Document Text Contents
Page 1
•
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/CONSULA T ANT/TRAINING/SIAlA/GASOUTLINES.doc
Page 2
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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 .... ..1r
-- The course has been :divided into 8 main chaplers as follows: ....
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/CONSULA T ANTITRAINING/SIALA/GASOUTLINES.doc 2
Page 107
Natural Gas Reservoir Engineering Dr. A. A. Urayet
Example Problem 4. 3
(Effect of Condensates on the Material Balance Calculations)
A gas reservoir was put on production in Nov.9, 1993. The following data was
available after one year of production:
Initial pressure (9/11/1993)
reservoir temperature
gas specific gravity
Zi
reservoir pressure (9/11/1994)
cumulative gas produced
Z factor at P= 4120 psia
cumulative hydr. liquids production
API of hydrocarbon liquids
Cumulative water production
4300 psi a
228 °F
0.71
0.945 0. ~ 'Z. t't l\-
4120 psia
23.5 MMM scf
0.936 t:J • 9 I 'r'+
865,000 STB
61
14,200 STBW
Assuming a Volumetric reservoir performance, calculate the Gas Initially In
Place,
a) assuming only gas is being produced
b) taking into consideration the liquid condensates, and assuming that all the
water produced is condensing from the water vapor
AAU/GASNOLUMETRICS.doc 33
Page 108
Natural Gas Reservoir Engineering Dr. A. A. Urayet
Solution
a) Calculate the Gas Initially In Place using Eq (4.19-A) as follows:
GllP = 719.8 MMMscf
~' ''t>"t
b) Correct the value of
9
. 4120 _ 4300 -(4300)23.5x10
0.936 0.945 0.945 G
0 ·~16r4' tJ•9l1-'-lf.f c·7t.'Pl''f
as follows:
G P(corrected) = G p(sep) + N p.(3003)(1.03-y0 ) + Wvp·(7390)
Gp (corrected) =
= 23.5xl09 + (865,000).(3003)(1.03- l 4t.5 + (14,200).(7390) = 24.37lxl09
131.5 + 61
- Calculate the Gas Initially In Place, using Eq (4.19-A), as follows:
p P: P:
-=-' - (-' ).GP
z zi G.zi
4120 = 4300 -(4300)24.371x10
9
0.936 0.945 0.945 G
~ O•'JJ,r't c5•'i2.l-'t-<t- 0·9'Z1-'t-c+
G = 746.49 MMM scf
t "l't. 't >?..
AAU/GASNOLUMETRICS.doc 34
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Page 213
TABLE C.2(1)
Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation
P, ReducectTemperature for B = 60.0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.6 1.9 2.0 2.2 2.4 2.6 2.8 3.0
0.20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.50 0.0019 0.0019 0.0019 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0017 0.0017 0.0017
1.00 0.0101 0.0093 0.0089 0.0087 0.0085 0.0084 0.0083 0.0082 0.0081 0.0081 0.0080 0.0080 0.0080 0.0079 0.0079
1.50 0.0277 0.0232 0.0215 0.0206 0.0200 0.0195 0.0192 0.0189 0.0188 0.0186 0.0184 0.0183 0.0181 0.0181 0.0180
2.00 0.0559 0.0443 0.0399 0.0376 0.0361 0.0351 0.0343 0.0338 0.0334 0.0331 0.0326 0.0323 0.0321 0.0319 0.0317
2.50 0.0870 0.0715 0.0637 0.0594 0.0566 0.0549 0.0535 0.0524 0.0518 0.0512 0.0504 0.0499 0.0494 0.0490 0.0487
~ 3.00 0.1189 0.1014 0.0913 0.0851 0.0808 0.0781 0.0760 0.0745 0.0734 0.0726 0.0714 0.0705 0.0698 0.0692 0.0687
3.50 0.1509 0.1325 0.1211 0.1135 0.1079 0.1043 0.1014 0.0993 0.0979 0.0966 0.0950 0.0989 0.0928 0.0920 0.0913
4.00 0.1831 0.1642 0.1521 0.1435 0.1369 0.1326 0.1291 0.1263 0.1245 0.1229 0.1209 0.1194 0.1181 0.1170 0.1161 1:1:1
4.50 0.2153 0.1962 0.1837 0.1745 0.1672 0.1624 0.1583 0.1551 0.1529 0.1510 0.1485 0.1466 0.1451 0.1438 0.1428 <::> -5.00 0.2475 0.2283 0.2157 0.2062 0.1984 0.1931 0.1887 0.1850 0.1826 0.1804 0.1775 0.1783 0.1736 0.1721 0.1709 ~
5.50 0.2798 0.2606 0.2479 0.2382 0.2301 0.2245 0.2198 0.2158 0.2132 0.2108 0.2075 0.2051 0.2032 0.2016 0.2003 ~
6.00 0.3120 0.2928 0.2801 0.2703 0.2620 0.2563 0.2515 0.2472 0.2444 0.2419 0.2383 0.2357 0.2337 0.2320 0.2306 ~
6.50 0.3443 0.3251 0.3124 0.3026 0.2942 0.2884 0.2834 0.2791 0.2761 0.2735 0.2697 0.2670 0.2648 0.2630 0.2616
;;-
7.00 0.3766 0.3574 0.3446 0.3348 0.3264 0.3206 0.3156 0.3111 0.3081 0.3<154 0.3015 0.2986 0.2964 0.2946 0.2932 ~
7.50 0.4088 0.3896 0.3769 0.3671 0.3587 0.3529 0.3478 0.3433 0.3403 0.3375 0.3336 0.3306 0.3284 0.3265 0.3251 ~
8.00 0.4411 0.4219 0.4091 0.3994 0.3910 0.3851 0.3801 0.3756 0.3725 0.3697 0.3651 0.3628 0.3605 0.3586 0.3572 !;
