Tuesday, July 9, 2013

Hillslope hydrology from the point of view of Richards equation

This is the lecture I gave at the second Summer school on Water Resources that I co-organised. The lecture was recorded in video and should be made available soon, together with the other lectures.
The lecture can be considered an expansion of the talk I gave at Ezio Todini's symposium with more details and, hopefully, more clear explanations. The presentation is, as usual, available on slideshare, and you can get it clicking on the image below.

The presentation mainly use a paper by Cordano and Rigon, 2008 to obtain various degree of simplification of the Richards equation. The scope (especially for what regards the Todini's symposium version) was to show that many approaches currently used derive, in fact, from a simplification of Richards, and there are not very much reasons to shoot to it as unphysical, at least if this is subsequently followed by the use of one of its simplifications.
However, the slides cover also a discussion over the time scales of flows in hillslopes, and the relative timing of vertical and lateral infiltration.

Here it is a short abstract:

The presentation covers Richards equation as applied to Hillslope Hydrology from its foundations. It is said that it assumes mass conservation and  the existence of the Darcy scale at which the soil medium can be treated as a continuum. Then it is specialised using some well known parameterisations (van-Genuchten Mualem), and subsequently is simplified to obtain other equations. In order: the 1-D Richards equation, the Boussinesq equation, the hillslope-storage Boussinesq equation, and finally its stationary approximations.  All of these are used in literature for various purposes, including soil moisture distribution and hillslope stability. The simplifications are based on the assumption that lateral (slope-parallel) flow is slower than slope-normal flow, which is subsequently shown not being necessarily true, true some simulation with a 3-D Richards equation solver. This eventuality is caused by hydraulic conductivity being (in some soils) high variable with water content. Eventually, a conceptual model is built on the knowledge acquired, in order to reduce the computational burden. Lastly some cases are discussed where Richards equation could fail using data from the Panola hillslope. It is shown that the fill-and-spill phenomena can be described properly, and that, on the contrary, the presence of macropores cannot.

The various schemes of simplification have a great effect on the identification of landslides' locations, and, in fact, many of the papers cited (and provided in the blog) are dealing with landslides hazard assessment. 

References cited in the presentation

Beven., K. J., M.J.. Kirkby, A physically based, variable contributing area model of basin hydrology, Hydrological Sciences bulletin-des Sciences Hydrologiques, 24, 1-3, 1979

Buckingham, E. 1907. Studies on the movement of soil moisture. Bulletin 38. USDA Bureau of Soils, Washington, DC.

Casulli, V,  Stelling GS, Semi‐implicit subgrid modelling of three‐dimensional free‐surface flows
International Journal for Numerical Methods in Fluids 67 (4), 441-449

Cordano, E., & Rigon, R. (2007). A perturbative view on the subsurface water pressure response at hillslope scale. Water Resources Research, 1–36.

Cordano, E., & Rigon, R. (2010). A mass-conservative method for the integration of two-dimensional groundwater (Boussinesq) equation. Water Resources Research, 1–24.
Dietrich, W. E., 1989, Slope morphology and erosion processes, in C. Wahrhaftig and D. Sloan (Eds.), Geology of San Francisco and Vicinity, Field Trip Guidebook T105, American Geophysical Union, p. 38-40.

D'Odorico, P., Fagherazzi, S., & Rigon, R. (2005). Potential for landsliding: Dependence on hyetograph characteristics. Journal of Geophysical Research, 110(F1), 1–10. doi:10.1029/2004JF000127

Iverson, R. M., Landslide triggering by rain infiltration, Water Resour. Res., Vol. 36, N0. 7,  1897-1910, 2000

 Lanni, C.; McDonnell, J. J.; Rigon, R., On the relative role of upslope anddownslope topography for describing water flow path and storage dynamics:a theoretical analysis, Hydrological Processes Volume: 25 Issue: 25 Pages: 3909-3923, DEC 15 2011, DOI: 10.1002/hyp.8263

 Lanni C., J. McDonnell JJ, Hopp L., Rigon R., "Simulated effect of soil depthand bedrock topography on near-surface hydrologic response and slope stability" in EARTH SURFACE PROCESSES AND LANDFORMS, v. 2012, (In press). - URL: http://onlinelibrary.wiley.com/doi/10.1002/esp.3267/abstract . - DOI: 10.1002/esp.3267

Lanni C., Borga M., Rigon R., and Tarolli P., Modelling catchment-scale shallowlandslide occurrence by means of a subsurface flow path connectivity index, Hydrol. Earth Syst. Sci. Discuss., 9, 4101-4134, 2013

Montgomery, D. R., and W. E. Dietrich, Where do channels begin?, 1988, Nature, v. 336, p. 232-234.

Mualem, Y., A New Model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res., vol 12, No 3, 1976

Narsilio, G. A., Buzzi, O., Fityus, S., Yun, T. S., & Smith, D. W. (2009). Upscaling Navier-Stokes Equations in porous media: Theoretical, numerical and experimental approach, Computers and Geotechnics, 36(7), 1200–1206. doi:10.1016/j.compgeo.2009.05.006

O'Loughlin, E.M., Prediction of Surface Saturation Zones in Natural Catchments by Topographic analysis, Water resour. Res., vol 22, no 5, 794-804, 1986

Orlandini, S., G. Moretti, M. Franchini, B. Aldighieri, and B. Testa (2003), Path-based methods for the determination of nondispersive drainage directions in grid-based digital elevation models, Water Resour. Res., 39(6), 1144, doi: 10.1029/2002WR001639.

Orlandini, S., P. Tarolli, G. Moretti, and G. Dalla Fontana (2011), On the prediction of channel heads in a complex alpine terrain using gridded elevation data, Water Resour. Res., 47(2), W02538, doi: 10.1029/2010WR009648.

Richards, L.A., Capillary conduction of liquids through porous mediums, Physics 1: 318-333, 1931

Troch P.A., Paniconi, C., van Loon E.E, Hillslope-storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristics response, Water Resour. Res., Vol 39, No 11, 1316, doi:10.1029/2002WR001728, 2003

Tromp-Van Meerveld, H. J., & Mcdonnell, J. J. (2006). Threshold relations in subsurface stormflow: 2. The fill and spill hypothesis. Water Resources Research, 42(2), W02411. doi:10.1029/2004WR003800

M. Th. van Genuchten, A Closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sc. Soc. of America, vol 44, no. 5, 1980

Whitaker, S., The Forcheimer equation: A theoretical development, Transport in Porous Media, October 1996, Volume 25, Issue 1, pp 27-61


  1. It seems that you have a great point of view on hydrology. What are the various fields where it can be used?

    Bruce Hammerson

    Hydraulic Hammers

  2. Hydrology, of course! Soil moisture distribution. Precision agricolture. Shallow landslides. Runoff formation ....