We present a fast technique for modeling convective displacements which aredominated by large scale reservoir heterogeneities. The direction of flow atany time during the displacement is mapped by streamtubes, which arerecalculated as the fluid mobility distribution changes. A one dimensionalsolution is then mapped along each streamtube as a Riemann solution, i.e. as anintegration from 0 to tD + tD rather than from tD to tD + tD, as inconventional time-stepping algorithms.
The Riemann approach allows for the rapid computation of flow using two tothree orders of magnitude fewer matrix inversions than traditional finitedifference simulators. The resulting two dimensional solutions are free fromnumerical diffusion and can include the effects of gravity, any type ofmultiphase, multicomponent compositional process and longitudinal physicaldiffusion, but cannot account for transverse physical diffusion or mixing dueto viscous or capillary cross-flow.
We test our techniques on immiscible and ideal miscible displacementsthrough a variety of two dimensional heterogeneous systems. We show that theRiemann technique is accurate and converges in less than 1% of the time takenby conventional finite difference simulators. Using multiple realizations ofpermeability fields with identical statistics we show that the nonlinearity ofthe displacement process and reservoir heterogeneity combine to define thepossible spread in recovery curves. For the ideal miscible case, we show thatthe stream-tube method is an example of how to nest physical phenomena thatdominate at different scales in order to capture the physical process ofinterest.