A major part of the world's remaining oil reserves is located in fractured carbonate reservoirs, which are dual-porosity (fracture-matrix) or multi-porosity (fracture-vug-matrix) in nature. Fractured reservoirs suffer from poor recovery, high water cut, and generally low performance. They are modelled using a dual-porosity approach, which assumes that the high-permeability fractures are mobile and low-permeability matrix is immobile. A single transfer function models the rate at which hydrocarbons migrate from the matrix into the fractures. As shown in many numerical, laboratory, and field experiments, a wide range of transfer rates occurs between the immobile matrix and mobile fractures. These arise, for example, from the different size of matrix blocks (yielding a distribution of shape factors), different porosity types, or the inhomogeneous distribution of saturations in the matrix blocks. Accurate models are hence needed that capture all the transfer rates between immobile matrix and mobile fracture domains, particularly to predict late-time recovery more reliably when the water cut is already high. In this work we propose a novel multi-rate mass transfer model for two-phase flow, which accounts for viscous dominated flow in the fracture domain and capillary flow in the matrix domain. It extends the classical (i.e., single-rate) dual-porosity model in that it allows us to simulate the wide range of transfer rates occurring in naturally fractured multi-porosity rocks. Using numerical simulations of water-flooding in naturally fractured rock masses at the grid-block scale we demonstrate that our multi-rate mass-transfer model matches the observed recovery curves more accurately compared to the classical dual-porosity model. We further discuss how tracer tests can be used to calibrate our multi-rate dual-porosity model before the water-flood commences and how our model could be employed in commercial reservoir simulation workflows.