This work considers cocurrent, three-dimensional, single-phase miscible and two-phase immiscible, hyperbolic flow in a general grid, structured or unstructured. A given grid block or control volume may have any number of neighbors. Heterogeneity, anisotropy, and viscous and gravity forces are included, while tensor considerations are neglected. The flow equations are discretized in space and time, with explicit composition and mobility used in the interblock flow terms (the Impes case).

Published stability analyses for this flow in a less general framework indicate that the CFL number must be < 1 or < 2 for stability. A recent paper reported non-oscillatory stability of one- and two-dimensional Buckley-Leverett two-phase simulations for CFL < 2. A subsequent paper claimed to predict this CFL < 2 limit from a stability analysis. This work gives a different reason for that stability up to CFL < 2.

This work shows that the eigenvalues of the stability matrix are equal to its diagonal entries, for any ordering scheme. The eigenvalues are in turn equal to 1-CFLi, which leads to a conclusion of an early paper that CFL < 1 is required for non-oscillatory stability. CFL values between 1 and 2 give oscillatory stability. In general, our Impes simulations require the non-oscillatory stability ensured by CFL < 1.

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