Determination of Heterogeneity Index(HI) First by Dykstra-Parsons Coefficient (Vk) and second Lorenz Coefficient
Mohammed Abdul Ameer Dhaher/Basra University For Oil and Gas/4th stage
Supervised by PhD Watheq J. Al-Mudhafar
2022-12-17
Dykstra-parsons Coefficient (Vk)
This method used Log-Normal distribution of permeability to define the Coefficient of permeability Variation, Vk.
Let’s upload some Real Data to work on it.
data<-read.csv("D:/mohammed/karpur.csv",header=T)
summary(data)## depth caliper ind.deep ind.med
## Min. :5667 Min. :8.487 Min. : 6.532 Min. : 9.386
## 1st Qu.:5769 1st Qu.:8.556 1st Qu.: 28.799 1st Qu.: 27.892
## Median :5872 Median :8.588 Median :217.849 Median :254.383
## Mean :5873 Mean :8.622 Mean :275.357 Mean :273.357
## 3rd Qu.:5977 3rd Qu.:8.686 3rd Qu.:566.793 3rd Qu.:544.232
## Max. :6083 Max. :8.886 Max. :769.484 Max. :746.028
## gamma phi.N R.deep R.med
## Min. : 16.74 Min. :0.0150 Min. : 1.300 Min. : 1.340
## 1st Qu.: 40.89 1st Qu.:0.2030 1st Qu.: 1.764 1st Qu.: 1.837
## Median : 51.37 Median :0.2450 Median : 4.590 Median : 3.931
## Mean : 53.42 Mean :0.2213 Mean : 24.501 Mean : 21.196
## 3rd Qu.: 62.37 3rd Qu.:0.2640 3rd Qu.: 34.724 3rd Qu.: 35.853
## Max. :112.40 Max. :0.4100 Max. :153.085 Max. :106.542
## SP density.corr density phi.core
## Min. :-73.95 Min. :-0.067000 Min. :1.758 Min. :15.70
## 1st Qu.:-42.01 1st Qu.:-0.016000 1st Qu.:2.023 1st Qu.:23.90
## Median :-32.25 Median :-0.007000 Median :2.099 Median :27.60
## Mean :-30.98 Mean :-0.008883 Mean :2.102 Mean :26.93
## 3rd Qu.:-19.48 3rd Qu.: 0.002000 3rd Qu.:2.181 3rd Qu.:30.70
## Max. : 25.13 Max. : 0.089000 Max. :2.387 Max. :36.30
## k.core Facies
## Min. : 0.42 Length:819
## 1st Qu.: 657.33 Class :character
## Median : 1591.22 Mode :character
## Mean : 2251.91
## 3rd Qu.: 3046.82
## Max. :15600.00Here we will find the distrirbutin of K (md),but first we should find the Frequency for each K and we will work on 100 samples to simplifed the resultant.
K_md<-kf$k.core
Vk<-tab1(K_md , sort.group = "decreasing", cum.percent = FALSE)
Vk## K_md :
## Frequency Percent
## 999.1555 1 0.1
## 999.09 1 0.1
## 991.5684 1 0.1
## 990.7814 1 0.1
## 9898.4785 1 0.1
## 980.9962 1 0.1
## 980.7972 1 0.1
## 97.3665 1 0.1
## 965.54 1 0.1
## 963.2323 1 0.1
## 959.0341 1 0.1
## 958.4979 1 0.1
## 957.25 1 0.1
## 956.3419 1 0.1
## 9533.3623 1 0.1
## 9458.1729 1 0.1
## 945.35 1 0.1
## 941.5269 1 0.1
## 939.43 1 0.1
## 927.2867 1 0.1
## 921.81 1 0.1
## 92.4887 1 0.1
## 917.3568 1 0.1
## 916.6622 1 0.1
## 913.8117 1 0.1
## 9120 1 0.1
## 910.1433 1 0.1
## 91.1881 1 0.1
## 906.6724 1 0.1
## 902.0577 1 0.1
## 896.8858 1 0.1
## 895.5803 1 0.1
## 895.5394 1 0.1
## 893.3821 1 0.1
## 884.5511 1 0.1
## 883.14 1 0.1
## 8820.1025 1 0.1
## 8820 1 0.1
## 881.0916 1 0.1
## 881.0173 1 0.1
## 8760 1 0.1
## 8742.7529 1 0.1
## 8689.8252 1 0.1
## 868.08 1 0.1
## 866.4542 1 0.1
## 865.2035 1 0.1
## 863.792 1 0.1
## 863.4496 1 0.1
## 863.37 1 0.1
## 862.5884 1 0.1
## 860.405 1 0.1
## 86.96 1 0.1
## 86.0763 1 0.1
## 859.5496 1 0.1
## 857.31 1 0.1
## 851.8912 1 0.1
## 85.0097 1 0.1
## 841.6011 1 0.1
## 8390 1 0.1
## 832.0446 1 0.1
## 831.6595 1 0.1
## 831.59 1 0.1
## 8306.0459 1 0.1
## 830.42 1 0.1
## 827.77 1 0.1
## 827.66 1 0.1
## 823.1191 1 0.1
## 822.9766 1 0.1
## 8190 1 0.1
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## 817.6682 1 0.1
