多孔介质溶质运移的分数弥散过程与Lévy分布

The Fractional Dispersion in Pore Medium and Lévy Distribution

在弥散核函数为负幂率函数的前提条件下,对传统的二阶对流—弥散方程进行非局域处理,推导出了分数阶对流—弥散方程,方程中的弥散项是分数阶微分.该方程柯西问题的格林函数解为一L啨vy分布密度函数,由此得到了一个包含3个参数的描述多孔介质中溶质运移行为的解.将所得到的L啨vy分布解用于模拟某一弥散试验中一空间点的溶质浓度的时间变化过程,模拟结果与实测结果吻合良好,很好地解释了实测结果的偏态和拖尾现象.而传统的二阶对流—弥散方程的高斯分布解却没有这些特征,不能解释偏态和拖尾现象.所得结果表明分数阶对流—弥散方程比传统的二阶对流—弥散方程能更好地描述多孔介质中的溶质运移行为.

On the basis of the hypothesis that diffusion kernel behaves as an exponent function, we develop a non_local method with spatial correlation to induce a fractional order advection_dispersion equation from the traditional local 2nd order advection_dispersion equation. In this equation, the dispersion is a fractional order derivative. The solution of this fractional order advection_dispersion equation is Lévy probability distribution density with three parameters. Then the Lévy stable probability distribution density is applied to simulate the temple variation of the solute concentration in the test. The result conforms to the tested data very well. It explains the property of skewness and long_tail in the temple variation of the solute concentration, while the Gauss probability distribution doesn't have these properties. So the result demonstrates that this fractional order advection_dispersion equation is better than the local 2nd order advection_dispersion equation in describing the movement of solute in pore medium.