Standard theory for liquid infiltration into porous media cannot explain saturation overshoot at an infiltration front. Based on a recent generalization of the Green-Ampt approach, a new theory for variably-saturated flow is presented that assumes that capillary pressure does not only depend on liquid content but also on the flow velocity. The Eulerian expression for the nonequilibrium capillary pressure is rotationally invariant and attempts to capture the conjecture that dynamic effects are more pronounced if flow is associated with moving fluid-fluid interfaces which cause a dynamic contact angle. The new theory correctly predicts how overshoot depends on the downstream and upstream liquid contents as well as on grain size. The theory also yields the hydrostatic pressure distribution in the liquid and allows for liquid contents exceeding the saturated liquid content of the main imbibition curve that may occur for overshoot.
 Flow of two immiscible fluids in porous media occurs when rainwater or liquid contaminants infiltrate into soil, when water is injected into oil reservoirs during secondary oil recovery, or when supercritical CO2is injected into geological formations for the purposes of carbon sequestration. The Darcy-scale capillary pressurepc critically determines the behavior of an infiltration front. Clearly pc affects the rate of infiltration. Maybe more importantly, the dynamics of pc affect the stability of an infiltration front [Nieber et al., 2005], which in turn affects the sweep efficiency in oil recovery and the formation of preferential flow paths that cause early breakthrough of an infiltrating liquid. In practice, two-phase flow in porous media is modeled by theories that assume capillary pressure to be in equilibrium. For liquid infiltration,Richards'  theory is typically employed.
 No Darcy-scale theory for variably-saturated flow has been presented that accounts for a capillary pressure that depends on the flow velocity, even though laboratory experiments have shown that the capillary pressure at an infiltration front depends on the flow velocity at the Darcy scale [Weitz et al., 1987; Geiger and Durnford, 2000; DiCarlo, 2004; Annaka and Hanayama, 2005]. Consistent with these experiments, Hsu and Hilpert recently generalized the Green-Ampt (GA) model [Green and Ampt, 1911] for water infiltration, which assumes a sharp wetting front and capillary pressure at the front to be constant. The modified GA approach describes the late stage of capillary rise experiments better than the traditional one [Hsu and Hilpert, 2011]. Current work in our laboratory has shown that this is also the case for downward infiltration, the topic of this paper. In this paper, we propose a velocity-dependent capillary pressure that can be incorporated into a Eulerian theory for variably-saturated flow, for which a wetting front may be diffuse.
 To test this new theory, we model simple 1D column water infiltration experiments which have shown that under certain circumstances “saturation overshoot” at the infiltration front (see Figure 1, left) occurs. Overshoot affects the stability of 3D infiltration fronts; however, it cannot be described by the traditional modeling approach based on Richards' equation [Eliassi and Glass, 2001, 2002]. Saturation overshoot is more pronounced if the initial water content θ0 ahead of the front is small, while infiltration into prewetted media produces lesser or no overshoot [DiCarlo, 2007]. DiCarlo  speculated that infiltration into significantly prewetted sands (θ0> 0.1) occurs through film flow which does not produce overshoot; on the contrary, infiltration into a completely dry porous medium involves piston flow, i.e., moving fluid-fluid interfaces.
 The 1D saturation overshoot experiments have been used as a test bed for emerging theories for multiphase flow. DiCarlo  has shown that Richards' theory augmented by Stauffer's rate law for the nonequilibrium capillary pressure [Stauffer, 1978; Gray and Hassanizadeh, 1991] is capable of describing overshoot for initial water contents θ0 = 0 and θ0 > 0. However, the parameterization for dynamic pc that yielded reasonable modeling results for θ0 = 0 did not work well for θ0 > 0. Sander et al.  also used Stauffer's rate model and additionally accounted for capillary hysteresis. This resulted in quite realistic overshoot simulations for an initial effective saturation of 0.4 (θ0 > 0). Also a phase field theory is capable of describing saturation overshoot for infiltration into dry porous media for which θ0 = 0 [Cueto-Felgueroso and Juanes, 2009]. Like these previous studies we use saturation overshoot to test our theory. We will do so for both θ0 = 0 and θ0 > 0. We will also address the fact that the liquid content may exceed the one at satiation, θs, from the equilibrium main imbibition curve and actually get quite close to the porosity ε (see Figure 1, right). These high θ-values cannot be modeled by Richards' theory and represent a significant challenge for emerging theories. Moreover we make sure that our theory predicts the hydrostatic pressure distribution.
