Exact and approximate analytical solutions for unsteady fully developed turbulent flow in porous media and fractures for time dependent boundary conditions

The problem of recharge of water to an initially dry, semi-infinite aquifer is investigated, where the occurrence of turbulent flow is considered. By applying a similarity transform an analytical solution is obtained by the Adomian’s decomposition method. The integration constant of the power series is determined by examining the mathematical structure of the solution at the downstream part of the front. It is demonstrated that for two specific cases the Adomian’s series are reduced to exact solutions. The analytical solutions above, derived on the basis of fully developed turbulent flow regime are compared to the water table profile computed on the basis of the Forchheimer equation. Four specific cases in real world coordinated are investigated: (a) the propagation of a linear traveling wave, (b) injection of fluid at constant flow-rate in a fracture or aquifer, (c) flow induced by a sudden raise of the piezometric head, (d) redistribution of a finite quantity of water in a porous medium. Potential applications of the above cited results are water body–aquifers interaction, and exploitation of hot dry rocks.