Nonpoint source transport models from empiricism to coherent theoretical frameworks

Abstract

Basin-scale transport of reactive solute species is studied through a class of stochastic models, termed mass response functions, which incorporate simplified concepts of chemical, physical or biological nonequilibrium kinetics into the theory of the hydrologic response. Here we examine the development of the field since its inception dealing with empirical approaches, a subject to which Giuseppe Bendoricchio actively contributed, and conclude that a coherent theoretical framework now exists that allows to address large-scale transport problems for catchment studies where geomorphological and hydrological complexity is not simply ignored.

Introduction

Nonpoint source transport driven by hydrologic runoff has been a longstanding interest of Giuseppe Bendoricchio, a scientist, a civil servant and a friend whom we sorely miss. It seems appropriate, therefore, to assess where we stand on matters of theory and application, while critically reviewing the development of the field.

Beppe started to study nonpoint source pollution models in the context of a field campaign carried out at a pumped outlet discharging into the Lagoon of Venice (Zuccarello, Italy) (Zingales et al., 1980, Zingales et al., 1981). Critical to his continued interest was the evaluation of the annual nutrient loads delivered to the lagoon from its mainland, an issue quite relevant to the (then, and even now) ongoing debate over the most suitable scheme for preservation of Venice and its environment (see e.g. Bendoricchio et al., 1985). That exercise requires modeling of the processes that originate nonpoint loads to hydrologic runoff. Modeling the individual role and the mutual interactions of hydrological and chemical processes leading to nonpoint load generation was in its infancy at that time. The best approach available was the so-called screening model (see e.g. Haith and Tubbs, 1981), in which one would essentially multiply runoff volumes from homogeneous areas (entailing a complex discussion on average patterns of rainfall in space and time, and on soil use) by an average flux concentration in the runoff, however abstract the concept might be. Estimates of loads to the lagoon of Venice in that context were indeed based on screening models (Rinaldo and Rinaldo, 1982, Bendoricchio et al., 1985, Rinaldo and Marani, 1986). Somewhat surprisingly, given the gross approximations involved, they yielded the correct order of magnitude of the phenomena at least for nitrates, as it later turned out through an elaborate field measurement campaign (Collavini et al., in press, Zonta et al., in press). It was clear from the onset, however, that a different approach would have been needed towards a real operational understanding of the transport processes relevant to the proper spatial and temporal scales: basin- and catchment-scales in space, from tens to hundreds of km possibly larger than any correlation scales of heterogeneous properties; and time scales comparable with the characteristic time of the hydrologic response. Notice that distributed models where detailed description of the relevant processes was enforced for basin units, were developed in parallel, following approaches already in use in the hydrological practice (e.g. Donigian and Crawford, 1976, Sonzogni et al., 1980, Knisel, 1980, Novotny, 1986; for a review, see Novotny and Chesters, 1981). These approaches showed, however, a rather marked difficulty in being used for predictions owing to the extraordinarily large number of parameters and processes to be controlled.

Notice that much of the fortune of screening and simplified empirical correlation models rested simply on their integral nature, owing to the fact that usually loads were operationally estimated on an annual basis (i.e. at time scales much larger than those of the hydrologic response of any of the catchments involved). Conceptual models coupling the hydrologic response with a quality component aimed at chemical or physical nonequilibrium models were developed at that time (Bendoricchio and Rinaldo, 1981, Bendoricchio and Rinaldo, 1986, Zingales et al., 1984). These models typically entailed cascades of well-mixed reactors (e.g. Jørgensen and Bendoricchio, 2001), in a somewhat underestimated mathematical scheme generalizing the so-called Nash model in hydrology (Rinaldo and Rodriguez-Iturbe, 1996).

Later work (Rinaldo and Marani, 1987; Rinaldo et al., 1989a, Rinaldo et al., 1989b) explicitly brought into the picture the formulation of reactive transport by travel- and life-time distributions, and connected to current theories of the hydrologic response, the so-called geomorphologic unit hydrograph and its extensions (e.g. Rodriguez-Iturbe and Valdes, 1979, Rodriguez-Iturbe et al., 1979, Gupta et al., 1980, Rinaldo and Rodriguez-Iturbe, 1996, Rodriguez-Iturbe and Rinaldo, 1997). The basic assumption therein was to consider the transport states well-mixed both in mobile and immobile phases owing to an embedded averaging of several sources of heterogeneities. The approach was termed after the transfer functions, MRFs from mass response functions, that characterize the (unit) response of the hydrologic system.

