Flow modeling in fractured reservoirs is largely confined to the so-called sugar cube model. Here, we consider a situation where matrix blocks are connected to neighboring blocks so that part of the global flow occurs only in the matrix domain. We call this a partially fractured reservoir (PFR). As opposed to the sugar cube model, global flow in the matrix blocks plays an important role in the PFR when the interconnections between the matrix blocks are sufficiently large. We apply homogenization to derive an upscaled model for partially fractured reservoirs that combines dual-porosity and dualpermeability concepts simultaneously. We formulate a well-posed fully implicit 3D upscaled numerical model and investigate oil recovery mechanisms for different dimensionless characteristic numbers. As we found previously for the sugar cube model, the Peclet number, defined here as the ratio between the capillary diffusion time in the matrix and the residence time of the fluids in the fracture, plays a crucial role. The gravity number and specific fracture-matrix interface area play a secondary role. For low Peclet numbers and high gravity numbers, the results are sensitive to gravity and water injection rates, but relatively insensitive to the specific fracture-matrix interface area, matrix block size, and reservoir geometry, i.e., sugar cube versus PFR. At low Peclet numbers and high gravity numbers, ECLIPSE simulations (Barenblatt or Warren and Root (BWR) approach) give poor predictions and overestimate the oil recovery, but initially show good agreement with the solution of the PFR model at intermediate Peclet numbers. At high Peclet numbers, the results are relatively insensitive to gravity, but sensitive to the other conditions mentioned above. In particular, when the specific fracture-matrix interface area is large, it enhances the imbibition and consequently leads to a higher oil production. If this fraction is low, it leads to a considerable retardation of the imbibition process, which leads to an earlier water breakthrough and lower oil recovery. The ECLIPSE simulations and the sugar cube model result in inaccurate predictions of the oil production rate at high Peclet numbers. This can be inferred from the discrepancy with respect to the PFR model for which we assert that it accurately predicts the oil recovery. We conclude that at low Peclet numbers and large gravity numbers it is advantageous to increase the water injection rate to improve the Net Present Value. However, at high Peclet numbers increasing the flow rate may lead to uneconomical water cuts.