A Similarity-Based Solution for Nonlinear Gas Fractional Diffusivity Equation with Application to Rate Transient Analysis of Unconventional Heterogeneous Reservoirs

Production characteristics of fractured wells in unconventional heterogeneous reservoirs have been shown to be effectively captured via anomalous diffusion model in which a partial differential equation (PDE) with fractional derivatives is solved. This paper presents a novel semianalytical solution of the nonlinear fractional diffusivity equation (FDE) applied to compressible fluid (gas) flow toward hydraulic fractures placed in heterogeneous and complex geological porous media. Self-similar theory and scaling transformation are used to solve the nonlinear PDE of fractional derivative written for real gas flow using density as the primary variable. The governing nonlinear partial gas FDE is transformed to ordinary nonlinear fractional differential equation after introducing similarity variables, which is later solved via shooting method coupled with Runge-Kutta integration. Pressure-dependent gas properties are captured straightforwardly in the solution without resorting to any further linearization via pseudopressure or pseudotime functions. The proposed similarity-based semianalytical solution is benchmarked against a Laplace transform-based analytical solution for linear, liquid FDE, and validated against a finely gridded numerical solution for the nonlinear, gas FDE. The proposed solution enables the diagnostic interpretation and characterization of production responses of unconventional gas wells exhibiting power-law behavior on the premise of anomalous diffusion during early transient period, which permits the estimation of important reservoir and fracture properties as shown in the case studies. Field and numerical examples are presented to showcase the capabilities of the proposed approach in the inverse, rate transient analysis.