Published on *PNG 301: Introduction to Petroleum and Natural Gas Engineering* (https://www.e-education.psu.edu/png301)

As we have discussed, multi-phase flow through tubing is typically performed using empirical, multi-phase flow correlations. These empirical correlations are developed on the observations made by the investigator in laboratory experiments, field measurements, or both. The differences in the correlations are based on many factors including:

- experimental technique
- tubing orientation
- vertical
- slanted
- horizontal

- tubing type
- smooth
- rough

- fluids used in the experiments
- air-water
- crude oil - natural gas

- pressure-temperature range used in the experiments

- tubing orientation
- flow regimes observed during the experiments and how they are incorporated into the correlations
- trade-off between the desired accuracy and the desired degree of mathematical simplicity or ease of application

In most cases, the multi-phase flow correlations are based on the flow regimes that we have already discussed. For use in a multi-phase flow correlation, these flow regimes are plotted as * Flow Pattern Maps*. Examples of the flow pattern maps are shown in

Source: Greg King © Penn State, licensed CC BY-NC-SA 4.0 [1]

Source: Greg King © Penn State, licensed CC BY-NC-SA 4.0 [1]

The example flow regime maps shown in **Figure 6.11** and **Figure 6.12** illustrate two important points. The first point is how the flow regimes are incorporated into the flow correlations. The variables on the x-axis and y-axis are calculated, and that point on the map indicates the flow regime that is occurring in a well segment based on the investigator’s experimental or field results. For the flow regime identified on the flow pattern map, different mathematical expressions are used to quantify the Liquid Hold-Up, Gas Hold-Up, the slip velocity, and the friction factors within that particular flow regime. These dynamic data are the data that I identified in **Table 6.05** with the blue cells. Once the dynamic data are calculated for a particular segment in a segmented well model, they are used in the energy balance equation to determine the pressure loss in that segment.

The second point illustrated by the example flow pattern maps is that the definitions of the x-axis and y-axis are different for each multi-phase flow correlation. This gets back to the empirical nature of the multi-phase flow correlations. Different investigators and different experimental/field procedures may result in different mathematical groups controlling the dynamics of the flow. The choice of these mathematical groups may also lie in the personal preference of the investigator.

For multi-phase flow, the fluid properties are typically calculated as mixture properties. These mixture properties are based on the * Phase Hold-Up*, ${H}_{l}$ and ${H}_{g}$. Hold-Up is the local fraction of the pipe volume occupied by the phase. In multi-phase flow correlations, the hold-up is determined from the map based on the mathematical expressions related to the flow regime. Once the hold-up is determined, fluid properties can be determined for the flowing mixture by:

$${\rho}_{m}={\rho}_{l}{H}_{l}+{\rho}_{g}\left(1-{H}_{l}\right)={\rho}_{l}{H}_{l}+{\rho}_{g}{H}_{g}$$

$${\mu}_{m}={\mu}_{l}{H}_{l}+{\mu}_{g}\left(1-{H}_{l}\right)={\mu}_{l}{H}_{l}+{\mu}_{g}{H}_{g}$$

for viscosity, some investigators prefer to use a different definition of mixture viscosity:

$${\mu}_{m}={\mu}_{l}{}^{{H}_{l}}+{\mu}_{g}{}^{\left(1-{H}_{l}\right)}={\mu}_{l}{}^{{H}_{l}}+{\mu}_{g}{}^{{H}_{g}}$$

As a practical point, a working production engineer typically is not expected to know the details of the multi-phase flow correlation used in a well model. There is no universal rule for selecting the proper correlation for use for a particular well, group of wells, or wells in a field. The production engineer simply performs * Flow Tests* on his/her wells to see the actual pressure drops at the current reservoir conditions (orange and yellow table entries in

There are many multi-phase correlations used in the oil and gas industry. It is beyond the scope of this course to discuss all of the correlations in use today. **Table 6.06** lists many of these correlations used in industry software along with notes describing their preferred applications.

