Treatment of Neumann boundaries in the complex variable boundary element method

Abstract

SUMMARYFor potential ow, the complex variable boundary element method (CVBEM) is formulated in terms of the velocity potential and the stream function . In actual ow problems, and @ =@n are given along Dirichlet and Neumann boundaries, respectively. In the CVBEM, the Neumann-type condition @ =@n is not directly handled, and, instead, is used to deÿne Neumann boundaries. Owing to this discrepancy, numerical di culties are raised along Neumann boundaries. The current study addresses two such di culties: (1) multiple Neumann boundaries and (2) branch cuts across Neumann boundaries. The ÿrst problem is due to the fact that along multiple boundaries cannot be speciÿed a priori, and the second problem is due to the discontinuous jump inherent in for sink=source singularities. To overcome these di culties, a new formulation of the CVBEM to solve for the unknown values and a proper way of branch-cut placement are proposed, and these techniques are veriÿed against example problems.

Advancement of sweep zones in waterflooding: conceptual insight based on flow visualizations of oil-withdrawal contours and waterflood time-of-flight contours using complex potentials

“…Internal anisotropy has been solved as a purely analytical expansion (Chirlin 1985) and can be incorporated in such semi-analytical streamline solvers (Sato and Abbaszadeh 1996), commonly referred to as the complex variable boundary element method (CVBEM;Johnson et al 2014;Sato 2015) but also as perturbation boundary element method (Sato 1992;Sato and Horne 1993a, b). The CVBEM is an alternative to real variable boundary element method (RVBEM) when stream functions and velocity potentials are of interest (Hromadka and Guymon 1984;Hromadka and Lai 1987;Sato and Watanabe 2004).…”