Capillary rise in Hele-Shaw models of disordered media

We study the capillary rise of a viscous liquid in large Hele-Shaw models of disordered media, both analytically and experimentally. Compared to the Fries-Dreyer and Lucas-Washburn solutions for capillary rise with and without gravity, our experimental data reveal a systematic deviation at short and intermediate times. The original pressure balance equation leading to Washburn's results is reformulated in order to include an additional resisting term, proportional to the mean velocity of the front h˙, which appears naturally as a result of the geometry of the cell. Analytical solutions h(t) are found for displacements with and without gravity. These new solutions reproduce the experimental results very accurately in Hele-Shaw cells of constant gap thickness, where the capillary pressure can be approximated by a constant. In cells of fluctuating gap thickness, where the capillary pressure fluctuates in space, a small additional pressure contribution is required. This correction that depends on h˙ is also studied.