Leonid L. Levitan,

Mark MurthaIntercon Westbury, N.Y.

*EQUATIONS AND NOMENCLATURE [41,389 bytes] PDF FORMAT*

_{b}) and formation volume factor (FVF).

The new correlations were used in a computer program developed by the authors for well bore hydraulics calculations. The resulting calculations provided a better description of the experimental data than existing models by Aziz^{1} and Orkiszewski.^{2}

The empirical correlations commonly used to determine P_{b} and FVF were reviewed to evaluate their accuracy and gain a better understanding of the qualitative and quantitative effects of the parameters on the fundamental components of oil and gas systems.

The new correlations for P_{b} and FVF for saturated and undersaturated systems include the same variables as the existing correlations. The new correlations also have the same qualitative agreement and acceptable quantitative agreement with the existing correlations.

The correlations are uniform at all ranges of variables, can be easily integrated and differentiated, can be used in theoretical investigations, and clearly show how each parameter affects P_{b}, B_{ob}, or c_{o}.

Future appropriate experimental data will allow the authors to extrapolate their results beyond the examined range of parameters.

### Bubblepoint pressure

Bubblepoint pressure is the pressure at which the first gas bubble appears when the pressure of an oil and gas system is decreased. The correlation between bubblepoint pressure and reservoir pressure indicates the nature of the reservoir volume. The reservoir has only single phase (liquid) if the reservoir pressure is greater than or equal to the P_{b}. Gas is present in the reservoir if the reservoir pressure is less than the P_{b}.

As the difference between the bubblepoint and reservoir pressures increases, the gas saturation in the reservoir also increases.

Relative permeabilities for gas and liquid phases depend on gas saturation. Therefore, the permeabilities depend on P_{b}. In the well bore, the gas volume along the tubing also depends on P_{b}.

P_{b} can be determined by laboratory analysis of a reservoir fluid sample. Retrieving representative samples is a complex process, however, and most oil scientists and engineers use empirical correlations based on experimental data for evaluating P_{b}.

The three correlations commonly used to evaluate P_{b} are:

- Standing,
^{3}with one equation (Equation 1 in the equation box). - Lasater,
^{4}with five equations (Equations 2a-2e). In the correlation, P_{b}is a function of parameter y_{g}. - Vasquez-Beggs,
^{5}with one equation (Equation 3).

Another correlation for bubblepoint pressure was recently proposed by M. Khairy, et al.^{6} It also has one equation (Equation 4).

For each correlation, Table 2 [27,258 bytes] shows the number of experimental points and the degree to which the measured P_{b} agrees with the calculated P_{b}. Because these equations are empirical, they are bounded by the range of their data Table 3 [38,724 bytes].

The new correlations were developed using the Standing, Lasater, and Vasquez-Beggs correlations. As will be shown, these correlations have defined disagreements. It is therefore impossible for the new correlation to agree with all of them. Using one correlation throughout, however, yields useful results.

The Vasquez-Beggs correlation was chosen because it is the most accurate and reliable. Fortunately, the Standing correlation is close to the Vasquez-Beggs correlation. Agreement with both of these correlations was sought. When they had considerable disagreement, the Lasater correlation was used for next step checking.

Analysis shows that each of the correlations represents P_{b} as a function of reservoir temperature, oil gravity (ý_{API}), gas gravity (ý_{g}), and gas/oil ratio (G) as shown in Equation 5a.

These shared variables have the same qualitative effect on P_{b}. P_{b} increases with the increase of G and T, and the decrease of ý_{API} and ý_{g}.

Because the existing correlations share the same parameters (Equation 5a) and the parameters have the same qualitative effect on the bubblepoint pressure, the same variables were used in the new correlation but only in dimensionless form, as shown in Equation 5b.

Because G is the ratio of produced phase volumes, it should be dimensionless. In the metric system, G has the units cu m/cu m, which actually makes it dimensionless. In the English system, G has the traditional units of scf/st-tk bbl, which is also a dimensionless criterion. G can be easily converted to a dimensionless form by dividing the G by 5.615, the number of scf/st-tk bbl.

As a result, only in the English unit system does the correlation contain a constant with units.

In the English system, Equation 6a is the new correlation. Equation 6b is the metric system equivalent.

Fig. 1 [228,136 bytes] compares Pb values calculated with Equation 6a and the existing correlations. The disagreement percentage is defined by Equation 7.

To determine the influence of oil density, Pb values calculated at the same temperature and gas gravity were compared (Figs. 1a-1c).

Equation 6a, Standing, and Vasquez-Beggs agree within ±10% at the ranges G = 0 to 1,500 scf/st-tk bbl and ý_{API} = 20 to 60° API.

The Lasater correlation deviates significantly, giving substantially lower values for heavy oils (Fig. 1a), and substantially higher values for light oils (Fig. 1c).

