One of the major causes for uncertainty and erroneous understanding of causal relationships and the magnitude of parameters and trends has been identified as being the ‘scale problem’. Different levels of heterogeneity are encountered when passing from the microscopic to the macroscopic scale. With regard to complex soil and groundwater systems, the question of a conceptual basis for combining different sources is of concern. The scale problem is due to the spatiotemporal (i.e., in space and in time) variability of the systems of interest: statements that concern a particular scale may (and often will) not hold at other scales. Hence, extrapolation of understanding to a larger or to a smaller scale may require additional knowledge at these larger or smaller scales. If this need for additional knowledge is not recognized, the implicit assumption of ‘scale invariance’ is made and if this assumption is false, the interpretation of measurements or of model exercises may be erroneous.
The scale problem permeates
most theoretical and experimental research of the environmental compartments
air, water, and soil (which each have biotic and abiotic sub-compartments). For
this reason, a comprehensive overview of all processes and parameters involving
scale issues is impossible. As a first step, the methodological approach is
depicted and a number of important issues are discussed such as how to choose
relevant scales explicitly accounting for governing processes and
characteristic time scales.
The key issue of the scale problem, is that at different scales different processes are important. For this reason, different scale classifications have been proposed, where a well known classification has been given by Bouma and co-workers. In this classification, two major diagnostic properties or (axes) are distinguished, i.e., space and time. Obvious classes to be considered are:
Space: nm (molecular, double layer), µm (pore size), mm (soil organisms), meter (soil profile, pedon), field, region (watersheds, river basins), country, continent, mondial scale, where illustrations for a soil’s perspective are given between brackets.
Though illustrative, such a classification remains somewhat arbitrary. For instance, one may be interested in studying river basin scales, which may be ranging from regional to continental sizes, yet are physically well distinguishable systems.
A more process oriented classification distinguishes the following scales:
1. scale of the natural medium and processes
2. scale of observation/experimentation
3. scale of modelling
4. scale of management and policy
This approach allows for anticipation of the difference of scales with respect to the system (1), activities (2 and 3), and demand (4). As modelling often is involved in ‘interpretation’ of experimental data, (2) and (3) may be combined. Here, this is not done, because of transparency reasons.
In the following paragraphs, scale issues are briefly discussed for different compartments.
Explicit attention for scaling issues has been predominantly given to the flow and transport of water and dissolved chemicals (solutes). Therefore, the experimental data base and the theoretical tools to jump between scales has advanced most for these issues.
A subject of study has been
the moving between the scales of the individual pore to the continuum of a
porous medium sample (a case of upscaling). Using different theoretical
approaches (regularisation, averaging, normalisation, etc.) it has been shown,
that the Cozeny-Karman equation that holds for the pore scale ‘degenerates’ to
an expression that maintains only the first (order) term. Its applicability is
limited (though still powerful for practice). Choosing different solutions for
the closure problem, the higher order terms as well as the cross coupling
effects can be addressed, at least in principle. The simplifications involved
in this upscaling exercise, are again encountered when the Darcy equation needs
to be upscaled from one continuum to a larger one with additional features:
directional effects such as anisotropy may be the result of such additional
The hydraulic properties (saturated hydraulic conductivity, permeability function and retention function) have been most commonly assumed to conform to scale invariance (Miller similitude).
As in most cases the scaling of hydraulic properties is pragmatic, rather than theoretically founded (on scale similarity), extrapolations to larger scales remain statistical extrapolations with inherent mechanistic uncertainties.
Besides laboratory approaches to assess hydraulic properties, also field scale measurements are possible. In principle, those are complicated by the same scale dependencies. For instance, the infiltrometer/permeameter equipments may provide strongly different data if transposed over a short distance, e.g. if measurements are made in a fractured clay soil. Likewise, water-extraction based measurements such as with suction cups in the vadose zone, or with pumping wells in ground water, seldom identify the origin of the extracted water. To be able to assess which proportion of the water is derived from more or less permeable sub-domains often remains obscure and would inevitably require additional information to make an educated guess. This implies, that the scale of the measurement tool is less than the scale of the typical medium variability, and the implications for interpretation of the raw data as a base for management decisions, have hardly been brought into perspective yet.
