Impacts of Non-Darcian Flow in the Fracture on Flow Field and Solute Plumes in a Fracture-Aquifer System
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摘要: 以基岩渗流方向与裂隙轴向呈45度角为例，探讨了当裂隙轴向与基岩水流斜交时，裂隙流的非达西程度对流场及溶质运移的影响.使用Comsol Multiphysics多物理场仿真软件构建了一个在中部包含水平单裂隙的正方形多孔介质模型，裂隙中的非达西流用Izbash方程去刻画，裂隙的上游基岩中存在持续的溶质源.随着裂隙水流的非达西程度逐渐增强，流场及污染物分布表现出如下特征：（1）裂隙中水流流速逐渐增大；流线在裂隙与基岩界面处的折射逐渐偏离折射定律；（2）裂隙水流的流向逐渐偏向基岩水流的渗流方向；（3）溶质羽宽度变宽但对称性逐渐降低；（4）溶质在水平方向上的浓度峰值逐渐降低，右侧浓度逐渐升高；（5）裂隙产生的回弥散对溶质运移作用逐渐增强，使裂隙中更多的溶质运移到了上层基岩中.总体而言，裂隙流的非达西程度对流场及污染物分布有着显著的影响.Abstract: In this study, a mathematical model with the intersection angle between the flow direction and the x axial direction of 45degree was developed to investigate the effect of non-Darcy parameter n on flow field and solute migration when flow is oblique crossing the fracture. The Comsol Multiphysics software was used to do such numerical simulation. The simulation area was assumed to be square with a horizontal fracture embedded in the middle.The flow in the fracture was assumed to be non-Darcian and can be described by the Izbash equation. A constant solute source has been assigned in the upper matrix. The results indicate the following phenomena as the power index in the Izbash equation(n) increases:(1) the fracture flow velocity increases, and the flow line at the matrix-fracture interfaces gradually deviates from the classical fraction law; (2) the flow direction in fracture gradually turns to the direction in the matrix; (3) the width of solute plume increases, while the symmetry of solute plume reduces; (4) the peak solute concentration in the horizontal section diminishes, and the solute concentration on the right sight of the model increases; (5) the effect of back dispersion caused by fracture on solute plume becomes stronger, which results in more solutes in the fracture migrated to the upper matrix. Overall, the non-Darcy flow fracture has a significant impact on the flow field and solute distribution.
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图 2 裂隙下边界流线位移（Dfl）和溶质峰值位移（Dc）示意图
Fig. 2. Schematic diagram of flow line displacement (Dfl) and peak solute concentration displacement (Dc)
图 3 流场及溶质分布示意图
a.模型不存在裂隙；b.存在裂隙且n=1时
Fig. 3. Schematic diagrams of flow field and solute distribution
图 4 不同n取值时，裂隙中流速（V）及其在x和y方向上的分量值（Vx；Vy）
Fig. 4. The velocity and the component of velocity on x and y axes in the fracture for different n values
图 5 不同情况下流线在裂隙与基岩界面处的折射示意图
Fig. 5. Schematic diagram of flow line refraction at the matrix-fracture interfaces for different conditions
图 6 溶质羽的分布
a.不存在裂隙；b. n=1.0；c. n=1.1；d. n=1.2；e. n=1.3；f. n=1.4时
Fig. 6. Solute plume distributions
图 7 不同n取值时裂隙上边界浓度分布
Fig. 7. Solute distributions at the upper boundary of the fracture for different n values
图 8 不同n取值时裂隙下边界浓度分布
Fig. 8. Solute distributions at the bottom boundary of the fracture for different n values
表 1 模型参数及默认值
Table 1. Parameters used in this study and default values
参数名称 符号 默认取值 高渗透性裂隙 PFF 参考等值线浓度 C0 0.001 (mol/m3) 溶质源浓度 Cs 1 (mol/m3) 基岩孔隙度 θm 0.1 基岩渗透系数 Km 1×10-5 (m/s) 纵向弥散度 αl 5×10-3 (m) 横向弥散度 αt 5×10-4(m) 裂隙隙宽 2b 0.01 (m) 裂隙孔隙度 θf 0.2 裂隙渗透系数 Kf 1×10-4 (m/s) 基岩渗流速度 Vm ≈1.44×10-6 (m/s) 裂隙渗流速度 V - 非达西系数 n 1.0~1.4 基岩渗流角度 δ 45° 裂隙渗流角度 θ - 流线位移 Dfl - 裂隙下边界浓度峰值位移 Dc - 
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表 2 Comsol Multiphysics数值解与Robinson and Werner (2017)解析解对比
Table 2. Comparison between numerical results of Comsol Multiphysics and analytical solutions of Robinson and Werner(2017)
2b(cm) 浓度峰值位移Dc (m) 浓度峰值Cb-max (mol/m3) Robinson and Werner (2017) Comsol Multiphysics Robinson and Werner (2017) Comsol Multiphysics 0.25 0.000 95 0.000 95 0.112 88 0.111 44 0.50 0.002 20 0.002 33 0.104 17 0.103 30 1.00 0.040 90 0.041 00 0.028 36 0.028 38 2.00 0.102 86 0.103 00 0.016 46 0.016 46
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表 3 不同情况下裂隙流场的结果.
Table 3. Characteristics of flow field in the fracture
n 不存在裂隙 1.0 1.1 1.2 1.3 1.4 V (m/s) 1.44×10-6 1.00×10-5 2.90×10-5 6.98×10-5 1.46×10-4 2.74×10-4 Vx(m/s) 1.02×10-6 9.95×10-6 2.88×10-5 6.93×10-5 1.45×10-4 2.71×10-4 Vy(m/s) 1.02×10-6 9.95×10-7 3.20×10-6 8.40×10-6 1.90×10-5 3.84×10-5 tanθ 1.00 10.00 9.00 8.25 7.62 7.07 θ(°) 45.0 84.2 83.7 83.0 82.3 81.9
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表 4 不同情况下Cb-max、Dc和Dfl的值
Table 4. Values of Cb-max, Dc and Dfl for different conditions
n 无裂隙 1.0 1.1 1.2 1.3 1.4 Cb-max(mol/m3) 0.089 5 0.023 6 0.015 25 0.009 0 0.005 24 0.003 18 Dfl(m) 0.010 0 0.110 0 0.100 00 0.092 5 0.086 20 0.080 70 Dc(m) 0.010 0 0.050 0 0.060 00 0.070 0 0.078 50 0.086 00
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