Core Permeability Correction to Log Scale using Multiple Linear Regression with Logarithmic Transformation
“Zainab Mohammed”
“Basra University For Oil and Gas”
“29/11/2024”
In this correlation we will use Facies to correct core PermeabilityUpload Data .
data<-read.csv("C:/Users/Dell/Desktop/rdata/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. :15600.00At First , we should find the correlation between phi.core and phi.log
model1 <- lm(data$phi.core/100 ~ data$phi.N)
summary(model1)##
## Call:
## lm(formula = data$phi.core/100 ~ data$phi.N)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.135237 -0.030779 0.009432 0.033563 0.104025
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.30962 0.00485 63.846 <2e-16 ***
## data$phi.N -0.18207 0.02080 -8.753 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04368 on 817 degrees of freedom
## Multiple R-squared: 0.08573, Adjusted R-squared: 0.08462
## F-statistic: 76.61 on 1 and 817 DF, p-value: < 2.2e-16plot(data$phi.N,data$phi.core,xlab="phi.log",ylab="phi.core",axes = F)
axis(2,col = "darkgreen",col.axis="black")
axis(1,col = "darkgreen",col.axis="red")
abline(model1, lwd=3, col='green')
Plot a histogram to the measured core permeabilit, From the below histograms, we see it has a non-normal distribution.
par(mfrow=c(1,1))
hist(data$k.core, main='Histogram of Permeability', xlab='Permeability, md', col='darkblue')
Multiple Linear Regression of Core Permeability as a function of other data
model1<-lm(k.core~.-1 ,data=data)
par(mfrow=c(2,2))
plot (model1)
phi.corel <- predict(model1,data)
#cbind(karpur$phi.core/100,phi.corel)Construction a relationship between permeability calculated from core and core porosity corrected to the log scale in order to get core premeability corrected to the log scale
model2<-lm(k.core~phi.corel+Facies-1,data=data)
summary(model2)##
## Call:
## lm(formula = k.core ~ phi.corel + Facies - 1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5585.6 -568.9 49.2 476.5 8928.4
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## phi.corel 1.000e+00 4.690e-02 21.32 <2e-16 ***
## FaciesF1 -2.353e-11 1.617e+02 0.00 1
## FaciesF10 4.619e-11 1.026e+02 0.00 1
## FaciesF2 2.128e-11 4.433e+02 0.00 1
## FaciesF3 1.105e-10 1.766e+02 0.00 1
## FaciesF5 2.318e-11 2.795e+02 0.00 1
## FaciesF7 -2.737e-11 4.261e+02 0.00 1
## FaciesF8 -6.261e-11 1.851e+02 0.00 1
## FaciesF9 -1.092e-11 1.046e+02 0.00 1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1253 on 810 degrees of freedom
## Multiple R-squared: 0.8457, Adjusted R-squared: 0.844
## F-statistic: 493.2 on 9 and 810 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(model2)
We note from the above plot that there are some inaccurate (anomalous) or ineffective points that may cause errors in the model, so they should be reduced and then reconstruct the relationship.
m1.red <- lm(k.core ~ depth+gamma+R.deep+R.med+SP+density+phi.core+Facies-1, data = data)
summary(m1.red)##
## Call:
## lm(formula = k.core ~ depth + gamma + R.deep + R.med + SP + density +
## phi.core + Facies - 1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5461.7 -545.5 37.0 505.0 9072.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## depth 8.088 1.059 7.639 6.23e-14 ***
