Relative permeability and capillary pressure estimation via physics-informed machine learning and reinforcement learning

Abstract

Accurately predicting multiphase fluid flow in oil and gas reservoirs is crucial to optimizing production and minimizing costs. However, traditional numerical reservoir simulations are computationally expensive, while innovative data-driven models may not adhere to physical laws. Building on established physics-informed machine learning (PIML) formulations, we present an integrated PIML–reinforcement learning (RL) workflow that embeds the governing fluid-flow equations and uses RL to solve the inverse problem of estimating relative-permeability model parameters from average water-saturation measurements. Using the inferred parameters, the forward physics-consistent model predicts saturation dynamics and enables inference of capillary-pressure trends during unsteady-state waterflooding. The proposed model accurately predicts the average water saturation over time and estimates trends in capillary pressure across three laboratory experiments. Additionally, a sensitivity analysis is performed to understand the impact of the estimated parameters on the model predictions.

Keywords

Relative permeability / Capillary pressure / Reinforcement learning / Physics-informed machine learning / Waterflooding

CRediT authorship contribution statement

Ramanzani Kalule: Writing – original draft, Visualization, Software, Methodology, Investigation, Conceptualization. Hamid Ait Abderrahmane: Writing – review & editing, Supervision, Conceptualization. Shehzad Ahmed: Writing – review & editing, Supervision, Methodology, Conceptualization. Waleed Alameri: Writing – review & editing, Supervision, Funding acquisition. Emad W. Al-Shalabi: Writing – review & editing, Supervision, Conceptualization.

Data availability

The authors do not have permission to share the data.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported by the Research and Innovation Center on CO₂ and Hydrogen, Khalifa University of Science and Technology (KU-RICH). The authors acknowledge the use of Khalifa University’s High-Performance Computing and Research Computing facilities (KU-HPC) in support of the computational aspects of this research.

References

[1] N. Wenck, S.J. Jackson, S. Manoorkar, A. Muggeridge, S. Krevor, Simulating core floods in heterogeneous sandstone and carbonate rocks, Water Resour. Res. 57 (2021) 1-11, https://doi.org/10.1029/2021WR030581.

[2] U. Alfazazi, W. AlAmeri, M.R. Hashmet, Experimental investigation of polymer flooding with low-salinity preconditioning of high temperature-high-salinity carbonate reservoir, J. Pet. Explor. Prod. Technol. 9 (2019) 1517-1530, https://doi.org/10.1007/s13202-018-0563-z.

[3] S. Ahmed, A.S. Hanamertani, W. Alameri, E.W. Al-Shalabi, M.R. Hashmet, Experimental investigation of flow diversion and dynamic retention during polymer flooding in high salinity fractured carbonates using CT imaging, Geoenergy Sci. Eng. 221 (2023) 211349, https://doi.org/10.1016/j.geoen.2022.211349.

[4] Y. Jing, A. Uthaia, D. Stimson, K. Mardon, L. Najdovski, C. Barkhausen, R.T. Armstrong, P. Mostaghimi, Visualisation of [11C]CO₂ storage in coal with positron emission tomography imaging, Fuel 356 (2024) 129511, https://doi.org/10.1016/j.fuel.2023.129511.

[5] S. Kameswaran, L.T. Biegler, G.H. Staus, Dynamic optimization for the core-flooding problem in reservoir engineering, Comput. Chem. Eng. 29 (2005) 1787-1800, https://doi.org/10.1016/j.compchemeng.2005.02.038.

[6] G.E. Karniadakis , I.G. Kevrekidis , L. Lu , P. Perdikaris , S. Wang , L. Yang , Physics-informed machine learning, Nat. Rev. Phys. 3 (2021) 422-440, https://doi.org/10.1038/s42254-021-00314-5.

[7] T. Farhana Faisal, A. Islam, M.S. Jouini, R.S. Devarapalli, M. Jouiad, M. Sassi, Numerical prediction of carbonate elastic properties based on multi-scale imaging, Geomechanics for Energy and the Environment 20 (2019) 100125, https://doi.org/10.1016/j.gete.2019.100125.

[8] M.S. Jouini , S. Vega , A. Al-Ratrout , Numerical estimation of carbonate rock properties using multiscale images, Geophys. Prospect. 63 (2015) 405-421, https://doi.org/10.1111/1365-2478.12156.

[9] M.D. Swinson , Statistical Modeling of High-Dimensional Nonlinear Systems: a Projection Pursuit Solution, Georgia Institute of Technology, 2005.

