 1.3851 l.4232 1.4613 1.4993 1.5373 1.5752 l.6130 1.6509 1.6887 1.7264 1.7641 1.8018 1.8394 1.8771 l.9146 l.9522 1.9897 2 0272 2.0647 2.1021 2.1395 2.1769 2.2142 2.2516
1.3 0.0000 0.0032 0.0152 0.0361 0.0657 0.1022 0.1425 0.1844 0.2266 0.2688 0.3106 0.3522 0.3934 0.4344 0.4752 0.5156 0.5558 0.5959 0.6357 0.6753 0.7147 0.7539 0.7930 0.8319 0.8607 0.9094 0.9479 0.9863 1.0246
1.0627 1.1008 1.1388 1.1767 1.2144 1.2521 1.2898 1.3273 1.3648 l.4022 l.4395 l.4768 l.5140 1.5511 1.5882 1.6252 l.6622 1.6991 1.7360 1.7729 1.8097 1 8464 l.8831 1.9198 l.9564 1.9930 2 0295 2.0661 2.1025 2.1390 2.1754 2.2118
1.4 0.0000 0.0031 0.0148 0.0346 0.0623 0.0965 0.1350 0.1759 0.2179 0.2601 0.3023 0.3442 0.3857 0.4270 0.4679 0.5085 0.5487 0.5888 0.6285 6.6681 0.7073 0.7464 0.7852 0.8239 0.8623 0.9006 0.9386 0.9765 1.0143
1.0519 1.0893 1.1266 1.1638 1.2008 1.2378 1.2746 1.3113 1.3479 1.3844 l.4208 l.4571 l.4933 1.5294 1.5655 1.6014 1.6373 l.6732 1.7089 1.7446 1.7802 1.8158 l.8513 l.8867 1.9221 1.9574 l.9927 2.0279 2.0631 2.0983 2.1333 2.1684
1.5 0.0000 0.0031 0.0145 0.0336 0.0601 0.0925 0.1295 0.1694 0.2108 0.2529 0.2951 0.3373 0.3791 0.4207 0.4618 0.5026 0.5431 0.5832 0.6230 0.6625 0.7017 0.7406 0.7793 0.8177 0.8559 0.8939 0.9317 0.9693 1.0067
1.0439 1.0809 1.1178 1.1549 1.1911 1.2275 1.2638 l.2999 1.3359 1.3718 1.4075 l.4432 l.4788 1.5142 1.5495 1.5848 1.6199 1.6550 1.6900 I. 7249 1.7597 1.7944 1.8291 1.8637 1.8982 1.9326 l.9670 2.0014 2.0356 2.0698 2.1040 2.1381
Reduced Temperature for 8 = 35.0 1.6 1.7 1.8 1.9 2.0 0.0000 0.0031 0.0143 0.0329 0.0585 0.0900 0.1259 0.1650 0.2059 0.2477 0.2899 0.3321 0.3742 0.4159 .0.4573 0.4983 0.5390 0.5792 0.6191 0.6586 0.6978 0.7367 0.7753 0.8136 0.8517 0.8895 0.9271 0.9645 1.0017
1.0386 1.0754 1.1120 1.1484 1.1846 1.2207 l.2566 1.2923 l.3280 1.3634 1.3988 l.4340 1.4691 1.5041 1.5390 l.5738 l.6084 1.6430 1.6775 1.7118 1.7461 1.7803 1.8144 1.8484 1.8824 l.9163 1.9501 1.9838 2.0175 2.0511 2.0846 2.1180
0.0000 0.0031 0.0141 0.0323 0.0573 0.0879 0.1230 0.1613 0.2017 0.2433 0.2854 0.3276 0.3698 0.4117 0.4532 0.4944 0.5352 0.5756 0.6156 0.6552 0.6945 0.7334 0.7719 0.8102 0.8481 0.8858 0.9232 0.9604 0 9973
1.0340 1.0704 1.1066 1.1426 1.1784 1.2140 1.2494 l.2846 1.3197 1.3546 1.3893 1.4239 1.4584 l.4927 l.5269 1.5609 1.5948 l.6286 1.6623 1.6959 l. 7294 1.7627 1.7960 l.8291 1.8622 1.8951 l.9280 1.9608 1.9935 2.0261 2.0587 2.0912
0.0000 0.0030 0.0139 0.0320 0.0564 0.0864 0.1208 0.1585 0.1984 0.2396 0.2816 0.3238 0.3660 0.4080 0.4498 0.4912 0.5822 0.5727 0.6129 0.6526 0.6919 0.7308 0.7694 0.8076 0.8655 0.8831 0.9204 0.9574 0.9941
1.0305 l.0667 l.1027 1.1384 1.1739 l.2092 1.2443 1.2792 1.3139 l.3484 1.3828 l.4170 l.4510 1.4849 1.5186 1.5522 1.5856 1.6189 l.6521 l.6851 17180 1.7508 1.7835 1.8161 l.8486 l.8810 1.9133 l.9454 1.9775 2.0095 2.0414 2.0732
0.0000 0.0030 0.0139 0.0317 0.0559 0.8055 0.1194 0.1567 0.1962 0.2372 0.2790 0.3211 0.3634 0.4055 0.4473 0.4889 0.5300 0.5707 0.6109 0.6507 0.6901 0.7291 0.7677 0.8059 0.8438 0.8813 0.9185 0.9554 0.9920
1.0282 1.0642 1.0999 1.1354 1.1705 1.2055 1.2402 1.2747 1.3089 1.3430 1.3769 1.4105 1.4440
1Ai73 1.5104 l.5434 1.5762 l.6088 1.6413 1.6736 1.7058 1.7379 1.7698 l.8016 l.8333 1.8649 1.8963 l.9277 1.9589 1.9900 2.0210 2.0519
0.0000 0.0030 0.0139 0.0315 0.0554 0.0847 0.1182 0.1550 0.1942 0.2350 0.2766 0.3187 0.3610 0.4032 0.4451 0.4867 0.5280 0.5688 0.6091 0.6490 0.6885 0.7275 0.7661 0.8043 0.3422 0.8797 0.9168 0.9535 0.9900
1.0261 1.0618 1.0973 1.1325 1.1674 1.2020 1.2364 1.2705 1.3044 1.3380 1.3714 1.4046 1.4376 1.4704 1.5030 1.5355 1.5677 1.5998 l.6317 1.6634 1.6950 1.7264 1.7577 1.7888 l.8198 1.8506 l.8814 1.9119 1.9424 l.9726 2.0030 2.0331
2.2 0.0000 0.0030 0.0137 0.0311 0.0546 0.0834 0.1165 0.1528 0.1916 0.2320 0.2734 0.3153 0.3576 0.3998 0.4418 0.4836 0.5247 0.5657 0.6062 0.6462 0.6858 0.7250 0.7637 0.8019 0.8398 0.8771 0.9141 0.9507 0.9869
1.0226 l.0580 1.0931 1.1278 1.1622 1.1962 1.2300 1.2634 1.2966 1.3294 1.3620 1.3944 1.4265 1.4583 1.4900 1.5214 1.5525 1.5835 1.6143 1.6448 1.6752 1.7054 1.7354 1.7652 1. 7949 1.8244 1.8537 1.8829 l.9119 1.9408 l.9696 l.9982
2.4 0.0000 0.0030 0.0136 0.0309 0.0542 0.0826 0.1153 0.1513 0.1897 0.2298 0.2710 0.3128 0.3550 0.3972 0.4394 0.4812 0.5227 0.5638 0.6044 0.6445 0.6842 0.7234 0.7621 0.8004 0.8381 0.8755 0.9124 0.9483 0.9848
1.0205 1.0557 1.0905 l.1247 1.1590 1.1928 1.2262 1.2592 1.2970 1.3245 1.3566 1.3885 1.4201 1.4515 l.4826 L5134 1.5440 1.5744 J.6046 1.6345 1.6642 1.6937 I. 7231 1.7522 I. 7812 1.8100 1.8386 1.8670 1.8953 1.9234 1.9513 l.9791
2.6 0.0000 0.0030 0.0136 0.0307 0.0537 0.0819 0.1142 0.1499 0.1880 0.2279 0.2690 0.3107 0.3529 0.3951 0.4373 0.4792 0.5208 0.5619 0.6026 0.6428 0.6825 0.7217 0.7604 0.7987 0.8364 0.8737 0.9106 0.9470 0.9829
