Discrete fracture network seepage analysis based on MODFLOW
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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:
Karst bedrock fractures are difficult to model in hydrological research due to their heterogeneous and anisotropic. Groundwater can mainly flow in the fractures and conduits of the karst aquifer system, and the bedrock voids and micro-fractures mainly play the role of water storage, so the ideal hydrological model of the karst area is the continuum-fracture (conduit) model. As the main path of groundwater flow, the connectivity of fractures has a great influence on the seepage of fracture media. The fracture-continuum model (FC) generated by combining the merits of the continuum model and the discrete fracture network model (DFN) can realistically describe the binary structure of karst aquifer system in principle, and the diffusion of matrix as well as the water exchange between matrix and fracture can also be considered, but there is a problem that the water exchange between matrix and fracture medium is difficult to determine. This study is based on the statistical distribution of fracture geometry and hydraulic parameters, the two-dimensional fracture network original maps were generated using Monte Carlo random simulation technology and MATLAB program with the same distribution as actual rock fractures. The adjacency matrix of undirected graph of graph theory is used to characterize the intersecting relationship between fractures in the fracture network. Using percolation theory to remove fracture segment which cannot conduct fluid, such as the 1st connected maps were generated by removing the isolated fractures (fracture clusters), and the 2nd connected maps generated by removing fractures with single nodes, as well as the 3rd connected maps generated by means of cutting fracture dead ends. The DFN model in a single-phase, saturated, steady-state flow system was established based on the connected graph of fracture network, applying the cubic law and the continuity equation of seepage. Mapping of single fractures of the connected graph onto finite difference grids, FC seepage model was built by combining DFN approach and MODFLOW software. When deleting the fracture dead ends, some nodes may be lost due to computer trade-off, as a result, the 2nd connected graph and the 3rd connected graph are used for DFN and FC approaches, respectively. The corresponding simulation programs were developed, with which reproducing cross fracture test conditions, and constructing two multi-fractures cases to analyze the impact of finite difference grid resolutions and fracture dead end on stable seepage simulation accuracy. The DFN flow solution is used as a metric to evaluate the correctness and effectiveness of FC model code writing. Research results show that the simulated total flow and head of the fine grid are in good agreement with the theoretical values of DFN under the two test conditions. The node head and total grid flow simulated by the FC model of two cases are overestimated or underestimated in the coarse grid. Although a longer path of grid flow Lg from cell to cell is corrected by increasing cell hydraulic conductivity, the coarse grid based on MODFLOW cannot accurately analyze the inclined fracture length Lf on the grid. If the selected coarse grid size is greater than the distance between two adjacent fractures, a part of the two adjacent fractures will merge in the grid, thereby shortening the actual flow path. Secondly, the overestimation or underestimation of the inclined fracture length Lf.. Inaccurate analysis of the flow path will lead to an increase in the estimation error of flow rate and head, especially in the densely fractured areas near the center of the model domain. Refining the grid can greatly eliminate this error. The fracture dead-end has non-negligible effect on MODFLOW grid flow. In this study, the MODFLOW-based FC model can be used to solve for pressure distributions both within the interconnected network of fractures and the rock matrix, and could also be a solution for the water exchange between the rock matrix and the fractures is difficult to determine. This study can achieve the expression of heterogeneity and anisotropy of fracture hydraulic conductivities in numerical calculation units and will promote the understanding of MODFLOW simulation of fracture 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 maps 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) two fractures merged 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, horizontal hatching represents Type I fracture cells, and inclined hatching represents Type II fracture cells
图 7 裂隙网络的某一次随机实现
Figure 7. Stochastic generation of fracture networks (one of the realizations for Case 1 (7a) and Case 2 (7b))
图 8 图7a的连通图（算例1）
Figure 8. Connectivity maps of Fig. 7a for Case 1: (a) 2nd-round cutting, and (b) 3rd -round cutting
图 9 图7b的连通图（算例2）
Figure 9. Connectivity maps of Fig. 7b for Case 2: (a) 2nd-round cutting, (b) 3rd -round cutting, (c) subdomain 1 for Fig. 9(a), and (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 QMOD/cm2·s−1 Q有限元/cm2·s−1 1 11.1792 11.6667 (4.36%)Δ=6.300 cm 12.1960 (9.10%)11.1111 (0.61%)Δ=3.150 cm 11.4499 (2.42%)2 5.2789 Δ=5.625 cm 5.5743 (5.60%)Δ=4.500 cm 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 head
计算工况 精确解HDFN/cm 有限差分网格分辨率 HMOD/cm 绝对误差/cm 1 39.219 Δ=6.300 cm 39.193 −0.026 Δ=3.150 cm 39.218 −0.001 2 8.066 Δ=5.625 cm 8.088 0.022 Δ=4.500 cm 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 fracture 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 fracture length, stair step length and DFN flow path (Case 2, 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 (0.20 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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