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Inference of relative permeability curves in reservoir rocks with ensemble Kalman method

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Abstract

Multiphase flows through reservoir rocks are a universal and complex phenomenon. Relative permeability is one of the primary determinants in reservoir performance calculations. Accurate estimation of the relative permeability is crucial for reservoir management and future production. In this paper, we propose inferring relative permeability curves from sparse saturation data with an ensemble Kalman method. We represent these curves through a series of positive increments of relative permeability at specified saturation values, which guarantees monotonicity within, and boundedness between 0 and 1. The proposed method is validated by the inference performances in two synthetic benchmarks designed by SPE and a field-scale model developed by Equinor that includes certain real field features. The results indicate that the relative permeability curves can be accurately estimated within the saturation intervals having available observations and appropriately extrapolated to the remaining saturations by virtue of the embedded constraints. The predicted well responses are comparable to the ground truths, even though they are not included as the observation. The study demonstrates the feasibility of using ensemble Kalman method to infer relative permeability curves from saturation data, which can aid in the predictions of multiphase flow and reservoir production.

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Code availability

The code is available on GitHub [49], which can be used by the readers for reproducing the results and further development.

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Acknowledgements

The authors thank Xin-Lei Zhang (Chinese Academy of Sciences), Hongsheng Wang (University of Texas at Austin), and Tor Harald Sandve (NORCE Norwegian Research Centre AS) for their valuable discussions and suggestions.

Funding

This work was financially supported by the University Coalition for Fossil Energy Research (UCFER) Program under the US Department of Energy’s National Energy Technology Laboratory through the Award No. DE-FE0026825 and SubAward No. S000038-USDOE.

Author information

Authors and Affiliations

  1. Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA, 24060, USA

    Xu-Hui Zhou

  2. Thomas Lord Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA

    Haochen Wang

  3. National Security Institute, Virginia Tech, Blacksburg, VA, 24060, USA

    James McClure

  4. Department of Civil, Environmental and Ocean Engineering, Stevens Institute of Technology, Hoboken, NJ, 07030, USA

    Cheng Chen

  5. Stuttgart Center for Simulation Science, University of Stuttgart, 70569, Stuttgart, Baden-Württemberg, Germany

    Heng Xiao

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  1. Xu-Hui Zhou
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Contributions

JM, CC, and HX supervised the project. XZ, HW, and JM performed the research. XZ and HW wrote the manuscript. JM, CC, and HX provided insightful feedback and helped shape the research, analysis, and manuscript.

Corresponding author

Correspondence to Cheng Chen.

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The authors have no relevant financial or non-financial competing interests.

Additional information

T.I.: Quantitative AI in Complex Fluids and Complex Flows: Challenges and Benchmarks. Guest editors: Luca Biferale, Michele Buzzicotti, Massimo Cencini.

Appendix A: Inference with production data

Appendix A: Inference with production data

In this paper, we conduct two more numerical experiments with SPE 1 benchmark to highlight the ill-posedness induced by using production data only, as is the norm in many research of this nature.

Fig. 12
figure 12

Evaluation of the inferred relative permeability curve in SPE 1 benchmark: a comparison of the inferred curve with the ground truth, b comparison of the simulated gas saturation fields at four distinct years based on the inferred and true curves, and c comparison of the time series of well responses over 10 years based on the baseline, inferred and true curves. Here, the oil production rate time series for the first three years are used as the observation data

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Fig. 13
figure 13

Evaluation of the inferred relative permeability curve in SPE 1 benchmark: a comparison of the inferred curve with the ground truth, b comparison of the simulated gas saturation fields at four distinct years based on the inferred and true curves, and c comparison of the time series of well responses over 10 years based on the baseline, inferred, and true curves. Here, the oil production rate time series for all 10 years are used as the observation data

Full size image

In both experiments, the inferred gas relative permeability curve diverges significantly from the ground truth. The first experiment is designed for a fair comparison with that described in Sect. 4.1. We infer the relative permeability curve using the oil production rate in the first three years (January 2015 to December 2017) as the observation data. The inference result is shown in Fig. 12a. The sample curves do not converge after iterations and their mean is nearly equal to the initial guess. Consequently, the predicted gas saturation fields and well responses are significantly different from the corresponding ground truths, as shown in Fig. 12b, c, respectively. The performance is expected because the initial guess can yield the same oil production rate for the first three years, rendering the inference by definition ill-posed.

In contrast to the first experiment, the second experiment infers the relative permeability curve using a 10-year time series of oil production rate as the observation. The ill-posedness is reduced by using data in a longer time period. Nonetheless, the inferred relative permeability curve still differs from the true curve (Fig. 13a), although the predicted oil production rates are consistent with the ground truths (Fig. 13c). The predicted gas saturation fields and the other three well responses are significantly different from the corresponding ground truths, as illustrated in Fig. 13b, c, respectively.

From the above results, we can observe that the relative permeability curve may not be uniquely determined by the time series of production data alone. With sparse saturation data, the ill-posedness can be regularized and the estimation of relative permeability curve can be significantly improved.

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Zhou, XH., Wang, H., McClure, J. et al. Inference of relative permeability curves in reservoir rocks with ensemble Kalman method. Eur. Phys. J. E 46, 44 (2023). https://doi.org/10.1140/epje/s10189-023-00296-5

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