Modelling Subsurface Heterogeneity by Coupled Markov Chains: Directional Dependency, Walther’s Law and Entropy

This paper is an extension of the two-dimensional coupled Markov chain model developed by Elfeki and Dekking (2001) supplemented with extensive simulations. We focus on the development of various coupled Markov chains models: the so-called fully forward Markov chain, fully backward Markov chain and forward–backward Markov chain models. We addressed many issues such as: sensitivity analysis of optimal sampling intervals in horizontal and lateral directions, directional dependency, use of Walther’s law to describe lateral variability, effect of conditioning on number of boreholes on the model performance, stability of the Monte Carlo realizations, various implementation strategies, use of cross validation techniques to evaluate model performance and image division for statistically non-homogeneous deposits are addressed. The applications are made on three sites; two sites are located in the Netherlands, and the third is in the USA. The purpose of these applications is to show under which conditions the Markov models can be used, and to provide some guidelines for the practice. Entropy maps are good tools to indicate places where high uncertainty is present, so can be used for designing sampling networks to reduce uncertainty at these locations. Symmetric and diagonally dominant horizontal transition probabilities with proper sampling interval show plausible results (fits with geologists prediction) in terms of delineation of subsurface heterogeneous structures. Walther’s law can be utilised with a proper sampling interval to account for the lateral variability.