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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,

Based on the mass balance at steady-state conditions, the superficial phase velocity is linked to the in-situ phase velocity as follows,

where *ρ* is the phase density, *A* is the cross-sectional area of the pipe and *H*_{L} is the liquid holdup. Substituting Eqs. 2 and 3 into Eq. 1, the following expression for *H*_{L} is found as follows,

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 *ρ*_{m} is defined as follows,

Based on the same weighting approach, some authors define the "Mixture Viscosity" *μ*_{m} as follows,

Trallero, J L, Intevep, S A, Sarica, C, and Brill, J P. A study of oil-water flow patterns in horizontal pipes. United States: N. p., 1996. Web.

Torres, C. Modeling of oil-water flow in horizontal and near-horizontal pipes. 2005. Ph.D Thesis, The University of Tulsa, Tulsa, OK.