Old water contribution to streamflow: Insight from a linear Boussinesq model

This is a paper by Aldo Fiori one one of the most interesting issues in Hydrology, just appeared in Water Resources Research (Fiori, 2012). I already talk on this issue in the blog one year ago or so. Now Aldo comes back with a new paper which I will start to read avidily.

This is an excerpt from his  introduction, with some references:

"The understanding of the main physical processes which rule runoff generation in catchments is limited by the so-called ‘‘old water paradox.’’ The latter states that a (sometimes significant) fraction of the runoff volume after a rainfall event is pre-event, or ‘‘old.’’ Experiments with passive tracers suggest that most of the water contributing to stormflow is pre-event [Neal and Rosier, 1990; Sklash, 1990; McDonnell, 2003; Kirchner, 2003; Botter et al., 2010], with percentages often close to 75% of the total flow [Buttle, 1994]. This experimental evidence, which seems to invalidate most of the existing rainfall-runoff models, have been explained by means of a few mechanisms [Beven, 2002]. Among the latter is the propagation of pressure waves with high celerity [e.g., Beven, 1981], the capillary fringe-ridging hypothesis [see, e.g., Sklash and Farvolden, 1979 ; Gillham, 1984 ; McDonnell, 1990 ; Cloke et al., 2006; Fiori et al., 2007], and transmissivity feedback or macropore flow [McDonnell and Buttle, 1998]. The issue is still a matter of debate [McDonnell et al., 2010], and the principal physical processes controlling the release of old water and the partition between old and new water are still poorly understood. This problem has a crucial impact on several processes of interest in hydrology, for example, the development of meaningful rainfall-runoff models and the analysis of solute transport in catchments, which is often performed in terms of travel time distribution [e.g., McGuire and McDonnell, 2006]. Among the processes which may control the age of water we point here at the ‘‘potentially under-appreciated importance of old ground- water input to streams,’’ and ‘‘we thus need to have a better understanding of where and when old groundwater inputs are important’’ (both statements by McDonnell et al. [2010])."

On the side of pressure wave, I would add the reference to the work by Rasmussen et al., (2000) which refers to an experiment of percolation through saprolite, and the  paper by Torres et al. (1998) where these pressure waves are seen in the field (see also the Commentary by Torres 2002).

Certainly in producing a retardation in travel times concurs also the slowness of flow in unsaturated conditions (e.g. Lanni et al., 2012a,b which apparently talk about shallow landslides, but, in fact, talk also about hillslopes' residence time) but still they do not explain enough of the very large age of water in streams.

However,  measuring travel times and interpreting them is not all that easy (e.g. Rinaldo et al. 2011, with an important reference to Niemi, 1977), and maybe some measurements should be rethought.

Talking about vague references (to me, obviously), some work by Jean Yves Parlange, on fast propagation of water in soils, could be interestingly related to this topic.  But this is just a stub for future literature investigations (for instance the paper on sound waves referred here should be related to the fast propagation of pressure waves).

In any case, again a lot of stuff to read.

References

Beven, K. (1981), Kinematic subsurface stormflow, Water Resour. Res., 17, 1419–1424.

Beven, K. J. (2002), Rainfall-Runoff Modelling, The Primer, 360 pp., John Wiley, Hoboken, N. J.

Botter, G., E. Bertuzzo, and A. Rinaldo (2010), Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox, Water Resour. Res., 46, W03514, doi:10.1029/ 2009WR008371.

Buttle, J. M. (1994), Isotope hydrograph separations and rapid delivery of pre-event water from drainage basins, Prog. Phys. Geogr., 18(1), 16–41.

Cloke, H. L., M. G. Anderson, J. J. McDonnell, and J. P. Renaud (2006), Using numerical modelling to evaluate the capillary fringe groundwater ridging hypothesis of streamflow generation, J. Hydrol., 316, 141–162.

Fiori, A., and D. Russo (2007), Numerical analyses of subsurface flow in a steep hillslope under rainfall: The role of the spatial heterogeneity of the formation hydraulic properties, Water Resour. Res., 43, W07445, doi:10.1029/2006WR005365.

