Pressure and Rate-Transient Performance Behaviors of a Horizontal Well Intercepting Multiple Hydraulic Fractures within a Shale Reservoir

The primary goal of this work is to develop semi-analytical models to predict the pressure and rate performance behaviors of unconventional reservoirs — specifically, shale reservoirs. In these types of reservoirs, there are multi-scale heterogeneities that can hinder the modeling and the diagnostic analyses. Additionally, the use of large stimulation treatments can further complicate the modeling of these heterogeneous reservoir systems. Our approach is to extend the existing models for a multi-fractured horizontal well (MFHW) in a homogenous reservoir to consider the fractal reservoir scenario. This work presents the detailed derivation of the model of a horizontal well intercepting a single finite-conductivity fracture within a fractal reservoir. The solution of this model is semi-analytical. This is developed by discretizing the hydraulic fracture, which defines a system of equations, the solution of which provides the pressure at any position inside the fracture. The shape of the imposed hydraulic fracture can be either circular or rectangular. By modifying the solution in the Laplace domain of the diffusivity equation for the reservoir, we have investigated different reservoir conditions, such as single porosity fractal reservoirs with typical or anomalous diffusions and double porosity reservoirs with typical diffusion. We have extended the semi-analytical solution for a horizontal well intercepting a single finiteconductivity fracture to the MFHW case. For this purpose, we have used the principle of superposition in space. We show that the pressure transient response of a MFHW within a fractal reservoir can exhibit a maximum of four (4) distinct periods of flow — (1) fracture (dominated) flow, (2) early fracture-reservoir interaction, (3) late fracture-reservoir interaction, and (4) reservoir-dominated flow. To provide an alternative explanation to the anomalous diffusion phenomenon in petroleum reservoirs, we have also developed a double porosity model considering matrix blocks with fractal geometry and a fractal fracture network. We assumed transient interporosity transfer conditions and we modeled it using the classical convolution scheme given in the literature. Under particular conditions, the resulting model acquires a similar mathematical shape to the so-called anomalous diffusion equation