The foam drainage equation for unsaturated flow in porous media
Department of Environmental Systems Science (D-USYS), Institute of Terrestrial Ecosystems (ITES), Soil and Terrestrial Environmental Physics (STEP), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
Corresponding author: D. Or, Department of Environmental Systems Science (D-USYS), Institute of Terrestrial Ecosystems (ITES), Soil and Terrestrial Environmental Physics (STEP), Swiss Federal Institute of Technology (ETH), CH-8092 Zurich, Switzerland. (firstname.lastname@example.org)
 A class of capillary flows in unsaturated porous media is characterized by quasi steady viscous flow confined behind curved air-water interfaces and within liquid bodies held by capillary forces along crevices and grain contacts. The geometry of the connected capillary liquid network within the pore space resembles channels that form between adjacent bubbles in foam (Plateau borders) with solid grains representing gas bubbles in foam. For simplified channel geometry, we combine expressions for viscous flow with continuity considerations to describe the evolution of the channels cross-sectional area during gravity drainage. This formulation enables modeling of unsaturated flow without invoking the Richards equation and associated hydraulic functions. We adapt a formalism originally developed for foam “free drainage” (drainage under gravity) or “forced drainage” (infiltration front motion) to a class of unsaturated flows in porous media that require a few input parameters only (mean channel corner angle, air entry value, and porosity) for certain initial and boundary conditions. We demonstrate that the reduction in capillary channel cross section yields a consistent description of self-regulating internal fluxes toward attainment of the so-called “field capacity” in soil and provides an alternative method for interpretation of outflow experiments for prescribed pressure boundary conditions. Additionally, the geometrically explicit formulation provides a more intuitive picture of capillary flows across textural boundaries (changes in channel cross section and number of channels). The foam drainage methodology expands the range of tools available for analyses of unsaturated flow processes and offers more realistic links between liquid configuration and flow dynamics in unsaturated porous media.
 Since its inception in the landmark monograph by Buckingham , the theory of unsaturated flow in porous media is based on the conceptual tenet whereby the configuration and size of capillary-held water control flow rates. Moreover, Buckingham's  own depiction of the unsaturated hydraulic conductivity function considers capillary flows behind air-water interfaces (his Figure 15), which stand in stark contrast with flow in closed conduits as certain models postulate [Burdine, 1953; Mualem, 1976]. Given modern observational capabilities, it comes as no surprise that various unsaturated flow regimes such as in the so-called “transmission zone” for infiltration or drainage, or during steady state unsaturated flows, take place through a network of menisci-bounded capillary channels along grain surfaces and crevices. If we focus our attention on the capillary held aqueous phase only (by abstractly removing the solid phase), we may envision the structure of the liquid channels as resembling channels forming between interacting bubbles in foam (these channels are known as Plateau borders) [Plateau, 1873; Weaire et al., 1997]. The prospects for analogy between liquid networks in unsaturated porous media and liquid foams (Figure 1) offers a potential for using geometrically explicit concepts from foam theory to obtain new insights into key aspects of unsaturated flows that are presently overlooked in continuum models.
Verbist et al.  insightfully commented that: “Given the intensive research currently under way on transport in porous media, it is natural to ask how our subject relates to it. In a porous medium one may also have a random network of channels, just as we have here, and Poiseuille flow may be assumed.” Similarly, Weaire et al.  stated “The flow of liquid through a foam may be related to soil science, in which wetting of soil by surface water proceeds in an analogous manner. It is by no means so simple to analyze, foam being a more tractable material than mud for theoretical analysis.” Clearly, flow in natural unsaturated porous media is far more complex than flow in foam; nevertheless, the foam flow framework offers an attractive alternative to shortcomings of continuum models, and provides more direct interpretation of flow and transport pathways (e.g., dispersion and colloids) in unsaturated porous media.
