Fluid Flow

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

Segregated Flow

CL < 0.01 and Frm < L1OR CL >= 0.01 and Frm < L2

Intermittent Flow

0.01 <= CL < 0.4 and L3 < Frm <= L1OR CL >= 0.4 and L3 < Frm <= L4

Distributed Flow

CL < 0.4 and Frm >= L4OR CL >= 0.4 and Frm > L4

Transition Flow

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
Flow Regime a b c
Segregated 0.98 0.4846 0.0868
Intermittent 0.845 0.5351 0.0173
Distributed 1.065 0.5824 0.0609

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 )
Uphill d e f g
Segregated 0.011 -3.768 3.539 -1.614
Intermittent 2.96 0.305 -0.4473 0.0978
Distributed β = 0
Downhill d e f g
All 4.7 -0.3692 0.1244 -0.5056

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))

(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, 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.

Resources

References

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