Non-ergodic solute transport in self-similar porous formations: the effect of conditioning

We analyze the influence of conditioning to hydraulic transmissivity measurements on transport of a conservative tracer in a two-dimensional evolving-scale formation with a large-scale cutoff of the hydraulic transmissivity. First-order analytical solutions of ensemble and effective plume moments, conditional to a single measurement of hydraulic transmissivity, are provided for an instantaneous release of solute within a linear source normal to the mean flow direction. The proposed solutions show that both ensemble and effective moments are significantly affected by the additional information brought into flow and transport models through conditioning to available data. We conclude that the effect of conditioning to available measurements is more pronounced on the ensemble moments than on the effective moments, and uncertainty is not reduced in the same proportion for longitudinal and transverse effective second moments, since the former shows the largest reduction of uncertainty for a given measurement location. This result confirms the experimental findings showing larger uncertainty in the estimation of the transverse moments. Furthermore, for all moments the impact of the measurement reduces with the distance of the measurement point from the mean plume trajectory.