Advective Transport in Heterogeneous Formations: The Impact of Spatial Anisotropy on the Breakthrough Curve

Water flow and solute transport take place in formations of spatially variable conductivity K. The logconductivity Y = ln K is modeled as a random stationary space function, of normal univariate pdf (of mean In K_G and variance σ_Y²) and of axisymmetric autocorrelation of integral scales I_h,I_v (anisotropy ratio f = I_v/I_h < 1). The head gradient and the velocity are uniform in the mean, parallel to bedding, and of constant and given as J and U, respectively. Transport is ruled by advection, which typically overwhelms pore scale dispersion in the breakthrough curve (BTC) determination. In the present study we analyze the impact of anisotropy f on the BTC of a passive solute, which is related to the mass flux μ (t, x) at a control plane at x. While a considerable body of literature dealt with weakly heterogeneous formations (σ_Y² < 1), the present study addresses the case of σ_Y² > 1, which is of interest for many aquifers and is more difficult to solve either numerically or by approximations. We approach the three dimensional problem by modeling the structure as an ensemble of densely packed oblate spheroids of semi-major and semi-minor axis R and f R, respectively, and independent lognormal K, submerged in a matrix of uniform conductivity K_ef, the effective conductivity of the ensemble. The detailed numerical simulations of transport show that the BTC is insensitive to the value of the anisotropy ratio f, i. e., μ (t, x) I_h/U depends only on σ_Y² (except for small differences in the tail). This important result implies that transport, as quantified by BTCs or spatial longitudinal mass distributions, can be modeled accurately by the much simpler solutions developed in the past for isotropic media, like e. g., the semi-analytical self-consistent approximation.