Slugging in multiphase flow as a mixed initial-boundary value problem for a quasilinear hyperbolic system Florent Di Meglio, Glenn-Ole Kaasa, Nicolas Petit, Vidar Alstad Abstract— This paper studies the multiphasis slugging flow phenomenon occurring in oil wells and flow lines. The main con- tribution is a low-dimensional distributed parameters model, comprising the gas mass fraction, the pressure, and gas velocity as states. Along with appropriate boundary conditions, on the one-dimensional space domain, it constitutes a well-posed mixed initial-boundary value problem for a quasilinear hyperbolic system. Numerical simulation results obtained with a presented characteristics method solver stress the validity of the approach and the fair representativeness of the model. In particular, the period of simulated oscillations and their overall shape is in accordance with reference results from the literature. Controllability and observability open problems are formulated for future works. I. Introduction In this article, we propose a model of the slugging phe- nomenon taking the form of a low-dimensional hyperbolic system of conservation laws. Slugging is a two-phase flow regime occurring during the process of oil production. In cer- tain circumstances, the inhomogeneous repartition of gas and liquid into the long transport pipes leads to this oscillating flow pattern, which is detrimental to the overall production and is at the source of severe issues concerning safety of operations. The physical description of this phenomenon is as follows. Elongated bubbles of gas, separated by “slugs” travel from one end of a pipe to the other. This results in large pressure oscillations and an intermittent flow. A main negative effect is that the average (over time) production of oil is decreased compared to steady flow regimes. Modeling this phenomenon is a difficult task, because its origins are not completely understood yet. Early models have focused on the transitions between flow patterns [31] or the prediction of the flow characteristics (e.g. average liquid hold-up, pressure drop...) [1]. More recently, distributed parameters models have been developed in commercial sim- ulation softwares, such as OLGA TM or TACITE TM . They are based on nonlinear Partial Differential Equations (PDEs), and reproduce with a good accuracy the dynamical behavior of slugging wells. However, even if they rely on well- documented physics and modeling assumptions, their “black- box” nature (for the end-user) and the high dimensionality F. Di Meglio (corresponding author) is a PhD candidate in Mathematics and Control at MINES ParisTech, 60, Bd St-Michel, 75272 Paris, Cedex 06, France G.O. Kaasa is Research Engineer at StatoilHydro ASA, Research Center Porsgrunn, Heroya Forksningspark 3908 Porsgrunn, Norway N. Petit is Professor at MINES ParisTech, 60, Bd St-Michel, 75272 Paris, Cedex 06, France V. Alstad is Research Engineer at Statoil ASA, Research Center Pors- grunn, Heroya Forksningspark 3908 Porsgrunn, Norway of the state equations make these softwares hardly usable for mathematical analysis, let alone control design. Conversely, reduced models have been developed [8], [24], [30]. They are based on nonlinear Ordinary Differential Equations (ODEs) and capture the main features of the slugging oscillations. Their relative simplicity makes them suitable for control (and observer) design, at the expense of sometimes tedious tuning procedures aimed at reproducing field data (see, e.g., [10]). These models rely on restrictive modeling assumptions which, in turn, might seem inappropri- ate from a physical modeling view-point. The Jansen model [24] is designed specifically for gas-lifted wells, whereas the Storkaas model [30] corresponds to risers with a low- point. The model proposed in [8] assumes the existence of an irregularity in the pipe geometry at the birth of instability. In this article, we propose a low-dimensional model which is minimal, in the sense that no assumptions are made on the geometry or setup of the system, and that it reproduces with a fair accuracy observed behaviors. Following many other modeling works (e.g. [1], [3], [12], [16]), the drift-flux approach is used. This implies that the momentum equations for the gas and the liquid are combined into a single one, and that an affine slip relation with constant parameters relates the velocities of the two phases. Importantly, this is the only empirical relation used in the model. The approach is very similar to the density wave model of Sin` egre [28], which was first described and illustrated by OLGA simulations in [22]. In [28], a distributed parameters model was provided, along with a thorough stability analysis, describing the phenomenon. Yet, the analysis relies on simplifications which preserve the stability properties, but may hurt the physical interpretation 1 . The main contribution of this article is a low-dimensional model of slugging phenomenon taking the form of a hy- perbolic system of conservation laws, with a one-sided boundary actuation. The advantages of such a formulation are two-fold. First, it is consistent with recent mathematical tools of analysis of PDE control systems, e.g. results that guarantee well-posedness of the problem. Similarly, theoret- ical controllability and observability results might be used. Such problems of well-posedness and boundary control of hyperbolic systems have been widely studied [6], [25], [26]. Second, the method of characteristics (see [2], [5], [27] for application to hyperbolic control systems) can be used to numerically solve the equations, which reduces the compu- 1 In particular, the gas velocity is assumed constant in time and space, which is not realistic in practice. No such assumption is made here. 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 978-1-4577-0079-8/11/$26.00 ©2011 AACC 3589