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Polymer translocation through a nanopore in a membrane is investigated theoretically. Recent experiments on voltage-driven DNA and RNA translocations through a nanopore indicate that the size and geometry of the pore are important factors in polymer dynamics. A theoretical approach is presented which explicitly takes into account the effect of the nanopore length and diameter for polymer motion across the membrane. It is shown that the length of the pore is crucial for polymer translocation dynamics. The present model predicts that for realistic conditions (long nanopores and large external fields) there are two regimes of translocation depending on polymer size: for polymer chains larger than the pore length, the velocity of translocation is nearly constant, while for polymer chains smaller than the pore length the velocity increases with decreasing polymer size. These results agree with experimental data.