## Sunday, November 13, 2016

### The Soil Water Retention Curves

When dealing with soils you are forced to implement mass conservation dependent on two variables, the dimensionless water content, usually named $\theta$ and suction, $\psi$, i.e. The energy contained in a volume of soil per unit mass. Therefore, to solve the budget, you need (at least) to get a new relationship which connects them.  This relation is called soil water retention curve. The plural in the title means that there are many. At least one for any soil type.

In fact,  the  relationship, and precisely $\theta(\psi)$, is dependent on soil types and structure (and some other factor probably, like temperature, organic content etc). It is a statistical quantity, which averages the behavior of many pores, and an ensamble of water injecting/extracting possibilities.
The figure below from Lu (GS) and Godt (GS) book (2013) is a clear visualisation of the problem.

The same Ning Lu, in a recent paper (2015) tried to disentangle the various forces acting on water when in pores, and obtained what is shown below.

As expected, the forces acting are not all of the same type, at varying suction values. At very high suction, adsorption forces act in which single water molecules adhere to soils. When more layers of water molecule add, water constitute  thermodynamics compounds, whose equilibrium is globally determined in between adhesion forces, bulk water weights, surface of water and air gas interactions, and which is usually known as capillarity.
Laws governing capillarity are described by Young-Laplace and  Kelvin laws.  Some insight of the therodynamics of these phenomena (an excellent explanation, indeed) can be found in the first pages of Steudle (2001) review about plant-root suction.
At this stage liquid water seems to, constitute a disconnected phase, while air gas is continuous inside the medium pores.
Increasing the water content water becomes a continuos medium and usual hydrodynamics laws become valid.  A recent review of parameterizations of the soil water retention curves (not particularly deep or brilliant though) is given by Too et al. (2014) that cites other older reviews.

When pressure increase, however, we can have two effect which partially depends on how wetting happens. If wetting happens through some sort of flooding then air can stay trapped in pores and decrease the space available for water. The net effect is  associable to a decrease of porosity. However, when water fills all the space (i.e. the soil is saturated) the soil matrix cannot be considered anymore rigid.

Assume it would be rigid. Then water content could not increase, any pressure applied to the saturated soil would transmit instantaneously through the water volume and water would be expelled where pressure is not applied or there is less pressure in a sort of piston flow.
Instead, because the medium is not rigid, any pressure is transmitted with a certain speed, and pressure waves can be measured. This fact implies that after saturation, the system behaves as porosity increases and, at the same time pressure varies.

From a practical point of view, soil water retention curves can be extended to positive pressure (negative suctions) adding a term which is well known in groundwater literature and is called specific specific storage
These qualitative descriptions do not end the complex phenomenology of water retention curves.

As Nunzio Romano (GS) and coworkers noticed, and Kosugi (1994) before them, soil water retention curves shape depend directly on the pores' distribution. This, however, is not necessarily a unimodal distribution but can be multimodal because of soil structure and soil "disturbances" in form of macropores due to animal or roots decay. In this case soil water retention curves (their integral) can be more complex than expected, as shown in Figure below.

This opens to a series of generalisation, but it would be the topic of some other post (and actually was already the topic of several posts on soil freezing).

References

Kosugi, K. 1994. Three-parameter log-normal distribution model for soil water retention. Water Resour. Res. 30:891–901.

Lu N, Godt JW. Hillslope Hydrology and Stability. Cambridge: Cambridge University Press; 2013.

Lu, N. (2016). Generalized Soil Water Retention Equation for Adsorption and Capillarity. Journal of Geotechnical and Geoenvironmental Engineering, 142(10), 04016051–15. http://doi.org/10.1061/(ASCE)GT.1943-5606.0001524

Romano, N., Nasta, P., Severino, G., & Hopmans, J. W. (2011). Using Bimodal Lognormal Functions to Describe Soil Hydraulic Properties. Soil Science Society of America Journal, 75(2), 468. http://doi.org/10.2136/sssaj2010.0084

Steudle, E. (2001). The Cohesion-Tension Mechanism and the Acquisition of Water by Plant Roots. Annual Review of Plant Physiology-Plant Molecular Biology, 847–877.

Too, V. K., Omuto, C. T., Biamah, E. K., & Obiero, J. P. (2014). Review of Soil Water Retention Characteristic (SWRC) Models between Saturation and Oven Dryness. Open Journal of Modern Hydrology, 04(04), 173–182. http://doi.org/10.4236/ojmh.2014.44017