Stability of gravity-driven multiphase flow in porous media: 40 Years of advancements


  • D. A. DiCarlo

    Corresponding author
    1. Department of Petroleum and Geosystems Engineering, University of Texas at Austin, Austin, Texas, USA
    • Corresponding author: D. A. DiCarlo, Department of Petroleum and Geosystems Engineering, University of Texas at Austin, Austin, TX 78712-0228, USA. (

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[1] Gravity-driven multiphase flow in porous media is ubiquitous in the geophysical world; the classic case in hydrology is vertical infiltration of precipitation into a soil. For homogenous porous media, infiltrations are sometimes observed to be stable and laterally uniform, but other times are observed to be unstable and produce preferential flow paths. Since Saffman and Taylor (1958), researchers have attempted to define criteria that determine instability. Saffman and Taylor's analysis consisted of two regions of single phase flow, while Parlange and Hill (1976) integrated this analysis with the multiphase flow equations to provide testable predictions. In the subsequent 40 years, great advances have been made determining the complex interactions between multiphase flow and instability. Theoretically, the stability of the standard multiphase flow equations has been verified, showing the necessity of extensions to the multiphase flow equations to describe the observed unstable flow. Experimentally, it has been shown that the instability is related to a phenomena in 1-D infiltrations called saturation or pressure overshoot. In this review, the connection between overshoot and instability is detailed, and it is described how models of overshoot can simplify the analysis of current and future models of instability and multiphase flow.

1. Introduction

[2] In hydrology, the classic case of gravity-driven multiphase flow in porous media is infiltration of water in soil. Stability of the flow field is a key question for infiltration, as the formation of preferential flow paths can create large consequences on the transport of contaminants to ground and surface waters. The filtering capacity is lost when infiltration occurs through a few preferential pathways, effectively bypassing the soil. For example, during the 1999 NY state fair, hundreds of people were sickened and one toddler perished after a summer storm caused E. coli from animal feces to enter the supposedly safe ground water drinking supply through preferential flow paths [Yarze and Chase, 2000]. There are many implicated specific causes of preferential vertical flow of water, for example, natural heterogeneities in the soil (wormholes and roots) or soil layering [Gerke et al., 2010; Steenhuis et al., 1998], but the general cause is that the water is heavier than the gas that it is displacing.

[3] This review, and the associated papers in this issue (R. Wallach and Q. Wang, On using Miller similarity to scale sorptivity by contact angle, submitted to Water Resources Research, 2013), concentrates on gravity-driven preferential flow in uniform porous media, i.e., with no macroscopic heterogeneities [Glass et al., 1991]. Here preferential flow occurs when the initially flat flow front becomes unstable and breaks up into many discrete flow paths, and thus is often referred to as a flow instability problem. In addition to ground water protection, correct models of gravity-driven flow instability are crucial to predicting subsurface flows such as transport of radionuclides [Bundt et al., 2000], recovery of hydrocarbons through water and/or gas flooding [Lake, 1989], or injection of CO2 into saline aquifers [Szulczewski et al., 2009].

[4] Figure 1 shows cartoons of a water front infiltrating into a porous media. On the left, the infiltration is stable with a laterally uniform front where the small frontal perturbations do not grow. On the right, the infiltration is unstable, where the small perturbations have grown into long fingers. Since a seminal paper by Parlange and Hill [1976], gravity-driven unstable flow has often been a problem in which many soil physicists cut their teeth. During this time much progress has been made on when and why the flow becomes unstable, how it depends on the particular porous media and initial conditions, and how an understanding of unstable flow is intimately connected with new models of unsaturated flow. We start with the traditional fluid-fluid instability analysis originally formulated by Saffman and Taylor [1958].

Figure 1.

Cartoon of (left) a stable infiltration and (right) an unstable infiltration.

2. Saffman-Taylor Instability Analysis

2.1. Description and Stability Criterion

[5] In 1958, Saffman and Taylor [1958] presented a stability analysis of a moving front between two immiscible fluids inside a porous medium of permeability K. They assumed a distinct interface between the invading phase (subscript i) and the defending phase (subscript d), and that upstream of the interface consists of entirely the invading phase and downstream consists entirely of the defending phase. They also assumed incompressibility of the phases so that the Laplace equation ( inline image) would apply for the flow potential inline image for each phase j. The flow potential is given by

display math(1)

where Pj is the pressure of phase inline image is the density, g is the gravitational constant, z is the flow direction (positive downward), and θ is the angle of the flow with respect to vertical.

[6] The stability analysis consists of adding sinusoidal perturbations (of wavelength λ) to the flat interface, solving the Laplace equation with the boundary condition that the phase pressures are equal at the interface, and determining whether the perturbations would grow or diminish with time. If the perturbations grow in time, the front would be unstable and preferential flow paths would form. The condition for instability that they obtained was

display math(2)

where μ is the viscosity of each phase, inline image is the medium's porosity, and U is the frontal velocity. Table 1 shows the conditions for stability in terms of the density differences and viscosity differences of the fluids for vertical flow, inline image.

Table 1. Saffman Taylor Stability Criteria for the Invading Phase Displacing the Defending Phase From Abovea
  inline image inline image
  1. a

    The case for water invading a dry soil is in the top left-hand quadrant.

inline imageUnstable for inline imageUnconditionally stable
inline imageUnconditionally unstableUnstable for inline image

[7] For the specific case of gravity driven flow of water into an unsaturated soil, the stability analysis occupies the top left-hand quadrant, with a dense viscous fluid (water), displacing a less dense, less viscous fluid (air). In this case, the Saffman-Taylor analysis suggests that the front should be stable at high velocities (or applied fluxes) and unstable at low velocities. In terms of fluxes, and converting the permeability to a conductivity the condition for instability becomes simply

display math(3)

where q is the infiltrating flux, and inline image is the saturated conductivity of the porous medium [Glass and Nicholl, 1996]. Again note for water-air inline image and inline image.