8.50 0.4734 0.4541 0.4413 0.4316 0.4232 0.4174 0.4123 0.4079 0.4048 0.4020 0.3976 0.3951 0.3928 0.3909 0.3894 ~
9.00 0.5056 0.4863 0.4735 0.4637 0.4554 0.4496 0.4445 0.4401 0.5370 0.4343 0.4297 0.4273 0.4251 0.4231 0.4217
9.50 0.5378 0.5185 0.5056 0.4958 0.4875 0.4817 0.4767 0.4722 0.4692 0.4665 0.4619 0.4596 0.4573 0.4554 0.4539
10.00 0.5701 0.5507 0.5377 0.5279 0.5195 0.5137 0.5087 0.5043 0.5013 0.4985 0.4940 0.4917 0.4894 0.4875 0.4861
e 10.50 0.6023 0.5828 0.5698 0.5599 0.5515 0.5457 0.5407 0.5363 0.5333 0.5305 0.5260 0.5237 0.5215 0.5196 0.5181 1.00 0.6344 0.6149 0.6018 0.5718 0.5833 0.5775 0.5725 0.5581 0.5651 0.5624 0.5579 0.5556 0.5534 0.5515 0.5500
11.50 0.6666 0.6469 0.6337 0.6237 0.6151 0.6093 0.6042 0.5998 0.5988 0.5942 0.5896 0.5873 0.5851 0.5832 0.5818
12.00 0.6987 0.6790 0.6656 0.6555 0.6469 0.6409 0.6359 0.6314 0.6284 0.6257 0.6212 0.6189 0.6166 0.6148 0.6133
12.50 0.7309 0.7110 0.6975 0.6872 0.6785 0.6725 0.6674 0.6629 o.Q599. 0.6571 0.6526 0.6503 0.6480 0.6461 0.6446
13.00 0.7630 0.7429 0.7293 0.7189 0.7101 0.7040 0.6986 0.6943 0.6912 0.6884 0.6838 0.6815 0.6792 0.6773 0.6758
13.50 0.7951 0.7749 0.7611 0.7505 0.7415 0.7354 0.7301 0.7255 0.7224 0.7196 0.7149 0.7125 0.7101 0.7032 0.7067
14.00 0.8272 0.8068 0.7929 0.7820 0.7730 0.7667 0.7613 0.7566 0.7534 0.7505 0.7457 0.7482 0.7409 0.7389 0.7374
14.50 0.8592 0.8387 0.8246 0.8135 0.8043 0.7979 0.7924 0.7876 0.7843 0.7813 0.7764 0.7738 0.7714 0.7694 0.7679
15.00 0.8913 0.8705 0.8562 0.8449 0.8355 0.8291 0.8233 0.8184 0.8151 0.8120 0.8069 0.8042 0.8017 0.7997 0.7982
15.50 0.9233 0.9024 0.8879 0.8763 0.8667 0.8601 0.8542 0.8492 0.8457 0.8425 0.8371 0.8343 0.8318 0.8298 0.8282
16.00 0.9554 0.9342 0.9195 0.9076 0.8978 0.8911 0.8850 0.8798 0.8762 0.8728 0.8672 0.8643 0.8617 0.8596 0.8580
16.50 0.9874 0.9660 0.9510 0.9389 0.9288 0.9219 0.9156 0.9103 0.9065 0.9030 0.8971 0.8940 0.8914 0.8892 0.8876
17.00 . 1.0194 0.9977 0.9826 0.9701 0.9598 0.9527 0.9462 0.9408 0.9368 0.9331 0.9269 0.9236 0.9208 0.9186 0.9170
17.50 1.0514 1.0295 1.0141 1.0012 0.9907 0.9835 0.9767 0.9711 0.9668 0.9630 0.9564 0.9529 0.9501 0.9478 0.9461
18.00 1.0834 1.0612 1.0455 1.0323 1.0215 1.0141 1.0070 1.0013 0.9968 0.9928 0.9858 0.9820 0.9791 0.9768 0.9751
e·tS.50 1.1153 1.0929 1.0769 1.0634 1.0523 1.0447 1.0373 1.0313 1.0267 1.0224 1.0150 1.0!10 1.0080 1.0056 1.0038
;;;2 1.1473 1.1246 l.!083 1.0944 1.0830 1.0752 1.0675 1.0613 1.0564 1.0519 1.0440 1.0398 1.0366 1.0342 1.0324 19.00
~
19.50 1.1792 1.1562 l.!397 1.1253 1.1137 1.1056 1.0976 1.0912 1.0860 1.0812 1.0728 1.0683 1.0651 1.0626 1.0607 ti:>
20.00 l.2112 1.1879 l.!711 1.1562 1.1443 1.1360 1.1277 1.1210 1.1155 1.1104 1.1015 1.0967 1.0933 1.0908 1.0889 ~ 20.50 1.2431 1.2195 1.2024 1.1871 1.1748 1.1663 1.1576 1.1507 1.1449 1.1395 1.1301 1.1250 1.1214 1.1188 1.1168
21.00 1.2750 1.2511 1.2337 l.2179 1.2053 1.1965 1.1875 1.1803 1.1741 1.1685 1.1584 1.1530 1.1493 l.!466 1.1446
...
t 21.50 1.3069 1.2827 1.2650 1.2487 1.2357 1.2267 1.2173 1.2099 1.2033 1.1974 1.1867 1.1800 1.1770 1.1743 1.1721 Cl..
22.00 1.3388 1.3143 1.2962 l.2795 1.2661 1.2568 1.2470 1.2393 l.2324 1.2261 1.2147 1.2086 1.2046 1.2018 1.1995 g
22.50 1.3707 1.3458 1.3274 1.3102 1.2964 1.2869 l.2766 1.2667 l.2614 1.2547 1.2427 1.2361 l.2319 1.2291 1.2268 ii!
23.00 1.4026 1.3774 1.3586 1.3409 1.3267 1.3169 1.3062 1.2979 1.2902 1.2832 1.2705 1.2635 l.2592 l.2562 1.2538 ~
23.50 1.4344 1.4089 1.3898 1.3715 1.3569 1.3469 1.3357 1.3190 1.3116 1.2981 l.2908 1.2862 1.2832 1.2807
::::
1.3271
~ 24.00 l .4663 l.4404 1.4210 1.4021 1.3871 1.3768 1.3652 1.3563 1.3477 1.3399 1.3256 1.3179 1.3131 1.3100 1.3074
24.50 1.4982 1.4719 1.4521 1.4327 1.4173 1.4066 1.3945 1.3853 1.3763 1.3681 1.3530 1.3448 1.3399 1.3366 1.3340 s. <::>
25.00 1.5300 1.0534 1.4832 1.4632 J.4474 1.4364 1.4238 1.4143 1.4048 1.3962 1.3803 1.3716 1.3664 1.3631 1.3604 Q.
25.50 1.5619 1.5349 1.5143 1.4937 1.4774 1.4662 1.4531 1.4432 1.4332 1.4242 1.4074 1.3983 1.3929 1.3895 1.3867
26.00 1.5937 1.5664 1.5454 1.5242 1.5075 1.4959 1.4823 1.4721 1.4616 1.4521 1.4344 1.4248 1.4192 1.4157 1.4128
...
