One-dimensional model equations for unsteady axisymmetric free-boundary potential flow in porous media

Carmine Di Nucci, Mohammad Afzal Shadab , Piera Fischione, Daniele Celli, Davide Pasquali, Marcello Di Risio

This paper deals with the derivation of one-dimensional model equations for unsteady axisymmetric free-boundary potential flow in saturated, homogeneous and isotropic porous media. An approximation of the irrotational and solenoidal velocity field, obtained by means of a Picard iteration, is employed to derive a momentum equation accounting for vertical-flow effects. The model is completed by the one-dimensional continuity equation. The axisymmetric Cauchy integral relation is also derived. The model is applied to flow toward a fully penetrating well in a non-deformable porous medium with a horizontal impermeable bed, neglecting capillary-fringe effects. The resulting equations are numerically integrated to determine the free-boundary evolution, the flow-rate distribution, the water level in the well, and the seepage-face development. The numerical results show satisfactory agreement with Boundary Integral Equation Method results available in the literature and with the VarSatFlow simulations under prescribed well-water-level conditions. The results indicate that the proposed model captures the main features of the transient flow; some differences occur near the well.