Parametric Representation of Boundary Flux
in Multi-Region Fluid Flow Problem
Yiteng Zhang, UiS/The National IOR Centre of Norway
* The work presented here is adapted from the author’s previous thesis submitted for Master of
Science in Petroleum Engineering at The University of Tulsa, supervised by Dr. Randy D. Hazlett.
The copyright is reserved by the author.
Motivation
Boundary flux distribution has a significant impact on computed well pressure with increasing significance for decentralized wells and
oriented flows. Hazlett and Babu introduced a parametric representation of boundary flux that adapts to well position and trajectory to
solve for coupling between analytic solutions for pressure. For equal cell size problems, only two unknowns per interface need to be
introduced with arbitrary choice of two surface locations for pressure matching. The current methodology is limited to cell shapes for which
there exists a closed-form, computable solution. This currently comprises rectangles, circles, circular sectors, and special case triangles in 2D
and their prismatic counterparts. Such problems are for illustrative purposes and lend themselves to generalization.
Objective
The work starts with showing the boundary flux structure is exact for cells of identical size and permeability. The work further extends the
solution of equal-sized regions to those of unequal size. Since lengths are scaled with respect to transport properties, heterogeneous transport
property problems can be recast as homogeneous problems of unequal cell size using prolongation.
Development
Applications
The governing equation in the Cartesian system in this work,
(Uniform Pressure)
Anisotropy
(Zero Flux)
Prolongation
One Dimensional Stretching
(Prolongation)
(Extra Uniform Flux)
(Circulation)
Discussions
Selected References
For isotropic systems of uniform cell size, the unknown flux is the
combination of zero flux and uniform pressure.
For heterogeneous systems, prolongation needs an extra uniform flux
and a circulation element to represent the unknown boundary flux.
Prolongation is the possible treatment for heterogeneous media.
R. D. Hazlett, D. K. Babu, Influence of cell boundary flux distribution on well
pressure, Procedia Computer Science 18 (2013) 2013–2146.
R. D. Hazlett, D. K. Babu, Readily computable greens and Neumann
functions for symmetry-preserving triangles, Quarterly of Applied Mathematics
67 (3) (2009) 579–592.
M. Prager, Eigenvalues and eigenfunctions of the Laplace operator on an
equilateral triangle, Applications of Mathematics 43 (4) (1998) 311–320.
Acknowledgement
The author acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, BP Norge AS, Det
Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE
AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS of The National IOR Centre of Norway for support.