# Density and pressure drop relationship trust

### Pressure Loss Calculations

In physics, the Young–Laplace equation is a nonlinear partial differential equation that The Young–Laplace equation relates the pressure difference to the shape of the surface . For a fluid of density ρ: . which reproduced in symbolic terms the relationship described earlier by Young. Available on-line at: Hathi Trust. Keywords: hydraulic manifold; pressure losses; Computational Fluid Dynamic ( CFD) . Moreover, some results about a connection with double elbows were introduced. . density of kg/m3; it is the same fluid available in the test rig .. ANOVA assumptions are checked and we can trust the following. system, A theoretical relation has been derived by means Already with a pressure difference of 1 atm over the leak, .. v/here ρ is the vapour pressure in atm, and p-, the density Belgian American Bank and Trust Company - New York.

These in-situ velocities depend on the density and viscosity of each phase.

Typically the phase that is less dense flows faster than the other. This causes a "slip" effect between the phases. As a consequence, the in-situ volume fractions of each phase under flowing conditions will differ from the input volume fractions of the pipe. If the slip condition is omitted, the in-situ volume fraction of each phase is equal to the input volume fraction. Because of slippage between phases, the liquid holdup EL can be significantly different from the input liquid fraction CL.

In other words, the liquid slip holdup EL is the fraction of the pipe that is filled with liquid when the phases are flowing at different velocities. It can be defined as follows: We can also write them in function of the superficial velocities as: QL is the liquid rate at the prevailing pressure and temperature. Similarly, QGBg is the gas rate at the prevailing pressure and temperature.

The input volume fractions, CL and EL, are known quantities, and are often used as correlating variables in empirical multiphase correlations. Actual Velocities Once the liquid holdup has been determined, the actual velocities for each phase can be determined as follows: Note that this is in contrast to the way density is calculated for friction pressure loss. Mixture Density Mixture density is a measure of the in-situ density of the mixture, and is defined as follows: Mixture density is defined in terms of in-situ volume fractions ELwhereas no-slip density is defined in terms of input volume fractions CL.

No-Slip Density "No-slip" density is the density that is calculated with the assumption that both phases are moving at the same in-situ velocity. No-slip density is therefore defined as follows: No-slip density is defined in terms of input volume fractions CLwhereas the mixture density is defined in terms of in-situ volume fractions EL.

Mixture Viscosity Mixture viscosity is a measure of the in-situ viscosity of the mixture and can be defined in several different ways. In general, unless otherwise specified, is defined as follows: Mixture viscosity is defined in terms of in-situ volume fractions ELwhereas no-slip viscosity is defined in terms of input volume fractions CL.

No-Slip Viscosity "No-slip" viscosity is the viscosity that is calculated with the assumption that both phases are moving at the same in-situ velocity.

There are several definitions of "no-slip" viscosity. However, a value is required for use in calculating certain dimensionless numbers used in some of the pressure drop correlations. For intermediate temperatures, linear interpolation is used. The dead oil interfacial tension is corrected for this by multiplying by a correction factor: Friction Component In pipe flow, friction pressure loss is the component of total pressure loss caused by viscous shear effects.

Friction pressure loss always acts against the direction of flow.

It is combined with the hydrostatic pressure difference which may be positive or negative, depending on whether the flow is upward also known as uphill or downward downhill to give the total pressure loss. Friction pressure loss is calculated from the Fanning friction factor equation as follows: Each multiphase flow correlation finds the friction factor differently.

This calculation depends, in part, on the gas and liquid flow rates but also on the standard Fanning single-phase friction factor chart. When evaluating the Fanning friction factor, there are many ways of calculating the Reynolds number depending on how the density, viscosity, and velocity of the two-phase mixture are defined. For example, the Beggs and Brill calculation of the Reynolds number uses mixture properties that are calculated by prorating the property of each individual phase in the ratio of the input volume fraction and not of the in-situ volume fraction.

It is of importance only when there are differences in elevation from the inlet end to the outlet end of a pipe segment. This pressure difference can be positive or negative depending on the reference point inlet higher vertically than outlet, or outlet higher than inlet. Under ALL circumstances, irrespective of what sign convention is used, the contribution of the hydrostatic pressure calculation must be such that it tends to make the pressure at the vertically-lower end higher than that at the upper end.

The hydrostatic pressure difference is calculated as follows: In the equation above, the problem lies in finding an appropriate value for density, as discussed below: For a single-phase liquid, the density of the mixture is equal to the liquid density.

For a single-phase gas, density varies with pressure, and the calculation must be done sequentially in small steps to allow density to vary with pressure. For multiphase flow, density is calculated from the in-situ mixture density, which in turn is calculated from the liquid holdup.

The liquid holdup is obtained from multiphase flow correlations, such as Beggs and Brill, and depends on the gas and liquid rates, pipe diameter, etc. Flow Correlations Many of the published multiphase flow correlations are applicable for vertical flow only, while others apply for horizontal flow only. Other than the Beggs and Brill correlation and the Petalas and Aziz mechanistic model, there are not many correlations that were developed for the whole spectrum of flow situations that can be encountered in oil and gas operations — namely, uphill, downhill, horizontal, inclined, and vertical flow.

