Introduction

In 1955, Miller and Miller created a new avenue for research in soil hydrology when they presented their pioneering concepts for scaling capillary-flow phenomena. Their description of the self-similar microscopic soil-particle structure and its implications for the retention and transport of soil water stimulated many studies to test how well laboratory-measured soil-water retention curves could be coalesced into a single-scale mean function (e.g., Klute and Wilkinson, 1958). Because the results from ensuing tests were not particularly encouraging, except for soils composed of graded sands, their scaling concepts lay idle for several years. At that time, when the pressure-outflow method (Gardner, 1956) and other transient methods for estimating the value of the hydraulic conductivity in the laboratory were still in their infancy, few attempts were being made to scale the hydraulic-conductivity function, owing to the paucity of data from its quantitative observation. It was during that same period that the classic works of Philip (1955, 1957), describing a solution for the Richards equation, shifted attention to infiltration. With field measurements of soil-water properties only beginning to be reported (e.g., Richards, Gardner, and Ogata, 1956), little information was available on the reliability of their measurements or on the variations in their magnitudes to be expected within a particular field or soil mapping unit.

During the 1960s, the development and acceptance of the portable neutron meter to measure soil-water content spurred research into field-measured soil-water properties. Because of its availability, combined with the well-known technology of tensiometry, field studies of soil-water behavior accelerated in the 1970s.

Introduction

Flow and transport processes in natural soils and rocks traditionally have been described by means of partial differential equations (PDEs). These equations generally are taken to represent basic physical principles (conservation and constitutive laws) that operate on some macroscopic scale (theoretical support volume) at which the geologic medium can be viewed as a continuum. The precise nature of this the-oretical macroscopic support scale remains generally unclear, though some derive comfort from associating it in the abstract with a “representative elementary volume” (REV). Unfortunately, the concept of an REV is equally difficult to define without ambiguity and to apply in practice. Flow and transport PDEs are local in the sense that all quantities (parameters; forcing functions, including initial, boundary, and source terms; dependent variables) that enter into them are defined at a single point (x, t) in space–time. Parameters such as permeability, porosity, and dispersivity are generally regarded as macroscopic medium properties that are well defined and thus can be determined (at least in principle) experimentally, and more or less uniquely, at any point x in the flow domain.

In reality, geologic media are heterogeneous and exhibit both discrete and continuous spatial variations on a multiplicity of scales, and therefore it can be anticipated that the flow and transport properties of these media will exhibit similar variations. Indeed, one manifestation of such variations is the observed and well-documented dependence of permeabilities and dispersivities on their scale of measurement (support volume). Figure 13.1 shows two superimposed profiles for hydraulic conductivity K as determined by packer tests in a single borehole that penetrated fractured granitic rocks at the Finnsjön site in Sweden.

Problems of Scale and Regionalization in River Basins

The occurrence of the hydrologic cycle covers a very wide range of space and time scales, and involves physical, chemical, and ecological processes. Therefore, in modeling and making predictions, one is required to understand how various properties and measurements behave under a change of scale. At the spatial and temporal scales of interest in river basins, space–time variability and fluctuations are displayed in input, output, and storage elements of the components of the hydrologic cycle and in their interactions. This variability is part of the physics and in this sense is different from the measurement noise. We can use the term physical-statistical or statistical-dynamical to describe such systems. An understanding of the physics of these systems in the presence of variability and fluctuations through mathematical notions of randomness has been and continues to be one of the central challenges of hydrology and constitutes the main theme of this chapter.

A river basin contains a channel network, as shown in Figure 4.1, and systems of hills on both sides of the channels in the network. Rainfall and/or snowmelt are transformed into runoff, and sediments are eroded over hills, and these in turn are fed into a channel network for their journey toward an ocean. The hydrologic cycle on a hillside involves transformation of rainfall to surface runoff, infiltration through the near-surface unsaturated soils, and evapotranspiration from the soil surface into the atmosphere. The infiltrated water goes into recharging the soil moisture in the unsaturated zone, the aquifers, and some of it also appears as subsurface runoff in a channel network.

