Adaptive stacked ensemble fusion: Bayesian porosity inference in alluvial aquifers when the errors are not...

Effective porosity in alluvial aquifers is routinely estimated by combining two indirect, grain-size-based methods-an empirical grain-size correlation (Vukovic-Soro) and a hydraulic-conductivity inversion (Kozeny-Carman)-under the textbook assumption that their errors are independent. Because both methods read the same grain-size information, that assumption is generally false, and the fused uncertainty is misstated. We show that the error correlation between these two estimators is the net of two physically distinct, oppositely signed channels: a negative measurement channel, because the estimators respond to the shared characteristic diameter with opposite signs so that shared grain-size measurement error anti-correlates their errors; and a positive shared-proxy channel because both miss the same porosity variation not explained by grain size. The net correlation can therefore be negative or positive, and we derive a closed-form critical value of the unexplained-porosity variance at which it changes sign. This partitions the fusion problem into three operationally distinct regimes-anti-correlated, in which the classical interval is needlessly wide; correlated-interior, in which it is overconfident; and exterior, in which correlation-aware fusion is essentially mandatory-and yields a pre-data diagnostic, computable from a grain-size curve, its measurement reproducibility, and a per-site bound on the porosity variance unexplained by grain size, that returns the likely sign of the error correlation, the regime, and the direction of bias in the reported uncertainty before any fusion is performed. The correlation-aware combination weight and the regime boundary it relies on are classical results of the estimator-combination literature; the contribution is applied: the error correlation, normally an assumed input, becomes derivable from grain-size sensitivities, and the circularity of a grain-size-derived conductivity becomes detectable in advance. We apply the framework to the Tirana-Fushë-Kuqe-Lezhë alluvial aquifer (Albania), where classical inverse-variance fusion is found to understate the effective-porosity variance by up to 57% in the fine, clay-bearing facies, so that its nominal 95% intervals attain only about 91% coverage overall, restored to the nominal 95% by correlation-aware fusion-and use an independent borehole-NMR porosity log at East Falmouth, Cape Cod (USA), as a reality check, which places that site near the sign-flip boundary, where the two grain-size pathway errors approximately cancel (ρ ≈ 0) once an independent hydraulic conductivity is used-whereas rebuilding the conductivity from the same grain-size curve, the common field practice, manufactures a spurious correlation of +0.89, so that the apparent two-method agreement corroborates the shared inputs rather than the porosity; the exterior regime is demonstrated on a controlled synthetic benchmark. Effective-porosity uncertainty propagates directly into solute travel time, so getting its sign and magnitude right is consequential for contaminant-transport and protection-zone predictions.