1.0184 1.0536 1.0883 l.1226 1.1566 1.1901 1.2234 1.2563 1.2889 1.3212 1.3531 1.3848 1.4162 1.4473 1.4782 1.5088 1.5391 1.5693 l.5992 1.6288 1.6583 1.6875 1.7165 1.7454 1.7740 1.8025 1.8306 1.8589 1.8868 1.9146 1.9422 1.9696
2.8
3.0
0.0000 0.0030 0.0135 0.0305 0.053-1 0.0813 0.1134 0.1487 0.1866 0.2263 0.2672 0.3089 0.3910 0.3932 0.4354 0.4774 0.5190 0.5602 0.6009 0.6412 0.6809 0. 7201 0.7589 0.7971 0.8349 0.8721 0.9089 0.9453 0.9812
1.0167 1.0517 1.086-1 1.1206 1.1545 l.1880 1.2212 1.2540 1.2865 l.3187 1.3506 1.3822 1.4135 1.4445 1.4752 1.5057 1.5360 1.5660 1.5957 1.6253 l.6546 1.6837 l.7126 l.7413 l.7698 l.7981 1.8262 l.8542 l.8820 1.9096 1.9370 1.9643
0.0000 0.0030 0.0135 0.0304 0.0531 0.0808 0.1127 0.1478 0.1855 0.2250 0.2658 0.3074 0.3495 0.3918 0.4339 0.4759 0.5175 0.5588 0.5996 0.6398 0.6796 0.7189 0.7576 0.7958 0.8336 0.8708 0.9076 0.9439 0.9798
1.0153 1.0503 1.0849 1.1191 l.1529 l.1864 1.2195 l.2522 1.2847 1.3168 1.3485 1.3800 1.4112 l.4422 1.4720 1.5032 l.5333 1.5632 l.5929 1.6223 l.6515 1.6805 1.7093 1.7378 1.7662 1.7944 1.8224 1.8502 1.8779 1.9053 l.9327 1.9598
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TABLE C.2(i) Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation P,
e
e
Reduced Temperature for 8 = 40.0 1.6 1.7 1.8 1.9 2.0
1.1
1.2
1.3
1.4
1.5
0.20 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00
0.0000 0.0029 0.0150 0.0403 0.0776 0.1170 0.1565 0.1958 0.2351 0.2743 0.3133 0.3523 0.3912 0.4300 0.4687 0.0573 0.5458 0.5843 0.6227 0.6609 0.6991 0.7372 0.7753 0.8132 0.8511 0.8890 0.9268 0.9645 1.0022
0.0000 0.0028 0.0139 0.0341 0.0640 0.1005 0.1393 0.1787 0.2182 0.2576 0.2969 0.3360 0.3750 0.4138 0.4525 0.4910 0.5294 0.5677 0.6059 0.6439 0.6818 0.7196 0.7573 0.7949 0.8324 0.8696 0.9072 0.9445 0.9816
0.0000 0.0028 0.0133 0.0318 0.0582 0.0912 0.1281 0.1668 0.2062 0.2457 0.2851 0.3244 0.3634 0.4023 0.4410 0.4795 0.5179 0.5560 0.5940 0.6319 0.6696 0.7071 0.7446 0.7819 0.8190 0.8561 0.8931 0.9299 0.9667
0.0000 0.0027 0.0129 0.0305 0.0551 0.0858 0.1208 0.1584 0.1973 0.2367 0.2762 0.3156 0.3549 0.3939 0.4328 0.4714 0.5097 0.5479 0.5859 0.6237 0.6612 0.6987 0.7359 0.7729 0.8098 0.8466 0.8832 0.9196 0.9559
0.0000 0.0027 0.0127 0.0296 0.0530 0.0821 0.1156 0.1520 0.1901 0.2292 0.2686 0.3081 0.3476 0.3868 0.4258 0.4646 0.5031 0.5413 0.5793 0.6171 0.6546 0.6919 0.7290 0.7659 0.8026 0.8391 0.8755 0.9117 0.9477
0.0000 0.0027 0.0125 0.0290 0.0517 0.0798 0.1122 0.1477 0.1853 0.2240 0.2633 0.3028 0.3423 0.3816 0.4208 0.4597 0.4983 0.5367 0.5747 0.6125 0.6500 0.6873 0.7243 0.7611 0.7977 0.8341 0.8703 0.9063 0.9421
0.0000 0.0027 0.0123 0.0284 0.0505 0.0779 0.1095 0.1442 0.1812 0.2195 0.2586 0.2980 0.3376 0.3770 0.4163 0.4553 0.4941 0.5325 0.5707 0.6085 0.6461 0.6833 9.7203 0.7571 0.7936 0.8299 0.8659 0.9017 0.9373
0.0000 0.0027 0.0122 0.0281 0.0497 0.0765 0.2074 0.1416 0.1780 0.2159 0.2548 0.2941 0.3336 0.3731 0.4124 0.4516 0.4905 0.5290 0.5673 0.6052 0.6429 0.6802 0.7172 0.7539 0.7903 0.8265 0.8624 0.8981 0.9335
0.0000 0.0027 0.0122 0.0279 0.0493 0.0756 0.1061 0.1398 0.1758 0.2135 0.2521 0.2913 0.3308 0.3703 0.4097 0.4490 0.4879 0.5266 0.5650 0.6030 0.6407 0.6780 0.7150 0.7517 0.7882 0.8243 0.8602 0.8957 0.9310
l-f.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 2-1.00 24.50 25.00 25.50 26.00 26.50 27.00 27.50 28.00 28.50 29.00 29.50 30.00
1.0398 1.0774 1.1149 1.1525 1.1899 1.2274 1.2648 1.3021 1.3395 1.3768 1.4140 1.4513 1.4885 1.5257 1.5629 1.6001 1.6372 1.6743 1.7114 1.7485 1.7855 1.8226 1.8596 1.8966 1.9336 1.9705 2.0075 2.0444 2.0813 2.1182 2.1551 2.1920
0.0188 1.0558 1.0928 1.1297 1.1666 1.2034 1.2402 1.2769 1.3136 1.3502 1.3868 1.4233 1.4598 1.4963 1.5327 1.5691 1.6054 1.6417 1.6780 1.7143 1.7505 1.7867 1.8229 1.8591 1.8952 1.9313 1.9674 2.0034 2.019-1 2.0755 2.1114 2.1474
1.0034 1.0400 1.0765 1.1129 1.1492 1.1855 1.2217 1.2579 1.2940 1.3300 1.3659 1.4019 1.4377 1.4735 1.5093 1.5450 1.5807 1.6163 1.6519 1.6874 1.7229 1.7584 1.7938 1.8292 1.8645 1.8999 1.9352 1.9704 2.0057 2.0409 2.0761 2.1112
0.9921 1.0282 1.0641 1.1000 1.1357 1.1713 1.2068 1.2422 1.2776 1.3128 1.3480 1.3831 1.4181 1.4530 1.4879 1.5227 1.5574 1.5920 1.6266 1.6612 1.6957 1.7301 1.7645 1.7988 1.8331 1.8673 1.9015 1.9356 1.9697 2.0038 2.0378 2.0717
0.9835 1.0193 1.0548 1.0903 1.1255 1.1607 1.1958 1.2307 1.2655 1.3002 1.3349 1.3694 1.4038 1.4381 1.4723 1.5065 1.5406 1.5746 1.6085 1.6423 1.6761 1.7098 1.7434 1.7770 1.8105 1.8439 1.8773 1.9107 1.9439 1.9771 2.0103 2.0434
0.9778 1.0133 1.0486 1.0837 1.1187 1.1536 1.1884 1.2230 1.2574 1.2918 1.3261 1.3602 1.3942 1.4281 1.4620 1.4957 1.5293 1.5629 1.5963 1.6297 1.6630 1.6962 1.7293 1.7624 1.7954 1.8283 1.8612 1.8940 1.9267 1.9594 1.9920 2.0246
