HW.2

Petroleum reservoirs are typically heterogeneous, requiring measurement of this heterogeneity. Dykstra and Parsons introduced a method using permeability data from core analysis, assuming a log-normal distribution and yielding a variation coefficient (0-1)

fisrt load the data:

data=read.csv("karpur.csv")
head(data)

NA

Arranging permeability values in descending order and identifying samples with values greater than or equal to k portions.

data=data[order(data$k.core, decreasing = TRUE),]
data$k.core

  [1] 15600.0000 14225.3135 13544.9785 13033.5283 11841.7432 11117.4023 10860.0000 10649.9580
  [9] 10540.0000  9898.4785  9533.3623  9458.1729  9120.0000  8820.1025  8820.0000  8820.0000
 [17]  8760.0000  8742.7529  8689.8252  8390.0000  8306.0459  8190.0000  8030.0000  7956.4258
 [25]  7930.0000  7918.8984  7862.0825  7740.0000  7725.9600  7525.2207  7517.4834  7430.0000
 [33]  7260.0000  7200.0000  7180.0000  7141.1079  7119.8560  7020.0000  6950.0000  6917.3823
 [41]  6911.2437  6879.2051  6860.0000  6844.0718  6697.7793  6690.1787  6680.0000  6447.5181
 [49]  6389.9697  6388.7993  6280.0000  6262.2485  6232.8345  6186.6069  6180.0000  6171.7534
 [57]  6138.8628  6070.0000  6046.8418  6027.3105  6008.5645  5963.4404  5954.8203  5950.0000
 [65]  5880.0000  5855.1367  5810.0000  5750.0000  5640.0000  5630.0000  5595.5122  5480.0000
 [73]  5479.2744  5467.9937  5430.0000  5412.3223  5363.0371  5338.7207  5266.6743  5202.5576
 [81]  5200.0000  5190.0000  5067.7798  5048.1919  5020.0947  4969.6675  4954.9937  4900.0000
 [89]  4876.0210  4832.6045  4830.0000  4810.0000  4804.8022  4786.5928  4760.0000  4749.9390
 [97]  4730.0000  4703.0840  4678.7793  4630.0000  4561.4106  4560.0000  4490.0000  4451.9521
[105]  4420.0000  4390.0000  4390.0000  4336.2285  4328.6606  4320.0000  4318.5918  4267.1182
[113]  4243.5298  4210.0000  4203.3662  4154.2788  4135.5439  4119.3125  4111.0464  4075.8799
[121]  4070.0000  4047.0544  4020.0000  4014.4380  4011.0640  3998.8315  3997.4805  3988.5752
[129]  3983.7849  3982.3992  3970.0000  3956.0886  3950.7856  3940.0000  3936.2375  3890.0000
[137]  3885.8645  3882.9058  3872.8757  3840.0000  3840.0000  3820.7578  3820.0000  3795.9231
[145]  3750.0000  3750.0000  3743.4775  3720.0000  3700.0000  3680.0000  3677.3889  3660.6824
[153]  3660.0000  3650.0000  3650.0000  3630.0000  3630.0000  3610.0000  3580.1563  3561.8772
[161]  3526.3140  3520.5999  3473.3303  3460.0000  3459.5876  3435.5005  3430.0000  3428.3037
[169]  3427.3506  3420.0000  3405.0000  3385.5239  3380.0000  3370.0000  3361.7825  3350.0000
[177]  3340.3401  3340.0000  3340.0000  3338.0410  3328.5642  3320.0000  3310.0000  3302.7673
[185]  3293.4705  3290.0000  3250.0000  3250.0000  3230.0000  3220.0000  3210.3184  3181.6086
[193]  3180.0000  3170.0000  3169.1899  3150.3203  3147.1648  3128.8020  3110.4392  3110.0000
[201]  3100.0000  3090.0657  3090.0000  3066.8655  3053.2048  3040.4338  3040.0000  3040.0000
[209]  3035.1306  3033.0540  3006.9888  3000.0000  2997.9072  2990.0000  2985.1362  2981.4578
[217]  2970.0000  2968.5474  2963.2549  2960.0000  2947.7891  2928.3137  2928.1562  2918.3340
[225]  2913.8325  2895.5957  2890.0000  2865.1624  2853.9758  2846.9758  2844.1299  2832.4385
[233]  2830.0000  2818.8896  2805.6685  2802.1687  2797.1609  2790.4949  2760.0000  2759.9543
[241]  2759.4919  2750.0000  2749.5947  2736.5945  2730.0000  2707.6108  2700.0000  2698.5298
[249]  2697.3630  2686.7124  2660.0000  2652.5637  2648.6399  2640.0000  2638.9934  2620.0000
[257]  2613.3105  2610.9341  2602.4993  2593.2590  2590.4043  2573.2031  2570.3201  2570.0000
[265]  2563.3101  2558.1506  2550.4099  2534.7722  2532.9214  2531.0762  2531.0000  2530.7581
