A method for determining characteristic pressure parameters during hydraulic fracturing based on linearized fitting
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摘要: 为了提高水压致裂试验中瞬时闭合压力（ps）与重张压力（pr）等关键参数的识别精度，解决传统方法（如单切线法、平移pb法以及马斯卡特法等）在处理非线性压力–时间曲线中易受噪声干扰、判定主观性强、识别精度不足的问题，提出了一种基于线性化拟合的参数识别方法，通过多项式平滑与分段线性回归将非线性压力–时间曲线转化为若干局部线性段，利用相邻段之间的斜率突变点自动识别ps与pr。该方法分别在室内真三轴水压致裂试验（花岗岩试样）与天津蓟州抽水蓄能电站野外实测（深度范围为75～277 m）中进行了验证。结果表明，线性化拟合方法在ps与pr识别方面具有较高的准确性与稳定性，相较于单切线法、平移pb法和马斯卡特法，平均偏差明显降低，且在存在数据扰动、非线性响应较强的条件下仍表现出良好的鲁棒性。该方法有效提升了参数识别的客观性、一致性和抗干扰能力，适用于硬脆性岩体地应力测量，为水压致裂试验的数据智能识别与解释提供了实用工具，具有良好的工程实用价值和推广前景。Abstract:
Objective Hydraulic fracturing is a fundamental technique for in-situ stress measurement, yet conventional methods of interpreting the instantaneous shut-in pressure (ps) and reopening pressure (pr) are often sensitive to noise, strongly subjective, and inadequate for facing nonlinear pressure–time responses. To address the lack of objective, robust, and high-accuracy identification methods, this study proposes a linearized curve-fitting approach capable of automatically determining key characteristic pressures from complex fracturing curves. Methods The method transforms a nonlinear pressure–time curve into multiple locally linear segments through polynomial smoothing, adaptive sliding-window regression, and statistical slope-change detection. Significant slope mutations are used to automatically identify ps during the shut-in decay stage and pr during the re-pressurization stage. The method is validated using true-triaxial hydraulic fracturing laboratory tests on granite and field tests at the Jizhou pumped-storage power station (75–277 m depth). Results Across six granite specimens (HF2–HF9), the proposed method consistently produced pr values between those obtained by the single-tangent and shifted-pb methods, avoiding the low–high systematic bias of the two techniques. For ps, the method yielded similar or more conservative results than traditional methods, with most absolute deviations <0.40 MPa. The method demonstrated strong consistency across varying stress states and significantly reduced subjective scatter. The principal stresses calculated from ps and pr showed physically reasonable trends. The σ1 errors were <30% in most cases and the behavior was stable without random jumps, indicating improved objectivity of the automated identification. In the four tested depth intervals of the LFZK02 borehole, pr and ps determined by traditional methods exhibited large spreads (e.g., pr deviations >1.5 MPa and ps deviations up to 0.74 MPa). In contrast, the linearized method consistently produced values close to the multi-method averages and with much smaller dispersion. For ps, deviations relative to Muskat and derivative methods were typically <0.20 MPa. Using the automatically identified ps and pr, the derived principal stresses showed a clear horizontally compressive regime (SH=6.55–9.85 MPa; Sh=3.54–6.21 MPa), matching regional stress data and confirming the reliability of the method in actual field conditions. Conclusion The linearized curve-fitting method effectively overcomes the subjectivity, noise sensitivity, and model-dependence of conventional