The simulation of unstable invasion patterns in porous media flow is very challenging because small perturbations are amplified, so that slight differences in geometry or initial conditions result in significantly different invasion structures at later times. We present a detailed comparison of pore-scale simulations and experiments for unstable primary drainage in porous micromodels. The porous media consist of HeleShaw cells containing cylindrical obstacles. By means of soft lithography, we have constructed two experimental flow cells, with different degrees of heterogeneity in the grain size distribution. As the defending (wetting) fluid is the most viscous, the interface is destabilized by viscous forces, which promote the formation of preferential flow paths in the form of a branched finger structure. We model the experiments by solving the NavierStokes equations for mass and momentum conservation in the discretized pore space and employ the Volume of Fluid (VOF) method to track the evolution of the interface. We test different numerical models (a 2-D vertical integrated model and a full-3-D model) and different initial conditions, studying their impact on the simulated spatial distributions of the fluid phases. To assess the ability of the numerical model to reproduce unstable displacement, we compare several statistical and deterministic indicators. We demonstrate the impact of three main sources of error: (i) the uncertainty on the pore space geometry, (ii) the fact that the initial phase configuration cannot be known with an arbitrarily small accuracy, and (iii) three-dimensional effects. Although the unstable nature of the flow regime leads to different invasion structures due to small discrepancies between the experimental setup and the numerical model, a pore-by-pore comparison shows an overall satisfactory match between simulations and experiments. Moreover, all statistical indicators used to characterize the invasion structures are in excellent agreement. This validates the modeling approach, which can be used to complement experimental observations with information about quantities that are difficult or impossible to measure, such as the pressure and velocity fields in the two fluid phases.

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A systematic analysis is presented of the undulations appearing on the surface of long bubbles in confined gas-liquid flows. CFD simulations of the flow are performed with a self-improved version of the open-source solver ESI OpenFOAM (release 2.3.1), for Ca = 0.002−0.1 and Re = 0.1−1000, where Ca = µU/σ and Re = 2ρU R/µ, with µ and ρ being, respectively, the viscosity and density of the liquid, σ the surface tension, U the bubble velocity and R the tube radius. A model, based on an extension of the classical axisymmetric Bretherton theory, accounting for inertia and for the curvature of the tube's wall, is adopted to better understand the CFD results. The thickness of the liquid film, and the wavelength and decay rate of the undulations extracted from the CFD simulations, agree well with those obtained with the theoretical model. Inertial effects appear when the Weber number of the flow We = Ca Re = O(10 −1 ) and are manifest by a larger number of undulation crests that become evident on the surface of the rear meniscus of the bubble. This study demonstrates that the necessary bubble length for a flat liquid film region to exist between the rear and front menisci rapidly increases above 10R when Ca > 0.01 and the value of the Reynolds number approaches 1000.

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