Conditional mean, effective, and realizations of hydraulic conductivity fields

This study first discusses the conditional mean, realizations, and effective hydraulic conductivity in a theoretical framework. It then introduces Monte Carlo simulation (MCS) algorithms for constraining the outcome by either hydraulic conductivity (K) samples or hydraulic head (h) measurements from the hydraulic tomographic survey (HT). It demonstrates that kriging using K measurements leads to a conditional mean K field, while inverse modeling using successive linear estimator (SLE) with head measurements of HT yields the conditional effective K field. The effects of conditioning using K measurements are different from those using heads. Besides, the conditional effective K leads to the unbiased prediction of the head that honors the observed head at measurement locations. More importantly, the study reveals that the harmonic and geometric means of conditional realizations of K fields of MCS, given head measurements, are equivalent to the conditional effective K in one- and two-dimensional flows, respectively. The first-order approximation in the SLE results in a conditional covariance similar to that from MCS with smaller magnitudes. Despite the difference, all approaches predict unbiased conditional mean head behaviors.