Application of boundary element method to reservoir engineering problems

Abstract

This paper presents the theory of the Boundary Integral Equation Method in general and discusses the application of this method in problems related to reservoir engineering. Integral equation methods have existed for quite some time in the mathematical literature but have been popularized only recently in engineering applications within such divers field as aeronautics, heat transfer, elasticity and groundwater hydrology. Reservoir engineering applications of this method are starting to be realized. The usefulness of the methodology lies in the fact that the solutions obtained are highly accurate and do not suffer from the usual drawbacks of the other domain type numerical schemes. Also complex reservoir geometries with multiple wells can be handled with ease due to the good boundary conformance obtained with the elements. Such desirable features are realized because the analytical nature of the solution is preserved due to the use of free space Green's function of the governing differential operator as the weighting function in the weighted residual approach. A collocation type method is used for the solution of the resulting integral equations. Also, since the method is a boundary procedure, the dimensions of the problem are reduced by one. This reduction in dimensionality is obtained in cases where there are no distributed sources/sinks in the problems domain and the initial conditions are homogeneous. The potential applications in reservoir engineering include rapid generation of streamlines and isochrones for steady-state single phase flow problems. Front tracking in the steady-state case can be done quickly and effectively for multi-well situations with arbitrarily shaped boundaries and a variety of boundary conditions. Application to pressure transient testing of arbitrary shaped, multi-well multi-rate reservoirs is also possible, under two different approaches. Both the above applications are demonstrated with a variety of examples, some of which are difficult to solve by analytical means.