HESS Opinions: Unsaturated infiltration – the need for a reconsideration of historical misconceptions

Briggs (1897) deduced capillary flow from deviation of the equilibrium between capillarity and gravity. Richards (1931) raised capillary flow to an unproven soil hydrological dogma. Attempts to correct the dogma led to concepts of non-equilibrium flow, macropore flow, and preferential flow during infiltration. Viscous film flow is proposed as an alternative approach to capillarity-driven flow during unsaturated infiltration.

1 Introduction

Why does water know/where to flow?/Is the strongest force/dictating the course?/Is the weakest resistance/controlling the distance?/Or is soil hydrology nothing but mental strength/fiddling with mass, time, and length?

(Germann, 2014)

2 How capillary flow got its position in unsaturated porous media flow

The success of terrestrial plants relies, among other phenomena, on the simultaneous supply of oxygen and water to the root tips in the size range of <5 to >50 µm. A mandatory prerequisite is the inferior water pressure relative to the air pressure around the tips. That again is the result of the water's surface tension against the air and its affinity towards the solid particles, resulting in the capillary potential ψ. As a physicist with the US Bureau of Soils, Lyman Briggs (1897) formalized the relationships mathematically and physically. He made use of data from a sand column experiment, reported in Franklin King (1892), 42 in. (1.067 m) high, that was completely water saturated and then left to drain for 40 d. He qualitatively expressed an equilibrium between the gravitational and capillary forces, represented by height and pressure drop across the water meniscus, deducing that moisture content will decrease with height in the column.

Briggs also provides a qualitative description for flow when there is any deviation from this static equilibrium in terms of a capillary gradient and further postulated a variable permeability concept in partially water-saturated soils. He certainly introduced these concepts to Edgar Buckingham, who joined the Bureau of Soils to work under Briggs in 1902. Buckingham further developed these concepts, and his reports on the movement of soil moisture are frequently viewed as the origin of modern concepts of capillary flow (Buckingham, 1904, 1907): he extended a first-order gradient flux law to unsaturated soils as analogous to thermal and electrical fluxes (though does not explicitly mention Darcy's law), and he proposed hydraulic conductivity, diffusivity and the water capacity as quantitative functions of soil moisture. However, “Buckingham did not manage to formulate a clear-cut physical-mathematical flow theory that quickly inspired other soil physicists” (Raats and Knight, 2018, p. 11).

This seminal work at the Bureau of Soils was set back by the Chief of the Bureau, Milton Whitney (who was chief from 1860 to his retirement in 1927). Whitney dismissed King in 1904 because of his belief that soil nutrient status was more important to crop growth than the soil physical properties favoured by Whitney. He also encouraged Briggs to leave in 1905 because his training as a physicist was resulting in the application of “rigid mathematical demonstration” to soil physics that Whitney felt could not be justified (see Landa and Nimmo, 2003). After the departure of Briggs, Buckingham worked under Frank Kenneth Cameron, who was not totally convinced by Buckingham's developments, suggesting that there might be inconsistencies in Buckingham's approach to water and vapour transport. Buckingham's response was not totally convincing, but Cameron decided not to hold back the publication as “you seem to be convinced that it is all sound and safe” (quoted in Nimmo and Landa, 2005). Buckingham's 1907 report was delayed as a consequence until after he had also left the Bureau of Soils to join the National Bureau of Standards in August 1906. He never returned to the study of soil physics. He became much better known as the author of the Π theorem for dimensional analysis (Buckingham, 1914), which was only much later applied to soil moisture characteristics (e.g. Miller and Miller, 1956).

It was more than two decades before there was further significant progress with Lorenzo K. Richards' (1931) paper “Capillary Conduction of Liquids Through Porous Mediums”, where he presented the three-dimensional form of the second-order convection–diffusion equation, which bears his name. Note that, despite the comment of Raats and Knight (2018) quoted above, all the elements of the Richards equation were already present in the report of Buckingham (1907), but there he did not combine them into a single flow equation. In fact, Richardson (1922) had already done so before Richards but outside the context of the soil physics community (see Raats and Knight, 2018). Richards did, however, also present an experimental procedure for the determination of the parameters of his equation, including a demonstration of hysteresis in the soil water characteristics. This combination of theory and experiment based on that experiment made the approach attractive for practical applications.

However, Richards was also instrumental in introducing some misconceptions into soil physics. He authoritatively stated the following:

If there is a steady flow of liquid through a porous medium which is only partially saturated, then the larger pore space containing air and the effective cross-sectional area of the water conducting region is reduced. If these air spaces could in some way be filled with solid, the condition of the flow would be unchanged and the proportionality between the flow and the water-moving force would still hold because Darcey's [sic] law is independent of the size of particles or the state of packing.

