New correlations for prediction of saturated and undersaturated oil viscosity of Arabian oil fields
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Abstract
New correlations for saturated and undersaturated oil viscosity were developed for Saudi Arabian crude oil. The data consist of 79 and 71 experimental measurements of saturated and undersaturated crude oil viscosity, respectively, at reservoir conditions. Other PVT measurements above and below bubble point pressure are also included. The new correlations were developed using genetic programming approach. The new models were developed and tested using linear genetic programming (GP) technique. The models efficiency was compared to existing correlations. Average absolute relative deviation, coefficient of correlation, and crossplots were used to evaluate the proposed models, and their outputs indicate the accuracy of the GP technique and the superiority of the developed models in comparison with the commonly utilized models tested.
Keywords
Genetic programming Oil viscosity Saturated Undersaturated CorrelationIntroduction
Crude oil viscosity is an important physical property that controls and influences the flow of oil through porous media and pipelines. The viscosity, in general, is defined as the internal resistance of a fluid to flow. Oil viscosity is a strong function of many thermodynamic and physical properties such as pressure, temperature, solution gas–oil ratio (GOR), bubble point pressure, gas gravity, and oil gravity. Viscosity of crude oil is a fundamental factor in simulating reservoirs, forecasting production as well as planning thermal enhanced oil recovery methods that make its accurate determination necessary.
Saturated oil viscosity
Numerous correlations have been proposed to calculate the oil viscosity. These correlations predict viscosities from available fieldmeasured variables including reservoir temperature, oil API gravity, solution gas–oil ratio, pressure, and saturation pressure.
Chew and Connally (1958) presented their crude viscosity correlation as a function of dead oil viscosity and solution gas–oil ratio under reservoir conditions. Viscosity was measured for data of 457 crude oil samples gathered from different areas of USA, Canada, and South America. Measurements were conducted within the ranges of 72–292 °F, 132–5645 psia, and 51–3544 cu ft/bbl for reservoir temperature, bubble point pressure, and solution gas–oil ratio at bubble point, respectively.
Beggs and Robinson (1975) developed fairly accurate and simple crude oil viscosity model based on API gravity temperature and solution gas–oil ratio. Measurements of 600 samples dataset were used to derive the correlation with pressure range of 0.0–5250 psig, solution GOR of 20–2070 scf/STB, oil gravity of 16–58 °API, and temperature of 70–295 °F. They limit their correlation on data that do not have crude composition and suggest using different correlations for better accuracy if composition is available.
Later, Khan et al. (1987) published their empirical correlation using Saudi Arabian crude oil viscosity measured using rolling ball viscometer at various pressures and temperatures. The study utilized viscosity data of 75 bottomhole samples taken from 62 Saudi oil reservoirs. A total of 1691 data measurements below the bubble point pressure were used to derive the correlation which is simply based on crude bubble point viscosity, pressure, and bubble point pressure. They compared their model with Begs and Robinson and Chew and Connally and claimed that their own correlation was the most accurate for Saudi crudes.
Naseri et al. (2005) used PVT experimental data of 472 series of Iranian oil reservoirs in developing their empirical correlation. These data include oil API gravity, reservoir temperature, saturation pressure, solution gas–oil ratio, and PVT measurements at reservoir temperature. Out of the total dataset, 250 were used to develop the empirical model and the rest was spared for validation purposes. Dead viscosity and bubble point pressure were used as input parameter and model developed was of good accuracy exceeding that of the models they compared with average absolute error of 26.31%.
Undersaturated oil viscosity
Many correlations have been proposed to calculate the undersaturated oil viscosity. These correlations predict viscosities from available field samples including reservoir temperature, oil API gravity, solution gas–oil ratio, pressure, and saturation pressure. Vazquez and Beggs (1977) used more than 600 laboratory PVT analyses from fields of different geographical locations. The data encompassed very wide ranges of pressure, temperature, and oil properties and included more than 6000 measurements of gas solubility, oil formations volume factor, and oil viscosity at various pressures. Regression analysis techniques were used to correlate the laboratory data.
Khan et al. (1987) utilized viscosity data of 75 bottomhole samples taken from 62 Saudi oil reservoirs. A total of 1503 data measurements above the bubble point pressure were used to derive the correlation which is simply based on crude bubble point viscosity, pressure, and bubble point pressure. They compared their model with Beal’s (1946) correlation, and it gives close estimates for undersaturated crude oil viscosity.
Kartoatmodjo and Schmidt (1991) used widespread data collected from PVT reports and literature. A set of 5392 data points was used to develop their correlation. These data represent 740 different crude oil samples. For the development of undersaturated oil properties correlations, a total of 3588 data points collected from 661 different crude oil samples were used. The functional form of Sutton’s was used in this study to develop the undersaturated oil viscosity correlation. They developed a crude oil viscosity model based on API gravity, temperature, and solution gas–oil ratio. The used data have the following ranges: oil gravity of 14.4–59 °API, pressure of 14.7–6054.7 psia, temperature of 75–320 °F, and solution–gas ratio of 0–2890 scf/stb.
Hossain et al. (2005) presented their empirical correlations for dead, saturated, and undersaturated heavy oil utilizing three databanks. The databanks consist of heavy oil data from various parts of the world with wide ranges of temperature, pressure, and fluid compositions. A total of 361 data points were used to develop the undersaturated oil viscosity correlation. With temperature range of 118–218.7 °F, solution–gas ratio of 19.4–493 scf/bbl, bubble point pressure of 121–6272 psia, and pressure of 300–6400 psia.
Bergman and Sutton (2006) developed their correlation which provides a wider range of bubble point viscosity and pressure differentials than other existing correlations for undersaturated oil viscosity. This model derives undersaturated viscosity using only bubble point viscosity and pressure differential. The correlation can be satisfactorily used on gas free oils and oil with solution gas. The data used to derive the correlation included samples with bubble point viscosity from less than 0.1–14,000 cp. Accuracy is maintained over this wide range of values.
Genetic programming
Genetic programming (GP) is a development in the field of evolutionary algorithms extending the classical genetic algorithms (GA) to a symbolic optimization technique and overcoming GA limitation of being a fixedlength representation scheme requiring encoding of the variables and its nondynamic variability requiring the string length to be defined in advance (Koza 1992). Unlike common optimization methods, GP is able to work with a coding of the design variables as opposed to the design variables themselves. It is a problemindependent application working with a population of points as opposed to a single point. In addition, it requires the objective function value only, not the derivatives. Finally, GP is considered highly exploitative family of probabilistic (nondeterministic) search approach (Alvarez 2000).
The generated potential solutions in the form of a tree structure during the GP operation may have better and worse terms (subtrees) that contribute more or less to the accuracy of the model represented by the tree structure. Orthogonal least squares (OLS) algorithm is used to estimate the contribution of the tree branches to the accuracy of the model, and hence, terms having the smallest error reduction ratio could be eliminated from the tree. Figure 2 illustrates an example of elimination of a subtree based on OLS.
Results and discussion
Saturated oil viscosity model
Minimum and maximum values for the used data in building and validating the new saturated crude viscosity model
Building and testing data  Validation data  

