Transitional Waves in Three-Phase Flows in Heterogeneous Petroleum Reservoirs

Natural reservoirs are porous and permeable rocks containing deposits of hydrocarbons (oil reservoirs) or pure water (aquifers). Porosity and permeability are the most important physical properties of reservoirs that affect flow and transport. In general these geological parameters are stochastically modeled (see [?, ?] and references therein). As a consequence the equations governing flow and transport in porous media are stochastic and predictions based on them (e.g. hydrocarbon production forecasts in petroleum reservoirs or prediction of contaminant movement in groundwater) are uncertain. In order to assess the influence of geological uncertainty in fluid flow in porous media, stochastic numerical simulators are commonly used. In this manuscript we are concerned with the numerical simulation of three-phase flow (water, gas and oil) in petroleum reservoirs. In immiscible three-phase oil, gas and water flows, typical of petroleum reservoirs, the leading oil front can split into two, a classical Buckley-Leverett front followed by a new type of shock wave [?]. This new shock wave is related to the existence of an elliptic region or an umbilic point for the system of nonlinear conservation laws describing the convective transport of the fluid phases. Unlike classical shock waves (e.g., BuckleyLeverett fronts), this non-classical ”transitional” shock wave is very sensitive to the form of the parabolic terms in the equations that arise from diffusive effects. It is then imperative that capillary pressure effects be modeled accurately in order to calculate the physically correct transitional waves [?, ?]. We describe a numerical procedure, based on a two-level operator splitting technique, for three-phase flows that takes into account capillary pressure (dispersive forces) [?, ?, ?, ?]. This numerical procedure has been used to indicate the existence of nonclassical transitional waves in multidimensional heterogeneous flows [?, ?], thereby ex-