Parallel to the above developments, and largely independently of them, the Lagrangian formulation of transport by travel time distributions had been studied as eminently suited to describe the dispersion of reactive solutes through different hydrologic heterogeneous media. Exact solutions now exist under a few simplifying assumptions, (Cvetkovic and Dagan, 1994, Cvetkovic and Dagan, 1996) which allowed applications applied to relevant case studies related to basin hydrology (e.g. Destouni and Graham, 1995, Simic and Destouni, 1999, Gupta and Cvetkovic, 2000, Lindgren et al., 2004) for the case corresponding to a ‘localized’ solute injection. Mathematically, this corresponds to a Dirac-delta distributed initial concentration in mobile phases (the hydrologic runoff).

However, many situations of environmental interest are concerned with rainfall-driven nonpoint source transport where flushing of solutes originates mainly from matter stored within immobile phases (e.g. hillslopes in unchanneled transport volumes; bed sediments in channeled transport volumes; dead end zones in both; agricultural soils where fertilizations are routinely applied). The carrier flow is that driving the hydrologic response and the typical spatial scales embed the geomorphic heterogeneities characteristic of the fluvial system—the ‘basin’ scales.

The above conditions seem hardly tackled by Lagrangian steady-flow models. Current Lagrangian models describe the spreading of a solute through heterogeneous media through a Lagrangian variable, τ, that represents the travel (or residence) time of a labeled water particle to an absorbing barrier corresponding to a given (and arbitrary) initial position within the control volume. The elegant writing of the Eulerian mass balance along a single streamline in a Lagrangian form (Cvetkovic and Dagan, 1994) allows for exact solutions for the solute flux concentration expressed in terms of the residence time distribution of the carrier flow. In particular, the solution obtained for zero initial conditions both in the fixed and in the mobile phases, and forced by an impulse of mass only in the mobile phase, has been employed for modeling basin scale transport of natural tracers or pollutants both in the surface (Gupta and Cvetkovic, 2000, Gupta and Cvetkovic, 2002) and in the subsurface water systems (Destouni and Graham, 1995, Simic and Destouni, 1999, Fossereau et al., 2000, Cvetkovic and Haggerty, 2002). A scheme possibly interpreting transitions from the subsurface to stream systems has also been proposed within that framework (Lindgren et al., 2004).

A feature of the existing exact solution that needs be carefully considered is the long-term removal of all mass injected, whether in mobile or immobile phases and however retarded by local equilibrium or chromatographic effects, owing to the basic assumption of steady-state carrier flow. Most of the compounds transported by the hydrological waters (like nutrients, pesticides or heavy metals), however, originate from the leaching of soluble matters stored within fixed phases in contact with intermittent carrier flows, particularly within hillslopes where the dynamics of saturated/unsaturated soil moisture shifts the driving forces from pressure gradients to gravity. Patterns of precipitation in space and time render the matter even more complex, adding further mathematical demands to the tools at hand. In fact, because rainfall is usually carrying negliglible concentrations of solutes, a ‘local’ injection in mobile phases at the inlet of each streamtube must be replaced by a spatially-distributed initial distribution of mass within the fixed phase of the system. The presence of a non-uniform initial distribution of solute within the fixed phase may affect significantly the resulting breakthrough curves at the control section of the system where arrival time distributions of the carrier and of solute matter are determined. Therefore, the development of a general solution of the reactive steady state problem holding for general (non-zero and arbitrary) initial conditions is argued to be of some hydrological interest (Botter et al., 2004). Thus, it is noteworthy that the practical applicability of the steady-state flow model of Cvetkovic and Dagan (1994) (and related papers) for large-scale nonpoint source transport has been assessed by Botter et al. (2004). Of particular interest is the fact that comparative analyses of the Lagrangian steady-state model and of ‘well-mixed’ MRF models yield a clear range of practical interest where the two are indistinguishable. This fact, anticipated (Rinaldo et al., 1989a) in a rather obscure (though correct) fashion as the case where the maximum eigenvalue of the eigenfunction expansion solving the passive transport case being much larger than the dimensionless reaction coefficient, yields to extended uses of MRFs for hydrologic purposes. Hence, where mass exchange reasonably depends mostly on contact time between fixed and mobile phases (easily identified with the residence time distributions), one expects MRFs to hold.

This note, sadly dedicated to Beppe's memory, shows the results of the application of a few of the theoretical results mentioned above in the case of the Dese river basin, a tributary of the lagoon of Venice, to which he dedicated much field and theoretical work in his brief and intense life-time.