2.2. Finger Size and Boundary Conditions

[8] The Saffman-Taylor stability criteria has been shown to predict instability for fluids in a Hele-Shaw cell [Homsy, 1987], and for fluid pairs in porous media that are viscously unstable [Chuoke et al., 1959]. If the flow is predicted to be unstable, one would also like to predict the width of the preferential flow path or finger caused by the instability. Saffman and Taylor [1958] obtained the finger width from the stability analysis by analyzing the growth of modes of a front. They assumed that the fastest growing wavelength, λ, of the perturbation dominates; and the resultant finger width is half of this wavelength, inline image.

[9] In this analysis, the fastest growing width depends on the boundary condition between the invading and defending phases. As will be shown later, this boundary condition is key in more ways than one.

[10] For the boundary condition of equal pressures across the interface,

display math(4)

the growth rate of each mode depends inversely on the wavelength, i.e., smaller wavelengths grow at faster rates. Thus in this limit, the fastest growing fingers would be of infinitesimal size and inline image. This leads to a highly curved interface between the phases, and this curved interface needs to be accounted for in determining the characteristics of the instability.

[11] For a curved interface between distinct phases, the boundary condition is modified by the interfacial tension, σ, between the phases through the Young-Laplace equation

display math(5)

where R1 and R2 are the principal radii of curvature. Saffman and Taylor [1958] studied the instability in a Hele-Shaw cell, a cell consisting of two parallel plates a small distance apart [Homsy, 1987]. In such a cell, one of the radii of curvature of the interface is out of plane, and one is in plane. With the addition of capillary forces, Saffman and Taylor [1958] showed that the perturbation wavelength must be greater than a critical wavelength inline image to be unstable, and the critical finger width is

display math(6)

[12] Thus as long as the lateral size of the cell is greater than wcrit, the flow will be unstable. For a displacement confined to two dimensions (e.g., a Hele-Shaw cell), the maximum growth rate will take place at a finger width of

display math(7)

[13] Importantly, the revised boundary condition did not change the overall stability. Thus the Saffman-Taylor analysis, says that for a displacement that meets the instability criterion of equation (2), capillary forces cannot stabilize the front, they can only provide a minimum size of the eventual fingers.

[14] In a Hele-Shaw cell the interface between the phases is distinct. The macroscopic interface is not quite as clear for a displacement in a porous medium. In a porous medium, the macroscopic interface between the two phases consists of many microscopic interfaces between the grain surfaces. Still, Chuoke et al. [1959] proposed that there is a “macroscopic” interfacial tension (called inline image) for the macroscopic interface, with the macroscopic interfacial tension linearly related to the microscopic interfacial tension. With this generalization, the physics is predicted to be the same as in a Hele-Shaw cell; the predicted finger width in a slab of porous medium is the same as in equations (6) and (7) with inline image replacing σ. In terms of water displacing air, this can be written as

display math(8)

[15] The dependence of the width on the flux q is seen in the last term. As the flux approaches the saturated conductivity, the driving force of the instability decreases (equations (3) and (6)), and the width of the fingers approaches infinity.

[16] Unlike a Hele-Shaw cell, flow in porous media is inherently three-dimensional. Peters and Flock [1981] extended the Chuoke et al.'s [1959] analysis to 3-D and obtained the same functional form for the width, but with a slightly different prefactor of 4.8 instead of π (the first zero of the Bessel function instead of the first zero of the sine function) [Glass et al., 1991].

3. Multiphase Description

3.1. Multiphase Equations

[17] A porous medium differs from a Hele-Shaw cell in the sense that there is no distinct macroscopic boundary between the invading and defending phases. Instead, the phase saturation S can be introduced as the fraction of a phase in the pore space, in two-phase situations this is usually taken as the water saturation. Above a certain length scale, called the representative elementary volume (REV), the saturation is considered to be a continuous variable [Bear, 1972], and thus this is also called the continuum scale. At the continuum scale the two phases are modeled as coexisting in this region, with the capillary pressure inline image being the difference in phase pressures. The capillary pressure can be measured as a function of S, depends on the particular porous medium, and is a hysteretic function. Finally, for flow the Darcy-Buckingham flux equation is used for each phase j,

display math(9)

with the relative permeability of the medium to a certain phase inline image also dependent on the local saturation.

[18] For water displacing air systems, the air is usually considered inviscid ( inline image) and thus there are no gradients in the air pressure (Pa). The Darcy-Buckingham flux equation for the water phase can be combined with conservation of mass to give the Richards equation [Richards, 1931; Bear, 1972] for flow of water in the vertical direction (z positive downward)

display math(10)

where inline image is the porosity of the medium, and the capillary pressure is in units of cm of water. This equation describes the evolution of the saturation on a macroscopic scale, and is used for the basis for almost all simulations of unsaturated flow [Bear, 1972] where the air is inviscid with free exits. Often the saturation dependence of Pc is incorporated into the Richards equation, resulting in an equation that is only saturation dependent with a nonlinear diffusivity given by inline image [Richards, 1931].

3.2. Multiphase Boundary Condition

[19] Instead of the concept of macroscopic interfacial tension, Parlange and Hill [1976] took a multiphase approach to the physics at the macroscopic front in porous media. In porous media, the front is not distinct like in a Hele-Shaw cell; instead capillary forces cause the saturation of the invading phase to change over a short distance. The distance over which the saturation changes is related to the sorptivity (usually given by the symbol S but here inline image is used to not confuse it with saturation) of the porous medium. Sorptivity is a scalar that is essentially the cumulative effect of the diffusivity at the wetting front on infiltration [Philip, 1957a; Parlange, 1975]. Parlange and Hill [1976] showed that the effect of this diffusivity on a curved interface was to slow the advancement of the front, with higher curvatures yielding lower velocities. Thus they came up with the revised boundary condition of

display math(11)

where U is the overall velocity of the interface and V is the local velocity of the interface.

