• 中文核心期刊
  • 中国科技核心期刊
  • RCCSE中国核心学术期刊

卸荷砂岩渐进破坏及渗透特性的中间主应力效应研究

杜佳慧, 李文璞, 常悦, 王泽, 王涛

杜佳慧,李文璞,常悦,等. 卸荷砂岩渐进破坏及渗透特性的中间主应力效应研究[J]. 煤矿安全,2024,55(6):19−29. DOI: 10.13347/j.cnki.mkaq.20230008
引用本文: 杜佳慧,李文璞,常悦,等. 卸荷砂岩渐进破坏及渗透特性的中间主应力效应研究[J]. 煤矿安全,2024,55(6):19−29. DOI: 10.13347/j.cnki.mkaq.20230008
DU Jiahui, LI Wenpu, CHANG Yue, et al. Study of intermediate principal stress effect on progressive damage and permeability characteristics of unloading sandstone[J]. Safety in Coal Mines, 2024, 55(6): 19−29. DOI: 10.13347/j.cnki.mkaq.20230008
Citation: DU Jiahui, LI Wenpu, CHANG Yue, et al. Study of intermediate principal stress effect on progressive damage and permeability characteristics of unloading sandstone[J]. Safety in Coal Mines, 2024, 55(6): 19−29. DOI: 10.13347/j.cnki.mkaq.20230008

卸荷砂岩渐进破坏及渗透特性的中间主应力效应研究

基金项目: 国家自然科学基金青年科学基金资助项目(51804211, 52204105);吕梁市重点研发资助项目(2022GXYF13)
详细信息
    作者简介:

    杜佳慧(1998—),女,山西太原人,硕士研究生,研究方向为安全工程及岩石力学。E-mail:415498364@qq.com

    通讯作者:

    李文璞(1986—),男,山西孝义人,副教授,博士,从事煤与瓦斯共采方面的教学与研究工作。E-mail:liwenpu@tyut.edu.cn

  • 中图分类号: TD315+.1

Study of intermediate principal stress effect on progressive damage and permeability characteristics of unloading sandstone

  • 摘要:

    为了探究中间主应力对卸荷砂岩渐进破坏及渗流特性的影响,基于多功能真三轴流固耦合试验系统,以含气砂岩为研究对象,进行不同中间主应力条件下的卸荷试验;得到与裂纹体应变、损伤变量相关的指数型渗透率模型。结果表明:随着中间主应力的增加,裂纹闭合应力、起裂应力、损伤应力均呈现增加趋势,弹性段与微小裂纹扩展阶段在峰前应变中的占比增加;在微小裂纹扩展阶段,中间主应力方向上的变形受到约束,对最小主应力方向产生的压缩效应随着中间主应力的增大而增强;在宏观裂纹扩展阶段,损伤应力与峰值应力比较接近,试样破坏呈脆性特征;随着中间主应力的增加,初期渗透率减少量呈下降趋势,渗透率升高点发生后移。

    Abstract:

    In order to investigate the effect of intermediate principal stress on the progressive damage and permeability characteristics of unloading sandstone, the unloading test was carried out under different intermediate principal stress conditions based on a multifunctional true triaxial fluid-solid coupling test system with gas-bearing sandstone as the research object; the exponential permeability model related to the cracked volume strain and damage variable were obtained. The results show that the crack closure stress, crack initiation stress and damage stress increase with the increase of intermediate principal stress, and the percentage of elastic section and microcrack extension stage in the pre-peak strain increases. The deformation in the direction of the intermediate principal stress is restrained in the microcrack extension stage, and the compression effect on the direction of the minimum principal stress is enhanced with the increase of the intermediate principal stress; in the macroscopic crack extension stage, the damage stress is closer to the peak stress, and the damage of the specimen is brittle in character. With the increase of the intermediate principal stress, the initial permeability reduction shows a decreasing trend and the point of elevated permeability shifts backward.