In other words, Lasater is less dependent on oil gravity than Standing and Vasquez-Beggs.

The Khairy correlation also deviates significantly. It gives considerably higher bubblepoint pressure values for G < 500 scf/st-tk bbl, and much lower values for G > 500 scf/st-tk bbl. G has less influence on Khairy than Standing and Vasquez-Beggs.

Oil density, on the other hand, has greater influence on Khairy than Standing and Vasquez-Beggs. Since Khairy based his correlation on 39 experimental samples, however, the correlation is only useful for a narrow range of parameters.

For the middle range of oil densities, ý_{API} = 40° API (Fig. 1b), all correlations agree within 䔮% at the range G = 500 to 1,500 scf/st-tk bbl, but the disagreement between Lasater and the other correlations dramatically increases for low gas/oil ratios (Fig. 1b).

The same dramatic increase in disagreement occurs at all temperatures, oil gravities, and gas gravities. The reason for this marked increase is the different limits as G approaches 0.

According to Standing, Vasquez-Beggs, Khairy, and Equation 6a, the limit of P_{b} = 0 is G = 0. According to Lasater, however, P_{b} does not equal 0 for G = 0.

If G = 0 then y_{g} = 0 (Equation 2c), and from Equation 2a one obtains Equation 8, where P_{blim} is the lowest bubblepoint pressure at G = 0, according to Lasater.

For example, if the oil temperature is 80° F. and ý_{g} = 0.7, then P_{blim} = 274.63 psia. The higher the temperature and the lower the ý_{g}, the higher the P_{blim}.

A pressure less than or equal to the P_{blim} implies that the oil does not contain any gas. This is an erroneous conclusion, because in nature, there is always gas present in the system. For this reason, correlations for which P_{b} = 0 at G = 0 were preferred, namely, the Standing and Vasquez-Beggs correlations.

Because the ratio between Lasater and the other correlations at G = 0 is undefined, calculating the percent disagreement at low G values would not be useful. Such comparisons ended for Lasater at G = 500 scf/st-tk bbl (Fig. 1).

To calculate the amount of dissolved gas in oil under a pressure less than Pb, versions of Equations 6a and 6b were used to obtain Equation 9a and 9b.

It is clear that if the pressure drops and the rest of the parameters remain the same, the R-value will decrease. Thus, G is the maximum possible amount of gas dissolved in oil at the given parameters T, ý_{g}, and ý_{o}. This maximum can be reached when PoP_{b}.

### Formation volume factor

The formation volume factor (FVF or B_{o}) is the volume occupied at the pressure and temperature of interest by a unit of stock-tank oil under standard conditions plus the volume of gas dissolved in the oil under the same pressure and temperature conditions.

The FVF value is necessary for reservoir and production calculations. Similar to bubblepoint pressure, the FVF value can be obtained from pressure-volume-temperature (PVT) analysis of formation samples, but the empirical correlations are usually used.

It is important to distinguish the FVF values for saturated (PoP_{b}) and undersaturated (P P_{b}) oil systems. Therefore, the Pb value should be defined first (Equations 6a or 6b).

### Saturated systems

In saturated systems (PoP_{b}), FVF is defined as a function of R, oil and gas gravities, and temperature (Equation 10).

In other words, B_{ob} is determined by the same parameters as P_{b} (Equation 5a).

There are two commonly used empirical correlations for B_{ob} at pressures less than P_{b}. The Standing correlation is shown as Equation 11, and the Vasquez-Beggs correlation is Equation 12.

Khairy also proposed a correlation to determine B_{ob} for saturated systems (Equation 13).

The Standing and Vasquez-Beggs equations are very similar. Their structures are identical, both use the same variables, and both have the same limit, B_{ob} = 1 at G = 0 and T = 60° F.

The only significant difference is the effect of gas gravity (ý_{g}) on FVF. If ý_{g} increases, B_{ob} increases according to Standing and decreases according to Vasquez-Beggs. According to Vasquez-Beggs' experimental data, however, ý_{g}'s effect on B_{ob} is not always negative.

To reconcile their formula with their experimental data, Vasquez and Beggs made constant C_{3} negative for heavy oils Table 4 [29,846 bytes]. Introducing a negative constant creates new problems, however. At high G ( 1,000), a negative C_{3} makes the entire term, (T - 60)(ý_{API}/ý_{g})(C_{2} + C_{3}R), negative.

Thus, the influence of T and ý_{API} becomes negative, which is also inconsistent with the experimental data. Therefore, using the negative constant to counteract the gas gravity's effect on FVF in the Vasquez-Beggs correlation is ineffective.

Analysis determined that the influence of ý_{g} on FVF varies according to the parameters. It is positive for high G and low temperatures and negative for low G and high temperatures. This concept was incorporated into the new correlation.

The correlation for FVF was developed in a manner similar to the P_{b} correlation, by using dimensionless variables or criteria based on existing correlations (Equations 10 and 11).

