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# Homogeneous Two-Phase Flow

## Introduction

In homogeneous two-phase flow, both phases are assumed to move at the same in-situ velocity. This assumption enables the simple calculation of liquid holdup and pressure drop in pipes and wellbores. It was found experimentally that this assumption of homogeneous flow is representative in "Dispersed Flow" of gas-liquid (Shoham 2006) and liquid-liquid flow (Torres 2005) and in "Mist Flow" of gas-liquid. Moreover, It was found that it is a representative assumption for flow inside liquid slugs (Zhang et al. 2003).

## Derivation of Steady-State Homogeneous Two-Phase Flow Equations

The condition of homogeneous flow is called the "no-slip" condition, where the slip is the difference between gas and liquid in-situ velocities. The "no-slip" and "homogeneous" terminologies are used interchangeably in the literature and they refer to the same concept. Based on the "no-slip" conditions, ....................(1)

Based on the mass balance at steady-state conditions, the superficial phase velocity is linked to the in-situ phase velocity as follows, , and....................(2) ,....................(3)

where ρ is the phase density, A is the cross-sectional area of the pipe and HL is the liquid holdup. Substituting Eqs. 2 and 3 into Eq. 1, the following expression for HL is found as follows, ,....................(4)

where λ becomes the liquid holdup at the "no-slip" conditions. The pressure gradient dp/dL of the "no-slip" flow is calculated from the momentum balance based on averaged fluid properties. E.g. the Mixture Density ρm is defined as follows, ....................(5)

Based on the same weighting approach, some authors define the "Mixture Viscosity" μm as follows, ....................(6)

In the case of three-phase flow, the interfacial tension is of oil and water mixture is estimated as follows, ....................(7)

where fw is the no-slip liquid holdup in the liquid phase and is defined, similar to λ as follows, ....................(8)

From the mass balance of the steady-state flow, the pressure gradient is then estimated as follows, ....................(9)

where vm is mixture velocity or the sum of the superficial velocities of the flow phases, while fn is the no-slip friction factor calculated from the following definition of Reynolds number Ren, ....................(10)