The mass transfer performance of an Intravenous Membrane Oxygenator is investigated by computational simulations of the conservation of mass, momentum and species equations. The Intravenous Membrane Oxygenator (IMO), is a device developed experimentally to provide consistent and reproducible oxygen and carbon dioxide exchange. The IMO is composed by an elastic and non-permeable pulsating balloon positioned within the vena cava, and micro-porous-membrane fibers that transport oxygen and carbon dioxide, located longitudinally between the balloon and vena cava. During the operation regime, the blood flow motion is originated by a blood pressure gradient and a pulsating balloon motion. A three-dimensional physical-computational model consisting of equally-spaced fibers and a Newtonian and time-dependent incompressible flow is used for the simulations. The numerical simulation results for the stationary balloon configuration, obtained using the spectral element method, demonstrate that the flow remains parallel, laminar and with absence of secondary flows in the whole domain. Evaluations of the mass transfer characteristics and parameters, such as the oxygen concentration profile around the fiber and the Sherwood number, for increasing Reynolds numbers, indicate that the parabolic flow regime increase the oxygen transfer rate until an asymptotic limit in the oxygen transfer capability is reached. A further increase in the Reynolds number beyond this asymptotic limit does not increase the oxygen transfer rate.