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

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 dispersion.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 homogeneous 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 obtained 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 sat isfactory.