In the English unit system, Equation 14a is the new equation and in metric it is Equation 14b.

Values for R or Rm can be found by using Equations 9a or 9b.

Equation 14a and the Standing and Vasquez-Beggs correlations are in agreement within 10% (Fig. 2 [172,665 bytes]). The disagreement percentage for the FVF correlations was calculated in the same manner as the disagreement for the Pb correlations (Equation 7).

### Undersaturated systems

In undersaturated systems (P P_{b}), the FVF value always decreases with pressure increase. In other words, such a system has a maximum FVF value at P

_{b}. Equation 15 calculates the FVF.

The B value can be calculated from Equations 14a or 14b for saturated systems.

Two correlations were used to define oil compressibility. Vasquez and Beggs proposed their correlation based on 2,000 experimental data gathered from 600 oil fields and used exactly the same variables for P_{b} and FVF in saturated systems (Equation 16).

The second correlation for oil compressibility was developed by Trube7 (Equations 17a, 17b, and 17c).

Trube used three graphs to determine c_{o}.^{7 8} To provide a wide range of parameters for the numerical experiments, Trube's graphs were converted into Equations 18, 19, 20a, and 20b.

Because all constants in Equations 18-20 are dimensionless, they will stay the same in metric.

No comparison of these two methods is available using independent data. Analysis shows, however, that the Vasquez-Beggs and Trube c_{o} correlations depend on the same variables T, ý_{API}, ý_{g}, and G (P_{b}60 includes G and ý_{g})^{7 8}. And according to Vasquez-Beggs, co increases when G, T, and ý_{API} increase, and ý_{g} and P decrease.

The authors' numerical experiments in the recommended ranges of parameters have shown that the qualitative dependence is also the same. Quantitatively, the c_{o} values calculated according to Trube and Vasquez-Beggs are close only at low Gs (100-500 scf/st-tk bbl), so that they were evaluated as reliable data.

At a high G, where the deviation is considerable (50% or more), the new correlation is closer to Vasquez-Beggs because it is based on reliable values of compressibility.

The new correlation was developed with dimensionless variables and the authors' conception of gas gravity's effect on FVF for saturated systems (Equation 14a). According to Vasquez-Beggs, c_{o} becomes negative at G = 0 and T_{r} = T_{o} (limit transaction). However, a negative c_{o} is impossible. Therefore, c_{o} = 0 at G = 0 and T_{r} = T_{o} was provided.

In the English unit system our new correlation for oil compressibility is Equation 21, and in the metric system it is Equation 21b.

Fig. 3 [186,175 bytes] compares oil isothermal compressibility values calculated from Vasquez-Beggs, Trube, and the new equation (Equation 21a). The c_{o} is calculated at the pressure P = 10,000 psi, which exceeds P_{b} at the chosen parameter ranges. The disagreement percentage between FVF values calculated by Equation 14a is displayed on the same graphs.

Normally, the deviation in B_{ob}, using the c_{o} value calculated from Vasquez-Beggs, Trube, and Equation 21a, is within ?1%. The deviation increases only at high G (1 000 scf/st-tk bbl). Of course, the disagreement between c_{o} values is much greater.

### References

- Aziz, K., Govier, G.W., and Fogarasi, M., "Pressure Drop in Wells Producing Oil and Gas," Journal of Canadian Petroleum Technology, July-September 1972, pp. 38-48.
- Orkiszewski, J., "Predicting Two-Phase Pressure Drops in Vertical Pipe," JPT, June 1976, pp. 829-38.
- Standing, M.B., "A Pressure-Volume-Temperature Correlation for Mixtures of California Oil and Gases," Drilling & Production Practices, API, 1974.
- Lasater, J.A., "Bubble Point Pressure Correlation," Trans. AIME, Vol. 213, 1958, pp. 379-81.
- Vasquez, M., and Beggs, H.D., "Correlations for Fluid Physical Property Prediction," JPT, June 1980, PP. 968-70.
- Khairy, M., El-Tayeb, S., Hammdallah, M., "PVT correlations developed for Egyptian crudes," OGJ, May 4, 1998, pp. 114-16.
- Trube, A.S., "Compressibility of Undersaturated Hydrocarbon Reservoir Fluids," Transactions AIME, Vol. 210, 1957, pp. 355-57.
- Petroleum Engineering Handbook, SPE, February 1992.

### The Authors

Leonid L. Levitanis an hydraulics engineer for Intercon, an international consulting firm that specializes in management strategy and technical services. He also serves as lead researcher of Intercon's enhancement technology division where he heads software and product development. Levitan holds an MS in hydrodynamics and thermophysics from the Moscow Power Institute and a PhD from the All Union Heat Engineering Institute.

Mark Murthawas an assistant programmer for this project.

*Copyright 1999 Oil & Gas Journal. All Rights Reserved.*