Flow in random porous media : mathematical formulation , variational principles , and rigorous bounds
The problem of the slow viscous flow of a fluid through a random porous medium is considered. The macroscopic Darcy's law, which defines the fluid permeability k , is first derived in an ensemble-average formulation using the method of homogenization. The fluid permeability is given explicitly in terms of a random boundary-value problem. General variational principles, different to ones suggested earlier, are then formulated in order to obtain rigorous upper and lower bounds on k . These variational principles are applied by evaluating them for four different types of admissible fields. Each bound is generally given in terms of various kinds of correlation functions which statistically characterize the microstructure of the medium. The upper and lower bounds are computed for flow interior and exterior to distributions of spheres.
SUMMARY ::: ::: In this paper six distribution-free (nonparametric) stereological methods for the solution of Wicksell's corpuscle problem—i.e. the determination of the distribution of diameters of spheres embedded in an opaque specimen from the diameters of their profiles on plane sections—are compared as regards their numerical stability, sensitivity to underlying distributions and certain error criteria. The study is based on the results of simulation studies for several types of distribution (one-point, normal, exponential, logarithmic normal) of sphere diameters. Recommendations are suggested for the choice of methods, sample size and the optimal number of classes for grouping sample data.