 Continuum models for unsaturated flow employ volume averaged fluxes and conservation laws supported by macroscopic transport functions (e.g., the water retention curve (WRC) and the unsaturated hydraulic conductivity function (HCF)), as used in the Richards equation [Richards, 1931] formulation. The weakest ingredient in these widely used models is their reliance on the HCF, K(ψ), a highly nonlinear function of the matric potential ψ, often spanning many orders of magnitude. K(ψ) is not only notoriously difficult to determine experimentally, but its formal definition tacitly assumes standard steady state conditions. Such assumption may introduce inconsistencies for transient flows that regularly invoke and use K(ψ). The proper HCF for describing flows across textural discontinuities and other heterogeneities is poorly defined. In practice, the conceptual pore scale basis for K(ψ) often relies on interpretation of flow through conduits (e.g., bundles of parallel capillaries) further constraining progress and limiting insights into important physical processes underlying unsaturated flow [Lenormand et al., 1983; Blunt and Scher, 1995; Or and Tuller, 2000].
 The main objective of this study is to adapt concepts from the foam drainage equation formalism to modeling unsaturated flows in porous media with applications to internal drainage dynamics resulting in the so-called field capacity. Additionally, we will lay the basis for using the flow geometry explicit in the foam equation to describe flow pathways across soil layers and across other textural contrasts. The paper is organized as follows: the basics of the foam drainage equation are presented with the specific adaptation to flow resistances along solid channels; next, we develop and apply approximate solutions for internal drainage and compare results with field capacity predictions based on the Richards equation; we implement the predictions for drainage dynamics in well-controlled column studies; and finally, we propose a foam drainage equation heuristic framework for explicit consideration of unsaturated flows and viscous resistances emerging from adjustment of flow pathways across textural contrasts (Table 1).
Table 1. Nomenclature and Primary Symbols Used in the Derivations
Capillary channel liquid cross section
First derivative of A with regard to time
First derivative of A with regard to elevation
Second derivative of A with regard to elevation
Liquid cross section in coarse sand
Liquid cross section in medium sand
A constant for the Plateau border geometry of foam C=√(√3-π/2)
Notational prefactor in equation (13)b = (σ√F/ρg)2
Geometrical resistance factor
Acceleration due to gravity
Index of the sand layers
Constant related to the sharpness of textural change
Meniscus height with minimum interface curvature
Radius of curvature (function of capillary head ψ)
Flow velocity in the coarse sand layer
Flow velocity in the medium sand layer
Capillary channel corner angle
Nondimensional flow resistance for corner angle α
Apparent liquid viscosity: η*= ε(α)η
Soil water capillary head
Liquid surface tension
2. The Soil Foam Drainage Equation (SFDE)
 Foams are formed by injection or entrainment of air in liquid forming dense packing of bubbles [Verbist et al., 1996; Weaire et al., 1997]. At the meeting of three neighboring bubbles capillary channels form (known as Plateau borders), and a vertex forms at the meeting of four bubbles (Figure 2). The liquid content in the foam may vary by expansion of the channels when liquid is added (or subsequently by drainage and bubble growth and consolidation). The initially high liquid content in freshly formed foam may drain through the capillary channels that, in turn, gradually reduce their cross section until capillary and gravity forces are balanced. The foam drainage dynamics and the resulting liquid content equilibrium profiles are described by the foam drainage equation [Weaire et al., 1997; Koehler et al., 2004]:
 The foam drainage equation describes explicitly the evolution of capillary channel cross section A(z, t), which for well-established Plateau border geometry, defines simultaneously capillary curvature through the parameter C, and the resistance to viscous flow with viscosity η, surface tension σ, density ρ, and acceleration of gravity g through the geometrical resistance factor f. We defer detailed explanation and derivation for the soil foam drainage analogous equation next.
 Consider an element of draining channel with liquid area A(z, t) and thickness dz (with z the vertical coordinate pointing downward—see Figure 2). The liquid area A(z, t) is a function of the channel's spanning angle α and the (local) interface radius of curvature r(ψ) which in turn is determined by the local matric potential ψ (or capillary pressure) according to:
with geometrical factor F(α) for corner angle α given by [Tuller et al., 1999]:
 Ignoring inertial effects, we may consider a simple form of balance on a vertical channel element A(z, t)dz where gravitational (Fg) driving force is balanced by capillary (Fc) and viscous (Fv) forces:
where, for notational simplicity, A(z, t) is expressed as A, Pa is the pressure in the gas phase, η* is an apparent viscosity that combines liquid viscosity and channel (geometrical) resistance, and u is the drainage velocity. Note that we have used equation (2) to define . The apparent viscosity, η*, is expressed in the form:
with ε(α) is a nondimensional flow resistance as function of corner angle originally derived numerically by Ransohoff and Radke  and later expanded analytically by Zhou et al. , and was subsequently incorporated into the estimation of soil unsaturated hydraulic conductivity function by Or and Tuller . For illustration, the value of viscous resistance for capillary flow in a square corner is ε(90°) = 91, and for triangular corner: ε(60°) = 34.