 Our theory is based on a mass conservation equation,
and the Buckingham-Darcy law for unsaturated flow,
where q is the Darcy velocity, ψ is pressure head, K(θ) is the hydraulic conductivity, and the z-axis is oriented downward. In Richards' theory, the pressure headψ is assumed to be in equilibrium and to depend on θ, ψ = ψeq(θ) (neglecting capillary hysteresis).
where D is an average grain diameter, γ is interfacial tension, is a function of θ0 and porosity ε, Ca = μ∣q∣/(θγ) is the capillary number, μ is the dynamic viscosity of the liquid, and α and β are nondimensional constants that quantify capillary nonequilibrium. Values of α, β, and for a given porous medium system can be inferred from variable-rate column infiltration experiments. It has been speculated that the observed Darcy-scale velocity dependence stems from a microscopic contact angle and capillary pressure that, at the pore scale, depend on the velocities of the fluid-fluid interfaces [Hsu and Hilpert, 2011].
Equation (3)cannot readily be used in theories for variably-saturated flow such as Richards' equation. To do so, the equation must be generalized in several respects: (1) A Eulerian expression forψ (or pc) is needed, because Richards' equation does not assume a sharp wetting front. Instead, θ and q can vary in both space and time. (2) Contrary to the GA approach for infiltration, Richards equation allows for q < 0 and, in 3D, for flow in any direction. Equation (3) does not account for the fact that is a vector.
 We propose the following 3D expression: ψ = ψeq + ψneq where
is the nonequilibrium capillary pressure, and β are nondimensional functions of ε and surface roughness, g is the gravitational acceleration, ϕ is the angle between and θ, and and θ are Eulerian fields. This model has the following features:
 1. The model is invariant under rotation and reflection. In 1D, ψneq is invariant under the coordinate transformation z′ = −z and allows for q < 0. The cos ϕ term makes sure that, no matter how the coordinate system is oriented, ψneq > 0 at an infiltration front where · θ < 0. This is consistent with column infiltration experiments and infiltration into capillary tubes where the contact angle in the liquid increases with the flow velocity. Even though capillary tubes do not account for the complex topology of granular porous media, we can expect ψneq > 0 behind the front where · θ > 0, because during liquid withdrawal from capillary tubes the contact angle in the liquid decreases with the flow velocity [Hilpert, 2010]. Figure 2 illustrates how the sign of ψneq can, at the pore scale, be explained by a dynamic contact angle.
 2. To ensure that ψneq is continuous and differentiable if θ = 0 (e.g., when θ = θm in case of overshoot), we write cos ϕ = ∣cos ϕ∣sgn(cos ϕ) and approximate the discontinuous sign function by sgn(x) = x/ where ϵ is a small number.
 3. To account for the conjecture by DiCarlo  that dynamic effects are less pronounced if pore filling at an infiltration front is due to film swelling rather than piston flow, we multiply Ca = ∣μ/(θγ)∣ by a factor 0 < ξ ≤ 1. This modification is consistent with the conjecture that ψneqis related to the velocity of fluid-fluid interfaces [Hsu and Hilpert, 2011] which is generally not equal to the average liquid velocity q/θ. Since the occurrences of film and piston flow are related to the initial water content, we make the probably simplifying assumption that ξ depends on θ. Our model for ξ mimics flow behavior in two limiting saturation regimes. At very high saturations, residual gas in the form of bubbles trapped due to capillary forces is present, i.e., ξ ≈ 0. For small saturations, piston flow occurs and ψneqtherefore depends on the flow velocity. The following one-parametric model captures these bounding limits:
where ξm is the maximum ξ-value, andθd is a decay constant which is a function of ε and surface roughness and which equals the water content at which ξ ≈ ξm/2.72. One can expect ξm ≤ 1. For a porous medium consisting of a parallel bundle of tubes and the z-axis aligned with the tube axes,ξm = 1. We set θd = 0.1, the value of θ0, above which DiCarlo did not observe overshoot. Figure 3 illustrates the ξ-function and its physical interpretation.
