Simulating Flow in Heterogeneous Systems Using Streamtubes and Streamlines
- M.R. Thiele (Stanford U) | R.P. Batycky (Stanford U) | M.J. Blunt (Stanford U) | F.M. Orr Jr. (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- February 1996
- Document Type
- Journal Paper
- 5 - 12
- 1996. Society of Petroleum Engineers
- 5.6.5 Tracers, 5.3.2 Multiphase Flow, 4.3.4 Scale, 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling, 5.5.3 Scaling Methods, 5.4.2 Gas Injection Methods, 5.4.9 Miscible Methods, 5.1.1 Exploration, Development, Structural Geology, 5.4.1 Waterflooding
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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.
Streamtubes have been used extensively in petroleum and groundwater modelingto characterize flow patterns in two-dimensional domains (Bear 1972). Someearly work was done by Higgins and Leighton (1961), Higgins (1964), and Martinand Wegner (1979). More recent ideas on streamtubes have been proposed byRenard (1990), Hewett and Behrens (1991) and King et al. (1993). In general,though, streamtubes have not been used as successfully in petroleumapplications as they have been in groundwater applications. Groundwaterproblems are generally single-phase, and their velocity field does not changewith time. Most petroleum engineering problems, on the other hand, such aswater and gas flooding, are multi-phase displacements with a significant changein the velocity field with time.
The main objective of this work is to use the streamtube approach to findrapid and accurate solutions to the more difficult nonlinear problems inmultiphase flow. We do this by allowing the streamtube geometries to changewith time and by mapping one-dimensional Riemann solutions (BuckleyLeverett,for example) along each streamtube.
Previous research using streamtubes focused almost exclusively on theimmiscible two-phase problem with an areal geometry. It is the weaknonlinearity of the two-phase problem that allows for the assumption ofconstant streamtube geometries, almost universally applied in the publishedliterature. A notable exception is the work of Renard (1990) . Here thestreamtubes are recalculated periodically, and the fluid is assigned to the newstreamtubes using a much finer mesh than that upon which the tubes arecalculated. In general though, the assumption of a fixed streamtube geometry iswidely used and reinforced by the areal geometry since, by continuity, astreamline must start and end at a source.
|File Size||6 MB||Number of Pages||8|