 The force balance may be rearranged to yield an expression for drainage velocity as a function of liquid cross-section area A:
 The drainage velocity may be substituted into the continuity equation:
to yield the foam drainage equation [Weaire et al., 1997]:
Verbist et al.  expanded the derivation to consider effects of random channel orientation (relative to the assumed vertical), leading to a corrected estimate of η* in equation (6) [η* = 3ε(α)η]. Equation (9) which is the corrected estimate of η* is the proposed soil foam drainage equation (SFDE). In a study focusing on imbibition into closed-end square capillary via corner flow, Dong and Chatzis  formulated and solve the transient imbibition problem using similar formalism as presented in equation (9) for constant curvature meniscus.
 An important feature of the SFDE is the direct link between drainage dynamics and liquid cross section (proportional to the water content) without invoking a hydraulic conductivity function (of course, information on viscous dissipation require certain geometrical simplifications as outlined above).
 In the following, we illustrate potential applications of the SFDE to two classical problems in unsaturated flow in soils: (1) using the SFDE to describe the dynamics associated with attainment of field capacity for prescribed (and simple) boundary condition; (2) for interpretation of outflow dynamics from laboratory columns. We also propose a preliminary framework for estimating effective hydraulic resistance for unsaturated flow across sharp textural contrasts (or layers of different media) under (quasi) steady conditions.
3. Linking the SFDE with Soil Parameters to Estimate Field Capacity Dynamics
 The concept of field capacity (FC) originated in agriculture to define the water content in a soil profile after the internal drainage following a heavy rainfall or a large irrigation becomes negligibly small. The attainment of nearly constant water content within a day or two defines the effective water storage for plant use (the so-called plant available soil water). The FC concept tacitly assumes separation of time scales for rapid drainage (1–2 days) and for plant water use (weeks), and is typically invoked to define a reference state and initial conditions of hydrological simulations involving plant water uptake. In the context of SFDE it is instructive to recall Richards et al.  description “field capacity is the moisture content of a soil 2 or 3 days after a heavy rain or irrigation when downward drainage has reduced the moisture content of the soil and the thickness of the moisture films to such an extent that the unsaturated permeability is no longer appreciable and further downward drainage of water is negligible.” Veihmeyer and Hendrickson  defined field capacity as the amount of soil moisture or water content held in soil after excess water has drained away and the rate of downward movement has materially decreased.
 A related phenomenon is the dynamics of aquifer yield following rapid drawdown of a water table. Similar to FC, the process involves gradual slowing down of internal drainage that, in turn, determines the amount of water extracted from the previously saturated aquifer above the water table [Acharya et al., 2012].
 The SFDE was implemented under the following assumptions and boundary conditions: a single vertical angular corner at equilibrium with a meniscus at height z0 above a reference was abruptly drawn down to height z1, and we seek a description of the flux rate and volume drained to the new equilibrium. For hydrostatic equilibrium |ψ| = z, hence:
 Taking the derivatives of the velocity expression in the SFDE (equation (9)) yields:
 We may use the relation in equation (10) (the hydrostatic profile approximation) to express A(z, t) as a function of the vertical coordinate z:
 For notational convenience, we denote the prefactor in equation (11) as and define another constant resulting in a simpler and explicit expression for equation (11) as:
 Substituting the explicit derivatives Az = −2bz−3 and Azz = 6bz−4 into equation (13), we obtain a closed-form expression for the SFDE in terms of z(t) (under the hydrostatic assumption):
 We may develop approximations for the rate of change in capillary channel cross section (saturation degree) in response to abrupt change in the curvature of the bottom meniscus by an amount equivalent to lowering the reference level by a prescribed distance, δ. The resulting change in liquid cross section in the channel at a level z is expressed by .
