Proper Spatial Average Operators in a Heterogeneous Porous Medium

In this work, we explore a wide range of kernel functions commonly used in Bayesian statistics to evaluate their functionality in attenuating geometrical fluctuations arising from sudden change in the porosity in a heterogeneous porous medium. Boxcar; single-, double-, and triple-convoluted boxcar; Epanechnikov; Tricube; Gaussian; Cosine; Logistic; Sigmoid; and Silverman are among weighting functions we examined for the cases where (1) a periodic homogeneous medium, (2) a quasi-periodic heterogeneous medium with a gradual porosity change, (3) a quasi-periodic heterogeneous material with a discontinuous jump in porosity, and (4) a disordered heterogeneous porous medium with a discontinuous jump in the porosity exist. We use an extended porous material’s diffusion model developed based on volume averaging method recently reported by Battiato et al 2019 [1]. The model retains the zero- and first-order terms in the closure problem and neglects the second-order Taylor series. Results are promising in a sense that mollified version of some of those weighting functions can improve divergence of Taylor series approximation of the average flux of mass through the porous material.
[1] Battiato, Ilenia, et al. Transport in Porous Media (2019): 1-72.