[20] Parlange and Hill [1976] used the velocity boundary condition in equation (11) combined with a constant entry pressure boundary condition of equation (4) directly into the Saffman-Taylor analysis. This resulted in a prediction of finger width of

display math(12)

[21] The dependence on the flux is slightly different than in the Chuoke et al. [1959] boundary condition. Again, assuming a macroscopic front, different boundary conditions give different finger widths, but cannot change the stability of the front. Incorporating 3-D instead of 2-D displacements again changes the prefactor from π to 4.8 [Glass et al., 1991].

[22] It is instructive to observe how the finger widths for both the Chuoke et al. [1959] and the Parlange and Hill [1976] scale in the flux limits. As the flux goes to zero, the flow is predicted to be unstable, with a finger width that is constant. As the flux approaches the saturated conductivity, the finger width approaches infinity. At and above the saturated conductivity, no modes grow with time, and the flow is stable.

[23] The Saffman-Taylor stability condition was put in other terms by Raats [1973] who argued that the flow will be unstable whenever the pressure of the invading water increases with depth (capillary pressure decreases with depth). When this is the case, any perturbation that moves a section of the front further downward will cause an increase of pressure of the invading phase, and in turn, displacing the air faster. Raats [1973] argued that the this condition can occur for other reasons such as air entrapment and when the hydraulic conductivity increases with depth; we will return to this idea later in the manuscript.

4. Preferential Flow Experiments

4.1. Slab Experiments and Finger Widths

[24] Unstable flow was first observed in field experiments [e.g., Parlange and Hill, 1976]. Some of the uniform glacial till soils of Long Island and Connecticut showed obvious preferential flow paths [Starr et al., 1977; Glass et al., 1988]. Early experiments were time consuming in that they involved identifying the parameters of the preferential flow paths through serial destruction of soil columns. Glass et al. [1989a] made a breakthrough in finger flow measurements through the use of light transmission measurements of slabs of porous media. Having a thickness of 1 cm, the porous media slabs were much easier to construct and perform experimental duplicates; more importantly, the water saturation could be monitored without destroying the columns, allowing quick in situ measurements. These slab experiments became the standard method of observing and measuring preferential flow in sandpacks.

[25] Early studies by Glass et al. [1989a] were set up to determine whether the width of preferential flow paths matched the Chuoke et al. [1959] or Parlange and Hill [1976] velocity dependence as described above. Water was uniformly applied at a constant flux rate at the top of the slab, and the finger widths were measured. The results were inconclusive in terms of determining which velocity dependence matched the finger width data. This was mainly because the finger width dependencies are only noticeably different when the flux approaches the saturated conductivity, inline image [Glass et al., 1989a]. Experimentally, the fingers were observed to merge at high fluxes, and thus the velocity dependence and the correct boundary condition could not be determined.

[26] In addition to the velocity dependence of the widths, Selker and Schroth [1998] studied how the finger widths scaled with the surface tension of the infiltrating fluid, and the grain size of the porous medium. They showed that the Parlange and Hill [1976] model predicted finger width scaling of

display math(13)

where D is the characteristic grain diameter of the medium. This is different than the dependence predicted by the Chuoke et al.'s [1959] model inline image, as inline image where inline image is some representative grain size. Selker and Schroth [1998] found the width scaling in equation (13) to be true for the mineral oils and different sized media in their study.

4.2. Wettability

[27] The oil and water results of Selker and Schroth [1998] did not scale together; this was interpreted to be a result of the water making a nonzero contact angle with the solid surfaces, while oil was observed to have a very small contact angle. Other experiments using water repellent and partially water repellent soils showed that repellency increased the occurrence of instability [Ritsema et al., 1996; Bauters et al., 1998]. Several soils were shown to exhibit preferential flow only when they were made water repellent (through natural processes or through laboratory addition of repellency). This was interesting as the Saffman-Taylor condition says nothing about the wettability of the medium. It only deals with the fluid properties and the flow rates.

4.3. Stable Flow?

[28] Infiltrations into other types of porous media have also been found to be stable. In particular, for finer grain sizes than those used by Selker and Schroth [1998] above, experiments at all fluxes showed no instabilities [Diment and Watson, 1985]. Other experiments with initial water in the sandy porous media also showed no instabilities [Diment and Watson, 1985; Glass and Nicholl, 1996; Bauters et al., 2000a]. Since in the Saffman-Taylor analysis, both finer grains and initial water are predicted to increase the finger widths, one possible explanation of the observed stability was that the unstable wavelengths became larger than the experimental chamber. Maybe wider experiments were needed as the finger width is limited by the size of the chamber. Unfortunately, field experiments of large spatial extent in the semiarid southwest [Glass and Nicholl, 1996] still showed no instability.

[29] This explanation that the columns were not wide enough was decisively disproved by experiments of Yao and Hendrickx [1996]. They infiltrated into large (30 cm diameter) 3-D soil columns at fluxes between inline image, ( inline image for the 0.9 mm (14/20 sieve size) sand in their study) and observed the subsequent flow patterns and finger widths through serial destruction. The results are depicted in Figure 2. At moderate flow rates inline image, they found that unstable flow with finger sizes consistent with those seen in slab experiments [Glass et al., 1989b] when scaled between 2-D and 3-D; these data are also shown in Figure 2. But when the very low fluxes were applied inline image), the finger widths increased until the flow was observed to be stable, i.e., the width of the infiltration was the size of the chamber. In essence, the finger widths decreased with decreasing flux near the saturated conductivity (as predicted by Chuoke et al. [1959] and Parlange and Hill [1976]), followed by a long flat valley of constant finger width with flux, finally followed by rapid increase in finger width, followed by stable flow. Thus, unstable flow was only observed for an applied flux within a certain range; in addition to the high flux limit predicted by the straightforward Saffman-Taylor analysis, there was a low flux limit. This low flux limit was not predicted from the Saffman-Taylor analysis using either the Chuoke et al. [1959] or Parlange and Hill [1976] boundary condition.

Figure 2.