## gamma -51.915 4.526 -11.471 < 2e-16 ***
## R.deep -22.467 6.261 -3.588 0.000353 ***
## R.med 48.859 8.730 5.597 2.99e-08 ***
## SP -6.490 3.118 -2.082 0.037671 *
## density 2013.607 1047.054 1.923 0.054818 .
## phi.core 188.002 21.791 8.628 < 2e-16 ***
## FaciesF1 -51703.918 5802.150 -8.911 < 2e-16 ***
## FaciesF10 -50893.923 5917.391 -8.601 < 2e-16 ***
## FaciesF2 -51009.565 5854.625 -8.713 < 2e-16 ***
## FaciesF3 -51322.213 5871.731 -8.741 < 2e-16 ***
## FaciesF5 -51174.044 6008.663 -8.517 < 2e-16 ***
## FaciesF7 -52302.690 5971.976 -8.758 < 2e-16 ***
## FaciesF8 -53362.629 6033.789 -8.844 < 2e-16 ***
## FaciesF9 -54796.231 6100.716 -8.982 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1266 on 804 degrees of freedom
## Multiple R-squared: 0.8436, Adjusted R-squared: 0.8407
## F-statistic: 289.1 on 15 and 804 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(m1.red)
now, we will predict by using the (m1) model to obtain predicted k.core
k.core.pred1 <- predict(m1.red,data)by using “cbind” function we can see the difference between the “measured kcore” and “predicted k.core”
#cbind(data$k.core ,k.core.pred1 )also we can see the difference between the “measured kcore” and “predicted k.core” from their plots
par(mfrow=c(1,3))
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(data$k.core),type="l", col="red", lwd = 5, pch=17, xlab='Measured',
ylab='Depth,m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2)
grid()
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(k.core.pred1),type="l", col="blue", lwd = 5, pch=17, xlab='Predicted',
ylab='Depth, m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2,)
grid()
# Matching the two curves for ease of comparison
par(mfrow=c(1,1))
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(data$k.core),type="l", col="red", lwd = 5, xlab='Permeability',
ylab='Depth,m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2)
grid()
par(new = TRUE)
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(k.core.pred1),type="l", col="blue", lwd = 5, xlab='',
ylab='Depth,m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2, main='R-sq=8436')
grid()
legend('topright', legend=c("Observed", "Predicted"), lty=c(1,1), col=c("red", "blue"))
addition adjusted R square & root mine square error for m1 to our matching plot after we calculat them.
AdjR.sq1 <- 1-sum((k.core.pred1 - data$k.core)^2)/sum((data$k.core-mean(data$k.core))^2)
AdjR.sq1## [1] 0.6847518mspe.model1 <- sqrt(sum((k.core.pred1 - data$k.core)^2)/nrow(data))
mspe.model1## [1] 1254.46par(mfrow=c(1,1))
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(data$k.core),type="p", col="red", lwd = 5, pch=16, xlab='Permeability',
ylab='Depth, m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2)
grid()
par(new = TRUE)
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(k.core.pred1),type="p", col="blue", lwd = 5, pch=15, xlab='',
ylab='Depth, m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2, main='AdjR.sq1=0.6847 & RMSE=1254')
grid()
legend('topright', legend=c("Observed", "Predicted"), pch=c(16,15), col=c("red", "blue"))
Construct core permeability modeling as log transformation
hist(log10(data$k.core) ,main='Histogram of log Permeability', xlab='log Permeability, md', col='red')
m2<-lm(log10(k.core) ~ .-1,data=data)
summary(m2)##
## Call:
## lm(formula = log10(k.core) ~ . - 1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5804 -0.1138 0.0322 0.1529 0.7384
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## depth 0.0007425 0.0004718 1.574 0.1160
## caliper -0.4605945 0.2693103 -1.710 0.0876 .
## ind.deep -0.0007951 0.0006222 -1.278 0.2017
## ind.med 0.0007137 0.0006833 1.044 0.2966
## gamma -0.0091269 0.0015885 -5.746 1.30e-08 ***
## phi.N -1.7628155 0.3901024 -4.519 7.16e-06 ***
## R.deep -0.0025878 0.0016620 -1.557 0.1199
## R.med 0.0044073 0.0023960 1.839 0.0662 .
## SP -0.0016935 0.0008312 -2.037 0.0419 *
## density.corr 1.4462633 1.2712045 1.138 0.2556
## density 1.6148374 0.3100921 5.208 2.44e-07 ***
## phi.core 0.0948634 0.0060329 15.724 < 2e-16 ***
## FaciesF1 -2.3461877 4.6532000 -0.504 0.6143
## FaciesF10 -2.2675417 4.6652182 -0.486 0.6271
## FaciesF2 -2.3646211 4.6544047 -0.508 0.6116
## FaciesF3 -2.3769425 4.6697172 -0.509 0.6109
## FaciesF5 -2.2367684 4.6697432 -0.479 0.6321
## FaciesF7 -2.0650257 4.6623059 -0.443 0.6579
## FaciesF8 -2.4438111 4.6776859 -0.522 0.6015
## FaciesF9 -2.7023993 4.7023453 -0.575 0.5657
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3335 on 799 degrees of freedom
## Multiple R-squared: 0.9891, Adjusted R-squared: 0.9888
## F-statistic: 3613 on 20 and 799 DF, p-value: < 2.2e-16# Plot the model (m2)
par(mfrow=c(2,2))
plot(m2)
We note from the above plot that there are some inaccurate (anomalous) or ineffective points that may cause errors in the model, so they should be reduced and then reconstruct the relationship.But this model”m2” is more accurate than model “m1”.
