Water pulse migration through semi-infinite vertical unsaturated porous column with special relative-permeability functions: Exact solutions

- We derive exact solutions for water migration in unsaturated vertical porous column.
- They correspond to linear, quadratic and cubic relative-permeability functions.
- They are useful to predict the post-infiltration flow of a finite water quantity.
- Exact expressions of the masses of water leaving a given depth are also derived.
- The proposed solutions are also useful for checking numerical schemes.

SummaryThe paper presents certain exact solutions describing the vertical movement of a water pulse through a semi-infinite unsaturated porous column. The saturation-based form of the Richards' equation is used with special power law relative-permeability functions. Both capillary and gravity effects are taken into account. Three exact solutions are derived corresponding to three relative-permeability functions, linear, quadratic and cubic. The Richards' equation is nonlinear for the three cases. The solutions are obtained by applying a general similarity transformation. They are explicit in space and time variables and do not contain any approximation. They describe the evolution of the water saturation in the vertical column and they can be used to predict the post-infiltration movement of a finite quantity of water. Exact expressions of the masses of water leaving a given depth are also derived for the three cases. We analyze the effect of relative-permeability and capillary pressure. The proposed solutions are also useful for checking numerical schemes. One of the exact solutions is used to validate numerical solution obtained from an arbitrary initial condition. Results show that the numerical solution converges to the exact solution for large times.