hydraulic-fracturing interpretation approaches. This method provides stable, accurate, and repeatable identification of ps and pr in both laboratory and field environments and maintains good performance under nonlinear responses, data disturbance, and multi-cycle loading–unloading conditions. [Significance] This study offers a robust, automated, and universally applicable tool for interpreting hydraulic-fracturing pressure curves, significantly enhancing the reliability of in-situ stress measurements and supporting the development of intelligent, standardized stress-testing systems for underground engineering. -
图 1 水压致裂过程中典型压力–时间曲线及关键参数（pb、ps、pr）示意图
Figure 1. Schematic diagram of a typical pressure–time curve and key parameters (pb, ps, pr) during hydraulic fracturing
图 2 单切线法识别特征压力的原理示意图
a—降压阶段识别瞬时闭合压力（ps）；b—升压阶段识别重张压力（pr）
Figure 2. Schematic diagrams illustrating the principle of identifying characteristic pressures using the single tangent methode
(a) Identification of the instantaneous shut-in pressure ps during the pressure decline stage; (b) Identification of the reopening pressure pr during the pressurization stage
图 3 基于不同方法识别瞬时闭合压力（ps）的原理示意图
p1—压力衰减趋近的稳定值或残余压力a—马斯卡特法的非线性回归拟合；b—导数分析法的dp/dt–压力曲线
Figure 3. Schematic diagrams illustrating the principle of identifying the instantaneous shut-in pressure using different methods (a) Nonlinear regression fitting based on the Muskat method; (b) The dp/dt–p curve based on the derivative analysis method
p1 – the stabilized or residual pressure that the pressure drawdown tends toward
图 4 平移pb法识别重张压力（pr）的原理示意图
Figure 4. Schematic diagram illustrating the principle of identifying the reopening pressure pr using the shifted pb method
图 5 线性化拟合的整体实现流程图
Figure 5. Flowchart of the overall implementation of the linearized curve-fitting method
图 6 基于线性化拟合识别特征压力的原理示意图
ai表示第i个数据窗口的斜率，如a1、a2、a3分别表示第1、2、3个数据窗口的斜率a—重张压力pr；b—瞬时闭合压力ps
Figure 6. Schematic diagrams illustrating the principle of identifying characteristic pressures based on the linearized fitting method(a) The reopening pressure pr; (b) The instantaneous shut-in pressure ps
ai denotes the slope of the i-th data window, i.e., a1, a2, and a3 correspond to the slopes of the first, second, and third data windows, respectively.
图 7 室内水压致裂试验试样结构示意图
σ1—最大主应力；σ2—最小主应力；σ3—中间主应力；P—注水孔位置，用于施加孔隙压力
Figure 7. Schematic diagram of the specimen structure used in the hydraulic fracturing laboratory test
σ1—maximum principal stress; σ2—minimum principal stress; σ3—intermediate principal stress; P—location of the injection hole for applying pore pressure
图 8 花岗岩试样压裂前后状态对比
Figure 8. Comparison of granite specimens before and after hydraulic fracturing
(a) HF2; (b) HF3; (c) HF5; (d) HF7; (e) HF8; (f) HF9
图 9 水压致裂全过程的压力–时间曲线
Figure 9. Pressure–time curve of the entire hydraulic fracturing process
图 10 线性化拟合法与传统方法平均值之间的绝对偏差对比图
a—重张压力pr；b—瞬时闭合压力ps
Figure 10. Comparison of absolute deviations between the linearized fitting method and the averaged values of the traditional methods
(a) Reopening pressure pr; (b) Instantaneous shut-in pressure ps
图 11 天津蓟州抽水蓄能电站水压致裂测试段岩芯照片
Figure 11. Core samples from hydraulic-fracturing test intervals at the Jizhou Pumped-Storage Power Station, Tianjin (a) 75.2–79.0 m; (b) 84.0–90.0 m; (c) 124.0–131.0 m; (d) 182.7–189.4 m