[273]  2523.2039  2520.0000  2519.8000  2495.1360  2495.1101  2492.1201  2485.9099  2483.5156
[281]  2478.3799  2473.3740  2468.4099  2464.1960  2463.9275  2457.0735  2455.0183  2450.0000
[289]  2448.0698  2445.8406  2442.5901  2436.6628  2434.4473  2433.3545  2427.4851  2423.3625
[297]  2423.1160  2421.8501  2417.5913  2393.9656  2383.2690  2371.0400  2368.7905  2367.5498
[305]  2348.8201  2346.7214  2335.4399  2331.9900  2322.1902  2309.5901  2303.7000  2302.9199
[313]  2299.0537  2295.7800  2294.7261  2285.0200  2284.4299  2283.9041  2276.3159  2274.8657
[321]  2273.6499  2270.0000  2270.0000  2264.6938  2262.0811  2250.9197  2234.4399  2225.8699
[329]  2216.6179  2214.3201  2202.3936  2200.3501  2200.1130  2196.3706  2192.6060  2183.5400
[337]  2182.8186  2173.0312  2163.2437  2155.4399  2155.4353  2150.3301  2149.6299  2147.4971
[345]  2137.3313  2126.4495  2114.7256  2100.5769  2091.1111  2089.7009  2084.4082  2067.3738
[353]  2056.2000  2049.1858  2047.1400  2032.8934  2030.7120  2029.4900  2023.9753  2023.0345
[361]  2019.7881  2014.7900  2011.7695  2010.0000  1994.3353  1990.0415  1978.5601  1975.0400
[369]  1969.0728  1956.1077  1952.0089  1931.9360  1928.4800  1922.0161  1917.3143  1913.9399
[377]  1907.3199  1899.3845  1884.6757  1876.8239  1873.8779  1868.6801  1866.9200  1866.0350
[385]  1862.8700  1859.2261  1852.2700  1828.3900  1821.4403  1820.0200  1804.5400  1790.6000
[393]  1789.1849  1777.6891  1743.1687  1736.2400  1722.9451  1712.4135  1708.2007  1679.6505
[401]  1652.2738  1648.4301  1641.8264  1634.0441  1628.7540  1619.7656  1605.7228  1603.0649
[409]  1599.2813  1591.2185  1585.5100  1584.9600  1574.8315  1562.2338  1556.9391  1552.5758
[417]  1550.0200  1538.3466  1530.4865  1528.5027  1510.6265  1508.5300  1498.7391  1481.5601
[425]  1479.8101  1479.3800  1474.6252  1467.6908  1466.9917  1454.2400  1441.6909  1439.0200
[433]  1437.5200  1435.2444  1433.7565  1411.6324  1408.8621  1405.6776  1403.4971  1383.6300
[441]  1380.1622  1377.0441  1371.7496  1371.3700  1366.1600  1362.4678  1361.9829  1348.7238
[449]  1340.0023  1330.3890  1312.6384  1309.5500  1308.2550  1306.1700  1302.1428  1300.7900
[457]  1291.5699  1291.2828  1290.3817  1280.9800  1276.5076  1274.9500  1271.3800  1269.6000
[465]  1261.7902  1259.5554  1257.2810  1254.4161  1249.7059  1248.4808  1244.7603  1237.4061
[473]  1227.8700  1217.8118  1213.6443  1213.0129  1209.9685  1205.7067  1201.6453  1196.2823
[481]  1183.6799  1181.8900  1181.2655  1181.2034  1178.3425  1171.3900  1149.5182  1142.8588
[489]  1138.5061  1137.1615  1117.7709  1114.6503  1109.5569  1103.1545  1102.0452  1099.4039
[497]  1089.8239  1089.4352  1086.0234  1074.0900  1066.3792  1064.6899  1064.5748  1061.7000
[505]  1059.9347  1054.2761  1053.1267  1053.0554  1042.6234  1040.1243  1036.0116  1022.5287
[513]  1020.4655  1015.6563  1006.8500  1006.0170   999.1555   999.0900   991.5684   990.7814
[521]   980.9962   980.7972   965.5400   963.2323   959.0341   958.4979   957.2500   956.3419
[529]   945.3500   941.5269   939.4300   927.2867   921.8100   917.3568   916.6622   913.8117
[537]   910.1433   906.6724   902.0577   896.8858   895.5803   895.5394   893.3821   884.5511
[545]   883.1400   881.0916   881.0173   868.0800   866.4542   865.2035   863.7920   863.4496
[553]   863.3700   862.5884   860.4050   859.5496   857.3100   851.8912   841.6011   832.0446
[561]   831.6595   831.5900   830.4200   827.7700   827.6600   823.1191   822.9766   817.8306
[569]   817.6682   815.7291   813.0876   809.4100   800.2973   797.7479   794.9600   793.8796
[577]   783.6499   780.4501   770.6907   770.6147   769.9286   768.5499   767.2200   762.7256