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## 809.41 1 0.1
## 8030 1 0.1
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## 7930 1 0.1
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## 7260 1 0.1
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## 7200 1 0.1
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## 1641.8264 1 0.1
## 1634.0441 1 0.1
## 1628.754 1 0.1
## 162.94 1 0.1
## 1619.7656 1 0.1
## 1605.7228 1 0.1
## 1603.0649 1 0.1
## 1599.2813 1 0.1
## 1591.2185 1 0.1
## 1585.51 1 0.1
## 1584.96 1 0.1
## 158.13 1 0.1
## 1574.8315 1 0.1
## 1562.2338 1 0.1
## 1556.9391 1 0.1
## 1552.5758 1 0.1
## 1550.02 1 0.1
## 1538.3466 1 0.1
## 1530.4865 1 0.1
## 1528.5027 1 0.1
## 152.9717 1 0.1
## 152.1418 1 0.1
## 1510.6265 1 0.1
## 1508.53 1 0.1
## 1498.7391 1 0.1
## 149.2137 1 0.1
## 1481.5601 1 0.1
## 1479.8101 1 0.1
## 1479.38 1 0.1
## 1474.6252 1 0.1
## 1467.6908 1 0.1
## 1466.9917 1 0.1
## 146.7933 1 0.1
## 146.079 1 0.1
## 1454.24 1 0.1
## 1441.6909 1 0.1
## 1439.02 1 0.1
## 1437.52 1 0.1
## 1435.2444 1 0.1
## 1433.7565 1 0.1
## 143.33 1 0.1
## 14225.3135 1 0.1
## 1411.6324 1 0.1
## 1408.8621 1 0.1
## 1405.6776 1 0.1
## 1403.4971 1 0.1
## 140.6149 1 0.1
## 140.07 1 0.1
## 1383.63 1 0.1
## 1380.1622 1 0.1
## 1377.0441 1 0.1
## 1371.7496 1 0.1
## 1371.37 1 0.1
## 1366.16 1 0.1
## 1362.4678 1 0.1
## 1361.9829 1 0.1
## 13544.9785 1 0.1
## 1348.7238 1 0.1
## 1340.0023 1 0.1
## 134.4366 1 0.1
## 1330.389 1 0.1
## 133.6028 1 0.1
## 1312.6384 1 0.1
## 131.3576 1 0.1
## 1309.55 1 0.1
## 1308.255 1 0.1
## 1306.17 1 0.1
## 13033.5283 1 0.1
## 1302.1428 1 0.1
## 1300.79 1 0.1
## 1291.5699 1 0.1
## 1291.2828 1 0.1
## 1290.3817 1 0.1
## 1280.98 1 0.1
## 128.2582 1 0.1
## 1276.5076 1 0.1
## 1274.95 1 0.1
## 1271.38 1 0.1
## 1269.6 1 0.1
## 1261.7902 1 0.1
## 126.0047 1 0.1
## 1259.5554 1 0.1
## 1257.281 1 0.1
## 1254.4161 1 0.1
## 1249.7059 1 0.1
## 1248.4808 1 0.1
## 1244.7603 1 0.1
## 124.71 1 0.1
## 1237.4061 1 0.1
## 1227.87 1 0.1
## 122.0799 1 0.1
## 1217.8118 1 0.1
## 1213.6443 1 0.1
## 1213.0129 1 0.1
## 1209.9685 1 0.1
## 1205.7067 1 0.1
## 1201.6453 1 0.1
## 1196.2823 1 0.1
## 119.7091 1 0.1
## 11841.7432 1 0.1
## 1183.6799 1 0.1
## 1181.89 1 0.1
## 1181.2655 1 0.1
## 1181.2034 1 0.1
## 1178.3425 1 0.1
## 1171.39 1 0.1
## 116.5507 1 0.1
## 115.9015 1 0.1
## 1149.5182 1 0.1
## 1142.8588 1 0.1
## 114.3747 1 0.1
## 1138.5061 1 0.1
## 1137.1615 1 0.1
## 112.6725 1 0.1
## 1117.7709 1 0.1
## 1114.6503 1 0.1
## 11117.4023 1 0.1
## 1109.5569 1 0.1
## 1103.1545 1 0.1
## 1102.0452 1 0.1
## 1099.4039 1 0.1
## 109.7232 1 0.1
## 1089.8239 1 0.1
## 1089.4352 1 0.1
## 10860 1 0.1
## 1086.0234 1 0.1
## 1074.09 1 0.1
## 1066.3792 1 0.1
## 10649.958 1 0.1
## 1064.6899 1 0.1
## 1064.5748 1 0.1
## 1061.7 1 0.1
## 1059.9347 1 0.1
## 10540 1 0.1
## 1054.2761 1 0.1
## 1053.1267 1 0.1
## 1053.0554 1 0.1
## 1042.6234 1 0.1
## 1040.1243 1 0.1
## 1036.0116 1 0.1
## 103.5448 1 0.1
## 1022.5287 1 0.1
## 1020.4655 1 0.1
## 102.8927 1 0.1
## 1015.6563 1 0.1
## 101.8555 1 0.1
## 1006.85 1 0.1
## 1006.017 1 0.1
## 0.81 1 0.1
## Total 807 100.0Then after finding the Frequency we will find the Cumlative Frequency Distirpution for X-values then after that ploting K (md) with Cumlative Frequency Distirpution.
