Semi-analytical solution for a hyperbolic system modeling 1D polymer slug flow in porous media

摘要

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 computational algorithms, used for the interpretation of laboratorial experiments, and in streamline simulators.