连续时间随机行走理论在地层反常输运过程中的应用

The phenomenon of anomalous transport in geological formations has attracted much interest recently among researchers in the fields of petroleum engineering, soil science, hydrology, etc, mainly due to the growing number of experiments that directly or indirectly verify such a common anomalous scenario, both field and laboratory, where the breakthrough curve have systematic errors compared with the predictions that the conventional advection-dispersion equation would make, or the pressure response of a reservoir qualitatively deviates from the standard exponential decay form and rather manifests a power law behavior. Continuous time random walk (CTRW) formulations have been demonstrated to successfully capture the most relevant characteristic of transport in complex porous media, say, the divergent average waiting time of a passive random walker, which naturally gives rise to the anomalous transport. Numerical and theoretical results within the CTRW framework show extremely good agreement with the experimental data. Also, the development of the visualization of the internal pore and throat structure of rocks helps to establish a representative complex pore network model of rocks, the parameters of which reflect the geometric properties of the original rock specimen. Thus the CTRW model should be modified so as to take the pore network’s geometrical properties into account. In this work, our goal is to establish, based on the modified CTRW model, a series of governing equations of different types of anomalous transport, including the mass transfer, heat conduction, and pressure wave spread processes, each of which can be of paramount importance in petroleum engineering and groundwater sciences. We will try to solve these equations analytically, or find their asymptotical solutions, in certain coordinate systems and with various sets of boundary conditions, which are the major concerns from an engineering-based perspective. Moreover, we will numerically solve them, mainly using the de Hoog algorithm for numerical inverse Laplace transform, and plot the profiles of the probability density function of the particles. As a comparative study, we will systematically examine the theories for anomalous transport, such as the fractal media theory, the non-constant coefficients theory (where the transport coefficients are position-dependent), and the CTRW theory. The physical origins of each theory and whether they are in accord with basic physical laws are checked, and we intend to by doing this work to show the advantages of the CTRW theory over other existing methods when the anomalous transport in geological media is concerned. Our work is expected to be helpful in analyzing experimental data and in identifying the underlying mechanism of the observed phenomenon.

地层中流体的输运过程对于石油工程和地下水科学研究具有重要的意义，近年来一系列现场和实验室实验，以及数值模拟的结果均表明地层中普遍存在反常输运行为，这种输运过程明显偏离了经典的对流扩散方程所描述的情况。连续时间随机行走(CTRW)理论是特别适合处理该问题的理论工具。本项目将主要研究两个方面的问题：1)利用CTRW理论推导地层中反常输运的输运方程，具体针对物质输运、热传导、压力传播等重要的实际物理过程，对可能出现的不同类型的反常输运过程进行分析建模，并在具有明确工程实际背景的边界条件和坐标系下对方程进行理论求解和数值模拟；2)系统的比较利用CTRW理论得到的输运方程与文献中其它的反常输运方程，包括考虑空间分形和输运系数为位置的函数等情况所得到的输运方程，我们将着重研究各方程的物理基础和适用范围，进而论证CTRW理论的优越性，以期指导今后的理论工作和实验数据分析。

本课题基本按照之前制定的研究计划，对地层中的反常输运（包括溶质分子输运、热传递、压力传播）这一问题进行了研究，获得如下阶段性结果：1. 基于连续时间随机行走理论，我们推导出了一个明确的分数阶对流扩散方程，它很可能适用于描述异质性不是很强的地层中的反常输运过程。这类反常输运过程表现为粒子的均方位移随时间的标度指数介于1和2之间，该类过程很可能在以砂岩为代表的地层介质中普遍存在，而砂岩是地壳表层数千米范围内最常见的岩石类型。我们的这个工作对于地下水污染颗粒、油气分子等的运移和预测有望起到比较重要的作用。该工作结果发表于Physical Review E。2. 连续时间随机行走模型是研究反常输运的出发点之一，我们提出了一个有效的Monte Carlo算法直接对该过程进行模拟。我们工作的核心内容是针对Berkowitz等人提出的粒子等待时间分布提出了一个精确的，高效率的抽样方法。基于该算法，我们成功重现了三种反常输运粒子均方位移随时间的标度关系和粒子位置随时间的分布。该工作发表于《计算物理》杂志。3. 我们还研究了一般复杂网络上的输运过程与网络结构的关系，该部分内容的相关结果在整理中，拟于近期内投稿。综上，本课题进展较为顺利，完成了预期结果（“形成2篇左右的论文”）。