An approach for rock density field inversion and spatial reconstruction of mechanical parameters
CHEN Yanzhao1, ,2, ZHOU Hao1, 2, YAO Xulong1, 2**, TAO Zhigang2, 3, LUAN Liming4, RONG Hui5, ZHAO Jizhong6
(1. College of Mining Engineering, North China University of Science and Technology, Tangshan，Hebei 063210, China;
2. Hebei Mine Green Intelligent Mining Technology Innovation Center, Tangshan, Hebei 063210, China; 3. School of Mechanics and Civil Engineering, China University of Mining and Technology, Beijing 100083, China; 4. Shandong Gold Group Co., Ltd., Jinan, Shandong 250102, China; 5. HBIS Mining Co., Ltd., Tangshan, Hebei 066501, China;
6. Shougang Luannan Macheng Mining Co., Ltd, Tangshan, Hebei 063500,, China)
Abstract：As conventional methods for obtaining mechanical parameters neglect the evolution of internal damage within rock masses, rely heavily on fundamental mechanical experiments, and struggle to accurately reflect the overall mechanical response of rock masses in engineering contexts due to the challenges of obtaining localised mechanical parameters on-site, a density field and mechanical parameter inversion method based on Gardner′s coupledintegrated damage variable is proposed. Precise inversion of rock mass density is achieved by incorporating damage variables into Gardner′s empirical formula, combined with mass conservation constraints and a density range iterative correction mechanism, enabling the further derivation of rock mechanical parameters. Verification was achieved through combined experiments involving sandstone wave velocity imaging and cutting sample density measurements. The average relative error between the inverted density and the measured density was 1.13%, indicating that the two were essentially consistent. Three-dimensional spatial reconstruction of rock mechanical parameter fields is achieved by the inversion methods. The mechanical parameter field exhibits pronounced spatial coupling with the density field. High-density zones correspond to mechanically strengthened zones (featuring peak elastic moduli reaching 20.1 GPa, shear moduli 30.6% above the mean, and bulk moduli 1.9 GPa above the mean), while low-density zones correspond to damage-prone zones (where microcrack development results in damage levels 2.14 times the rock mass mean). The statistical reliability of the reconstructed mechanical parameter field was validated using the Weibull distribution. This approach overcomes the limitations of conventional empirical formulas by employing multi-physics collaborative inversion, thereby providing a quantitative basis for analysing rock mass fracture mechanisms and assessing engineering stability.
陈燕朝1，2，周 浩1，2，姚旭龙1，2**，陶志刚2，3，栾立明4，荣 辉5，赵继忠6. 岩石密度场反演及其力学参数空间重构方法[J]. 岩石力学与工程学报, 2026, 45(S1): 213-225.
CHEN Yanzhao1,,2, ZHOU Hao1, 2, YAO Xulong1, 2**, TAO Zhigang2, 3, LUAN Liming4, RONG Hui5, ZHAO Jizhong6. An approach for rock density field inversion and spatial reconstruction of mechanical parameters. , 2026, 45(S1): 213-225.
