Parametric Representation of Boundary Flux in Multi-Region Fluid Flow Problem

Yiteng Zhang, UiS/The National IOR Centre of Norway

Boundary flux distribution has a significant impact on computed well pressure with increasing significance for decentralized wells andoriented flows. Hazlett and Babu introduced a parametric representation of boundary flux that adapts to well position and trajectory tosolve for coupling between analytic solutions for pressure. For equal cell size problems, only two unknowns per interface need to beintroduced with arbitrary choice of two surface locations for pressure matching. The current methodology is limited to cell shapes for whichthere exists a closed-form, computable solution. This currently comprises rectangles, circles, circular sectors, and special case triangles in 2Dand their prismatic counterparts. Such problems are for illustrative purposes and lend themselves to generalization.

The governing equation in the Cartesian system in this work,

Motivation

The work starts with showing the boundary flux structure is exact for cells of identical size and permeability. The work further extends thesolution of equal-sized regions to those of unequal size. Since lengths are scaled with respect to transport properties, heterogeneous transportproperty problems can be recast as homogeneous problems of unequal cell size using prolongation.

Objective

The author acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, BP Norge AS, DetNorske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGEAS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS of The National IOR Centre of Norway for support.

Acknowledgement

Development

Selected ReferencesDiscussions

Applications

(Uniform Pressure)(Uniform Pressure) (Zero Flux)(Zero Flux)

One Dimensional StretchingOne Dimensional Stretching

AnisotropyAnisotropy ProlongationProlongation

(Extra Uniform Flux)(Extra Uniform Flux) (Circulation)(Circulation)(Prolongation)(Prolongation)

For isotropic systems of uniform cell size, the unknown flux is thecombination of zero flux and uniform pressure.

For heterogeneous systems, prolongation needs an extra uniform fluxand 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 Mathematics67 (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.

* 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.