R _{s} (cf/bbl)  µ _{d} (cp)  µ _{o} (cp)  R _{s} (cf/bbl)  µ _{ob} (cp)  µ _{o} (cp)  
Minimum  0  1.085  0.37  0  1.085  0.38 
Maximum  1020  6.72  5.06  895  6.72  6.72 
a_{1}  1.9244  a_{9}  0.9592 
a_{2}  0.0026  a_{10}  0.6221 
a_{3}  0.6189  a_{11}  0.0023 
a_{4}  2.8541  a_{12}  0.1979 
a_{5}  0.9404  a_{13}  0.6342 
a_{6}  1.0895  a_{14}  0.6617 
a_{7}  1.4270  a_{15}  1.3709 
a_{8}  1.0632  a_{16}  0.9974 

\(\mu_{Actual}\) = measured viscosity value, cp.

\(\mu_{Forecast}\) = correlated viscosity value, cp.

\(\overline{{\mu_{Actual} }}\) = average measured viscosity value, cp.

\(\overline{{\mu_{Forecast} }}\) = average correlated viscosity value, cp.
Accuracy of developed saturated crude oil viscosity model in comparison with different published correlation
Models  AARE (%)  COC (%) 

GPbased model  9.37  99.35 
Beggs and Robinson  16.79  98.90 
Chew and Connally  25.15  97.07 
Khan et al.  17.82  98.40 
Naseri et al.  34.34  93.75 
Undersaturated oil viscosity model
Minimum and maximum values for the used data in building and validating the new undersaturated crude viscosity model
Building data  Validation data  

Pressure (psi)  P _{b} (psi)  µ _{ob} (cp)  µ _{o} (cp)  Pressure (psi)  P _{b} (psi)  µ _{ob} (cp)  µ _{o} (cp)  
Minimum  400  317  0.37  0.38  1000  317  0.37  0.38 
Maximum  3495  2530  4.43  5.6  3200  2530  4.43  5.73 
b_{1}  0.1317  b_{4}  1.0529  b_{7}  0.0086 
b_{2}  1.7892  b_{5}  0.3579  b_{8}  0.3055 
b_{3}  3.4466  b_{6}  0.1323  b_{9}  0.0099 
Accuracy of developed model in comparison with different methods in predicting the undersaturated crude oil viscosity
Models  AARE (%)  COC (%) 

GPbased model  1.64  99.78 
Vazquez and Beggs  4.97  99.34 
Khan et al.  1.73  99.86 
Kartoatmodjo and Schmidt  5.96  99.84 
Hossain et al.  2.18  99.91 
Bergman and Sutton  1.84  99.89 
Sensitivity analysis
Conclusion

Saturated viscosity model developed using solution gas–oil ratio (R _{s}) and dead crude viscosity (µ _{d}) as input variables provided good accuracy in predicting the experimental measurements and outperforms the other tested correlations with AARE of 9.37%.

Undersaturated viscosity model developed using reservoir pressure (P) and crude bubble point pressure (P _{ob}) and crude viscosity at bubble point pressure (µ _{ob}) as inputs provided good accuracy in predicting the experimental measurements and outperforms the other tested correlations with AARE of 1.64%.

The developed saturated model sensitivity analysis indicates the equivalent impact of dead crude viscosity (µ _{d}) and solution gas–oil ratio (R _{s}) but in opposite trend.

The developed undersaturated model sensitivity analysis indicates the high positive impact of crude viscosity at bubble point (µ _{ob}) and small negative impact of bubble point pressure (P _{ob}) and trivial positive impact of reservoir pressure (P).
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