Catchment travel time distributions and water flow in soils

Catchment travel time distributions and water flow in soils

Abstract
[1] Many details about the flow of water in soils in a hillslope are unknowable given current technologies. One way of learning about the bulk effects of water velocity distributions on hillslopes is through the use of tracers. However, this paper will demonstrate that the interpretation of tracer information needs to become more sophisticated. The paper reviews, and complements with mathematical arguments and specific examples, theory and practice of the distribution(s) of the times water particles injected through rainfall spend traveling through a catchment up to a control section (i.e., “catchment” travel times). The relevance of the work is perceived to lie in the importance of the characterization of travel time distributions as fundamental descriptors of catchment water storage, flow pathway heterogeneity, sources of water in a catchment, and the chemistry of water flows through the control section. The paper aims to correct some common misconceptions used in analyses of travel time distributions. In particular, it stresses the conceptual and practical differences between the travel time distribution conditional on a given injection time (needed for rainfall‐runoff transformations) and that conditional on a given sampling time at the outlet (as provided by isotopic dating techniques or tracer measurements), jointly with the differences of both with the residence time distributions of water particles in storage within the catchment at any time. These differences are defined precisely here, either through the results of different models or theoretically by using an extension of a classic theorem of dynamic controls. Specifically, we address different model results to highlight the features of travel times seen from different assumptions, in this case, exact solutions to a lumped model and numerical solutions of the 3‐D flow and transport equations in variably saturated, physically heterogeneous catchment domains. Our results stress the individual characters of the relevant distributions and their general nonstationarity yielding their legitimate interchange only in very particular conditions rarely achieved in the field. We also briefly discuss the impact of oversimple assumptions commonly used in analyses of tracer data.

1. Introduction
[2] It has long been the case that hydrologists have been content with the analysis and prediction of hydrographs. This is (still) a challenging problem given the limited information content of hydrological measurements. However, the hydrograph does not provide a full picture of the hydrological response of catchments. A full hydrological theory of catchment response needs to be able to treat the analysis and prediction of travel time distributions and residence time distributions in both unsaturated and saturated zones (and surface runoff) in understanding variations in water quality from point and diffuse sources [e.g., McDonnell et al., 2010; Beven, 2010]. This raises, however, an interesting issue. Travel and residence times on hillslopes will be strongly controlled by water flow processes in the soil and regolith. The details of those processes, including the heterogeneities of soil properties, preferential flow pathways and bypassing, details of root extraction etc., are essentially unknowable using current measurement techniques, including modern geophysics. Such details, however, my be important in the response of the hillslopes [see, e.g., Beven, 2006, 2010]. One way of learning about the bulk effects of such complexity at the hillslope and catchment scales is the interpretation of tracer observations to provide information about Lagrangian velocity distributions. This information differs from that given by the hydrograph response because of the differences between the mechanisms controlling wave celerities and water velocities. This paper will demonstrate that the interpretation of tracer information needs to become more sophisticated and, in particular, to consider more explicitly the variation in time of travel and residence time distributions, precisely defined in what follows.

[3] There are dynamic models that deal with both flow and transport that have been developed primarily within the groundwater field. Such models can be purchased with complete graphical interfaces and, as well as flow and transport calculations, and be linked to chemical reaction codes. The theory on which such models are based is, however, restrictive. It (mostly) assumes that flow is Darcian and dispersion is Fickian and requires spatial distributions of flow and transport parameters to be specified. Experiments and theory suggest that those parameters are both place and scale dependent [e.g., Dagan, 1989]. The patterns of effective parameter values are not easy to either specify a priori or infer by calibration. In addition, the problem of groundwater transport has often been simplified, for convenience in obtaining analytical solutions, to transport and dispersion in steady flows in appropriately defined stream tubes [e.g., Destouni and Cvetkovic, 1991; Cvetkovic and Dagan, 1994; Destouni and Graham, 1995; Gupta and Cvetkovic, 2000, 2002; Lindgren et al., 2004, 2007; Botter et al., 2005; Darracq et al., 2010]. While applicable to regional groundwater systems, this approach is not well suited to the type of catchment hillslopes with relatively shallow flow pathways that is of interest here where the flow system may include transient saturation, macropores and other types of small‐scale preferential flows [Beven and Germann, 1982; Hooper et al., 1990; Peters and Ratcliffe, 1998; Burns et al., 1998, 2001; Freer et al., 2002; Seibert et al., 2003].