  • 煤储层含气量是煤层气资源勘探测试的关键参数[1-4],也是表征煤储层开发潜力和确保矿井瓦斯安全的关键参数之一[4-11];煤储层含气量测试可为煤层气资源量、储量估算和煤层气开发设计提供重要依据[4]。煤储层含气量测值的不准确性是导致煤层气资源计算出现差异的重要原因[12]。现阶段,我国煤炭资源勘探和煤层气资源勘探开发煤层含气量测试方法主要采用GB/T19559—2021《煤层气含量测定方法》提供的解吸法及其矫正计算方法[13-17],其中损失气量计算采用最初10个地面实测解吸气量数据,由损失气时间与解吸时间和的平方根与累计解吸气量之间线性关系倒推零时间解吸气量得出。

    应用上述方法,多年来我国在多个煤层气勘探开发区块获取了大量含气量数据。然而,目前煤层含气量计算仍有以下不足:吸附气欠饱和储层现场含气量测试的可靠性缺乏数值模拟验证,特别是损失气量估算的准确程度尚未有数值模型验证;另外,上述方法对含气饱和储层(吸附气、游离气均饱和,下同)含气量测试的准确性尚不清楚,饱和储层煤心含气量测试过程模拟尚未开展。基于此,选取新疆低煤化煤(本次为长焰煤)煤心为研究对象,构建了储层煤心含气量解吸-扩散数值模型,并通过模型计算分析了吸附气欠饱和与含气饱和储层煤心解吸动态,对比分析了数值模型与现场测试损失气量、解吸气量、残余气量构成的差异性,以期为我国低煤化储层煤层气勘探提供含气性分析新思路。

    按照GB/T 19559—2021《煤层气含量测定方法》国家标准提供的方法,开展自然煤心采样,记录采样及装样时间,开展自然解吸气量连续测试和残余气测试;根据GB/T 19560—2008《煤的高压等温吸附试验方法》、GB/T 212—2008《煤的工业分析方法》等开展煤等温吸附实验和工业分析。现场煤心采样及上述相关工作开展于新疆某长焰煤储层煤层气开发先导试验区,煤心样品为长焰煤(属于低煤化度煤),且煤心对应的实测含气饱和度分别为54.77%、77.51%、99.79%。工业分析及煤岩组分分析成果见表1

    表  1  煤心工业分析及煤岩组分分析成果
    Table  1.  Coal core industrial analysis and coal rock composition analysis results
    样品编号 直径/mm 长度/cm 宏观煤岩类型 平均镜质组最大反射率/% 水分/% 灰分/% 挥发分/% 镜质组组分/% 惰质组组分/% 壳质组组分/%
    BF-1 63.0 26.8 半亮煤 0.60 0.90 11.77 43.79 84.2 14.8 1.0
    BF-2 60.1 30.0 半亮煤 0.61 0.86 22.08 40.45 79.6 19.4 1.0
    BF-3 61.0 34.0 半亮煤 0.67 1.12 13.40 37.17 78.6 25.5 1.0
    下载: 导出CSV 
    | 显示表格

    煤是一种复杂的多孔介质,为了方便求解,通常对煤心样品做出以下假设:①煤屑由球形颗粒组成;②煤颗粒为均质、各向同性体;③CH4解吸-扩散遵从连续性原理;④扩散系数与浓度、时间和坐标无关;⑤煤屑瓦斯解吸为等温条件下的解吸过程;⑥煤心含气量测试过程中孔隙度不变。

    单位体积吸附气欠饱和储层煤心基质CH4质量可表述为(CH4全部吸附在煤基质表面,在储层压力条件下吸附解吸平衡,孔隙表面外不含游离CH4):

    $$ {m}_{\mathrm{m}\mathrm{u}}={\varphi }_{\mathrm{u}\mathrm{n}}(1-{\phi }_{\mathrm{m}})\left(\dfrac{100-{M}_{\mathrm{a}\mathrm{d}}-{A}_{\mathrm{a}\mathrm{d}}}{100}\right)\dfrac{{V}_{\mathrm{L}}{p}_{\mathrm{m}}}{{p}_{\mathrm{m}}+{p}_{\mathrm{L}}}{\rho }_{\mathrm{c}\mathrm{o}\mathrm{a}\mathrm{l}}{\rho }_{\mathrm{s}\mathrm{c}} $$

