Difficulties and uncertainty in mathematical/numerical modelling of fluid flow in fractured media

The ability to numerically model single-phase and multiphase flow of fluids in porous or fractured media is extremely important in developing an understanding of the complex phenomena governing the flow. The flow is complicated by the presence of heterogeneities in the reservoir at many different length scales by special flow features such as fractures or faults and by phenomena such as diffusion and dispersion. These effects must be effectively modelled by terms in coupled systems of non-linear partial differential equations which form the basis of the simulator. The simulator must be able to model both single and multiphase flows and the transition regimes between the two in unsaturated flow applications. A discussion of some of the aspects of modelling unsaturated and multiphase flows in the presence of heterogeneities and severe channelling is presented along with directions for future work. Simulators are severely hampered by the lack of knowledge of reservoir properties, heterogeneities, fracture dimension and orientation, and relevant length scales and other important mechanisms. Simulations can be performed either deterministically, to predict the outcome of a single realization of reservoir and flow properties, or via stochastic techniques to incorporate uncertainties of flow directly. Due to the extreme difficulties in using stochastic differential equation models for non-linear multiphase flows, we will concentrate on the potential of deterministic models. Recent developments have been made in homogenization, scaled averaging, and the use of the simulator as an experimental tool to develop methods to model the interrelations between localized and larger-scale media effects. Monte Carlo techniques using simulators with effective parameters can generate statistics for multiphase flow.