# 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-E_{k})

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}).

`Fr`_{m} = v_{m}²/ g.D

where, v_{m} is mixture velocity, D is pipe inside diameter and g is gravitational constant.

`C`_{L} = Q_{L}/ (Q_{L} + Q_{G})

where, Q_{L} is liquid volumetric flow and Q_{G} is gas volumetric flow.

The transition lines for correlation are defined as follows:

`L`

_{1}= 316 C_{L}^{0.302}`L`

_{2}= 0.0009252 C_{L}^{-2.4684}`L`

_{3}= 0.1 C_{L}^{-1.4516}`L`

_{4}= 0.5 C_{L}^{-6.738}

#### Segregated Flow

`C`

_{L}< 0.01 and Fr_{m}< L_{1}`OR C`

_{L}>= 0.01 and Fr_{m}< L_{2}

#### Intermittent Flow

`0.01 <= C`

_{L}< 0.4 and L_{3}< Fr_{m}<= L_{1}`OR C`

_{L}>= 0.4 and L_{3}< Fr_{m}<= L_{4}

#### Distributed Flow

`C`

_{L}< 0.4 and Fr_{m}>= L_{4}`OR C`

_{L}>= 0.4 and Fr_{m}> L_{4}

#### Transition Flow

`L`_{2} < Fr_{m} < L_{3}

### Liquid Holdup, E_{L}(θ)

Once flow type has been determined, liquid holdup for horizontal flow E_{L}(0) is calculated.

`E`_{L}(0) = a C_{L}^{b} / Fr_{m}^{c}

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 |

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).

`E`_{L}(θ) = B(θ) x E_{L}(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 - C`_{L})ln( d.C_{L}^{e}.N_{LV}^{f}.Fr_{m}^{g} )

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, N_{LV} is given by:

`N`_{LV} = 1.938 V_{sl}(ρ_{L}/ (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,

`E`_{L}(θ)_{transition} = AE_{L}(θ)_{segregated} + BE_{L}(θ)_{intermittent}

where A and B are as following –

`A = ( L`

_{3}- Fr_{m})/(L_{3}- L_{2})`B = 1- A`

Liquid holdup, E_{L}(θ) is used to calculate mixture density ρ_{m}.

`ρ`_{m} = ρ_{L}.E_{L}(θ) + ρ_{G}.(1-E_{L}(θ))

### (dP/dZ)_{Elevation}

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

`(dP/dZ)`_{Ele.} = ρ_{m}.g.sin(θ)/(144.g_{c})

### (dP/dZ)_{Friction}

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

`Re`_{NS} = ρ_{NS}.V_{m}.D/μ_{NS}

No slip friction factor, f_{NS} is then calculated using Colebrook-White equation.

Ratio of friction factor is defined as

`f`_{TP}/ f_{NS} = e^{S}

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 = C`_{L} / E_{L}(θ)²

Pressure loss due to friction is:

`(dP/dZ)`_{Fric.} = 2.f_{TP}.V_{m}².ρ_{NS} /(144.g_{c}.D)

Pressure loss due to acceleration, factor E_{k} is given by:

`E`_{k} = ρ_{m}.V_{m}.V_{sg}/(g_{c}.P)

where, V_{sg} is no slip gas velocity and P is gas pressure.

Spreadsheet for Beggs & Brill Method

### References

- Pressure Loss Calculations at Fekete.com
- Standard Handbook of Petroleum & Natural Gas Engineering, William C Lyons, Volume 2