Measured finger widths as a function of applied flux. Yao and Hendrickx [1996] measurements were performed in 3-D chambers, and Glass et al. [1989b] measurements were performed in 2-D slab chambers; the 2-D data has been scaled by inline image for ease of comparison. The stability arguments [Chuoke et al., 1959; Parlange and Hill, 1976] predict the increase in finger widths at high applied fluxes, but do not predict the increase in finger width at low fluxes.

[30] Hendrickx and Yao [1996] conceptually explained this limit as the capillary forces were greater than the gravitational forces and this stabilized the flow. Attempts were made to perform the instability analysis using a full multiphase flow description, but the analysis was difficult as it involved perturbations in multiple dimensions and nonlinear flow equations [Du et al., 2001]. The results were inconclusive.

4.4. Finger Dynamics and Saturation Overshoot

[31] Concurrent with the experiments concerning widths and overall stability of the flow, slab experiments looked into the dynamics within a preferential flow path. Figure 3 shows a false colored snapshot of preferential flow paths measured using light transmission. Here the fingers on the left and right have reached the bottom of the column, while the finger in the middle is progressing downward at constant velocity. The hotter colors in Figure 3 correspond to higher light transmission and higher water saturations.

Figure 3.

Preferential flow paths seen in a slab chamber using light transmission. The colored region are the flow paths, and the hotter colors corresponding to higher water saturation.

[32] The light transmission slab experiments of Glass et al. [1989a] showed that the fingers had a unique water saturation signature. When preferential flow occurred, the tips of the fingers (the downstream end) always showed a higher water saturation, SK, than the saturation at the tail, SJ (the upstream end); this can be seen in the central finger in Figure 3. This nonmonotonic variation in saturation versus space within a flow path is depicted in Figure 4, and has been dubbed simply “saturation overshoot,” as the saturation overshoots its final asymptotic value. Saturation overshoot was observed in all gravity driven fingers measured using light transmission and X-rays [Liu et al., 1993]. These fingers were also observed to move downward with a constant velocity.

Figure 4.

Saturation overshoot seen within a flow path.

[33] Further measurements by Selker et al. [1992] using miniature tensiometers showed that along with saturation overshoot, there existed pressure overshoot in the paths, where the water pressure was greater at the tip than the tail. This matched the pressure inversion as a cause for preferential flow as predicted by Raats [1973].

4.5. Numerical Simulation of Instability

[34] Attempts were made to model preferential flow using the multiphase flow equations. Nieber [1996] was the first to create preferential flow paths in 2-D models; one of the key insights obtained was this required an nonstandard way of weighting relative permeability across grid blocks [Nieber et al., 2000]. Eliassi and Glass [2001] argued that this was not an appropriate method of dealing with the relative permeability on a numerical grid. This debate on the numerical modeling brought to the forefront that the entire flow instability, and not just the finger width, was contingent on the exact boundary condition physics [Eliassi and Glass, 2003a, 2003b; Braddock and Norbury, 2003; Deinert et al., 2003].

5. Saturation Overshoot and Instability

5.1. Numerical Simulation and Overshoot

[35] As a result of the numerical simulation studies [Nieber, 1996; Nieber et al., 2000; Eliassi and Glass, 2001], Eliassi and Glass [2001] formulated a prescient hypothesis. Is saturation overshoot the cause rather than the effect of gravitational driven instability?

[36] This hypothesis was backed up by numerical experiments of Eliassi and Glass [2003c]. When they altered the multiphase flow equations such that saturation overshoot was observed in their 1-D model, they found unstable flow in their 2-D models. If there was no saturation overshoot in their 1-D model, there was no instability in their 2-D models. This was suggestive, but experiments were necessary.

5.2. Experiments and Overshoot

[37] Simultaneously, Geiger and Durnford [2000] took up this challenge experimentally. They infiltrated water into soil columns that were smaller in diameter than the observed preferential flow paths, thus keeping the flow 1-D and eliminating preferential flow. They measured the water pressure at a point in the column during the infiltration. They found a pressure inversion (or overshoot) for coarse sands at all infiltration rates, but no pressure overshoot for fine sands at low infiltration rates. They proposed a mechanistic model on how pressure inversions create unstable flow, similar to the arguments of Raats [1973].

[38] Bauters et al. [2000a] performed point source infiltrations into a slab container and measured the saturation profile using light transmission. They kept the flux and the porous media the same for their experiments, and varied the initial water saturation of the sand pack. They found that for an initial water saturation SI less than 0.06, ( inline image), the profiles exhibited overshoot and exhibited widths consistent with a preferential flow path. For inline image, the profiles had no overshoot, and exhibited widths consistent with diffusive flow rather than that seen for preferential flow paths.

[39] Deinert et al. [2002, 2003] obtained capillary pressure and saturation profiles in the wetting front region of a preferential flow field with 1 mm spatial resolution. Similar to Selker et al. [1992] and Geiger and Durnford [2000], the resulting profiles showed a change in sign for inline image within the wetting front region.

[40] Subsequently, DiCarlo [2004] performed constant flux infiltrations into sands confined into 1-D columns and measured the water saturation using light transmission. The sands were identical to those used in the 2-D slab experiments for easy comparison. In this case, the flux of the infiltrations was varied. Figure 5 shows snapshots of the water saturation profiles during infiltration at 6 different fluxes.

Figure 5.

Saturation profiles during infiltration for six different applied fluxes. Overshoot behavior is seen at intermediate fluxes, but not at the highest flux ( inline image) nor the lowest flux ( inline image).

[41] At the highest (11.8 cm/min) and lowest ( inline image) flux the profiles are monotonic with distance and no saturation overshoot was observed. At intermediate fluxes overshoot was observed. From these measurements, the tip saturation (SK) and the tail saturation (SJ) can be obtained.

[42] Figure 6 shows the tip and tail saturations for 35 different infiltrations as a function of flux into dry porous media of grain size 0.7 mm (20/30 sand). The solid symbols are the measured saturations behind the front (SJ), and the open symbols are the measured saturations at the front (SK). At a particular applied flux, if inline image the profile exhibited overshoot, if inline image there is no overshoot. In this experimental set, we find that the profile exhibits overshoot between fluxes of inline image and inline image.

