土壤和裂隙介质中反常扩散的分数阶变导数建模

Contaminant transport in natural soil and fractured media is an important topic of applied research in pollution prediction and control.Numerous experimental and field observations have found that such transport in general do not obey Fick's second law and is of anomalous diffusion. In recent decades, fractional derivative diffusion models have been successfully applied to anomalous diffusion of this type. However, these fractional models only consider the heterogeneity of medium and do not involve the influence of temporal and spatial variations of medium structures or porosity. Moreover, little experiment has been done and reported concerning their mechanism in literature.Based on my research work in recent five years, the objective of this proposal is to establish a variable-order fractional diffusion model to describe contaminant transport in such varying soils or fractured media. We will combine lab experiments and numerical modeling to explore underlying physical mechanism. The key point is the analysis of non-Fickian feature via the physical significance of fractional variable-order derivative and then investigates mechanism underlying solute transport in soils and fractured media. The purpose is to develop a variable-order fractional advection-diffusion equation model with clear physical mechanism and easy-to-get parameters to characterize the evoluation trend and main features of contaminant concentration transport. This study will offer a mechanical theoretical model for prediction and control of water pollution in complex environments.

自然环境土壤和裂隙介质中污染物迁移过程是污染预测和治理的一个重要应用基础研究课题。大量的实验和现场观测发现这些迁移过程一般不符合Fick第二定律，是一类反常扩散。近年来分数阶导数扩散方程已成功应用于描述此类反常扩散现象。但是常规的分数阶扩散模型仅考虑介质的非均质性，未涉及介质结构或孔隙率随时空变化的影响，此外，有关的机理实验也鲜有报道。基于申请者近5年来的研究工作，本项目拟发展分数阶变导数扩散模型，反映介质非均质性的时空变化对扩散行为的影响；采用溶质迁移机理实验与数值模拟相结合的方法，探明模型阶数与介质结构的关系。从变导数的物理意义入手，结合非Fick扩散特征分析，研究土壤和裂隙介质中溶质迁移演化的机理。本项目的目标是建立物理机理明确，参数估计简单的变导数对流-扩散方程模型，刻画污染物浓度的演化趋势和主要特征，评价污染物对水环境的长期影响，为复杂环境中水污染预测和治理提供力学理论模型。

土壤和地下含水层中溶质迁移过程中介质结构或孔隙率往往随时空变化，因而溶质的扩散行为也随时空变化。本项目发展了分数阶变导数扩散模型，刻画介质结构时空变化对扩散行为的影响，数值模拟结果与实验数据吻合，达到了预期目标。通过分数阶导数扩散方程模型数值结果分析与人工分形结构中粒子随机运动模拟，初步掌握了模型参数、扩散统计规律与介质结构的关系。项目团队经多年探索，引入了分形导数尺度变换的概念，提出了分形导数Richards 方程模型，解释了非饱和介质中水分扩散的非Boltzmann 尺度律现象，准确刻画了水分扩散过程的时间尺度效应。为了提高数值模拟的精度和速度，建立了求解时间分数阶导数扩散方程的半离散有限单元法，实现了模型的长时间高精度模拟，数值结果和实验数据的对比表明新算法能够准确地描述溶质扩散过程，并预测长时间演化趋势。项目还开发了复杂介质中溶质粒子随机运动模型的Matlab工具箱，为深入研究和工程应用提供了基础。成果方面，团队成员基于上述研究内容，发表了学术论文12篇，其中SCI检索论文9篇，综述论文1篇,软件著作权1项。人才培养方面，依托项目培养了博士研究生4名，硕士研究生2名，其中2名研究生已顺利获得博士学位。项目也极大地促进了项目组成员的学术交流，共资助学术会议10次，学术访问2次。综上所述，项目在溶质反常扩散分数阶导数建模方面达到了预期目标，促进了科研团队的发展，培养了科研人才。

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