\Q
26.50 1.6255 1.5978 1.5765 1.5547 1.5374 1.5255 1.5114 1.5008 1.4898 1.4799 1.4613 1.4512 1.4454 1.4417 1.4388
(;.I
27.00 1.6574 1.6292 1.6075 1.5851 1.5674 1.5552 1.5405 1.5295 1.5180 J.5076 1.4881 1.4775 1.4714 1.4677 1.4646
27.50 1.6892 1.6607 1.6385 1.6155 1.5973 1.5847 1.5695 1.5582 1.5461 J.5353 1.5148 1.5036 1.4973 1.4935 1.4903
28.00 1.7210 1.6921 1.6695 1.6459 1.6272 1.6143 1.5985 1.5868 1.5742 1.5628 J.5413 1.5296 1.5231 1.5191 1.5159
28.50 1.7528 l.7235 1.7005 l.6762 1.6570 1.6438 1.6274 1.6153 1.6021 J.5903 1.5678 1.5555 1.5407 1.5447 l .5413
29.00 1.7846 1.7549 1.7315 1.7065 J.6868 1.6732 1.6563 1.6438 1.6300 J.6176 1.5941 1.5813 1.5742 1.5701 1.5666
29.50 1.8164 1.7863 1.7625 1.7168 1.7166 1.7026 1.6851 1.6722 l.6579 1.6449 l.6204 1.6070 1.5997 1.5954 1.5918
30.00 1.8482 1.8177 1.793.J 1.7671 1.7463 1.7320 1.7139 1.7005 1.6856 1.6722 1.6465 1.6325 1.6249 1.6205 1.6168
Page 214
TABLE C.2(m)
Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation
P, Reduced·Temperature for 8 = 70.0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0
0.20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.50 0.0017 0.0016 0.0016 0.0016 0.0016 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015
1.00 0.0087 0.0080 0.0077 0.0074 0.0073 0.0072 0.0071 0.0070 0.0070 0.0070 0.0069 0.0069 0.0068 0.0068 0.0068
1.50 0.0240 0.0199 0.0185 0.0177 0.0172 0.0168 0.0165 0.0163 0.0161 0.0160 0.0158 0.0157 0.0156 0.0155 0.0154
2.00 0.0491 0.0385 0.0345 0.0325 0.0312 0.0303 0.0296 0.0291 0.0288 0.0285 0.0281 0.0278 0.0276 0.0274 0.0273
2.50 0.0772 0.0625 0.0554 0.0515 0.0490 0.0475 0.0462 0.0453 0.0448 0.0443 0.0435 0.0431 0.0426 0.0423 0.0420
3.00 0.1063 0.0894 0.0799 0.0742 0.0703 0.0679 0.0660 0.0646 0.0637 0.0629 0.0618 0.0611 0.0604 0.0599 0.0595
"""' 3.50 0.1356 0.1175 0.1066 0.0994 0.0943 0.0910 0.0884 0.0864 0.0851 0.0840 0.0825 0.0815 0.0806 0.0798 0.0792 'g
4.00 0.1651 0.1464 0,1346 0.1264 0.1202 0.1162 0.1129 0.1104 0.1087 0.1073 0.1054 0.1040 0.1029 0.1018 0.1010
4.50 0.1947 0.1756 0.1634 0.1545 0.1475 0.1429 0.1391 0.1360 0.1340 0.1322 0.1299 0.1282 0.1268 0.1256 0.1246 b:I <:> 5.00 0.2243 0.2050 0.1926 0.1833 0.1756 0.1706 0.1664 0.1629 0.1606 0.1585 0.1558 0.1538 0.1522 0.1508 0.1497 -S' 5.50 0.2540 0.2347 0.2221 0.2125 0.2045 0.1991 0.1946 0.1907 0.1881 0.1859 0.1827 0.1805 0.1787 0.1772 0.1760 Eil 6.00 0.2838 0.2644 0.2517 0.2420 0.2337 0.2281 0.2233 0.2192 0.2164 0.2140 0.2106 0.2081 0.2061 0.2045 0.2032 ~ 6.50 0.3135 0.2941 0.2815 0.2716 0.2632 0.2574 0.2525 0.2482 0.2453 0.2427 0.2390 0.2363 0.2343 0.2326 0.2313 ;--7.00 0.3433 0.3239 0.3113 0.3014 0.2929 0.2870 0.2820 0.2775 0.2745 0.2718 0.2680 0.2652 0.2630 0.2613 0.2599
41 7.50 0.3732 0.3538 0.341l 0.3312 0.3226 0.3167 0.3116 0.3071 0.3040 0.3013 0.2973 0.2944 0.2922 0.2904 0.2890
8.00 0.4030 0.3836 0.3710 0.3611 0.3525 0.3465 0.3414 0.3368 0.3337 0.3309 0.3262 0.3239 0.3217 0.3198 0.3184 ~ 8.50 0.4328 0.4135 0.4009 0.3909 0.3824 0.3764 0.3713 0.3667 0.3635 0.3607 0.3560 0.3536 0.3514 0.3495 0.3481 ~ 9.00 0.4627 0.4434 0.4307 0.4208 0.4122 0.4063 0.4011 0.3965 0.3934 0.3905 0.3858 0.3834 0.3812 0.3793 0.3779
9.50 0.4926 0.4733 0.4606 0.4507 0.4421 0.4362 0.4310 0.4264 0.4233 0.4204 0.4157 0.4133 0.4110 0.4092 0.4077
10.00 0.5225 0.5031 0.4905 0.4805 0.4720 0.4660 0.4609 0.4563 0.4531 0.4503 0.4456 0.4432 0.4409 0.4390 0.4376
10.50 0.5523 0.5330 0.5203 0.5104 0.5018 0.4958 0.4907 0.4861 0.4830 0.4801 0.4754 0.4730 0.4708 0.4689 0.4675 e \1.00 0.5822 0.5629 0.5502 0.5402 0.5316 0.5256 0.5204 0.5159 0.5127 0.5099 0.5052 0.5028 0.5005 0.4987 0.4972 ll.50 0.6121 0.5927 0.5800 0.5700 0.5613 0.5553 0.5502 0.5456 0.5424 0.5396 0.5349 0.5325 0.5303 0.5284 0.5270
12.00 0.6420 0.6226 0.6098 0.5997 0.5910 0.5850 0.5798 0.5i52 0.5721 0.5692 0.5645 0.5621 0.5599 0.5580 0.5566
12.50 0.6718 0.6524 0.6396 0.6294 0.6207 0.6146 0.6094 0.60.t7 0.6016 0.5987 0.5940 0.5916 0.5893 0.5875 0.5860
13.00 0.7017 0.6822 0.6693 0.6591 0.6503 0.6442 0.6389 0.6.3-i2 0.63II 0.6282 0:6234 0.6210 0.6187 0.6168 0.6154
