Finite Element Method For The Fractional Advection-Dispersion Equation And Its Scale-dependent Dispersion

The fractional advection-dispersion equation (FADE) is a new theory for simulating solute transport. However, 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. Meanwhile, the analytical solution of FADE only can be obtained under specific conditions, i.e. steady state flow with step input. Using the numerical methods to solve FADE is the best choice. In this dissertation, the scale-dependent dispersion was investigated, and then a finite element scheme for FADE was developed.First, the dispersion coefficient was calculated by fitting the analytic solution of FADE to the laboratory data with long homogeneous and heterogeneous columns, and the relationship between the dispersion coefficient of FADE and the transport scale was then analyzed. It was found that the fractional dispersion coefficient of FADE increased with the transport scale, and the higher of the heterogeneity was, the more significant of the scale dependent of the dispersion would be. Except for the two nonlinear scale-dependent dispersion coefficients, another two linear dispersion coefficient functions, i.e. the time-dependent and distance-dependent dispersion coefficients were used, and then three types of modified FADE with their explicit finite difference approximations were established to simulate the scale-dependent transport in homogeneous and heterogeneous columns respectively. Parameters in the later two dispersion coefficient functions were calculated by using the finite difference schemes with the measured transport data at the location of 100cm both for homogeneous and heterogeneous columns. Thus we used the finite difference schemes with obtained scale-dependent dispersion coefficients to simulate and predict the transport in other locations. Results indicated that the predicted concentration agreed with the measured data reasonably well both for homogeneous and heterogeneous columns. It then implied that the FADE with the three proposed scale-dependent dispersion coefficients could be used to simulate the transport with scale effect of dispersion in homogeneous and heterogeneous columns respectively.Second, we developed a finite element scheme for numerically solving FADE, and a corresponding FORTRAN based program was established. The finite element method and the program were used to simulate the NaCl transport in long homogeneous and heterogeneous columns with step input and Br" transport in homogeneous soil with pulse input. The effect of a on the long tail of the breakthrough was also analyzed. Results indicated that simulation results were in agreement with the measurements, the difference between the simulation results and the measurements may be attributable to the errors brought by the approximations, i.e. using linear interpolation function and so on in the finite element method.