Convective mixing in porous media is triggered by a Rayleigh-Bénard-type hydrodynamic instability as a result of an unstable density stratification of fluids. While convective mixing has been studied extensively, the fundamental behavior of the dissolution flux and its dependence on the system parameters are not yet well understood. Here, we show that the dissolution flux and the rate of fluid mixing are determined by the mean scalar dissipation rate. We use this theoretical result to provide computational evidence that the classical model of convective mixing in porous media exhibits, in the regime of high Rayleigh number, a dissolution flux that is constant and independent of the Rayleigh number. Our findings support the universal character of convective mixing and point to the need for alternative explanations for nonlinear scalings of the dissolution flux with the Rayleigh number, recently observed experimentally.

Existing continuum models of multiphase flow in porous media are unable to explain why preferential flow (fingering) occurs during infiltration into homogeneous, dry soil. Following a phase-field methodology, we propose a continuum model that accounts for an apparent surface tension at the wetting front and does not introduce new independent parameters. The model reproduces the observed features of fingered flows, in particular, the higher saturation of water at the tip of the fingers, which is believed to be essential for the formation of fingers. From a linear stability analysis, we predict that finger velocity and finger width both increase with infiltration rate, and the predictions are in quantitative agreement with experiments.

Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid [1]. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. We present a fluid dynamics video of the mixing process in a viscously unstable flow, generated from a high-resolution numerical simulation using a computational strategy that is stable for arbitrary viscosity ratios. We develop a two-equation dynamic model of concentration variance and mean dissipation rate to quantify the degree of mixing in such a displacement process [2]. The model reproduces accurately the evolution of these two quantities as observed in highresolution numerical simulations and captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.

Multiphase flows in porous media are important in many natural and industrial processes. Pore-scale models for multiphase flows have seen rapid development in recent years and are becoming increasingly useful as predictive tools in both academic and industrial applications. However, quantitative comparisons between different pore-scale models, and between these models and experimental data, are lacking. Here, we perform an objective comparison of a variety of state-of-the-art pore-scale models, including lattice Boltzmann, stochastic rotation dynamics, volume-of-fluid, level-set, phase-field, and pore-network models. As the basis for this comparison, we use a dataset from recent microfluidic experiments with precisely controlled pore geometry and wettability conditions, which offers an unprecedented benchmarking opportunity. We compare the results of the 14 participating teams both qualitatively and quantitatively using several standard metrics, such as fractal dimension, finger width, and displacement efficiency. We find that no single method excels across all conditions and that thin films and corner flow present substantial modeling and computational challenges.

1] We present a phase field model of infiltration that explains the formation of gravity fingers during water infiltration in soil. The model is an extension of the traditional Richards equation, and it introduces a new term, a fourth-order derivative in space, but not a new parameter. We propose a scaling that links the magnitude of the new term to the relative strength of gravity-to-capillary forces already present in Richards' equation. We exploit the thermodynamic framework to design a flow potential that constrains the water saturation to be between 0 and 1, its physically admissible values. The model predicts a saturation overshoot at the wetting front, which is in good agreement with experimental measurements. Two-dimensional numerical simulations predict gravity fingers with the appearance and characteristics observed in visual laboratory experiments. A linear stability analysis of the model shows that there is a direct relation between saturation overshoot and the strength of the front instability. Therefore our theory supports the conjecture that saturation overshoot, a pileup of water at the wetting front, is a prerequisite for gravity fingering.Citation: Cueto-Felgueroso, L., and R. Juanes (2009), A phase field model of unsaturated flow, Water Resour. Res.,