[10] Y. Wang, G. Lin, Efficient deep learning techniques for multiphase flow simulation in heterogeneous porous media, J. Comput. Phys. 401 (2020), https://doi.org/10.1016/j.jcp.2019.108968.

[11] E.S. Mathew , M. Tembely , W. AlAmeri , E.W. Al-Shalabi , A.R. Shaik , Application of machine learning to interpret steady state drainage relative permeability experiments, in: Abu Dhabi International Petroleum Exhibition & Conference to be Held in Abu Dhabi, UAE, 2021, https://doi.org/10.2118/207877-ms.

[12] R. Kalule, H.A. Abderrahmane , W. Alameri, M. Sassi, Stacked ensemble machine learning for porosity and absolute permeability prediction of carbonate rock plugs, Sci. Rep. 13 (2023) 1-17, https://doi.org/10.1038/s41598-023-36096-2.

[13] M.M. Almajid , M.O. Abu-Alsaud , M.M. Almajid , M.O. Abu-Al-Saud , Prediction of porous media fluid flow using physics informed neural networks, J. Pet. Sci. Eng. 208 (2022) 1-31, https://doi.org/10.1016/j.petrol.2021.109205.

[14] P. Tang, D. Zhang, H. Li, Predicting permeability from 3D rock images based on CNN with physical information, J. Hydrol. (Amst.) 606 (2022) 127473, https://doi.org/10.1016/j.jhydrol.2022.127473.

[15] R. Pires de Lima, D. Duarte, C. Nicholson, R. Slatt, K.J. Marfurt, Petrographic microfacies classification with deep convolutional neural networks, Comput. Geosci. 142 (2020) 104481, https://doi.org/10.1016/j.cageo.2020.104481.

[16] M.I.K. Haq , I.N. Yulita , I.A. Dharmawan , A study of transfer learning in digital rock properties measurement, Mach Learn Sci Technol 4 (2023) 1-19.

[17] M. Raissi, Open problems in applied deep learning, ArXiv arXiv 2301 (2023) 1-52.

[18] P. Shokouhi, V. Kumar, S. Prathipati, S.A. Hosseini, C.L. Giles, D. Kifer, Physics-informed deep learning for prediction of CO₂ storage site response, J. Contam. Hydrol. 241 (2021) 1-30, https://doi.org/10.1016/j.jconhyd.2021.103835.

[19] J.D. Willard , X. Jia , S. Xu , M. Steinbach , V. Kumar , Integrating physics-based modeling with machine learning: a survey, ArXiv 1 (2020) 1-34.

[20] M. Raissi, A. Yazdani, G.E. Karniadakis, Hidden Fluid Mechanics: Learning Velocity and Pressure Fields from Flow Visualizations, 2020, https://doi.org/10.1126/science.aaw4741.

[21] L. Von Rueden, S. Mayer, K. Beckh, B. Georgiev, S. Giesselbach, R. Heese, B. Kirsch, J. Pfrommer, A. Pick, R. Ramamurthy, M. Walczak, J. Garcke, C. Bauckhage, J. Schuecker, Informed machine learning - a taxonomy and survey of integrating knowledge into learning systems, ArXiv (2019) 1-20.

[22] M. Raissi, G.E. Karniadakis, Hidden physics models: machine learning of nonlinear partial differential equations, J. Comput. Phys. 357 (2018) 125-141, https://doi.org/10.1016/j.jcp.2017.11.039.

[23] A. Karpatne, G. Atluri, J.H. Faghmous, M. Steinbach, A. Banerjee, A. Ganguly, S. Shekhar, N. Samatova, V. Kumar, Theory-guided data science: a new paradigm for scientific discovery from data, IEEE Trans. Knowl. Data Eng. 29 (2017) 2318-2331, https://doi.org/10.1109/TKDE.2017.2720168.

[24] M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics informed deep learning (Part II): data-Driven discovery of nonlinear partial differential equations, ArXiv (2017) 1-19.

[25] Y. Wang, G. Lin, Efficient deep learning techniques for multiphase flow simulation in heterogeneous porous media, J. Comput. Phys. 401 (2020), https://doi.org/10.1016/j.jcp.2019.108968.

[26] D.S. Oliver , Y. Chen , Recent progress on reservoir history matching: a review, Comput. Geosci. 15 (2011) 185-221, https://doi.org/10.1007/s10596-010-9194-2.

[27] K. Johansen, General Rights Statistical Methods for History Matching, 2007. www.kt.dtu.dk.

[28] S. Bryant, S. Srinivasan, A. Barrera, Y. Kim, S. Yadav, History Matching in Parallel Computational Environments Final Report, 2006. Texas, https://www.netl.doe.gov/. (Accessed 25 December 2025).

