Analytical expressions for macrodispersion in three-dimensional Sub-Gaussian hydraulic conductivity fields

Subsurface flow and transport settings are typically characterized by spatial variability of the underlying hydro-geological attributes (e.g., permeability and porosity) and a high degree of uncertainty associated with data and model parameter estimates. In this context, we rely on a stochastic approach and analyse conservative solute transport taking place within three-dimensional, sub-Gaussian domains with isotropic, exponential correlation structure of the associated log-conductivity fields. The flow is uniform in the mean and driven by an imposed average head gradient. We present an analytical solution based on a small perturbation approach that allows assessing the temporal evolution of longitudinal and transverse macrodispersion. Similar to what is observed for Gaussian log-conductivity domains, these are seen to attain a (Fickian) asymptotic regime after the solute plume has travelled a sufficient number of conductivity correlation scales. We also derive closed-form analytical expressions for other statistical moments of interest (e.g., seepage velocity and particle displacement covariance) and benchmark these solutions against numerical Monte Carlo simulations for various degrees of domain heterogeneity. This enables us to assess the extent at which a small perturbation approximation can embed the key features of macrodispersion within three-dimensional sub-Gaussian conductivity fields of increasing heterogeneity levels. Our results suggest that, similar to what already observed for Gaussian fields, the analytical formulation fully captures the trend of longitudinal macrodispersion for values of log-conductivity variance smaller than the unity, the goodness of the results becoming worse as the variance increases. Our formulation also captures directional displacement and seepage velocity covariances, even though the degree of agreement with their numerical Monte Carlo counterparts rapidly deteriorates with increasing conductivity variance. Particularly refined numerical grids are required to capture the nugget effect exhibited by the analytical longitudinal velocity covariance, thus posing a challenge to assess the system behaviour at short distances.