Understanding and modeling the liver biomechanics represent a significant challenge due to its complex nature. While many studies have been performed to fit hyperelastic constitutive laws on rheological experiments, they tend to agree about the importance of strain rate in the liver mechanical behavior. Furthermore, as the liver is heavily perfused with blood, its constitutive behavior is greatly porous. Supported by these observations, we developed a porous visco-hyperelastic model as a liver parenchyma material. More precisely, visco-hyperelasticity is obtained through Prony series while the mechanical effect of liver perfusion is represented with a linear Darcy's law. Since this mechanical model is developed in the context of real time surgery simulation, a compromise between biomechanical accuracy and computational efficiency must be found. We propose the Multiplicative Jacobian Energy Decomposition method (MJED) to obtain a fast assembly of stiffness matrices on linear tetrahedral elements. Finally, the relative effects of the hyperelastic, viscous and porous components on the proposed liver model are discussed and compared to some rheological experiments.