A refined evaluation method for in situ stresses in orthotropic shale formations
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摘要: 针对正交各向异性地应力评价方法缺乏的问题，研究通过考虑岩石的正交各向异性（ORT）特征，开发了一种完备的页岩地层原位应力评估方法。该方法首先建立了各向异性岩石的动−静态力学参数转换关系；在此基础上，基于有效应力与应变的关系，推导出Biot系数的通解表达式，给出不同情况下的具体解；同时构建基于测井资料的ORT页岩岩石物理模型，获得用于计算ORT弹性参数和Biot系数的刚度矩阵，实现从各向异性的广义胡克定律出发推导出正交各向异性地层地应力的完整解析表达式。该方法在x1井的应用表明，ORT页岩地层水平向和垂向岩石弹性参数的显著差异，通过按方向建立的动−静态转换模型提高了静态岩石弹性参数的转换精度；垂直于层理面内的有效应力系数（Biot系数）α11明显小于α33，平均相对差异达13.4%；而在平行于层理面内，α11和α22的差异相对较小，Biot系数的各向异性特征显著。ORT模型不仅提供了详细的应力剖面，清晰反映了因黏土含量变化引起的应力“V”形波动，而且预测的最小和最大水平主应力误差分别低至1.9%和4.0%，优于传统横观各向同性（VTI）模型的5.8%、5.2%和各向同性（ISO）模型的8.2%、10.9%。ORT地应力评估方法提供了更准确、详细的页岩地层原位应力评价方法，这对钻井液密度窗口精细设计、压裂参数工艺制定以及压裂后的闷井时间的确定具有重要意义。Abstract:
Objective To address the scarcity of methods for evaluating in situ stress in orthotropic media, this study aims to develop a comprehensive in situ stress evaluation method in shale formations by incorporating the orthotropic (ORT) characteristics of rocks. Methods The method first establishes the conversion relationship between the dynamic and static mechanical parameters of anisotropic rocks. Based on the relationship between effective stress and strain, a general analytical expression for the Biot coefficient is derived and specific solutions for different conditions are presented. An orthorhombic shale rock physics model is constructed using well-log data to obtain the stiffness matrix required for calculating orthorhombic elastic parameters and Biot coefficients. Starting from the generalized Hooke’s law for anisotropic media, a complete analytical expression for in situ stress in orthorhombic formations is ultimately derived. Results The application of this method to well x1 demonstrates that: (1) the significant differences in elastic parameters between the horizontal and vertical ORT shale formation directions are addressed by establishing direction-specific dynamic and static parameter conversion models, enhancing the conversion accuracy of static rock elastic parameters; (2) the Biot coefficient α11 perpendicular to the bedding plane is significantly smaller than α33, with an average relative difference of 13.4%, while the differences between α11 and α22 parallel to the bedding plane are relatively small, indicating remarkable anisotropic characteristics of the Biot coefficients; (3) the ORT model not only provides a detailed stress profile that clearly reflects the V-shaped stress fluctuations caused by changes in clay content but also reduces the prediction errors of the minimum and maximum horizontal principal stresses to 1.9% and 4.0%, respectively. These values are lower than those of the traditional vertical-transverse-isotropy (VTI) model (5.8%, 5.2%) and the isotropy (ISO) model (8.2%, 10.9%). Conclusion The ORT in-situ stress evaluation method offers a more accurate and detailed assessment of in situ stress in shale formations, which is significant for refining the design of drilling fluid density windows, developing fracturing parameters and processes, and determining well shutdown time after fracturing. -
Key words:
- shale formation /
- orthotropic /
- biot coefficients /
- in situ stress evaluation
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图 1 岩石弹性参数的动−静态转换关系
E1s—静态水平向杨氏模量；E1d—动态水平向杨氏模量；E3s—静态垂向杨氏模量；E3d—动态垂向杨氏模量；Es—静态不考虑各向异性杨氏模量；Ed—动态不考虑各向异性杨氏模量；ν1s—静态水平向泊松比；ν1d—动态水平向泊松比；ν3s—静态垂向泊松比；ν3d—动态垂向泊松比；νs—静态不考虑各向异性泊松比；νd—动态不考虑各向异性泊松比；N—岩芯测试数量；R—相关系数
Figure 1. Relationship between the dynamic and static conversion of rock elastic parameters
