Heterogeneity Quantification Using Dykstra-Parson Cofficient Method

Ali Abdulhussein Abdulazeez ,BUOG,Basrah,Iraq

2024/10/21

Petroleum reservoirs are nearly always heterogeneous. Therefore, it is necessary to be able to quantify the degree of heterogeneity of a reservoir.

Dykstra-Parson coefficient

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

For simplicity, sorting permeability values in a descending order.

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

Since values are sorted, number of samples \>= k is actually the row indices

sample = c(1: length(k))

Calculating %\>=k

k_percent=(sample * 100)/length(k)

When plotting the output, a straight line is observed.

xlab = "Portion of Total Samples Having Larger or Equal K" 
ylab = "Permeability (md)" 
plot(k_percent, k, log= 'y', xlab = xlab, ylab = ylab, pch = 10, cex = 0.5,col = "#001c49")

A linear model is fitted to the data.

log_k = log(k) 
model = lm(log_k ~ k_percent) 
plot(k_percent, log_k, xlab = xlab , ylab = ylab, pch = 10, cex = 0.5, col = "#001c49") 
abline(model, col = 'red' , lwd =2)

Brief description of the model coefficients and evaluation metrics.

summary(model)

Call:
lm(formula = log_k ~ k_percent)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.8697 -0.2047  0.1235  0.3150  0.4280 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  9.2584172  0.0377994  244.94   <2e-16 ***
k_percent   -0.0425617  0.0006541  -65.07   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5404 on 817 degrees of freedom
Multiple R-squared:  0.8382,    Adjusted R-squared:  0.838 
F-statistic:  4234 on 1 and 817 DF,  p-value: < 2.2e-16

calculating the HI mathematically.

new_data = data.frame(k_percent = c(50, 84.1)) 
predicted_values = predict(model, new_data) 
heterogeneity_index = (predicted_values[1] - predicted_values[2]) /
predicted_values[1] 
heterogeneity_index

        1 
0.2035464 

So, the heterogeneity index = 0.2 , which means slightly heterogeneous reservoir.