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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塔里木盆地轮古西油田是典型的碳酸盐岩岩溶缝洞型油气藏。可溶性岩层内部不同大小和规模的洞穴及其网络系统、洞穴充填和垮塌作用以及不同地貌和构造背景下表层岩溶带的差异性发育特征决定了储集空间发育及其连通状况在空间上的强烈非均质性，并由此导致了油藏流体的流动和分布样式的复杂性。因此如何从不同尺度上来认识这类油藏的储集空间结构和流体分布样式是揭示其油气成藏规律和高效开发这类油藏的关键。从油藏宏观结构上，目前研究者们[1-2]逐步趋向于将其定义为由若干互不连通的缝洞单元组成的潜山油藏，早前也有学者[3-5]将这种油气藏称为“网络状油气藏”，指出该类油气藏没有统一油水界面，流体压力变化无常，油气富集于大断裂、背斜轴及古地貌“山梁”等，油气聚集单体规模很小，群体可构成大规模含油区的特征；而将单个缝洞单元称为“油元”[6]。这为宏观上整体认识该类油藏的层次性格架奠定了基础。但多年的规模性开发生产发现潜山高部位，并不意味着具有高的油气柱高度。一些学者[7-11]通过现场资料统计法、实验室测定法以及间接测定法等方法对单井油水界面的量化计算，发现在潜山油藏内油井油气水产出深度差异明显，油柱高度往往超过局部残丘的闭合幅度，远大于圈闭的油气充满度，并不受潜山圈闭的控制[12]，但总体具有“山高水高”的特点[13]；为此韩剑发等[14]明确提出了轮南奥陶系碳酸盐岩次生残余“准层状油气藏模式”，突出了潜山风化壳缝洞型储集体的准层状特点对油气分布的控制，并通过对单井烃类底界做趋势面分析，认识到了烃类底界面与潜山顶面具有正相关性。刘静江等[15]也证实了轮南潜山油水界面呈倾斜状的分布特征；韩杰等[16]则进一步指出地表岩溶沟谷趋势面可以指示油水界面分布，并阐述了不同井区油水界面存在差异性。可以看出，随着研究工作的不断深化，研究者们逐步认识到了潜山油藏整体区域性油水界面具有准层状、倾斜分布特征，而且与地表岩溶残丘和岩溶沟谷存在内在联系，这表明尽管不同地区单井油水界面差异比较大，却仍有规律可寻，但对油水界面遵循何种规律，特别是与岩溶水文地貌之间的具体关联性，以及相应的机理仍缺乏深入研究。
针对这一科学问题，基于现代岩溶理论，笔者拟通过对岩溶地表水系的详细分级刻画，以及单井油水界面的确定，采用趋势面分析方法建立不同级次残丘与油水界面的关系，并运用油气圈闭理论阐述这种分布规律的流体动力学机理。
1. 油田地质概况
轮古西地区位于塔里木盆地塔北隆起轮南低凸起西斜坡，东临轮古西大断裂，西接哈拉哈塘凹陷，北侧以轮台断裂为界，南侧为塔河油田，是3个凹陷油气的长期运移指向区，勘探面积近200 km2(图 1)。自加里东中期—海西早期，经历了多期构造运动。其中，海西早期运动是最重要的一期构造运动，地层遭受抬升剥蚀，下奥陶统鹰山组地层大范围暴露地表，遭受大气淡水岩溶改造，形成典型岩溶缝洞储层，是主要产油层系之一[17]。古地貌上，海西早期整体呈现一个西北倾向的宽缓斜坡，但由东至西可大致可划分为岩溶高地、岩溶斜坡和岩溶洼地3个三级岩溶地貌单元[18]。整个宽缓斜坡因地表岩溶沟谷切割而发育了若干个大小不同形态各异的岩溶残丘。
图 1 轮古西奥陶系潜山油田构造位置图Figure 1. Map showing the tectonic location of Lunguxi buried hill oil field轮古西奥陶系油水关系复杂，各类建产井59口，开井40口，其中试采井试采初期(90 d)无水和低含水井比例占59.5%。截止至2017年6月，所有油井已全部含水生产，90%以上的油井为高含水井和特高含水井。地面原油总体表现为超重质普通稠油特征，平均密度1.08 g/cm3，地层水呈弱酸性，水型为CaCl2型，平均密度1.12 g/cm3[19]。该区复杂的地质条件决定了该油藏油水关系的复杂性，含水变化的规律性差，油水界面标定更加复杂。
2. 单井油水界面的确定
轮古西油田奥陶系油藏储层复杂，确定油水界面的难度较大。但工区内存在直接钻遇水体的井(如LG421井)和直接钻遇油水界面的井(如LG15-2井)，因此可以获得较丰富的水体压力资料。借鉴前人对于缝洞单元划分的结果，采用直接法、静压交会法以及原始地层压力法综合推算研究区油水界面。
