Time-fractional nonlinear gas transport equation in tight porous media: An application in unconventional gas reservoirs

The prospects of meeting the future's high energy demands lie in the exploration of unconventional hydrocarbon reservoirs, of which the shale gas and the tight gas are two important resources. The deep understanding of such reservoirs is crucial to the economical recovery of such energy resources. With the advancement in the technological sides, such as, hydraulic fracturing and horizontal drilling, new mathematical models are needed that can precisely capture the complexity of the physical phenomena and can describe the flow of gas through the natural and induced fractures. The performance and future behavior of such reservoirs can be enhanced through careful modeling. We develop a new mathematical model based on time fractional derivative combined with the consideration of various flow regimes and a nonlinear treatment of reservoir parameters. The model describes the transport of gas in tight porous media (such as shale formations). The derivation of the model is done by using the mass balance equation and momentum conservation equation (basically modified time-fractional form of Darcy's law) which incorporates the properties of tight porous media and accounts on the previous behavior. We find the pressure equation by considering that the rock properties, such as, permeability, viscosity, porosity, are pressure dependent. The pressure equation can be used to study the pressure distribution in the reservoir.