    式中:mmu为单位体积煤基质中赋存的瓦斯质量,g;φun为欠饱和储层含气饱和度,%;ϕm为基质孔隙度;Mad为空气干燥基水分含量,%;Aad为空气干燥基灰分产率,%;VL为朗缪尔体积,单分子层最大的吸附量,cm3/g;pm为基质孔隙中的CH4压力,MPa;pL为朗缪尔压力,吸附量为最大吸附量1/2时间的吸附平衡压力,MPa;ρcoal为煤视密度,kg/m3ρsc为标准状态下的CH4密度,g/cm3

    煤心解吸至大气环境控制方程:

    $$\begin{array}{c} \dfrac{\partial p}{\partial t}\left({\varphi }_{{\mathrm{un}}}\left(1-{\phi }_{{\mathrm{m}}}\right)\left(\dfrac{100-{{M}}_{\mathrm{a}\mathrm{d}}-{{A}}_{\mathrm{a}\mathrm{d}}}{100}\right)\dfrac{{V}_{{\mathrm{L}}}{p}_{{\mathrm{L}}}}{{\left(p+{p}_{{\mathrm{L}}}\right)}^{2}}{\rho }_{{\mathrm{coal}}}\dfrac{{M}_{{\mathrm{c}}}}{{V}_{{\mathrm{M}}}}\right)=\\ \nabla \left(\dfrac{{M}_{{\mathrm{c}}}}{RT}D\nabla p\right) \end{array} $$

    式中:t为时间,s;p为煤心孔隙气体压力,MPa;Mc为甲烷分子摩尔质量,kg/mol;VM为CH4分子摩尔体积,m3/mol;R为理想气体常数,J/(mol·K);T为煤层温度,K;D为扩散系数,10−9 m2/s。

    单位体积饱和储层煤心基质CH4质量可表述为(CH4在煤基质表面吸附饱和,且在储层压力条件下游离CH4充满孔隙):

    $$ \begin{array}{c} {m}_{{\mathrm{mo}}}=(1-{\phi }_{{\mathrm{m}}})\left(\dfrac{100-{{M}}_{\mathrm{a}\mathrm{d}}-{{A}}_{\mathrm{a}\mathrm{d}}}{100}\right)\dfrac{{V}_{{\mathrm{L}}}{p}_{{\mathrm{m}}}}{{p}_{{\mathrm{m}}}+{p}_{{\mathrm{L}}}}{\rho }_{{\mathrm{coal}}}{\rho }_{{\mathrm{sc}}}+\\ {\phi }_{{\mathrm{m}}}\dfrac{{M}_{{\mathrm{c}}}}{RT}{p}_{{\mathrm{m}}} \end{array} $$

    式中:mmo为初始条件下单位体积饱和储层煤心基质CH4质量,g。

    饱和储层煤心含气饱和度:

    $$ {\varphi }_{{\mathrm{m}}}=\dfrac{{m}_{{\mathrm{mo}}}}{(1-{\phi }_{{\mathrm{m}}})\left(\dfrac{100-{{M}}_{\mathrm{a}\mathrm{d}}-{{A}}_{\mathrm{a}\mathrm{d}}}{100}\right)\dfrac{{V}_{{\mathrm{L}}}{p}_{0}}{{p}_{0}+{p}_{{\mathrm{L}}}}{\rho }_{{\mathrm{coal}}}{\rho }_{{\mathrm{sc}}}} $$

