## 1. Introduction

[2] Multiphase flows in porous media have a wide array of engineering, geological, and environmental applications. The contamination of water reservoirs during the downward migration of dense nonaqueous phase liquids (DNAPL) has, for example, been subject to numerous studies [*Ewing and Berkowitz*, 1998, 2001; *Berglund*, 1997]. The transport and storage ability of porous reservoirs hosting two immiscible fluid phases are controlled by their respective spatial distributions. Fingering instabilities, the process by which the distribution of each fluid phases becomes heterogeneous during the injection of the invading fluid, are expected to arise as a consequence of pore-scale heterogeneities (capillary fingering) or of a difference in viscosity between a low viscosity invading fluid and a more viscous defending fluid [*Lenormand et al*., 1988; *Løvoll et al*., 2004, 2011]. The development of fingering instabilities at the invading front affects greatly the storage and transport of each fluid phase and is therefore a big concern for remediation processes or carbon sequestration strategies. In this study, we focus on the buoyant migration of a nonwetting invading phase in a saturated porous medium. Several experimental and numerical studies have shown that, at low injection rate, a buoyant invading fluid (either a dense NAPL migrating downward or a light NAPL migrating upward) forms fingers because of a capillary instability [*Ewing and Berkowitz*, 1998; *Berglund*, 1997; *Yortsos et al*., 1997; *Nsir et al*., 2012]. In this study, we focus on the development of and evolution of capillary instabilities during the migration of the invading fluid, using a macroscale scaling argument based on energy optimization and numerical calculations that solve for the multiphase dynamics at the pore scale. We show cases where the capillary coupling between the two fluids, rather than the pore geometry, controls the redistribution of invading fluid among competing capillary channels.

[3] Numerical studies can be generally divided in two groups: (1) macroscale models derived from the multiphase Darcy equation [*Nieber et al*., 2005; *Cueto-Felgueroso and Juanes*, 2009] and (2) pore-scale approaches such as pore-network and stochastic models [*Ewing and Berkowitz*, 1998, 2001; *Lenormand et al*., 1988; *Yortsos et al*., 1997; *Blunt*, 2001]. Continuum models such as those based on variants of the Richards equation [*Nieber et al*., 2005; *Cueto-Felgueroso and Juanes*, 2009] solve for the migration of the invading fluid at a scale much greater than the pore. The multiphase permeability and capillary effects are therefore introduced as constitutive equations in these macroscopic models. The Richards equation was shown to be intrinsically stable and the generation of capillary fingering required the introduction of higher order terms in the governing equation [*Cueto-Felgueroso and Juanes*, 2009; *Ewing and Berkowitz*, 2001]. These macroscopic models do not solve for the force balance between capillary, buoyancy, and viscous forces at the pore scale, but rather compute an average over a large number of pores from empirically derived relationships between the capillary forces and the invading fluid saturation *S*. Alternatively, pore-scale models are often limited to the study of a small number of pores or require, in the case of pore-network models, some geometrical simplifications to model the flow over volumes where continuous properties computed from averages become meaningful. In the present study, we use a highly parallel multiphase lattice Boltzmann model to benefit from a realistic treatment of the pore-scale force balance while solving the dynamics in a volume large enough to relate our results to macroscale continuous properties such as porosity, fluid saturation, and discharge.