Semi-analytical solution for a hyperbolic system modeling 1D polymer slug flow in porous media Bruno J. Vicente, Viatcheslav I. Priimenko, Adolfo P. Pires n North Fluminense State University Darcy Ribeiro, Macaé, Brazil article info Article history: Received 5 February 2013 Accepted 1 February 2014 Available online 14 February 2014 Keywords: flow in porous media chemical flooding polymer slug flow hyperbolic systems splitting procedure semi-analytical solutions abstract In this paper we study one-dimensional displacement of oil by a polymeric solution in porous medium. The model uses the two phase extension of the Darcy law, and does not include capillarity and gravity effects. Under these conditions the mathematical model is composed of a 2  2 hyperbolic system. The boundary condition is defined by a piecewise function representing a particular case of polymer slug injection. The polymer adsorption is modeled by the Langmuir isotherm and water viscosity is considered as a linear function of the concentration. The differential equations are decoupled using a new variable associated with the conservation of water phase instead of the time variable. Such a replacement allows splitting the original system into two independent equations: one depending on the thermodynamic model of solid–liquid equilibrium only and other equation depending on the transport properties. The resulting system is solved using the method of characteristics. We present the complete semi-analytical solution to the formulated problem in detail, describing the characteristic waves that may arise. The solution constructed in this work presents good agreement compared to a commercial numerical reservoir simulator based on finite difference schemes (implicit in pressure and explicit in saturation) and to the simpler case of a finite slug with constant concentration. The analytical development presented here can be used for the construction of efficient computa- tional algorithms, used for the interpretation of laboratorial experiments, and in streamline simulators. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Oil reservoirs, that do not exhibit satisfactory recovery factors when produced by natural mechanisms or by water flooding, are natural candidates for the use of enhanced oil recovery (EOR) methods to improve oil production. Among the most important EOR techniques are the chemical methods (Green and Willhite, 1998), and within this category, polymer flooding is suitable for application in highly heterogeneous porous medium and with large contrast between phase viscosities (Littmann, 1988; Sorbie, 1991). Analytical and semi-analytical solutions, describing the performance of production, become an important tool for the selection of the most effective methods and for the experimental determination of the main parameters affecting the final results. Some mathematical models based on the theory of water fractional flow were developed to describe several methods of EOR (Pope, 1980; Helfferich, 1981). The hydrodynamic behavior of the displacement of oil by a polymeric solution in porous medium can be modeled by a 2  2 quasilinear hyperbolic system (Fayers and Perrine, 1958). The main phenomena occurring in this process are the adsorption of the injected chemicals in the rock surface that can be represented by an adsorption isotherm and the increase in water viscosity as function of polymer concentration. The solution of this system subjected to constant initial and boundary conditions (a classical Riemann problem) is self-similar and well known (Rhee et al., 1989). For practical purposes, usually a finite volume of polymer is injected (polymer slug) followed by water drive. In this case a discontinuity in the boundary condition arises, and the solution of the problem is not self-similar, owing to interactions between waves of different families (Bedrikovetsky, 1982). Introducing a potential function related to the conservation of water volume as an independent variable instead of time makes it possible to decouple the hyperbolic system into two equations (Pires et al., 2004; Pires and Bedrikovetsky, 2005; Ribeiro and Pires, 2008). One equation contains only the thermodynamic function representing a purely chromatographic process (Rhee et al., 1970) whereas the other contains thermodynamic functions and transport properties. We use this approach to construct the 1D semi-analytical solution to the initial boundary-value problem formulated in the paper, representing the injection of polymer into four stages. During the first stage polymer is injected at Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/petrol Journal of Petroleum Science and Engineering http://dx.doi.org/10.1016/j.petrol.2014.02.005 0920-4105 & 2014 Elsevier B.V. All rights reserved. n Correspondence to: North Fluminense State University Darcy Ribeiro, P.O. Box 119562, 27910-970 Macaé, Brazil. Tel.: þ55 22 27656565. E-mail addresses: adolfo.puime@gmail.com, puime@lenep.uenf.br (A.P. Pires). Journal of Petroleum Science and Engineering 115 (2014) 102–109