## 1. Introduction

[2] The displacement of one fluid by another in a porous medium can produce very different patterns over many different length scales [*Lenormand*, 1986]. A classic example is a liquid (typically water) displacing air in a porous medium under gravity [*Hill and Parlange*, 1972]. On the macroscale this infiltration is observed to be either stable with a laterally uniform front or unstable with preferential flow paths (fingers) [*Glass and Nicholl*, 1996; *Yao and Hendrickx*, 1996]. Since the liquid is denser and more viscous than the gas, the *Saffman and Taylor* [1958] instability condition predicts unstable flow for fluxes less than the saturated conductivity of the medium. Experimentally, the stability of the displacement is observed to change with small changes in any of the following: the initial condition (the volume of initial liquid) [*Bauters et al.*, 2000; *Diment and Watson*, 1985], the boundary condition (applied flux) [*Yao and Hendrickx*, 1996], the fluid properties (viscosity and surface tension) [*Aminzadeh and DiCarlo*, 2010], and the porous media properties (permeability and wettability) [*Bauters et al.*, 1998].

[3] Sorting out the correct physics of the instability is difficult because of the multidimensionality of the flow and the nonlinearities in the flow equations. Insight comes from early experimental observations of preferential flow in slab experiments [*Selker et al.*, 1992]. These experiments found that within a preferential flow path the liquid saturation *S* (the locally averaged fraction of pore space occupied by liquid) showed a characteristic profile depicted in Figure 1. This profile has been named saturation overshoot as the finger tip (directly behind the wetting front) has a higher saturation than the finger tail. During a stable infiltration, the overshoot profile is not observed (Figure 1) [*Bauters et al.*, 2000;*Yao and Hendrickx*, 1996].

[4] Additionally, identical overshoot is observed when the flow is confined laterally to be less than the characteristic finger width [*Geiger and Durnford*, 2000; *DiCarlo*, 2004]; thus, overshoot is independent of the dimensionality of the flow. Finally, the Saffman-Taylor instability condition predicts that the flow will be unstable if saturation overshoot occurs and stable if no saturation overshoot occurs [*Raats*, 1973]. Thus, theoretically and experimentally, saturation overshoot is a necessary and sufficient condition for a gravity-driven instability, or, in other words, a description of the simpler phenomena of overshoot is sufficient to describe the more complicated instability phenomena [*Eliassi and Glass*, 2002].

[5] Both gravity-driven fingering and saturation overshoot are forbidden in the classic continuum two-phase flow equations [*Egorov et al.*, 2003; *van Duijn et al.*, 2004]. For liquid and air phases, these equations assume that the liquid saturation *S* is a continuous variable for length scales greater than a multiple of the grain size [*Bear*, 1972]. The saturation develops in space and time through the conservation equation ϕ∂*S*/∂*t* = −∇ · *q*, where ϕ is the porosity and the liquid flux *q* is given by the Darcy-Buckingham equation *q* = −(*kk*_{r}/*μ*)∇(*P* − *ρgz*), where *z* is positive downward, *k* is the intrinsic permeability of the medium, *μ* is the fluid viscosity, and *ρ* is the fluid density [*Bear*, 1972]. The relative permeabilities, *k*_{r}, are monotonic (though hysteretic) functions of saturation. Finally, the air pressure is assumed to be constant as the air is inviscid compared to the liquid and the air is allowed to have free egress in the experiments; thus, *P* = −*P*_{c}. To remedy the failure of the flow equations to describe the observed behavior, continuum extensions with higher-order derivatives have been proposed, with varying success at reproducing the observed dependencies [*Cueto-Felgueroso and Juanes*, 2008; *Eliassi and Glass*, 2002; *DiCarlo*, 2005; *Nieber et al.*, 2005].

[6] Here we hypothesize that saturation overshoot occurs when the pore-scale filling pattern is compact, and a nonovershoot profile occurs when the pore-scale pattern is dendritic; these pore-scale patterns are mediated by the competition between capillary and viscous forces [*Lenormand*, 1986]. We develop a semicontinuum model that uses both continuum and pore-scale physics and find that the proposed model reproduces the observed overshoot dependence.