Spatial Correlation of Permeability in Cross-Stratified Sediment with Hierarchical Architecture

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Cross-stratified deposits can give rise to a hierarchy of permeability modes, across scales, corresponding to a hierarchy of sedimentary unit types. The shape of the sample semivariogram for permeability can be largely controlled by the shape of the cross-transition probabilities of unit types having the greatest contrast in permeability. The shape of those cross-transition probabilities can be, in turn, largely determined by the variance of the lengths of those unit types. A sufficient condition for an exponential-like semivariogram is the repeated occurrence of unit types having both a contrast in permeability and a large length variance. These relationships are shown through writing the identities for spatial correlation of permeability in a hierarchical and multimodal form and as a function of the transition probabilities for the sedimentary unit types. These relationships are also illustrated through analyzing data representing cross-stratified sediments within a point bar deposit.