## Summary

Fluid evidence shows that prediction of water breakthrough and oil recovery from fractured reservoirs cannot be performed accurately without upscaled relative permeability functions. Relative permeability is commonly assumed to be a scalar quantity, although the justification of that—specifically for naturally fractured reservoirs (NFRs)—is rarely attempted. In this study, we investigate the validity of this scalar-quantity assumption and how it affects fracture/matrix equivalent relative permeabilities, $kri(Sw)$, achieved by a numerical simulation of unsteady‐state waterflooding of discrete‐fracture/matrix models (DFMs).

Numerical determination of relative permeability requires a realistic model, a spatially adaptive simulation approach, and a sophisticated analysis procedure. To fulfil these requirements, we apply the discrete‐fracture/matrix modeling to well‐characterized outcrop analogs at the hectometer to kilometer scale. These models are parameterized with aperture and capillary entry pressure data, taking into account variations from fracture segment to segment, trying to emulate in‐situ conditions. The finite‐element‐centered finite‐volume method is used to simulate two‐phase flow in the fractured rock, while also considering a range of wettability conditions from water‐wet to oil‐wet.

Our results indicate that the fracture/matrix equivalent relative permeability is a weakly anisotropic property. The tensors are not necessarily symmetric, and the absolute‐permeability tensor is the most influential factor, determining the level of anisotropy of $kri$. The anisotropy ratio (AR) changes with saturation, is influenced by the fracture/matrix‐interface wetted area (Awf), and differs for each phase. In addition, the diagonal terms of the equivalent relative permeability tensor ($krii$), determined using our novel approach, can be different from those obtained using the assumption that $kri$ is scalar. The magnitude of the difference is controlled by the absolute permeability, wettability, flow rate, and orientation of the fractures in the model. It is worth mentioning that the type and direction of imbibition can be determined by off‐diagonal terms of the $kri$ tensor. Furthermore, $krii$ largely depends on the direction of the waterflood along the i‐axis.