Complexity, Heterogeneity, and Scale: Relationships Between Real and Modeled Systems

A common description of subsurface heterogeneity as a function of scale takes the form of ‘nested’ heterogeneities.  This conceptual model recognizes that atom scale (10^-10m) heterogeneities of mineral interfaces exist, and that this atom scale heterogeneity is a subset of the total heterogeneity at the next higher scale of interest, the pore scale (~10^-8-10^-6m).  The pore scale itself exhibits heterogeneous distributions of mineral phases and aqueous complexes; both the atom and pore scale heterogeneities are incorporated in the next highest scale, the batch scale (~10^-3-10^-2m).  This process continues up to the field scale (~10²m) which has heterogeneity length scales from 10^-10-10² depending on which characteristic is considered.  This is the source of complexity in subsurface systems.

Models which perform upscaling calculations are often based on the conceptual model of nested heterogeneity either explicitly or through model assumptions.  This approach relies heavily on advanced computing technologies and complex mathematical relationships; the complexity of the real system is mirrored in model formulation.  Another possible approach is to change the conceptual model to include physical and chemical behaviors observed as a function of scale.  This may lead to simpler mathematical models which can still describe complex systems.  This latter approach is supported by recent evidence from 2-D and 3-D uranium transport experiments performed at the intermediate scale (2.4m).  In these experiments the amount of heterogeneity in terms of particle sizes used, physical packing orientations, chemical characteristics of the influent water, and dimensionality all varied widely and yet many chemical and physical behaviors are similar or even identical.  Thus despite the complexity of subsurface systems, they may exhibit simplified behavior as local behaviors are integrated over increasing length scales.