Miscible viscous fingering in three dimensions: fractal-to-compact crossover and interfacial roughness

Using our standard pore-level model, we have extended our earlier study of the crossover from fractal viscous fingering to compact /linear flow at a characteristic crossover time, tau , in three dimensions to systems with as many as a 10(6) pore bodies. These larger systems enable us to investigate the flows in the fully compact/well-past-crossover regime. The center of mass of the injected fluid exhibits basically the same behavior as found earlier but with an improved characteristic time. However, our earlier study of much smaller systems was unable to study the interfacial width in the important well-past-crossover regime, ttau. Now, we can study both the time evolution and roughness of the interfacial width. The interfacial width exhibits the same fractal-to-compact crossover as the center of mass, with the same characteristic time. In the fully compact regime, ttau, the interfacial width grows approximately linearly with time so that the standard growth exponent is approximately unity, beta=1.0+/-0.1. We find that neither is the interface self-affine nor is the roughness of the interface in the compact regime consistent with an effective long-range surface tension as assumed by various theories. In fact, similar to Lévy flights, the height variations across the interface appear to be random with occasional large height variations.