[29] M.M. Khaninezhad , B. Jafarpour , Sparse randomized maximum likelihood (SpRML) for subsurface flow model calibration and uncertainty quantification, Adv. Water Resour. 69 (2014) 23-37, https://doi.org/10.1016/j.advwatres.2014.02.005.

[30] S.L. Brunton, J.N. Kutz, Data-driven science and engineering: machine learning, Dyn. Syst. Control (2019), https://doi.org/10.1017/9781108380690. Second.

[31] C. Vignon, J. Rabault, J. Vasanth, F. Alcántara-Avila, M. Mortensen, R. Vinuesa, Effective control of two-dimensional Rayleigh-Benard convection: invariant multi-agent reinforcement learning is all you need, Phys. Fluids 35 (2023) 1-15, https://doi.org/10.1063/5.0153181.

[32] S. Cuomo, V.S. di Cola , F. Giampaolo, G. Rozza, M. Raissi, F. Piccialli, Scientific Machine Learning Through Physics-Informed Neural Networks: where We are and What's next, 2022, pp. 1-67.

[33] R. Kalule, U. Alfazazi, S. Ahmed, H.A. Abderrahmane , W. Alameri, M. Sassi, Estimating relative permeability and capillary pressure functions from unsteady-state core flooding data using dynamic optimization techniques, in: SPE Abu Dhabi International Petroleum Exhibition and Conference (ADIPEC), 2023, pp. 1-12, https://doi.org/10.2118/216279-MS.

[34] R. Kalule, A.H. Abderrahmane, S. Ahmed, W. Alameri, M. Sassi, Dynamic optimization for petrophysical property estimation in Unsteady-state coreflooding using pyomo, SPE J. SPE-219450-PA (2024) 1-13, https://doi.org/10.2118/219450-PA.

[35] F. Lomeland, E. Ebeltoft, W.H. Thomas, A New Versatile Relative Permeability Correlation, International Symposium of the Society of Core Analysts, Toronto, Canada, 2005, pp. 1-12.

[36] E. Ghoodjani, S.H. Bolouri , A novel two-parameter relative permeability model, J. Porous Media 15 (2012) 1061-1066.

[37] Azziz and Settari , Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979, pp. 135-139, https://doi.org/10.1016/c2018-0-04535-1.

[38] P.C. Richmond , A.T. Watson , Estimation of multiphase flow functions from displacement experiments, SPE Reserv. Eng. 5 (1990) 121-127, https://doi.org/10.2118/18569-pa.

[39] M. Yaralidarani, H. Shahverdi, Co-estimation of saturation functions (kr and Pc) from unsteady-state core-flood experiment in tight carbonate rocks, J. Pet. Explor. Prod. Technol. 8 (2018) 1559-1572, https://doi.org/10.1007/s13202-018-0452-5.

[40] H. Jian, L. Chenghui, C. Zhimin, M. Haiwei, Integration of deep neural networks and ensemble learning machines for missing well logs estimation, Flow Meas. Instrum. 73 (2020) 101748, https://doi.org/10.1016/j.flowmeasinst.2020.101748.

[41] T. Qin, K. Wu, D. Xiu, Data driven governing equations approximation using deep neural networks, J. Comput. Phys. 395 (2019) 620-635, https://doi.org/10.1016/j.jcp.2019.06.042.

[42] M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019) 686-707, https://doi.org/10.1016/j.jcp.2018.10.045.

[43] P. Garnier, J. Viquerat, J. Rabault, A. Larcher, A. Kuhnle, E. Hachem, A review on deep reinforcement learning for fluid mechanics, ArXiv 225 (2021) 1-23, https://doi.org/10.1016/j.compfluid.2021.104973.

[44] X.Y. Zhang , M.N. Trame , L.J. Lesko , S. Schmidt , Sobol sensitivity analysis: a tool to guide the development and evaluation of systems pharmacology models, CPT Pharmacometrics Syst. Pharmacol. 4 (2015) 69-79, https://doi.org/10.1002/psp4.6.

[45] J. Herman, W. Usher, Salib Documentation, 2019. Release 1.1.3.post0. dev82+ng5deeaf3.

[46] T. Bognø, Impacts on Oil Recovery from Capillary Pressure and Capillary Heterogeneities, University of Bergen, 2008.

[47] W.G. Anderson , Wettability literature survey - part 1: Rock/Oil/Brine interactions and the effects of core handling on Wettability, Journal of Petroleum Science and Technology 38 (1986) 1125-1144.