An analytical equation to predict gas/water relative permeability curves in fractures

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 twophase 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.