Glossary

Multiphase Flow Terms

Input Volume Fraction

The 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 ( + )

Note: is the liquid rate at the prevailing pressure and temperature. Similarly,* is the gas rate at the prevailing pressure and temperature.

The input volume fractions, and , are known quantities, and are often used as correlating variables in empirical multiphase correlations.

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 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 Effect

When 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 Density

The 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 ( ), whereas the no-slip density is defined in terms of input volume fractions ( ).

Mixture Velocity

Mixture 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 Viscosity

The 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.

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 ( ), whereas the no-slip viscosity is defined in terms of input volume fractions ( ).

No-Slip Density

The "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 (), whereas the mixture density is defined in terms of in-situ volume fractions ().

No-Slip Viscosity

The "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 Velocity

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 and the gas phase accounts for the solution gas going in and out of the oil as a function of pressure( ), the superficial velocities can be rewritten as:

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.

Surface Tension

The 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 Tension

The 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 Tension

The 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.

Nomenclature

a = ash content (dimensionless, mass %)

A = area (ft²) (F.A.S.T. CBM^TM uses acres: 1 acre = 43560ft²)

b = Langmuir pressure parameter (equal to )

B_g = gas formation volume factor (ft³/scf)

B_w = water formation volume factor (bbl/STB)

c_f = formation compressibility (/psia)

C_L = liquid input volume fraction

C_m = matrix swelling coefficient (microstrain*ton/scf).

c_m = matrix compressiblity (/psi) (note: this c_m is not the same as the one used in the Seidle formulation)

C_p = mechanical strain due to pressure

c_p = mechanical compliance coefficient, represents compressibility of coal matrix (/psia)

c_w = water compressibility (/psia)

C_f = formation compressibility (1/psia)

D = inside diameter of pipe (in)

D = nominal decline rate

D_e = effective decline rate

Depth = Landed depth of tubing or casing

Duration = Well Timing, duration of forecast, time interval from start date to end date (months)

E

E = Young’s modulus. Measures opposition of a substance to extensional stress

= horizontal liquid holdup

= inclined liquid holdup

End = Well Timing, end date of forecast (MM/DD/YYYY)

Excess Pres. Drop = Additional pressure loss (ie: to represent a the hydrostatic loss from a suspected column of liquid at the bottom of the wellbore)

f = Fanning friction factor (function of Reynolds number)

f = an undefined fraction (0 to 1, usually 0.5, used in calculating matrix shrinkage)

= no-slip friction factor

= Froude Mixture Number

= two phase friction factor

F(x) = probability distribution function

F_N(x) = normal distribution function

g = gravitational acceleration

g_c = conversion factor

G_c = gas content (scf/ton)

= recoverable gas content (scf/ton)

G_c,pure = gas content of pure coal (scf/ton)

G_p = gas produced (Bcf)

GIP = gas in place (scf)

h = net pay (ft)

ID = inside diameter of tubing or casing

IGIP = initial gas in place (Bcf)

k = permeability (md)

k = absolute roughness (in)

k/D = relative roughness (unitless)

K = bulk modulus. Mesures opposition of a substance to compressional stress.

k₀ = initial permeability at P₀ and S_w0 (md)

k_g = gas effective permeability (md)

k_w = water effective permeability (md)

k_rg = relative permeability to gas

k_rg0 = endpoint relative permeability to gas

k_rw = relative permeability to water

k_rw0 = endpoint relative permeability to water

L = length of pipe section (ft)

Layer = Layer name

m(P) = gas pseudopressure at pressure P (psi²/cp)

M = constrained axial modulus

n = exponent, used in relative permeability curves

n_w = exponent of the water relative permeability curve

n_g = exponent of the gas relative permeability curve

= Number of wells

N_VL = liquid velocity number

OD = outside diameter of tubing or casing

OGIP = original gas in place

= original gas in place, adsorbed

= original gas in place, free gas

OWIP = Original water in place

P = pressure, usually average reservoir pressure (psia)

= Abandonment pressure (psia)

P₀ = initial reservoir pressure (psia)

P_c = desorption pressure (psia)

P_i = initial reservoir pressure (psia)

= Aquifer productivity index (bbls/d/psi / m3/d/kPa)

= pressure loss due to friction effects (psi)

= pressure change due to hydrostatic head (psi)

P_L = Langmuir pressure parameter (psia)

P_sc = standard pressure (14.7 psia)

P_wf = bottomhole flowing pressure (psia)

= reservoir pressure at 50% matrix strain (psia)

q_g = gas rate (MCFd)

q_w = water rate (STB/day)

= = Gas rate at end of forecast (Mcf/day / m3/day)

= Maximum gas rate (Mcf/day / m3/day)

= Maximum water rate (bbl/day / m3/day)

r_e = external radius of reservoir (ft)

Re = Reynold’s number

RGIP = Recoverable gas in place

r_w = wellbore radius (ft)

% Rec = recovery factor

s = skin

S_g = gas saturation (% or fraction)

S_gc = irreducible gas saturation (% or fraction)

S_w = water saturation (% or fraction)

= average water saturation (% or fraction)

S_wc = irreducible water saturation (% or fraction)

= initial water saturation (% or fraction)

Start = well timing, Start date of forecast (MM/DD/YYYY)

Start = layer name, Start date of forecast (MM/DD/YYYY)

T = Temperature, usually reservoir temperature (F)

= flowing wellhead temperature

T_sc = standard temperature (60F = 519.67R)

V = average velocity (ft/s)

V_m = mixture velocity (ft/s)

V() = adsorbed gas (scf)

V(P) = amount of gas at P, also known as gas content (scf/ton)

V_L = Langmuir volume parameter (scf/ton)

= superficial liquid velocity (ft/s)

w_c= water content (dimensionless, mass %)

W_e = water encroached (bbls)

W_p = water produced (STB)

= fracture half length

Z

Z = gas compressibility factor (dimensionless)

Z_i = initial gas compressibility factor (dimensionless)

Z* = gas factor for unconventional gas reservoir (dimensionless)

Z*_i = Z* evaluated at initial conditions

Z_sc = standard gas compressibility factor (1)

= elevation change (ft)

Greek Symbols

= gas/liquid surface tension (dynes/cm)

= effective stress.

= effective stress at P₀.

= strain (dimensionless)

= net strain between overburden stress effect and matrix shrinkage as measured experimentally. This is the difference between strain caused by swelling due to gas adsorption and strain caused by applied stress. (strain, dimensionless)