Gas puffing with a gas pipe situated near the JET Lower Hybrid (LH) antenna increases the Scrape-Off Layer (SOL) electron density, ne,SOL, in the region magnetically connected to the gas pipe, which improves the LH wave coupling. Numerical modeling with the fluid code EDGE-2D suggested that enhanced edge radial plasma transport can play a role in the ne,SOL increase, but the agreement with the measured profiles was reached by using ad hoc modifications of the transport. The modeling also did not take into account direct ionization by the LH wave, which is thought to contribute, either because of the oscillatory motion of the LH wave, or due to the fast electrons created parasitically in front of the grill mouth, or both. In the work presented here, EDGE-2D was modified to include a simple representation of the LH wave and used to explore its effect on ne,SOL. In addition, the width of the grid of EDGE-2D was enlarged to model shots with a large distance dPL 12cm between the separatrix and the LH launcher, which are of interest for LH wave coupling in ITER. The effects of varying the LH heating and the gas puffing rate on ne,SOL are shown for shots with various dPL, and compared to experiments. For example, the slope and magnitude of the measured ne profile for a shot with dPL = 8cm, exhibiting hot spots on the inner divertor apron attributed to LH parasitic fast electrons, are reproduced by the modeling, with a SOL temperature locally elevated by LH power. Also, the ionization sources calculated for various LH heating and puffing rates will be compared to those measured in Tore Supra.