Dranchuk-Purvis-Robinson correlation

knitr::opts_chunk$set(echo=T, comment=NA, error=T, warning=F, message = F, fig.align = 'center', results="hold")

Why DPR is different

The Dranchuk-Purvis-Robinson correlation is different than the other correlations in the sense that the authors were looking at redrawing three of the isotherms of the original work by Standing-Katz: the curves at Tpr=1.05, Tpr=1.1 and Tpr=1.15. The authors were expecting that later the corrections to be adopted with experimental data matching those two new curves. We haven't found a paper confirming that yet. [See @Dranchuk1973].

Other than that, the DPR correlation behaves pretty well but not much better than the DAK correlation.

Get z at selected Ppr and Tpr

Use the corelation to calculate z. From the Standing-Katz chart we obtain z at a digitized point at the given Tpr and Ppr.

# get a z value
library(zFactor)

ppr <- 1.5
tpr <- 2.0

z.calc <- z.DranchukPurvisRobinson(pres.pr = ppr, temp.pr = tpr)

# get a z value from the SK chart at the same Ppr and Tpr
z.chart <- getStandingKatzMatrix(tpr_vector = tpr, 
                      pprRange = "lp")[1, as.character(ppr)]

# calculate the APE
ape <- abs((z.calc - z.chart) / z.chart) * 100

df <- as.data.frame(list(Ppr = ppr,  z.calc =z.calc, z.chart = z.chart, ape=ape))
rownames(df) <- tpr
df
# HY = 0.9580002; # DAK = 0.9551087

Get z at selected Ppr and Tpr=1.1

From the Standing-Katz chart we obtain z at a digitized point:

library(zFactor)

ppr <- 1.5
tpr <- 1.1
z.calc <- z.DranchukPurvisRobinson(pres.pr = ppr, temp.pr = tpr)

# From the Standing-Katz chart we obtain a digitized point:
z.chart <- getStandingKatzMatrix(tpr_vector = tpr, 
                      pprRange = "lp")[1, as.character(ppr)]

# calculate the APE
ape <- abs((z.calc - z.chart) / z.chart) * 100

df <- as.data.frame(list(Ppr = ppr,  z.calc =z.calc, z.chart = z.chart, ape=ape))
rownames(df) <- tpr
df

At lower Tpr we can see that there is some difference between the values of z from the DPR calculation and the value read from the Standing-Katz chart.

Get values of z for combinations of Ppr and Tpr

In this example we provide vectors instead of a single point. With the same ppr and tpr vectors that we use for the correlation, we do the same for the Standing-Katz chart. We want to compare both and find the absolute percentage error.

# test HY with 1st-derivative using the values from paper 
ppr <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5) 
tpr <- c(1.05, 1.1, 1.7, 2) 

# calculate using the correlation
z.calc <- z.DranchukPurvisRobinson(ppr, tpr)

# With the same ppr and tpr vector, we do the same for the Standing-Katz chart
z.chart <- getStandingKatzMatrix(ppr_vector = ppr, tpr_vector = tpr)
ape <- abs((z.calc - z.chart) / z.chart) * 100

# calculate the APE
cat("z.correlation \n"); print(z.calc)
cat("\n z.chart \n"); print(z.chart)
cat("\n APE \n"); print(ape)

You can see errors of r round(ape["1.05", "1.5"],2)% and r round(ape["1.05", "2.5"],2)% in the isotherm Tpr=1.05. Other errors, greater than one, can also be found at the isotherm Tpr=1.1: r round(ape["1.1", "2.5"],2)%. Then, the rest of the Tpr curves are fine.

Analyze the error at the isotherms

Applying the function summary over the transpose of the matrix:

sum_t_ape <- summary(t(ape))
sum_t_ape

We see that the errors at Tpr=1.05 in z are considerable with a r sum_t_ape[1,1]% and r sum_t_ape[6,1]%, and a r sum_t_ape[1,2]% and r sum_t_ape[6,2]% for Tpr=1.10. We will explore later a comparative tile chart where we confirm these early calculations.

Analyze the error for greater values of Tpr

library(zFactor)
# enter vectors for Tpr and Ppr
tpr2 <- c(1.2, 1.3, 1.5, 2.0, 3.0) 
ppr2 <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5) 
# get z values from the SK chart
z.chart <- getStandingKatzMatrix(ppr_vector = ppr2, tpr_vector = tpr2, pprRange = "lp")

# We do the same with the DPR correlation:
# calculate z values at lower values of Tpr
z.calc <- z.DranchukPurvisRobinson(pres.pr = ppr2, temp.pr = tpr2) 
ape <- abs((z.calc - z.chart) / z.chart) * 100

# calculate the APE
cat("z.correlation \n"); print(z.calc)
cat("\n z.chart \n"); print(z.chart)
cat("\n APE \n"); print(ape)

We observe that at Tpr above or equal to 1.2 the DAK correlation behaves very well.