Figure 6.

Tip (SK) and tail (SJ) saturation as a function of flux. Overshoot is seen when inline image. Overshoot occurs at the same flux range as when fingers are observed (Figure 2).

[43] The results showed that the flux range of 1-D saturation overshoot matched the flux range of 3-D preferential flow [Yao and Hendrickx, 1996] for roughly the same sized sand (see Figure 2). Also, when the infiltration took place into sand with an initial saturation, the amount of saturation overshoot measured in the 1-D experiments matched the amount of overshoot seen in 2-D experiments [Bauters et al., 2000a].

[44] Saturation overshoot was also seen in experiments by Shiozawa and Fujimaki [2004] and pressure overshoot by Annaka and Hanayama [2005] and Bottero et al. [2011]. It was also realized that measurements of pressure overshoot in coarse sands were reported previously by Stonestrom and Akstin [1994]. At the time, the observed pressure overshoot was regarded as a potentially important oddity; its importance and connection to preferential flow would not be clear until later.

[45] Subsequent pressure overshoot measurements [DiCarlo, 2007] were taken on the same sands, flux ranges, and initial water contents giving a full accounting of the water state when overshoot occurs. Further experiments to delineate the overshoot were taken using different fluids, e.g., single chain hydrocarbons and alcohols instead of water [Aminzadeh and DiCarlo, 2010]. Finally, measurements showed the flux range for which saturation overshoot takes place decreased with decreasing grain size; with the range eventually disappearing for fine sands, also matching that of the pressure overshoot and preferential flow measurements [Geiger and Durnford, 2000; DiCarlo, 2004; DiCarlo et al., 2011].

5.3. Saturation Overshoot Boundary Condition

[46] As Eliassi and Glass [2001] hypothesized, experiments show a connection between saturation overshoot and the gravity driven instability. To understand this connection, we delve into the details of a 1-D overshoot profile and a 1-D nonovershoot profile. These profiles are depicted side by side in Figure 7.

Figure 7.

Saturation profile during an (left) infiltration that is described by the Richards equation, and (right) overshoot infiltration.

[47] Both of these potential profiles are for a 1-D constant flux infiltration into a medium with a uniform initial saturation (SI). They are also fully developed, in the sense that the transients that occur due to the top boundary have all been dissipated; experiments show this to happen very quickly (less than 10 cm from the top boundary). In both cases, a traveling wave will result, with the final saturation (SJ) is set by the applied flux q0 through

display math(14)

and the velocity of the traveling wave determined by the Rankine-Hugoniot condition,

display math(15)

[48] Typically the initial saturation is low enough such that inline image and equation (15) becomes

display math(16)

[49] We call the nonovershoot profile the Richards profile as it is what results when the Richards equation is solved for monotonic P − S and inline image curves [Philip, 1957b]. The Richards profile monotonically increases from the initial saturation SI to the final saturation SJ.

[50] The overshoot profile is what is obtained from experiments with certain initial and boundary conditions; it has the same initial and final saturations, but instead of moving monotonically to the final saturation, there is a jump in saturation at the front,

display math(17)

[51] This is essentially the boundary condition for an overshoot infiltration. After this saturation jump, the profile follows the Richards equation to monotonically decrease from SK to SJ [Selker et al., 1992].

[52] Combining these saturation profiles with the hysteretic Pc curves, we can observe how the saturation overshoot is combined with pressure overshoot. Figure 8 depicts the path through the inline image space for both the Richards and overshoot profile. For the Richards profile, the saturation only increases, and thus the profile moves smoothly down the imbibition curve to SJ. For the overshoot profile, at the front the saturation jumps over intermediate saturations to a value inline image (Point A to Point B). Understanding the physics of this jump or boundary condition is the key, if one postulates this jump as a boundary condition at the front, the rest of the profile follows.

Figure 8.

Paths through the pressure-saturation curve for (left) a Richards infiltration and (right) an overshoot infiltration.

5.4. Overshoot and Instability

[53] In terms of pressure it is seen that the Richards case, the water pressure increases upstream, and in the overshoot case, the water pressure decreases upstream. Raats [1973] argued that when the water pressure decreases (capillary pressure increases) in the upstream direction, the overall flow will be unstable. Here we give a review of his argument in terms of Richards and overshoot profiles.

[54] In terms of 2-D stability, from the 1-D Richards profile, we note that the conductivity of the medium to water ( inline image) is not the saturated conductivity. In fact, since the conductivity is monotonic with respect to saturation, the conductivity of the medium to water is less than or equal to the applied flux q0 for the entire profile. Using this actual conductivity, as opposed to the saturated conductivity, in the Saffman-Taylor analysis, the stability criterion for unsaturated flow becomes

display math(18)

and since inline image, this criterion is met for diffuse fronts as inline image at all times and positions. Thus the flow is predicted to be stable, and the expected 2-D Richards flow pattern is shown in the left-hand side of Figure 9 for a 1-D Richards profile.

Figure 9.

2-D flow patterns that are observed during (left) a Richards infiltration and (right) an overshoot infiltration.

[55] In contrast, for a 1-D overshoot profile, the conductivity of the medium to water behind the front is now greater than the applied flux. Now the criterion for stability in equation (18) is not met, as inline image at the front. An instability is created leading to fingered flow shown in the right-hand side of Figure 9.

[56] As initially pointed out by Glass et al. [1989a] and reaffirmed in models and experiments by Nieber [1996], DiCarlo et al. [1999], and Rezanezhad et al. [2006], the hysteresis in the capillary pressure curve allows the fingers to persist for long times. In the finger, the pressure profile goes from the imbibition to the drainage curve through points B-D as depicted by Figure 8. In the tail Pc is on the drainage curve (point D); at an equivalent Pc on the imbibition curve (point E), the saturation is very low. The lateral movement of water out of the tail is very slow as the capillary pressure gradient is small [Glass et al., 1989a], and the conductivity of the medium is very small at the low saturation [DiCarlo et al., 1999]. Even when the pressure equilibrates laterally, the hysteresis in the Pc curve allows a large difference in saturation to persist between the finger and the surrounding medium.

