A Matched Asymptotic Expansion Analysis of Highly Unsteady Porous Media Flows

The method of matched asymptotic expansions is used to analyse a mixture of wave and diffusive behaviours governing flow in a saturated porous medium inside an elastic pipe that is suddenly subjected to a large hydraulic gradient at its entrance. At early times and near the entrance, the head is a diffusing wave that can be reduced to the linear and non-linear telegrapher equations for the laminar and partially developed turbulent flows, respectively. At later times, laminar flows are diffusive and partially developed turbulent flows follow a ‘fast diffusion’ behaviour. In the case of fully developed turbulence, flows at later times follow a fast diffusion form which is complicated by advection at extremely high gradients. A matched asymptotic expansion approach is used to match flows at early times and near the entrance, with complementary forms that are away from the entrance and which occur at later times.