直接法是指钻井钻遇了油藏的油水界面，通过录井分析、测井资料解释、试油等方法来确定出油水界面的海拔深度[7]，这种方法的优势在于确定出来的油水界面相对准确。静压交会法的理论核心是油层和水层在油水界面处压力相等，利用随钻压力测试或者生产后的静压测试建立储层油层和水层深度与压力的关系，根据此关系，求取相同压力下水层(或油层)海拔即预测的油水界面[8]。原始地层压力法通过生产井压力恢复曲线确定得出原始地层压力，以及结合取样测试得到的地层流体密度资料综合分析，建立油层中深的压力与中深静水压力差和油藏厚度的关系，测算出油水界面所处深度[9]。
根据单井是否钻遇油水界面、无水采油期的长短，将生产井分为以下三类：①单井钻遇油水界面，则可以通过直接法来推算油水界面位置；②单井未钻遇油水界面，又可分为两类：a.投产即产水，油水界面定在产层井底位置；b.有无水采油期，使用静压交会法、原始地层压力法综合推算油水界面；③单井未钻遇有效储层，则无法确定油水界面位置。计算结果如表 1。
表 1 轮古西部分单井油水界面计算Table 1. Calculated results of oil-water interface for some wells in Lunguxi oil field井号 潜山面深度/m 综合预测油水界面/m 距离潜山面深度/m LG15 -4 785.5 -4 928.05 142.55 LG15-1 -4 754.76 -4 928.05 173.29 LG15-2 -4 822.55 -4 928.05 105.5 LG15-4 -4 805.63 -4 865.63 60 LG15-4-1 -4 736.28 -4 783.28 47 LG15-19 -4 722.73 -4 774.73 52 LG15-22 -4 709.42 -4 771.42 62 LG15-26 -4 733.47 -4 771.97 38.5 LG15-33 -4 725.14 -4 816.14 91 LG41 -4 602.91 -4 692.91 90 LG43 -4 634.54 -4 703.54 69 LG45 -4 828.07 -4 880.57 52.5 LG9 -4 608.3 -4 668.3 60 LG902 -4 636.49 -4 686.49 50 3. 海西早期岩溶残丘级次划分
3.1 岩溶地表古水系刻画及分级
基于高精度的三维地震数据的精细解释结果，采用印模法[20]对岩溶古地貌进行恢复，再通过趋势面分析、最大负曲率及精细相干分析以及混频切片等多项技术[21]对研究区古水系进行了精细刻画(图 2)。水系分级采用Strahler[22]提出的规则定义，但级序标定正好相反(图 3)，即将河源不再分支的小支流作为第四级河道，2支第四级河道相汇合后，汇合点以下为三级河道，同理，2支第三级支流汇合点以下河段为第二级河道，2支第二级支流汇合点以下河段为不再汇流的第一级河道。研究区水系具体标定结果见图 4。
图 2 轮古西中上奥陶系古地貌及古水系恢复图Figure 2. Geomorphic units and fluviokarstic surface river system on the top of Lower to Middle Ordovican in Lunguxi area
图 4 轮古西中-下奥陶统顶面河道分级图Figure 4. Distribution of different grade of river valley on the top of Lower to Middle Ordovican in Lunguxi area可以看出，轮古西地区奥陶系古岩溶地表水系受海西早期岩溶古地貌东高西低格局的影响，整体由东向西汇流，平面上主要呈树枝状结构。干流与支流几乎呈直角相交，具有明显的四级分级特征。河道的级次可反映水域面积、河道长短、流速等水文特征，对应形成的沟谷在面积、下切深度及宽度也具明显差异。高级次的水系水域面积较大，河道较长、水流量大，形成的沟谷下切深度大、延伸长、沟谷较宽。若从三维角度观视各级沟谷水系的谷底构成的底面，所构成的二维曲面则由西向东呈逐步抬升趋势，并且宏观上与古地貌的抬升呈协变关系。
3.2 岩溶残丘级次划分