[57] It is important to note that although hysteresis stabilizes the fingers once they are formed, hysteresis is not the cause of the instability in the first place. As shown by the Richards profile, even with hysteresis, the flow is completely stable as the medium remains on the imbibition curve and thus hysteresis is immaterial.

5.5. Stability of the Richards Equation

[58] At roughly the same time of the experiments, the Richards equation was being shown to be unconditionally stable [Egorov et al., 2003; van Duijn et al., 2004]. In particular, the work by Egorov et al. [2003] is very revealing for understanding stability. First, it detailed errors in previous work on the Richards equation stability [Kapoor, 1996; Du et al., 2001]. Second, it showed that when an overshoot front is postulated in the Richards equation, the flow will be unstable. Finally, it showed mathematically that nonmonotonicity (overshoot) is a requisite condition for flow instability, affirming the physical arguments in the previous section.

6. Saturation Overshoot Paradigm

6.1. Richards or Overshoot?

[59] Both experimentally and mathematically, the evidence and logic are clear that if a Richards profile is observed in 1-D infiltrations, the 2-D flow is predicted to be stable [Raats, 1973; Egorov et al., 2003; van Duijn et al., 2004]. If an overshoot profiles is observed in a 1-D infiltration, the 2-D flow is predicted to be unstable [Saffman and Taylor, 1958; Raats, 1973; Egorov et al., 2003; Nieber et al., 2005]. This correlation is exactly what is seen between the multidimensional experiments [Bauters et al., 2000b; Yao and Hendrickx, 1996] and the 1-D experiments [DiCarlo, 2004]. In essence, the hypothesis of Eliassi and Glass [2001] was confirmed: the cause of the gravitational flow instability is the saturation and pressure overshoot. Thus the unanswered question for gravity driven preferential flow becomes: Why do some infiltrations have Richards profiles, and why do some infiltrations have overshoot profiles? Or, as discussed in section 5.3., this question can be rephrased as: Why are some intermediate saturations jumped over at an infiltrating front? This greatly simplifies the gravity driven preferential flow problem as now the question has moved from a nonlinear multidimensional flow problem to a nonlinear 1-D flow problem.

6.2. Overshoot Experiments

[60] To gain insight to the physics of overshoot, we first review the experiments on when an overshoot profile is observed and when a Richards profile is observed. In general, overshoot profiles are much more prevalent for: sandy soils (in particular sorted sands) [Glass and Nicholl, 1996], initially dry porous media [Diment and Watson, 1985; Bauters et al., 2000a; DiCarlo, 2004], water repellent soils [Bauters et al., 1998; Wallach and Jortzick, 2008], and infiltrations at moderate to high infiltration fluxes [Yao and Hendrickx, 1996; DiCarlo, 2004].

6.3. Conceptional and Pore-Scale Models of Overshoot Profiles

[61] Eliassi and Glass [2001] produced a conceptual model of overshoot (i.e., why intermediate saturations are jumped over) that they labeled the hold-back pile-up effect. For overshoot infiltrations, at the initial wetting front, the physics of the flow slightly prevents the water from entering the dry porous medium. This causes the front to be “held back”; with this hold back the water from above “piles up” at the interface. The pile up necessarily requires an abrupt interface, and it causes the requisite high saturation and a pressure inversion behind the main front. Once enough water is piled up, and the entry pressure is high enough, the water can advance. For constant flux infiltrations, this process eventually reaches steady state, and the overshoot profile is observed. Again, the Richards equation cannot produce this “hold back—pile up” behavior due to its diffusive nature.

[62] A potential understanding of the “hold back—pile up” effect comes from observations of pore-scale filling processes. Using micromodels of porous media, Lenormand and Zarcone [1984] and Lenormand [1986] observed that at high frontal velocities the wetting front is very sharp with almost all pores filled directly behind the front. At very low frontal velocities, the front is diffuse with many pores and throats remaining unfilled behind the front. Lenormand and Zarcone [1984] explained this phenomenon in terms of a competition between piston-like collective filling at the main front and filling ahead of the front due to conduction through water layers in the corners of the pore space [see also Blunt and Scher, 1995]. At high frontal velocities, piston-like collective filling preferentially occurs as conduction through the water layers cannot match the speed of the front, leading to a sharp jump in saturation at the front. At low frontal velocities, conduction through the layers can match the speed of the front allowing pores to fill ahead of the main front, and on a macroscale, a relatively smooth increase of the saturation at the front. This picture has been observed in pore-scale observations on infiltration into glass bead porous media [Lu et al., 1994a, 1994b].

[63] This pore-scale, saturation-jump, behavior can be mimicked using rule-based network models. These lattice-based models have elements (pores and throats) that can only have discrete saturation states, they are either filled or empty of water. The element state changes discretely based on rules derived by capillary forces and the states of nearby elements. Using these models, Glass and Yarrington [1996] and Glass et al. [1998, 2000a, 2000b] have shown finger patterns with drying behind the finger tip that mimic those observed in experiments for moderate flow rates. In essence, these pore-scale processes create a sharp front with a jump in water saturation, which would be akin to water being held back at a moving front.

[64] Unfortunately these models predict instability for all fluxes, which is not seen in experiments. Network models with rules that allow partially filled (but capillary stable) elements allow water to be transported in layers in the partially filled elements [Valvatne and Blunt, 2004; Valvatne et al., 2005]. Incorporating viscous pressure drops into the filling rules of these network models, the transition between Richards profiles and overshoot profiles as a function of flux has been observed [Hughes and Blunt, 2000, 2001; DiCarlo, 2006]. This further suggests that the pore-scale physics controls the saturation jump, although the comparisons have only been qualitative. Perhaps with the new generation of dynamic network models [Al-Gharbi and Blunt, 2005; Joekar-Niasar et al., 2010], and accurate descriptions of the pore-space through microtomographic scanning [Al-Raoush and Willson, 2005] the connection can be made quantitative.