Fiori, A. (2012), Old water contribution to streamflow: Insight from a linear Boussinesq model, Water Resour. Res., 48, W06601, doi:10.1029/2011WR011606.

Gillham, R. W. (1984), The capillary fringe and its effect on water table response, J. Hydrol., 67, 307–324.

Kirchner, J. W. (2003), A double paradox in catchment hydrology and geo- chemistry, Hydrol. Processes, 17, 871–874.

Lanni C., J.J. McDonnell, L. Hopp, R. Rigon, 2012. Hydrological controls on shallow landslide triggering: the role of soil depth and bedrock topography. Earth Surface Processes and Landforms, in print (see http://abouthydrology.blogspot.it/2012/01/simulated-effect-of-soil-depth-and.html)

C. Lanni, M. Borga, R. Rigon, and P. Tarolli, Modelling catchment-scale shallow landslide occurrence by means of a subsurface flow path connectivity index, Hydrol. Earth Syst. Sci. Discuss., 9, 4101-4134, 2012, www.hydrol-earth-syst-sci-discuss.net/9/4101/2012/doi:10.5194/hessd-9-4101-2012

McDonnell, J. J. (1990), A rationale for old water discharge through macro- pores in a steep, humid catchment, Water Resour. Res., 26 (11), 2821– 2832.

McDonnell, J. J., and J. M. Buttle (1998), Comment on ‘‘A deterministic empirical model of the effect of the capillary-fringe on nearstream area runoff. 1. description of the model’’ by Jayatilaka CJ, Gillham RW (J. Hydrol. 184 (1996) 299–315), J. Hydrol., 207, 280–285.

McDonnell, J. J. (2003), Where does water go when it rains? Moving beyond the variable source area concept of rainfall-runoff response, Hydrol. Processes, 17(9), 1869–1875.

McDonnell, J., et al. (2010), How old is the water ? Open questions in catchment transit time conceptualization, modelling and analysis, Hydrol. Processes, 24(12), 1745–1754.

McGuire, K. J., and J. J. McDonnell (2006), A review and evaluation of catchment transit time modelling, J. Hydrol., 330, 543–563.

Niemi, A. J. (1977), Residence time distribution of variable flow processes, Int. J. Appl. Radiat. Isotopes, 28, 855–860.

Neal, C., and P. T. W. Rosier (1990), Chemical studies of chloride and sta- ble oxygen isotopes in two conifer afforested and moorland sites in the British uplands, J. Hydrol., 115(1–4), 269–283.

Rasmussen, T.C., Baldwin, R.H. Jr.,  Dowd, J. F., and  Williams, A.G., Tracer vs. Pressure Wave Velocities through Unsaturated Saprolite, Soil Sci. Soc. Am. J. 64:75–85 (2000).

Rinaldo, A. Beven, K. J., Bertuzzo, E., Nicotina, L., Davies, J., Fiori, A., Russo D. and G. Botter G., Catchment travel time distributions and water flow in soils, Water Resources Research, vol. 47, p. -, 2011.

Sklash, M. G., and R. N. Farvolden (1979), The role of groundwater in storm runoff, J. Hydrol., 43, 45–65.

Sklash, M. G. (1990), Environmental isotope studies of storm and snowmelt runoff generation, in Process Studies in Hillslope Hydrology, edited by M. G. Anderson and T. P. Burt, pp. 401–435, John Wiley, N. Y.

Torres R, Dietrich WE, Montgomery DR, Anderson SP, Loague KM. 1998. Unsaturated zone processes and the hydrologic response of a steep, unchanneled catchment. Water Resources
Research 34(8): 1865– 1879.

Torres R., A threshold condition for soil-water transport, Hydrol. Process. 16, 2703–2706 (2002) DOI: 10.1002/hyp.5060

A question a day ... about infiltration

One of my colleagues asked this morning the following:  I would like to insert in my model a module for infiltration and redistribution of water in soil. I obviously was thinking to a simple solution. For instance inspired by the paper by Iverson (2000) on which you work times ago. What do you suggest ?

