MODFLOW-based analysis on seepage in discrete fissure networks
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摘要: 岩溶基岩裂隙水文研究考虑裂隙非均质和各向异性建模较难。裂隙−连续介质模型（FC）从原理上能够比较真实地刻画实际的渗流形态，是岩溶地区较理想的水文模型。基于裂隙空间形态及分布的统计特性，应用Monte Carlo模拟技术和图论，采用MATLAB 程序生成二维裂隙网络原图及三级连通图，将第三级连通图映射到有限差分网格，建立离散裂隙网络（DFN）与MODFLOW相结合的FC渗流模型，编制相应的渗流模拟程序。复现交叉裂隙试验工况，及通过2个多裂隙算例分析有限差分网格分辨率以及裂隙死端对稳定渗流模拟精度的影响，以DFN渗流为准则，验证FC模型代码编写的正确性和有效性。结果表明：FC模型模拟的节点水头、总网格流、裂隙长度在粗网格的高估或低估，通过细化网格可大大消除这种影响，裂隙死端对FC模型的MODFLOW网格流模拟具有不可忽略的影响。本研究实现了裂隙渗透性非均质和各向异性在数值计算单元的表达，将促进对基于连通性的裂隙流的MODFLOW模拟理解。Abstract:
Studies on karst bedrock fissures face challenge in modeling due to the heterogeneity and anisotropy of fissures. Groundwater primarily flows through fissure and pipe networks within karst aquifer systems, while the bedrock pores and micro-fissures mainly serve as water storage. Therefore, the ideal hydrological model for karst areas is the continuum–fissure–pipe model. Fissures, as the main pathway for groundwater flow, significantly influence seepage in fissure media due to their connectivity. The fissure–continuum model (FC), which integrates the advantages of continuum models and discrete fissure network (DFN) models, can realistically characterize the dual texture of karst aquifers. The FC model can also account for matrix diffusion and water exchange between the matrix and fissures. However, this model remains challenging to accurately determine water exchange between rock matrix and fissure media. This study was based on the statistical distribution of fissure geometry and hydraulic parameters. It utilized the Monte Carlo stochastic simulation technique and MATLAB program to generate a two-dimensional fissure network diagram that mirrored the distribution of actual rock fissures. The adjacency matrix of an undirected graph of graph theory was employed to represent the intersection relationships between fissures in the fissure network. In addition, this study utilized percolation theory to eliminate isolated fissures and clusters of isolated fissures that cannot conduct fluid. This approach facilitated the generation of a first-level connectivity diagram. Fissures with single nodes were then eliminated to create a second-level connectivity diagram, and dead-ends in fissures were eliminated to generate a third-level connectivity diagram. Based on the connectivity diagrams of fissure networks, the cubic law and the continuity equation for seepage were applied to establish a single-phase, saturated, and stable DFN model. Each fissure in the connectivity diagram was mapped onto a finite difference grid to develop an FC model that combines DFN with MODFLOW. When eliminating the fissure dead-ends, computational rounding may result