We have investigated experimentally the competition between viscous, capillary, and gravity forces during drainage in a two-dimensional synthetic porous medium. The displacement of a mixture of glycerol and water by air at constant withdrawal rate has been studied. The setup can be tilted to tune gravity, and pressure is recorded at the outlet of the model. Viscous forces tend to destabilize the displacement front into narrow fingers against the stabilizing effect of gravity. Subsequently, a viscous instability is observed for sufficiently large withdrawal speeds or sufficiently low gravity components on the model. We predict the scaling of the front width for stable situations and characterize it experimentally through analyses of the invasion front geometry and pressure recordings. The front width under stable displacement and the threshold for the instability are shown, both experimentally and theoretically, to be controlled by a dimensionless number F which is defined as the ratio of the effective fluid pressure drop (i.e., average hydrostatic pressure drop minus viscous pressure drop) at pore scale to the width of the fluctuations in the threshold capillary pressures.

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 Hele‐Shaw 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 Navier‐Stokes 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.

We present in this paper an experimental study of the invasion activity during unstable drainage in a two-dimensional random porous medium, when the (wetting) displaced fluid has a high viscosity with respect to that of the (nonwetting) displacing fluid, and for a range of almost two decades in capillary numbers corresponding to the transition between capillary and viscous fingering. We show that the invasion process takes place in an active zone within a characteristic screening length $\lambda$ from the tip of the most advanced finger. The invasion probability density is found to only depend on the distance $z$ to the latter tip and to be independent of the value for the capillary number $\mathrm{Ca}$. The mass density along the flow direction is related analytically to the invasion probability density, and the scaling with respect to the capillary number is consistent with a power law. Other quantities characteristic of the displacement process, such as the speed of the most advanced finger tip or the characteristic finger width, are also consistent with power laws of the capillary number. The link between the growth probability and the pressure field is studied analytically and an expression for the pressure in the defending fluid along the cluster is derived. The measured pressure is then compared with the corresponding simulated pressure field using this expression for the boundary condition on the cluster.