13.50 0. 7316 0.7121 0.6991 0.6887 0.6798 0.6737 0.6683 0.6636 0.6604 0.6575 0.6527 0.6502 0.6479 0.6460 0.6445
14.00 0.7615 0.7419 0.7288 0.7183 0.7093 0.7031 0.6977 0.6929 0.6897 0.6867 0.6818 0.6793 0.6770 0.6750 0.6736
14.50 0.7913 0.7717 0.7585 0.7479 0.7388 0.7325 0.7270 0. i222 0.7189 0.7158 0.7108 0.7062 0.7059 0.7039 0.7024
15.00 0.8212 0.8014 0.7881 0.7774 0.7682 0.7619 0.7562 O.i513 0.7479 0.7448 0.7397 0.7370 0.7346 0.7326 0.7311
15.50 0.8510 0.8312 0.8178 0.8069 0.7976 0.7911 0.7854 O.iS04 0.7769 0.7737 0.7684 0.7656 0.7632 0.7612 0.7597
16.00 0.8809 0.8609 0.8474 0.8363 0.8269 0.8203 0.8145 0.8094 0.8058 0.8025 0.7969 0.7941 0.7916 0.7898 0.7880
16.50 0.9107 0.8907 0.8770 0.8658 0.8662 0.8495 0.8435 0.8363 0.8345 0.8311 0.8254 0.8224 0.8198 0.8178 0.8162
17.00 · o.9406 0.9204 0.9066 0.8951 0.8854 0.8786 0.8724 0.8671 0.8632 0.8597 0.8537 0.8505 0.8479 0.8458 0.8442
17.50 0.9704 0.9501 0.9362 0.9245 0.9146 0.9076 0.9013 0.8958 0.8918 0.8881 0.8818 0.8765 0.8758 0.8737 0.8721
18.00 1.0002 0.9798 0.9657 0.9538 0.9437 0.9366 0.9300 0.9245 0.9203 0.9164 0.9098 0.9064 0.9036 0.9014 0.8997
e 18.50 1.0300 1.0095 0.9953 0.9831 0.9728 0.9656 0.9588 0.9530 0.9486 0.9446 0.9377 0.9340 0.93II 0.9289 0.9272 19.00 1.0599 1.0392 1.0248 1.0123 1.0018 0.9945 0.9874 0.9815 0.9769 0.9727 0.9654 0.9615 0.9586 0.9563 0.9545 ~
19.50 1.0897 1.0689 1.0543 1.0415 1.0308 1.0233 1.0160 1.0099 1.0051 1.0007 0.9930 0.9889 0.9858 0.9835 0.9817 "
20.00 1.1195 1.0985 1.0837 1.0707 1.0597 1.0521 1.0445 1.0383 1.0332 1.0286 1.0204 1.0181 1.0129 1.0105 1.0087 ~
20.50 1.1493 1.1282 1.1132 1.0999 1.0886 1.0808 1.0730 1.0665 1.0612 1.0564 1.0478 1.0432 1.0398 1.0374 1.0355 ~ 21.00 1.1791 1.1578 1.1426 1.1290 1.1175 1.1095 1.1014 1.0947 1.0892 1.0841 1.0749 1.0701 1.0666 1.0641 1.0622
21.50 1.2089 1.1874 1.1721 1.1581 1.1463 1.1381 1.1297 LJ229 1.1170 1.1116 1.1020 1.0968 1.0933 1.0907 1.0887 ~
22.00 1.2387 1.2170 1.2015 1.1871 1.1751 1.1667 1.1580 1.1509 1.1448 1.1391 1.1289 1.1235 1. II98 1.1171 1.1151
Q,
22.50 1.2685 1.2466 1.2309 1.2162 1.2039 1.1953 1.1862 1.1789 1.1724 1.1665 1.1558 1.1500 1.1461 1.1434 1.1413 g
23.00 1.2982 1.2762 1.2602 1.2452 1.2326 1.2238 1.2144 1.2069 1.2000 1.1938 1.1825 1.1763 1.1723 1.1695 1.1674 3
" 23.50 1.3280 1.3058 1.2896 1.2742 1.2613 1.2522 1.2425 1.23-H 1.2276 1.2210 1.2090 1.2026 1.1984 1.1955 1.1933 :::::
24.00 1.3578 1.3354 1.3190 1.3031 1.2899 1.2807 1.2706 1.2625 1.2550 1.2482 1.2355 1.2287 1.2243 1.2214 1.2191 ~ 24.50 l.3876 l.3650 1.3483 1.3321 l.3185 l.3090 1.2986 r.:903 1.2824 l.2752 1.2619 1.2546 1.2501 1.2471 1.2447 Er
25.00 1.4173 1.3946 1.3776 1.3610 1.3471 1.3374 1.3265 1.3180 1.3097 1.3022 1.2881 1.2805 1.2758 1.2727 1.2702 0 Q,
25.50 1.4471 1.4241 1.4069 1.3899 1.3757 1.3657 1.3544 J.3.156 1.3369 1.3290 1.3142 1.3062 1.3013 1.2981 1.2956
26.00 1.4769 1.4537 1.4362 1.4187 1.4042 1.3940 1.3823 1.3';32 1.3641 1.3658 1.3403 1.3318 1.3267 l.3235 1.3209
"""' \C) 26.50 1.5066 1.4832 1.4655 1.4476 1.4327 1.4222 1.4101 1.4007 1.3912 1.3625 1.3662 l.3573 1.3520 l.3487 1.3460 Ul
27.00 1.5364 1.5127 1.4948 1.4764 1.4611 1.4504 1.4379 1.4282 1.4182 1.4092 l.3920 l.3827 1.3772 1.3738 1.3710
27.50 1.5661 1.5423 1.5240 1.5052 1.4895 1.4786 1.4656 l.-l556 1.4452 1.4357 1.4178 1.4079 1.4023 1.3987 1.3759
28.00 1.5959 1.5718 1.5533 1.5340 1.5179 1.5067 1.4933 l.-l829 1.4721 1.4622 1.4434 1.4331 1.4272 1.4236 1.4206
28.50 1.6256 1.6013 1.5825 1.5627 1.5463 1.5348 1.5209 1.5102 1.4989 1.4886 1.4690 1.4581 1.4520 1.4483 1.4452
29.00 1.6554 1.6308 1.6117 1.5915 1.5747 1.5629 1.5485 1.5375 1.5257 1.5150 1.4944 1.4831 1.4768 1.4729 1.4698
29.50 1.6851 1.6603 1.6410 1.6202 1.6030 1.5909 1.5761 1.55-n 1.5524 1.5412 1.5198 1.5079 l.5014 1.4974 1.4942
30.00 1.7148 1.6898 1.6702 1.6489 1.6313 1.6189 1.6036 1.5919 1.5791 1.5675 1.5450 1.5327 1.5259 1.5218 1.5165
•
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/CONSULA T ANT/TRAINING/SIAlA/GASOUTLINES.doc
Page 2
• ..