in the loss of some nodes. Therefore, the second-level and third-level connectivity diagrams were separately used for DFN seepage simulation and FC seepage simulation, respectively. Simulation programs were developed to reproduce cross-fissure test scenarios, and two multi-fissure cases were analyzed to investigate the effects of finite difference grid resolution and fissure dead-ends on the accuracy of stable seepage simulations. The DFN was used as a metric to evaluate the correctness and validity of FC model code writing. This study was based on the statistical distribution of fissure geometry and hydraulic parameters. It utilized the Monte Carlo stochastic simulation technique and MATLAB program to generate a two-dimensional fissure network diagram that mirrored the distribution of actual rock fissures. The adjacency matrix of an undirected graph of graph theory was employed to represent the intersection relationships between fissures in the fissure network. In addition, this study utilized percolation theory to eliminate isolated fissures and clusters of isolated fissures that cannot conduct fluid. This approach facilitated the generation of a first-level connectivity diagram. Fissures with single nodes were then eliminated to create a second-level connectivity diagram, and dead-ends in fissures were eliminated to generate a third-level connectivity diagram. Based on the connectivity diagrams of fissure networks, the cubic law and the continuity equation for seepage were applied to establish a single-phase, saturated, and stable DFN model. Each fissure in the connectivity diagram was mapped onto a finite difference grid to develop an FC model that combines DFN with MODFLOW. When eliminating the fissure dead-ends, computational rounding may result in the loss of some nodes. Therefore, the second-level and third-level connectivity diagrams were separately used for DFN seepage simulation and FC seepage simulation, respectively. Simulation programs were developed to reproduce cross-fissure test scenarios, and two multi-fissure cases were analyzed to investigate the effects of finite difference grid resolution and fissure dead-ends on the accuracy of stable seepage simulations. The DFN was used as a metric to evaluate the correctness and validity of FC model code writing. Results show that the total flow simulation values and head simulation of the fine grid under the two test conditions are in good agreement with the theoretical values of DFN. However, the node heads and total grid flow simulated by the FC model in two cases are either overestimated or underestimated in the coarse grid. Although elongated paths of grid flow Lg between cells are corrected by increasing the