Water Resources Research

Semicontinuum model of saturation overshoot and gravity-driven fingering in porous media



[1] Gravity-driven multiphase flow in porous media is observed to be unstable under certain conditions. This Saffman-Taylor instability is driven by an inversion of the water saturation profile known as saturation overshoot that standard continuum models forbid. We hypothesize that the overshoot and the large-scale instability are created by different pore-filling mechanisms and patterns at the discrete pore scale. We present a new model that bridges continuum-scale and pore level physics which predicts the observed dependence of overshoot on grain size and fluid properties of the invading liquid.

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].

Figure 1.

Snapshots of the ethanol saturation profile versus depth for four different applied fluxes in a confined column. (top right) At high fluxes, the 1-D overshoot profile is the same as seen in 3-D preferential flow paths and the overshoot is the driving force for the instability. (top left) At low fluxes, the 1-D profile does not show overshoot and no instability is observed for 3-D infiltrations.

[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 = −(kkr/μ)∇(Pρgz), 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, kr, 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 = −Pc. 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.

2. Experimental Results

[7] We conducted vertical infiltration in slim tubes (9 mm diameter) packed with spherical sands of varying sizes (Unimin Corporation) [Schroth et al., 1996; DiCarlo, 2004]. A liquid (either water, an alcohol, or an alkane) was injected through a needle at a constant flux from the top. A light transmission system measured the saturation profile within the sand column as a function of space and time [Selker et al., 1992]. The slim tubes were used to keep the infiltration one dimensional as the tube diameter was less than the size of the preferential flow paths; further details on the experimental setup can be found elsewhere [Aminzadeh and DiCarlo, 2010].

[8] Figure 1 shows a snapshot of saturation as a function of vertical distance for four different applied fluxes (q0) of ethanol into a sand of grain size D = 0.5 mm. Once away from the top boundary, the saturation profile is observed to translate downward at a constant velocity; the profiles shown are offset in time to prevent overlap. Saturation overshoot occurs when the tip saturation (Stip) exceeds the tail saturation (Stail).

[9] Figure 2 shows the measured tip and tail saturation as a function of applied flux for ethanol. For each flux, the tail saturation is the saturation that can carry the applied flux under gravitational gradient, kr(Stail) = μq0/ρgk, as the capillary pressure gradient must asymptote to zero. Three different regions of behavior are observed for the tip saturation. At the lowest fluxes in region 1, the tip saturation equals the tail saturation and there is no observed overshoot. Above a certain flux, which we label the overshoot flux qover, the tip saturation rises rapidly with increasing applied flux; this is region 2. Again there is a transition flux, above which the tip saturation is relatively constant with applied flux (region 3). Region 3 ends at the flux corresponding to the saturated conductivity as both the tip and tail are completely saturated.

Figure 2.

Measured tip and tail saturation as a function of flux for ethanol. The overshoot flux marks the onset of overshoot and is found to be qover = 2.05 × 10−6 m/s.

[10] The crucial flux for pattern development is the overshoot flux as below this flux no overshoot is observed and the classic multiphase model works and above this flux overshoot is observed and the classic model fails. To obtain the overshoot and transition fluxes, we fit the tip saturation data in each region with a straight line on a semilog plot and the tail saturation with a power law; the overshoot flux is determined by the intersection of the fitted tip and tail saturations [Aminzadeh and DiCarlo, 2010].

[11] Figure 3 shows the overshoot flux, transition flux, and hydraulic conductivity, K = ρgk/μ, as a function of grain size with water as the imbibing fluid. As expected, the hydraulic conductivity scales with grain size D as D2; however, we find that the overshoot flux scales as D−3. Since overshoot only occurs between the overshoot flux and the hydraulic conductivity, this graph implies that when these lines intersect, there will be no fluxes for which overshoot will occur. This intersection occurs at a grain size of 0.13 mm; experiments with sands of grain size 0.12 mm show no overshoot.

Figure 3.

Measured overshoot flux, transition flux, and hydraulic conductivity as a function of grain size with water as the invading liquid. Hydraulic conductivity scales as D2, while the overshoot flux scales as D−3. The inset shows measured overshoot flux versus σ/μ of the invading liquid.

[12] In the inset of Figure 3 we plot the overshoot flux as a function of surface tension divided by viscosity σ/μ for eight experimental liquids in dry 0.5 mm sand. Except for water (which is discussed in section 3), there is a strong linear correlation between qover and σ/μ, corresponding to a capillary number of NCa = 1.7 × 10−6.

3. Semicontinuum Model

[13] To explain these results, we develop a model of overshoot that solves for the continuum dynamics of the wetting front by explicitly including the discreteness of the porous media. Following Lenormand [1986], we conceptualize the open space of a porous medium to be a set of large voids (pores) connected by smaller voids (throats). The boundaries between these elements (pores or throats) are where fluid-fluid interfaces are capillary stable. Within each element, capillary forces dictate that the primary volume can be either liquid or gas filled (unfilled); that is, their states are binary. Assuming that the liquid wets the solid surfaces, gas-filled elements will have capillary-stabilized conductive layers of liquid where the grains touch (corners) and smaller-sized layers on the surface roughness of the grains.

[14] During infiltration (imbibition), capillary forces act to fill small elements (throats, characteristic size rt) with liquid before large elements (pores, characteristic size rp). These elements are filled through two different processes, piston-like and snap-off [Lenormand, 1986; Blunt and Scher, 1995]. Snap-off filling is when wetting layers coalesce and fill an element from the edges (occurring in throats at a snap-off pressure Pcs = σ/rt). Piston-like filling is when the liquid fills an element by entering from a connected filled element. Importantly for overshoot, piston-like filling is collective: (1) throats are immediately filled if an adjacent pore is filled because of the throats being smaller than the pore, and (2) for a particular pore, the more connecting throats that are filled, the smaller the amount of energy (higher capillary pressure) that is needed to fill that element. For simplicity let the characteristic capillary pressure to fill a pore through a piston-like process be given by Pcp = 2σ/rp. For media with a grain size D, both rp, rt scale with the grain size D. Since the aspect ratio, rp/rt, is ≈3 for sands (and greater than this for consolidated media), PcsPcp and snap-off is usually energetically preferred for filling throats [Lenormand, 1986; Blunt and Scher, 1995].