0.9727 1.0079 1.0429 1.0777 1.1123 1.1468 1.1811 1.2152 1.2492 1.2831 1.3168 1.3504 1.3838 1.4171 1.4503 1.4834 1.5164 1.5492 1.5820 1.6146 1.6472 1.6797 1.7120 1.7443 1.7765 1.8086 1.8406 1.8726 1.904-1 1.9362 1.9680 1.9996
0.9588 1.0037 1.0385 1.0731 1.1075 1.1417 1.1757 1.2095 1.2432 1.2767 1.3101 1.3433 1.3763 1.4093 1.4421 1.4747 1.5072 1.5396 1.5719 1.6041 1.6362 1.6582 1.7000 1.7318 1.7634 1.7950 1.8265 1.8579 1.8892 1.9204 J.9516 1.9826
0.9661 1.0009 1.0355 1.0698 1.1039 1.1378 1.1714 1.2049 1.2382 1.2713 1.3042 1.3369 1.3695 1.4019 1.4341 1.4662 1.4982 1.5300 1.5617 1.5932 1.6246 1.6559 1.6871 1.7181 1.7491 1.7799 1.8106 1.8412 1.8717 1.9021 1.9325 1.9627
2.2
2.4
2.6
0.0000 0.0026 0.0121 0.0276 0.0488 0.0749 0.1050 0.1383 0.1740 0.2113 0.2498 0.2889 0.3283 0.3678 0.4073 0.4466 0.4856 0.5244 0.5628 0.6009 0.6386 0.6760 0.7130 0.7498 0.7862 0.8223 0.8580 0.8935 0.9287'
0.0000 0.0026 0.0120 0.0273 0.0482 0.0738 0.1034 0.1362 0.1714 0.2084 0.2465 0.2854 0.3247 0.3642 0.4037 0.4431 0.4819 0.5208 0.5593 0.5975 0.6353 0.6728 0.7099 0.7466 0.7830 0.8190 0.8547 0.8900 0.9250
0.0000 0.0026 0.0119 0.0271 0.0477 0.0730 0.1023 0.1348 0.1696 0.2063 0.2442 0.2829 0.3221 0.3616 0.4011 0.4405 0.4797 0.5187 0.5573 0.5955 0.6334 0.6710 0.7081 0.7448 0.7812 0.8171 0.8527 0.8879 o.m8
0.0000 0.0026 0.0119 0.0270 0.0473 0.0724 0.1013 0.1335 0.1681 0.2045 0.2422 0.2808 0.3199 0.3594 0.3989 0.4383 0.4776 0.5166 0.5553 0.5936 0.6315 0.6690 0.7052 0.7429 0.7792 0.8152 0.8507 0.8859 0.9207
0.0000 0.0000 0.0026 0.0026 0.0118 0.Dl 18 0.0268 0.0267 0.0471 0.0468 0.0718 0.0714 0.1005 0.0999 0.1324 0.1315 0.1667 0.1656 0.2029 0.2017 0.2405 0.2391 0.2790 0.2775 0.3181 0.3166 0.3575 0.3560 0.3970 0.3955 0.4365 0.4350 0.4758 0.4743 0.5148 0.5133 0.5535 0.5521 0.5918 0.5904 0.6298 0.6284 0.6673 0.6660 0.7045 0.7031 0.7412 0.7398 0.7775 0.7762 0.8134 0.8121 0.8490 0.8476 0.8841 0.8827 0.9188 0.9174
0.9636 0.9982 1.0328 1.0667 1.1005 1.1341 1.1675 1.2006 1.2336 1.2663 1.2988 1.3311 1.3633 l.3952 1.4270 1.4586 1.4900 1.5213 1.5525 1.5834 1.6143 1.6-150 1.6755 1.7059 1.7362 1.7664 1.7965 1.8264 1.8562 1.8859 1.9155 1.9450
0.9596 0.9939 1.0279 1.0516 1.0949 1.1280 1.1608 1.1934 1.2256 1.2577 1.2894 1.3210 1.3523 1.3834 1.4143 1.4449 1.4754 1.5057 1.5358 1.5657 1.5954 1.6249 1.6543 1.6836 1.7126 1.7415 1.7703 1.7989 1.8274 1.8557 1.8840 1.9120
0.9572 0.9914 1.0251 1.0586 1.0917 1.1245 1.1570 1.1892 1.2211 1.2528 1.2842 1.3153 1.3462 1.3768 1.4072 1.4373 1.4673 1.4970 1.5265 1.5559 1.5850 1.6139 1.6427 1.6713 1.6997 1.7279 1.7560 1.7839 1.8116 1.8393 1.8667 1.8940
0.9551 0.9891 1.0228 1.0561 1.0891 1.1218 1.1541 1.1862 1.2180 1.2494 1.2806 1.3116 1.3422 1.3727 1.4028 1.4328 1.4625 1.4920 1.5213 1.5503 1.5792 1.5078 1.6363 1.6646 1.6927 1.7207 1.7484 1.7760 1.8035 1.8308 1.8579 1.8849
0.9532 0.9872 1.0208 1.0541 1.0870 1.1196 1.1519 1.1839 1.2155 1.2469 1.2780 1.3089 1.3395 1.3698 1.3999 1.4297 1.4593 1.4887 1.5178 1.5468 1.5755 1.6041 1.632-1 1.6606 1.6886 1.7164 1.7440 1.7715 1.7988 1.8259 1.8529 1.8797
2.8
3.0
0.9517 0.9856 1.0192 1.0525 1.0853 1.1179 1.1501 1.1820 1.2136 1.2450 1.2760 1.3068 1.3373 1.3675 1.3975 1.4272 1.4567 1.4860 1.5151 1.5439 1.5725 1.6010 1.6292 1.6572 1.6851 1.7128 1.7403 1.7676 1.7946 1.8216 1.8487 1.875-1
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TABLE C.2(j) Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation P,
e
e
Reduced Temperature for B = 45.0 2.0 1.9 1.6 1.7 1.8
1.1
1.2
1.3
1.4
1.5
0.20 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 I0.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00
0.0000 0.0026 0.0134 0.0362 0.0707 0.1016 0.1449 0.1821 0.2193 0.2565 0.2936 0.3306 0.3676 0.4045 0.4414 0.4782 0.5150 0.5517 0.5883 0.6248 0.6613 0.6978 0.7342 0.7705 0.8068 0.8430 0.8792 0.9153 0.9514
0.0000 0.0025 0.0124 0.0305 0.0576 0.0912 0.1273 0.1643 0.2015 0.2388 0.2760 0.3131 0.3501 0.3871 0.4239 0.4607 0.4973 0.5339 0.5704 0.6067 0.6430 0.6792 0.7153 0.7514 0.7874 0.8233 0.8591 0.8949 0.9306
0.0000 0.0025 0.0119 0.0284 0.0522 0.0823 0.1163 0.1523 0.1892 0.2264 0.2637 0.3009 0.3380 0.3750 0.4118 0.4486 0.4852 0.5216 0.5580 0.5942 0.6304 0.6664 0.7023 0.7381 0.7738 0.8094 0.8449 0.8804 0.9157
0.0000 0.0024 0.0115 0.0272 0.0494 0.0772 0.1093 0.1441 0.1803 0.2172 0.2544 0.2917 0.3289 0.3660 0.4029 0.4397 0.4763 0.5128 0.5492 0.5853 0.6214 0.6573 0.6930 0.7286 0.7641 0.7994 0.8347 0.8698 0.9048
0.0000 0.0024 0.0113 0.0264 0.0475 0.0738 0.10-13 0.1378 0.1732 0.2096 0.2466 0.2838 0.3211 0.3583 0.3954 0.4323 0.4690 0.5055 0.5419 0.5780 0.6140 0.6498 0.6854 0.7209 0.7562 0.7914 0.8264 0.8613 0.8961
0.0000 0.0024 0.0111 0.0258 0.0462 0.0716 0.1012 0.1338 0.1685 0.2045 0.2412 0.2783 0.3158 0.3528 0.3900 0.4270 0.4638 0.5004 0.5368 0.5730 0.6090 0.6447 0.6803 0.7157 0.7509 0.7860 0.8209 0.8556 0.8902