 The total drainable volume for a channel section of length (L − l), where l is the meniscus height with interface curvature corresponding to minimum curvature (e.g., air entry value for the capillary or the porous medium) and L is an arbitrary height (preferably a height at which the curvature does not vary too much with δ, which may correspond to residual water content for the medium) is easily computed from the initial and final equilibrium profiles. The drainable section length (L − l) may be limited by the distance to the soil surface for shallow water tables. In the case of the applications under interest, namely, attainment of field capacity or aquifer yield estimation, the drainable volume is approximated by integrating the difference between initial and final equilibrium profiles of liquid filled channel cross section (hatched area in Figure 3):
 Finally, we may approximate the associated drainage dynamics by considering the average drainage rate for abrupt variation in menisci curvature by (z + δ), , integrating the right-hand side of equation (14) along the channel segment (L − l) according to
 The time required for draining the volume element expressed in equation (15) at the mean rate estimated using equation (16) is simply: . This approximation enables representation of the drainage flux (δ) as a function of time t(δ) for a series of values of δ from 0 to L. To evaluate the utility of the derivations above, we compared the approximation with an analytical solution of Nachabe  for drainage flux in different soils for certain initial soil wetting to a prescribed depth. The comparison results depicted in Figure 4 were obtained based on the following assumptions and parameters:
 1. The limits of integration l and L were treated as fitting parameters to match calculated drainage rates by Nachabe's expression. We observed that the value of l is related to air entry value, however, more studies are needed to ascertain the exact relation.
 2. The number of channels per cross section (N) was also considered as a fitting parameter. As a first approximation, N can be estimated from the ratio of areal water content to channel liquid filled cross section at the minimum curvature A(z = l).
 3. Corner angle was maintained constant at 60°. We noted that the results were insensitive to a range of corner angle values.
 We thus varied values of l, L, and N to obtain a reasonable match with Nachabe's calculations without invoking automatic fitting procedure. The number of channels (Plateau borders) in the cross section fitted to drainage dynamics from sand was N = 1.0 × 108 and for sandy clay N = 3.0 × 108 (approximately 1/4 and 30 times the values from first approximation obtained by dividing areal water content by channel area at air entry value for sand and sandy clay, respectively). The overall agreement in terms of flux absolute values and temporal dynamics is encouraging and suggests that the proposed SFDE approximation captures the salient features of internal drainage in real porous media and the approximations proposed herein are reasonable. It is interesting to note that, while the definition of Richards et al.  regarding field capacity is valid for the sand, it is far from corresponding to the 20 days or so needed to the sandy clay to reach a negligible drainage flux.
3.1. Application of the SFDE for One-Step Outflow Experiments
 The relatively good agreement obtained by using the SFDE to describe drainage flux dynamics under natural (gravity) gradient toward attainment of field capacity inspire confidence in application of the expressions for describing the dynamics of single step and multistep outflow laboratory experiments. These well-controlled experiments are often used for the estimation soil hydraulic parameters by inverse methods [Parker et al., 1985]. We reanalyzed the experimental results of Wildenschild et al.  focusing on the data for one-step experiment in Lincoln sand for two different pressure steps (0–125 mbar and 0–250 mbar). The SFDE fitting parameters to outflow measurements depicted in Figure 5 required l ≈ −0.3 m (close to air entry value, see Figure 3 of Wildenschild et al. ) and the lower pressure boundary parameter L = −1.3 m marking the dry end of the characteristic curve for Lincoln sand. The number of channels (Plateau borders) in the cross section was slightly different for the two pressure steps ranging from N = 1.3 × 107 for the low large pressure step (0–250 mbar) to N = 1.35 × 107 for the 0–125 mbar pressure step. The discrepancy and sensitivity of the overall fit to values of l and L reflect of our limited understanding of the proper strategy for parameter selection and interpretation, and how should one implement known macroscopic constraints to optimize SFDE parameters (l, L, and N). At this exploratory phase of proposing this new framework, we limit the discussion to identification of these key parameters that require a more systematic exploration using different soils, initial, and boundary conditions.