permeability coefficients of the cells, the coarse grid based on MODFLOW cannot accurately analyze the length of inclined fissures Lf on the grid. If the size of selected coarse grid is larger than the distance between two adjacent fissures, part of the two adjacent fissures will be merged in the grid, thereby shortening the actual flow path. Secondly, the length of inclined fissure Lf. is either overestimated or underestimated. The inaccurate analysis of flow paths will lead to an increase in the estimation error of flow rates and heads, especially in dense fissure zones near the center of the model domain. This effect can be greatly alleviated by refining the grid. The fissure dead-ends have non-negligible effects on MODFLOW grid flow. In this study, the MODFLOW-based FC model can be used to solve the pressure distributions between interconnected network of fissures and the rock matrix, overcoming the difficulty of determining water exchange between rock matrix and fissure media. This study can realize the expression of heterogeneity and anisotropy of fissure permeability in numerical computational units, which will promote the understanding of MODFLOW simulation of fissure flow based on connectivity. Furthermore, The size of the MODFLOW grid should be determined in combination with the size of the study area, which should be small enough to capture the detailed features in the flow, and large enough to reduce the computational time and cost. Generally, the grid size is determined by a trial algorithm. When FC approach is applied to the actual watershed, the spatial variability of topography and landform factors should also be considered, such as coupling surface elevation DEM data. -
Key words:
- DFN /
- connectivity /
- MODFLOW /
- seepage /
- fracture-continuum model
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图 3 图2的连通图
Figure 3. Connectivity diagram of Fig. 2. (a) 1st- round cutting, (b) 2nd-round cutting, (c) 3rd -round cutting
图 4 沿网格的流动路径示意图
(a) 两条裂隙融合在一个单元格；(b) 倾斜裂隙的垂直分量$ L_{e} \sin (\theta) $ 小于阶梯模式的垂直分量$ L_{f} \sin (\theta) $；(c) 倾斜裂隙的垂直分量$ L_{e} \sin (\theta) $大于阶梯模式的垂直分量$ L_{f} \sin (\theta) $
Figure 4. Sketch map of flow path along the grid
(a) mergence of two fractures within a grid, (b) actual vertical component of the inclined fracture $ L_{e} \sin (\theta) $ shorter than that of the ‘‘stair step’’ pattern $ L_{f} \sin (\theta) $, and (c) actual vertical component of the inclined fracture $ L_{e} \sin (\theta) $ longer than that of the ‘‘stair step’’ pattern $ L_{f} \sin (\theta) $
图 5 水平线、倾斜线分别表示I型、II型裂隙单元格
Figure 5. Flow path along the grid,the horizontal line represents fissure cell of Type I,and the inclined line represents fracture cells of Type II
