From Single Phase to Compositional Flow Applicability of Mixed Finite Elements

In this paper we discuss the formulation of the governing equations that describe ow of uids in porous media Various types of uid ow ranging from single phase ow to compositional ow are considered It is shown that all the di erential equations governing these types of ow can be e ectively rewritten in a fractional ow formulation i e in terms of a global pressure and saturation or saturations and that mixed nite element methods can be accurately exploited to solve the pressure equation Extensive numerical results are presented to see the performance of the mixed methods for the ow equations using di erent mixed nite elements Introduction Multiphase ow of uids in porous media is physically and chemically complex It in volves heterogeneities in the porous media at many di erent length scales and complicated processes such as di usion and dispersion Numerical simulation of these phenomena is a critical step in understanding multicomponent multiphase ow in porous media see e g and the bibliography therein There are two critical factors that determine the choice of discretization techniques for numerically solving the governing di erential equations describing ow of uids in porous media the mathematical properties of the di erential equations and the geometry of the porous media It is well known that the transport and di usion dispersion terms in the ow and transport equations are governed by uid velocities Thus accurate numerical simulation requires accurate approximations for these velocities Often the ow properties of the porous media change abruptly with sharp changes in lithology Also the viscosity changes rapidly in space across uid interfaces Consequently the coe cient in the ow equations is quite rough In order for uids to ow smoothly pressures change extremely rapidly Standard nite di erence and nite element methods for solving these ow equations determine a pressure and then a velocity by di erencing or di erentiating the resulting pressure and multiplying it by the rough coe cient This approach generates a rough and inaccurate velocity which then reduces the accuracy of numerical simulation of the uid ow in porous media Also the geologic structure and topography of porous media can have a signi cant impact on the ow Finite volume methods with harmonic Mathematics Subject Classi cation Primary N S