Towards a filtered density function approach for reactive transport in groundwater
Towards a filtered density function approach for reactive transport in groundwater
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地下水反应输运的过滤密度函数方法
DOI：
10.1016/j.advwatres.2016.02.016
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发表时间：
2016
影响因子：
4.7
通讯作者：
P. Knabner
中科院分区：
文献类型：
--
作者：
L. Schüler;S. Attinger;C. Vamoş;P. Knabner
将守恒标量加权的随机物种浓度的概率密度函数（pdf）和过滤密度函数（fdf）的演化方程表述为描述浓度-位置空间中随机等效过程的Fokker-Planck方程。这种方法提供了一致的数值PDF/FDF解，由相关Itô方程控制的计算粒子系综的浓度-位置空间中的密度给出。采用全局随机游走（GRW）算法求解，该算法稳定，不受数值扩散的影响，对粒子数量的增加几乎不敏感。一般的FDF方法和GRW数值解说明了一个降低复杂性的问题，包括一个单一的标量在地下水中的输送。随机是由水力电导率的随机参数化引起的，具有短距离相关和小方差的特点。目的是推断在羽流质量中心采样的随机浓度的统计数据，在二维空间域的横向维度上进行积分。因此，PDF/FDF问题也可以在二维域、一个空间维度和一个集中空间中表述。描述物理空间中PDF输运的上尺度漂移和扩散系数由于其自平均性质，在速度场中具有短程相关的扩散单轨迹上估计。描述浓度空间中PDF传输的混合系数由模拟浓度时间序列集合的统计分析得出的趋势和噪声以及经典混合模型参数化。将高斯空间滤波器应用于Kraichnan速度场发生器，构建了FDF问题的粗粒度模拟（CGS）。CGS模拟的目的有两个：首先，从实际角度理解FDF方法的意义及其与PDF方法的关系；其次，研究本文所考虑的混合模型的局限性以及地下水系统混合模型的理想特征。
Evolution equations for probability density functions (PDFs) and filtered density functions (FDFs) of random species concentrations weighted by conserved scalars are formulated as Fokker–Planck equations describing stochastically equivalent processes in concentration-position spaces. This approach provides consistent numerical PDF/FDF solutions, given by the density in the concentration-position space of an ensemble of computational particles governed by the associated Itô equations. The solutions are obtained by a global random walk (GRW) algorithm, which is stable, free of numerical diffusion, and practically insensitive to the increase of the number of particles. The general FDF approach and the GRW numerical solution are illustrated for a reduced complexity problem consisting of the transport of a single scalar in groundwater. Randomness is induced by the stochastic parameterization of the hydraulic conductivity, characterized by short range correlations and small variance. The objective is to infer the statistics of the random concentration sampled at the plume center of mass, integrated over the transverse dimension of a two-dimensional spatial domain. The PDF/FDF problem can therefore be formulated in a two-dimensional domain as well, a spatial dimension and one in the concentration space. The upscaled drift and diffusion coefficients describing the PDF transport in the physical space are estimated on single-trajectories of diffusion in velocity fields with short-range correlations, owing to their self-averaging property. The mixing coefficients describing the PDF transport in concentration spaces are parameterized by the trend and the noise inferred from the statistical analysis of an ensemble of simulated concentration time series, as well as by classical mixing models. A Gaussian spatial filter applied to a Kraichnan velocity field generator is used to construct coarse-grained simulations (CGS) for FDF problems. The purposes of the CGS simulations are two-fold: first to understand the significance of the FDF approach from a practical point of view and its relation to the PDF approach; second to investigate the limits of the mixing models considered here and the desirable features of the mixing models for groundwater systems.
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