Representing Aquifer Architecture in Macrodispersivity Models with an Analytical Solution of the Transition Probability Matrix

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The multi-dimensional transition probability model represents hydrofacies architecture in modeling aquifer heterogeneity. The structure of the aquifer architecture is mathematically characterized by a canonical representation of the transition probability matrix, solved by an eigenvalue decomposition method. Whereas the eigenvalue decomposition has been numerically solved previously, we show here that it can be analytically solved under the assumptions that cross-transition probabilities are dictated by facies proportions and that the juxtapositional tendencies of the facies are symmetric. Although limited by the assumptions, analytical solutions provide more immediate insights about the relationships between transition probability and facies proportion and mean length. The analytical solution is first tested by comparison with the numerical solutions and then used to represent hydrofacies architecture within expressions for the spatial covariance of conductivity and the macrodispersivity. The relationship between the longitudinal macrodispersivity and integral scale, the indicator correlation length, and the facies proportion is represented in an equation for estimating the field-scale dispersivity. An example is used to show how sedimentary structures, conductivity contrasts, and facies mean lengths affect the scales of the macrodispersivity.