Finsler geometry for nonlinear path of fluids flow through inhomogeneous media

Abstract

Fluids flow followed by Darcy’s law through inhomogeneous porous media is studied by the theory of Finsler geometry. According to Fermat’s variational principle, the nonlinear paths of fluids flow called Darcy’s flow are described by geodesics in a Finsler space. For inhomogeneous media, the direction dependence of Darcy’s flow gives a Finsler metric called Kropina metric. Then, the influence of direction dependence on the Darcy’s flow is shown by the differences between Riemannian geodesics and Finslerian geodesics. In this case, the deviation curvature tensor implies that the trajectory of Darcy’s flow is Jacobi unstable for the deviation of geodesics. Moreover, similar to Darcy’s flow, the seismic ray path in anisotropic media can be defined in Finsler space, and the metric of seismic ray path is given by the m th root metric. It is shown that the relationship between the variational problems of Darcy’s flow and seismic ray path in Finsler space.