# SPE Production & Operations Volume 21, Number 1, February 2006, pp. 98-106

SPE-90583-PA

### Including Nonequilibrium Relaxation in Models for Rapid Multiphase Flow in Wells

View full textPDF ( 415 KB )

DOI  10.2118/90583-PA http://dx.doi.org/10.2118/90583-PA

### Citation

• Civan, F. 2006. Including Nonequilibrium Relaxation in Models for Rapid Multiphase Flow in Wells. SPE Prod & Oper21 (1): 98-106. SPE-90583-PA.

### Discipline Categories

• 4.5.1 Piping Design
• 6.3.2 Multi-phase Flow
• 6.6.5 Well Performance Monitoring, Inflow Performance
• 5.6.3 Slurry Flow and Erosion
• 4.5 Pipelines, Flowlines and Risers

### Summary

The modeling of rapid flow of multiphase reservoir fluids in wells involving nonequilibrium partial separation of dissolved gases from the metastable oil and water phases is reviewed and formulated. The multiphase-fluid flow is described by means of the differential mass, momentum, and energy-balance equations along the well. The rate of gas transfer from the metastable liquid phases to the gas phase is expressed in terms of the relaxation time determined by the prevailing gas-phase volume fraction and the pressure and temperature conditions. The differential equations are solved numerically under a typical scenario involving the constant gas-, oil-, and water-production rates at the wellhead. It is demonstrated that the nonequilibrium saturation and pressure distributions in wells deviate from the equilibrium results, depending on the variation of the local relaxation time along the well. It is concluded that inclusion of nonequilibrium relaxation effects in models for rapid multiphase flow in wells is required.

### Introduction

When the reservoir fluid system invades the wellbore, it is usually at an equilibrium state at the prevailing pressure and temperature conditions. As this fluid system moves along the well from the bottom hole toward the wellhead, the pressure of the fluid system decreases owing to the decline of the hydrostatic pressure and the loss of mechanical energy by wall shear. Simultaneously, an exchange of heat may take place between the fluid system flowing through the well and the surrounding geological formation. A fraction of the dissolved gas separates from the liquid phases to form and/or join the gas phase.

The separation of the dissolved gas from the liquid phases (oil and water) occurs under nonequilibrium conditions when the flow of the multiphase-fluid system is sufficiently rapid. This is because the separation of the gas component from the liquid phases involves the transport of the gas to the interface located between the liquid and gas phases by diffusion, which is a much slower process than the transport of the well fluids by convection (Civan 1994; Civan and Rasmussen 2003). Such conditions may not allow enough time for the fluid system to attain a local equilibrium state, referred to as the saturation condition, along the well. Therefore, as described in Fig. 1, the equilibrium gas saturation of the multiphase-fluid system may be higher than the actual gas saturation, the difference of which constitutes the driving force for the interface gas transfer between the liquid and gas phases.

As stated by Bilicki and Kestin (1990) and Downar-Zapolski et al. (1996) about a similar process concerning the phenomenon of vapor production associated with the flashing (rapid evaporation) of liquid flows, the liquid system attains a metastable condition and, therefore, the flushing of the liquid phase at saturation delays. The retardation depends on the local fluid conditions (including the vapor fraction in the multiphase-fluid system) and the pressure and temperature conditions. Consequently, the prevailing metastable condition creates nonequilibrium between the liquid and vapor phases and causes the interface transfer of the gas from the liquid phases to the vapor phase to occur gradually at a finite rate. Bilicki and Kestin (1990) and Downar-Zapolski et al. (1996) describe the nonequilibrium interface mass transfer by means of a first-order relaxation equation, similar to Einstein (1920). Badur and Banaszkiewicz (1998) modified the conventional relaxation equation to include the capillary effects based on the Ginzburg-Landau model. In contrast, the frequently used equilibrium-state models assume an infinitely fast interface mass transfer and, therefore, an instantaneous gas separation at saturation. Inherently, such equilibrium models, including the popular OLGA model [an extended mechanistic two-fluid equilibrium model by Bendiksen et al. (1991)], are not adequate for describing the rapid fluid flow in wells experiencing an incomplete interface gas transfer.

In this paper, the relevant modeling approaches for description of rapid multiphase flow of reservoir fluids in wells from the bottom hole to the wellhead, by considering the effect of the nonequilibrium incomplete separation of dissolved gases from the oil and water phases, are reviewed and discussed. The multiphase-fluid flow is described by means of the differential mass, momentum, and energy-balance equations along the well. The mathematical modeling considers transient-state phase transition and partitioning of the gas between the gas and liquid (oil and water) phases. The partitioning occurs partially depending on the multiphase-flow rates and the relaxation time, determined by the prevailing local gas-phase fraction, and the multiphase-fluid pressure, temperature, and mixing conditions.

The model is simplified for flow under steady-state production conditions and applied for a typical case involving the well operation at constant gas-, oil-, and water-production rates at the wellhead. A solution of the governing differential equations is obtained numerically, and the simulation results are presented. The extent of the nonequilibrium effect on the deviation of the pressure distribution along the wells from the equilibrium-model predictions is delineated. The results obtained for nonequilibrium conditions are compared with those obtained under equilibrium conditions, the latter of which is based on conventional modeling assuming equilibrium between the various fluid phases. This study demonstrates that the nonequilibrium gas saturation and multiphase-fluid-pressure profiles along the well deviate from the equilibrium results depending on the relaxation time involving the rate of gas transfer from the metastable liquid phases to the gas phase.

View full textPDF ( 415 KB )

### History

• Original manuscript received: 5 June 2004
• Revised manuscript received: 29 July 2005
• Manuscript approved: 2 August 2005
• Version of record: 20 February 2006