A second order alternating direction implicit scheme for time-dependent Riesz space distributed-order advection-dispersion equations is applied to higher dimensions with the Tensor-Train decomposition technique. The solutions are solved in compressed format, the Tensor-Train format, and the errors accumulated due to compressions are analyzed to ensure convergence. Problems with low-rank data are tested, the results illustrated a steeper growth in the ranks of the numerical solutions than that in related works.