Advective Transport Through Three-Dimensional Anisotropic Formations of Bimodal Hydraulic Conductivity

Abstract

Transport of a non-reactive solute in anisotropic bimodal porous formations is investigated through the evaluation of the breakthrough curve. The hydraulic conductivity $$K$$ structure is modelled as an ensemble of anisotropic and non-overlapping inclusions, with given hydraulic conductivity $$K_{i}$$ , randomly placed in an homogeneous matrix with hydraulic conductivity $$K_{0}$$ . The conductivity contrast $$\kappa =K_{i}/K_{0}$$ between the inclusions and the matrix represents the main source of heterogeneity. The resulting $$K$$ structure is a theoretical representation of natural porous formations where inclusions and matrix play the role of two distinct hydrofacies. Semi-analytical solutions for flow and transport are carried out in both 3D and 2D domains following a Lagrangian approach by adopting the dilute limit approximation. The procedure, formally valid for media with low inclusions–matrix relative volume fraction, is extended to dense media by a simplified formulation of the self-consistent approach. Natural porous formations are intrinsically anisotropic due to the depositional process; hence, particular emphasis is addressed to the quantification of the effects of the anisotropy on solute transport. The solution of our physically based model depends on three structural parameters: the hydraulic conductivity contrast $$\kappa $$ , the relative volume fraction $$n$$ , and the statistical anisotropy ratio $$e$$ . The procedure allows for the physical representation of high-velocity channels and stagnation zones, which can be encountered in bimodal media, and it is particular advantageous for a quick estimation of the solute breakthrough curve. Our analysis shows different transport dynamics, as function of the conductivity contrast $$\kappa =K_{i}/K_{0}$$ , leading to fast preferential flow ( $$\kappa >1$$ ) or mass retention ( $$\kappa <1$$ ), as a consequence of the lack of symmetry of the flow and transport parameters (velocity and travel time residuals). High dispersion is typically found when $$\kappa <1$$ , i.e. for low conductive inclusions, where the latter typically trap the solute particles, determining large travel times. Conversely, when $$\kappa >1$$ flow in the inclusions is constrained by the outer flow, limiting the solute velocity inside the inclusions, which results is a rather limited dispersion. Anisotropy typically enhances the solute spreading for $$\kappa <1,$$ as a consequence of the slower velocity predicted for more anisotropic inclusions, while its impact on formations with high $$\kappa $$ is rather limited.