Abstract

This paper presents the mathematical basics of the Boundary Element Method (BEM) as they are implemented in the first reservoir modeling software of its kind. Multiple real-world examples illustrate how it works for oil and gas fields with different boundary conditions (flux or pressure specified), how it handles heterogeneity and how material balance calculations are performed.

Perhaps the greatest advantages of the BEM are its ability to arrive at a solution without gridding the space, as is necessary for finite difference or finite element methods, and the achievement of analytical accuracy when handling different types of singularities (vertical, horizontal and fractured wells) with virtually no limitations on their orientation to each other or the shape of the solution domain.

These features enabled the development of a fully graphical interface that dramatically reduces the time needed to set up a case, and the short run times on most types of scenarios make the interactive process of history matching much faster and intuitively clear. The program operates on all Windows 32 bit platforms.

A large section of the paper is devoted to the comparison of model results with various well known analytical solutions, including steady state and transient solutions for "standard" reservoirs, compatibility with the method of images and with well established results in the area of well test analysis. The results of modeling vertical, fractured and horizontal wells (oil, gas and water) compare very well with standard numerical solutions that are documented within reservoir engineering community. The stability and convergence of this new model is verified by numerous tests that vary boundary element size and time step duration. Our analysis of the BEM implementation provides significant insight into how reservoir analysis and the entire process of oil/gas field development may be dramatically changed in the foreseeable future.

Introduction

The Boundary Element Method (BEM) now does not need to be defended as a powerful tool in reservoir engineering research. Scores of papers utilizing advantages of this method are being published. A long list of monographs and textbooks on the subject are also available.

The advantages of the method are well known1–9. They are:

  • Reduction of the dimensionality of the problem by one.

  • Elimination of the grid orientation effects and numerical dispersion.

  • Easy accommodation to arbitrary reservoir shapes with different types of boundary conditions and elimination of gridding.

  • Analytical accuracy in handling singularities (radial, horizontal and fractured wells).

  • Ability to handle moving and infinite boundaries.

At the same time, the BEM has not yet reached the masses of practicing engineers in the oil and gas industry. The purpose of this project was to make this powerful method available to engineers who do not specialize in numerical methods of modeling. Achieving this goal would require developing a familiar looking and easy to use graphical interface, and making choices for the brand of the BEM to use, the types of approximations adopted in the mathematical model, and the extent to which reservoir characteristics are detailed.

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