The Problem of Scaling of Unsaturated Water Flow

Model studies of water transfer in saturated and unsaturated zones within soils have shown that there is an urgent need to assess the adequacy and usefulness of scale matching for the different flow phenomena involved in the interactions among the atmosphere, the land surface, vadose zones, and aquifers. Correct specifications of soil hydrologic processes for both small and large scales are directly dependent on the possibility of parameterizing soil-water fluxes for scales compatible with the grid size of the field scales involved. The difficulty of parameterization for soil-water dynamics arises not only from the nonlinearity of the equations for saturated and unsaturated flows but also from the mismatch between the scale of field measurements and the scale of model predictions. All of the standard measurement methods, such as the entire range of techniques to monitor in situ soil-moisture content and soil-water pressure, provide only point information and highlight the underlying variability in the hydrologic characteristics of soils. The efficiency of parameterization of soil characteristics at different scales depends on clear definitions of the functional relationships and parameters to be measured.

One of the first studies to apply the concept of scaling to the theory of water flow in unsaturated soils was that by Miller and Miller (1955a, b, 1956). They derived scale factors for soil-water properties such as soil-water pressure, hydraulic conductivity, diffusivity, and soil-water transport coefficients by the use of the similar-media concept.

Introduction

This chapter discusses some research problems associated with the scaling of land-surface hydrologic processes involved in the land-surface water-and-energy budget. As pointed out by Dubayah, Wood, and Lavallee (1996), the term “scaling” has come to have multiple definitions, depending not only on the general discipline (e.g., hydrology, meteorology, geography, physics) but also on the application within a discipline. Dubayah et al. (1996) refer to a process as exhibiting scaling “if no characteristic length scale exists; i.e. the statistical spatial properties of the field do not exhibit scale-dependent behavior.” Bloschl and Sivapalan (1995) refer to scaling as the transfer of information between different spatial (or temporal) lengths. The “transfer of information” may consist in mathematical relationships, statistical relationships, or observations describing physical phenomena. This definition is consistent with that of Dubayah et al. (1996), albeit more qualitative. The term “upscaling” is often used by the remote-sensing community to describe going from observations on a small spatial (or temporal) scale to observations on larger scales, whereas “downscaling” consists in going from large-scale observations to smaller scales.

Interest in scaling as it relates to the land-surface water-and-energy budget arose from the more general problem of land-surface parameterization in climate models. In the 1970s, a series of Atmospheric General Circulation Model (AGCM) climate studies demonstrated the importance of land hydrology for the earth's climate: the sensitivity of albedo to climate (Charney et al., 1977), and the influence of soil-moisture anomalies (e.g., Walker and Rowntree, 1977).

Introduction

Fractures result from mechanical breaks in intact geologic media such as rocks or compacted glacial tills. Although a fracture that is completely filled by minerals is still considered a fracture in the geologic sense, within the context of subsurface fluid flow we think of a fracture as a mechanical break that results in void space between the fracture walls. This void space is more or less planar – one of its dimensions (the aperture or distance between fracture walls) is much smaller than the other two (the extension of the fracture plane). When interconnected, fractures provide pathways for fluid flow through geologic media that would be significantly less permeable if the media were unfractured.

The geometry of subsurface fractures varies greatly, with fracture lengths ranging from less than a millimeter (e.g., a microcrack in a rock grain) to thousands of kilometers (e.g., a fault along a tectonic-plate boundary). Fracture apertures vary from minute “hairline” cracks, nearly imperceptible to the naked eye, to solution-enlarged channels wide enough for human exploration. Fractures can be highly interconnected in a densely fractured rock, or isolated and poorly connected in a sparsely fractured rock. Some fracture networks exhibit a nested pattern, with smaller fractures bounded by larger ones (Barton and Hsieh, 1989). Studies of the processes that create fractures over this broad range of scales constitute an active area of research in the earth sciences.

In a study of fragmentation, Turcotte (1986) found that the fragmentation process often results in a power-law or fractal distribution of fragment sizes.