[585]   759.7528   745.9776   744.1807   739.4100   736.8026   735.0908   733.3159   729.7252
[593]   729.5110   728.3082   722.0266   721.3600   718.6742   712.3636   710.9836   705.0552
[601]   704.7114   699.5100   691.0581   690.1351   683.6700   679.7400   677.5000   673.3079
[609]   672.4337   669.3860   665.2421   664.4689   662.5491   660.1379   654.5244   652.6051
[617]   651.9620   643.0700   641.5605   635.5146   625.7729   619.6802   614.1521   613.7889
[625]   609.8132   609.7765   603.8804   601.2100   599.3371   597.4316   595.5261   593.6207
[633]   591.7152   589.8097   588.2900   586.3036   583.4219   582.1041   578.0658   575.6158
[641]   572.5554   564.8300   562.5400   558.2472   546.8344   546.3184   537.4426   534.7041
[649]   526.1400   522.3776   520.6862   514.5711   512.2100   512.1923   508.1339   507.4758
[657]   507.3651   499.3700   485.5200   482.8237   480.6394   470.2819   467.8958   467.5622
[665]   465.0500   459.3311   458.8995   458.2000   454.5060   452.0254   451.0764   449.4991
[673]   447.7827   440.9626   437.7937   433.0881   431.1000   428.4266   421.6921   419.3290
[681]   418.6400   417.1300   412.7467   406.2400   399.1000   398.5300   397.6351   395.8943
[689]   392.7800   388.9573   387.5817   386.1200   383.8092   371.8761   369.7330   369.4849
[697]   368.3500   358.7004   358.4117   357.9313   355.8343   350.6500   349.4881   339.4923
[705]   332.4404   324.0869   322.6981   321.5066   316.3239   310.0188   308.4382   303.5096
[713]   303.1158   300.3600   300.2600   298.5033   292.3396   287.3767   282.9963   282.9518
[721]   282.0900   279.4883   273.4974   271.5998   271.4900   270.5496   263.7113   260.5923
[729]   258.1300   257.7916   255.8228   248.3004   247.9343   246.6400   244.4739   242.9852
[737]   240.2611   240.0458   231.2015   231.0803   228.8449   228.5700   226.0300   223.7040
[745]   216.6000   210.2700   204.7999   197.0975   193.4600   191.6111   190.5800   185.0957
[753]   181.7682   173.4400   168.2400   165.3502   162.9400   158.1300   152.9717   152.1418
[761]   149.2137   146.7933   146.0790   143.3300   140.6149   140.0700   134.4366   133.6028
[769]   131.3576   128.2582   126.0047   124.7100   122.0799   119.7091   116.5507   115.9015
[777]   114.3747   112.6725   109.7232   103.5448   102.8927   101.8555    97.3665    92.4887
[785]    91.1881    86.9600    86.0763    85.0097    76.8100    74.1400    73.2072    73.2033
[793]    70.1081    69.2599    63.8231    55.5300    54.2100    52.4436    50.1500    48.6200
[801]    44.5030    43.9438    39.6700    39.4679    35.8600    35.6272    34.9919    33.7340
[809]    32.5300    30.6700    22.0900     7.3800     5.8300     4.8748     3.5809     3.2300
[817]     2.2869     0.8100     0.4200

#Calculating Number of Samples >= k
sample =c(1: length(K))
# Calculating % >= k
k_percent <- (sample * 100) / length(K)

we can recognize the line in the plot showing below:

plot(k_percent, K, log='y', xlab="Portion of total sample having higher permeability", ylab="Sample permeability, md", pch=10, cex=1, col="green")

A linear model is applied to the data:

log_k <- log(K)
slr <- lm(log_k ~ k_percent)
plot(k_percent, log_k, xlab="Portion of total sample having higher permeability", ylab="Sample permeability, md", pch=10, cex=1, col="yellow")
abline(slr, col='blue', lwd=2)

now we calculating the Heterogeneity Index

Hdata <- data.frame(k_percent = c(50, 84.1))
pred.values <- predict(slr, Hdata)
Heterogeneity.index <- (pred.values[1] - pred.values[2]) / pred.values[1]
Heterogeneity.index

        1 
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