Single-PhaseFlow |
Gas |
Darcy-Weisbach |
Theoretical Energy Balance. Uses a general friction factor. Requires an iteritive solution for copressible fluids. Can be used for wells (verticle, inclined, or horizontal), flow lines, or transmission lines. |

Weymouth |
Empiricla Energy Balance. Uses an equation specific friction factor. A non-iterative solutionis possible if inlet and outlet ressures are specified. An iterative solution is required otherwise. Can be used for wells, flow lines, or transmission lines. | ||

Panhandle "A" |
Empiricla Energy Balance. Uses an equation specific friction factor. A non-iterative solutionis possible if inlet and outlet ressures are specified. An iterative solution is required otherwise. Can be used for wells, flow lines, or transmission lines. | ||

Panhandle "B" |
Empiricla Energy Balance. Uses an equation specific friction factor. A non-iterative solutionis possible if inlet and outlet ressures are specified. An iterative solution is required otherwise. Can be used for wells, flow lines, or transmission lines. | ||

Cullender and Smith |
Empirical Energy Balance. Uses an equationspecific friction factor. Requires an iterative solution. | ||

Liquid |
Darcy-Weisbach |
Theoretical Energy Balance. Uses a general friction factor. Requires an iterative solution for slightly compressible liquids. Can be used for wells (vertical, inclined, or horizontal), flow lines, or transmission lines. | |

Hazen-Williams |
Empirical. The friction factor is replaced with material specific constant. Requires an iterative solution for slightly compressible liquids. Used for water disposal, water source, water injection, or light hydrocarbon wells (possible use in condensate reservoirs). | ||

Multi-Phase Flow |
Vertical Flow |
Fancher and Brown(no slip and no flow pattern map) |
The no slip assumption and no pattern map imply that the correlation is not generally applicable.The no slip assumption is only applicable in flow regimes where liquid and gas velocities are the same. |
---|---|---|---|

Hagedorn and Brown |
Developed from experiments on 1,500 ft experimental well using 1 inch to 4 inch tubing. Experiments included thre-phase flow. One of the most commonly used multi-phase flow correlations for vertical or near vertical wells. | ||

Beggs and Brill(no slip) |
The no slip assumption is only applicable in flow regimes where liquid and gas velocities are the same. One of the few multi-phase flow corrections capable of modeling vertical, inclined, or horizontal flow. Assumes smooth pipe. | ||

Beggs and Brill(with Darcy-Weisbach friction factor) |
The no slip assumption is only applicable in flow regimes where liquid and gas velocities are the same. One of the few multi-phase flow corrections capable of modeling vertical, inclined, or horizontal flow. Pipe is allowed to include roughness. | ||

Orkiszewski |
Took existing correlations and compared them to field results. Selected the best correlations for different regimes and developed a single correlation. This is apopular multi-phase flow correlation, but may exhibit discontinuities when crossing regime boundaries. | ||

Gray |
Developed for gas condensate reservoirs (most accurate for these reservoirs). Uses non-compositional approach. It is based on the observation that hold-up is not as great in condensate wells as in oil wells. Roughness is ignored, but uses an efficiency instead. | ||

Gray(with Darcy-Weiabach friction factor) |
Developed for gas condensate reservoirs (most accurate for these reservoirs). Similar to the standard Gray correlation, but roughness is incorporated through the Moody Diagram. | ||

Duns and Ros |
Uses combined experimental and field measurements. The first multi-phase flow correlation to use flow pattern mapping. A popular multi-phase flow correlation. | ||

Horizontal Flow |
Eaton-Flanigan |
This correlation is a hybrid correlation of the Eaton hold-up and friction loss correlations and the Flanigan inclined pipe correlation | |

Eaton-Dunkler-Flanigan |
This correlation is another hybrid correlation of the Eaton hold-up correlation, the Dukler friction correlation, and the Flanigan inclined pipe correlation. | ||

Beggs and Brill(no slip) |
The no slip assumption is only applicable in flow regimes where liquid and gas velocities are the same. One of the few multi-phase flow correlations capable of modeling vertical, inclined, or horizontal flow. Assumes smooth pipe. | ||

Beggs and Brill(with Darcy-Weisbach friction factor) |
The no slip assumption is only applicable in flow regimes where liquid and gas velocities are the same. One of the few multi-phase flow correlations capable of modeling vertical, inclined, or horizontal flow. Pipe is allowed to include roughness. |