在沟谷分级的基础上，采用趋势面拟合技术，将不同级次的沟谷底界进行大网格的平滑处理，形成的平滑趋势面即为不同级次的残丘趋势面，再将Tg52(奥陶系鹰山组顶部不整合面)与其相减，正地形代表岩溶残丘(图 5)。以二级残丘为例，将工区内所有二级沟谷按照沟谷海拔拟合出二级沟谷趋势面，所对应的二级残丘的范围为二级沟谷的限定区域。通过该方法，分别拟合出一级残丘对应的一级沟谷趋势面和二级残丘对应的二级沟谷趋势面。因三级、四级沟谷下切深度较低，本文主要讨论一级、二级沟谷趋势面对油水界面的控制作用。
图 5 沟谷趋势面拟合技术原理[52]Figure 5. Technic principle about trend surface analysis of river valley按照沟谷分级结果，对残丘进行分级(图 6)。可以看出轮古西主要分布5个一级残丘，每个一级残丘之内存在若干个二级残丘。其中分布范围最广的一级残丘为LG9残丘，包含7个二级残丘；生产井数量最多的一级残丘为LG15残丘，包含7个二级残丘和23口生产井。
图 6 轮古西残丘分级图Figure 6. Distribution of first- and second-grade monadnock of buried hill in Lunguxi area4. 轮古西油水分布规律
4.1 残丘级次与油水界面分布的关系
通过合成地震记录进行井震标定，将各产能井单井计算的油水界面标定至地震剖面，并与一、二级沟谷趋势面线进行对比。可以发现位于各级残丘内的油井油水界面与一、二级残丘趋势面具有明显相关性。
在LG15一级残丘上(图 6, 7, 9)，二级残丘Ⅲ中LG15-40、LG15-11、LG15-18三口井的油水界面深度略高于一级残丘趋势面，LG15、LG15-1、LG15-2、LG15-3、LG15-6和LG15-12井油水界面深度略低于一级残丘趋势面，只有LG15-8井油水界面略高于二级残丘趋势面油水界面。因此整体上看，大部分生产井油水界面十分靠近一级残丘趋势面，油水界面分布主要受一级残丘控制。
图 7 轮古15井区生产井连井地震剖面图(剖面位置见图 9)Figure 7. Seismic section integrated oil-water intersurface of wells and the two grade trend surface of river valley in Lungu 15 wells area
图 9 轮古西地区奥陶系油藏油水界面受控残丘趋势面级次分布图Figure 9. Distribution of oil-water interfaces in different grade monadnock of buried hill in Lunguxi area而在LG9一级残丘中(图 6, 8, 9)，二级残丘Ⅰ内LG15-20、LG902-1和LG902三口井的油水界面深度落在二级残丘趋势面之上，LG422井为一口干井，无法计算油水界面。除LG42井油水界面深度高于一级残丘趋势面外，该残丘内生产井整体油水界面基本高于二级残丘趋势面，因此该残丘内油水界面分布主要受二级残丘控制，但残丘内油气充注度不高。
图 8 轮古9井区生产井连井地震剖面图(剖面位置见图 9)Figure 8. Seismic section integrated oil-water intersurface of wells and the two grade trend surfaces of river valley in Lungu 9 wells area通过上述井震结合分析，可以得到轮古西地区奥陶系油藏油水界面受控残丘趋势面级次分布图(图 9)。全区残丘次级对油水界面的控制显现出一定的规律性，西南部残丘主要为一级，中部主要为二级残丘，东北部目前尚无钻井。控油残丘级次自SW向NE方向递减。这种特征同时也展现出与古地貌的对应性，从西南向北东，随着岩溶古地貌的逐步抬升，油水界面是逐步抬升的。同时油井产能与残丘级次分布密切相关：产量高、油柱高度大的生产井油水界面往往受一级残丘趋势面控制，是油气的富集区域。
研究过程中，同时还发现该区尚有一部分高产井(如LG15-1、LG15-2、LG15-3、LG15-6、LG15、LG15-12、LG15-17、LG42)的油水界面并不完全受控于一级沟谷趋势面，均要深于一级沟谷趋势面，说明对油水界面的控制另有原因。通过研究发现油水界面深于残丘厚度的井主要沿着一条NE-SW方向的断层分布，而且整体上油水界面顺着断层延伸方向自SW方向朝NE方向逐渐升高。初步判断，这些油井深的油水界面可能主要与北东向延伸的深断裂有关，说明这条断层可能是一条油源断层，它不仅控制了该区油的垂向运移，使得断裂带附近发育的缝洞系统具有比周围邻区更深的油水界面外，同时还存在自南向北的横向运移，使得油水界面向北逐步抬升。