图 7 裂隙网络的某一次随机实现
Figure 7. Stochastic generation of fissure networks (one realizations for Case 1 (7a) and Case 2 (7b))
图 8 图7a的连通图（算例1）
Figure 8. Connectivity diagram of Fig. 7a for Case 1: (a) 2nd-round cutting;(b)3rd-round cutting
图 9 图7b的连通图（算例2）
Figure 9. Connectivity diagram of Fig. 7b for Case 2: (a) 2nd-round cutting; (b) 3rd -round cutting; (c) subdomain 1 for Fig. 9(a); (d) subdomain 2 for Fig. 9(a)
表 1 交叉流总流量的实测值、有限差分值、有限元值与理论值对比
Table 1. Comparison of measured values, finite difference values, finite element values, with theoretical values of cross flow
计算工况 精确解QDFN/cm2·s−1 Q实测/cm2·s−1 有限差分网格分辨率/cm QMOD/cm2·s−1 Q有限元/cm2·s−1 1 11.1792 11.6667 (4.36%)Δ=6.300 12.1960 (9.10%)11.1111 (0.61%)Δ=3.150 11.4499 (2.42%)2 5.2789 Δ=5.625 5.5743 (5.60%)Δ=4.500 5.3619 (1.57%)
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表 2 交叉节点水头的有限差分值与理论值对比
Table 2. Comparison of finite difference values and theoretical values of intersection heads
计算工况 精确解HDFN/cm 有限差分网格分辨率/cm HMOD/cm 绝对误差/cm 1 39.219 Δ=6.300 39.193 −0.026 Δ=3.150 39.218 −0.001 2 8.066 Δ=5.625 8.088 0.022 Δ=4.500 8.049 −0.017
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表 3 裂隙特征参数的统计分布（算例1）[22]
Table 3. Statistical distribution of characteristic parameters of fractures for Case 1[22]
裂隙组号 密度 方向 θ/° 迹长 l/ m 中心点
位置导水系数Tf/m2·s−1 N·m−2 分布 μ σ 分布 μ σ 分布 μ σ 分布 1 0.10 Poisson 15.0 8.0 正态 5.5 0.5 负指数 均匀分布 10−6 100.5 对数正态 2 0.10 Poisson 126.0 21.0 正态 6.5 0.5 负指数 均匀分布 10−6 100.5 对数正态
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表 4 裂隙节点坐标和模拟水头（算例1，单位: m）
Table 4. Node coordinates and simulated heads by DFN and MODFLOW for Case 1 (Unit: m)
节点 ⑨ ⑩ ⑪ ⑫ ⑬ ⑭ ⑮ ⑯ ⑰ ⑱ ⑲ ⑳ ㉑ X 0.74 3.17 4.88 7.33 7.87 8.74 8.93 6.83 1.27 4.28 6.76 3.43 5.40 Y 2.93 3.38 2.52 1.28 1.64 2.22 2.34 3.19 7.28 5.45 3.94 7.80 5.73 HDFN 9.75 8.24 7.54 6.79 6.62 6.31 6.14 6.66 6.83 7.37 6.61 6.70 6.77 HMOD, 0.02 9.75 8.23 7.51 6.76 6.60 6.28 6.14 6.63 6.82 7.36 6.58 6.68 6.75 HMOD, 0.05 9.76 8.21 7.49 6.71 6.55 6.25 6.13 6.59 6.79 7.31 6.55 6.66 6.72 HMOD, 0.1 9.75 8.19 7.44 6.63 6.48 6.16 6.10 6.52 6.79 7.29 6.49 6.68 6.68 Err. HMOD, 0.02 0.00 −0.01 −0.03 −0.03 −0.02 −0.03 0.00 −0.03 −0.01 −0.01 −0.03 −0.02 −0.02 Err. HMOD, 0.05 0.01 −0.03 −0.05 −0.08 −0.07 −0.06 −0.01 −0.07 −0.04 −0.06 −0.06 −0.04 −0.05 Err. HMOD, 0.1 0.00 −0.05 −0.10 −0.16 −0.14 −0.15 −0.04 −0.14 −0.04 −0.08 −0.12 −0.02 −0.09 节点 ㉒ ㉓ ㉔ ㉕ ㉖ ㉗ ㉘ ㉙ ㉚ ㉛ ㉜ ㉝ X 7.00 7.58 6.72 7.72 8.16 7.08 8.88 8.97 9.13 9.23 8.96 9.79 Y 4.05 3.45 6.06 4.37 3.64 8.68 5.63 5.48 5.54 5.04 0.00 5.28 HDFN 6.54 6.47 6.58 6.32 6.28 5.88 5.08 5.07 5.06 5.14 6.62 5.02 HMOD, 0.02 6.52 6.44 6.57 6.31 6.27 5.88 5.08 5.07 5.06 5.14 6.60 5.02 HMOD, 0.05 6.49 6.41 6.54 6.29 6.25 5.87 5.07 5.07 5.06 5.13 6.55 5.02 HMOD, 0.1 6.43 6.35 6.50 6.24 6.22 5.86 5.08 5.07 5.06 5.12 6.48 5.01 Err. HMOD, 0.02 −0.02 −0.03 −0.01 −0.01 −0.01 0.00 0.00 0.00 0.00 0.00 −0.02 0.00 Err. HMOD, 0.05 −0.05 −0.06 −0.04 −0.03 −0.03 −0.01 −0.01 0.00 0.00 −0.01 −0.07 0.00 Err. HMOD, 0.1 −0.11 −0.12 −0.08 −0.08 −0.06 −0.02 0.00 0.00 0.00 −0.02 −0.14 −0.01