(a) Dynamic and static conversion of the horizontal Young's modulus; (b) Dynamic and static conversion of the vertical Young's modulus; (c) Dynamic and static conversion of the horizontal Poisson's ratio; (d) Dynamic and static conversion of the vertical Poisson's ratio; (e) Dynamic and static conversion of Young's modulus without considering anisotropy; (f) Dynamic and static conversion of Poisson's ratio without considering anisotropy E1s—Static horizontal Young's modulus; E1d—Dynamic horizontal Young's modulus; E3s—Static vertical Young's modulus; E3d—Dynamic vertical Young's modulus; Es—Static Young's modulus without considering anisotropy; Ed—Dynamic Young's modulus without considering anisotropy; ν1s—Static horizontal Poisson's ratio; ν1d—Dynamic horizontal Poisson's ratio; ν3s—Static vertical Poisson's ratio; ν3d—Dynamic vertical Poisson's ratio; νs—Static Poisson's ratio without considering anisotropy; νd—Dynamic Poisson's ratio without considering anisotropy; N—Number of core tests; R—Correlation coefficient
图 2 ORT岩石物理建模流程
VTI—横观各向同性；ORT—正交各向异性；CL系数—黏土成层系数
Figure 2. ORT rock physics modeling workflow
VTI—transversely isotropic; ORT—orthorhombic anisotropic; CL coefficient—clay lamination coefficient
图 3 基于岩石物理建模的地应力评价流程
Figure 3. Stress evaluation workflow based on rock physics modeling
图 4 ORT静态岩石弹性参数
E1—沿x轴方向的杨氏模量；E2—沿y轴方向的杨氏模量，GPa；E3—沿z轴方向的杨氏模量，GPa；ν12—垂直于x轴、沿y轴的泊松比；ν13—垂直于x轴、沿z轴的泊松比；ν21—垂直于y轴、沿x轴的泊松比；ν23—垂直于y轴、沿z轴的泊松比；ν31—垂直于z轴、沿x轴的泊松比；ν32—垂直于z轴、沿y轴的泊松比
Figure 4. ORT static rock elastic parameters
E1—Young's modulus along the x-axis; E2—Young's modulus along the y-axis, GPa; E3—Young's modulus along the z-axis, GPa; ν12—-Poisson's ratio perpendicular to the x-axis and along the y-axis, dimensionless; ν13—Poisson's ratio perpendicular to the x-axis and along the z-axis, dimensionless; ν21—Poisson's ratio perpendicular to the y-axis and along the x-axis, dimensionless; ν23—Poisson's ratio perpendicular to the y-axis and along the z-axis, dimensionless; ν31—Poisson's ratio perpendicular to the z-axis and along the x-axis, dimensionless; ν32—Poisson's ratio perpendicular to the z-axis and along the y-axis.
图 5 不同模型计算的Biot系数
αiso—各向同性Biot系数；α11vti—水平向Biot系数；α33vti—垂向Biot系数；α11ort—x方向Biot系数；α22ort—y方向Biot系数；α33ort—z方向Biot系数
Figure 5. Biot coefficients calculated by different models
αiso—Isotropic Biot coefficient; α11—vti-Horizontal Biot coefficient; α33vti—Vertical Biot coefficient; α11ort—Biot coefficient in the x-direction; α22ort—Biot coefficient in the y-direction; α33ort—Biot coefficient in the z-direction
图 6 x1井龙马溪组微注施工曲线
Figure 6. Micro-injection operation curve of the Longmaxi Formation in Well x1
图 7 各类模型地应力计算结果
Vclay—黏土含量；σH实测/推算—微注测试推算或声发射实测最大水平主应力；σHiso—iso模型预测最大水平主应力；σHvti—VTI模型预测最大水平主应力；σHort—ORT模型预测最大水平主应力；σh实测—微注测试或声发射实测最小水平主应力；σhiso—iso模型预测最小水平主应力；σhvti—VTI模型预测最小水平主应力；σhort—ORT模型预测最小水平主应力；A1—微注测试反算最大水平主应力点位；A2—微注测试最小水平主应力点位；B1—声发射测试最大水平主应力点位；B2—声发射测试最小水平主应力点位
Figure 7. In situ stress calculated by various models
Vclay—Clay content; σH实测/推算—Maximum horizontal principal stress calculated from micro-injection test or measured from acoustic emission; σHiso-Maximum horizontal principal stress predicted by the iso model; σHvti—Maximum horizontal principal stress predicted by the VTI model; σHort-Maximum horizontal principal stress predicted by the ORT model; σh实测—Minimum horizontal principal stress measured from the micro-injection or the acoustic emission test; σHiso—Minimum horizontal principal stress predicted by the iso model; σHvti—Minimum horizontal principal stress predicted by the VTI model; σHort—Minimum horizontal principal stress predicted by the ORT model; A1—Position of the maximum horizontal principal stress point calculated from the micro-injection test; A2—Position of the minimum horizontal principal stress point; B1—Maximum horizontal principal stress point measured from the acoustic emission test; B2—Minimum horizontal principal stress point measured from the acoustic emission test
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