基质酸化过程中的一类数模求解

The Solution of a Sort of Mathematical Model during Matrix Acidizing

在基质酸化过程中,经常遇到一类带移动边界条件问题的二维半线性抛物型方程,该类方程的求解对现场施工十分关键,由于带自由移动的边界条件,给求解过程增添了不少难度。文中通过引进时变网格法,转换自由移动边界条件,用有限差分格式将模型离散,求得模型的数值解,并作出了解的变化关系曲线,通过曲线变化关系图可以大致确定三次开采过程,即酸化过程中注聚合物溶液要求对时间和地点的控制(什么时候、什么地点是最佳时刻或地方注入聚合物),给现场施工提供了理论指导。同时,可以将该边界变换方法推广到一般的带自由移动边界条件的数学模型。

During the matrix acidizing, we often encounter a sort of two dimension semi-linear parabola equations with moving and free boundary, and the solution of this kind of equations is very important to the spot construction. However, the existing of moving and free boundary condition enhanced the difficulty of solution greatly. In this paper, through introducing time-variant grid method and converting moving and (ree boumiarv condition, we dispersed the, model with finite difference. then worked out the numerical solution of the model and plot variation relation curve. From the plot, we can approximately identify the tertiary recovery process, namely during the matrix acidizing how to control time and site of the injection of polymer (when is the best time and where is the best site to inject polymer) , so provide the theory guidance for the spot construction. At the same time, we can spread this method to common mathematical model with moving and free boundary.