Practical application of the fractional flow approach to multi-phase flow simulation

Fractional flow formulations of the multi-phase flow equations exhibit several attractive attributes for numerical simulations. The governing equations are a saturation equation having advection-diffusion form, for which characteristic methods are attractive, and a global pressure equation whose form is elliptic. While numerical advantages may be demonstrated for a variety of model problems, general implementation requires careful consideration of both the form of the governing equations and the choice of numerical methods. Difficulties with current methods include implementation of general boundary conditions that are appropriate for hydrological application, inclusion of general material heterogeneities, application to problems with multiple infiltration and drainage fronts, and inclusion of both gravity and compressibility in general, multi-dimensional problems. Numerical approaches continue to be developed to address specific aspects of these difficulties, but a number of outstanding numerical issue remain to be solved.