Cumlative_Frequency_Distirpution## # A tibble: 807 x 4
## k.core Frequency Num.samples cumulative_frequency
## <dbl> <dbl> <dbl> <dbl>
## 1 14225. 1 1 0.122
## 2 13545. 1 2 0.244
## 3 13034. 1 3 0.367
## 4 11842. 1 4 0.489
## 5 11117. 1 5 0.611
## 6 10860 1 6 0.733
## 7 10650. 1 7 0.856
## 8 10540 1 8 0.978
## 9 9898. 1 9 1.10
## 10 9533. 1 10 1.22
## # ... with 797 more rowsNow we are Ready to plot our Graph
df = read_xlsx("D:/mohammed/mm.xlsx")
model1 <- lm( df$k.core ~ df$cumulative_frequency)
plot(df$cumulative_frequency,df$k.core,xlab="Cumlative Frequency Distirpution",ylab="k (md)")
abline(model1, lwd=4, col='red', untf=FALSE)
Now by using this relationship,
Vk= (K50-k84.1)/K50
we will be able to quantify Heterogeneity Index (HI)
Vk= (1591.2185-392.78)/1591.2185
Vk= 0.7531577
Lorenz Coefficient
Therefore from this Value, we know that this reservoir is Extremely Heterogeneous.
data = read.csv('karpur.csv', header = TRUE)
head(data)## depth caliper ind.deep ind.med gamma phi.N R.deep R.med SP
## 1 5667.0 8.685 618.005 569.781 98.823 0.410 1.618 1.755 -56.587
## 2 5667.5 8.686 497.547 419.494 90.640 0.307 2.010 2.384 -61.916
## 3 5668.0 8.686 384.935 300.155 78.087 0.203 2.598 3.332 -55.861
## 4 5668.5 8.686 278.324 205.224 66.232 0.119 3.593 4.873 -41.860
## 5 5669.0 8.686 183.743 131.155 59.807 0.069 5.442 7.625 -34.934
## 6 5669.5 8.686 109.512 75.633 57.109 0.048 9.131 13.222 -39.769
## density.corr density phi.core k.core Facies
## 1 -0.033 2.205 33.9000 2442.590 F1
## 2 -0.067 2.040 33.4131 3006.989 F1
## 3 -0.064 1.888 33.1000 3370.000 F1
## 4 -0.053 1.794 34.9000 2270.000 F1
## 5 -0.054 1.758 35.0644 2530.758 F1
## 6 -0.058 1.759 35.3152 2928.314 F1summary(data)## depth caliper ind.deep ind.med
## Min. :5667 Min. :8.487 Min. : 6.532 Min. : 9.386
## 1st Qu.:5769 1st Qu.:8.556 1st Qu.: 28.799 1st Qu.: 27.892
## Median :5872 Median :8.588 Median :217.849 Median :254.383
## Mean :5873 Mean :8.622 Mean :275.357 Mean :273.357
## 3rd Qu.:5977 3rd Qu.:8.686 3rd Qu.:566.793 3rd Qu.:544.232
## Max. :6083 Max. :8.886 Max. :769.484 Max. :746.028
## gamma phi.N R.deep R.med
## Min. : 16.74 Min. :0.0150 Min. : 1.300 Min. : 1.340
## 1st Qu.: 40.89 1st Qu.:0.2030 1st Qu.: 1.764 1st Qu.: 1.837
## Median : 51.37 Median :0.2450 Median : 4.590 Median : 3.931
## Mean : 53.42 Mean :0.2213 Mean : 24.501 Mean : 21.196
## 3rd Qu.: 62.37 3rd Qu.:0.2640 3rd Qu.: 34.724 3rd Qu.: 35.853
## Max. :112.40 Max. :0.4100 Max. :153.085 Max. :106.542
## SP density.corr density phi.core
## Min. :-73.95 Min. :-0.067000 Min. :1.758 Min. :15.70
## 1st Qu.:-42.01 1st Qu.:-0.016000 1st Qu.:2.023 1st Qu.:23.90
## Median :-32.25 Median :-0.007000 Median :2.099 Median :27.60
## Mean :-30.98 Mean :-0.008883 Mean :2.102 Mean :26.93
## 3rd Qu.:-19.48 3rd Qu.: 0.002000 3rd Qu.:2.181 3rd Qu.:30.70
## Max. : 25.13 Max. : 0.089000 Max. :2.387 Max. :36.30
## k.core Facies
## Min. : 0.42 Length:819
## 1st Qu.: 657.33 Class :character
## Median : 1591.22 Mode :character
## Mean : 2251.91
## 3rd Qu.: 3046.82
## Max. :15600.00The Lorenz coefficient of permeability variation is obtained by plotting a graph of cumulative kh versus cumulative phi*h, sometimes called a flow capacity plot.
