Capillary rise in porous media: analytical and numerical solutions

Capillary rise in porous media is frequently described using the Green-Ampt model (the Washburn equation). Recent accurate measurements of the advancing front clearly illustrate the failure of the GA model to describe the phenomenon in the long time. The observed under-prediction in the position of the front is due to the neglect of dynamic saturation gradients in the formulation of the Green-Ampt model. Here we consider the full governing macroscopic equation, retaining these gradients. An approximate solution is derived and gives new analytical formulae for the position of the advancing front, its speed of propagation and the cumulative uptake. A numerical solution is also developed based on the Kirchhoff integral transform. Comparison between the analytical and numerical solutions demonstrated that the former solution is very accurate despite its simple form. These solutions are found to describe properly the observed behaviour of the capillary rise in the long time. It is shown that the Washburn equation can be recovered as a special case of the new analytical solution.