Unraveling the Complex Dynamics of Fluid Flow in Porous Media: Effects of Viscosity, Porosity, and Inertia on the Motion of Fluids

This study investigates novel solitary wave solutions of the Gilson–Pickering ([Formula: see text]) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The [Formula: see text] equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences.