A Gaussian-Mixture based stochastic framework for the interpretation of spatial heterogeneity in multimodal fields

We provide theoretical formulations enabling characterization of spatial distributions of variables (such as, e.g., conductivity/permeability, porosity, vadose zone hydraulic parameters, and reaction rates) that are typical of hydrogeological and/or geochemical scenarios associated with randomly heterogeneous geomaterials and are organized on various scales of heterogeneity. Our approach and ensuing formulations embed the joint assessment of the probability distribution of a target variable and its associated spatial increments, DY, taken between locations separated by any given distance (or lag). The spatial distribution of Y is interpreted through a bimodal Gaussian mixture model. The modes of the latter correspond to an indicator random field which is in turn related to the occurrence of different processes and/or geomaterials within the domain of observation. The distribution of each component of the mixture is governed by a given length scale driving the strength of its spatial correlation. Our model embeds within a unique theoretical framework the main traits arising in a stochastic analysis of these systems. These include (i) a slight to moderate asymmetry in the distribution of Y and (ii) the occurrence of a dominant peak and secondary peaks in the distribution of DY whose importance changes with lag together with the moments of the distribution. This causes the probability distribution of increments to scale with lag in way that is consistent with observed experimental patterns. We analyze the main features of the modeling and parameter estimation framework through a set of synthetic scenarios. We then consider two experimental datasets associated with different processes and observation scales. We start with an original dataset comprising microscale reaction rate maps taken at various observation times. These are evaluated from AFM imaging of the surface of a calcite crystal in contact with a fluid and subject to dissolution. Such recent high resolution imaging techniques are key to enhance our knowledge of the processes driving the reaction. The second dataset is a well established collection of Darcy-scale air-permeability data acquired by Tidwell and Wilson (1999) [Water Resour Res, 35, 3375-3387] on a block of volcanic tuff through minipermeameters associated with various measurement scales.