Beggs and Brill (1973) correlation, is one of the few correlations capable of handling all flow directions encountered in oil and gas operations, namely uphill, downhill, horizontal, inclined and vertical flow for two phase fluid.

Total pressure gradient is described by following relation.

dP/dZ = [(dP/dZ)Fric. +(dP/dZ)Ele.]/(1-Ek)

where, (dP/dZ)_Fric. is pressure gradient due to friction, (dP/dZ)_Ele. is hydrostatic pressure difference and E_k estimates pressure loss due to acceleration.

Flow Pattern Map

A flow regime is identified based on the Froude number of the mixture (Fr_m) and input liquid content (no slip liquid holdup C_L).

Frm = vm²/ g.D

where, v_m is mixture velocity, D is pipe inside diameter and g is gravitational constant.

CL = QL/ (QL + QG)

where, Q_L is liquid volumetric flow and Q_G is gas volumetric flow.

The transition lines for correlation are defined as follows:

L1 = 316 CL0.302
L2 = 0.0009252 CL-2.4684
L3 = 0.1 CL-1.4516
L4 = 0.5 CL-6.738

Segregated Flow

CL < 0.01 and Frm < L1
OR CL >= 0.01 and Frm < L2

Intermittent Flow

0.01 <= CL < 0.4 and L3 < Frm <= L1
OR CL >= 0.4 and L3 < Frm <= L4

Distributed Flow

CL < 0.4 and Frm >= L4
OR CL >= 0.4 and Frm > L4

Transition Flow

L2 < Frm < L3

Liquid Holdup, E_L(θ)

Once flow type has been determined, liquid holdup for horizontal flow E_L(0) is calculated.

EL(0) = a CLb / Frmc

E_L(0) must be greater than C_L, if E_L(0) is smaller than C_L, then E_L(0) is assigned a value of C_L. Actual liquid volume fraction is obtained by multiplying E_L(0) by a correction factor, B(θ).

EL(θ) = B(θ) x EL(0)

B(θ) is obtained as -

B(θ) = 1 + β(sin(1.8θ) - (1/3)sin³(1.8θ))

where θ is the angle of inclination of pipe with horizontal.

Correction factor β is calculated as following -

β = (1 - CL)ln( d.CLe.NLVf.Frmg )

Liquid velocity number, N_LV is given by:

NLV = 1.938 Vsl(ρL/ (g.σ))1/4

V_sl is no slip liquid velocity, ρ_L is liquid density, g is gravitational constant and σ is surface tension.

For transition flow,

EL(θ)transition = AEL(θ)segregated + BEL(θ)intermittent

where A and B are as following -

A = ( L3 - Frm)/(L3 - L2)
B = 1- A

Liquid holdup, E_L(θ) is used to calculate mixture density ρ_m.

ρm = ρL.EL(θ) + ρG.(1-EL(θ))

(dP/dZ)_Elevation

Pressure change due to the hydrostatic head of the vertical component of the pipe is given by:

(dP/dZ)Ele. = ρm.g.sin(θ)/(144.gc)

(dP/dZ)_Friction

Calculate no slip Reynold's number using no slip mixture density and viscosity.

ReNS = ρNS.Vm.D/μNS

No slip friction factor, f_NS is then calculated using Colebrook-White equation.

Ratio of friction factor is defined as

fTP/ fNS = eS

Value of S is governed by following conditions -

S = ln(2.2y - 1.2)

if 1 < y < 1.2, otherwise -

S = ln(y)/(-0.0523 + 3.182.ln(y) - 0.8725.(ln(y))2 + 0.01853.(ln(y))4 )

where y is defined as

y = CL / EL(θ)²

Pressure loss due to friction is:

(dP/dZ)Fric. = 2.fTP.Vm².ρNS /(144.gc.D)

Pressure loss due to acceleration, factor E_k is given by:

Ek = ρm.Vm.Vsg/(gc.P)

where, V_sg is no slip gas velocity and P is gas pressure.

Resources

1. Web based calculation available at checalc.com
2. Spreadsheet for Beggs Brill Method

References

1. Standard Handbook of Petroleum & Natural Gas Engineering, William C Lyons, Volume 2