Where the above scale issues have consequences for assessing water budgets for soils, ecosystems, and in agrometeorology, etc., they may have even larger consequences with regard to solute transport and the processes affected by solute transport.
The scale problem in solute transport allows for an illustration that has been well considered in the scientific literature. It is attractive to distinguish two scales, i.e., the scale of heterogeneity (X) and the scale of the considered system or domain (L). We may distinguish the following situations:
1. X and L are of similar size: then, the boundaries between different subdomains and the interaction between them must be deterministically known. Separate transport equations have to be formulated for each subdomain.
2. X < L: the heterogeneities may be described with simplified geometries (e.g. cubes, spheres) with average properties, and exchange between each ‘aggregate’ and macropores by diffusion has to be considered; sometimes a simple first order mass transfer may be appropriate.
3. X << L: the heterogeneities are visualised as a separate subdomain (or unspecified geometry) that exchanges with first order with the other subdomain (where fastest transport occurs).
4. X <<< L: the heterogeneity gives rise to additional dispersion but need not be considered as a subdomain (equilibrium assumption is valid for solute transport).
With the above distinguished four situations, it does not matter whether a flow in laboratory columns is considered, or flow and transport in an aquifer or a fracture rock system or even in a karst region. However, to assess which of the four cases holds, is not always simple. The main reason is that the scales as such are not the only factor of importance, since the rate of interaction/exchange between different subdomains also affects when X is considered small, very small, et cetera.
variability of chemical properties and processes has received considerably less
attention than e.g. hydrological and soil physical variability. Nevertheless,
in the past two decades, attention has been given to the effect of spatial
variabilility on chemical transport, taking both physical and chemical
parameter variability into account. This has resulted in two main approaches of
stochastic transport modeling, where either the temporal development of spatial
moments or the temporal development of fluxes at a control plane were
considered. One of the main conceptual approaches is called the stochastic
convective SC model, in which local (small scale) dispersion is neglected,
which leads to a parallel streamtube approximation.
The large scale transport of reactive chemicals still has major unresolved conceptual issues, which need to be urgently addressed in view of the public demand to provide model answers at aquifer, or even watershed/riverbasin scales.
Compared with soil physics and soil chemistry (including adjacent areas of hydrology, reservoir engineering, and geochemistry), the study of spatial variability in biology, ecology, ecotoxicology, is old. No doubt, that this is true because this research area has had an early focus on classification and mapping, as is the case for traditional soil science and for geological mapping. Similar to those areas, however, the study of spatial variability was mainly implicit, as different ecotypes and ecotopes may have been recognised, and major processes may have been identified, but variability as such in relationship with that of the environment (soil, hydrology, etc.) was rarely considered.
For this reason, perhaps, there is much and advanced understanding of why certain vegetation is found on particular soils or under particular physico-chemical conditions, but causal quantitative relations with physico-chemical properties, processes and their variability have not been developed. In part, this may be due to soil physics and chemistry not being adjusted to help in unravelling such relationships. Nevertheless, this interaction between physical, chemical and biological processes and their variability should become a major and challenging focus for the next decades. As recent soil biological studies revealed, the concept of relevant scale may have to be developed first, because (geostatistical) tools may already be available, but the scaling-concepts are still lacking.
In view of contamination with biodegradable compounds, much work has been made of understanding, assessing, and quantifying the biodegradation rate of such compounds as a function of their concentration, bioavailability, and to a lesser extent the environmental conditions such as trophic status. This has provided useful information, it does, however, not yet combine well with the physico-chemical aspects involved. These links require further attention.
Authors: Sjoerd van der Zee, Kevin Jones; Michel Jauzein, Tomas Vogel
SOWA – Integrated Soil and water Protection: Risks from diffuse pollution