4.2 油水界面分布模式
根据Gussow[23]提出的油气差异聚集原理，油气在区域倾斜背景上向上倾方向的系列圈闭运移过程中，自上倾方向的空圈闭向下倾方向变为纯油藏-油气藏-纯气藏，系列圈闭中每个圈闭的油气充注受局部圈闭闭合点的控制。轮古西油田原油性质主要为重质油，溶解气油比低。因此由西南向北东方向的残丘圈闭中充注的油气类型上并没有显现出差异。主要的不同在于尽管油水界面也是逐渐向上抬升的，但自轮古15残丘向轮古9残丘分别由一级沟谷趋势面控制向二级沟谷趋势面控制变化，2个一级残丘中油水界面并没有受更高级次的残丘趋势面控制，这显然不能单纯用经典油气差异聚集原理进行解释。其原因主要是因为岩溶储层的横向非均质性造成了油气聚集单元的相对独立性。原油从运移近源低部位向远源高部位运移过程中，近源低部位发育的低级次深沟谷作为溢出点控制了油水界面分布，分别进入各个相对独立的缝洞单元，控制的油柱高度大。向上游运移远源方向，随谷底抬升变浅，沟谷级次升高，控制的相应溢出点上升，导致各缝洞单元内的油水界面也抬升，同时原油充满度逐步降低(图 10)。因此这种差异聚集特征主要是由岩性非均质性和岩溶沟谷联合作用造成的，虽然本质上仍属于差异聚集，但有别于经典理论中受单一局部构造溢出点的控制。
图 10 轮古西地区油水界面与残丘级次关系模式图Figure 10. Schematic model showing the relationship between oil-water interfaces to trend surfaces in Lunguxi area5. 结论
(1) 轮古西地区奥陶系古潜山油藏岩溶残丘依古岩溶地表水系分级可以划分出多个级次，残丘级次与油水界面密切相关，油水界面分布自西南的轮古15残丘向中部的轮古9残丘逐步抬升，分别由一级沟谷趋势面控制向二级沟谷趋势面控制变化。
(2) 油水界面的这种变化规律有别于经典差异聚集原理，并非受单一局部构造溢出点控制，而是由岩性非均质性和逐步抬升的岩溶沟谷溢出点联合作用造成。
(3) 联合作用造成的差异聚集不仅控制了油水界面的系统变化，更与油井产能密切相关，其中产量高、油柱高度大的生产井油水界面往往更受一级残丘趋势面的控制，是油气富集的主要区域。
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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\{ {bh=b2mJRC−2.50,us<0.75uspbh=b1/2mJRCmob,us⩾usp} \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\{ {bh=bm1+Z2.252,Re<1bh=bm1+Z2.252+(6×10−5+4×10−3)(Re−1),Re⩾1} \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\{ {bh=α+βbm,us<uspbh=αeβbm,us⩾usp} \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] $ bh=(−0.327Q+0.5)+(0.311Q−0.5)⋅σb+(0.144Q−0.232)⋅R+(−0.546Q+0.761)⋅bm $ 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] $bh=0.177 3Q−0.263 5(−0.307 1ln(Z2)+1.428 7)×(0.133 3bm+0.094 6) $ 数值模拟 甘磊等[36] $bh=−135.6B2e+24.332Be−5.231 4lnt−1 396.9B2e−185.86Be+63.709 $ 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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