Long-wave-induced flows in an unsaturated permeable seabed

We present both analytical and numerical solutions describing seepage flows in an unsaturated permeable seabed induced by transient long waves. The effects of compressibility of pore water in the seabed due to a small degree of unsaturation are considered in the investigation. To make the problem tractable analytically, we first focus our attention on situations where the horizontal scale of the seepage flow is much larger than the vertical scale. With this simplification the pore-water pressure in the soil column is governed by a one-dimensional diffusion equation with a specified pressure at the water-seabed interface and the no-flux condition at the bottom of the seabed. Analytical solutions for pore-water pressure and velocity are obtained for arbitrary transient waves. Special cases are studied for periodic waves, cnoidal waves, solitary waves and bores. Numerical solutions are also obtained by simultaneously solving the Navier-Stokes equations for water wave motions and the exact two-dimensional diffusion equation for seepage flows in the seabed. The analytical solutions are used to check the accuracy of the numerical methods. On the other hand, numerical solutions extend the applicability of the analytical solutions. The liquefaction potential in a permeable bed as well as the energy dissipation under various wave conditions are then discussed.