first, let’s calculate the thickness from the depths.
the differences in depths gives the thickness:
d = data$depth
n = length(d)
h = d[2:n]-d[1:n-1] #differences in depths
head(h)## [1] 0.5 0.5 0.5 0.5 0.5 0.5we need one more value to have the same number of data samples, since most of the values are 0.5, this what I’m going to add.
h = append(h, 0.5)
head(h)## [1] 0.5 0.5 0.5 0.5 0.5 0.5second, lets calculate the cumulative kh, and cumulative phi*h
df = data.frame(h) # we will use data frames to store the data
df['h'] = h
k = data$k.core
df['k'] = k
df['phi'] = data$phi.core
df = df[order(k, decreasing = TRUE),] #sort data such that k is in descending order
df['kh'] = df['k']*df['h'] #multiply k and h
df['sum_kh'] = cumsum(df['kh']) #calculate the cumulative sum
df['sum_kh/total'] = df['sum_kh']/df[n, 'sum_kh'] #divid by the total
# same for phi and h
df['phih'] = df['phi']*df['h']
df['sum_phih'] = cumsum(df['phih'])
df['sum_phih/total'] = df['sum_phih']/df[n, 'sum_phih']
head(df)## h k phi kh sum_kh sum_kh/total phih sum_phih
## 809 0.5 15600.00 27.9000 7800.000 7800.00 0.008237623 13.95000 13.95000
## 808 0.5 14225.31 28.5071 7112.657 14912.66 0.015749339 14.25355 28.20355
## 810 0.5 13544.98 27.7563 6772.489 21685.15 0.022901802 13.87815 42.08170
## 807 0.5 13033.53 29.0333 6516.764 28201.91 0.029784192 14.51665 56.59835
## 806 0.5 11841.74 29.5596 5920.872 34122.78 0.036037257 14.77980 71.37815
## 811 0.5 11117.40 27.5865 5558.701 39681.48 0.041907832 13.79325 85.17140
## sum_phih/total
## 809 0.001241781
## 808 0.002510583
## 810 0.003745967
## 807 0.005038189
## 806 0.006353836
## 811 0.007581664scatter.smooth(df['sum_phih/total'][1:n,1] ,df['sum_kh/total'][1:n,1],
xlab = "fraction of total volume (phih)",
ylab = "fraction of total flow capacity (kh)")
we need to calculate the area under the above graph, I wrote the fallowing code to find the area, basically this is an implementation of the trapezoid method to fined AUC.
note: I’m familiar with python, and still new to R, writing my own code actully took less time than searching for some libraries that would calculate the area for me.
x = c(df['sum_phih/total'][1:n,1])
dx = x[2:n] - x[1:n-1]
y1 = c(df['sum_kh/total'][1:n,1])[1:n-1]
y2 = c(df['sum_kh/total'][1:n,1])[2:n]
lower_area = Reduce("+", y1*dx)
upper_area = Reduce("+", y2*dx)
area = (lower_area + upper_area)/2
print(area)## [1] 0.7247318finally we calculate Lorenz coefficient as fallows:
Lorenz coefficient = (Area - 0.5)/0.5
Lorenz_coefficient = (area - 0.5)/0.5
print(Lorenz_coefficient)## [1] 0.4494636The Lorenz coefficient of permeability variation also varies from zero to one. Unfortunately, the Lorenz coefficient is not a unique measure of reservoir heterogeneity. Several different permeability distributions can give the same value of Lorenz coefficient. For log-normal permeability distributions, the Lorenz coefficient is very similar to the Dykstra-Parsons coefficient of permeability variation.