Isothermal two-phase flow of a vapor–liquid system with non-negligible inertial effects Iacopo Borsi, Lorenzo Fusi ⇑ , Fabio Rosso Alessandro Speranza Dipartimento di Matematica, ‘‘U. Dini’’, Viale Morgagni 67/a, 50134 Firenze, Italy article info Article history: Received 15 July 2010 Accepted 5 May 2011 Available online xxxx Keywords: Flow in porous media Liquid–gas two-phase flows Nonlinear constitutive equations Numerical simulation abstract In this paper we consider the problem of gas/liquid extraction near the bottom well in the context of geothermal energy exploitation. In particular we develop a mathematical model for the isothermal two-phase flow of a mono-component fluid in an undeformable porous media taking into account inertial effects. We use the so-called Forchheimer’s equation to model the relation between the fluid velocity and the pressure gradient in the region of co- existence of the two phases. We formulate the problem in cylindrical geometry assuming steady state and isothermal conditions. We take into account capillary pressure and we study its influence on the whole system. We derive important formulas that allow to pre- dict the main thermodynamical quantities in the region of co-existence of the liquid and gaseous phase and we determine constraints on the physical parameters in order to predict the behavior of the fluid in the domain of the problem. Finally, we perform some numerical simulations to investigate the dependence on the physical parameters involved in the model. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction It is known that flows through porous media can be modeled within the theory of mixture that allows to take into account the numerous mechanisms occurring during the flow (see Rajagopal, 2007, Rajagopal & Tao, 1995). Popular models such as Brinkman (1947a, 1947b), Biot (1956a, 1956b), and Forchheimer (1901) and the most famous Darcy (1856) can be obtained from classical balance laws under specific assumptions in the context of mixture theory. Recently Rajagopal (2007) has developed a hierarchy of approximate models in which Darcy’s law represents the simplest of the models considered. Depending on which effects one wants to model (frictional effects, drag effects, inertial effects, compressibility, etc.) a series of assumptions which lead to particular constitutive equations must be made. For example, Darcy’s law is obtained assuming that the porous matrix is rigid, that the interaction forces between the solid and the fluid are only due to the friction between the fluid and the pore surface (expressed by a drag-like term proportional to the velocity of the fluid), that the frictional ef- fects within the fluid due to its viscosity can be neglected, that the flow is sufficiently slow that inertial terms can be ignored and that the partial stress is the one of an Euler fluid. When relaxing some assumptions one gets more complex models. For instance, assuming that the frictional effects in the fluid are not negligible one gets the Brinkman model (Brinkman, 1947a, 1947b). Further, if one relaxes the hypothesis of neg- ligible inertial effects one gets a series of nonlinear models (Dupuit–Forchheimer of second or third grade or power law, see Aulisa, Bloshanskaya, Hoang, & Ibragimov (2009)), depending on the way interaction forces are modeled. In other situations (e.g. oil recovery) the effects of pressure on viscosity cannot be neglected or, as in the case developed by Munaf, Lee, and 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.05.003 ⇑ Corresponding author. E-mail address: fusi@math.unifi.it (L. Fusi). International Journal of Engineering Science xxx (2011) xxx–xxx Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci