低渗非达西渗流运动方程研究

The study on non-Darcy seepage equation of low velocity flow

低速非达西渗流现象广泛存在于高放废物地质处置库及能源开发领域,描述这种非线性过程具有重要的意义.本文利用分数阶微积分方法对幂函数型、指数型两种常用的非线性运动方程进行了研究.首先从达西定律出发,通过拉普拉斯变换得到了与Hansbo幂函数型方程相一致的渗流运动方程.进而对Swartzendruber指数型方程进行研究,得到了分数阶导数意义下的Swartzendruber运动方程及其解析表达式.验证了在求导阶次为1的情况下,原Swartzendruber运动方程是分数阶Swartzendruber方程的一种特例.基于非饱和渗流实验数据,采用最小二乘法拟合获得了新运动方程中的相关参数,对参数变化趋势进行了分析.由曲线拟合结果可知,分数阶Swartzendruber方程精度较高,可以更好地描述非线性渗流过程.

Many experiments have confirmed that the seepage flow in low permeability porous media is nonlinear. Mathematical model research for nonlinear seepage flow is of great significance for guiding engineering practice in geological repository and energy exploration. Aiming at modifying the typical power and exponential seepage equation, this paper attempts to study nonlinear seepage motion by using of fractional calculus. Based on the classical Darcy＇s Law, a seepage equation is derived via Laplace transformation which is similar to the Hansbo＇s power law equation. Moreover, by applying fractional derivative to Swartzendruber exponential seepage equation, we obtain a new three parameters mathematical model of continuous function with the same procedure. The parametric sensitivity analysis is conducted to investigate the effects of model parameters on the relationship between water flux and hydraulic gradient. The parametric analysis also shows that the fractional Swartzendruber seepage equation reduces to be the classical Swartzendruber seepage equation when the fractional order equals to one. All the parameters of the fractional Swartzendruber seepage equation are determined by the Levenberg-Marquardt algorithm based on the unsaturated seepage experimental data. The fitting results show that fractional Swartzendruber seepage equation has better effects than the original model, i.e. the proposed model can characterize nonlinear seepage process more precisely.