Analytical Study of Instability Phenomenon With and Without Inclination in Homogeneous and Heterogeneous Porous Media Using Fractional Approach

This paper derives the mathematical model of the instability phenomenon in the heterogeneous porous media. The equation of continuity and some standard relation give rise to a nonlinear partial differential equation. To study the anomalous behavior of the saturation rate versus time at different inclination angles and porosity, we considered the time-fractional partial order derivative in the Caputo sense. The fractional reduced differential transform method (FRDTM) is used to obtain the approximate analytical solution of the time-fractional-order model. Finally, the accuracy of FRDTM is studied numerically by comparing with the numerical solutions for integer order α = 1. The result shows that FRDTM is accurate, reliable, efficient, and equivalent to physical phenomena.