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Data assimilation for real-time subsurface flow modeling with dynamically adaptive meshless node adjustments

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Abstract

Over the past few decades, various inverse modeling and data assimilation techniques have been proposed to integrate observed data into subsurface flow models for optimal parameter estimation. In practice, subsurface flow models are often constructed based only on preliminary hydrogeological surveys. Additional data will be collected from supplementary hydrogeological surveys afterward and will therefore be difficult, if not impossible, to incorporate into predefined numerical nodes. Grid refinement or model reconstruction is repetitively required whenever new or additional data are assimilated into physical models. A novel data assimilation method that uses a dynamically adaptive node adjustment (DANA) scheme was proposed in this paper. DANA avoids laborious remeshing to assimilate real-time data. It combines the meshless method, interpolation method, and fast-node-placement algorithm to automatically update the layouts of computational nodes according to the newly available data over time. The meshless generalized finite difference was chosen to develop the DANA framework, and the ensemble Kalman filter (EnKF) was used as the data assimilation approach. The accuracy and computational efficiency of the proposed methods were investigated, and the applicability of DANA was demonstrated by solving a hypothetical assimilation problem. The results indicate that DANA can efficiently cooperate with the EnKF to achieve real-time updating for subsurface modeling. The DANA-based assimilation model can flexibly handle randomly distributed additional data, efficiently reduce parameter uncertainty, and provide versatile dynamical modeling.

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Data availability

The data that support the findings of this study are openly available in Mendeley Data at https://doi.org/10.17632/gtsxgxrh9t.1.

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Acknowledgements

This study was financially supported by the National Science and Technology Council (NSTC), Taiwan (Project Nos. 111-2116-M-006-015, 112-2811-E-006-018-MY3, and 112-2221-E-006 -050 -MY3).

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  1. Department of Resources Engineering, National Cheng Kung University, No. 1, University Road, Tainan City, 701, Taiwan (R.O.C.)

    Shang-Ying Chen, Jian-Yu Wei & Kuo-Chin Hsu

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  1. Shang-Ying Chen
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Appendices

Appendix

Influences of interpretation methods and complex boundaries

In this appendix, we present an additional case study to demonstrate the applicability of the proposed DANA-based assimilation model to complex domains with concave boundaries, which are often encountered in practical applications. The case study was designed to investigate the performance of the novel tool combined with different interpolation methods. The three interpolation methods used in the case study are kriging, inverse distance weighting (IDW), and natural neighbor interpolation [78]. The boundary geometry and hydraulic head boundary conditions for the hypothetical case were obtained from the research published by Gaitanaru [79] focusing on a site located near Podari, Romania, with a concave boundary. The irregular geometry of the aquifer was addressed using the proposed node-placement algorithm. The boundary conditions are defined as depicted in Fig. 23. In the present case, the concave part is associated with the no-flow boundary, whereas the example presented in Fig. 6(a) is associated with the Dirichlet boundary.

Fig. 23
figure 23

Computational nodes and boundary conditions of the concave case

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The hydraulic conductivity was determined using the SGS employed in this study. The variogram model had a range of 1500 m. The heterogeneity in hydraulic conductivity obtained using SGS is illustrated in Fig. 24. For simplicity, it was assumed that no pumping wells or other injection operations affected the aquifer. The simulated spatial distributions of the hydraulic heads are shown in Fig. 25.

Fig. 24
figure 24

Background conductivity field and sampling schedule of the concave case

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Fig. 25
figure 25

Background head field of the concave case

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These representations of the concave case, including the boundary geometry, hydraulic conductivity field, and hydraulic head distribution, were treated as true fields for data assimilation evaluations and subsequent analyses. To obtain the necessary data for the assimilation process, a dataset comprising the hydraulic head and hydraulic conductivity information was collected from the background true field. Initially, the data assimilation update scheme did not include any data. Subsequently, a stratified sampling approach was adopted to collect daily pairs of hydraulic head and conductivity data within a grid-based domain. The sampling locations and timing are depicted as red diamond-shaped markers in Fig. 24. The data collected through the process of the proposed DANA-EnKF were dynamically assimilated into the numerical model to update the estimated parameter field and estimated state of hydraulic heads. Prior to data acquisition, the logarithm of the hydraulic conductivity (Y) was assumed to be homogeneous with a value of − 7.