6.4. Boundary Condition at the Wetting Front

[65] Both the large-scale (hold back pile up) and pore-scale (collective filling) explanations of overshoot suggest that the boundary condition at the wetting front needs to be modified for continuum-scale models of overshoot. Importantly, neither of these explanations require any changes to the flow physics behind the front. This is backed up by experimental observations [Selker et al., 1992; Deinert et al., 2002]. They found that the saturation profile behind the tip could be well described as a translating hanging water column. With the measurement of the water pressure at the finger tip, the Richards equation predicts the trailing pressure profile and saturation profile [Liu et al., 1993].

[66] If one simply postulated the water saturation or pressure at the front, i.e., the boundary condition at the wetting front, the entire overshoot profile is calculable using the Richards equation [Selker et al., 1992] in the regions ahead of and behind the front. Using this idea, Egorov et al. [2003] showed that this postulate would make the flow field unstable.

[67] Experiments show that the saturation jump at the front is observed to be a function of the applied flux q0, the initial saturation SI, the advancing contact angle of water inline image, the grain size of the porous media D, and likely the spread of grain sizes inline image (as noticed in loamy soils),

display math(19)

[68] This concept of a certain saturation jump at the wetting interface is general, it can be used if there is overshoot or not (a Richards profile occurs if inline image). Currently, this functional form of the front saturation is only obtained from experiments. Ideally one would like to obtain the functional form from first principles of fluid flow in porous media (for instance, either judicious Lattice Boltzmann or network modeling), or using continuum extensions as described in the next section. In this issue, R. Wallach and Q. Wang (submitted manuscript, 2013) give reasons how the observed overshoot dependence on the flux is due to the advancing contact angle dependence on the flux.

[69] In terms of finger widths and spacings, the Saffman-Taylor stability models of Chuoke et al. [1959] and Parlange and Hill [1976] assumed a completely saturated region behind the front, e.g., inline image. Liu et al. [1994] modified the theory of Parlange and Hill [1976] to calculate finger widths in the case for general SK. They obtained

display math(20)

where η is a Corey exponent for the permeability [Brooks and Corey, 1964]. This model was shown to work well for point source infiltrations, but it needed the slope of the wetting curve inline image to be modified depending on the initial saturation. Still, it may provide a way forward on calculating the widths and spacings once the front saturation is known.

6.5. Continuum Extensions of Richards Equation

[70] Instead of an interface boundary condition that is determined through experiments and discrete models of porous media, there is a great preference for a continuum-scale model that would produce the saturation jump at the interface naturally. Such a mathematical model would be useful for calculation and prediction. As the derivation of the Richards equation only requires the multiphase Darcy-Buckingham equation and conservation of mass (and the fact that the Richards equation works in almost all cases!), most continuum models that are proposed are extensions of the Richards equation.

[71] Eliassi and Glass [2002] proposed several potential extensions to the Richards equation in the context of overshoot. They proposed three different types of additional terms that can be added: a hypodiffusive term (second order in space), a hyperbolic term (second order in time), and mixed or relaxation term (second order in space, first order in time). Each of these terms can act to hold back water at the infiltration front and produce saturation overshoot. Using the hypodiffusive term, Eliassi and Glass [2003c] were able to produce overshoot profiles in 1-D numerical simulations and fingers in 2-D simulations.

[72] Preceding the work of Eliassi and Glass [2002], other experimental measurements suggested the need for additional terms in the continuum equations [e.g., Weitz et al., 1987]. The most common of these additions is the introduction of a relaxation time to take into account the redistribution of fluids within the pore space. In particular, Hassanizadeh and Gray [1993] advocated a dynamic capillary pressure, where the capillary pressure depends not only on the saturation and saturation direction but also on the rate of saturation change. Barenblatt [1971] and Barenblatt et al. [2003] suggested using an effective saturation for calculating relative permeability and capillary pressure, where the effective saturation contains a relaxation term.

[73] Cuesta et al. [2000] first studied mathematically the effect of dynamic capillary pressure, and showed that it can produce overshoot and oscillations for traveling waves; relaxation parameters were subsequently estimated by comparing to experimentally measured overshoot [DiCarlo, 2005]. Egorov et al. [2003] showed that relaxation terms produce instability, and reasonable looking fingers. Recently, van Duijn et al. [2007, 2013] analyzed flows where the displaced fluid has viscosity; and determined how the overshoot saturation SK is related to inline image, and the dynamic parameter. Since relaxation models predict oscillations, hysteresis was included in these model calculations by Rätz and Schweizer [2013].

[74] Cueto-Felgueroso and Juanes [2008] proposed a phase field model extension to the multiphase equations. Taking a cue from heuristic descriptions of phase transitions, they associated an energy with the macroscopic boundary that appears on infiltration. Mathematically this results in a fourth-order spatial derivative in the flow equations. Like the other higher-order terms, overshoot profiles and instabilities are created with an appropriate magnitude of the phase field term [Cueto-Felgueroso and Juanes, 2008, 2009a, 2009b].

[75] Other models have been proposed based on the fact that the contact angle of the invading water varies with the speed of the microscopic wetting front [Hoffman, 1975; de Gennes, 1985; Hilpert, 2009]. A velocity-dependent contact angle can produce a dynamic effect similar to that proposed by Hassanizadeh and Gray [1993] and Barenblatt [1971], but with potential differences [Hilpert, 2012]. These models also produce overshoot [Hilpert, 2012]. This agrees with experimental evidence with changes in the contact angle and overshoot [Wallach and Jortzick, 2008].

[76] In addition to producing overshoot profiles, all the models above show an overshoot flux; there is a transition from a Richards profile to an overshoot profile only above a certain applied flux. Again, this is not seen in stability analysis of a sharp front, nor in the Richards equation description. The magnitude of this overshoot flux depends on the magnitude of the additional term that is added to the equations. This universality suggests that continuum models may be an appropriate method to model overshoot and preferential flow. In terms of the moving macroscopic interface, all of the potential extensions alter the boundary condition to essentially artificially hold back the wetting front.

