Characterization model for the equivalent hydraulic aperture of a nonmatching fracture based on the MIC
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摘要:
等效水力开度可以定量表征粗糙裂隙在达西流态下的导流能力，等效水力开度的精准预测对很多实际应用工程具有重要的意义。粗糙裂隙的等效水力开度受控于复杂的壁面形貌和开度分布等几何特征，综合考虑裂隙几何信息，通过最大互信息系数（MIC）的方法确定了等效水力开度的主控因子，并基于主控因子建立了粗糙裂隙等效水力开度的表征模型。首先，基于Barton 10条标准曲线构造了900组非吻合粗糙裂隙，通过壁面几何信息得到13个几何参数并采用数值模拟获取了所有裂隙的等效水力开度试验值，然后，采用最大互信息系数方法分析了等效水力开度与13个几何参数之间的相关性，共确定了4个主控因子，并基于主控因子建立了粗糙裂隙等效水力开度的表征模型。基于900个粗糙裂隙数据，选取已有的2个等效水力开度表征模型进行了对比分析。分析结果显示本研究提出的水力开度预测模型具有更好的表征性能。最后，研究了尺寸效应对建立等效水力开度表征模型的影响，并讨论了将该模型推广至三维裂隙的方法。
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关键词:
- 粗糙裂隙 /
- 等效水力开度 /
- 壁面形貌 /
- 裂隙开度 /
- 最大互信息系数（MIC）
Abstract:Equivalent hydraulic aperture can quantitatively characterize hydraulic conductivity in rough fractures under Darcy flow conditions, making significance for various engineering applications.
Objective The equivalent hydraulic aperture of rough fractures is influenced by complex geometric features such as wall topography and aperture distribution. This study comprehensively considers fracture geometry, applies the maximal information coefficient (MIC) method to identify key factors influencing equivalent hydraulic aperture, and develops a characterization model based on it.
Methods 900 sets of nonmatching rough fractures were generated through Barton's 10 standard curves. Geometric information from fracture walls provided 13 parameters, and numerical simulations were used to obtain the equivalent hydraulic apertures. MIC considers the correlations between equivalent hydraulic aperture and 13 geometric parameters.
Results Four key factors were identified to form the basis of a characterization model for the equivalent hydraulic aperture of rough fractures.
Conclusion 900 rough fracture datasets validated the model's performance against two existing models, with the proposed model being more advanced than the current representations. The study also discussed the impact of size effects on hydraulic aperture models and the application for three-dimensional cases.
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图 1 非吻合裂隙样本构建示意图
No.1，···，No.10分别为标准曲线编号；bmin为最小开度；bi为关键点A、B之间的垂直距离
Figure 1. Schematic diagram of nonmatching fracture sample construction
图 3 裂隙No.72网格无关性分析结果
Figure 3. Results of the irrelevance analysis for No.72 fracture mesh
图 5 基于MIC建立bh表征模型流程图
Figure 5. Flowchart of establishing the bh characteristic model based on the MIC
图 8 不同模型计算值与预测值的对比
Figure 8. Comparison of calculated values and predicted values based on different models
图 11 不同尺寸模型试验值与预测值的对比
Figure 11. Comparison of experimental and predicted values based on different size models
表 1 等效水力开度预测方程
Table 1. Prediction equations for the equivalent hydraulic aperture
