Interface roughening in Hele--Shaw flows with quenched disorder: experimental and theoretical results

We study the forced fluid invasion of an air-filled model porous medium at constant flow rate, in 1+1 dimensions, both experimentally and theoretically. We focus on the non-local character of the interface dynamics, due to liquid conservation, and its effect on the scaling properties of the interface upon roughening. Specifically, we study the limit of large flow rates and weak capillary forces. Our theory predicts a roughening behaviour characterized at short times by a growth exponent $\beta_1 = 5/6$, a roughness exponent $\alpha_1=5/2$, and a dynamic exponent $z_1=3$, and by $\beta_2=1/2$, $\alpha_2=1/2$, and $z_2=1$ at long times, before saturation. This theoretical prediction is in good agreement with the experiments at long times.The ensemble of experiments, theory, and simulations provides evidence for a new universality class of interface roughening in 1+1 dimensions.