Time-dependent transport in heterogeneous formations of bimodal structures: 1. The model

Flow of uniform mean velocity U takes place in a heterogeneous medium made up from a matrix of conductivity K₀ and inclusions of a different conductivity K. The inclusions of given shape are implanted at random and independently in the medium, without overlapping. The aim of the study is to derive simple, approximate solutions of advective transport of solutes in such heterogeneous formations for arbitrary permeability ratio k = K/K₀ and inclusions volume fraction n. Transport is characterized by the spatial moments, which in turn are equal to the one particle trajectory statistical moments for ergodic plumes. The flow and transport problems are solved for isotropic media, for circular (2D) and spherical (3D) inclusions by using the model of composite inclusions of Hashin and Shtrikman [1962]. The results tend to the dilute limit analyzed in the past by Eames and Bush [1999] for n = o(1). Asymptotic, analytical results are derived for weak heterogeneity (k ≃ 1); they coincide with those of Rubin [1995] for a similar structure. Similarly, simple asymptotic expressions of the macrodispersivity are derived for low-permeability inclusions, k = o(1). The theoretical developments are applied to effectively computing trajectories moments in part 2 [Fiori and Dagan, 2003].