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表 5 裂隙节点坐标和模拟水头（算例2，单位: m，部分节点）
Table 5. Node coordinates and simulated heads by DFN and MODFLOW for Case 2 (Unit: m，partial nodes )
节点 ⑭ ⑮ ⑯ ⑰ ⑱ ⑲ ⑳ ㉑ ㉒ ㉓ ㉔ ㉕ ㉖ X 3.17 3.25 3.28 3.30 3.66 3.74 3.88 3.98 4.68 4.75 5.03 6.19 6.22 Y 12.56 12.58 16.53 12.48 14.24 12.18 16.09 18.02 13.34 18.34 13.03 10.56 11.16 HDFN 8.88 8.86 8.78 8.86 8.64 8.83 8.71 8.65 8.58 8.65 8.55 8.55 8.42 HMOD, 0.04 8.87 8.86 8.77 8.87 8.64 8.83 8.70 8.64 8.58 8.64 8.55 8.55 8.42 HMOD, 0.1 8.90 8.89 8.80 8.88 8.68 8.86 8.73 8.67 8.60 8.67 8.58 8.57 8.44 HMOD, 0.2 8.94 8.93 8.85 8.93 8.75 8.90 8.80 8.74 8.67 8.74 8.65 8.60 8.55 Err. HMOD, 0.04 −0.01 0.00 −0.01 0.01 0.00 0.00 −0.01 −0.01 0.00 −0.01 0.00 0.00 0.00 Err. HMOD, 0.1 0.02 0.03 0.02 0.02 0.04 0.03 0.02 0.02 0.02 0.02 0.03 0.02 0.02 Err. HMOD, 0.2 0.06 0.07 0.07 0.07 0.11 0.07 0.09 0.09 0.09 0.09 0.10 0.05 0.13 节点 ㉗ ㉘ ㉙ ㉚ ㉛ ㉜ ㉝ ㉞ ㉟ ㊱ ㊲ ㊳ ㊴ X 6.56 7.02 7.05 7.05 7.07 7.22 7.39 7.82 7.82 7.94 7.96 7.99 8.00 Y 10.68 11.27 9.99 11.28 11.23 10.90 10.95 8.91 9.48 11.13 11.41 10.42 10.29 HDFN 8.46 8.38 8.40 8.38 8.38 8.37 8.35 8.26 8.28 8.30 8.33 8.29 8.29 HMOD, 0.04 8.47 8.38 8.39 8.38 8.38 8.36 8.35 8.26 8.28 8.30 8.34 8.29 8.29 HMOD, 0.1 8.49 8.40 8.41 8.40 8.40 8.39 8.38 8.28 8.30 8.33 8.36 8.32 8.31 HMOD, 0.2 8.55 8.49 8.44 8.49 8.49 8.47 8.47 8.35 8.37 8.44 8.45 8.42 8.41 Err. HMOD, 0.04 0.01 0.00 −0.01 0.00 0.00 −0.01 0.00 0.00 0.00 0.00 0.01 0.00 0.00 Err. HMOD, 0.1 0.03 0.02 0.01 0.02 0.02 0.02 0.03 0.02 0.02 0.03 0.03 0.03 0.02 Err. HMOD, 0.2 0.09 0.11 0.04 0.11 0.11 0.10 0.12 0.09 0.09 0.14 0.12 0.13 0.12
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表 6 MODFLOW网格倾斜长度、阶梯长度与DFN流动路径对比（算例1，单位: m，部分节点）（括号里的值表示Lf和Le的误差）
Table 6. Comparison of MODFLOW inclined fissure length, stair step length and DFN flow path (Case 1, Unit: m，partial nodes) (The bracket value indicates error between Lf and Le)
节点 DFN 倾斜裂隙长度 阶梯长度 Le 粗网格
Lf (0.10 m)细网格
Lf (0.05 m)精细网格
Lf (0.02 m)粗网格
Lg (0.10 m)细网格
Lg (0.05 m)精细网格