My 2 C:

among the simplest solution I would use the Philip's one, or just a little more complicate, the analytical solution from D'Odorico et al., 2003. That solution derives from a linearization, and in the case you use variable precipitations, you must convolve the kernel solution with them, as indicated in the paper (the same as in the IUH). 

Increasing the complexity of what you could do, next step would be to numerically solve the Richards 1-D equation, and integrate this solution with a simple solution of the water table movement. For a rigorous derivation see the paper Cordano and Rigon, 2008. The model TRIGRS uses more or less this strategy. 

With Emanuele Cordano I also implemented a model for solving the Boussinesq equation with a rigorous and fast method. This could be a solution for the water table movements after infiltration, if you are not satisfied with steady state solutions (http://abouthydrology.blogspot.com/2012/01/solving-boussinesqs-groundwater.html). This last one does not have yet a module for infiltration.

Other would suggest to use Green-Ampt solutions: but I do not see any advantage to use it with respect with the above ones. Perhaps some gain in speed but certainly no gain in clarity. Obviously the simpler the solution, more the parameters must be considered variables to be calibrated ex-post. The assumptions on which they (the parameters) are assumed constant are, in fact,  not really supported by reality, and made for being able to use analytical solutions. 

P.S. - Many in the past (And in the present) use the Curve Number or SCS method. This is a case of pure mimesis of the behavior of infiltration. However, when calibrated, and in certain conditions, the method "works". I hate solutions that just work, but this in particular works, in my opinion for two  reasons: someway it accounts for soil cover (when we usually say infiltration, in reality we have the coupling of interception,   through-flow and,  eventually, infiltration), and, besides, the introduction of an initial abstraction accounts for the fact that the lateral flow is triggered only when there is a little of water table over the bedrock  (http://abouthydrology.blogspot.com/2011/09/on-relative-role-of-upslope-and.html, and http://abouthydrology.blogspot.com/2012/01/simulated-effect-of-soil-depth-and.html)

Simulated effect of soil depth and bedrock topography on near-surface hydrologic response and shallow landslide triggering by Lanni, McDonnell, Hopp and Rigon

We have just submitted a paper  that, looked from a certain perspective, can be thought on the evolution of soil moisture content in presence of variable soil depth. In fact, variable soil depth, jointly with the fact that increases in hydraulic conductivity follows the increase in water pressure, and that hydraulic conductivity itself can often be considered negligible when the soil is unsaturated, delays the formation of a widespread water table in a hillslope. Therefore the effective contributing area above a point of a catchment is usually  not the total upstream area but just a part of it.  This obviously has consequences on the propagation of instabilities along a slope.

The Abstract of the paper:

This paper explores the effect of hillslope hydrological behavior on slope stability in the context of transient subsurface saturation development and landslide triggering. We perform a series of virtual experiments to address how subsurface topography affects the location and spatial pattern of slip surface development and pore pressure dynamics. We use a 3D Darcy-Richards equation solver (Hydrus 3-D) combined with a cellular automata slope stability model to simulate the spatial propagation of the destabilized area. Our results showed that the soil-bedrock interface and in particular, bedrock depressions, played a key role on pore pressure dynamics, acting as an impedance for the downslope drainage of perched water. Filling and spilling of depressions in the bedrock surface microtopography induced localized zones of increased pressure head such that the development of pore-pressure fields—not predictable by surface topography—lead to rapid landslide propagation. Our work suggests that landslide models should consider the subsurface topography in order to include a connectivity component in the mathematical description of hydrological processes operating at the hillslope scale. Quantitative soil- landscape methods combined with physically-based landslide models may improve our ability to predict shallow landslide potential. 

Among the original stuff presented in this paper, there is a tentativ to move away from the concept of instable points to the one of instable regions.  The traditional (simplified) approach based on the infinite slope stability (Ning Lu is discussing it here) is in fact used in modern GIS based program like SHALSTAB (here on ARCGIS or here on opensource GIS) or SINMAP to determine the instabilities of single points, which we try here to generalize a little. 

The paper draft is available here, if you are interested in. A related discussion can also be found in the previous papers by Lanni et Al., 2011 and in the draft by Cordano and Rigon, 2012.