Predictions of Two-Phase Critical Flow Boundary and Mass Flow Rate Across Chokes
Authors: Alsafran Eissa M. , Kelkar Mohan Gajanan 
Published: December 2007

Abstract
The objective of this study is to develop and validate a theoretical slip model for two-phase flow through chokes. As opposed to the models1,2 currently used by the industry, the present model accounts for slippage between the liquid and gas phases as they pass through the choke. The theoretical basis of the model is a 1D balance equation of mass, momentum and energy with the assumptions of constant quality and incompressible liquid phase. The present slip model is capable of predicting the critical/sub-critical flow boundary and the critical and sub-critical mass flow rates. A model validation study demonstrated the capability of the slip models to predict the critical flow boundary with an average error and standard deviation of 5.2% and 15.5% respectively. Furthermore, the present slip model predicted the sub-critical mass flow rate with an average error and standard deviation of 2.7% and 12.5%, respectively. Compared to existing no-slip models1,2 commonly used by the industry, the present slip model predictions outperformed their predictions, indicating the importance of the slippage phenomenon.

Introduction
The two common restrictions encountered in a production system are surface chokes and subsurface safety valves (SSSV). The purposes of installing chokes and SSSV's are different; however, the basic equations which govern the flow through these restrictions are very similar. There are two types of flow behavior across chokes, namely, critical and sub-critical. The critical flow occurs when the fluid velocity at the smallest cross section in the restriction is equal to the velocity of sound in that medium. When the velocity is less than the velocity of sound, we call it a sub-critical flow; if the velocity is greater than the velocity of sound, we call it a critical flow. If the flow is sub-critical, the flow rate is related to the pressure drop across the restriction. On the other hand, if the flow is critical, the rate is only related to the upstream pressure, thus reduction in down stream pressure does not affect the rate since the reduction can never be transmitted upstream. Predicting the flow pattern, critical/sub-critical boundary, as well as the flow rate across the choke is crucial for well productivity and optimization.
Extensive studies were conducted and several models were developed on two-phase flow across chokes which mainly fall into two categories, namely empirical and theoretical models. The empirical models such as Gilbert3, Ros4, Achong5, Pilehvari6, Ashford and Pierce7, Osman and Dokla8, and Omana et al.9 were all developed on specific range of data and cannot be extended beyond their range. The second category comprises the theoretical approaches, basically derived from mass, momentum and energy balances such as Sachdeva et al.1, Perkins2 and Selmer-Olsen et al.9 The theoretical models are mostly used by the industry because of their ability to simulate the physical phenomena, thus they are considered more accurate. The following is a brief description of these theoretical models.
Sachdeva et al.1 acquired experimental data for critical, sub-critical flows and the boundary between them using air/water and air/kerosene system for different sizes of choke diameters ranging from 0.25 to 0.5 in. A theoretical model is developed from the 1D mass, momentum and energy balance equations for a two-phase gas liquid mixture to predict the critical and sub-critical mass fluxes, which is considered comprehensive since fluid properties are also involved. Perkins2 presented theoretical equations that describe is entropic multiphase flow across a choke valid for critical sub-critical flows. The model incorporates the oil, gas and water properties correlations to predict the critical/sub-critical boundary. An expression for total mass flow rate across the choke was developed through a combined equation of conservation of mass and is entropic expansion of a homogeneous multiphase mixture. Selmer-Olsen et al.10 acquired experimental data and developed the "Hydro" model which uses a control volume approach for the choke orifice and its downstream to mechanistically account for irreversible loss process instead of using discharge coefficient only. The model is derived from the local cross-sectional averaged balance equations of mass, momentum and energy for steady-state flow of a multiphase mixture. Contrary to Sachdeva et al.1 and Perkins2 models, the "Hydro" model accounts for slippage between the phases. Although the "Hydro" model is more accurate, it is complex in formulation and solution for the boundary.