资料来源 表达式 方法 符号描述 PATIR等[10] ${b_{\text{h}}} = {b_{\text{m}}}{\left( {1 - 0.9{\xi ^{ - 0.56/{C_v}}}} \right)^{1/3}}$ 理论分析 bh为等效水力开度；bm为力学开度平均值；ξ为绝对突起高度；Cv为力学开度变异系数；Δb为力学开度增量；f为0.5～1.0之间的常数；$\eta $为经验常数；ε为裂隙表面积比常数；JRC为节理粗糙度系数；σb为力学开度均方根；τ为曲折系数；m为粗糙影响系数；JRC0为初始粗糙度系数；us为剪切位移；usp峰值处剪切位移；JRCmob为动粗糙度系数；Z2为一阶导数均方根；$\alpha '$为指数函数常数；L为二维粗糙裂隙剖面线的投影长度；D为分形维数；C为接触比；D*为相对分形维数；α和β均为拟合系数；Q为流量；R为粗糙度；（−αi+αi*）为拉格朗日乘子；Xi为支持向量；X为输入变量；A0/A为初级粗糙度归一化面积；C*为无量纲拟合参数；$ \sigma_\Pi $为次级粗糙度标准差；σc为静水压力；Re为雷诺数；t为时间；Be为充填隙宽，下同 WITHERSPOON等[11] ${b_{\text{h}}} = {b_{\mathrm{m}}} + f\Delta b$ 室内试验 WALSH[12] ${b_{\text{h}}} = {b_{\text{m}}}{\left[ {\left( {1 + \eta \varepsilon } \right)\left( {1 - \varepsilon } \right)} \right]^{ - 1/3}}$ 理论分析 BARTON等[13] ${b_{\text{h}}} = b_{\mathrm{m}}^2JR{C^{ - 2.5}}$ 室内试验 AMADEI等[14] ${b_{\text{h}}} = {b_{\text{m}}}{\left[ {1 + 0.6\left( {{\sigma _b}/{b_{\text{m}}}} \right)} \right]^{ - 1/3}}$ 数值模拟 RENSHAW[15] ${b_{\text{h}}} = {b_{\text{m}}}\exp \left( { - \sigma _b^2/2} \right)$ 理论分析 ZIMMERMAN等[16] $ {b_{\text{h}}} = {b_{\text{m}}}{\left[ {\left( {1 - 1.5\sigma _b^2/b_{\text{m}}^2} \right)\left( {1 - 2\varepsilon } \right)} \right]^{1/3}} $ 理论分析 WAITE等[17] $ {b_{\text{h}}} = {b_{\mathrm{m}}} \cdot {\tau ^{ - 1/3}} $ 理论分析 OLSSON等[18] $ \left\{ {\begin{array}{*{20}{l}} {{b_{\text{h}}} = b_{\text{m}}^2JRC_0^{ - 2.5},{u_{\mathrm{s}}} < 0.75{u_{{\mathrm{sp}}}}} \\ {{b_{\text{h}}} = b_{\text{m}}^{1/2}JR{C_{{\text{mob}}}},{u_{\mathrm{s}}} \geqslant {u_{{\mathrm{sp}}}}} \end{array}} \right. $ 室内试验 LIU[19] ${b_{\text{h}}} = {b_{\text{m}}}{\left[ {1 + \left( {\sigma _b^2/b_{\mathrm{m}}^2} \right)} \right]^{ - 1/2}}$ 理论分析 SCESI等[20] $ {b_{\text{h}}} = b_{\mathrm{m}}^{2/3}{\left[ {1 + 8.8\left( {0.5 - {b_{\mathrm{m}}}/2JR{C^{2.5}}} \right)} \right]^{ - 1/2}} $ 理论分析 QIAN等[21] ${b_{\text{h}}} = {b_{\mathrm{m}}}{\left( {\tau \cdot m} \right)^{1/3}}$ 理论分析 LI等[22] $ \left\{ {\begin{array}{*{20}{l}} {{b_{\text{h}}} = \dfrac{{{b_{\mathrm{m}}}}}{{1 + Z_2^{2.25}}},{Re} < 1} \\ {{b_{\text{h}}} = \dfrac{{{b_{\mathrm{m}}}}}{{1 + Z_2^{2.25} + \left( {6 \times {{10}^{ - 5}} + 4 \times {{10}^{ - 3}}} \right)\left( {{Re} - 1} \right)}},{Re} \geqslant 1} \end{array}} \right. $ 数值模拟 LIU等[23] ${b_{\text{h}}} = {\left( {4/\pi \alpha '} \right)^{4 - 2{D}}}{L^{{D} - 1}}$ 理论分析 ZOORABADI等[24] ${b_{\text{h}}} = {b_{\mathrm{m}}}\left( {0.991\ 2 - 4.53 \times {{10}^{ - 6}} \times JR{C^{3.303}}} \right)$ 室内试验 XIE等[25] $ {b_{\text{h}}} = {b_{\text{m}}}{\left[ {0.94 - 5{{\left( {{\sigma _b}/{b_{\mathrm{m}}}} \right)}^2}} \right]^{1/3}} $ 数值模拟 王报等[26] ${b_{\text{h}}} = {b_{\mathrm{m}}}\left( {0.991\ 2 - 1.525 \times {{10}^{ - 6}}JR{C^{3.76}}} \right)$ 数值模拟 CHEN等[27] $ {b_{\text{h}}} = {b_{\mathrm{m}}}{\left( {1 - 1.1C} \right)^4}{\left( {1 + 2/{D^*}} \right)^{3/5}} $ 室内试验 CAO等[28] $ \left\{ {\begin{array}{*{20}{l}} {{b_{\text{h}}} = \alpha + \beta {b_{\mathrm{m}}},{u_{\mathrm{s}}} < {u_{{\mathrm{sp}}}}} \\ {{b_{\text{h}}} = \alpha {e^{\beta {b_{\mathrm{m}}}}},{u_{\mathrm{s}}} \geqslant {u_{{\mathrm{sp}}}}} \end{array}} \right. $ 室内试验 鲁俊杰[29] $ {b_{\text{h}}} = {b_{\text{m}}}\left[ {1 - {\text{exp}}\left( { - 0.388\ 96{b_{\mathrm{m}}}/{\sigma _b}} \right)} \right] $ 室内试验 XIAO等[30] $ \begin{aligned} {b_{\text{h}}} = &\left( { - 0.327Q + 0.5} \right) + \left( {0.311Q - 0.5} \right) \cdot {\sigma _b} + \\ &\left( {0.144Q - 0.232} \right) \cdot R + \left( { - 0.546Q + 0.761} \right) \cdot {b_{\mathrm{m}}} \end{aligned} $ 数值模拟 SUN等[31] $ {b_{\text{h}}} = \displaystyle\sum\limits_{i = 1}^{206} {\left( { - {\alpha _i} + \alpha _i^*} \right){{\mathrm{e}}^{ - 0.939\left\| {{X_i},X} \right\|}} + 0.247\ 3} $ 数值模拟 TAN等[32] $ {b_{\text{h}}} = {\left[ {{A_0}/A\left( {1 - 1/{C^*}} \right)} \right]^{0.4}}\left[ {{b_{\mathrm{m}}} - 1.7\left( {\sigma _\Pi ^2/{b_{\mathrm{m}}}} \right)} \right] $ 理论分析 YIN等[33] $ {b_{\text{h}}} = {b_{\mathrm{m}}}\exp \left( { - 0.686\ 5Z_2^{0.579\ 8}/b_{\mathrm{m}}^{ - 0.887\ 6}} \right) $ 室内试验 赵鹏等[34] $ {b_{\text{h}}} = 16.7 + 72.54\exp \left( { - {\sigma _{\mathrm{c}}}/10.2} \right) $ 室内试验 ZHANG等[35] $\begin{aligned} {b_{\text{h}}} = &0.177\ 3{Q^{ - 0.263\ 5}}\left( { - 0.307\ 1\ln \left( {{Z_2}} \right) + 1.428\ 7} \right)\times \\&\left( {0.133\ 3{b_{\mathrm{m}}} + 0.094\ 6} \right) \end{aligned} $ 数值模拟 甘磊等[36] $\begin{aligned} {b_{\text{h}}} =& { - 135.6B_{\text{e}}^2 + 24.332{B_{\mathrm{e}}} - 5.231\ 4} \ln t - \\ & {1\ 396.9B_{\text{e}}^2 - 185.86{B_{\mathrm{e}}} + 63.709} \end{aligned} $ 室内试验 
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表 2 不同模型预测性能对比结果
Table 2. Comparison result of the predictive performance of different models
模型 预测公式 使用参数 NOF 本研究 $ {b_{\mathrm{h}}} = \left( {{Z_2} + 3.139} \right)\sqrt {{b_{\mathrm{m}}}{b_{{\mathrm{min}}}}} /\sqrt {5.469 + {R_{\mathrm{t}}}} $ Rt，Z2，bm，bmin 0.280 YIN等[33] $ {b_{\mathrm{h}}} = {b_{\mathrm{m}}}\exp \left( { - 0.686\ 5Z_2^{0.579\ 8}/b_{\mathrm{m}}^{ - 0.887\ 6}} \right) $ Z2，bm 0.531 RENSHAW[15] ${b_{\text{h}}} = {b_{\text{m}}}\exp \left( { - \sigma _b^2/2} \right)$ σb，bm 0.616
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表 3 LIU等[44]裂隙模型的几何参数和水力参数
Table 3. Geometric and hydraulic parameters of the fracture model of LIU et al
模型序号 JRC 力学开度/mm bh试验值/mm bh预测值/mm 1 0～2 0.80 0.775 0.716 2 0～2 1.00 0.956 0.945 3 0～2 1.32 1.365 1.311 4 8～10 0.56 0.596 0.641 5 8～10 0.80 0.811 0.916 6 8～10 1.20 1.182 1.174 7 18～20 0.42 0.393 0.479 8 18～20 0.70 0.697 0.599 9 18～20 0.95 0.885 1.084
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表 4 Synfrac软件生成裂隙的输入参数
Table 4. Input parameters for fracture generation via Synfrac software
序号 L/mm ML TL/mm σb/mm WLmax WLmin D An 1 150 2.5 90 0.517 0.99 −0.06 2.1 1.07 2 200 2.5 90 0.617 0.99 −0.06 2.2 1.07 3 250 2.5 90 0.717 0.99 −0.06 2.3 1.07 注：L代表裂隙的物理尺寸；ML代表裂隙的不匹配波长；TL代表转变长度；σb代表力学开度标准差；WLmax代表最大匹配因子；WLmin代表最小匹配因子；D代表分形维数；An代表各向异性系数
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