Naturally fractured reservoirs (e.g., carbonates) and hydraulically fractured reservoirs (e.g., shale gas) contain a major part of the world's remaining hydrocarbon reserves yet suffer from low recovery. The accurate perdiction of multi-phase flow in fractures is hence highly important. In 1966, E. S. Romm proposed an equation to calculate oil-water relative permeability curves in fractures based on experimental results using kerosene and water. He suggested that relative permeability is a linear function of saturation (krw = Sw, krnw = Snw). However, it is not clear if linear relative permeability curves can be used for gas-water flow in fractures because there is mounting experimental evidence that in this case relative permeability curves in fractures are non-linear function of saturation. Despite this fact, Romm's equations continue to be widely used in most reservoir simulators to calculate relative permeability curves in fractures for gas recovery processes, which may lead to significant errors when predicting gas recovery from fractured reservoirs, particularly unconventional reservoirs such as shale gas. In this work, we hence use the concept of shell momentum balance, Newton's law of viscosity and the cubic law for flow in fractures to derive a new analytical equation to calculate relative permeability curves in fractured systems for gas-water two-phase flow. This derivation shows that the relative peremeability curves should be non-linear functions of not only saturation but also viscosity. Our proposed equations, which can be implemented straightforwardly in commercial reservoir simulators, are validated with laboratory data measurements and show much better agreement compared to Romm's equation. We further demonstrate the impact of Romm's and our relative permeability models on gas recovery and time to water breakthrough: Romm's equation will overestimate gas recovery and time to water breakthrough by a factor of two, which is significant considering the low recovery typically encountered in naturally and hydraulically fractured reservoirs.

E. S. Romm in 1966 (Romm 1966) proposed an equation to calculate the relative permeability curves in fractured systems based on experimental results. His research concluded that relative permeability is a linear function of saturation (krw = Sw, krnw=Snw). Currently most fractured reservoir simulators use E. S. Romm's equation to estimate relative permeability curves in fractures. Later in 1994, D. A. Pieters and R. M. Graves (Pieters & Graves 1994) demonstrated experimentally that relative permeability curves are not a linear function of saturation. In this work, we use the concepts of shell momentum balance (Bird, Stewart & Lightfoot 2002), the equation of change for isothermal systems, Newton's law of viscosity and Darcy's law in order to derive an analytical equation to calculate relative permeability curves in fractured systems for two phase flow. These proposed equations assume gravitational segregation between the two phases (Rossen & Kumar 1992), Newtonian fluids, incompressible fluids, Laminar flow, steady-state and constant temperature. These proposed equations in this paper could be applied to calculate relative permeability curves in two-phase flow systems in fractures. Further, the two-phase relative permeability curves can be used to predict the behavior of naturally fractured reservoirs. This work also includes a comparison of the calculated data between E. S. Romm's equation, Pieters and Graves's Laboratory data and the equations proposed in this paper. As a result of this research we demonstrated analytically that relative permeability curves in two-phase flow in fractures are not a linear function of saturation. Also we confirmed the non linearity of the fracture's relative permeability curves using a reservoir and fluid data with the proposed equation.