Beggs & Brill Method
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 Ek estimates pressure loss due to acceleration.
Flow Pattern Map
A flow regime is identified based on the Froude number of the mixture (Frm) and input liquid content (no slip liquid holdup CL).
Frm = vm²/ g.D
where, vm is mixture velocity, D is pipe inside diameter and g is gravitational constant.
CL = QL/ (QL + QG)
where, QL is liquid volumetric flow and QG 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
CL < 0.01 and Frm < L1
OR CL >= 0.01 and Frm < L2
0.01 <= CL < 0.4 and L3 < Frm <= L1
OR CL >= 0.4 and L3 < Frm <= L4
CL < 0.4 and Frm >= L4
OR CL >= 0.4 and Frm > L4
L2 < Frm < L3
Liquid Holdup, EL(θ)
Once flow type has been determined, liquid holdup for horizontal flow EL(0) is calculated.
EL(0) = a CLb / Frmc
EL(0) must be greater than CL, if EL(0) is smaller than CL, then EL(0) is assigned a value of CL. Actual liquid volume fraction is obtained by multiplying EL(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 )
|Distributed||β = 0|
Liquid velocity number, NLV is given by:
NLV = 1.938 Vsl(ρL/ (g.σ))1/4
Vsl is no slip liquid velocity, ρL 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, EL(θ) is used to calculate mixture density ρm.
ρm = ρL.EL(θ) + ρG.(1-EL(θ))
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)
Calculate no slip Reynold's number using no slip mixture density and viscosity.
ReNS = ρNS.Vm.D/μNS
No slip friction factor, fNS 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 Ek is given by:
Ek = ρm.Vm.Vsg/(gc.P)
where, Vsg is no slip gas velocity and P is gas pressure.
- Standard Handbook of Petroleum & Natural Gas Engineering, William C Lyons, Volume 2