14.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00 25.50 26.00 26.50 27.00 27.50 28.00 28.50 29.00 29.50 30.00
0.9875 l.0235 1.0595 1.0955 1.1315 1.1674 1.2032 1.2391 1.2749 1.3107 1.3465 1.3823 1.4180 1.4538 1.4895 15251 1.5608 1.5965 1.6321 1.6677 1.7033 1.7389 l.7745 1.8100 1.8456 1.8811 1.9166 1.9521 1.9876 2.0231 2.0586 2.0941
0.9663 1.0019 1.0374 1.0729 1.1084 1.1438 1.1791 l.2145 1.2497 1.2850 1.3202 1.3554 l.3905 1.4256 1.4607 1.4958 1.5308
0.9510 0.9863 l.02H 1.0565 1.0915 1.1265 1.1614 1.1962 1.2310 1.2658 1.3005 1.3351 1.3697 1.4043 1.4388 1.4733 1.5077 1.5421 1.5765 l.6108 1.6451 1.6794 1.7136 1.7478 1.7820 l.8162 1.8503 1.8844 1.9184 1.9525 1.9865 2.0205
0.9396 0.9744 l.0091 1.0437 1.0782 1.1126 1.1469 l.1811 1.2153 1.2494 1.2834 1.3173 1.3512 l.3850 1.4187 1.4524 1.4860 1.5196 1.5531 1.5866 1.6200 1.6534 1.6867 1.7200 1.7532 1.7864 1.8195 1.8526 l.8857 1.9187 1.9517 1.9847
0.9307 0.9652 0.9995 1.0338 1.0679 1.1019 1.1358 1.1696 1.2033 1.2370 1.2705 1.3039 1.3373 l.3706 1.4038 1.4369 1.4699 1.5029 1.5358 1.5687 1.6015 1.6342 1.6668 1.6995 l.i320 1.7645 1.7969 1.8293 1.8617 1.8940 1.9262 1.9584
0.9246 0.9589 0.9931 1.0271 1.0609 1.0947 1.1283 l.1619 1.1953 1.2286 1.2618 1.2949 1.3279 1.3608 1.3937 1.4264 1.4591 1.4916 1.5242 1.5566 1.5890 1.6212 1.6535 1.6856 1.7177 1.7498 1.7817 l.8136 l.8455 1.8773 1.9091 1.9408
1.5658 1.6008 1.6357 1.6706 1.7055 1.7404 !. 7752 1.8101 1.8449 1.8797 1.9144 1.9492 1.9839 2.0186 2.0533
0.0000 0.0000 0.0024 . 0.0024 0.0110 0.0109 0.0250 0.0254 0.0452 0.0445 0.0586 0.0699 0.0967 0.0986 0.1304 0.1279 0.1614 0.1645 0.1966 0.2001 0.2366 0.2327 0.2735 0.2695 0.3107 0.3066 0.3480 0.3439 0.3852 0.3811 0.4182 0.4223 0.4552 0.4592 0.4959 0.4920 0.5286 0.5323 0.5686 0.5649 0.6009 0.6046 0.6367 0.6606 0.6759 0.6723 0.7113 0.7076 0.7427 0.7464 0.7776 0.7814 0.8122 0.8161 0.8507 0.8467 0.8809 0.8851
0.9193 0.9533 0.9872 1.0209 1.0544 1.0878 1.1211 1.1542 l.1872 1.2200 1.2528 1.2854 1.3179 1.3503 1.3825 1.4147 1.4468 1.4788 1.5106 1.5424 1.5741 l.6057 1.6373 1.6687 l. 7001 1.7314 1.7626 1.7937 l.8248 1.8558 1.8868 1.9176
0.9150 0.9489 0.9825 1.0160 1.0494 1.0825 1.1155 1.1484 1.1811 l.2136 1.2460 1.2783 1.3105 1.3425 1.3744 1.4062 1.4379 1.4694 1.5009 1.5323 1.5635 1.5947 l.6257 l.6567 l.6876 1.7184 1.7491 1.7798 1.8103 1.8408 l.8712 l.9016
2.2
2.4
2.6
2.8
3.0
0.0000 0.0023 0.0106 0.0240 0.0423 0.0648 0.0910 0.1203 0.1520 0.1855 0.220..i 0.2562 0.2928 0.3297 0.3669 0.4042 0.4414 0.4785 0.5153 0.5519 0.5882 0.6242 0.6598 0.6952 0.7302 0.7649 0.7992 0.8332 0.8669
0.0000 0.0023 0.0105 0.0239 0.0420
0.0000 0.0023 0.0105 0.0238 0.0418 0.0640 0.0897 0.1185 0.1496 0.1828 0.2174 0.2530 0.2895 0.3264 0.3635 0.4008 0.4360 0.4751 0.5120 0.5486 0.5650 0.6210 0.6566 0.6920 0.7270 0.7616 0.7959 0.8299 0.8635
0.0000 0.0024 0.0108 0.0248 0.0440 0.0678 0.0955 0.1263 0.1594 0.1942 0.2301 0.2667 0.3038 0.3410 0.3782 0.4154 0.4525 0.4893 0.5259 0.5623 0.5984 0.6342 0.6698 0.7051 0.7402 0.7751 0.8097 0.8440 0.8782
0.0000 0.0024 0.0108 0.0247 0.0436 0.0671 0.0944 0.1248 0.1576 0.1921 0.2278 0.2643 0.3012 0.3384 0.3757 0.4129 0.4500 0.4869 0.5235 0.5599 0.5961 0.6320 0.6676 0.7029 0.7380 0.7728 0.8073 0.8416 0.8756
0.0000 0.0023 0.0107 0.0244 0.0430 0.0661 0.0930 0.1229 0.1552 0.1893 0.2246 0.2608 0.2976 0.3347 0.3719 0.4092 0.4459 0.4828 0.5196 0.5561 0.5923 0.6283 0.6639 0.6993 0.7343 0.7690 0.8035 0.8376 0.8715
0.0000 0.0023 0.0106 0.0242 0.0426 0.0654 0.0919 0.1215 0.1534 0.1872 0.2223 0.2583 0.2949 0.3319 0.3692 0.4064 0.4436 0.4806 0.5174 0.5540 0.5903 0.6262 0.6619 0.6977 0.7323 0.7670 0.8013 0.8354 0.8691
0.9121 0.9458 0.9793 1.0125 1.0456 1.0785 1.1112 1.1437 1.1761 l.2082 1.2403 l.2721 1.3038 1.3354 l.3668 1.3981 1.4292 1.4603 1.4912 1.5219 1.5526 1.5831 l.6136 1.6439 1.6741 1.7042 1.7343 1.7642 1.7940 1.8238 1.8534 l.8830
0.9094 0.9429 0.9762 1.0093 1.0422 1.0748 1.1072 1.139..t 1.1715 l.2033 1.2350 1.2665 1.2978 1.3290 1.3599 1.3908 1.4215 1.4520 l.4824 1.5127 1.5428 1.5728 1.6027 1.6324 1.6621 l.6916 1.7210 1.7503 1.7795 1.8086 1.8376 1.8664
0.9050 0.9382 0.9712 1.0039 1.0364 1.0685 1.1005 1.1321 l.1636 1.1948 1.2258 1.2566 1.2871 1.3175 1.3477 1.3776 l.4074 1.4371 1.4665 1.4958 1.5249 1.5538 1.5826 1.6112 1.6397 1.6681 1.6963 1.7244 1.7523 1.7801 1.8078 1.8354
0.9025 0.9355 0.9684 1.0009 1.0331 1.0650 1.0967 1.1281 l.1592 1.1901 1.2207 1.2511 1.2812 1.3112 1.3409 1.3704 1.3997 1.4288 1.4577 1.4865 1.5150 1.5436 1.5716 1.5996 1.6275 1.6552 1.6828 1.7102 1.7375 1.7646 1.7916 1.8184
0.9002 0.9332 0.9660 0.9984 1.0305 1.0623 1.0938 1.1250 l.1560 1.1867 1.2172 1.2474 1.2774 1.3071 1.3367 1.3660 1.3951 1.4239 1.4526 1.4811 1.5094 J.5375 1.5655 1.5933 1.6209 1.6483 1.6756 1.7027 1.7297 1.7565 1.7832 1.8097