6.6. Evaluating Continuum Model Additions

[77] As mentioned above, continuum multiphase models with higher order derivatives can produce overshoot profiles. Since each model has different proponents, there has been interest in evaluating the models in terms of the observed overshoot and fingering.

[78] In evaluating which model is “best” or “most appropriate,” one would look to see which is the most parsimonious. In particular, the model should:

[79] 1. Have a minimum of adjustable parameters, and the parameter(s) should have a defined and natural scaling as a function of grain size, fluid pairs, and initial conditions,

[80] 2. Reduce to the Richards equation in nonovershoot and static profiles,

[81] 3. Produce a good match of the observed 1-D profiles, not just the magnitude of the overshoot,

[82] 4. Can produce predictions of the 2-D and 3-D preferential flow in terms of finger widths and finger spacings.

[83] There have been detailed studies into evaluating different models in terms of the first three overshoot items above [DiCarlo, 2005, 2007; Cueto-Felgueroso and Juanes, 2010]. One of the main advantages of evaluating the models in terms of overshoot rather than full field preferential flow is that the 1-D traveling wave nature of the problem lets one break the multidimensional nonlinear partial differential equations down into a nonlinear ordinary differential equation. This greatly simplifies issues with numerical simulation artifacts [Nieber et al., 2000; Eliassi and Glass, 2003a, 2003b; Braddock and Norbury, 2003]. The simulated 1-D profiles can be obtained quickly as a function of the input parameters, and how the models change the effective boundary condition can be determined. Judging parsimony of the different models can be subjective, and is currently much debated [DiCarlo, 2005, 2007; Cueto-Felgueroso and Juanes, 2010; DiCarlo, 2010a, 2010b].

[84] Going forward, here we concentrate on the connections between overshoot models and unstable flow. One interesting argument is that 1-D overshoot is necessary to produce unstable flow, but under what conditions is the overshoot sufficient to produce unstable flow? Also, when overshoot is sufficient to produce preferential flow, what are the properties of the resultant preferential flow paths? In particular, what are the predicted spacing between the flow paths, what are the widths of these flow paths, and how does these properties scale with different fluids, wettability, and porous media properties?

[85] For instance, in response to the first question, Egorov et al. [2003] showed that dynamic capillary pressure model Hassanizadeh and Gray [1993] produces overshoot, but the 2-D flow field only becomes unstable only when the 1-D overshoot is above a certain magnitude. Instead, Egorov et al. [2003] also showed that if one postulated only the overshoot boundary condition, the flow would be unstable, but the fastest growing fingers would be of infinitesimal width. Stability arguments, such as [Saffman and Taylor, 1958] and those in section 2, start with a flat front and the finger width is the fastest growing wavelength. But as soon as the perturbation grows, the flat front is lost, and thus the stability arguments cannot predict the final preferential structure of the flow field in terms of the finger spacings, the stability of the widths after flow field development, and the structure of the saturation field inside and outside of the finger cores.

[86] One way to overcome this difficulty is to input the proposed analytic extensions into multiphase simulators and observing the dynamics of the flow field structure. Simulations that test 2-D patterns using the analytic extensions are not trivial. Full descriptions of fingering require an accurate description of the hysteresis to prevent lateral expansions of the fingers (see earlier arguments in section 5.4. and Glass et al. [1989a]). Again, the first attempt was by Eliassi and Glass [2003c] which showed structures that compare favorably to experiments. The model of Cueto-Felgueroso and Juanes [2008] produces very physical looking fingering displacements. Nieber et al. [2005], Chapwanya and Stockie [2010], Kissling et al. [2012], and Rätz and Schweizer [2013] have also produced physical looking fingering displacements using the dynamic extensions with the latter including hysteresis.

[87] Again, these qualitative results show the connection between overshoot and fingering; one potential next step is determining how the width and the spacing of fingers vary with the magnitude of the higher order extension terms, and how this compares with experiments. Early preferential flow experiments were compared to scaling predictions from stability theory and a sharp front (in particular the scaling with interfacial tension and flux, see section 2). It will be instructive to see how the extensions, without postulating the frontal boundary condition, quantitatively compare to the widths and spacing of the observed fingers. These comparisons and predictions in the 2-D realm, will provide additional evaluations of the most parsimonious model, either of the boundary condition or the complete flow equations.

7. Summary

[88] For many decades, the stability of gravity-driven flow has been one of the great unsolved problems in multiphase flow. Alongside the issues with unstable flow, there is a wide spread belief (mainly as a result of anomalous experimental results) that the standard multiphase model is incomplete. Through observations like overshoot, these core multiphase flow issues have revealed how intertwined they are. To correctly model unstable flow, additional physics is needed, in particular, the boundary condition at the moving front needs to be determined. Thus the need for extensions to the multiphase model. In turn, unstable flow has become the ideal testing ground for proposed multiphase model extensions. Thus, both unstable flow and the multiphase extensions are intimately tied with the pore-scale dynamics of the moving fronts.

[89] The delineation of this connection has rejuvenated unstable flow, and added excitement that this once intractable problem can now be solved. Further experiments (in terms of full 2-D and 3-D flow fields, 1-D columns, and microscopic and macroscopic saturation and pressure profiles) combined with theoretical arguments and multiphase modeling now hold the promise of providing a complete description of unstable flow. Simple evaluations of models can be made comparing 1-D overshoot profiles, and more complex evaluations can be made in comparing 2-D flow field structures. Finally, a complete description of unstable flow will be a big driver, if not the main driver, in the development of more accurate and appropriate multiphase models.


[90] I wish to thank J. Selker and T. Steenhuis for the opportunity to review the last 40 years of preferential flow, and B. Aminzadeh, M. Mirzaei, R. Wallach, and M. Deinert for helpful discussions of the ideas in this review. The comments of J. Nieber and two anonymous reviewers were much appreciated. This work was supported by the Center for Frontiers of Subsurface Energy Security (CFSES), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award DE-SC0001114.