[1] 戴兴旺. 铀尾矿库坝体震害失稳机理与风险评估研究[博士学位论文][D]. 衡阳：南华大学，2024.(DAI Xingwang. The mechanism of seismic destabilization and risk assessment of uranium tailing pond dams[Ph. D. Thesis][D]. Hengyang：University of South China，2024.(in Chinese))
[2] 殷跃平，王鲁琦，赵 鹏，等. 三峡库区高陡岸坡溃屈失稳机理及防治研究[J]. 水利学报，2022，53(4)：379–391.(YIN Yueping，WANG Luqi，ZHAO Peng，et al. Crashed failure mechanism and prevention of fractured high-steep slope in the Three Gorges Reservoir，China[J]. Journal of Hydraulic Engineering，2022，53(4)：379–391.(in Chinese))
[3] 孙艺丹. 深部开采覆岩断裂动载下巷道围岩失稳机理及控制研究[博士学位论文][D]. 阜新：辽宁工程技术大学，2020.(SUN Yidan. Study on the mechanism and control of surrounding rock instability under the dynamic load of overburden fracture in deep mining[Ph. D. Thesis][D]. Fuxin：Liaoning Technical University，2020.(in Chinese))
[4] 刘之喜，赵光明，孟祥瑞，等. 岩石蠕变应变能密度演化时效特征与损伤本构模型研究[J]. 采矿与岩层控制工程学报，2025，7(1)：13–21.(LIU Zhixi，ZHAO Guangming，MENG Xiangrui，et al. Study on aging characteristics of creep strain energy density evolution and damage constitutive model of rock[J]. Journal of Mining and Strata Control Engineering，2025，7(1)：13–21.(in Chinese))
[5] 彭立志. 常规三轴压缩条件下砂岩能量演化规律及其破坏准则研究[硕士学位论文][D]. 淮南：安徽理工大学，2024.(PENG Lizhi. Research on energy evolution law and failure criterion of sandstone under conventional triaxial compression conditions[M.S. Thesis][D]. Huainan：Anhui University of Science and Technology，2024.(in Chinese))
[6] 李丽丽，曹 锐，马国庆，等. 井物性约束下的布格重力异常反演方法与应用[J]. 世界地质，2023，42(4)：714–723.(LI Lili，CAO Rui，MA Guoqing，et al. Bouguer gravity anomaly inversion method constrained by well physical property and its application[J]. Global Geology，2023，42(4)：714–723.(in Chinese))
[7] 姜 勇. 多参数拟合测井曲线环境校正方法研究及应用[J]. 海洋石油，2013，33(3)：70–75.(JIANG Yong. Multi-parameter matching well log environmental calibration method and application[J]. Offshore Oil，2013，33(3)：70–75.(in Chinese))
[8] 牛 聪，刘志斌，张益明，等. 基于密度分析的含气饱和度预测研究应用[J]. 中国石油勘探，2012，17(3)：52–55.(NIU Cong，LIU Zhibin，ZHANG Yiming，et al. Research and application of gas saturation prediction based on density analysis[J]. China Petroleum Exploration，2012，17(3)：52–55.(in Chinese))
[9] 姚旭龙，张艳博，孙 林，等. 基于区域相关性的岩石损伤声发射探测与成像方法研究[J]. 岩石力学与工程学报，2017，36(9)：2 113–2 123.(YAO Xulong，ZHANG Yanbo，SUN Lin，et al. Research on acoustic emission inspecting and imaging method for rock damage based on regional correlation[J]. Chinese Journal of Rock Mechanics and Engineering，2017，36(9)：2 113–2 123.(in Chinese))
[10] 刘 臻，姚旭龙，张艳博，等. 区域化结构与破裂耦合作用下的岩石破裂演化规律[J]. 矿业研究与开发，2023，43(5)：120–130.(LIU Zhen，YAO Xulong，ZHANG Yanbo，et al. Evolution law of rock fracture under the coupling effect of regionalized structure and fracture[J]. Mining Research and Development，2023，43(5)：120–130.(in Chinese))
[11] 张 显，侯振隆，赵福权，等. 基于模糊C均值聚类的空–地–井垂直重力梯度数据反演方法[J]. 石油地球物理勘探，2024，59(3)：629–639.(ZHANG Xian，HOU Zhenlong，ZHAO Fuquan，et al. Inversion of airborne-ground-borehole vertical gravity gradient data based on fuzzy C-means clustering[J]. Oil Geophysical Prospecting，2024，59(3)：629–639.(in Chinese))
[12] 朱雨菲，王武星，曹建玲，等. 基于深度学习的2D–3D地下密度异常体反演方法[J]. 地球物理学报，2023，66(12)：4 958–4 972.(ZHU Yufei，WANG Wuxing，CAO Jianling，et al. A new 2D–3D deep learning based inversion method for underground density anomalous body[J]. Chinese Journal of Geophysics，2023，66(12)：4 958–4 972.(in Chinese))
[13] LIANG L F，ZHANG H B，DAN Z W，et al. Prestack density inversion using the Fatty equation constrained by the P- and S-wave impedance and density[J]. Applied Geophysics，2017，14(1)：133–141.
