Artificial Neural Network correlation

In zFactor: Calculate the Compressibility Factor 'z' for Hydrocarbon Gases

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

How do we find the limits of accuracy in the ANN10 correlation

Get z at selected Ppr and Tpr

# get a z value using DPR correlation
library(zFactor)
# library(rJava)

z.Ann10(pres.pr = 1.5, temp.pr = 2.0)
# HY = 0.9580002

From the Standing-Katz chart we obtain a digitized point at the same Ppr and Tpr:

# get a z value from the SK chart at the same Ppr and Tpr
library(zFactor)

tpr_vec <- c(2.0)
getStandingKatzMatrix(tpr_vector = tpr_vec, 
                      pprRange = "lp")[1, "1.5"]

It looks pretty good.

Get z at selected Ppr and Tpr

library(zFactor)
z.Ann10(pres.pr = 1.5, temp.pr = 1.1)

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

library(zFactor)
tpr_vec <- c(1.1)
getStandingKatzMatrix(tpr_vector = tpr_vec, 
                      pprRange = "lp")[1, "1.5"]

At lower Tpr there is some error. We see a difference between the values of z from the ANN10 calculation and the value read from the Standing-Katz chart.

Get values of z for several Ppr and Tpr

# 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(1.3, 1.5, 1.7, 2) 

dpr <- z.Ann10(ppr, tpr)
print(dpr)

# From Hall-Yarborough
#    0.5       1.5       2.5       3.5       4.5       5.5       6.5
# 1.3 0.9176300 0.7534433 0.6399020 0.6323003 0.6881127 0.7651710 0.8493794
# 1.5 0.9496855 0.8581232 0.7924067 0.7687902 0.7868071 0.8316848 0.8906351
# 1.7 0.9682547 0.9134862 0.8756412 0.8605668 0.8694525 0.8978885 0.9396353
# 2   0.9838234 0.9580002 0.9426939 0.9396286 0.9490995 0.9697839 0.9994317

# From Dranchuk-AbouKassem
#  0.5       1.5       2.5       3.5       4.5       5.5       6.5
# 1.3 0.9203019 0.7543694 0.6377871 0.6339357 0.6898314 0.7663247 0.8499523
# 1.5 0.9509373 0.8593144 0.7929993 0.7710525 0.7896224 0.8331893 0.8904317
# 1.7 0.9681353 0.9128087 0.8753784 0.8619509 0.8721085 0.9003962 0.9409634
# 2   0.9824731 0.9551087 0.9400752 0.9385273 0.9497137 0.9715388 1.0015560

With the same ppr and tpr vectors, we do the same for the Standing-Katz chart:

library(zFactor)

sk <- getStandingKatzMatrix(ppr_vector = ppr, tpr_vector = tpr)
print(sk)

Subtract the two matrices and find the difference:

err <- round((sk - dpr) / sk * 100, 2)
err

# DAK
# 0.5   1.5  2.5   3.5   4.5   5.5   6.5
# 1.30 -0.47  0.22 0.03 -0.15 -0.85 -0.97 -0.71
# 1.50 -0.31 -0.04 0.13 -0.14  0.05  0.34  0.18
# 1.70 -0.01  0.13 0.07 -0.58 -0.94 -0.38  0.11
# 2.00 -0.05  0.09 0.10 -0.16 -0.50 -0.26  0.14

Error by Ppr and by PPr

print(colSums(err))

print(rowSums(err))

Analyze the error for smaller values of Tpr

library(zFactor)

tpr2 <- c(1.05, 1.1) 
ppr2 <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5) 

sk2 <- getStandingKatzMatrix(ppr_vector = ppr2, tpr_vector = tpr2, pprRange = "lp")
sk2

We do the same with the DPR correlation:

# calculate z values at lower values of Tpr
library(zFactor)

dpr2 <- z.Ann10(ppr2, tpr2)
print(dpr2)

Subtract the matrices and calculate the error in percentage:

err2 <- round((sk2 - dpr2) / sk2 * 100, 2)
err2

# DAK
# 0.5    1.5    2.5   3.5   4.5   5.5
# 1.05 -0.13 -12.15 -12.78 -7.49 -4.34 -1.68
# 1.10 -0.36  -4.79  -4.97 -3.56 -2.14 -1.21

Transposing the matrix with Tpr as columns and Ppr as rows:

t_err2 <- t(err2)
t_err2

A statistical summary by Tpr curve:

sum_t_err2 <- summary(t_err2)
sum_t_err2

# We can see that the errors in `z` with `ANN10` are less than `HY` with a `r sum_t_err2[1,1]`% and `r sum_t_err2[6,1]`% for `Tpr = 1.05`, and a `r sum_t_err2[1,2]`% and `r sum_t_err2[6,2]`% for `Tpr = 1.10`. 

Prepare to plot SK chart vs ANN10 model.

library(zFactor)
library(tibble)

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_dpr_2 <- createTidyFromMatrix(ppr2, tpr2, correlation = "N10")
as.tibble(sk_dpr_2)

library(ggplot2)

p <- ggplot(sk_dpr_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)

Analysis at the lowest Tpr

Extract only values at Tpr = 1.05.

sk_dpr_3 <- sk_dpr_2[sk_dpr_2$Tpr==1.05,]
sk_dpr_3

p <- ggplot(sk_dpr_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=.4,
                  position=position_dodge(0.05))
print(p)

summary(sk_dpr_3)

 #         dif ANN10      
 # Min.   :-0.048404  
 # 1st Qu.:-0.035300  
 # Median :-0.025978  
 # Mean   :-0.023178  
 # 3rd Qu.:-0.009960  
 # Max.   : 0.002325

Analyzing performance of the ANN10 correlation for all the Tpr curves

library(ggplot2)
library(tibble)

# get all `lp` Tpr curves
tpr_all <- getCurvesDigitized(pprRange = "lp")
ppr <- c(0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5) 
sk_corr_all <- createTidyFromMatrix(ppr, tpr_all, correlation = "N10")
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)

# MSE: Mean Squared Error
# RMSE: Root Mean Sqyared 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, 
          #mean=mean(z.calc), 
          #sd=sd(z.calc), 
          RMSE= sqrt(mean((z.chart-z.calc)^2)), 
          MSE = sum((z.calc - z.chart)^2) / n(), 
          # rmse2 = sqrt(sum((z.chart-z.calc)^2)/n()),
          RSS = sum((z.calc - z.chart)^2),
          ARE = sum((z.calc - z.chart) / z.chart) * 100 / n(),
          AARE = sum( abs((z.calc - z.chart) / z.chart)) * 100 / n()
          )

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

Looking numerically at the errors in DKP vs SK chart

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

# calculate HY for the given Tpr
all_dak <- z.Ann10(pres.pr = ppr, temp.pr = tpr)
cat("Calculated from the correlation \n")
print(all_dak) 

cat("\nStanding-Katz chart\n")
all_sk <- getStandingKatzMatrix(ppr_vector = ppr, tpr_vector = tpr)
all_sk

# find the error
cat("\n Errors in percentage \n")
all_err <- round((all_sk - all_dak) / all_sk * 100, 2)  # in percentage
all_err

cat("\n Errors in Ppr\n")
summary(all_err)

# for the transposed matrix
cat("\n Errors for the transposed matrix: Tpr \n")
summary(t(all_err))

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