3.2. Formalism for Unsaturated Hydraulic Resistance Across Textural Contrasts
 A topic of significant theoretical and practical importance is the quantification of unsaturated flows across layers or textural discontinuities in the soil profile that may have very different transport properties. The particular capillary behavior at such interfaces has been used as a basis for designing capillary barriers and flux diversion for waste isolation [Ross, 1990; Oldenburg and Pruess, 1993]. Additionally, textural interfaces have been proposed as a means to disrupt capillary continuity and suppress evaporation [Shokri et al., 2010]. Such textural contrasts present a challenge to numerical models whereby the effective unsaturated hydraulic conductivity governing flow across a textural interface is estimated as a linear or geometric combination of the domains own hydraulic conductivities [Assouline and Or, 2006]. Various homogenization theories that preserve certain global behavior of the system have been proposed and implemented [Neuweiler et al., 2012]. Nevertheless, such averaging approaches offer little insight into the origins of the flow resistance and continuity of flow pathways. In this section, we propose using the SFDE formalism to clarify geometrical and interfacial adjustments required for maintenance of flow across hydraulically connected domains with sharp textural contrast.
 We build on the analysis of Cox et al.  originally developed for quantifying flow through nonuniform foam consisting of regions with bubbles of different sizes. Cox et al.  defined a general continuity foam equation that accounts for spatial and temporal variations in the number of capillary pathways N(z, t), such variations corresponding in the soil analogue to abrupt changes in grain sizes (and associated number of channels per cross-sectional area):
 Considering N(z) only (no temporal variations in N), expansion of equation (17) recovers the original foam equation (for a single channel) and a correction resulting from the variation in the number of channels, N:
 The geometrical picture at the interface for steady flow (Figure 6) dictates geometrically explicit continuity as shown above. The requirement of interfacial curvature (capillary pressure) continuity across the textural interface limits the range of possible outcomes in terms of mean velocity, water content, and interfacial curvature above and below the interface. Certain scaling relationships could be invoked in analogy with foam flow, for example, liquid cross section ratio is expected to be proportional to the squared mean grain size ratio [Cox et al., 2000]:
where Ai denotes the liquid cross section and Di is the mean grain size of layer i (i = 1, 2). More studies are needed to establish the validity of this ratio for porous media, and this scaling would certainly be questionable for porous media with significantly different pore shapes (e.g., channel corner angles), and during flow conditions where capillary adjustment for flow accommodation may exert influence as in the example from Yeh and Harvey  illustrated below. A more straightforward relationship is the ratio of the number of channels across the textural contrast. This ratio is related to the pore forming number of grains that, in turn, is proportional to mean grain size and porosity [Wu et al., 1993] assuming that the coordination number and other topological parameters are similar across the interface:
where φ denotes the porosity of each layer. Because natural textural interfaces are seldom abrupt, one may invoke a certain transition function to provide continuity across the textural interface. One such function was proposed by Cox et al.  for flow across foam interfaces with abrupt change in bubble sizes
where u1 and u2 are the mean channel velocities in the two “layers,” with the textural interface located at z0, and the parameter k defines the “sharpness” of the transition (k may be derived from solution of the steady flow problem). The proposed adjustment for a sharp interface (equation (21)) was derived for foam based on the linearized steady state drainage solution of Verbist et al. [1996, equation (8)]. The experimental results of Yeh and Harvey  (discussed next) clearly show emergence of such transition zone across the sharp interface during steady flow. For illustration of some of the concepts presented above, we use the experimental results of Yeh and Harvey  that provide some interesting insights on steady flow behavior across textural contrasts. The experiment was conducted in a column filled with alternating layers of coarse and medium sand with steady state water fluxes of different rates established and capillary pressure measured in the layers (using tensiometers). The mean sand grain diameters for the medium and coarse sands were 0.4 and 0.7 mm, respectively. The bulk density of the medium sand was 1460 kg/m3 and of the coarse 1500 kg/m3, a similarity mirrored also by the mean porosities of 0.449 and 0.435 for the medium and coarse sands, respectively. Based on equation (20), the estimated ratio between the number of channels in the medium (NM) and the coarse (NC) sand layers is:
 We focus on results from the wetting experiment at high flux rate of 0.0065 cm/s, noting that for steady flow conditions continuity requires:
 At the interface between the coarse and medium sand layers, continuity of the capillary pressure head implies equal curvatures, hence: AC = AM. Consequently, at the interface,
where uM and uC are the mean channel velocities in the medium and coarse sand layers, respectively. Consequently, the flow velocity must adjust by the ratio NC/NM when flow crosses from the coarse to the medium layer and by the ratio NM/NC when it crosses the medium-coarse interface. One can estimate the cross-sectional area, A, and the related capillary head, ψ, that would accommodate the flow velocity, u, within each respective sand layer. Assuming that A at the top of the experimental column corresponds to the measured ψ value by the upper tensiometer, application of the relationships presented above allow the estimate of the distribution of the hydraulic head, Φ, considering elevation z along the column and steady flow, as Φ(z) = z − ψ(z). It is thus possible to reproduce the hydraulic head gradient, ΔΦ/Δz driving the flow, and compare it with the measured gradient during the experiment. The results of this comparison are depicted in Figure 7. The agreement between estimated and measured gradients is reasonable. It is interesting to see that (1) although the expected mean hydraulic head gradient should approach unity (steady flow), it is >1.0 after the coarse-medium interfaces and <1.0 after the medium-coarse interfaces; (2) flow from coarse to medium sand layers requires an increase of Φ while flow from medium to coarse sand causes a decrease in Φ; and (3) the accommodation of the hydraulic head gradient, and consequently, the flow characteristics, to the sharp textural contrast begins before the interface and, according to Figure 7, is completed midway between the interfaces, within the homogeneous sand layers. This indicate that textural interfaces are transition zones whose effects on flow processes may extend well beyond the physical dimension of the textural change.
4. Summary and Conclusions
 The conceptual picture of water filled pores may hinder a more realistic description of capillary-retained water in unsaturated porous media. We adopt an alternative geometrical description of capillary water in crevices behinds curved air-water interfaces that enables derivation of flow equations governing gravity drainage of the retained liquid in a manner similar to foam drainage. Interestingly, such a geometrical picture of water retention and flow was anticipated in the pioneering work of Buckingham . The main result in this paper is a formulation of unsaturated flow based on the adaptation of the soil foam drainage equation (SFDE) that provides self-consistent macroscopic flow representation without requiring hydraulic conductivity as a separate macroscopic function. In this formulation, the viscous resistance to flow emerges from geometrical adjustment of air-water interfaces that confine water pathways through interconnected crevices. The resulting self-regulating flux is a characteristic of the drainage equation and could be used to describe a class of soil processes such as drainage dynamics leading to attainment of field capacity, or interpretation of outflow experiments with prescribed pressure boundary conditions. Additionally, the SFDE offers a means for explicitly accounting for pathways and interfacial adjustments occurring across textural contrasts that are essential for establishing unsaturated flow. Such adjustments may yield new insights into the effective hydraulic conductivity that govern flow across textural discontinuities, and better understanding of the function of capillary barriers used in waste isolation designs. Based on preliminary analysis of drainage processes in coarse-textured soils, the parameter requirements for SFDE application are relatively modest (l, L, and N); nevertheless, more studies are needed to establish these requirements more rigorously. A central and yet unresolved issue pertains to the process and rate by which interfaces attain equilibrium and assume the “foam” geometries postulated in the SFDE [Tallakstad et al., 2009]. Additionally, it is not clear which of the components of the well-researched classical foam equation are transferable to porous media, especially pertaining to establishing links with macroscopic hydraulic conductivity [Lorenceau et al., 2009].
 The authors gratefully acknowledge funding by the German Research Foundation DFG of project (FOR 1083) Multiscale interfaces in Unsaturated Soil (MUSIS), the assistance of Peter Lehmann (ETH Zurich) with Figure 1 and with many helpful discussions, and numerous insightful comments by three anonymous reviewers.