图 12 天津蓟州龙潭沟LFZK02钻孔现场水压致裂试验压力–时间曲线
Figure 12. Pressure–time curve of the hydraulic fracturing test in the LFZK02 borehole at Longtangou, Jizhou, Tianjin field site
表 1 花岗岩试样室内水压致裂试验加载与注水速率条件
Table 1. Loading conditions and injection rate of granite specimens in hydraulic fracturing laboratory tests
试样编号 $ {\textit{σ}}_{\text{1}} $/MPa $ {\textit{σ}}_{\text{2}} $/MPa $ {\textit{σ}}_{\text{3}} $/MPa 注水速率/(mL/min) HF2 2.40 0.90 1.80 2.20 HF3 3.60 2.00 1.20 2.25 HF5 2.40 1.50 0.90 2.30 HF7 4.80 1.80 2.40 2.00 HF8 3.60 1.20 1.80 2.40 HF9 4.80 3.60 2.40 2.10 注：σ1—最大主应力；σ2—最小主应力；σ3—中间主应力 
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表 2 各方法判读重张压力（pr）与瞬时闭合压力（ps）的结果
Table 2. Reopening pressure (pr) and instantaneous shut-in pressure (ps) determined by various methods
试样编号 pr识别结果/MPa ps识别结果/MPa 单切法 平移pb法 线性化拟合法 单切法 马斯卡特法 导数分析法 线性化拟合法 HF2 0.91 1.50 1.12 0.86 1.03 0.74 0.68 HF3 1.30 1.67 1.61 1.63 2.02 1.81 1.52 HF5 0.95 1.41 1.78 0.98 1.04 0.74 0.99 HF7 3.83 4.87 4.72 3.70 4.02 3.97 3.57 HF8 1.33 2.02 1.77 1.24 1.43 1.34 1.53 HF9 4.05 5.38 4.68 4.16 4.48 4.48 3.98
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表 3 基于线性化拟合法计算的最大主应力（σ1）和最小主应力（σ2）与试验设定值的对比及误差分析
Table 3. Comparison and error analysis for the maximum principal stress σ1 and the minimum principal stress σ2 calculated with the linearized fitting method using the experimental preset values
试样编号 σ1计算结果/MPa σ1试验设定值/MPa σ1相对误差 σ2计算结果/MPa σ2试验设定值/MPa σ2相对误差 HF2 2.33 2.40 3.00% 0.68 0.90 24.44% HF3 2.71 3.60 24.72% 1.52 1.20 26.67% HF5 1.69 2.40 29.58% 0.99 0.90 10.00% HF7 6.48 4.80 35.00% 3.57 1.80 98.33% HF8 3.64 3.60 1.11% 1.53 1.20 27.50% HF9 5.62 4.80 17.08% 3.98 2.40 65.83%
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表 4 现场水压致裂试验中不同方法识别重张压力（pr）与瞬时闭合压力（ps）的结果
Table 4. Reopening pressure (pr) and instantaneous shut-in pressure (ps) identified by different methods in the hydraulic fracturing field test
深度/m pr判读数据/MPa ps判读数据/MPa 单切法 平移pb法 平均值 线性化拟合法 单切法 马斯卡特法 导数分析法 平均值 线性化拟合法 75.2 1.48 3.16 2.32 2.61 1.07 1.11 0.96 1.05 1.09 87.2 2.47 4.33 3.40 3.72 2.28 2.39 2.06 2.24 2.12 124.6 3.52 4.92 4.22 4.44 3.82 4.22 3.75 3.93 4.33 186.1 3.80 3.51 3.66 3.29 3.80 3.18 3.20 3.39 3.06
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表 5 线性化拟合法与各判读方法的重张压力（pr）与瞬时闭合压力（ps）偏差比较
Table 5. Comparison of the deviations in reopening pressure (pr) and instantaneous shut-in pressure (ps), as determined by various interpretation methods relative to the linearized fitting method
参数 对比方法 平均偏差/MPa 标准差/MPa 最大偏差/MPa 重张压力（pr） 单切法 0.953 0.282 1.25 平移pb法 0.465 0.148 0.61 瞬时闭合压力（ps） 单切法 0.356 0.284 0.740 马斯卡特法 0.130 0.090 0.270 导数分析法 0.228 0.206 0.580
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表 6 基于线性化拟合法计算的各深度段主应力（ $ {{S}}_{\text{H}} $、$ {{S}}_{\text{h}} $和$ {{S}}_{\text{v}} $）
Table 6. Principal stress values SH, Sh, and Sv at various depth intervals calculated based on the linearized curve fitting method
深度/m 最大水平主应力
$ {{S}}_{\text{H}} $/MPa最小水平主应力
$ {{S}}_{\text{h}} $/MPa垂直主应力
$ {{S}}_{\text{v}} $/MPa75.2 6.55 3.54 5.76 87.2 9.85 6.21 6.62 124.6 8.03 6.19 6.53 186.1 8.93 5.56 5.97
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