The experimental observations of sequential double ionization (SDI) of Ar [A. N. Pfeiffer {\it et al.}, Nature Phys. {\bf 7}, 428 (2011)], such as the four-peak momentum distribution and the ionization time of the first and second electrons, are investigated and reproduced with a quantum model by including and excluding the $e$-$e$ correlation effect. Based on the comparison of experiment and simulation, the role of $e$-$e$ correlation in SDI is discussed. It is shown that the inclusion of $e$-$e$ correlation is necessary to reproduce the momentum distribution of electrons. ; Comment: 14 pages, 5 figures, submitted