However, we have adapted all of the correlations as appropriate so that they apply to all flow situations. Following is a list of the multiphase flow correlations that are available: Beggs and Brill — one of the few published correlations capable of handling all of the flow directions.

## Young–Laplace equation

It was developed using sections of pipeline that could be inclined at any angle. Gray — developed for vertical flow in wet gas wells. We have modified it so that it applies to flow in all directions by calculating the hydrostatic pressure difference using only the vertical elevation of the pipeline segment and the friction pressure loss based on the total length of the pipeline.

Hagedorn and Brown — developed for vertical flow in oil wells. We have modified it so that it applies to flow in all directions by calculating the hydrostatic pressure difference using only the vertical elevation of the pipe segment and the friction pressure loss based on the total pipeline length. Petalas and Aziz — developed to overcome the limitations imposed by using previous correlations. It applies to all pipe geometries, fluid properties, and flow in all directions.

A mechanistic approach is combined with empirical closure relationships to provide a model that is more robust than other models and can be to used predict pressure drop and holdup in pipes over a more extensive range of conditions. Each of these correlations was developed for its own unique set of experimental conditions or designed using a mechanistic modeling approach, and accordingly, results vary between them.

For multiphase flow in essentially vertical wells, the available correlations are Beggs and Brill, Petalas and Aziz, Gray and Hagedorn and Brown. If used for single-phased flow, these four correlations devolve to the Fanning gas or Fanning liquid correlation as needed.

When creating a new wellbore, Harmony sets a default multiphase correlation depending upon the type of well that exists in the Entity Viewer. This default correlation is based on our expected use cases, and thus may not apply to every wellbore. Of course, the correlation for the wellbore configuration can be changed at any time. Beggs and Brill Correlation The Beggs and Brill correlation is one of the few published correlations capable of handling all these flow directions.

The Beggs and Brill multiphase correlation deals with both friction pressure loss and hydrostatic pressure difference. First, the corresponding flow pattern for the particular combination of gas and liquid rates segregated, intermittent, or distributed is determined.

The liquid holdup, and hence, the in-situ density of the gas-liquid mixture, is then calculated according to the identified flow pattern to obtain the hydrostatic pressure difference.

A two-phase friction factor is calculated based on the input gas-liquid ratio and the Fanning friction factor.

### Pressure Drop Through a Packed Bed – Neutrium

The Modified Flanigan Correlation is an extension of the Modified Panhandle single-phase equation to multiphase flow. It incorporates the Flanigan correction of the Flow Efficiency for multiphase flow and a calculation of hydrostatic pressure difference to account for uphill flow. There is no hydrostatic pressure recovery for downhill flow. In this software, the Flanigan multiphase correlation is also applied to the Panhandle and Weymouth correlations. The Petalas and Aziz Model is a correlation that was developed to overcome the limitations imposed by using previous correlations.

It applies to all pipe geometries, fluid properties and flow in all directions. A mechanistic approach fundamental laws are combined with empirical closure relationships to provide a model that is more robust than other models and can be to used predict pressure drop and holdup in pipes over a more extensive range of conditions.

Each of these correlations was developed for its own unique set of experimental conditions or designed using a mechanistic modeling approach, and accordingly, results will vary between them. For multiphase flow in essentially vertical wells, the available correlations are Beggs and Brill, Petalas and Aziz, Gray and Hagedorn and Brown.

If used for single-phased flow, these four correlations devolve to the Fanning Gas or Fanning Liquid correlation. When switching from multiphase flow to single-phase flow, the correlation will default to Fanning. When switching from single-phase to multiphase flow, the correlation will default to Beggs and Brill.

### Pressure Loss Correlations

The Flanigan, Modified-Flanigan and Weymouth Multiphase correlations can give erroneous results if the pipe described deviates substantially more than 10 degrees from the horizontal. The Gray and Hagedorn and Brown correlations were derived for vertical wells and may not apply to horizontal pipes. Beggs and Brill Correlation For multiphase flow, many of the published correlations are applicable for "vertical flow" only, while others apply for "horizontal flow" only.

Few correlations apply to the whole spectrum of flow situations that may be encountered in oil and gas operations, namely uphill, downhill, horizontal, inclined and vertical flow. The Beggs and Brill correlation, is one of the few published correlations capable of handling all these flow directions. The Beggs and Brill multiphase correlation deals with both the friction pressure loss and the hydrostatic pressure difference.

First the appropriate flow regime for the particular combination of gas and liquid rates Segregated, Intermittent or Distributed is determined. The liquid holdup, and hence, the in-situ density of the gas-liquid mixture is then calculated according to the appropriate flow regime, to obtain the hydrostatic pressure difference. A two-phase friction factor is calculated based on the "input" gas-liquid ratio and the Fanning friction factor.

From this the friction pressure loss is calculated using "input" gas-liquid mixture properties.