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Summary

This paper outlines a Boundary Element Method (BEM) for a piece-wise analytic solution of the Laplace (Poisson) equation for pseudosteady-state, single-phase flow on unstructured, rectangular grids.  The method models flow through a reservoir that has been segmented into interacting homogeneous rectangular regions; no further discretization of the solution space analogous to grid refinement in numerical schemes is required for improved accuracy.  Rather, boundary discretization allows for continuation of pressure and flux.  Previous work on pressure distribution modeling is extended to analytically capture the stream function. Stream-function solutions can then form the basis for other performance measures, such as improved oil recovery efficiency estimation or tracer flow analysis. Moving beyond structured grids into unstructured grid geometry allows for advanced flexibility in problem development and improved efficiency in solution construction.  The analytic approach avoids the need for numerical differentiation of the pressure field and particle tracking methods to recover streamlines.  Capturing flow in highly heterogeneous media, without local grid refinement, is demonstrated to showcase the robustness of the technique in handling complex reservoir architecture, of particular interest in optimal well positioning and optimal well-pattern development.

Introduction

The solution to fluid flow problems is typically a map of the driving force (i.e., potential or pressure).  A more intuitive result is a map that shows actual trajectories of fluid elements, the stream function (Muskat 1937).  Potential, F, and stream-function, Ψ, are related in 2D by their spatial gradients.

(Eq. 1)

Curves of constant stream-function value are the so-called streamlines.  Stream-function indexing is associated with integration, as the difference in stream-function indices represents the amount of fluid flowing between streamlines of those fixed values.  By definition, no fluid convection occurs across a streamline.  Unfortunately, the concept of stream function is restricted to two dimensions.  While an orthogonal mesh to the pressure field can certainly be constructed in 3D, an equivalent of stream function is found in three dimensions only for flows exhibiting symmetry properties that effectively reduce the dimensionality of the flow problem.