    煤心扩散、解吸控制方程:

    $$ \begin{array}{c} \dfrac{\partial p}{\partial t}\left(\left(1 - {\phi }_{{\mathrm{m}}}\right)\left(\dfrac{100 - {{M}}_{\mathrm{a}\mathrm{d}} - {{A}}_{\mathrm{d}}}{100}\right)\dfrac{{V}_{{\mathrm{L}}}{p}_{{\mathrm{L}}}}{{\left(p + {p}_{{\mathrm{L}}}\right)}^{2}}{\rho }_{{\mathrm{coal}}}\dfrac{{M}_{{\mathrm{c}}}}{{V}_{{\mathrm{M}}}} + {\phi }_{{\mathrm{m}}}\dfrac{{M}_{{\mathrm{c}}}}{RT}\right) =\\ \nabla \left(\dfrac{{M}_{{\mathrm{c}}}}{RT}D\nabla p\right) \end{array} $$

    几何模型为实际圆柱体煤心物理模型,几何尺寸与现场测量一致;初始煤心各点孔隙压力为根据煤层气井试井储层压力推算得出,由于解吸时煤心已与大气接触,认为边界条件煤心柱面和断面表面压力为大气压力0.1 MPa。煤心几何模型如图1

    图  1  煤心几何模型 (单位:m)
    Figure  1.  Geometric model of coal core

    采用与煤心参数近似数据开展模拟并拟合解吸体积曲线,还原现场解吸参数和煤样特性参数。储层煤心参数实测值与拟合值对比见表2,煤心xy中心截面单位体积煤心含气性变化如图2。BF-1煤心累计解吸体积与解吸时间关系如图3

    表  2  储层煤心参数实测值与拟合值对比
    Table  2.  Comparison between measured values and fitting values of coal core parameters of reservoir
    参数 实测值(BF1/BF2/BF3) 拟合值(BF1/BF2/BF3) 误差率/%
    (BF1/BF2/BF3)
    储层压力/ MPa 9.88/9.25/8.96 9.88/9.25/8.96 0/0/0
    兰氏压力/ MPa 1.92/2.05/1.65 1.92/2.05/1.65 0/0/0
    兰氏体积/
    (cm3·g−1
    19.23/22.05/15.40 18.60/20.10/13.80 3.28/8.84/10.39
    水分含量
    (空干基)/%
    0.90/0.86/1.12 0.93/0.93/1.08 3.33/8.14/3.57
    灰分产率
    (干燥基)/%
    11.77/22.08/13.40 11.72/23.18/14.20 0.42/4.98/5.97
    煤心密度/
    (g· cm−3
    1.36/1.44/1.44 1.36/1.44/1.44 0/0/0
    孔隙度 0.10/0.09/0.08 0.10/0.08/0.072 0/11.1/10.0
    扩散系数/
    10−9(m2·s−1
    4×10−9/—/— 3.70/1.00/0.80 7.50/—/—
    含气
    饱和度/%
    54.77/77.51/99.79 54.77/77.51/99.79 0/0/0
    下载: 导出CSV 
    | 显示表格
    图  2  煤心xy中心截面单位体积煤心含气性变化
    Figure  2.  Change of gas content per unit volume of coal core at each position of xy central section of coal core

    图2可知:解吸开始24 h,含气量快速下降,各时刻煤心中心位置含气量最高而边缘最低。

    图  3  煤心累计解吸体积与解吸时间关系
    Figure  3.  Relationship between cumulative desorption volume and desorption time

    图3可知:BF-1扣除损失气时间拟合累计解吸气量(6 427.66 cm3)与现场煤心累计解吸气量(6 346.73 cm3)接近,相差2.00%;未扣损失气时间模拟累计解吸量(7 361.4 cm3)大于上述二者,显示损失气的存在;同理,BF-2和BF-3也有类似结果。

    吸附气饱和储层煤心现场解吸与数值模拟含气量构成对比见表3,BF-1、BF-2、BF-3煤心损失气时间(T)与解吸时间(t)和的平方根和解吸初期累计解吸气量回归关系如图4

