Two-phase flow in porous media: Crossover from capillary fingering to compact invasion for drainage

It had been predicted that the capillary fingering observed at small capillary numbers should change or cross over to compact invasion at larger capillary numbers or longer times [D. Wilkinson, Phys. Rev. A 34, 1380 (1986)]. We present results from pore-level modeling in two dimensions for the average position (related to the position of the interface) of the injected fluid as well as the width of the interface between the injected, nonwetting fluid and the defending, wetting fluid. These results are entirely consistent with the predicted crossover from the fractal flow characterized by invasion percolation with trapping (IPWT) to compact/linear/stable flow, where the position of the injected fluid advances linearly with time and where the width of the interface is constant. Furthermore, our results for the characteristic time, at which the crossover occurs, agree with the predictions of Wilkinson. To focus on the effect of capillary number, we are considering only viscosity-matched flows where both fluids have the same viscosities. To our knowledge, these are the first pore-level modeling results that quantitatively test the general predictions of Wilkinson for this capillary crossover in the case of drainage. Our modeling results are used to provide closed form expressions predicting the dependence of average position and interfacial width upon capillary number and time, regardless of the size of the system. The size scaling inherent in the crossover combined with our results locating the upper and lower bounds of the crossover regime enable us to predict the location of the crossover for two-dimensional systems of different size. These predictions are compared with flow patterns from experiments in the literature. The agreement between our predictions and the experimental flow patterns indicates that the experiments exhibit the same IPWT to compact crossover observed in our modeling.