Solute Transport in a Doublet‐Type Flow Configuration Through a Weakly Heterogeneous Porous Formation

Abstract

AbstractSteady flow generated by an injecting and a pumping well (doublet) takes place in a porous formation where the spatially variable hydraulic conductivity K is modeled as a stationary, lognormal, random field with anisotropic two‐point autocorrelation. The latter is characterized by a vertical integral scale, that is, Iv, smaller than the horizontal one, that is, I. A solute, either passive or reactive, is injected in the medium, and we aim at computing the breakthrough curve (BTC) and its moments not only at the recovery (pumping) well, but also at any location between the two wells. The strong coupling between K and the nonuniformity of the flow renders the problem very difficult. Nevertheless, a simple (analytical) solution is obtained by adopting a few assumptions: (a) wells are replaced by lines of singularity, (b) a perturbation solution which regards the variance of the log‐conductivity Y = ln K as a perturbation parameter is employed, (c) the study is limited to strongly anisotropic heterogeneous formations (for which the anisotropy ratio λ = Iv/I is much smaller than one), and (d) the impact of pore‐scale dispersion is neglected. Central for the computation of the BTC is the statistics of the travel time of a fluid particle released at the injecting well and reaching a control plane located at any position x1 along the distance connecting the two wells. It is shown that the spatial variability of Y acts de facto like a dispersion mechanism: it enhances spreading, especially in the early arrivals. Useful closed form expressions for moments of the travel time along the central trajectory are also obtained. Finally, the theoretical framework presented in this study is applied to two transport experiments in order to compute the second‐order (temporal) moment as function of x1, and therefore to quantify dispersion occurring in the zone delimited by the two wells.