水文地質參數的資料稀缺與空間劇烈變動是地下資源建模的重要挑戰。數十年來，已經發展了許多反算方法與資料同化技術，其可整合觀測到的狀態資料進入模式系統，以求得最佳參數分布。然而，建構地質資源模式時，往往需配合多層且循序的水文地質調查。由循序多次的調查中，彙整並加入新增資料於模式之中。由於這些新增資料的位置可能與原建構的模式網格位置不一致，造成使用原傳統的數值方法，難以精準處理新增資料，造成模式之不確定性增加。因此，本研究提出一個動態自動調適節點 (DANA) 方法，其結合無網格法和快速節點放置法來緩解即時數據同化遭遇的問題。本研究利用無網格廣義有限差分 (GFD) 來開發 DANA 架構，並使用系集卡門濾波器 (EnKF) 作為資料同化方法。本研究驗證所提出之方法的準確性和計算效率，並通過一個假設問題展示了 DANA 的應用性。研究結果顯示使用 GFD 下的 DANA 有效與 EnKF 結合，可實現地下水流建模之即時更新，並提升模式之準度與精度。

The data scarcity of direct measurement of heterogeneous hydrogeological parameters is one of the critical challenges for geo-resources modeling. Over the past decades, various inverse modeling and data assimilation techniques have been proposed to integrate observed data into modeling systems for optimal parameter estimations. However, subsurface flow models are usually built based on only preliminary hydrogeological surveys to minimize the cost of assessments. In many circumstances, sequential and additional data will be collected from supplementary hydrogeological surveys. Because the locations of these additional data may not coincide with the positions of the pre-defined meshes, the additional data may not be exactly handled by original numerical methods. This study developed a dynamically adaptive node adjustment (DANA) scheme to maximize the utilization of available data. The DANA was combined with a meshless method and a fast node placement algorithm to alleviate the problem that real-time data assimilation is usually encountered. This study utilized the meshless Generalized Finite Difference (GFD) to develop the DANA framework, and the Ensemble Kalman Filter (EnKF) served as the data assimilation approach. The accuracy and computational efficiency of the proposed methods were investigated, and the applicability of the DANA was demonstrated by solving a hypothetical data assimilation problem. The results showed that the DANA under the GFD was efficaciously coupled with the EnKF to achieve real-time updating for geo-resources modeling and improve the accuracy and precision of the model.

Alcolea, A., J. Carrera, and A. Medina, 2006, Pilot points method incorporating prior information for solving the groundwater flow inverse problem, Advances in Water Resources, 29, 1678-1689.
Bai, B., 2013, Thermal response of saturated porous spherical body containing a cavity under several boundary conditions, Journal of Thermal Stresses, 36, 1217-1232.
Benito, J.J., F. Urena, and L. Gavete, 2001, Influence of several factors in the generalized finite difference method, Applied Mathematical Modelling, 25, 1039-1053.
Carrera, J., and P. Neuman, 1986, Estimation of aquifer parameters under transient and steady state conditions: maximum likelihood method incorporating prior information, Water Resources Research, 22, 199-210.
Cattaneo, L., A. Comunian, G. de Filippis, M. Giudici, and C. Vassena, 2016, Modeling groundwater flow in heterogeneous porous media with YAGMod, Computation, 4, 2.
Chang, L.C., 2012, Application of data assimilation method for regional groundwater utilization study, project report in Chinese, Water Resources Planning Institute, Water Resources Agency, Ministry of Economic Affairs.
Chang, L.C., J.P. Tsai, and Y.C. Chen, 2019, Estimating hydraulic conductivity and specific yield by time-lapse gravity survey and hydraulic tomography, American Geophysical Union Fall Meeting 2019 Abstracts, H53Q-2062
Chavez-Negrete, C., F.J. Dominguez-Mota, and D. Santana-Quinteros, 2018, Numerical solution of Richards' equation of water flow by generalized finite differences, Computers and Geotechnics, 101, 168-175.
Chen, J.S., M. Hillman, and S.W. Chi, 2017, Meshfree methods: progress made after 20 years, Journal of Engineering Mechanics, 14, 04017001.
Chen, S.Y., K.C. Hsu, and C.M. Fan, 2021, Improvement of generalized finite difference method for stochastic subsurface flow modeling, Journal of Computational Physics, 429, 110002.
