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Wellbore Calculations

Home > Theory and Equations > Wellbore Calculations

Wellbore CalculationsMultiphase Flow DefinitionsInput Volume FractionThe input volume fractions are defined as:

We can also write this as:

Where: = gas formation volume factor = input gas volume fraction = input liquid volume fraction = gas flow rate (at standard conditions) = liquid flow rate (at prevailing pressure and temperature) = superficial gas velocity = superficial liquid velocity = mixture velocity ( + ) * is the gas rate at the prevailing pressure

Note: is the liquid rate at the prevailing pressure and temperature. Similarly, and temperature. The input volume fractions, multiphase correlations. and

, are known quantities, and are often used as correlating variables in empirical

In-Situ Volume Fraction (Liquid Holdup)The in-situ volume fraction, (or ), is often the value that is estimated by multiphase correlations. Because of "slip" between phases, the "holdup" ( ) can be significantly different from the input liquid fraction ( ). For example, a single-phase gas can

http://www.fekete.com/software/cbm/media/webhelp/c-te-calculations.htm[24/5/2011 3:33:11 PM]

Wellbore Calculations

percolate through a wellbore containing water. In this situation = 0 (single-phase gas is being produced), but > 0 (the wellbore contains water). The in-situ volume fraction is defined as follows:

Where: = cross-sectional area occupied by the liquid phase A = total cross-sectional area of the pipe

Liquid Holdup EffectWhen two or more phases are present in a pipe, they tend to flow at different in-situ velocities. These in-situ velocities depend on the density and viscosity of the phase. Usually the phase that is less dense will flow faster than the other. This causes a "slip" or holdup effect, which means that the in-situ volume fractions of each phase (under flowing conditions) will differ from the input volume fractions of the pipe.

Mixture DensityThe mixture density is a measure of the in-situ density of the mixture, and is defined as follows:

Where: = in-situ liquid volume fraction (liquid holdup) = in-situ gas volume fraction = mixture density = liquid density = gas density Note: The mixture density is defined in terms of in-situ volume fractions ( input volume fractions ( ). ), whereas the no-slip density is defined in terms of

Mixture VelocityMixture Velocity is another parameter often used in multiphase flow correlations. The mixture velocity is given by:

Where: = mixture velocity = superficial liquid velocity = superficial gas velocity

Mixture ViscosityThe 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, m is defined as follows.

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Wellbore Calculations

W here: = in-situ liquid volume fraction (liquid holdup) = in-situ gas volume fraction = mixture viscosity = liquid viscosity = gas viscosity Note: The mixture viscosity is defined in terms of in-situ volume fractions ( of input volume fractions ( ). ), whereas the no-slip viscosity is defined in terms

No-Slip DensityThe "no-slip" density is the density that is calculated with the assumption that both phases are moving at the same in-situ velocity. The no-slip density is therefore defined as follows:

Where: = input liquid volume fraction = input gas volume fraction = no-slip density = liquid density = gas density Note: The no-slip density is defined in terms of input volume fractions ( in-situ volume fractions ( ). ), whereas the mixture density is defined in terms of

No-Slip ViscosityThe "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. In general, unless otherwise specified, is defined as follows.

Where: = input liquid volume fraction = input gas volume fraction = no-slip viscosity = liquid viscosity = gas viscosity

Superficial Velocityhttp://www.fekete.com/software/cbm/media/webhelp/c-te-calculations.htm[24/5/2011 3:33:11 PM]

Wellbore Calculations

The superficial velocity of each phase is defined as the volumetric flow rate of the phase divided by the cross-sectional area of the pipe (as though that phase alone was flowing through the pipe). Therefore:

and

Where: = gas formation volume factor D = inside diameter of pipe = measured gas flow rate (at standard conditions) = liquid flow rate (at prevailing pressure and temperature) = superficial gas velocity = superficial liquid velocity

Since the liquid phase accounts for both oil and water solution gas going in and out of the oil as a function of pressure( as:

and the gas phase accounts for the ), the superficial velocities can be rewritten

Where: = oil flow rate (at stock tank conditions) = water flow rate in (at stock tank conditions) = gas flow rate (at standard conditions of 14.65psia and 60F) = liquid flow rate (oil and water at prevailing pressure and temperature) = oil formation volume factor = water formation volume factor = gas formation volume factor = solution gas/oil ratio WC = water of condensation (water content of natural gas, Bbl/MMscf) The oil, water and gas formation volume factors ( , and ) are used to convert the flow rates from standard (or stock tank) conditions to the prevailing pressure and temperature conditions in the pipe. Since the actual cross-sectional area occupied by each phase is less than the cross-sectional area of the entire pipe the superficial velocity is always less than the true in-situ velocity of each phase.

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Wellbore Calculations

Surface TensionThe surface tension (interfacial tension) between the gas and liquid phases has very little effect on two-phase pressure drop calculations. However a value is required for use in calculating certain dimensionless numbers used in some of the pressure drop correlations. Empirical relationships for estimating the gas/oil interfacial tension and the gas/water interfacial tension were presented by Baker and Swerdloff, Hough and by Beggs.

Gas/Oil Interfacial TensionThe dead oil interfacial tension at temperatures of 68 F and 100 F is given by:

Where: = interfacial tension at 68 F (dynes/cm) = interfacial tension at 100 F (dynes/cm) API = gravity of stock tank oil (API) If the temperature is greater than 100 F, the value at 100 F is used. If the temperature is less than 68 F, the value at 68 F is used. For intermediate temperatures, linear interpolation is used. As pressure is increased and gas goes into solution, the gas/oil interfacial tension is reduced. The dead oil interfacial tension is corrected for this by multiplying by a correction factor.

Where: P = pressure (psia) The interfacial tension becomes zero at miscibility pressure, and for most systems this will be at any pressure greater than about 5000 psia. Once the correction factor becomes zero (at about 3977 psia), 1 dyne/cm is used for calculations.

Gas/Water Interfacial TensionThe gas/water interfacial tension at temperatures of 74 F and 280 F is given by:

Where: = interfacial tension at 74 F (dynes/cm) = interfacial tension at 280 F (dynes/cm) P = pressure (psia) If the temperature is greater than 280 F, the value at 280 F is used. If the temperature is less than 74 F, the value at 74 F is used. For intermediate temperatures, linear interpolation is used.

Wellbore CorrelationsBeggs and Brill CorrelationFor multiphase flow, many of the published correlations are applicable for "vertical flow" only, while others apply for "horizontal flow" only. Not many 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 (1973) correlation, is one of the few published correlations capable of handling all these flow directions. It was developed using 1" and 1-1/2" sections of pipe that could be inclined at any angle from the horizontal.

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Wellbore Calculations

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 frictional pressure loss is calculated using "input" gas-liquid mixture properties.

Flow Pattern MapThe Beggs and Brill correlation requires that a flow pattern be determined. Since the original flow pattern map was created, it has been modified. We have used this modified flow pattern map for our calculations. The transition lines for the modified correlation are defined as follows:

Where: = liquid input volume fraction The flow type can then be readily determined either from a representative flow pattern map or according to the following conditions, where

. Where: D = inside pipe diameter (ft) = Froude Mixture Number (unitless) g = acceleration of gravity (32.2 ft/s 2 ) = mixture velocity (ft/s)

SEGREGATED flowif and

or

and

INTE