Dilution enhancement in hierarchical and multiscale heterogeneous sediments

We quantify the dilution of solute clouds in geological formations displaying hierarchical and multiscale sedimentary architecture through the use of a Lagrangian framework. This allows for a fundamental understanding of how dilution arises from the hierarchical architecture of sedimentary facies, and enables for a quantitative decomposition of dilution into facies-related contributions at different scales within the hierarchy. The new developed semi-analytical solution that quantify the dilution index is based on hierarchical expressions of the spatial covariance of the log-conductivity. The spatial correlation structure in the covariance expression is defined by the probability of transitioning within and across facies types of different scales, which are parameterized by independent and physically quantifiable attributes of sediments (including geostatistical properties of the conductivity field) and the proportions and lengths of facies types. We illustrate the applicability of the model in the well-documented tracer test at the Borden research site (Canada). We show how different scales of sedimentary architecture contribute to the dilution enhancement of the solute cloud at Borden aquifer. By quantitatively decomposing the dilution index into facies-related contributions, we gain fundamental insights on the key factors controlling the time-dependent rate of dilution. The results also illustrate that the probability of transitioning from one facies to another (i.e. cross-transitions) at different scales defines the temporal scaling for enhanced dilution. Additionally, we perform sensitivity analyses to investigate the effects of mean difference in conductivity, volume proportion, the statistical anisotropy ratio, and the scale of heterogeneity on the overall solute dilution.