Multiphase flow in heterogeneous porous media from a stochastic differential geometry viewpoint

Multiphase flow of immiscible fluids is studied by means of a stochastic flow path approach. This approach is based on a differential geometric formulation that replaces the partial differential equations (PDEs) of flow by a set of ordinary differential equations (ODEs) that determine the flow paths and impose conservation of flux. It is shown that flux conservation along the flow paths involves a space transform. Other formulations of the multiphase flow equations involve Jacobian mappings. Flow realizations as well as statistical flow moments can be derived by means of the stochastic flow path method. Advantages of the stochastic flow path method include: reduction of a PDE to an ODF system, independence from perturbation approximations and Green's functions, and the freedom to use random initial conditions at the boundary. Using the stochastic flow path method, closed-form expressions are obtained for two-phase flow in uniaxially heterogeneous media. Two-phase flow in a heterogeneous two-dimensional medium is also investigated using numerical simulations.