## 1. Introduction

[2] Since its inception in the landmark monograph by *Buckingham* [1907], 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 [1907] 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.

[3] *Verbist et al*. [1996] 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*. [1997] 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.

[4] 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].

[5] 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).

Symbol | Quantitiy | Dimensions |
---|---|---|

A(z, t) | Capillary channel liquid cross section | [L^{2}] |

A_{t} | First derivative of A with regard to time | [L^{2}T^{−1}] |

A_{z} | First derivative of A with regard to elevation | [L] |

A_{zz} | Second derivative of A with regard to elevation | [−] |

A_{C} | Liquid cross section in coarse sand | [L^{2}] |

A_{M} | Liquid cross section in medium sand | [L^{2}] |

C | A constant for the Plateau border geometry of foam C=√(√3-π/2) | [−] |

D | Mean sand grain size | [L] |

F(α) | Geometrical factor for corner angle α | [−] |

F_{c} | Capillary force | [MLT^{−2}] |

F_{g} | Gravitational force | [MLT^{−2}] |

F_{v} | Viscous force | [MLT^{−2}] |

K(ψ) | Soil hydraulic conductivity function | [LT^{−1}] |

L | Maximum height of integration | [L] |

P_{a} | Air pressure | [ML^{−1}T^{−2}] |

N | Number of flow channels | [−] |

N_{C} | Number of flow channels in coarse sand | [−] |

N_{M} | Number of flow channels in medium sand | [−] |

V | Drainable volume | [L^{3}] |

V_{t} | Drainage rate | [L^{3}T^{−1}] |

a | Notational prefactor in equation (13) a = σ^{2}F/ρg | [MT^{−2}] |

b | Notational prefactor in equation (13) b = (σ√F/ρg)^{2} | [MT^{2}L^{−1}] |

f | Geometrical resistance factor | [−] |

g | Acceleration due to gravity | [LT^{−2}] |

i | Index of the sand layers | [−] |

k | Constant related to the sharpness of textural change | [−] |

l | Meniscus height with minimum interface curvature | [L] |

r(ψ) r(h) | Radius of curvature (function of capillary head ψ) | [L] |

t | Time | [T] |

u | Flow velocity | [LT^{−1}] |

u_{C} | Flow velocity in the coarse sand layer | [LT^{−1}] |

u_{M} | Flow velocity in the medium sand layer | [LT^{−1}] |

z | Elevation | [L] |

α | Capillary channel corner angle | [−] |

δ | Height increment | [L] |

ε(α) | Nondimensional flow resistance for corner angle α | [−] |

φ | Soil porosity | [−] |

Φ | Hydraulic head | [L] |

ρ | Liquid density | [ML^{−3}] |

η | Liquid viscosity | [ML^{−1}T^{−1}] |

η^{*} | Apparent liquid viscosity: η^{*}= ε(α)η | [ML^{−1}T^{−1}] |

ψ | Soil water capillary head | [L] |

σ | Liquid surface tension | [MT^{−2}] |