approximation of nonlinear terms in the pressure-based solution of multiphase flow equations in porous media

 simultaneous flow of two or three immiscible fluids in porous media is modeled by a system of coupled and highly nonlinear partial differential equations. these equations are reduced to a system of algebraic equations by the use of numerical methods. selecting the proper primary variables is a critical step in efficiently solving of multiphase subsurface flow problems. one of the common selections is to select the phase pressures as the primary variables (pressure-based solution). this type of solution consists of strong nonlinear terms and thus encounters some difficulties, such as ill conditioning of the system of algebraic equations under dry flow condition. here, we present a systematic study to deal with these difficulties in the solution of algebraic systems. it is believed that the solution techniques presented here may also prove useful for other strongly nonlinear partial differential equations.