    表  3  吸附气饱和储层煤心现场解吸与数值模拟含气量构成对比
    Table  3.  Comparison of adsorbed gas saturated reservoirs between on-site desorption and numerical simulation of cores
    样品 含气饱和度/% 含气量构成 实测体积/cm3 占比/实测/% 拟合体积/cm3 占比(拟合)/% 实测-拟合误差/%
    BF-1 54.77 损失气 979.43 13.28 880.26 11.95 10.12
    解吸气 6 346.73 86.06 6427.66 87.32 2.00
    残余气 48.43 0.66 46.24 0.63 4.52
    总含气量 7 374.59 100.00 7361.40 100.00 0.18
    BF-2 77.51 损失气 823.04 7.80 755.37 7.03 8.22
    解吸气 9680.43 91.78 9937.60 92.52 2.66
    残余气 43.88 0.42 47.89 0.45 9.14
    总含气量 10290.96 100.00 10740.86 100.00 1.83
    BF-3 99.79 损失气 793.71 6.36 841.77 6.75 6.06
    解吸气 11638.39 93.26 11575.20 92.84 0.54
    残余气 47.56 0.38 50.32 0.40 5.80
    总含气量 12479.66 100.00 12467.29 100.00 0.10
    下载: 导出CSV 
    | 显示表格
    图  4  煤心损失气时间与解吸时间和的平方根和解吸初期累计解吸气量回归关系
    Figure  4.  Linear regression relationship between the square root of the sum of gas loss time and desorption time and cumulative amount of desorbed gas at initial stage of desorption

    对于BF-1煤心,现场测试BF-1煤心损失气量结果为979.43 cm3,与数值模拟结果前15 min解吸气量(880.26 cm3)(图4(a))接近。现场BF-1煤心解吸气量和残余气量与数值模型计算的解吸气量和残余气量也较接近。

    BF-2和BF-3煤心现场测试煤心损失气量结果分别为823.04、793.71 cm3,略低于与数值模拟结果前15 min解吸气量(952.40、1015.80 cm3) (图4(b)、图4(c)),说明吸附气饱和度较高煤心解吸初期损失气量时间平方根法计算结果可能略偏小。这可能是吸附气饱和度较高,煤心内CH4浓度高,扩散作用更显著导致。用损失气时间与解吸时间和的平方根与解吸初期累计解吸气量的线性回归关系计算BF-2和BF-3煤心损失气量,发现基于数值模拟计算的损失气量结果(755.37、841.77 cm3)与基于现场实测的损失气量计算结果(832.04、793.71 cm3)较为接近。说明基于数值模拟数据利用时间平方根法计算的损失气量与实测时间平方根法估算的损失气量较为接近。

    现场实测BF-1、BF-2和BF-3煤心解吸气量和残余气量与数值模型计算出的解吸气量和残余气量也较为接近,误差不超过10.12%(表3)。说明数值模型可以近似反映吸附气欠饱和储层煤心真实解吸过程和损失气量、解吸气量、残余气量特征。

    以BF-1煤心为例,采用现场实测参数数据(表2),假设煤心处于饱和状态(即吸附气饱和、游离气在储层压力条件下充满孔隙也达到饱和状态),对现阶段现场含气量测试难度较大的含气饱和煤心解吸-扩散过程进行模拟。含气饱和煤心xy中心截面各位置单位体积煤心含气性变化如图5

    图  5  含气饱和煤心xy中心截面各位置单位体积煤心含气性变化
    Figure  5.  Change of gas content per unit volume of coal core at each position of xy center section of gas bearing saturated coal core