O.OW 0.0903 0.1193 0.1507 0.1840 0.2187 0.254-1 0.2909 0.3278 0.3650 0.4023 0.4395 0.4766 0.5135 0.5501 0.586-t 0.6224 0.6580 0.6934 0.7284 0.7630 0.7974 0.8313 0.8650
0.8983 0.9312 0.9639 0.9963 1.0283 1.0600 1.0915 1.1227 1.1536 1.1842 1.2146 1.244"7 1.2746 1.3043 1.3337 1.3629 1.3919 1.4201.4493 1.4776 l.505S 1.5338 1.5617 1.5893 1.6168 l.6441 1.6712 1.6982 1.7251 1.7518 1.7783 1.80-17
0.8968 0.9297 0.9623 0.99-16 1.0266 1.0583 1.0697 1.1208 1.1517 1.1823 1.2126 1.2426 1.2724 1.3020 1.3314 1.3605 1.3894 1.4181 1.4466 1.4748 1.5029 1.5308 1.5585 1.5861 1.6134 1.6406 1.6677 1.6945 1.7212 1.7478 1.7742 1.8005
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TABLE C.2(k) Extended Sukkar-Cornell _Integral for Bottom-hole Pressure Calculation P,
e
e
0.20 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00
14.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00 25.50 26.00 26.50 27.00 27.50 28.00 28.50 29.00 29.50 30.00
Reduced Temperature for 8 = 50.0 1.6 1.7 1.8 1.9 2.0
1.1
1.2
1.3
1.4
1.5
0.0000 0.0023 0.0121 0.0328 0.0649 0.0997 0.1350 0.1703 0.2057 0.2410 0.2763 0.3116 0.3469 0.3821 0.4173 0.4525 0.4876 0.5227 0.5577 0.5927 0.6277 0.6626 0.6974 0.7323 0.7670 0.8018 0.8365 0.8712 0.9059
0.0000 0.0023 0.0111 0.0276 0.0524 0.0835 0.1173 0.1521 0.1873 0.2226 0.2579 0.2933 0.3285 0.3638 0.3990 0.4341 0.4692 0.5042 0.5391 0.5739 0.6087 0.6435 0.6781 0.7127 0.7473 0.7818 0.8163 0.8507 0.8850
0.0000 0.0022 0.0107 0.0257 0.0474 0.0750 0.1066 0.1402 0.1749 0.2101 0.2454 0.2807 0.3161 0.3513 0.3865 0.4216 0.4567 0.4916 0.5264 0.5612 0.5959 0.6304 0.6649 0.6994 0.7337 0.7680 0.8022 0.8363 0.8704
0.0000 0.0022 0.0104 0.0246 0.0447 0.0702 0.0998 0.1322 0.1660 0.2008 0.2359 0.2712 0.3066 0.3419 0.3772 0.4123 0.4474 0.4823 0.5171 0.5518 0.5864 0.6209 0.6553 0.6896 0:1237 0.7578 0.7917 0.8256 0.8594
0.0000 0.0022 0.0102 0.0238 0.0430 0.0670 0.0951 0.1261 0.1591 0.1933 0.2281 0.2632 0.2985 0.3339 0.3692 0.4044 0.4395 0.4745 0.5093 0.5440 0.5786 0.6130 0.6473 0.6815 0.7155 0.7494 0.7832 0.8169 0.8504
0.0000 0.0022 0.0100 0.0233 0.0418 0.0650 0.0921 0.1222 0.1545 0.1882 0.2227 0.2577 0.2929 0.3282 0.3636 0.3989 0.4340 0.4690 0.5039 0.5386 0.5732 0.6076 0.6418 0.6759 0.7099 0.7437 0.7774 0.8109 0.8443
0.8839 0.9172 0.9504 0.9836 1.0166 1.0495 1.0824 1.1151 1.1478 1.1804 1.2129 1.2453 1.2777 1.3100 1.3422 1.3743 1.4064 1.4385 1.4704 1.5024 1.5342 1.5660 1.5978 1.6295 1.6611 1.6927 1.7243 1.7558 1.7872 1.8187 1.8500 1.8814
0.8776 0.9108 0.9438 0.9768 1.0096 1.0423 1.0749 1.1074 1.1398 1.1721 1.2044 1.2365 1.2686 1.3005 1.3324 1.3643 1.3960 1.4277 1.4593 1.4908 1.5223 1.5537 1.5851 1.6164 1.6476 1.6788 1.7100 1.7410 1.7721 1.8030 1.8340 1.8649
0.9405 0.9751 1.0097 1.0442 1.0788 1.1133 1.1477 1.1822 1.2167 1.2511 1.2855 1.3199 1.3542 1.3886 1.4229 1.4573 1.4916 1.5259 1.5602 1.5944 1.6287 1.6629 1.6972 1.7314 1.7656 1.7998 1.8340 1.8882 1.9024 1.9366 1.9707 2.0049
0.9193 0.9536 0.9878 1.0220 1.0561 1.0902 1.1243 1.1583 1.1923 1.2263 1.2602 1.2942 1.3280 1.3619 1.3957 1.4295 1.4633 1.4971 1.5308 1.5646 1.5983 1.6319 1.6656 1.6992 1.7329 1.7665 1.8001 1.8337 1.8672 1.9008 1.93-11 1.9678
0.9044 0.9384 0.9722 1.0061 1.0399 1.0736 1.1073 1.1409 1.1745 1.2081 1.2416 1.2751 1.3085 1.3419 1.3753 1.4086 1.4419 1.4752 1.5084 1.5416 1.5748 1.6079 1.6410 1.6741 1.7072 1.7403 1.7733 1.8063 1.8393 1.8722 1.9052 1.9381
0.8930 0.9266 0.9601 0.9935 1.0269 1.0601 1.0933 1.1266 1.1595 1.1925 1.2254 1.2583 1.2911 1.3238 1.3565 1.3892 1.4218 1.4543 1.4868 1.5193 1.5517 1.5841 1.6164 1.6487 1.6809 1.7131 1.7453 1.7775 1.8096 1.8616 1.8787 1.9057
0.0000 0.0021 0.0099 0.0229 0.0409 0.0634 0.0897 0.1191 0.1507 0.1839 0.2181 0.2529 0.2880 0.3233 0.3587 0.3940 0.4292 0.4643 0.4992 0.5340 0.5685 0.6029 0.6372 0.6712 0.7051 0.7388 0.7724 0.8058 0.8391
0.8722 0.9051 0.9379 0.9706 1.0031 1.0355 1.0678 1.0999 1.1320 1.1639 1.1957 1.2274 1.2590 1.2905 1.3219 1.3532 1.3844 1.4155 1.4456 1.4775 1.5084 1.5392 1.5700 1.6006 1.6312 1.6617 1.6922 1.7226 1.7529 1.783 l 1.8133 1.8435
0.0000 0.0021 0.0098 0.0226 0.0402 0.0622 0.0879 0.1167 0.1477 0.1804 0.2143 0.2488 0.2838 0.3190 0.3544 0.3897 0.4250 0.4601 0.4951 0.5299 0.5645 0.5990 0.6332 0.6672 0.7011 0.7347 0.7582 0.8015 0.8347
0.8576 0.9004 0.9331 0.9656 0.9979 1.0301 1.0621 1.0940 1.1258 1.1575 1.1890 1.2204 1.2517 1.2829 1.3140 1.3449 1.3758 1.4066 1.4372 1.4678 1.4983 1.5287 1.5590 1.5892 1.6194 1.6494 1.6794 1.7094 1.7392 1.7690 1.7987 1.8284
0.0000 0.0000 0.0021 0.0021 0.0098 0.0097 0.0224 0.0222 0.0398 0.0395 0.0615 . 0.0608 0.0868 0.0858 0.1151 0.1138 0.1457 0.1440 0.1781 0.1761 0.2117 0.2094 0.2461 0.2436 0.2809 0.2784 0.3161 0.3135 0.3514 0.3488 0.3868 0.3841 0.4221 0.4194 0.4573 0.4547 0.4923 0.4897 0.5271 0.5246 0.5618 0.5593 0.5962 0.5938 0.6305 0.6280 0.6645 0.6621 0.6984 0.6959 0.7320 0.7295 0.7654 0.7629 0.7987 0.7960 0.8317 0.8290
