The Convergence of Upstream Collocation in the Buckley-Leverett Problem

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Abstract
Upstream collocation is a fast and accurate scheme for simulating multiphase flows in oil reservoirs. In contrast to standard orthogonal collocation, upstream collocation yields numerical solutions to the Buckley-Leverett problem that converge to correct solutions physically. The failure of standard orthogonal collocation is not surprising, since the Buckley-Leverett problem as commonly stated is posed incompletely. The equal-area rule of Buckley and Leverett and the Welge tangent construction both specify additional constraints needed to close the problem properly. An error analysis of upstream collocation shows that this method forces convergence through an artificial dissipative term analogous to the "vanishing viscosity" used in shock fitting. This constraint is mathematically equivalent to the more familiar constructions and should prove beneficial in stimulating EOR schemes based on frontal displacement.

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