    图5可知:解吸前24 h煤心含气量快速下降,各时刻煤心中心位置含气量最高而煤心边缘含气量最低。

    损失气时间与解吸时间和的平方根和累计解吸气量的线性回归关系如图6。饱和储层煤心解吸-扩散数值模拟含气量构成预测见表4

    图  6  损失气时间与解吸时间和的平方根和累计解吸气量的线性回归关系
    Figure  6.  Linear regression relationship between the square root of the sum of gas loss time and desorption time and cumulative amount of desorbed gas
    表  4  饱和储层煤心解吸-扩散数值模拟含气量构成预测
    Table  4.  Prediction of gas content composition of coal core desorption-diffusion numerical simulation in saturated core
    含气量构成 预测体积/cm3 占比/%
    损失气 3 792.50 18.64
    解吸气 16 460.70 80.90
    残余气 93.51 0.46
    总含气量 20 346.71 100.00
    下载: 导出CSV 
    | 显示表格

    数值模拟显示,损失气时间内(15 min)假设的饱和煤心解吸-扩散气量达到4 441.3 cm3,相同损失气时间的情况下,约为相应吸附气欠饱和储层煤心此时间段解吸气量(即本例损失气量)的4倍。

    损失气时间与解吸时间和的平方根与解吸初期累计解吸气量的线性回归关系截距为正数(R2=0.9988),暗示采用解吸时间平方根与解吸初期累计解吸气量线性回归方法并不适用于饱和煤心损失气量的估算;采用解吸时间平方根与解吸初期累计解吸气量多项式回归方法,计算得煤心损失气量仅为433.11 cm3R2=0.9988),甚至小于吸附气欠饱和储层煤心损失气量估算值,说明基于时间平方根与累计解吸气量的相关性分析方法可能无法准确估算损失气量。同时测试初期时间平方根与累计解吸气量的回归关系(图6蓝色标记)和损失气时间与累计气量的回归关系(图6红色标记,实测测不出部分)也存在较大差异,说明饱和煤心损失气时间解吸-扩散规律较为复杂,常规回归分析法可能难以实现对损失气量的准确估算。

    表4可知:饱和煤心损失气、解吸气、残余气占比分别为18.64%、80.90%、0.46%,其损失气占比高于吸附气欠饱和煤心损失气占比、解吸气占比低于吸附气欠饱和煤心解吸气占比。

    数值模型可以近似反映吸附气欠饱和储层煤心真实解吸过程和损失气量、解吸气量、残余气量特征。目前,含气量现场测试手段很难达到对饱和煤心含气量的准确测试和估算,希望通过数值模拟为含气饱和煤心含气量分析提供思路。

    1)构建的吸附气欠饱和煤心解吸数值模型计算的损失气量、解吸气量和残余气量与现场测试相应结果接近(误差<10.12%),可近似反映吸附气欠饱和储层煤心含气量构成。

    2)对于构建的含气饱和储层煤心数值模型,损失气时间与解吸时间和的平方根与解吸初期累计解吸气量的回归分析法预测损失气量误差较大。饱和煤心损失气时间解吸-扩散规律较为复杂,常规回归分析可能难以实现对损失气的准确估算。

    3)相同损失气时间条件下,本例饱和煤心损失气总含气量占比(18.64%)高于吸附气欠饱和煤心损失气占比(11.95%),饱和煤心解吸气占比(80.90%)低于吸附气欠饱和煤心解吸气占比(87.32%),饱和煤心残余气占比(0.46%)低于吸附气欠饱和煤心残余气占比(0.63%)。

  • 图  1   多功能真三轴流固耦合试验系统结构示意图

    Figure  1.   Structural schematic diagram of true triaxial fluid-solid coupling experiment system

    图  2   试验路径

    Figure  2.   Test path

    图  3   不同中间主应力条件下砂岩的应力-应变曲线

    Figure  3.   Stress-strain curves of sandstone under different intermediate principal stress conditions

    图  4   不同中间主应力条件下砂岩的渐进破坏分段曲线

    Figure  4.   Segment curves of progressive failure of sandstone under different intermediate principal stress conditions

    图  5   不同中间主应力条件下砂岩裂纹体应变-轴向应变曲线

    Figure  5.   Strain-axial strain curves of sandstone crack volume under different intermediate principal stress conditions