(Richards, 1931, p. 323)

This statement reveals the fundamental, but scientifically quite wrong, exclusivity he put on capillarity-driven flow during infiltration. Since then it has been raised to a soil hydrological dogma that we are still confronted with today. In the theory of Richards, increasing the flow rate in a partially saturated porous medium mandatorily requires the increase of the hydraulic gradient by locally either increasing or decreasing capillary potential that, in turn, has to equilibrate in either case with the water content. In Richards' context this is only feasible during infiltration when flow remains sequential: finer pores have to fill before coarser pores are allowed to (Germann, 2018a, b). Indeed, it has been suggested that the Richards' experiment, which relied on the application of air pressure to ensure that the larger pores were empty at each potential and flow rate, was simply the wrong experiment for flows under more natural conditions (Beven, 2014, 2018).

3 Alternative approach to infiltration and drainage

A reconsideration of the ubiquity of Richards' (1931) capillary approach to infiltration and drainage demands a concept that replaces this assumption of dominance by capillarity. In early attempts, Germann and Beven (1981), Beven and Germann (1981), and Germann (1985) successfully approached transient infiltration/drainage with kinematic wave theory that can be applied when gravity is the dominant driving force rather than the gradient of capillary potential, which will be the strongest driver under drier conditions. More recently, Germann and al Hagrey (2008) summarize the features of gravity-dominated viscous film flow¹ during transient infiltration and drainage in the Kiel sand tank: (i) constant velocity of the wetting front, (ii) collapse to atmospheric pressure of capillarity behind the wetting front, and (iii) infiltration and drainage follow the same simple rules of viscous film flow. Moreover, no pore classification or size distribution or consideration of heterogeneity of soil properties is required for the application of viscous film flow, since film thickness, flow velocity, and celerity during both wetting and drying can be related to flux rates.

4 What happens at the soil surface during infiltration?

Infiltration expresses how water arriving from above the ground, usually in the shape of drops, transitions to water flow in the soil below the ground surface. Drops have internal pressures higher than atmospheric pressure. According to Laplace–Young theory, pressures against the atmosphere in drops increase from about 15 to 750 Pa as their diameters reduce from 5 to 0.1 mm. When drops hit the ground surface, depending on their size and Bond number, they will often burst. The remnants then join and form local films at atmospheric pressure. There are two ways for water to continue. Either water has to follow the strongest gradient, as in capillary flow. Richards' (1931) forceful conjecture of the dominance of capillarity during infiltration in unsaturated porous media works well under steady-state conditions. However, increasing the flow rate causes the well-known phenomena of non-equilibrium flow, macropore flow, and preferential flow. Water then follows the way of least resistance, as a viscous film flow. This type of flow occurs under atmospheric pressure, regardless of the thickness and path widths of the films, as Flammer et al. (2001) have demonstrated with acoustic velocity measurements across soil columns. Germann (2018b) also demonstrates how acoustic velocity experiments suggest that preferential flows can proceed faster than capillary gradient-driven fluxes in unsaturated soils.

5 Where are we now?

On the one hand, the Richards' (1931) capillary gradient flow (and implied local equilibration of potentials) continues to be taught and used in models of soil hydrology in a rather dogmatic way. There have been modifications suggested to allow for dual permeability functions near saturation, but these are not really convincing (Beven and Germann, 2013; Beven, 2018). On the other hand, there are a plethora of review articles that discuss the hydro-mechanical inconsistencies with the Richards equation and observations of preferential, non-equilibrium, and macropore flows in both unsaturated and saturated soils (see again Beven, 2018, and references therein). This conflict remains to be resolved but is clearly important for the understanding of both flow and transport. It is, however, the case that there are few ongoing research efforts, either theoretical and experimental, on developing coherent descriptions of preferential flow in soil physics. The author's application of viscous film flow concepts to infiltration in unsaturated soils has been one such effort, and it is hoped that it will be taken on and further developed by others (see, for example, the recent papers of Germann, 2017, 2018a, b; Germann and Prasuhn, 2018; Germann and Karlen, 2016).

In conclusion, Briggs' (1897) convenient and, within his concept, correct definition of capillary flow led, via the work of Buckingham, to Richards' (1931) dogma of the unproven dominance of capillary gradient-driven fluxes during infiltration that is still widely used today. However, as it turns out, while capillary flow relies on the strongest force exerted on water in an unsaturated soil, the weaker force of viscosity can dominate during infiltration as a result of the formation of film flows at and near the soil surface.