A New Analytical Equation to Predict Gas-Water Two-Phase Relative Permeability Curves in Fractures
Authors: Lei Gang , Dong Pingchuan , Yang Shu , Li Yuansheng , Mo Shaoyuan , Gai Shaohua , Wu Zisen 
Published: May 2016

Abstract
Multiphase flow in naturally fractured reservoirs and hydraulically fractured reservoirs, which hold about a major part of the world's remaining hydrocarbon reservoirs, is strongly influenced by fractures in the geological formations. Fractures are the major flow channels for fluid flowing in fractured reservoirs. So, the accurate prediction of multiphase flow in fractures is highly important. In 1966, the earliest and simplest equation to calculate two-phase relative permeability curves in fractures was proposed by Romm based on experimental results using kerosene and water, which could be called Romm's model. Romm's model is widely used to calculate relative permeability curves in fractures for gas recovery processes. However, the model suggested relative permeability is a linear function of saturation which doesn't reflect nonlinear characteristic and has been proved not to be suitable for gas-water permeability curves in fractures. In 2012, an analytical equation to predict gas-water relative permeability curves in fractures has been derived by Chima using a perfectly smooth fracture model. Chima's model was validated with experimental data and shows superior agreement compared to Romm's model. However it never took the fracture surface roughness and irreducible water saturation in the fracture into consideration. And the model overestimated gas recovery. In this work, a new analytical gas-water relative permeability model of fractures considering irreducible water has been proposed, based on Chima's method, when the surface geometry of fracture is assumed to be two ideal parallel planes. Besides, another new analytical model has been derived to calculate relative permeability curves in fractures for gas-water two-phase flow in which fracture systems are rough and irreducible water saturation is non-uniform. The varying aperture of fracture systems and the heterogeneous distribution of irreducible water saturation have been taken into consideration. Furthermore, these two relative permeability models have been improved by considering the influence of tortuosity. The results show that the relative permeability models are nonlinear functions of mobile water, viscosity, irreducible water, the fracture pore characteristics and fracture distribution index. Our proposed equations are validated with laboratory data measurements and other original models and never overestimate gas recovery compared to other equations. The relative permeability models proposed in this work will be useful to professionals involved in modeling well performance, and gas production forecasting in fractured reservoirs.

Introduction
Naturally fractured reservoirs and hydraulically fractured reservoirs contain a major part of the world's remaining hydrocarbon reservoirs but are very difficult to produce. Fractures are the major flow channels for fluid flowing in these fractured reservoirs. A good understanding of fluid flow in fracture systems is a necessity. The relative permeability is one of the most common and useful parameters for understanding flow mechanisms in fractured reserviors.
In 1966, Romm[1] firstly established relative permeability models for fractures based on the kerosene/water experimental results and he suggested that two-phase flow in fractures can be modeled by straight-line, i.e. Krw=Sw, Krg=Sg and Krw+Krg=1. Romm's model is widely used to calculate relative permeability curves in fractures for gas recovery processes. However, the model assumed no phase interference existence between two-phase. And many scholars [2] [3] [4] [5] [6] [7] proved that the sum of gas-water relative permeability is less than 1 and relative permeability curves in fractures were not a simple linear function of saturation with experimental evidence, which implied that Romm's model was not suitable to describe relative permeability for fractures.