Crotogino, F., 2022, Large-scale hydrogen storage, Storing Energy with Special Reference to Renewable Energy Sources, 26, 613-632.
Dagan, G., 1989, Flow and Transport in Porous Formations, Springer-Verlag, New York.
Deutsch, C.V., and A.G. Journel, 1998, GSLIB Geostatistical Software Library and User's Guide, second edition, Oxford University Press, New York.
Erdal, D., and O.A. Cirpka, 2016, Joint inference of groundwater-recharge and hydraulic-conductivity fields from head data using the ensemble Kalman filter, Hydrology and Earth System Sciences, 20, 555-569.
Fan, C.M., C.N. Chu, B. Sarler, and T.H. Li, 2019, Numerical solutions of waves-current interactions by generalized finite difference method, Engineering Analysis with Boundary Elements, 100, 150-163.
Feng, S., and P.J. Vardanega, 2019, A database of saturated hydraulic conductivity of fine-grained soils: probability density functions, Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 13, 255-261.
Fornberg, B., and N. Flyer, 2015, Fast generation of 2-D node distributions for mesh-free PDE discretizations, Computers and Mathematics with Applications, 69, 531-544.
Freeze, R.A., 1975, A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media, Water Resources Research, 11, 725-741.
Fu, J., and J.J. Gomez-Hernandez, 2009, A blocking markov chain monte carlo method for inverse stochastic hydrogeological modeling, Mathematical Geosciences, 41, 105-128.
Gavete, L., F. Urena, J.J. Benito, M. Urena, and M.L. Gavete, 2018, Solving elliptical equations in 3d by means of an adaptive refinement in generalized finite differences, Mathematical Problems in Engineering, 9678473.
Gomez-Hernandez, J.J., 1991, A stochastic approach to the simulation of block conductivity fields conditioned upon data measured at a smaller scale, Ph.D. dissertation, Stanford University.
Gomez-Hernandez, J.J., A. Sahuquillo, and J.E. Capilla, 1997, Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data-I. Theory, Journal of Hydrology, 203, 162-174.
Ha, S.T., L.C. Ngo, M. Saeed, B.J. Jeon, and H. Choi, 2017, A comparative study between partitioned and monolithic methods for the problems with 3D fluid-structure interaction of blood vessels, Journal of Mechanical Science and Technology, 31, 281-287.
Hendricks Franssen, H.J., and W. Kinzelbach, 2008, Real-time groundwater flow modeling with the ensemble kalman filter: joint estimation of states and parameters and the filter inbreeding problem, Water Resources Research, 44, W09408.
Hsu, K.L., T. Bellerby, and S. Sorooshian, 2009, LMODEL: A satellite precipitation methodology using cloud development modeling. part ii: validation, Journal of hydrometeology, 10, 1096-1108.
Jiang, S., Y. Gu, C.M. Fan, and W. Qu, 2021, Fracture mechanics analysis of bimaterial interface cracks using the generalized finite difference method, Theoretical and Applied Fracture Mechanics, 113, 102942.
Kitanidis, P.K., 2015, Persistent questions of heterogeneity, uncertainty, and scale in subsurface flow and transport, Water Resources Research, 51, 5888-5904.
Konapala, G., A.K. Mishra, Y. Wada, and M.E. Mann, 2020, Climate change will affect global water availability through compounding changes in seasonal precipitation and evaporation, Nature Communications, 11, 3044.
Kumar, A.V., R. Burla, S. Padmanabhan, and L. Gu, 2008, Finite element analysis using nonconforming mesh, Journal of Computing and Information Science in Engineering, 8, 031005.
Lei, J., X. Wei, Q. Wang, Y. Gu, and C.M. Fan, 2022, A novel space–time generalized FDM for dynamic coupled thermoelasticity problems in heterogeneous plates, Archive of Applied Mechanics, 92, 287-307.
Li, L., and M. Zhang, 2017, Inverse modeling of interbed parameters and transmissivity using land subsidence and drawdown data, Stochastic environmental research and risk assessment, 32, 921-930.
Li, L., H.A. Tchelepi, and D. Zhang, 2003, Perturbation-based moment equation approach for flow in heterogeneous porous media: applicability range and analysis of high-order terms, Journal of Computational Physics, 188, 297-317.