m2.red <- lm(log10(k.core) ~ caliper+gamma+phi.N+SP+density+phi.core+Facies-1,data=data)
summary(m2.red)##
## Call:
## lm(formula = log10(k.core) ~ caliper + gamma + phi.N + SP + density +
## phi.core + Facies - 1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.60108 -0.11891 0.03253 0.15792 0.70925
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## caliper -0.840136 0.182354 -4.607 4.74e-06 ***
## gamma -0.010369 0.001337 -7.754 2.68e-14 ***
## phi.N -1.596764 0.359392 -4.443 1.01e-05 ***
## SP -0.001585 0.000793 -1.998 0.04600 *
## density 1.762145 0.287056 6.139 1.31e-09 ***
## phi.core 0.093920 0.005931 15.835 < 2e-16 ***
## FaciesF1 4.981486 1.783281 2.793 0.00534 **
## FaciesF10 5.101106 1.793029 2.845 0.00455 **
## FaciesF2 4.995149 1.796855 2.780 0.00556 **
## FaciesF3 4.967152 1.799799 2.760 0.00591 **
## FaciesF5 5.158638 1.760353 2.930 0.00348 **
## FaciesF7 5.272425 1.759300 2.997 0.00281 **
## FaciesF8 4.952389 1.753512 2.824 0.00486 **
## FaciesF9 4.687264 1.762588 2.659 0.00799 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3346 on 805 degrees of freedom
## Multiple R-squared: 0.9889, Adjusted R-squared: 0.9887
## F-statistic: 5128 on 14 and 805 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(m2.red)
now, we will predict by using the (m2) model to obtain predicted k.core as log transformation.
k.core.pred2 <- predict(m2.red,data)back transformation to the predicted core permeability
k.core.pred2 <- 10^(k.core.pred2) # this back transformation to log By using “cbind” function we can see the difference between the “measured kcore” and “predicted k.core as log transformation”
# cbind(data$k.core ,k.core.pred2 )Also we can see the difference between the “measured kcore” and “predicted k.core as log transformation” from their plots
par(mfrow=c(1,3))
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(data$k.core),type="l", col="red", lwd = 5, xlab='Measured',
ylab='Depth,m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2)
grid()
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(k.core.pred2),type="l", col="green", lwd = 5, xlab='Predicted',
ylab='Depth, m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2,)
grid()
# Matching the two curves for ease of comparison
par(mfrow=c(1,1))
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(data$k.core),type="l", col="red", lwd = 5, xlab='Permeability',
ylab='Depth,m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2)
grid()
par(new = TRUE)
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(k.core.pred2),type="l", col="green", lwd = 5, xlab='',
ylab='Depth,m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2, main='R-sq=8436')
grid()
legend('topright', legend=c("Observed", "Predicted"), lty=c(1,1), col=c("red", "green"))
addition adjusted R square & root mine square error for m1 to our matching plot after we calculat them.
AdjR.sq2 <- 1-sum((k.core.pred2 - data$k.core)^2)/sum((data$k.core-mean(data$k.core))^2)
AdjR.sq2## [1] 0.6366977mspe.model2 <- sqrt(sum((k.core.pred2 - data$k.core)^2)/nrow(data))
mspe.model2## [1] 1346.681par(mfrow=c(1,1))
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(data$k.core),type="p", col="red", lwd = 5, pch=16, xlab='Permeability',
ylab='Depth, m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2)
grid()
par(new = TRUE)
plot(y=y<-(data$depth),ylim=rev(range(data$depth)),x=x<-(k.core.pred2),type="p", col="green", lwd = 5, pch=17, xlab='',
ylab='Depth, m', xlim=c(0,16000), cex=1.5, cex.lab=1.5, cex.axis=1.2, main='AdjR.sq1=0.6367 & RMSE=1346')
grid()
legend('topright', legend=c("Observed", "Predicted"), pch=c(16,17), col=c("red", "green"))