A physical basis for calculating instrument spatial weighting functions in homogeneous systems

Abstract

Starting with a streamline-based argument, the theoretical basis for a spatial weighting function in a steady, homogenous, and isotropic flow system is developed for both compressible and incompressible flow. A physical interpretation of spatial weighting functions in terms of the ratio of steady state energy dissipation rate per unit volume of porous medium to total energy dissipation rate over the entire flow domain is formulated. Finally, applications of the spatial weighting functions are presented based on calculated gas minipermeameter mass flux fields for the conventional surface probe and a new small drill hole probe configuration. Spatial weighting function distributions indicate that with diverging flow field instruments, such as the gas minipermeameter, porous medium volumes in the inlet vicinity are heavily weighted, with volumes near the seal boundaries shown to be extremely important. The technique described allows one to quantify the size and shape of the subvolume of a domain contributing to a hydraulic conductivity or intrinsic permeability measurement. For heterogeneous systems it is suggested that the spatial weighting function for a given instrument will be heterogeneity dependent and therefore nonunique. However, extending the weighting function concept to heterogeneous and/or transient systems is an important research priority.