Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium
Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium
复制标题
化学非均质多孔介质中溶质云的时间行为
DOI：
10.1017/s0022112099004334
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发表时间：
1999
影响因子：
3.7
通讯作者：
W. Kinzelbach
中科院分区：
文献类型：
--
作者：
S. Attinger;M. Dentz;H. Kinzelbach;W. Kinzelbach
在本文中，我们研究了溶质云在非均质多孔介质中使用随机建模方法的时间行为。羽流从一个点状瞬时注入演化而来的行为，其特征在于其质心的速度和作为时间函数的弥散。在随机方法中，这些量表示为介质所有可能实现的集合上的适当平均值。我们发展了一种一般的摄动方法，它允许人们以系统和统一的方式计算各种量。我们在一个简化的含水层模型上证明了这种方法，其中只有由于线性瞬时化学吸附引起的延迟因子R(x)在空间上随机变化。我们分析了由此产生的质心速度和色散系数的两个概念上不同的定义：“有效”色散系数来自于每个实现中空间浓度分布的中心第二矩的平均值，以及“系综”色散系数，来自于平均浓度分布的第二矩。第一个量描述了典型介质中色散作为时间函数的特征，而第二个量描述了作为整体的系综的（正式的）色散特性。我们证明了对于有限时间，这两个量是不等价的，而对于t→∞和空间维度d[ges]2，它们是相同的。通常在文献中评估的系综色散系数大大高估了在一个给定介质实现中通常发现的色散。我们首次导出了这两个量作为时间函数的显式解析表达式。从这些，我们确定了两个相关的时间尺度，分离了定性和定量不同时间行为的制度：两个尺度中较短的是由一个无序相关长度上的溶质云的平流输送设定的，而第二个，更大的一个，与相同距离上的色散传播有关。只有在比这第二尺度大得多的情况下，并且在空间维度d[ges]2时，由于局部横向色散引起的混合，有效色散系数和系综色散系数才会相等。最后，将形式主义推广到一个扩展源。随着源尺寸的增大，有效色散系数向系综色散系数的收敛速度加快，因为扩展源已经代表了点源的系综。在非常大的源尺寸的限制下，在一个无序长度的平流输运时间尺度上发生收敛。我们为点源和扩展源在不同时间制度下的时间行为得出明确的结果。
In this paper we investigate the temporal behaviour of a solute cloud in a heterogeneous porous medium using a stochastic modelling approach. The behaviour of the plume evolving from a point-like instantaneous injection is characterized by the velocity of its centre-of-mass and by its dispersion as a function of time. In a stochastic approach, these quantities are expressed as appropriate averages over the ensemble of all possible realizations of the medium. We develop a general perturbation approach which allows one to calculate the various quantities in a systematic and unified way. We demonstrate this approach on a simplified aquifer model where only the retardation factor R(x) due to linear instantaneous chemical adsorption varies stochastically in space. We analyse the resulting centre-of-mass velocity and two conceptually different definitions for the dispersion coefficient: the ‘effective’ dispersion coefficient which is derived from the average over the centred second moments of the spatial concentration distributions in every realization, and the ‘ensemble’ dispersion coefficient which follows from the second moment of the averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium as a function of time, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We show that for finite times the two quantities are not equivalent whereas they become identical for t→∞ and spatial dimensions d[ges ]2. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. We derive for the first time explicit analytical expressions for both quantities as functions of time. From these, we identify two relevant time scales separating regimes of qualitatively and quantitatively different temporal behaviour: the shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, and spatial dimensions d[ges ]2, do the effective and the ensemble dispersion coefficients become equivalent due to mixing caused by the local transversal dispersion. Finally, the formalism is generalized to an extended source. With growing source size the convergence of the effective dispersion coefficient to the ensemble dispersion coefficient happens faster as the extended source already represents an ensemble of point sources. In the limit of a very large source size, convergence occurs on the time scale of advective transport over one disorder length. We derive explicit results for the temporal behaviour in the different time regimes for both point and extended sources.