We present a fast technique for modeling convective displacements which are dominated by large scale reservoir heterogeneities. The direction of flow at any time during the displacement is mapped by streamtubes, which are recalculated as the fluid mobility distribution changes. A one dimensional solution is then mapped along each stream tube as a Riemann solution, i.e. as an integration 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 to three orders of magnitude fewer matrix inversions than traditional finite difference simulators. The resulting two dimensional solutions are free from numerical diffusion and can include the effects of gravity, any type of multiphase, multicomponent compositional process and longitudinal physical diffusion, but cannot account for transverse physical diffusion or mixing due to viscous or capillary cross-flow.
We test our techniques on immiscible and ideal miscible displacements through a variety of two dimensional heterogeneous systems. We show that the Riemann technique is accurate and converges in less than 1% of the time taken by conventional finite difference simulators. Using multiple realizations of permeability fields with identical statistics we show that the nonlinearity of the displacement process and reservoir heterogeneity combine to define the possible spread in recovery curves. For the ideal miscible case, we show that the stream-tube method is an example of how to nest physical phenomena that dominate at different scales in order to capture the physical process of interest.
Streamtubes have been used extensively in petroleum and groundwater modeling to characterize flow patterns in two-dimensional domains (Bear 1972). Some early work was done by Higgins and Leighton (1961), Higgins (1964), and Martinand Wegner (1979). More recent ideas on streamtubes have been proposed by Renard (1990), Hewett and Behrens (1991) and King et al. (1993). In general, though, streamtubes have not been used as successfully in petroleum applications as they have been in groundwater applications. Ground water problems are generally single-phase, and their velocity field does not change with time. Most petroleum engineering problems, on the other hand, such as water and gas flooding, are multi-phase displacements with a significant change in the velocity field with time.
The main objective of this work is to use the streamtube approach to find rapid and accurate solutions to the more difficult nonlinear problems in multiphase flow. We do this by allowing the streamtube geometries to change with time and by mapping one-dimensional Riemann solutions (Buckley Leverett, for example) along each streamtube.
Previous research using streamtubes focused almost exclusively on the immiscible two-phase problem with an areal geometry. It is the weak nonlinearity of the two-phase problem that allows for the assumption of constant streamtube geometries, almost universally applied in the published literature. A notable exception is the work of Renard (1990) . Here the streamtubes are recalculated periodically, and the fluid is assigned to the new streamtubes using a much finer mesh than that upon which the tubes are calculated. In general though, the assumption of a fixed streamtube geometry is widely used and reinforced by the areal geometry since, by continuity, a streamline must start and end at a source.