The goal of this paper is to provide analytical assessments to a fluid flowing in a porous medium with a non-linear diffusion linked to a degenerate diffusivity. The viscosity term is formulated with an Eyring-Powell law, together with a non-homogeneous diffusion typical of porous medium equations (as known in the theory of partial differential equations). Further, the equation is supplemented with an absorptive reaction term of Darcy-Forchheimer, commonly used to model flows in porous medium. The work starts by analyzing regularity, existence and uniqueness of solutions. Afterwards, the problem is transformed to study travelling wave kind of solutions. An asymptotic expansion is considered with a convergence criteria based on the geometric perturbation theory. Supported by this theory, there exists an exponential decaying rate in the travelling wave profile. Such exponential behaviour is validated with a numerical assessment. This is not a trivial result given the degenerate diffusivity induced by the non-linear diffusion of porous medium type and suggests the existence of regularity that can serve as a baseline to construct numerical or energetic approaches.