------~-----------
This course offers a comprehensive review of all aspects related to
Natural-Gas Reservoir Engineering
from the Producing Formation to the surface:
i .... ..1r
-- The course has been :divided into 8 main chaplers as follows: ....
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/CONSULA T ANTITRAINING/SIALA/GASOUTLINES.doc 2
Page 107
Natural Gas Reservoir Engineering Dr. A. A. Urayet
Example Problem 4. 3
(Effect of Condensates on the Material Balance Calculations)
A gas reservoir was put on production in Nov.9, 1993. The following data was
available after one year of production:
Initial pressure (9/11/1993)
reservoir temperature
gas specific gravity
Zi
reservoir pressure (9/11/1994)
cumulative gas produced
Z factor at P= 4120 psia
cumulative hydr. liquids production
API of hydrocarbon liquids
Cumulative water production
4300 psi a
228 °F
0.71
0.945 0. ~ 'Z. t't l\-
4120 psia
23.5 MMM scf
0.936 t:J • 9 I 'r'+
865,000 STB
61
14,200 STBW
Assuming a Volumetric reservoir performance, calculate the Gas Initially In
Place,
a) assuming only gas is being produced
b) taking into consideration the liquid condensates, and assuming that all the
water produced is condensing from the water vapor
AAU/GASNOLUMETRICS.doc 33
Page 108
Natural Gas Reservoir Engineering Dr. A. A. Urayet
Solution
a) Calculate the Gas Initially In Place using Eq (4.19-A) as follows:
GllP = 719.8 MMMscf
~' ''t>"t
b) Correct the value of
9
. 4120 _ 4300 -(4300)23.5x10
0.936 0.945 0.945 G
0 ·~16r4' tJ•9l1-'-lf.f c·7t.'Pl''f
as follows:
G P(corrected) = G p(sep) + N p.(3003)(1.03-y0 ) + Wvp·(7390)
Gp (corrected) =
= 23.5xl09 + (865,000).(3003)(1.03- l 4t.5 + (14,200).(7390) = 24.37lxl09
131.5 + 61
- Calculate the Gas Initially In Place, using Eq (4.19-A), as follows:
p P: P:
-=-' - (-' ).GP
z zi G.zi
4120 = 4300 -(4300)24.371x10
9
0.936 0.945 0.945 G
~ O•'JJ,r't c5•'i2.l-'t-<t- 0·9'Z1-'t-c+
G = 746.49 MMM scf
t "l't. 't >?..
AAU/GASNOLUMETRICS.doc 34
•
Page 213
TABLE C.2(1)
Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation
P, ReducectTemperature for B = 60.0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.6 1.9 2.0 2.2 2.4 2.6 2.8 3.0
0.20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.50 0.0019 0.0019 0.0019 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0017 0.0017 0.0017
1.00 0.0101 0.0093 0.0089 0.0087 0.0085 0.0084 0.0083 0.0082 0.0081 0.0081 0.0080 0.0080 0.0080 0.0079 0.0079
1.50 0.0277 0.0232 0.0215 0.0206 0.0200 0.0195 0.0192 0.0189 0.0188 0.0186 0.0184 0.0183 0.0181 0.0181 0.0180
2.00 0.0559 0.0443 0.0399 0.0376 0.0361 0.0351 0.0343 0.0338 0.0334 0.0331 0.0326 0.0323 0.0321 0.0319 0.0317
2.50 0.0870 0.0715 0.0637 0.0594 0.0566 0.0549 0.0535 0.0524 0.0518 0.0512 0.0504 0.0499 0.0494 0.0490 0.0487
~ 3.00 0.1189 0.1014 0.0913 0.0851 0.0808 0.0781 0.0760 0.0745 0.0734 0.0726 0.0714 0.0705 0.0698 0.0692 0.0687
3.50 0.1509 0.1325 0.1211 0.1135 0.1079 0.1043 0.1014 0.0993 0.0979 0.0966 0.0950 0.0989 0.0928 0.0920 0.0913
4.00 0.1831 0.1642 0.1521 0.1435 0.1369 0.1326 0.1291 0.1263 0.1245 0.1229 0.1209 0.1194 0.1181 0.1170 0.1161 1:1:1
4.50 0.2153 0.1962 0.1837 0.1745 0.1672 0.1624 0.1583 0.1551 0.1529 0.1510 0.1485 0.1466 0.1451 0.1438 0.1428 <::> -5.00 0.2475 0.2283 0.2157 0.2062 0.1984 0.1931 0.1887 0.1850 0.1826 0.1804 0.1775 0.1783 0.1736 0.1721 0.1709 ~
5.50 0.2798 0.2606 0.2479 0.2382 0.2301 0.2245 0.2198 0.2158 0.2132 0.2108 0.2075 0.2051 0.2032 0.2016 0.2003 ~
6.00 0.3120 0.2928 0.2801 0.2703 0.2620 0.2563 0.2515 0.2472 0.2444 0.2419 0.2383 0.2357 0.2337 0.2320 0.2306 ~
6.50 0.3443 0.3251 0.3124 0.3026 0.2942 0.2884 0.2834 0.2791 0.2761 0.2735 0.2697 0.2670 0.2648 0.2630 0.2616
;;-
7.00 0.3766 0.3574 0.3446 0.3348 0.3264 0.3206 0.3156 0.3111 0.3081 0.3<154 0.3015 0.2986 0.2964 0.2946 0.2932 ~
7.50 0.4088 0.3896 0.3769 0.3671 0.3587 0.3529 0.3478 0.3433 0.3403 0.3375 0.3336 0.3306 0.3284 0.3265 0.3251 ~
8.00 0.4411 0.4219 0.4091 0.3994 0.3910 0.3851 0.3801 0.3756 0.3725 0.3697 0.3651 0.3628 0.3605 0.3586 0.3572 !;