Li, L., H.J. Hendricks Franssen, and J.J. Gomez-Hernandez, 2012, Groundwater flow inverse modeling in non-MultiGaussian media: performance assessment of the normal-score Ensemble Kalman Filter, Hydrology and Earth System Sciences, 16, 573-590.
Li, L., R. Puzel, and A. Davis, 2018, Data assimilation in groundwater modelling: ensemble Kalman filter versus ensemble smoothers, Hydrological Processes, 32, 2020-2029.
Linde, K., D. Ginsbouger, J. Irving, F. Nobile, A. Doucet, 2017, On uncertainty quantification in hydrogeology and hydrogeophysics, Advanced in Water Resources, 110, 166-181.
Lykkegaard, M.B., and T.J. Dodwell, 2022, Where to drill next? A dual-weighted approach to adaptive optimal design of groundwater surveys, Advanced in Water Resources, 104219.
Matos, C.R., J.F. Carneiro, and P.P. Silva, 2019, Overview of large-scale underground energy storage technologies for integration of renewable energies and criteria for reservoir identification, Journal of Energy Storage, 21, 241-258.
Metropolis, N.C., 1987, The Begining of the Monte Carlo Method, Los Alamos Science, Los Alamos National Laboratory, 15, 125-130.
Michael, I., T. Seifarth, J. Kuhnert, and P. Suchde, 2021, A meshfree generalized finite differencemethod for solution mining processes, Computational Particle Mechanics, 8, 561-574.
Mishra, P.K., 2019, NodeLab: A MATLAB package for meshfree node-generation and adaptive refinement, The Journal of Open Source Software, 4, 1173.
Moradkhani, H., K.-L. Hsu, H. Gupta, and S. Sorooshian, 2005, Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter, Water Resources Research, 41, W05012.
Nakamura, G., and R. Potthast, 2015, An introduction to the theory and methods of inverse problems and data assimilation, Inverse Modeling, IOP Publishing.
Ni, C.F., Y.J. Huang, J.J. Dong, and T.C.J. Yeh, 2015, Sequential hydraulic tests for transient and highly permeable unconfined aquifer systems - model development and field-scale implementat, Hydrology and Earth System Sciences Discussions, 12, 12567-12613.
Nonino, M., F. Ballarin, and G. Rozza, 2021, A monolithic and a partitioned, reduced basis method for fluid–structure interaction problems, Fluids, 6, 229.
Notton, G., M.L. Nivet, C. Voyant, C. Paoli, C. Darras, F. Motte, and A. Fouilloy, 2018, Intermittent and stochastic character of renewable energy sources: Consequences, cost of intermittence and benefit of forecasting, Renewable and Sustainable Energy Reviews, 87, 96-105.
Panzeri, M., M. Riva, A. Gaudagnini, and S.P. Neuman, 2015, EnKF coupled with groundwater flow moment equations applied to Lauswiesen aquifer, Germany, Journal of Hydrology, 521, 205-216.
Panzeri, M., M. Riva, A. Guadagnini, and S.P. Neuman, 2014, Comparison of ensemble kalman filter groundwater-data assimilation methods based on stochastic moment equations and monte carlo simulation, Advanced in Water Resources, 66, 8-18.
Raanes, P.M., 2015, Improvements to ensemble methods for data assimilation in the geosciences, A thesis submitted for Doctor of Philosophy, St Hugh's College University of Oxford.
Rao, X., H. Zhao, and Z. Wentao, 2022, An upwind generalized finite difference method for meshless solution of two-phase porous flow equations, Engineering Analysis with Boundary Elements, 137, 105-118.
Robertson, R., 2002, Direct sequential simulation algorithms in geostatistics, A thesis submitted to the faculty of communications, Health and Science Edith Cowan University Perth, Western Australia.
Suchde, P., 2018, Conservation and accuracy in meshfree generalized finite difference methods, A thesis submitted for Doctor of Philosophy, Department of Mathematics, University of Kaiserslautern, Germany.
Suchde, P., and J. Kuhnert, 2019, A meshfree generalized finite difference method for surface PDEs, Computers and Mathematics with Applications, 78, 2789-2805.
Sun, N.Z., and A. Sun, 2015, Data Assimilation for Inversion, Model Calibration and Parameter Estimation For Environmental and Water Resource Systems, Springer.