As the data assimilation process progressed, the DANA framework automatically adjusted the positions of the computational nodes to adapt to the incoming data (Fig. 26). After relocating the nodes, numerical interpolation is performed to estimate the values at the newly positioned nodes. In this case study, three different interpolation methods, namely kriging, natural neighbor interpolation, and inverse distance weighting, were compared. The results obtained using kriging as the interpolation method in DANA are presented in Fig. 27. The updated parameter field, specifically the hydraulic conductivity, was progressively refined as more data became available. The corrected parameter field gradually resembled the characteristics of the true background field (Fig. 24), and the discrepancies between the estimated field and true background field continually diminished. In addition, parameter uncertainty was dynamically quantified over time. Figure 28 displays the updated distribution of the hydraulic heads as a result of the DANA iterative relocation and assimilation process. Remarkably, without modifying the boundary conditions or forcing terms, the distribution of the hydraulic heads approached a stabilized state that closely resembled the true background field (Fig. 25). The accuracy and precision of the hydraulic head estimation were significantly enhanced by this iterative assimilation procedure.

This assimilation process enabled the integration of the observed data into the numerical model, facilitating the refinement and calibration of the parameter field and the estimation of hydraulic head distributions. By iteratively assimilating the collected data, the model gradually improved its representation of the aquifer system, leading to enhanced simulation accuracy and precision.

Fig. 26
figure 26

Node adjustment with respect to assimilated data for the concave case when t = 0, 9, 18, and 27 d

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Fig. 27
figure 27

a Estimate mean, b estimate error, and c estimate variance of conductivity field obtained using the kriging-based DANA with t = 0, 9, 18, and 27 d

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Fig. 28
figure 28

a Estimate mean, b estimate error, and c estimate variance of hydraulic head field obtained using the kriging-based DANA with t = 0, 9, 18, and 27 d

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Three interpolation methods were employed in the DANA framework. The performance was evaluated based on the accuracy and precision of the estimated parameters and model predictions, quantified using the RMSE and ES metrics. The RMSE results for the hydraulic conductivity and hydraulic head are shown in Figs. 29 and 30, respectively, whereas Fig. 32 shows the spread of the hydraulic conductivity. The IDW has a larger root-mean-square error fluctuation in conductivity. Figures 29, 30, 31, 32 illustrate that the trends obtained using the three interpolation methods were similar, and the differences were not significant. Notably, the kriging and natural neighbor interpolation methods exhibited higher stability than that observed with IDW. The resulting RMSE and ES values, which represent the quality of the optimized parameters and model predictions, are summarized in Table 5. The similarity in the trends and outcomes observed among the interpolation methods demonstrated that all three methods were effective within the DANA framework for achieving accurate parameter estimation and improving model predictions. The choice between kriging, natural neighbor, and IDW methods can be made based on considerations, such as computational efficiency, ease of implementation, and specific characteristics of the study. Overall, the results confirmed that the DANA framework, regardless of the interpolation method employed, yielded reliable and robust assimilation.

Fig. 29
figure 29

Root-mean-square-error for the conductivity of the concave case

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Fig. 30
figure 30

Root-mean-square-error for the head of the concave case

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Fig. 31
figure 31

Ensemble spread for the conductivity of the concave case

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Fig. 32
figure 32

Ensemble spread for the head of the concave case

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Table 5 RMSE and ES results after conducting data assimilation using different interpolation methods
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By comparing the results obtained from the concave case, we conclude that the various interpolation methods have no significant impact on the performance of our tool and the proposed DANA framework can be applied to complex domains with concave boundary problems. Based on the case study results, we provide additional evidence to support the versatility of the proposed DANA framework.

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Chen, SY., Wei, JY. & Hsu, KC. Data assimilation for real-time subsurface flow modeling with dynamically adaptive meshless node adjustments. Engineering with Computers (2023). https://doi.org/10.1007/s00366-023-01897-6

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