[15] Returning to the observed overshoot phenomenon, overshoot is simply when the saturation is higher at the front than the saturation needed to carry the applied flux under the gravitational gradient. The pattern of filling at the pore scale was described by Lenormand [1986]. If only collective piston-like filling is allowed, the collective nature will act to fill almost all (or all) of the pores at the main wetting front. Thus, this compact front will create overshoot for applied fluxes less than the saturated conductivity of the media. In contrast, if elements can fill ahead of the main wetting front through snap-off, a more dendritic front will occur and on the macroscale, a smooth increase of the saturation to its asymptotic saturation will occur. This is the main reason why overshoot and preferential flow are not observed in prewet media [DiCarlo, 2004; Bauters et al., 2000]. Thus, we hypothesize that if the front is slow enough, snap-off can happen ahead of the front and overshoot will not be observed and vice versa.

[16] We calculate this overshoot flux using a semicontinuum formulation as follows. We first calculate the capillary pressure gradient as a function of flux using a continuum formulation. Following the arguments above, the capillary pressure at the main wetting front zf is Pc(zf) = Pcp (as this is where the majority of the pores are filled), and snap-off will occur only if the capillary pressure a pore distance ahead of the front is low enough to create snap-off (Pc(zf + D) ≤ Pcs).

[17] Crucial to this development is a calculation of the local flux ahead of the main wetting front through the films, crevices, and layers in the initially dry sand. As fluxes through this initial saturation are lower than can be measured macroscopically, we forgo using the macroscopic Darcy flux equation and instead develop the flux equation heuristically. We assume that the sand grains are spherical of diameter D and have a surface roughness of length scale λ. As the wetting front advances, liquid will be conducted through capillary-stable layers in the pore corners where grains meet (also known as pendular rings) and at capillary-stable layers along the roughness of the grain surface. These roughness-stabilized layers conduct like films but are much larger than molecular films [Tokunaga et al., 2000], and thus we retain the name roughness-stabilized layers. Although the corners are much more conductive than the layers (Dλ and flow scales roughly as L4), the corners do not form a percolating network along the grain surfaces (unless the capillary pressure is very low). Thus, the limiting conduction is through the roughness-stabilized layers.

[18] Assuming that capillary forces will cover each grain with a layer of liquid on the scale of the roughness, we use the equation for laminar film flow (thickness λ) on a flat plane [Bird et al., 2007] to estimate the flux over one grain,

equation image

where the width of the flow over one grain is roughly the diameter of the grain D. Summing over all the grains gives a liquid flux q = Q/τMD2. Here τM is a modified tortuosity of order unity that contains geometrical factors corresponding to the ratio of the path length through the roughness layers per the representative elementary volume length [Lake, 1989], and the fact that the grain surfaces are not vertically oriented. Using P = −Pc, the fact that the capillary pressure gradient is much greater than the gravitational gradient (σ/D2ρg) at the main front and that the infiltrations are observed to be traveling waves (q(z) = ϕvS(z), where the front velocity is v = q0Stail) gives

equation image

or, in other words, the capillary pressure gradient ahead of the front depends on the velocity of the front. Thus, the velocity for which the snap-off will not be reached a pore distance ahead of the front can be found from the integration

equation image

[19] We integrate equation (3) by assuming that S is a constant equal to the corner saturation, Sc, and by assuming the empirical relation λ = λ0(Pc/Pc0)−α with α ≈ 0.37 that Tokunaga et al. [2000] measured using roughened glass plates. Performing the integration results in the right-hand side of equation (3) being equal to λ3Pcln(Pcs/Pcp) for α = 1/3. The logarithm term only depends on the aspect ratio and is roughly 0.5. This result scales the same if one assumes that α = 0 (the layer thickness is constant with Pc) but with a slightly different constant from the aspect ratio. Finally, using Pc ≈ 2σ/D gives the condition that there will be no overshoot for front velocities greater than

equation image

where we convert the velocity to a flux using the traveling wave observation.

[20] Comparing to the data presented in Figure 3, we find that this formulation matches the overshoot flux dependence on fluid parameters (σ/μ) and on media size (D−3) if the layer thickness does not vary with D (or Pc). The overshoot flux and equation (4) can also be used to estimate the layer thickness, and we find λ ≈ 1 μm. The viscous pressure drop due to flow through surface roughness also explains the discrepancy between water and the alkanes and alcohols in Figure 3; the alkanes and alcohols are completely wetting, but the water wets with a finite contact angle which decreases the size of the layers and overshoot flux [Aminzadeh and DiCarlo, 2010].

[21] In summary, we develop a semicontinuum model that connects the pore-scale physics to the observed 1-D phenomena of overshoot. The model reproduces the observed dependence on capillary number and grain size by invoking viscous forces, pore-scale displacement patterns, and the surface roughness. Since overshoot is the cause of instabilities and large-scale pattern formation for gravity-driven displacements, the implication is that the pore-scale displacement patterns directly control the large-scale displacement patterns.


[22] We wish to thank R. Wallach for helpful discussions and S. Vyas for experimental assistance. This work was supported by the Center for Frontiers of Subsurface Energy Security (CFSES), an Energy Frontier Research Center funded by the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, under award DE-SC0001114.