Abstract

This work presents a simple yet accurate model for two-phase flow that is easily incorporated in a heat transfer model. The fluid flow model uses a single expression for liquid holdup, with flow-pattern-dependent values for flow parameter and rise velocity. To avoid abrupt changes in gradients at flow pattern boundaries, we use an empirically determined exponential function for smooth transition of parameter values between flow patterns. Frictional and kinetic heads, whose contribution to total pressure loss are small, are estimated using the homogeneous model.

Our approach to modeling heat transfer is based on treating the wellbore as a source/sink of heat of finite diameter with the formation being an infinite sink/source of heat. The resulting heat loss/gain by the wellbore from the formation is used in an energy balance for the wellbore fluid to calculate steam quality and other fluid properties. Proper calculation of the resistances to heat transfer of the various elements of the wellbore - particularly the natural convective heat transfer of the fluid in the tubing/casing annulus - is shown to be important.

We used high quality field data to validate our model. Estimates of pressure loss and heat transfer were also made using such popular models as Beggs and Brill, Orkiszewisky, and Hagedorn and Brown. The model we present here outperforms the other models for the field data we used.

Introduction

Simultaneous flow of steam and water occurs in geothermal wells, in steam injection, in SAGD wells, and in many other industrial applications, such as boilers and evaporators. An accurate model of steam-water flow is, therefore, very important. Because of condensation/vaporization occurring in a steam-water system, modeling geothermal wells must account for the coupled nature of fluid flow and heat transfer.

Development of pressure-traverse computation for steam-water flow followed a trend similar to that for wells producing hydrocarbons. As the fluid moves along the well, depressurization usually leads to steam generation, similar to the case of gas coming out of solution in an oil well. The principal flow regimes - bubbly, slug, churn, and annular - are common to both systems. Most of the steam-water flow modeling effort has adopted a hybrid approach where the slip between steam and water phases is computed with a different, generally empirical, model in each flow regime. Regime delineations also are generally empirical and usually cause difficulty due to discontinuity near the transition between the flow regimes. Correlations proposed for geothermal wells by Gould (1974), Chierici et al. (1981), Ambastha and Gudmundsson (1986a, 1986b), Chadha et al. (1993), and other fall in this category. Popular two-phase flow correlations - Orkiszewski (1967), Hagedorn-Brown (1965), Beggs and Brill (1973) - have also been applied to steam-water flow in geothermal wells with mixed results. Hasan and Kabir (2009) present a detail review of these and other works for steam-water flow in geothermal wells.

Most of these pressure-traverse computation procedures are empirical in nature and run the risk of incurring significant estimation error for systems that are much different from the data-base basis of the correlation. The recent work of Garg et al. (2004a, 2004b) is a case in point. They had to modify the Hughmark's (1963) correlation (used along with Duns and Ros (1963) flow pattern map) to fit the high quality data they gathered. In this work we utilize a simple, physics based approach that accounts for the buoyancy of the vapor (gas) phase and its tendency to move through the channel center.

Most steam-water flow model, especially those for geothermal wells, uses a constant overall-heat-transfer coefficient (Ut) for the entire wellbore-formation system. The approach presented here separates the two (wellbore and the formation) systems for modeling heat transfer, which offers greater flexibility and versatility.

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