一维均质与非均质土柱中溶质迁移的分数微分对流-弥散模拟

Simulation of one-dimensional solute transport in homogeneous and heterogeneous soils with scale-dependent fractional advection-dispersion equation

分数微分对流-弥散方程(FADE)是模拟溶质迁移问题的新理论,但应用FADE来模拟溶质迁移时能否克服弥散的尺度效应尚待验证。利用长土柱实验资料结合FADE的解析解拟合推求FADE的弥散系数,并分析其与尺度之间的相关关系。研究结果表明,FADE的弥散系数具有随尺度增大而增大的现象,且均质土柱中FADE的弥散系数尺度效应小于非均质土柱中弥散系数尺度效应。在均质土柱中,弥散系数与尺度之间成指数相关关系,在非均质土柱中,弥散系数与尺度之间成幂相关关系。考虑了弥散系数分别与迁移时间和迁移距离呈线性递增两种相关关系,进而分别构建了3种考虑弥散尺度效应的FADE模型,并提出了求解的差分方法。利用上述3种考虑弥散尺度效应的FADE来模拟和预测不同空间位置处的溶质迁移过程。结果表明,对均质土柱中的溶质迁移可得到较好的模拟结果;对于非均质土柱,其模拟结果与实测结果仍然存在一定的差异。

The fractional advection-dispersion equation （FADE） is a new theory for simulating solute transport, but it needs to be validated whether the FADE can be directly used to simulate the scale-dependent transport without considering the scale effect of the disr}ersion. The dispersion coefficient is calculated by fitting the analytical solution of FADE to the laboratory data for long homogeneous and heterogeneous columns, and the relationship between the dispersion coefficient of FADE and the transport scale is then analyzed. It is found that the fractional dispersion coefficient of FADE increases with the transport scale,and the scale effect of the dispersion coefficient in the heterogeneous soil is much more significant comparing to that in the ho mogeneous soil. The relationship between the dispersion coefficient and the distance can be described using an exponential function for the homogeneous soil and a power law function for the heterogeneous soil, respectively. Except for the nonlinear scale-dependent dispersion coefficients, the linear time-dependent and distance-dependent dispersion coefficients are used,and then three types of the modified FADE with their explicit finite difference approximations are established to simulate the scale-dependent transport in both the columns. Parameters in the later two dispersion coefficient functions are fitted with the measured transport data at the location of 100 cm for both the columns. Thus we use the finite difference schemes with the ob tained scale-dependent dispersion coefficients to simulate and predict the transport in other locations. The results indicate that the simulated concentrations with the proposed three scale-dependant dispersion coefficients are in good agreement with the measured concentrations for the homogeneous soil, while the agreement for the heterogeneous soil is less satisfactory.