Lg (0.02 m)㉙-㉛ 0.52 0.50(−0.02) 0.51(−0.01) 0.53(+0.01) 0.70 0.70 0.72 ㉛-⑦ 1.52 1.48(−0.04) 1.50(−0.02) 1.51(−0.01) 2.00 2.05 2.06 ㉛-㉝ 0.62 0.54(−0.08) 0.60(−0.02) 0.62 0.70 0.80 0.82 ㉝-⑤ 0.22 0.22 0.22 0.22 0.30 0.30 0.28 ㉝-⑥ 0.22 0.20(−0.02) 0.21(−0.01) 0.22 0.20 0.25 0.28 ㉚-④ 0.93 0.85(−0.08) 0.92(−0.01) 0.92(−0.01) 1.10 1.20 1.18 ㉚-㉝ 0.71 0.67(−0.04) 0.70(−0.01) 0.70(−0.01) 0.90 0.90 0.90 ㉒-㉕ 0.79 0.76(−0.03) 0.78(−0.01) 0.79 1.00 1.05 1.04 ㉕-㉖ 0.84 0.81(−0.03) 0.87(+0.03) 0.83(−0.01) 1.10 1.20 1.14 ㉕-㉛ 1.64 1.66(+0.02) 1.63(−0.01) 1.64 2.20 2.15 2.16 ㉖-⑮ 1.52 1.53(+0.01) 1.50(−0.02) 1.52 2.10 2.05 2.08 ⑪-⑯ 2.06 2.09(+0.03) 2.06 2.06 2.60 2.60 2.62 ⑪-⑫ 2.74 2.82(+0.08) 2.75(+0.01) 2.73(−0.01) 3.80 3.70 3.66 ⑯-㉓ 0.78 0.76(−0.02) 0.79(+0.01) 0.78 1.00 1.00 1.00 ⑯-⑬ 1.86 1.80(−0.06) 1.87(+0.01) 1.87(+0.01) 2.50 2.60 2.60 ⑫-⑬ 0.64 0.64 0.65(+0.01) 0.64 0.90 0.90 0.88 ㉓-⑭ 1.69 1.70(+0.01) 1.66(−0.03) 1.71(+0.02) 2.40 2.35 2.42 ⑬-⑭ 1.05 1.08(+0.03) 1.04(−0.01) 1.05 1.50 1.45 1.46 ㉓-㉖ 0.61 0.63(+0.02) 0.63(+0.02) 0.61 0.80 0.80 0.78 ⑲-㉒ 0.26 0.32(+0.06) 0.27(+0.01) 0.26 0.40 0.35 0.34 ㉒-㉓ 0.83 0.78(−0.05) 0.81(−0.02) 0.82(−0.01) 1.10 1.15 1.16 ⑯-㉓ 0.78 0.76(−0.02) 0.79(+0.01) 0.78 1.00 1.00 1.00 ⑭-⑮ 0.22 0.22 0.22 0.23(+0.01) 0.30 0.30 0.32 ㉘-㉙ 0.18 0.22(+0.04) 0.18 0.17(−0.01) 0.30 0.25 0.24 ㉙-㉚ 0.17 0.22(+0.05) 0.16(−0.01) 0.16(−0.01) 0.30 0.20 0.20 ㉘-㉚ 0.27 0.32(+0.05) 0.27 0.28(+0.01) 0.40 0.35 0.36
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表 7 MODFLOW网格倾斜长度、阶梯长度与DFN流动路径对比（算例2，单位: m，部分节点）（括号里的值表示Lf和Le的误差）
Table 7. Comparison of MODFLOW inclined fissure length, stair step length and DFN flow path (Case 2, Unit: m，partial nodes) (The bracket value indicates the error between Lf and Le)
节点 DFN 倾斜裂隙长度 阶梯长度 Le 粗网格
Lf (0.20 m)细网格
Lf (0.10 m)精细网格
Lf (0.04 m)粗网格
Lg (0.20 m)细网格
Lg (0.10 m)精细网格
Lg (0.04 m)⑤-⑨ 0.89 0.89 0.89 0.88(−0.01) 1.20 1.20 1.16 ⑤-⑭ 1.20 1.28(+0.08) 1.22(+0.02) 1.19(−0.01) 1.80 1.70 1.64 ⑨-④ 0.60 0.63(+0.03) 0.63(+0.03) 0.62(+0.02) 0.80 0.80 0.76 ⑨-⑩ 0.60 0.63(+0.03) 0.63(+0.03) 0.59(−0.01) 0.80 0.80 0.76 ⑨-⑭ 0.67 0.60(−0.07) 0.61(−0.06) 0.66(−0.01) 0.60 0.70 0.80 ④-⑧ 0.72 0.85(+0.13) 0.71(−0.01) 0.74(+0.02) 1.20 1.00 1.04 ⑲-⑰ 0.53 0.57(+0.04) 0.58(+0.05) 0.52(−0.01) 0.80 0.80 0.72 ⑭-⑮ 0.08 0.20(+0.12) 0.10(+0.02) 0.08 0.20 0.10 0.08 ⑭-⑰ 0.15 0.20(+0.05) 0.14(−0.01) 0.17(+0.02) 0.20 0.20 0.24 ⑰-⑮ 0.11 0.00(−0.11) 0.10(−0.01) 0.13(+0.02) 0.00 0.10 0.16 ㉘-㉛ 0.06 0.00(−0.06) 0.00(−0.06) 0.06 0.00 0.00 0.08 ㉘-㉚ 0.03 0.00(−0.03) 0.00(−0.03) 0.04(+0.01) 0.00 0.00 0.04 ㉚-㉛ 0.05 0.00(−0.05) 0.00(−0.05) 0.04(−0.01) 0.00 0.00 0.04 ㉛-㉜ 0.36 0.45(+0.09) 0.45(+0.09) 0.36 0.60 0.60 0.48 ㉛-㉝ 0.43 0.45(+0.02) 0.42(−0.01) 0.43 0.60 0.60 0.60 ㉜-㉝ 0.18 0.00(−0.18) 0.14(−0.04) 0.16(−0.02) 0.00 0.20 0.20 ㉜-㉟ 1.54 1.52(−0.02) 1.52(−0.02) 1.56(0.02) 2.00 2.00 2.04 ㉝-㊱ 0.58 0.63(+0.05) 0.63(+0.05) 0.59(+0.01) 0.80 0.80 0.76 ㉝-㊳ 0.80 0.72(−0.08) 0.78(−0.02) 0.79(−0.01) 1.00 1.10 1.12 ㉙-㉞ 1.33 1.28(−0.05) 1.28(−0.05) 1.32(−0.01) 1.80 1.80 1.84 ㉙-㉟ 0.92 0.89(−0.03) 0.94(+0.02) 0.92 1.20 1.30 1.28
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