0.8645 0.8972 0.9297 0.9620 0.9941 1.0260 1.0578 1.0894 1.1209 1.1522 1.1834 1.2144 1.2453 1.2761 1.3067 1.3372 1.3676 1.3979 1.4280 1.4581 1.4880 1.5178 1.5476 1.5772 1.6068 1.6362 1.6656 1.6948 1.7240 1.7531 1.7821 1.8111
0.8617 0.8942 0.9265 0.9586 0.9906 1.0223 1.0538 1.0852 1.1164 1.1474 1.1783 1.2090 1.2395 I.2699 1.3001 1.3302 1.3602 1.3900 1.4197 1.4493 1.4788 1.5081 1.5373 1.5664 1.5954 1.6243 1.6531 1.6818 1.7104 1.7389 1.7673 1.7956
2.2
2.4
2.6
0.0000 0.0021 0.0096 0.0220 0.0389 0.0599 0.0844 0.1119 0.1417 0.1734 0.2063 0.2402 0.2747 0.3097 0.3450 0.3803 0.4151 0.4504 0.4855 0.5204 0.5552 0.5897 0.6240 0.6581 0.6919 0.7254 0.7587 0.7917 0.8245
0.0000 0.0021 0.0096 0.0218 0.0385 0.0593 0.0835 0.1100 0.1401 0.1714 0.2040 0.2377 0.2721 0.3069 0.3421 0.3774 0.4128 0.4481 0.4832 0.5182 0.5530 0.5875 0.6219 0.6558 0.6897 0.7232 0.7565 0.7894 0.8221
0.0000 0.0021 0.0095 0.0217 0.0382 0.0587 0.0827 0.1095 0.1387 0.1697 o.202i 0.2356 0.2700 0.3048 0.3399 0.3752 0.4105 0.4458 0.4810 0.5160 0.5508 0.5854 0.6197 0.6537 0.6875 0.7210 0.7542 0.7672 0.8198
0.8570 0.8893 0.9213 0.9531 0.9847 1.0160 1.0471 1.0779 1.1086 1.1390 1.1693 1.1993 1.2292 1.2589 1.2884 1.3177 1.3468 1.3758 1.4046 1.4333 1.4618 1.4902 1.5184 1.5465 1.5744 1.6022 1.6299 1.6574 1.68-19 1.7122 1.7394 1.7664
0.8545 0.8866 0.9185 0.9501 0.9814 1.0125 1.0434 1.0740 1.1043 1.1345 1.1644 1.1941 1.2236 1.2528 1.2819 1.3108 1.3395 1.3680 1.3964 1.4245 1.4525 1.4803 1.5080 1.5355 1.5629 1.5901 1.6172 1.6441 1.6709 1.6976 1.7241 1.7505
0.8521 0.8842 0.9160 0.9475 0.9788 1.0097 1.0405 1.0709 1.1012 1.1312 1.1609 1.1905 1.2198 1.2489 1.2778 1.3065 1.3350 1.3633 1.3914 1.4193 1.4471 1.4747 1.5021 1.5294 1.5565 1.5835 1.6103 1.6389 1.6634 1.6898 1.7160 1.7421
2.8
3.0
0.0000 0.0000 0.0021 0.0021 0.0095 0.0095 0.0216 0.0215 0.0380 0.0378 0.0583 0.0579 0.0820 0.0814 0.1085 0.1078 0.1375 0.1365 0.1683 0.1671 0.2006 0.1993 0.2339 0.2326 0.2681 0.2667 0.3029 0.3014 0.3380 0.3365 0.3733 0.3718 0.4086 0.4071 0.4439 0.4424 0.4791 0.4777 0.5142 0.5127 0.5490 0.5475 0.5835 0.5821 0.6179 0.6184 0.6519 0.6505 0.6857 0.6842 0.7192 0.7177 0.7523 0.7509 0.7852 0.7838 0.8178 0.8183
0.8502 0.8822 0.9139 0.9454 0.9766 1.0075 1.0382 1.0686 1.0988 1.1287 1.1584 1.1878 1.2171 1.2461 1.2749 1.3035 1.3319 1.3601 1.3881 1.4160 1.4436 1.4711 1.4984 1.5256 1.5526 1.5794 1.6061 1.6328 1.6590 1.6853 1.7114 1.7373
0.8486 0.8806 0.9123 0.9438 0.9749 1.0058 1.0364 1.0668 1.0969 1.1268 1.1564 1.1858 1.2149 1.2439 1.2726 1.3011 1.3295 1.3576 1.3855 1.4133 1.4408 1.4682 1.4954 1.5225 1.5494 1.5761 1.6027 1.6291 1.6553 1.6815 1.7075 1.7333
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TABLE C.2(1) Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation P,
ReducectTemperature for B = 60.0 1.6 1.7 2.0 1.6 1.9
1.1
1.2
1.3
1.4
1.5
0.0000 0.0019 0.0101 0.0277 0.0559 0.0870 0.1189 0.1509 0.1831 0.2153 0.2475 0.2798 0.3120 0.3443 0.3766 0.4088 0.4411 0.4734 0.5056 0.5378 0.5701 0.6023 0.6344 0.6666 0.6987 0.7309 0.7630 0.7951 0.8272
0.0000 0.0019 0.0093 0.0232 0.0443 0.0715 0.1014 0.1325 0.1642 0.1962 0.2283 0.2606 0.2928 0.3251 0.3574 0.3896 0.4219 0.4541 0.4863 0.5185 0.5507 0.5828 0.6149 0.6469 0.6790 0.7110 0.7429 0.7749 0.8068
0.0000 0.0019 0.0089 0.0215 0.0399 0.0637 0.0913 0.1211 0.1521 0.1837 0.2157 0.2479 0.2801 0.3124 0.3446 0.3769 0.4091 0.4413 0.4735 0.5056 0.5377 0.5698 0.6018 0.6337 0.6656 0.6975 0.7293 0.7611 0.7929
0.0000 0.0018 0.0087 0.0206 0.0376 0.0594 0.0851 0.1135 0.1435 0.1745 0.2062 0.2382 0.2703 0.3026 0.3348 0.3671 0.3994 0.4316 0.4637 0.4958 0.5279 0.5599 0.5718 0.6237 0.6555 0.6872 0.7189 0.7505 0.7820
0.0000 0.0018 0.0085 0.0200 0.0361 0.0566 0.0808 0.1079 0.1369 0.1672 0.1984 0.2301 0.2620 0.2942 0.3264 0.3587 0.3910 0.4232 0.4554 0.4875 0.5195 0.5515 0.5833 0.6151 0.6469 0.6785 0.7101 0.7415 0.7730
0.0000 0.0018 0.0084 0.0195 0.0351 0.0549 0.0781 0.1043 0.1326 0.1624 0.1931 0.2245 0.2563 0.2884 0.3206 0.3529 0.3851 0.4174 0.4496 0.4817 0.5137 0.5457 0.5775 0.6093 0.6409 0.6725 0.7040 0.7354 0.7667
0.0000 0.0018 0.0083 0.0192 0.0343 0.0535 0.0760 0.1014 0.1291 0.1583 0.1887 0.2198 0.2515 0.2834 0.3156 0.3478 0.3801 0.4123 0.4445 0.4767 0.5087 0.5407 0.5725 0.6042 0.6359 0.6674 0.6986 0.7301 0.7613
0.0000 0.0018 0.0082 0.0189 0.0338 0.0524 0.0745 0.0993 0.1263 0.1551 0.1850 0.2158 0.2472 0.2791 0.3111 0.3433 0.3756 0.4079 0.4401 0.4722 0.5043 0.5363 0.5581 0.5998 0.6314 0.6629 0.6943 0.7255 0.7566
0.0000 0.0018 0.0081 0.0188 0.0334 0.0518 0.0734 0.0979 0.1245 0.1529 0.1826 0.2132 0.2444 0.2761 0.3081 0.3403 0.3725 0.4048 0.5370 0.4692 0.5013 0.5333 0.5651 0.5988 0.6284 o.Q599. 0.6912 0.7224 0.7534
