Averaging and upscaling: continuous vs “ultimate”
Abstract
Modeling of many natural and industrial processes requires their description on multiple scales. As an example, the description of flows in natural porous media involves the molecular scale, of nanometers (e.g. interactions of molecules and ions with the internal porous surface); the pore scale, of micrometers (flows in individual pores); the laboratory scale, of centimiters (flows in the laboratory samples); and the formation scale, of kilometers (the flows in geological formations). Often, the intermediate scales are also defined.
The transition between the different scales is carried out by employing a mathematical procedure of upscaling (averaging). Several such procedures have been suggested in the literature. Most of them suggest a “jumpwise” averaging between the distant scales.
Unlike the previous works, we have developed an approach to upscaling/averaging based on a continuous set of scales, ranging from the finest to the coarsest scale. The upscaling is thought of as a continuous procedure where the densities and fluxes describing the flows gradually become more and more smoothened as the system moves along the “scale axis”. The rules of such upscaling have been established.
A sample process to upscale is the process of diffusion, with or without convection, or with or without external sources. We have derived the equations for the evolution of the diffusion coefficient along the scale axis and solved it analytically and numerically. As the averaging progresses from a finer to a coarser scale, the fine-scale heterogeneities disappear, and the diffusion coefficient progresses toward its average value.
But now, the question arises, whether, in the limit of the large difference between the scales, the continuous upscaling gives the same results as the jump-wise upscaling. This question is studied in the present work. It is established that jumpwise upscaling cannot always be performed. In cases where it can, the continuous upscaling produces asymptotically the same result as the jumpwise upscaling. However, continuous upscaling can produce the averaged diffusion coefficient even if the jumpwise upscaling is impossible.