Laplace方程有孔椭圆区域边值问题的解析解

本文利用相似流动替换方法 ,解决了中心有圆孔的椭园形区域上 Laplace方程第一类边值问题 ;采用分区域解法 ,给出了中心有椭园孔的椭园形区域上 Laplace方程第一类边值问题的解析通解 .这一结果在许多工程领域有重要应用 ,本文给出了油藏工程实例

Analytic Solution for Laplace Equation on Elliptical Area with a Centre Hole

The first kind of boundary value proplem for Laplace Equation in an elliptical area with a circular centre hole is similar to the problem of the flow from a point source to two parallel straight line boundaries. It is solved by the method of substitution with the similar flow. The error of the substitution method is computed and analysed. It shows that the error is generally negligible. The same kind of the problem but with a elliptical centre hole is more complex. It can be divided into two problems in an inner and an outer area respectively. The inner area is between two confocal elliptical boundaries. The inner problem is solved by a conformal transformation. The outer area is an elliptical area with a circular centre hole. The outer problem can be solved by the method above in this paper. The whole analytic solution of the problem is obtained by combining the solutions of the inner and outer areas. The flow through anisotropic oil reservoirs is typical in petroleum engineering, but has never been solved before. This problem can be transformed to the problem above. The analytic solutions of pressure distribution and production rate are presented and compared with the isotropic reservoir results. The effects of reservoir anisotropy are respectable. This is very important for computations and designs in oil reservoir engineering.