8.50 0.4734 0.4541 0.4413 0.4316 0.4232 0.4174 0.4123 0.4079 0.4048 0.4020 0.3976 0.3951 0.3928 0.3909 0.3894 ~
9.00 0.5056 0.4863 0.4735 0.4637 0.4554 0.4496 0.4445 0.4401 0.5370 0.4343 0.4297 0.4273 0.4251 0.4231 0.4217
9.50 0.5378 0.5185 0.5056 0.4958 0.4875 0.4817 0.4767 0.4722 0.4692 0.4665 0.4619 0.4596 0.4573 0.4554 0.4539
10.00 0.5701 0.5507 0.5377 0.5279 0.5195 0.5137 0.5087 0.5043 0.5013 0.4985 0.4940 0.4917 0.4894 0.4875 0.4861
e 10.50 0.6023 0.5828 0.5698 0.5599 0.5515 0.5457 0.5407 0.5363 0.5333 0.5305 0.5260 0.5237 0.5215 0.5196 0.5181 1.00 0.6344 0.6149 0.6018 0.5718 0.5833 0.5775 0.5725 0.5581 0.5651 0.5624 0.5579 0.5556 0.5534 0.5515 0.5500
11.50 0.6666 0.6469 0.6337 0.6237 0.6151 0.6093 0.6042 0.5998 0.5988 0.5942 0.5896 0.5873 0.5851 0.5832 0.5818
12.00 0.6987 0.6790 0.6656 0.6555 0.6469 0.6409 0.6359 0.6314 0.6284 0.6257 0.6212 0.6189 0.6166 0.6148 0.6133
12.50 0.7309 0.7110 0.6975 0.6872 0.6785 0.6725 0.6674 0.6629 o.Q599. 0.6571 0.6526 0.6503 0.6480 0.6461 0.6446
13.00 0.7630 0.7429 0.7293 0.7189 0.7101 0.7040 0.6986 0.6943 0.6912 0.6884 0.6838 0.6815 0.6792 0.6773 0.6758
13.50 0.7951 0.7749 0.7611 0.7505 0.7415 0.7354 0.7301 0.7255 0.7224 0.7196 0.7149 0.7125 0.7101 0.7032 0.7067
14.00 0.8272 0.8068 0.7929 0.7820 0.7730 0.7667 0.7613 0.7566 0.7534 0.7505 0.7457 0.7482 0.7409 0.7389 0.7374
14.50 0.8592 0.8387 0.8246 0.8135 0.8043 0.7979 0.7924 0.7876 0.7843 0.7813 0.7764 0.7738 0.7714 0.7694 0.7679
15.00 0.8913 0.8705 0.8562 0.8449 0.8355 0.8291 0.8233 0.8184 0.8151 0.8120 0.8069 0.8042 0.8017 0.7997 0.7982
15.50 0.9233 0.9024 0.8879 0.8763 0.8667 0.8601 0.8542 0.8492 0.8457 0.8425 0.8371 0.8343 0.8318 0.8298 0.8282
16.00 0.9554 0.9342 0.9195 0.9076 0.8978 0.8911 0.8850 0.8798 0.8762 0.8728 0.8672 0.8643 0.8617 0.8596 0.8580
16.50 0.9874 0.9660 0.9510 0.9389 0.9288 0.9219 0.9156 0.9103 0.9065 0.9030 0.8971 0.8940 0.8914 0.8892 0.8876
17.00 . 1.0194 0.9977 0.9826 0.9701 0.9598 0.9527 0.9462 0.9408 0.9368 0.9331 0.9269 0.9236 0.9208 0.9186 0.9170
17.50 1.0514 1.0295 1.0141 1.0012 0.9907 0.9835 0.9767 0.9711 0.9668 0.9630 0.9564 0.9529 0.9501 0.9478 0.9461
18.00 1.0834 1.0612 1.0455 1.0323 1.0215 1.0141 1.0070 1.0013 0.9968 0.9928 0.9858 0.9820 0.9791 0.9768 0.9751
e·tS.50 1.1153 1.0929 1.0769 1.0634 1.0523 1.0447 1.0373 1.0313 1.0267 1.0224 1.0150 1.0!10 1.0080 1.0056 1.0038
;;;2 1.1473 1.1246 l.!083 1.0944 1.0830 1.0752 1.0675 1.0613 1.0564 1.0519 1.0440 1.0398 1.0366 1.0342 1.0324 19.00
~
19.50 1.1792 1.1562 l.!397 1.1253 1.1137 1.1056 1.0976 1.0912 1.0860 1.0812 1.0728 1.0683 1.0651 1.0626 1.0607 ti:>
20.00 l.2112 1.1879 l.!711 1.1562 1.1443 1.1360 1.1277 1.1210 1.1155 1.1104 1.1015 1.0967 1.0933 1.0908 1.0889 ~ 20.50 1.2431 1.2195 1.2024 1.1871 1.1748 1.1663 1.1576 1.1507 1.1449 1.1395 1.1301 1.1250 1.1214 1.1188 1.1168
21.00 1.2750 1.2511 1.2337 l.2179 1.2053 1.1965 1.1875 1.1803 1.1741 1.1685 1.1584 1.1530 1.1493 l.!466 1.1446
...
t 21.50 1.3069 1.2827 1.2650 1.2487 1.2357 1.2267 1.2173 1.2099 1.2033 1.1974 1.1867 1.1800 1.1770 1.1743 1.1721 Cl..
22.00 1.3388 1.3143 1.2962 l.2795 1.2661 1.2568 1.2470 1.2393 l.2324 1.2261 1.2147 1.2086 1.2046 1.2018 1.1995 g
22.50 1.3707 1.3458 1.3274 1.3102 1.2964 1.2869 l.2766 1.2667 l.2614 1.2547 1.2427 1.2361 l.2319 1.2291 1.2268 ii!
23.00 1.4026 1.3774 1.3586 1.3409 1.3267 1.3169 1.3062 1.2979 1.2902 1.2832 1.2705 1.2635 l.2592 l.2562 1.2538 ~
23.50 1.4344 1.4089 1.3898 1.3715 1.3569 1.3469 1.3357 1.3190 1.3116 1.2981 l.2908 1.2862 1.2832 1.2807
::::
1.3271
~ 24.00 l .4663 l.4404 1.4210 1.4021 1.3871 1.3768 1.3652 1.3563 1.3477 1.3399 1.3256 1.3179 1.3131 1.3100 1.3074
24.50 1.4982 1.4719 1.4521 1.4327 1.4173 1.4066 1.3945 1.3853 1.3763 1.3681 1.3530 1.3448 1.3399 1.3366 1.3340 s. <::>
25.00 1.5300 1.0534 1.4832 1.4632 J.4474 1.4364 1.4238 1.4143 1.4048 1.3962 1.3803 1.3716 1.3664 1.3631 1.3604 Q.
25.50 1.5619 1.5349 1.5143 1.4937 1.4774 1.4662 1.4531 1.4432 1.4332 1.4242 1.4074 1.3983 1.3929 1.3895 1.3867
26.00 1.5937 1.5664 1.5454 1.5242 1.5075 1.4959 1.4823 1.4721 1.4616 1.4521 1.4344 1.4248 1.4192 1.4157 1.4128
...
