Identifying spatial correlation structure of multimodal permeability in hierarchical media with Markov chain approach

Outcrop analogs for aquifers allow measurements of permeability with relatively high-resolution and mapping sedimentary unit types. The spatial bivariate structure of permeability is defined by aquifer architecture. The architecture is often organized into a hierarchy of unit types, and associated permeability modes, across different spatial scales. The composite covariance (or semivariogram) is a linear summation of the auto- and cross-covariances (or semivariograms) of unit types defined at smaller scales, weighted by the related proportions and transition probabilities. It is well-known that an appreciable fraction of the composite variance arises from differences in mean permeability across unit types defined at smaller scales. Previous work has shown that the transition probabilities usually define the spatial bivariate correlation structure. The composite spatial bivariate statistics for permeability (covariance or semivariogram) will not be representative unless data locations allow proper definition of the transition probabilities of the units. Quantification of the stratal architecture can be used to better interpret the transition probabilities and thereby improve a model for the sample covariance and semivariogram. In this paper, we use an inverse modeling algorithm to fit the components of the hierarchical model, written as nested functions, in developing a hierarchical spatial correlation models. Specifically, the least-squares criterion along with prior information and other weighted constraints are used as the objective function for the inverse problem, which is solved by the Gauss-Newton-Levenberg-Marquardt method. The estimated covariance and transition probability models provides accurate representation of the spatial correlation structure of permeability for field-measured data from in Espanola Basin, New Mexico.