A Lumped-Parameter Model for Transient Two-Phase Gas-Liquid Flow in a Wellbore
Authors: Fairuzov Y.V. , Gonzalez J. , Lobato G. , Fuentes F. , Camacho R. 
Published: December 2007

Abstract
In this paper a lumped-parameter model of transient two-phase gas-liquid flow in a wellbore was developed. The model is based on the assumption that the flow process is essentially one-dimensional so that area-average properties can be represented as function of one space variable and time. The wellbore is divided into a number of control volumes (nodes) that are connected by junctions (flow paths). Mass and energy equations are applied to each control volume and an approximate momentum equation is used to calculate flow rate at each junction. A numerical algorithm for coupling the wellbore model with a reservoir model is proposed. Calculations were carried out to determine the behavior of the wellbore flow during a pressure buildup test. The results obtained from this calculations were compared to theoretical results found in prior literature.

Introduction
Most of buildup tests are performed by shutting in the well at the surface. This complicates the interpretation of field data, because the effect of the wellbore on the pressure buildup at the sand face. When the shut-in is carried out at the surface, the fluid continues to flow from the reservoir into the wellbore for a period of time after shut-in because of the compressibility of the wellbore fluid. This phenomenon is commonly called "afterflow" or "wellbore storage". The transient bottomhole pressure response during the wellbore storage period is a result of a dynamic interaction between the reservoir and the wellbore. Furthermore, if the wellbore contains a two-phase mixture, the process of phase redistribution can cause anomalous pressure buildup behavior (a pressure hump in the early portion of the buildup curve). Thus, the understanding of physical processes occurring in the wellbore during the buildup test is important in well test analysis.
Several mathematical models have been proposed for analyzing the pressure buildup data. van Everdingen and Hurst developed a simple analytical model which assumes that the wellbore storage coefficient is constant. This assumption is valid for single-phase flow of liquid of constant compressibility. The use of constant wellbore storage concept for single-phase gas flow or two-phase flow in the wellbore may lead to a great error in the evaluation of the sandface rate because of changes in the fluid compressibility and of the phase redistribution effects. Agarwal et al. presented an analytical solution for a constant wellbore storage coefficient. Fair modified the equation proposed by van Everdingen and Hurst by adding a term to account for the pressure change due to phase redistribution. Hegeman et al. extended the method of Fair by using an error function to represent field data, Based on the model proposed by Fair and Hegeman, Vasquez and Camacho developed a methodology to analyze transient buildup data obtained during short tests. All these models are analytical.
With the progress in the development of two-phase flow models, a number of numerical models have been proposed to describe the fluid flow in wells. In contrast to analytical methods, these models are based on the fundamental equations describing flow process in the well. Winterfield presented a model which incorporates separate equations of continuity and momentum for each phase (a two-fluid model). Hasan and Kabir developed a mechanistic model for simulating phase segregation in the wellbore which is based on the equations describing the dynamics of a single bubble migrating upward through the liquid column in the tubing.