耦合多孔介质达西流与裂隙非达西流的有限节点闭合解析方法

Double porosity of porous medium and fractures is a typical aquifer formation worldwide. Analytical methods for double-porosity model assume uniform fracture distribution in aquifers. Numerical methods such as finite element method (FEM) have two problems of difficult grid development and much computing time although it can deal with irregular fracture distribution. Parallel computing efficiency is limited to data transfer between computers, which restricts development of inverse problems. This proposal develops a promising method called “finite nodal point closed-form solution” that can solve the two problems and release the assumption. Consider a pumped fractured confined aquifer where the formation can be divided into several porous zones and fractures. Closed-form well and kernel functions for non-Darcian flow are derived based on Izbash’s law and Padé approximation. A closed-form kernel function for Darcian flow is also derived. An analytical solution of drawdown for a porous zone can be expressed as a simple series expanded by the individual kernel functions. Similarly, an analytical solution of drawdown for a fracture can be expressed as the well function plus a simple series expanded by the individual kernel functions. Domains inside a porous zone and a fracture are meshless but the interface between them is discretized as finite nodal points. Each of the points provides two continuity requirements of drawdown and flux to couple the analytical solutions for porous zone and fracture. Compared with FEM, the proposed method can take even less computing time to predict accurate drawdown in fractured confined aquifers. This has not only guide to groundwater production but also implications for petroleum and natural gas extractions.

多孔介质和裂隙的双重孔隙含水层是全世界普遍的地质情况。解析方法假设裂隙是均匀分布的连续体，无法处理实际复杂的裂隙分布。有限元法等数值方法能考虑实际的裂隙分布，但存在网格建构困难和计算耗时两个问题，即使多台计算器并行计算，仍受限于计算器间的数据传输速度，限制了反问题的发展。为此，发展有限节点闭合解析方法，能克服上述问题。基于局限含水层抽水试验，应用Izbash定律和帕德近似法，发展裂隙非达西流井函数与基函数两个闭合解，每条裂隙的降深解析解表示为井函数和基函数展开的级数；多孔介质的降深解析解表示为达西流基函数展开的级数。多孔介质内部和裂隙内部无网格，交界处离散为有限节点，每个节点提供降深和流通量两个连续条件，来耦合多孔介质和裂隙的降深解析解。相较有限元法，本项目方法高效且精准模拟双重孔隙地下水流，是反问题求解的新途径。研究成果对地下水利用与保护、石油和天然气开采都具有重要理论意义和实践价值。

多孔介质和裂隙的双重孔隙含水层是全世界普遍的地质构造。解析方法假设裂隙是均匀分布的连续体，无法处理实际复杂的裂隙分布。有限元法等数值方法存在复杂网格建构、长计算时间、高内存需求等高计算成本的问题，即使多台计算机并行计算，仍受限于计算机间的数据传输速度，限制了反问题的发展。为此，本项目发展了耦合多孔介质达西流与裂隙非达西流的有限节点闭合解析方法（简称本方法），结合解析方法和数值方法的优点，克服了上述问题。针对承压、潜水、越流含水层的二维、三维、准三维地下水流，建构了标准、微水、振荡抽水试验的基函数（或称井函数）。此外，发展了河床非达西流与含水层达西流的耦合理论，导出了滞时、河床储水率、厚度、渗透系数的关系式，在既有的河床边界条件与源/汇项加入滯時，反映河床储水性的影响，提出更廣義的边界条件与源/汇项，大幅降低工程实践的计算成本。与传统数值方法相比，本方法高效且精准模拟地下水流，是反问题求解的新途径。研究成果对地下水利用与保护、石油和天然气开采均有重要的理论意义和实践价值。