Equations (1), (2), and (4) constitute three equations for the three unknowns θ, ψ, and q. If dynamic effects are absent, = 0, one can combine these equations to obtain Richards' equation, = , a single partial differential equation for the unknown θ. For the case of a wave traveling in the positive z-direction,θ, q and ψ become functions of the sole independent variable η = −(z − ct) where c is the wave speed. Then we obtain a system of three ordinary differential equations (ODEs):
where from now on a prime indicates a derivative with respect to η. Note that we used ∣cos ϕ∣ = 1 for any 1D flow (except if q or θ′ are zero). This equation is true even in case of overshoot when ϕ = 180° ahead of the front and ϕ = 0° behind the front (see Figure 2). Also note that in case of overshoot, sgn(qθ′) changes its value from 1 to −1 if θ = θm as θ′ = 0. We use characteristic system variables to define ϵ in the sign function, ϵ = cθsε/he where he is the entry pressure head. We solve the ODEs subject to the boundary conditions θ(η → −∞) = θ0, θ(η → ∞) = θ1, θ′(η → −∞) = 0, and ψ′(η → −∞) = 0.
 We now address the issue that Richards' theory cannot deal with θ > θs because then both ψeq and K cannot be evaluated. Given that this problem has been largely avoided, we make the tentative assumption that the actual liquid content can be related to an equilibrium liquid content through θeq = θ − θneq, where θneq = Δθ is the nonequilibrium liquid content, and Δθis a maximum nonequilibrium liquid content. This model is inspired by the Michaelis-Menten model for enzyme kinetics and ensures thatθneq is bounded. In equilibrium when q = 0, θ = θeq. The choice Δθ = (ε − θs) makes sure that 0 ≤ θeq ≤ θs for 0 ≤ θ ≤ ε. One may now determine θeq from q and the actual θ:
Indeed θeq → θθs/ε ≤ θs as q → ∞. The θeq is used to calculate ψeq and K in equation (7).
 To obtain a single ODE for θ, we first integrate the mass balance from the bottom boundary condition and obtain c(θ − θ0) − q + q0 = 0. Thus, for given initial conditions, q is a function of θ:
Therefore for a traveling wave, the q dependence in ψneq given by equation (8) turns into a dependence on θ. By equating equations (7) and (10) and making use of q0 = K0 = K(θ0) (because ψ′ → 0 for η→ −∞) we now obtain a single second-order ODE forθ(η):
where ψeq′ = θ′ and ψ′neq = ψneq. By evaluating equation (11) at the upstream boundary condition, we can see that the wave speed is given by c = (K1 − K0)/(θ1 − θ0) where K1 = K(θ1). To solve equation (11)numerically, we recasted it as a system of two first-order ODEs,
where y1 = θ and y2 = θ′.
 We performed simulations to mimic traveling wave experiments performed by DiCarlo  for a 20/30 Accusand. For this sand, ψeq(θ) was measured independently for primary imbibition and parameterized by the van Genuchten model, while K(θ) was estimated with the Mualem model (parameters listed in Table 1 of DiCarlo ).
 To arrive at a parameterization for the nonequilibrium capillary pressure given by equation (4), we built on a parameterization of dynamic capillary pressure at an infiltration front that was recently developed [Hsu and Hilpert, 2011] based on an analysis of water-infiltration experiments into dry porous media [Geiger and Durnford, 2000]: = Peq − fneq where Peq is the nondimensional equilibrium capillary pressure, and fneq = 315 and β = 0.31 are nondimensional parameters that quantify capillary nonequilibrium. It is reasonable to assume that the parameterizations of nondimensional dynamic capillary pressure of the DiCarlo  and Geiger and Durnford  (GD) experiments are similar, because they both used relatively pristine sand size fractions which can be expected to possess similar surface properties. Therefore we used β = 0.31. We crudely estimated the parameter combination ξmβ in equation (4) from the GD experiments, even though GD measured pc only at the infiltration front but not for the entire possible range of θ. We estimated the water content at the front, θf, to be the arithmetic mean of the water contents ahead and behind of the front. Since GD did not measure θ at all, we assumed a 20% residual gas saturation behind the front and therefore θ = 0.8 εGD behind the front where εGD ≈ 0.38 is the porosity of the GD soils. Since the initial water content was zero, θf ≈ 0.4 εGD. With these assumptions, we can now equate equation (4) and the parameterization for the GD experiments at the water content θf and obtain ξmβ ≈ fneq 281. We set the “half-saturation” constantKhequal to the sole Darcy-scale quantity that has units of a velocity, that is, the saturated hydraulic conductivityKs.