\Q
26.50 1.6255 1.5978 1.5765 1.5547 1.5374 1.5255 1.5114 1.5008 1.4898 1.4799 1.4613 1.4512 1.4454 1.4417 1.4388
(;.I
27.00 1.6574 1.6292 1.6075 1.5851 1.5674 1.5552 1.5405 1.5295 1.5180 J.5076 1.4881 1.4775 1.4714 1.4677 1.4646
27.50 1.6892 1.6607 1.6385 1.6155 1.5973 1.5847 1.5695 1.5582 1.5461 J.5353 1.5148 1.5036 1.4973 1.4935 1.4903
28.00 1.7210 1.6921 1.6695 1.6459 1.6272 1.6143 1.5985 1.5868 1.5742 1.5628 J.5413 1.5296 1.5231 1.5191 1.5159
28.50 1.7528 l.7235 1.7005 l.6762 1.6570 1.6438 1.6274 1.6153 1.6021 J.5903 1.5678 1.5555 1.5407 1.5447 l .5413
29.00 1.7846 1.7549 1.7315 1.7065 J.6868 1.6732 1.6563 1.6438 1.6300 J.6176 1.5941 1.5813 1.5742 1.5701 1.5666
29.50 1.8164 1.7863 1.7625 1.7168 1.7166 1.7026 1.6851 1.6722 l.6579 1.6449 l.6204 1.6070 1.5997 1.5954 1.5918
30.00 1.8482 1.8177 1.793.J 1.7671 1.7463 1.7320 1.7139 1.7005 1.6856 1.6722 1.6465 1.6325 1.6249 1.6205 1.6168
Page 214
TABLE C.2(m)
Extended Sukkar-Cornell Integral for Bottom-hole Pressure Calculation
P, Reduced·Temperature for 8 = 70.0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0
0.20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.50 0.0017 0.0016 0.0016 0.0016 0.0016 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015
1.00 0.0087 0.0080 0.0077 0.0074 0.0073 0.0072 0.0071 0.0070 0.0070 0.0070 0.0069 0.0069 0.0068 0.0068 0.0068
1.50 0.0240 0.0199 0.0185 0.0177 0.0172 0.0168 0.0165 0.0163 0.0161 0.0160 0.0158 0.0157 0.0156 0.0155 0.0154
2.00 0.0491 0.0385 0.0345 0.0325 0.0312 0.0303 0.0296 0.0291 0.0288 0.0285 0.0281 0.0278 0.0276 0.0274 0.0273
2.50 0.0772 0.0625 0.0554 0.0515 0.0490 0.0475 0.0462 0.0453 0.0448 0.0443 0.0435 0.0431 0.0426 0.0423 0.0420
3.00 0.1063 0.0894 0.0799 0.0742 0.0703 0.0679 0.0660 0.0646 0.0637 0.0629 0.0618 0.0611 0.0604 0.0599 0.0595
"""' 3.50 0.1356 0.1175 0.1066 0.0994 0.0943 0.0910 0.0884 0.0864 0.0851 0.0840 0.0825 0.0815 0.0806 0.0798 0.0792 'g
4.00 0.1651 0.1464 0,1346 0.1264 0.1202 0.1162 0.1129 0.1104 0.1087 0.1073 0.1054 0.1040 0.1029 0.1018 0.1010
4.50 0.1947 0.1756 0.1634 0.1545 0.1475 0.1429 0.1391 0.1360 0.1340 0.1322 0.1299 0.1282 0.1268 0.1256 0.1246 b:I <:> 5.00 0.2243 0.2050 0.1926 0.1833 0.1756 0.1706 0.1664 0.1629 0.1606 0.1585 0.1558 0.1538 0.1522 0.1508 0.1497 -S' 5.50 0.2540 0.2347 0.2221 0.2125 0.2045 0.1991 0.1946 0.1907 0.1881 0.1859 0.1827 0.1805 0.1787 0.1772 0.1760 Eil 6.00 0.2838 0.2644 0.2517 0.2420 0.2337 0.2281 0.2233 0.2192 0.2164 0.2140 0.2106 0.2081 0.2061 0.2045 0.2032 ~ 6.50 0.3135 0.2941 0.2815 0.2716 0.2632 0.2574 0.2525 0.2482 0.2453 0.2427 0.2390 0.2363 0.2343 0.2326 0.2313 ;--7.00 0.3433 0.3239 0.3113 0.3014 0.2929 0.2870 0.2820 0.2775 0.2745 0.2718 0.2680 0.2652 0.2630 0.2613 0.2599
41 7.50 0.3732 0.3538 0.341l 0.3312 0.3226 0.3167 0.3116 0.3071 0.3040 0.3013 0.2973 0.2944 0.2922 0.2904 0.2890
8.00 0.4030 0.3836 0.3710 0.3611 0.3525 0.3465 0.3414 0.3368 0.3337 0.3309 0.3262 0.3239 0.3217 0.3198 0.3184 ~ 8.50 0.4328 0.4135 0.4009 0.3909 0.3824 0.3764 0.3713 0.3667 0.3635 0.3607 0.3560 0.3536 0.3514 0.3495 0.3481 ~ 9.00 0.4627 0.4434 0.4307 0.4208 0.4122 0.4063 0.4011 0.3965 0.3934 0.3905 0.3858 0.3834 0.3812 0.3793 0.3779
9.50 0.4926 0.4733 0.4606 0.4507 0.4421 0.4362 0.4310 0.4264 0.4233 0.4204 0.4157 0.4133 0.4110 0.4092 0.4077
10.00 0.5225 0.5031 0.4905 0.4805 0.4720 0.4660 0.4609 0.4563 0.4531 0.4503 0.4456 0.4432 0.4409 0.4390 0.4376
10.50 0.5523 0.5330 0.5203 0.5104 0.5018 0.4958 0.4907 0.4861 0.4830 0.4801 0.4754 0.4730 0.4708 0.4689 0.4675 e \1.00 0.5822 0.5629 0.5502 0.5402 0.5316 0.5256 0.5204 0.5159 0.5127 0.5099 0.5052 0.5028 0.5005 0.4987 0.4972 ll.50 0.6121 0.5927 0.5800 0.5700 0.5613 0.5553 0.5502 0.5456 0.5424 0.5396 0.5349 0.5325 0.5303 0.5284 0.5270
12.00 0.6420 0.6226 0.6098 0.5997 0.5910 0.5850 0.5798 0.5i52 0.5721 0.5692 0.5645 0.5621 0.5599 0.5580 0.5566
12.50 0.6718 0.6524 0.6396 0.6294 0.6207 0.6146 0.6094 0.60.t7 0.6016 0.5987 0.5940 0.5916 0.5893 0.5875 0.5860
