We report computer simulations of semi-permeable membranes using the method of molecular dynamics, to study solutions undergoing osmosis and reverse osmosis. These studies have been carried out at steady state, by periodically recycling solvent molecules that permeate the semi-permeable membrane. The method has been used to investigate the effect of a range of molecular forces, state conditions, and membrane morphology on the mass transfer rate of solvents across the membrane. These results could be used to test the accuracy of continuum, micro-continuum and statistical mechanical theories for transport across membranes, apart from being of intrinsic fundamental interest.