A novel description of fluid flow in porous and debris materials

• New sub-advection/sub-diffusion model describes fluid flow in porous/debris material.
• Exact, analytical and high resolution numerical solutions are presented.
• Solutions reveal many essential physical phenomena and very special flow behavior.
• This includes evolution of forward advecting frontal bore head followed by thin tail.
• Application in environmental/engineering/industrial fluid flow in general porous media.

Based on a generalized two-phase mass flow model (Pudasaini, 2012) as a mixture of solid particles and interstitial fluid, here, we derive a novel dynamical model equation for sub-diffusive and sub-advective fluid flow in general porous media and debris material in which the solid matrix is stationary. We construct some exact analytical solutions to the new model. The complete exact solutions are derived for the full sub-diffusive fluid flows. Solutions for the classical linear diffusion and the new sub-diffusion with quadratic fluxes are compared, and the similarities and differences are discussed. We show that the solution to sub-diffusive fluid flow in porous and debris material is fundamentally different from the diffusive fluid flow. In the sub-diffusive process, the fluid diffuses slowly in time, and thus, the flow (substance) is less spread. Furthermore, we construct some analytical solutions for the full sub-diffusion and sub-advection equation by transforming it into classical diffusion and advection structure. High resolution numerical solutions are presented for the full sub-diffusion and sub-advection model, which is then compared with the solution of the classical diffusion and advection model. Solutions to the sub-diffusion and sub-advection model reveal very special flow behavior, namely, the evolution of forward advecting frontal bore head followed by a gradually thinning tail that stretches to the original rear position of the fluid. However, for the classical diffusion–advection model, the fluid simply advects and diffuses. Moreover, the full sub-diffusion and sub-advection model solutions are presented both for the linear and quadratic drags, which show that the generalized drag plays an important role in generating special form and propagation speed of the sub-diffusion–advection waves. We also show that the long time solution to sub-diffusive and sub-advective fluid flow through porous media is largely independent of the initial fluid profile. These exact, analytical and numerical solutions reveal many essential physical phenomena, and thus may find applications in modeling and simulation of environmental, engineering and industrial fluid flows through general porous media and debris materials.

Evolution of the sub-diffusion and sub-advection of a fluid through a debris and porous media. The initial condition is indicated by the dashed curve at t = 0.Figure optionsDownload as PowerPoint slide