13.00 0.7017 0.6822 0.6693 0.6591 0.6503 0.6442 0.6389 0.6.3-i2 0.63II 0.6282 0:6234 0.6210 0.6187 0.6168 0.6154
13.50 0. 7316 0.7121 0.6991 0.6887 0.6798 0.6737 0.6683 0.6636 0.6604 0.6575 0.6527 0.6502 0.6479 0.6460 0.6445
14.00 0.7615 0.7419 0.7288 0.7183 0.7093 0.7031 0.6977 0.6929 0.6897 0.6867 0.6818 0.6793 0.6770 0.6750 0.6736
14.50 0.7913 0.7717 0.7585 0.7479 0.7388 0.7325 0.7270 0. i222 0.7189 0.7158 0.7108 0.7062 0.7059 0.7039 0.7024
15.00 0.8212 0.8014 0.7881 0.7774 0.7682 0.7619 0.7562 O.i513 0.7479 0.7448 0.7397 0.7370 0.7346 0.7326 0.7311
15.50 0.8510 0.8312 0.8178 0.8069 0.7976 0.7911 0.7854 O.iS04 0.7769 0.7737 0.7684 0.7656 0.7632 0.7612 0.7597
16.00 0.8809 0.8609 0.8474 0.8363 0.8269 0.8203 0.8145 0.8094 0.8058 0.8025 0.7969 0.7941 0.7916 0.7898 0.7880
16.50 0.9107 0.8907 0.8770 0.8658 0.8662 0.8495 0.8435 0.8363 0.8345 0.8311 0.8254 0.8224 0.8198 0.8178 0.8162
17.00 · o.9406 0.9204 0.9066 0.8951 0.8854 0.8786 0.8724 0.8671 0.8632 0.8597 0.8537 0.8505 0.8479 0.8458 0.8442
17.50 0.9704 0.9501 0.9362 0.9245 0.9146 0.9076 0.9013 0.8958 0.8918 0.8881 0.8818 0.8765 0.8758 0.8737 0.8721
18.00 1.0002 0.9798 0.9657 0.9538 0.9437 0.9366 0.9300 0.9245 0.9203 0.9164 0.9098 0.9064 0.9036 0.9014 0.8997
e 18.50 1.0300 1.0095 0.9953 0.9831 0.9728 0.9656 0.9588 0.9530 0.9486 0.9446 0.9377 0.9340 0.93II 0.9289 0.9272 19.00 1.0599 1.0392 1.0248 1.0123 1.0018 0.9945 0.9874 0.9815 0.9769 0.9727 0.9654 0.9615 0.9586 0.9563 0.9545 ~
19.50 1.0897 1.0689 1.0543 1.0415 1.0308 1.0233 1.0160 1.0099 1.0051 1.0007 0.9930 0.9889 0.9858 0.9835 0.9817 "
20.00 1.1195 1.0985 1.0837 1.0707 1.0597 1.0521 1.0445 1.0383 1.0332 1.0286 1.0204 1.0181 1.0129 1.0105 1.0087 ~
20.50 1.1493 1.1282 1.1132 1.0999 1.0886 1.0808 1.0730 1.0665 1.0612 1.0564 1.0478 1.0432 1.0398 1.0374 1.0355 ~ 21.00 1.1791 1.1578 1.1426 1.1290 1.1175 1.1095 1.1014 1.0947 1.0892 1.0841 1.0749 1.0701 1.0666 1.0641 1.0622
21.50 1.2089 1.1874 1.1721 1.1581 1.1463 1.1381 1.1297 LJ229 1.1170 1.1116 1.1020 1.0968 1.0933 1.0907 1.0887 ~
22.00 1.2387 1.2170 1.2015 1.1871 1.1751 1.1667 1.1580 1.1509 1.1448 1.1391 1.1289 1.1235 1. II98 1.1171 1.1151
Q,
22.50 1.2685 1.2466 1.2309 1.2162 1.2039 1.1953 1.1862 1.1789 1.1724 1.1665 1.1558 1.1500 1.1461 1.1434 1.1413 g
23.00 1.2982 1.2762 1.2602 1.2452 1.2326 1.2238 1.2144 1.2069 1.2000 1.1938 1.1825 1.1763 1.1723 1.1695 1.1674 3
" 23.50 1.3280 1.3058 1.2896 1.2742 1.2613 1.2522 1.2425 1.23-H 1.2276 1.2210 1.2090 1.2026 1.1984 1.1955 1.1933 :::::
24.00 1.3578 1.3354 1.3190 1.3031 1.2899 1.2807 1.2706 1.2625 1.2550 1.2482 1.2355 1.2287 1.2243 1.2214 1.2191 ~ 24.50 l.3876 l.3650 1.3483 1.3321 l.3185 l.3090 1.2986 r.:903 1.2824 l.2752 1.2619 1.2546 1.2501 1.2471 1.2447 Er
25.00 1.4173 1.3946 1.3776 1.3610 1.3471 1.3374 1.3265 1.3180 1.3097 1.3022 1.2881 1.2805 1.2758 1.2727 1.2702 0 Q,
25.50 1.4471 1.4241 1.4069 1.3899 1.3757 1.3657 1.3544 J.3.156 1.3369 1.3290 1.3142 1.3062 1.3013 1.2981 1.2956
26.00 1.4769 1.4537 1.4362 1.4187 1.4042 1.3940 1.3823 1.3';32 1.3641 1.3658 1.3403 1.3318 1.3267 l.3235 1.3209
"""' \C) 26.50 1.5066 1.4832 1.4655 1.4476 1.4327 1.4222 1.4101 1.4007 1.3912 1.3625 1.3662 l.3573 1.3520 l.3487 1.3460 Ul
27.00 1.5364 1.5127 1.4948 1.4764 1.4611 1.4504 1.4379 1.4282 1.4182 1.4092 l.3920 l.3827 1.3772 1.3738 1.3710
27.50 1.5661 1.5423 1.5240 1.5052 1.4895 1.4786 1.4656 l.-l556 1.4452 1.4357 1.4178 1.4079 1.4023 1.3987 1.3759
28.00 1.5959 1.5718 1.5533 1.5340 1.5179 1.5067 1.4933 l.-l829 1.4721 1.4622 1.4434 1.4331 1.4272 1.4236 1.4206
28.50 1.6256 1.6013 1.5825 1.5627 1.5463 1.5348 1.5209 1.5102 1.4989 1.4886 1.4690 1.4581 1.4520 1.4483 1.4452
29.00 1.6554 1.6308 1.6117 1.5915 1.5747 1.5629 1.5485 1.5375 1.5257 1.5150 1.4944 1.4831 1.4768 1.4729 1.4698
29.50 1.6851 1.6603 1.6410 1.6202 1.6030 1.5909 1.5761 1.55-n 1.5524 1.5412 1.5198 1.5079 l.5014 1.4974 1.4942
30.00 1.7148 1.6898 1.6702 1.6489 1.6313 1.6189 1.6036 1.5919 1.5791 1.5675 1.5450 1.5327 1.5259 1.5218 1.5165