[14] 梁立锋，刘秀娟，但志伟，等. 利用叠前密度反演预测岩性的应用研究[J]. 海洋石油，2011，31(2)：53–58.(LIANG Lifeng，LIU Xiujuan，DAN Zhiwei，et al. Application research on using prestack density inversion to predict lithology[J]. Offshore Oil，2011，31(2)：53–58.(in Chinese))
[15] 刘 涛，方中于，但志伟，等. 叠前同步反演在涠西南凹陷储层预测中的应用[J]. 石油天然气学报，2011，33(9)：77–81.(LIU Tao，FANG Zhongyu，DAN Zhiwei，et al. Application of prestack simultaneous inversion of reservoir prediction in Weixinan depression[J]. Journal of Oil and Gas Technology，2011，33(9)：77–81.(in Chinese))
[16] 孙豪志，冯英浚，范祯祥. 声波介质中速度和密度反演[J]. 石油地球物理勘探，1993，28(4)：411–417.(SUN Haozhi，FENG Yingjun，FAN Zhenxiang. Inversions of velocity and density in acoustic medium[J]. Oil Geophysical Prospecting，1993，28(4)：411–417.(in Chinese))
[17] GARDNER G H F，GARDNER L W，GREGORY A R. Formation velocity and density–The diagnostic basics for stratigraphic traps[J]. Geophysics，1974，39(12)：770–780.
[18] 王 盼，刘 挺，唐 峰. 基于Gardner公式的岩石密度优化算法[J]. 录井工程，2016，27(4)：12–15.(WANG Pan，LIU Ting，TANG Feng. Rock density optimization algorithm based on Gardner formula[J]. Mud Logging Engineering，2016，27(4)：12–15.(in Chinese))
[19] ZHANG Y B，YAO X L，LIANG P，et al. Fracture evolution and localization effect of damage in rock based on wave velocity imaging technology[J]. Journal of Central South University，2021，28(9)：2 752–2 769.
[20] 刘世翔，胡艳飞，闫清华，等. 专家克里金插值法在空间插值中的应用[J]. 地质与资源，2011，20(4)：292–294.(LIU Shixiang，HU Yanfei，YAN Qinghua，et al. Application of expert kriging algorithm in the spatial interpolation[J]. Geology and Resources，2011，20(4)：292–294.(in Chinese))
[21] 杨正华. 岩石物性概论[M]. 西安：西安交通大学出版社，2021：35–40.(YANG Zhenghua. Introduction to rock physical properties[M]. Xi'an：Xi'an Jiaotong University Press，2021：35–40.(in Chinese))
[22] 陈 颙，黄庭芳，刘恩儒. 岩石物理学[M]. 合肥：中国科学技术大学出版社，2009：78–80.(CHEN Yong，HUANG Tingfang，LIU Enru. Rock physics[M]. Hefei：University of Science and Technology of China Press，2009：78–80.(in Chinese))
[23] YU C，JI S，LI Q . Effects of porosity on seismic velocities, elastic moduli and Poisson′s ratios of solid materials and rocks[J]. Journal of Rock Mechanics and Geotechnical Engineering，2016，8(1)：35–49.
[24] 徐佳慧，杨 予，刘 恒，等. 泊松比变化对基于波速混凝土损伤计算的影响[J]. 计量学报，2024，45(6)：881–889.(XU Jiahui，YANG Yu，LIU Heng，et al. Effects of the variation of Poisson′s ratio on the calculation of concrete damage based on wave velocity[J]. Acta Metrologica Sinica，2024，45(6)：881–889.(in Chinese))
[25] SUAREZ L，TOJAGA V，OLSSON E，et al. Multiscale modeling of rock fracture in comminution — A comparative study of FEM accuracy and DEM scalability[J]. Minerals Engineering，2025，232：109488.
2026, Vol. 45 
