On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory

Abstract

A generalized theory of multiphase transport is presented which combines the concepts of scale, instrumentation, stochastics, and time series with the development of transport equations. By defining the filtering process as a convolution of a measure P with a field property ψ, we are able to exploit the Fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment. An ideal instrument is defined which filters out high‐frequency noise (corresponding to short distances) and yet does not alter the structure of low frequencies. Using ideal instruments we successively filter out lower frequency noise in a multiscale, multiphase environment. Formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process. An equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed. Power spectra are developed which relate spectra on different scales to the measuring instrument used. By successively applying filtering theorems, a hierarchy of multiscale transport equations is developed. Filtered properties in the transport equations are mass averages. Different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases. The concept of a wide sense stationary, ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices.

Untitled

“…Upscaling is accomplished by spatially integrating the stochastic partial-differential equations derived from the fundamental conservation laws of continuum mechanics. The roots of permeability upscaling lie at the pore scale and hence significant effort has been made to predict the macroscopic permeability from the pore-scale characteristics of the porous medium (e.g., Cushman, 1984;Bear and Bachmat, 1990). This level of upscaling is generally averted due to the availability of permeability data measured on some macroscopic sample support (Le., sample volume).…”

What does an instrument measure? Empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports

“…Accordingly, they proposed the weighting function as a convenient means for relating characteristics of an instrument's measurement to the measured macroscopic field variable. Cushman [1984] exploited the filtering aspects of an instrument so as to place restrictions that make its measurement relevant to its physical environment. Cushman defined an ideal instrument as one that filters out small-scale features (noise) while preserving the large-scale structure.…”

Macroscopic modeling of columnar dendritic solidification

“…For small evolving heterogeneities, the dendritic layer can be viewed as homogeneous and therefore averaged properties are constant whatever the size of the averaging volume. For large τ , scale separation is not satisfied, and more theoretical work is needed to explore alternative descriptions using for instance deforming averaging volume [29] or jump boundary condition [30][31][32]. Between these two limiting situations, i.e.…”