Steady 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, I_v, 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 = I_v/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.

中文翻译：

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通过弱非均质多孔地层的双线型流动配置中的溶质传输

由注入井和抽水井（双峰）产生的稳定流动发生在多孔地层中，其中空间可变的水力传导率K被建模为具有各向异性两点自相关的平稳、对数正态、随机场。后者的特点是垂直积分尺度，即I _v，小于水平积分尺度I。一种溶质，无论是被动的还是反应性的，被注入介质中，我们的目标是计算突破曲线 (BTC) 及其不仅在恢复（泵送）井处的时刻，而且在两口井之间的任何位置。K之间的强耦合流动的不均匀性使问题变得非常困难。然而，通过采用一些假设获得了一个简单（解析）的解决方案：（a）井被奇点线代替，（b）将对数电导率Y  = ln  K的方差视为扰动参数的扰动解决方案被采用，（c）研究仅限于强各向异性非均质地层（各向异性比率λ  =  I _v / I远小于一），并且（d）忽略了孔隙尺度分散的影响。计算 BTC 的核心是流体粒子在注入井释放并到达位于沿连接两个井的距离的任意位置x _{1的控制平面的传播时间的统计数据。}结果表明，Y的空间变异性实际上就像一种分散机制：它增强了传播，尤其是在早期到达时。还获得了沿中心轨迹的旅行时间矩的有用闭合形式表达式。最后，将本研究中提出的理论框架应用于两个传输实验，以计算作为x ₁函数的二阶（时间）矩，因此要量化发生在由两口井划定的区域中的分散。