    图  6   不同中间主应力下砂岩的特征点应力

    Figure  6.   Characteristic point stress of sandstone under different intermediate principal stress conditions

    图  7   不同中间主应力下砂岩渗透率

    Figure  7.   Permeability of sandstone under different intermediate principal stress conditions

    图  8   不同中间主应力条件下砂岩的渗透率与裂纹体应变随最大主应变的演化规律

    Figure  8.   Evolution law of permeability and crack volume strain of sandstone with maximum principal strainunder different intermediate principal stress conditions

    图  9   σ2=30 MPa时渗透率-裂纹体应变拟合曲线

    Figure  9.   Fitting curves of permeability-crack volume strain at σ2=30 MPa

    图  10   不同中间主应力条件下砂岩损伤和裂纹体应变的演化规律

    Figure  10.   Evolution of sandstone damage and crack volume strain under different intermediate principal stress conditions

    图  11   不同中间主应力条件下砂岩损伤和渗透率的演化规律

    Figure  11.   Evolution of sandstone damage and permeability under different intermediate principal stress conditions

    图  12   σ2=30 MPa时渗透率-损伤变量拟合曲线

    Figure  12.   Fitting curves of permeability-damage variables at σ2=30 MPa

    表  1   不同中间主应力条件下裂纹体应变-渗透率拟合曲线

    Table  1   Crack volume strain-permeability fitting curves under different intermediate principal stress conditions

    中间主应力/MPa 阶段 曲线 相关系数R2
    30裂纹闭合阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}0.998{{\mathrm{e}}^{ - 0.888{\varepsilon _{{\text{cv}}}}}}$0.982
    裂纹扩展阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}1.080{{\mathrm{e}}^{ - 4.190{\varepsilon _{{\text{cv}}}}}}$0.972
    40裂纹闭合阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{ - 6.220{\varepsilon _{{\text{cv}}}}}}$0.973
    裂纹扩展阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}1.073{{\mathrm{e}}^{ - 3.980{\varepsilon _{{\text{cv}}}}}}$0.969
    50裂纹闭合阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}0.997{{\mathrm{e}}^{ - 0.595{\varepsilon _{{\text{cv}}}}}}$0.906
    裂纹扩展阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}1.056{{\mathrm{e}}^{ - 3.312{\varepsilon _{{\text{cv}}}}}}$0.863
    60裂纹闭合阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{ - 0.967{\varepsilon _{{\text{cv}}}}}}$0.905
    裂纹扩展阶段${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. }{{k_0} = }}1.112{{\mathrm{e}}^{ - 6.280{\varepsilon _{{\text{cv}}}}}}$0.935
    下载: 导出CSV

    表  2   不同中间主应力条件下损伤-渗透率拟合曲线

    Table  2   Damage-permeability fitting curves under different intermediate principal stress conditions

    中间主应力/MPa 阶段 曲线 相关系数R2
    30 裂纹闭合 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{ - 0.058D + 0.019{D^2}}}$ 0.994
    裂纹扩展 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{ - 0.003 + 0.325D + 0.016{D^2}}}$ 0.998
    40 裂纹闭合 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{0.002 - 0.029D + 0.010{D^2}}}$ 0.998
    裂纹扩展 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{ - 0.024 - 0.218D + 0.146{D^2}}}$ 0.985
    50 裂纹闭合 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{0.001 - 0.038D + 0.020{D^2}}}$ 0.998
    裂纹扩展 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{ - 0.030 - 0.149D + 0.260{D^2}}}$ 0.850
    60 裂纹闭合 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{0.006 - 0.048D + 0.024{D^2}}}$ 0.998
    裂纹扩展 ${k \mathord{\left/ {\vphantom {k {{k_0} = }}} \right. } {{k_0} = }}{{\mathrm{e}}^{0.002 - 0.204D + 0.370{D^2}}}$ 0.805
    下载: 导出CSV
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  • 收稿日期:  2023-01-05
  • 修回日期:  2023-03-05
  • 刊出日期:  2024-06-29

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