Tinoco-Guerrero, G., F.J. Dominguez-Mota, and J.G. Tinoco-Ruiz, 2020, A study of the stability for a generalized finite-difference scheme applied to the advection - diffusion equation, Mathematics and Computers in Simulation, 176, 301-311.
Tran, D.H., S.J. Wang, and Q.C. Nguyen, 2022, Uncertainty of heterogeneous hydrogeological models in groundwater flow and land subsidence simulations - A case study in Huwei Town, Taiwan, Engineering Geology, 298, 106543.
Tsai, J.P., T.C.J. Yeh, C.C. Cheng, Y. Zha, L.C. Chang, C. Hwang, Y.L. Wang, and Y. Hao, 2017, Fusion of time-lapse gravity survey and hydraulic tomography for estimating spatially varying hydraulic conductivity and specific yield fields, Water Resources Research, 53, 8554-8571.
Urena, F., L. Gavete, A Garcia, J.J. Benito, and A.M. Vargas, 2018, Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM), Journal of Computational and Applied Mathematics, 354, 221-241.
Urena, F., L. Gavete, A. Garcia, J.J. Benito, and A.M. Vargas, 2020, Solving second order non-linear hyperbolic PDEs using generalized finite difference method (GFDM), Journal of Computational and Applied Mathematics, 363, 1-21.
van der Sande, K., and B. Fornberg, 2021, Fast variable density 3-D node generation, SIAM Journal on Scientific Computing, 43, A242-A257.
Van Lent, T., and P.K. Kitanidis, 1996, Effects of first-order approximations on head and specific discharge covariances in high-contrast log conductivity, Water Resources Research, 32, 1.
Wang, S.J., and K.C. Hsu, 2009, The application of the first-order second-moment method to analyze poroelastic problems in heterogeneous porous media, Journal of Hydrology, 369, 209-221.
Xia, C.A., A. Guadagnini, B.X. Hu, M. Rive, and P. Ackerer, 2019, Grid convergence for numerical solutions of stochastic moment equations of groundwater flow, Stochastic Environmental Research and Risk Assessment, 33, 1565-1579.
Xia, C.A., X. Luo, B.X. Hu, M. Riva, and A. Guadagnini, 2021, Data assimilation with multiple types of observation boreholes via ensemble kalman filter embedded within stochastic moment equations, Hydrology and Earth System Sciences, 25, 1689–1709.
Yang, H. J., F. Boso, H. A. Tchelepi, and D. M. Tartakovsky, 2020, Method of distributions for quantification of geologic uncertainty in flow simulations, Water Resources Research, 56, e2020WR027643.
Yeh, T.C.J., and S. Liu, 2000, Hydraulic tomography: development of a new aquifer test method, Water Resources Research, 36, 2095-2105.
Yeh, T.C.J., M. Jin, and S. Hanna, 1996, An iterative stochastic inverse method: conditional effective transmissivity and hydraulic head fields, Water Resources Research, 32, 85-92.
Younger, P.L., 2014, Hydrogeological challenges in a low-carbon, Quarterly Journal of Engineering Geology and Hydrogeology, 47, 7-27.
Yu, H.L., Y.Z. Wu, and S.Y. Cheung, 2020, A data assimilation approach for groundwater parameter estimation under Bayesian maximum entropy framework, Stochastic Environmental Research and Risk Assessment, 34, 709-721.
Zhang, C., L. Zhang, X. Wang, S. Rao, Y. Zuo, R. Huang, P. Gao, R. Song, and Z. Wang, 2021, Effects of the spatial heterogeneity in reservoir parameters on the heat extraction performance forecast based on a 3D thermo-hydro-mechanical coupled model: a case study at the Zhacang geothermal field in the Guide basin, northeastern Tibetan plateau, Geothermics, 95, 102161.
Zhang, D., 1998, Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media, Water Resources Research, 34, 529-538.
Zhang, D., and C.L. Winter, 1999, Moment-equation approach to single phase fluid flow in heterogeneous reservoirs, SPE Journal, 4, 118-127.
Zhou, H., J.J. Gomez-Hernandez, and L. Li, 2014, Inverse methods in hydrogeology: Evolution and recent trends, Advances in Water Resources, 63, 22-37.