Governing equations of orthogonal nonlinear flow in porous media

Using vector field theory, the governing equation of orthogonal high-velocity nonlinear flow in porous media is obtained for the first time by expanding and simplifying the velocity-free equation, which is derived by substituting the explicit function from pressure gradient to macroscopic velocity of fluid flowing in porous media solved from Forchheimer equation into the continuity equation of fluid flowing in porous media. The governing equation of orthogonal high-velocity nonlinear flow in porous media and the governing equation of orthogonal low-velocity nonlinear flow in porous media will be collectively referred to as the governing equations of orthogonal nonlinear flow in porous media. While the symbolic solutions are out of reach because the governing equations of orthogonal nonlinear flow in porous media are rather cumbersome, the flow field geometry of orthogonal nonlinear flow in porous media is analyzed by using vector field theory and differential geometry knowledge. Then a correct understanding which has been missed for 21 years is demonstrated that the conformal mapping cannot be used for analyzing orthogonal nonlinear flow fields in porous media other than fields in which all streamlines are always straight. In order to bypass the boundary conditions in solving the governing equation of orthogonal nonlinear flow in porous media for symbolic solutions, applying the complex potential function and the metric tensor function from the “general kinematic formula for steady flow velocity field”, the governing equation of orthogonal nonlinear flow in porous media in the Cartesian coordinate system is transformed into governing equation of orthogonal nonlinear flow in porous media in the corresponding potential-stream coordinate system for a given problem. In view of the fact that the governing equations of orthogonal nonlinear flow in porous media in the potential-stream coordinate system are still cumbersome, that is, the symbolic solutions for variable-directions flow field are difficult to obtain directly, it is suggested to search possible mapping that can indirectly obtain the symbolic field functions of orthogonal nonlinear flow in porous media.