14.50 0.8592 15.00 0.8913 15.50 0.9233 16.00 0.9554 16.50 0.9874 17.00 . 1.0194 17.50 1.0514 18.00 1.0834 1.1153 e·tS.50 19.00 1.1473 19.50 1.1792 20.00 l.2112 20.50 1.2431 21.00 1.2750 21.50 1.3069 22.00 1.3388 22.50 1.3707 23.00 1.4026 23.50 1.4344 24.00 l .4663 24.50 1.4982 25.00 1.5300 25.50 1.5619 26.00 1.5937 26.50 1.6255 27.00 1.6574 27.50 1.6892 28.00 1.7210 28.50 1.7528 29.00 1.7846 29.50 1.8164 30.00 1.8482
0.8387 0.8705 0.9024 0.9342 0.9660 0.9977 1.0295 1.0612 1.0929 1.1246 1.1562 1.1879 1.2195 1.2511 1.2827 1.3143 1.3458 1.3774 1.4089 l.4404 1.4719 1.0534 1.5349 1.5664 1.5978 1.6292 1.6607 1.6921 l.7235 1.7549 1.7863 1.8177
0.8246 0.8562 0.8879 0.9195 0.9510 0.9826 1.0141 1.0455 1.0769 l.!083 l.!397 l.!711 1.2024 1.2337 1.2650 1.2962 1.3274 1.3586 1.3898 1.4210 1.4521 1.4832 1.5143 1.5454 1.5765 1.6075 1.6385 1.6695 1.7005 1.7315 1.7625 1.793.J
0.8135 0.8449 0.8763 0.9076 0.9389 0.9701 1.0012 1.0323 1.0634 1.0944 1.1253 1.1562 1.1871 l.2179 1.2487 l.2795 1.3102 1.3409 1.3715 1.4021 1.4327 1.4632 1.4937 1.5242 1.5547 1.5851 1.6155 1.6459 l.6762 1.7065 1.7168 1.7671
0.8043 0.8355 0.8667 0.8978 0.9288 0.9598 0.9907 1.0215 1.0523 1.0830 1.1137 1.1443 1.1748 1.2053 1.2357 1.2661 1.2964 1.3267 1.3569 1.3871 1.4173 J.4474 1.4774 1.5075 1.5374 1.5674 1.5973 1.6272 1.6570 J.6868 1.7166 1.7463
0.7979 0.8291 0.8601 0.8911 0.9219 0.9527 0.9835 1.0141 1.0447 1.0752 1.1056 1.1360 1.1663 1.1965 1.2267 1.2568 1.2869 1.3169 1.3469 1.3768 1.4066 1.4364 1.4662 1.4959 1.5255 1.5552 1.5847 1.6143 1.6438 1.6732 1.7026 1.7320
0.7924 0.8233 0.8542 0.8850 0.9156 0.9462 0.9767 1.0070 1.0373 1.0675 1.0976 1.1277 1.1576 1.1875 1.2173 1.2470 l.2766 1.3062 1.3357 1.3652 1.3945 1.4238 1.4531 1.4823 1.5114 1.5405 1.5695 1.5985 1.6274 1.6563 1.6851 1.7139
0.7876 0.8184 0.8492 0.8798 0.9103 0.9408 0.9711 1.0013 1.0313 1.0613 1.0912 1.1210 1.1507 1.1803 1.2099 1.2393 1.2667 1.2979 1.3271 1.3563 1.3853 1.4143 1.4432 1.4721 1.5008 1.5295 1.5582 1.5868 1.6153 1.6438 1.6722 1.7005
0.7843 0.8151 0.8457 0.8762 0.9065 0.9368 0.9668 0.9968 1.0267 1.0564 1.0860 1.1155 1.1449 1.1741 1.2033 l.2324 l.2614 1.2902 1.3190 1.3477 1.3763 1.4048 1.4332 1.4616 1.4898 1.5180 1.5461 1.5742 1.6021 1.6300 l.6579 1.6856
e
0.20 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 1.00 11.50 12.00 12.50 13.00 13.50 14.00
2.2
2.4
2.6
2.8
3.0
0.0000 0.0018 0.0081 0.0186 0.0331 0.0512 0.0726 0.0966 0.1229 0.1510 0.1804 0.2108 0.2419 0.2735 0.3
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TABLE C.2(m) Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation P,
e
e
0.20 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 \1.00 ll.50 12.00 12.50 13.00 13.50 14.00
Reduced·Temperature for 8 = 70.0 1.6 1.7 1.8 1.9 2.0
1.1
1.2
1.3
1.4
1.5
0.0000 0.0017 0.0087 0.0240 0.0491 0.0772 0.1063 0.1356 0.1651 0.1947 0.2243 0.2540 0.2838 0.3135 0.3433 0.3732 0.4030 0.4328 0.4627 0.4926 0.5225 0.5523 0.5822 0.6121 0.6420 0.6718 0.7017 0. 7316 0.7615
0.0000 0.0016 0.0080 0.0199 0.0385 0.0625 0.0894 0.1175 0.1464 0.1756 0.2050 0.2347 0.2644 0.2941 0.3239 0.3538 0.3836 0.4135 0.4434 0.4733 0.5031 0.5330 0.5629 0.5927 0.6226 0.6524 0.6822 0.7121 0.7419
0.0000 0.0016 0.0077 0.0185 0.0345 0.0554 0.0799 0.1066 0,1346 0.1634 0.1926 0.2221 0.2517 0.2815 0.3113 0.341l 0.3710 0.4009 0.4307 0.4606 0.4905 0.5203 0.5502 0.5800 0.6098 0.6396 0.6693 0.6991 0.7288
0.0000 0.0016 0.0074 0.0177 0.0325 0.0515 0.0742 0.0994 0.1264 0.1545 0.1833 0.2125 0.2420 0.2716 0.3014 0.3312 0.3611 0.3909 0.4208 0.4507 0.4805 0.5104 0.5402 0.5700 0.5997 0.6294 0.6591 0.6887 0.7183
0.0000 0.0016 0.0073 0.0172 0.0312 0.0490 0.0703 0.0943 0.1202 0.1475 0.1756 0.2045 0.2337 0.2632 0.2929 0.3226 0.3525 0.3824 0.4122 0.4421 0.4720 0.5018 0.5316 0.5613 0.5910 0.6207 0.6503 0.6798 0.7093
0.0000 0.0015 0.0072 0.0168 0.0303 0.0475 0.0679 0.0910 0.1162 0.1429 0.1706 0.1991 0.2281 0.2574 0.2870 0.3167 0.3465 0.3764 0.4063 0.4362 0.4660 0.4958 0.5256 0.5553 0.5850 0.6146 0.6442 0.6737 0.7031
0.0000 0.0015 0.0071 0.0165 0.0296 0.0462 0.0660 0.0884 0.1129 0.1391 0.1664 0.1946 0.2233 0.2525 0.2820 0.3116 0.3414 0.3713 0.4011 0.4310 0.4609 0.4907 0.5204 0.5502 0.5798 0.6094 0.6389 0.6683 0.6977
0.0000 0.0000 0.0015 0.0015 0.0070 0.0070 0.0163 0.0161 0.0291 0.0288 0.0453 0.0448 0.0646 0.0637 0.0864 0.0851 0.1104 0.1087 0.1360 0.1340 0.1629 0.1606 0.1907 0.1881 0.2192 0.2164 0.2482 0.2453 0.2775 0.2745 0.3071 0.3040 0.3368 0.3337 0.3667 0.3635 0.3965 0.3934 0.4264 0.4233 0.4563 0.4531 0.4861 0.4830 0.5159 0.5127 0.5456 0.5424 0.5i52 0.5721 0.60.t7 0.6016 0.6.3-i2 0.63II 0.6636 0.6604 0.6929 0.6897
0.0000 0.0015 0.0070 0.0160 0.0285 0.0443 0.0629 0.0840 0.1073 0.1322 0.1585 0.1859 0.2140 0.2427 0.2718 0.3013 0.3309 0.3607 0.3905 0.4204 0.4503 0.4801 0.5099 0.5396 0.5692 0.5987 0.6282 0.6575 0.6867