Analyze the error at the isotherms

Applying the function summary over the transpose of the matrix to observe the error of the correlation at each isotherm.

sum_t_ape <- summary(t(ape))
sum_t_ape
 # Hall-Yarborough
 #      1.2               1.3              1.5               2         
 # Min.   :0.05224   Min.   :0.1105   Min.   :0.1021   Min.   :0.0809  
 # 1st Qu.:0.09039   1st Qu.:0.2080   1st Qu.:0.1623   1st Qu.:0.1814  
 # Median :0.28057   Median :0.3181   Median :0.1892   Median :0.1975  
 # Mean   :0.30122   Mean   :0.3899   Mean   :0.2597   Mean   :0.2284  
 # 3rd Qu.:0.51710   3rd Qu.:0.5355   3rd Qu.:0.3533   3rd Qu.:0.2627  
 # Max.   :0.57098   Max.   :0.8131   Max.   :0.5162   Max.   :0.4338  
 #       3          
 # Min.   :0.09128  
 # 1st Qu.:0.17466  
 # Median :0.35252  
 # Mean   :0.34820  
 # 3rd Qu.:0.52184  
 # Max.   :0.59923  

We see that the errors in z with DPR are far lower than Hall-Yarborough with a r sum_t_ape[1,1]% and r sum_t_ape[6,1]% for Tpr=1.2, and a r sum_t_ape[1,2]% and r sum_t_ape[6,2]% for Tpr=1.3.

Prepare to plot SK vs DPR correlation

The error bars represent the difference between the calculated values by the DPR corrrelation and the z values read from the Standing-Katz chart.

library(zFactor)
library(tibble)
library(ggplot2)

tpr2 <- c(1.05, 1.1, 1.2, 1.3) 
ppr2 <- c(0.5, 1.0, 1.5, 2, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5) 

sk_corr_2 <- createTidyFromMatrix(ppr2, tpr2, correlation = "DPR")
as_tibble(sk_corr_2)

p <- ggplot(sk_corr_2, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) +
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=.4,
                  position=position_dodge(0.05))
print(p)

We observe that with Dranchuk-Purvis-Robinson there are still errors or differences between the z values in the Standing-Katz chart and the values obtained from the correlation but they are not so bad as in the HY correlation.

Analysis at the lowest Tpr

This is the isotherm where we see the greatest error.

library(zFactor)

sk_corr_3 <- sk_corr_2[sk_corr_2$Tpr==1.05,]
sk_corr_3

p <- ggplot(sk_corr_3, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) +
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=.2,
                  position=position_dodge(0.05))
print(p)

Analyzing performance of the DPR correlation for all the Tpr curves

In this last example, we compare the values of z at all the isotherms. We use the function getStandingKatzTpr to obtain all the isotherms or Tpr curves in the Standing-Katz chart that have been digitized. The next function createTidyFromMatrix calculate z using the correlation and prepares a tidy dataset ready to plot.

library(ggplot2)
library(tibble)

# get all `lp` Tpr curves
tpr_all <- getStandingKatzTpr(pprRange = "lp")
ppr <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5) 

sk_corr_all <- createTidyFromMatrix(ppr, tpr_all, correlation = "DPR")
as_tibble(sk_corr_all)

p <- ggplot(sk_corr_all, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) +
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=.4,
                  position=position_dodge(0.05))
print(p)

The greatest errors are localized in two of the Tpr curves: at 1.05 and 1.1.

Range of applicability of the correlation

# MSE: Mean Squared Error
# RMSE: Root Mean Squared Error
# RSS: residual sum of square
# ARE:  Average Relative Error, %
# AARE: Average Absolute Relative Error, %
library(dplyr)
grouped <- group_by(sk_corr_all, Tpr, Ppr)

smry_tpr_ppr <- summarise(grouped, 
          RMSE= sqrt(mean((z.chart-z.calc)^2)), 
          MPE = sum((z.calc - z.chart) / z.chart) * 100 / n(),
          MAPE = sum(abs((z.calc - z.chart) / z.chart)) * 100 / n(), 
          MSE = sum((z.calc - z.chart)^2) / n(), 
          RSS = sum((z.calc - z.chart)^2),
          MAE = sum(abs(z.calc - z.chart)) / n(),
          RMLSE = sqrt(1/n()*sum((log(z.calc +1)-log(z.chart +1))^2))
          )

ggplot(smry_tpr_ppr, aes(Ppr, Tpr)) + 
    geom_tile(data=smry_tpr_ppr, aes(fill=MAPE), color="white") +
    scale_fill_gradient2(low="blue", high="red", mid="yellow", na.value = "pink",
                         midpoint=12.5, limit=c(0, 25), name="MAPE") + 
    theme(axis.text.x = element_text(angle=45, vjust=1, size=11, hjust=1)) + 
    coord_equal() +
    ggtitle("Dranchuk-Purvis-Robinson", subtitle = "DPR")

The greatest errors are localized in two of the Tpr curves: 1.05 and barely at 1.1

Plotting the Tpr and Ppr values that show more error

library(dplyr)

sk_corr_all %>%
    filter(Tpr %in% c("1.05", "1.1")) %>%
    ggplot(aes(x = z.chart, y=z.calc, group = Tpr, color = Tpr)) +
    geom_point(size = 3) +
    geom_line(aes(x = z.chart, y = z.chart), color = "black") +
    facet_grid(. ~ Tpr) +
    geom_errorbar(aes(ymin=z.calc-abs(dif), ymax=z.calc+abs(dif)), 
                  position=position_dodge(0.5))

Looking numerically at the errors

Finally, the dataframe with the calculated errors between the z from the correlation and the z read from the chart:

as_tibble(smry_tpr_ppr)

References



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zFactor documentation built on Aug. 1, 2019, 5:04 p.m.