Unstable production behavior affects many offshore fields, particularly at their decline stages. "Casing heading" and "density wave" are two types of instability that have been identified and both can result in production loss. These instabilities have influence on each other, but the interaction has not been fully addressed yet. One contribution of this paper is to study the interplay of these two phenomena. For this purpose, key variable of density wave phenomenon; pressure at injection point in tubing that is governed by an integral equation; is considered as a fourth state variable added to 3-D dynamical model of casing heading. Another contribution is some modifications on the casing heading and density wave dynamic models.
Finally, the proposed model is simulated in MATLAB® and OLGA®, and the results are compared with well operation facts and data from field.
Modeling and analysis of two-phase flow through vertical pipes have gained considerable interest in recent years [1, 2].
Especially, simplified versions of two-phase flow model has application in stability analysis and stabilization of gas lifted oil wells [3–5]. Figure 1 shows a typical diagram of a gas-lifted oil well. In this technology, gas is routed from the surface into the annulus and then injected deep into the tubing in order to be mixed with the fluid form the reservoir. This reduces the density of the fluid in the bottom-hole and hence the production rate from the low pressure reservoir is increased .
A flow model is derived in  directly from equations of momentum and enthalpy balance laws in inclined pipes for calculation of pressure and temperature profile of gas production systems. This model has been applied to simulate the various types of pipe in gas production. This model has also been used for modeling of two-phase flow in gas lifted oil wells . In , a three-dimensional first-order linear model is obtained and solved with Laplace transform. The resulting characteristic equation is used for stability analysis. In , a linear three-dimensional state-space model is proposed in which tubing pressure, fluid flow rate from reservoir and gas flow rate constitute the state variables. In , model of gas injection system is described by Darcy-Weisbach equation and formula of orifice flow for adiabatic conditions. Then reservoir model is described using inflow relationship for a reservoir in which both single-phase and two-phase flow can occur. Finally reservoir model and well model are coupled to complete the mathematical model.
Gas-lifted oil wells often become unstable at their decline stages. There are several different phenomena to account for the instability behavior in oil and gas wells. The fluctuating and sometimes chaotic unstable production behavior affects many offshore fields. The hazardousness of the fluctuations to operation safety and smoothness has been warned and the production reduction due to the instabilities has been widely addressed. Two types of instabilities, "casing heading" and "density wave" instabilities, which both can result in production loss has been reported in . This work addresses this issue by attempting to derive a mathematical model that can well describe the instabilities.
A simplified model that describes the well behavior in casing heading phenomenon is presented in [6, 11]. It consists of a three dimensional nonlinear state space model. Mass of gas in annulus, mass of gas in tubing and mass of oil in tubing are the corresponding state variables, that we will discuss it. But, the second state variable is eliminated in  and a two dimensional model is used. For density wave instability, a distributed parameter model, that is described by an integral is presented in [5, 13]. Also, in  model of density wave and 2-D version of casing heading is used for predicting instabilities in gas lifted oil wells. But, interaction between the two types of instability is not fully considered. In the proposed density wave model, constant value of separator pressure is considered as well head pressure and pressure gradient is calculated based on it. In practical situation, well head pressure is not constant because of casing heading phenomenon.
In this work, we present some modifications on density wave model to be suitably jointed with a modified version of 3-D casing heading model.