0.7717 0.8014 0.8312 0.8609 0.8907 0.9204 0.9501 0.9798 1.0095 1.0392 1.0689 1.0985 1.1282 1.1578 1.1874 1.2170 1.2466 1.2762 1.3058 1.3354 l.3650 1.3946 1.4241 1.4537 1.4832 1.5127 1.5423 1.5718 1.6013 1.6308 1.6603 1.6898
0.7585 0.7881 0.8178 0.8474 0.8770 0.9066 0.9362 0.9657 0.9953 1.0248 1.0543 1.0837 1.1132 1.1426 1.1721 1.2015 1.2309 1.2602 1.2896 1.3190 1.3483 1.3776 1.4069 1.4362 1.4655 1.4948 1.5240 1.5533 1.5825 1.6117 1.6410 1.6702
0.7479 0.7774 0.8069 0.8363 0.8658 0.8951 0.9245 0.9538 0.9831 1.0123 1.0415 1.0707 1.0999 1.1290 1.1581 1.1871 1.2162 1.2452 1.2742 1.3031 1.3321 1.3610 1.3899 1.4187 1.4476 1.4764 1.5052 1.5340 1.5627 1.5915 1.6202 1.6489
0.7388 0.7682 0.7976 0.8269 0.8662 0.8854 0.9146 0.9437 0.9728 1.0018 1.0308 1.0597 1.0886 1.1175 1.1463 1.1751 1.2039 1.2326 1.2613 1.2899 l.3185 1.3471 1.3757 1.4042 1.4327 1.4611 1.4895 1.5179 1.5463 1.5747 1.6030 1.6313
0.7325 0.7619 0.7911 0.8203 0.8495 0.8786 0.9076 0.9366 0.9656 0.9945 1.0233 1.0521 1.0808 1.1095 1.1381 1.1667 1.1953 1.2238 1.2522 1.2807 l.3090 1.3374 1.3657 1.3940 1.4222 1.4504 1.4786 1.5067 1.5348 1.5629 1.5909 1.6189
0.7270 0.7562 0.7854 0.8145 0.8435 0.8724 0.9013 0.9300 0.9588 0.9874 1.0160 1.0445 1.0730 1.1014 1.1297 1.1580 1.1862 1.2144 1.2425 1.2706 1.2986 1.3265 1.3544 1.3823 1.4101 1.4379 1.4656 1.4933 1.5209 1.5485 1.5761 1.6036
0. i222 O.i513 O.iS04 0.8094 0.8363 0.8671 0.8958 0.9245 0.9530 0.9815 1.0099 1.0383 1.0665 1.0947 LJ229 1.1509 1.1789 1.2069 1.23-H 1.2625 r.:903 1.3180 J.3.156 1.3';32 1.4007 1.4282 l.-l556 l.-l829
0.7158 0.7448 0.7737 0.8025 0.8311 0.8597 0.8881 0.9164 0.9446 0.9727 1.0007 1.0286 1.0564 1.0841 1.1116 1.1391 1.1665 1.1938 1.2210 1.2482 l.2752 1.3022 1.3290 1.3658 1.3625 1.4092 1.4357 1.4622 1.4886 1.5150 1.5412 1.5675
14.50 0.7913 15.00 0.8212 15.50 0.8510 16.00 0.8809 16.50 0.9107 17.00 · o.9406 17.50 0.9704 18.00 1.0002 18.50 1.0300 19.00 1.0599 19.50 1.0897 20.00 1.1195 20.50 1.1493 21.00 1.1791 21.50 1.2089 22.00 1.2387 22.50 1.2685 23.00 1.2982 23.50 1.3280 24.00 1.3578 24.50 l.3876 25.00 1.4173 25.50 1.4471 26.00 1.4769 26.50 1.5066 27.00 1.5364 27.50 1.5661 28.00 1.5959 28.50 1.6256 29.00 1.6554 29.50 1.6851 30.00 1.7148
1.5102 1.5375 1.55-n 1.5919
0.7189 0.7479 0.7769 0.8058 0.8345 0.8632 0.8918 0.9203 0.9486 0.9769 1.0051 1.0332 1.0612 1.0892 1.1170 1.1448 1.1724 1.2000 1.2276 1.2550 1.2824 1.3097 1.3369 1.3641 1.3912 1.4182 1.4452 1.4721 1.4989 1.5257 1.5524 1.5791
2.2
2.4
2.6
2.8
3.0
0.0000 0.0015 0.0069 0.0158 0.0281 0.0435 0.0618 0.0825 0.1054 0.1299 0.1827 0.2106 0.2390 0.2680 0.2973 0.3262 0.3560 0.3858 0.4157 0.4456 0.4754 0.5052 0.5349 0.5645 0.5940 0:6234 0.6527 0.6818
0.0000 0.0015 0.0069 0.0157 0.0278 0.0431 0.0611 0.0815 0.1040 0.1282 0.1538 0.1805 0.2081 0.2363 0.2652 0.2944 0.3239 0.3536 0.3834 0.4133 0.4432 0.4730 0.5028 0.5325 0.5621 0.5916 0.6210 0.6502 0.6793
0.0000 0.0015 0.0068 0.0156 0.0276 0.0426 0.0604 0.0806 0.1029 0.1268 0.1522 0.1787 0.2061 0.2343 0.2630 0.2922 0.3217 0.3514 0.3812 0.4110 0.4409 0.4708 0.5005 0.5303 0.5599 0.5893 0.6187 0.6479 0.6770
0.0000 0.0015 0.0068 0.0155 0.0274 0.0423 0.0599 0.0798 0.1018 0.1256 0.1508 0.1772 0.2045 0.2326 0.2613 0.2904 0.3198 0.3495 0.3793 0.4092 0.4390 0.4689 0.4987 0.5284 0.5580 0.5875 0.6168 0.6460 0.6750
0.0000 0.0015 0.0068 0.0154 0.0273 0.0420 0.0595 0.0792 0.1010 0.1246 0.1497 0.1760 0.2032 0.2313 0.2599 0.2890 0.3184 0.3481 0.3779 0.4077 0.4376 0.4675 0.4972 0.5270 0.5566 0.5860 0.6154 0.6445 0.6736
0.7108 0.7397 0.7684 0.7969 0.8254 0.8537 0.8818 0.9098 0.9377 0.9654 0.9930 1.0204 1.0478 1.0749 1.1020 1.1289 1.1558 1.1825 1.2090 1.2355 1.2619 1.2881 1.3142 1.3403 1.3662 l.3920 1.4178 1.4434 1.4690 1.4944 1.5198 1.5450
0.7062 0.7370 0.7656 0.7941 0.8224 0.8505 0.8765 0.9064 0.9340 0.9615 0.9889 1.0181 1.0432 1.0701 1.0968 1.1235 1.1500 1.1763 1.2026 1.2287 1.2546 1.2805 1.3062 1.3318 l.3573 l.3827 1.4079 1.4331 1.4581 1.4831 1.5079 1.5327
0.7059 0.7346 0.7632 0.7916 0.8198 0.8479 0.8758 0.9036 0.93II 0.9586 0.9858 1.0129 1.0398 1.0666 1.0933 1. II98 1.1461 1.1723 1.1984 1.2243 1.2501 1.2758 1.3013 1.3267 1.3520 1.3772 1.4023 1.4272 1.4520 1.4768 l.5014 1.5259
0.7039 0.7326 0.7612 0.7898 0.8178 0.8458 0.8737 0.9014 0.9289 0.9563 0.9835 1.0105 1.0374 1.0641 1.0907 1.1171 1.1434 1.1695 1.1955 1.2214 1.2471 1.2727 1.2981 l.3235 l.3487 1.3738 1.3987 1.4236 1.4483 1.4729 1.4974 1.5218
0.1558
0.7024 0.7311 0.7597 0.7880 0.8162 0.8442 0.8721 0.8997 0.9272 0.9545 0.9817 1.0087 1.0355 1.0622 1.0887 1.1151 1.1413 1.1674 1.1933 1.2191 1.2447 1.2702 1.2956 1.3209 1.3460 1.3710 1.3759 1.4206 1.4452 1.4698 1.4942 1.5165
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