Two-phase flow in a rough fracture: experiment and modeling.

To develop and test theory-based procedures for modeling two-phase flow through fractures, it is important to be able to compare computational results for a fracture with experiments performed on the exact same fracture. Unfortunately for real fractures, any attempt to image the fracture and to produce a numerical model of the fracture accessible to computer modeling unavoidably results in a coarsening of the resolution, with the very small-scale features of the imaged fracture averaged to produce the numerical representation used in modeling. Contrary to the hope that these high-resolution features would be unimportant, several modeling efforts have shown that such changes in resolution do affect the flow. Therefore, the numerical representation is different from the real fracture because of this unavoidable coarsening of the resolution. To remove the problems caused by the use of different fractures in the experiment and in the model, the fracture used in our experiments was stereographically constructed from the same numerical representation used in the modeling so that the only difference between the experimental "fracture" and the modeling "fracture" is a manufacturing error of approximately 3% or less in the aperture sizes of the manufactured experimental model. Using several models not unlike others in the literature, we modeled injection of air into the water-saturated fracture. The modeling results are compared to experimental results for injection of air into the water-saturated stereolithographically constructed fracture. A comparison between modeling and experimental results for the essentially identical fractures shows a much better detailed agreement than obtained in other studies, which compared experimental flows on the real fracture with modeling results for a lower resolution representation of the real fracture. This suggests that many of the differences between experiment and modeling in previous work resulted from the differences between the experimental and modeling fractures. For our low capillary-number cases, the best agreement with experiment is for a modification of invasion percolation with trapping (IPwt) that included approximations to viscosity ratio effects and to the interfacial tension effects in reducing very short-range curvature.

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.

Crossover from fractal capillary fingering to compact flow: The effect of stable viscosity ratios.

Using a standard pore-level model, which includes both viscous and capillary forces, we have studied the injection of a viscous, nonwetting fluid into a two-dimensional porous medium saturated with a less viscous, wetting fluid, i.e., drainage with favorable viscosity ratios, M> or =1 . We have observed a crossover from fractal capillary fingering to standard compact flow at a characteristic time, which decreases with increased capillary number and/or viscosity ratio. We have tested an earlier prediction for the dependence of this crossover upon viscosity ratio and capillary number using our data for a wide-but-physical range of capillary numbers and viscosity ratios. We find good agreement between the predicted behavior and our results from pore-level modeling. Furthermore, we show that this agreement is not affected by changes in the random distribution of pore throat radii or by changes in the coordination number, suggesting that the prediction is universal, i.e., valid for any porous medium structure, as expected from the general nature of the derivation of the prediction. Furthermore, this agreement indicates that the prediction correctly accounts for dependence of the flow upon capillary number and viscosity ratios, thereby enabling predictions for interfacial advance and width as well as saturation and fractional flow profiles. Also this agreement supports the validity of the general theoretical development lending credence to the three-dimensional predictions.

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.

Crossover from capillary fingering to viscous fingering for immiscible unstable flow:Experiment and modeling.

Invasion percolation with trapping (IPT) and diffusion-limited aggregation (DLA) are simple fractal models, which are known to describe two-phase flow in porous media at well defined, but unphysical limits of the fluid properties and flow conditions. A decade ago, Fernandez, Rangel, and Rivero predicted a crossover from IPT (capillary fingering) to DLA (viscous fingering) for the injection of a zero-viscosity fluid as the injection velocity was increased from zero. [Phys. Rev. Lett. 67, 2958 (1991)]]. We have performed experiments in which air is injected into a glass micromodel to displace water. These experiments clearly demonstrate this crossover as the injection velocity of the air is increased. Furthermore, simulations, using our standard pore-level model, also support the predicted IPT-to-DLA crossover, as well as the predicted power-law behavior of the characteristic crossover length.

Pore-level modeling of drainage: crossover from invasion percolation fingering to compact flow.

A pore-level model of drainage, which has been quantitatively validated, is used to study the effect of increased injection rate (i.e., increased capillary number) upon the flow, with matched-viscosity fluids. For small enough capillary number, the flows from the model correctly reproduce the flows from the invasion percolation with trapping (IPWT) model. As the capillary number is increased, the early-time flows mimic those of the IPWT-model, but then deviate towards compact flow at a characteristic time that decreases as the capillary number increases. That is, the larger the capillary number, the sooner the flow crosses over from IPWT flows towards compact (linear) flows.