# 18.5: Volumetric Factors (Bo and Bg)

- Page ID
- 590

Due to the dramatically different conditions prevailing at the reservoir when compared to the conditions at the surface, we do not expect that 1 barrel of fluid at reservoir conditions could contain the same amount of matter as 1 barrel of fluid at surface conditions. Volumetric factors were introduced in petroleum and natural gas calculations in order to readily relate the *volume* of fluids that are obtained at the surface (stock tank) to the volume that the fluid actually occupied when it was compressed in the reservoir.

For example, the volume that a *live oil* occupies at the reservoir is *more* than the volume of oil that leaves the stock tank at the surface. This may be counter-intuitive. However, this is a result of the evolution of gas from oil as pressure decreases from reservoir pressure to surface pressure. If an oil had no gas in solution (i.e., a *dead oil*), the volume that it would occupy at reservoir conditions is less than the volume that it occupies at the surface. In this case, only liquid compressibility plays a role in the change of volume.

**The formation volume factor of a natural gas** (\(B_g\)) relates the volume of 1 lbmol of gas at reservoir conditions to the volume of the same lbmol of gas at standard conditions, as follows:

\[B_{g}=\frac{\text { Volume of } 1 \text { lbmol of gas at reservoir conditions, RCF }}{\text { Volume of } 11 \text { bmol gas at standard conditions, SCF }} \label{18.5}\]

Those volumes are, evidently, the specific molar volumes of the gas at the given conditions. The reciprocal of the specific molar volume is the molar density, and thus, Equation \ref{18.5} could be written:

\[B_{g}=\frac{\bar{v}_{g} / res}{\bar{v}_{g} /sc}=\frac{\bar{\rho}_{g} / s c}{\bar{\rho}_{g} / r e s}=\frac{\left.\left(\rho_{g} / M W_{g}\right)\right|_{sc}}{\left.\left(\rho_{g} / M W_{g}\right)\right|_{r e s}} \label{18.6)}\]

Introducing the definition for densities in terms of compressibility factor,

\[B_{g}=\frac{\frac{P_{s c}}{R T_{s c} Z_{s c}}}{\frac{P}{R T Z}} \label{18.7}\]

Therefore, recalling that \(Z_{s c} \approx 1\),

\[B_{g}=\frac{P_{s c}}{T_{s c}} \frac{Z T}{P}=0.005035 \frac{Z T}{P}_{[R C F / S C F]} \label{18.8}\]

Gas formation volume factors can be also expressed in terms of [RB/SCF]. In such a case, 1 RB = 5.615 RCF and we write:

\[B_{g}=0.005035 \frac{Z T}{P}_{[R C F / S C F]} \label{18.9}\]

The formation volume factor of an **oil or condensate** (B_{o}) relates the volume of 1 lbmol of liquid at reservoir conditions to the volume of that liquid once it has gone through the surface separation facility.

\[B_{o}=\frac{\text { Volume of lbmol of liquid at reservoir conditions, RB }}{\text { Volume of that lbmol after going through seperation, STB }} \label{18.10}\]

The total volume occupied by 1 lbmol of liquid at reservoir conditions (V_{o})_{res} can be calculated through the compressibility factor of that liquid, as follows:

\[\left(V_{o}\right)_{r e s}=\left(\frac{n Z_{o} R T}{P}\right)_{r e s} \label{18.11}\]

where \(n = 1 \,lbmol\).

Upon separation, some gas is going to be taken out of the liquid stream feeding the surface facility. Let us call “n_{st}” the moles of liquid leaving the stock tank per mole of feed entering the separation facility. The volume that 1 lbmol of reservoir liquid is going to occupy after going through the separation facility is given by:

\[\left(V_{o}\right)_{r e s}=\left(\frac{n_{s t} Z_{o} R T}{P}\right)_{s c} \label{18.12}\]

where \(n = 1 \,lbmol\).

Here we assume that the last stage of separation, the stock tank, operates at standard conditions. Introducing Equations \ref{18.12} and \ref{18.11} into \ref{18.10}, we end up with:

\[B_{o}=\frac{\left(\frac{n Z_{o} R T}{P}\right)_{r e s}}{\left(\frac{n_{s t} Z_{0} R T}{P}\right)_{s c}} \label{18.13}\]

or,

\[B_{o}=\frac{1}{n_{s t}} \frac{\left(Z_{o}\right)_{r e s}}{\left(Z_{o}\right)_{s c}} \frac{T}{P} \frac{P_{s c}}{T_{s c}} \label{18.14}\]

Please notice that \((Z_o)_{sc}\) - unlike \(Z_{sc}\) for a gas - is never equal to one. Oil formation volume factor can be also seen as the volume of reservoir fluid required to produce one barrel of oil in the stock tank.

## Contributors

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.