Simple stochastic model of spontaneous imbibition in Hele-Shaw cells

The imbibition of water in oil in Hele-Shaw cells is described as an asymme tric random walk of water in a globally disordered medium. The disorder is modeled by an external, nonwhite, dichotomic noise, and the resulting rando m walk is described in terms of an effective Markovian master equation. The solution to this equation is given, and the mean-squared-displacement (MSD ) of fluid particles in an effective ordered medium is calculated. This sam e quantity was also measured in squared cells with a fixed separation, wher e Soltrol 170 was displaced by twice-distilled water. The flow generated by spontaneous displacement of oil presented three sequential stages, each wi th a characteristic speed and advancing front structure. In the first and t hird stages viscous fingering was not detected, but in the intermediate dev elopment stage it was observed. From the recorded time evolution of the fro nt, its average displacement speed and the MSD of water in oil as a functio n of time were determined. Imbibition of water shows an enhanced diffusion regime in the first stage where MSD Varies with t(2), without any external noise effect. This is followed by a simple diffusion behavior in an effecti ve medium in the second stage, where the external noise effect is large. In the third stage Gaussian diffusion prevails again. Numerical regression te chniques were used to deters mine the best values of model parameters that fit experimental values for different stages, so that the sum of squared er rors between theoretical predictions and experimental measurements is minim ized. Good agreement was found with average errors of 14.3%, 5.9% and 0.6%, respectively.