 We only had to lower ξmβ, the parameter combination that we crudely estimated, to 110 in order to obtain numerical convergence for θ0 → 0. Figure 4 (top) shows the resulting simulated water content profiles for θ0 = 0.001. They capture the observation by DiCarlo that overshoot is strongest for intermediate values of the upstream water content θ1 which controls the capillary number Ca = μK1/(θ1γ). Consistent with the experiments, the length of the overshoot region increases with the upstream water content θ1.
 To test the predictive capability of the parametrization of dynamic capillary pressure, we used it to predict saturation overshoot for initial water contents θ0 > 0. Consistent with the DiCarlo experiments, overshoot disappears as θ0 increases (see Figure 4, middle). This is because then both θ0 and the maximum liquid content θm approach the same limit ε.
 We also modeled water infiltration experiments into dry media that showed that overshoot disappears if the grain size D < 0.13 mm, no matter which upstream water content θ1 (as controlled by the infiltration rate q1 = K1) was applied [DiCarlo et al., 2011]. We performed simulations in which we successively divided D from the simulations shown in Figure 4 (top) by a factor of 2 and used different values of θ1. We assumed saturated hydraulic conductivity to scale like D2 and entry pressure head like 1/D. Figure 4 (bottom) shows that overshoot, as indicated by θm/θ1 > 1, becomes less pronounced as D decreases and is experimentally not observable for D < 0.1 mm. This value agrees quite well with the measurements by DiCarlo et al. whose measurement accuracy seemed to be on the order of Δθ ≈ 10−3.
 The new model is quite robust in that simulations converge as long as does not exceed extreme values. Only if is too large does the simulated front become too steep, and simulations diverge. The model is also insensitive to the choice of Kh for low values of θ, while Kh significantly affects results when θ exceeds θs. This behavior is expected because Kh affects the difference between the actual and equilibrium liquid content, which is only significant if θ is on the order of θs or larger.
 We presented a theory that can predict an important feature of simple 1D liquid infiltration experiments, i.e., saturation overshoot at an infiltration front, which cannot be described by standard theory (Richards' equation). The new theory is based on a generalization of a relation for dynamic capillary pressure that was incorporated into the Lagrangian GA model for infiltration. The Eulerian expression for capillary pressure given by equation (4) describes variably saturated flow and accounts, through the ξ-function, for the fact that dynamic effects are more pronounced if flow is associated with moving fluid-fluid interfaces. Moreover,equation (4) is rotationally invariant such that flow can occur in any direction. We used independently performed column infiltration experiments in order to obtain rough estimates for the parameters in the relation for the nonequilibrium capillary pressure. Only the parameter combination ξmβ had to be slightly lowered in order to mimic realistic infiltration behavior. It is encouraging that the theory correctly predicts the occurrence of overshoot as a function of the initial water content θ0, the upstream water content θ1, and grain size. Furthermore, simulated lengths of the overshoot zones roughly match experimental data, even though we did not even attempt to fit such data.
 Richards' theory was generalized such that it still correctly predicts the hydrostatic pressure distribution in the liquid, as can be seen from equation (2). Hence the pressure potential ψ can be measured with a tensiometer.
 We also addressed the problem that the liquid content θ may exceed the saturated liquid content θs. While our proposed equation (9) solves the problem of evaluating ψeq and K for θ > θs, more research is needed to establish the correct relation for the equilibrium liquid content θeq.
 Saturation overshoot indeed offers a testing ground for emerging theories for multiphase flow [DiCarlo, 2005]. We initially tested general 1D expressions for the nonequilibrium capillary pressure of the form ψneq = ψneq(θ, q). However, this ansatz neither allows for overshoot (proof is not shown), nor does it allow for ψneq > 0 and ψneq < 0 ahead of and behind a wetting front, respectively.
 In the future, the new model for dynamic capillary pressure should further be tested for its ability to model true 3D flows such as gravity-driven fingering. Also capillary hysteresis should be considered as it affects overshoot [Sander et al., 2008].
 This work was supported by NSF grant EAR-0739038. The author thanks the anonymous reviewer for his/her useful comments.
 The Editor thanks the anonymous reviewer for his/her assistance in evaluating this paper.