Machine learning-enhanced fully coupled fluid–solid interaction models for proppant dynamics in hydraulic fractures

Wayo D.D.K. Irawan S. Wang L. Goliatt L.
December 2025 Nature Research

Scientific Reports
2025 #15 Issue 1

This study presents a hybrid modeling framework for predicting proppant settling rate (PSR) in hydraulic fracturing by integrating symbolic physics-based derivations, parametric simulations, and ensemble machine learning. Symbolic expressions were formulated using Stokes’ law, drag equations, and pressure-gradient dynamics. A symbolic dataset was synthetically generated by sampling realistic physical ranges: proppant density, fluid viscosity, and particle diameter. Complementary CFD-informed datasets were simulated to represent complex flow behavior. Both datasets were used to train stacked ensemble regressors comprising five base learners: Random Forest, Extra Trees, Gradient Boosting, XGBoost, and Support Vector Regression (SVR), combined with a RidgeCV meta-learner. Numerical analysis validated the physics consistency of the symbolic model. ODE-based simulations revealed terminal velocity of 0.39 m/s reached within 0.5 s, while parametric studies showed velocity reductions up to 40% for strain. Pressure-gradient analysis showed a 45% reduction in settling depth as increased from 0.1 to 1.0 bar/m. Model performance was evaluated across symbolic, CFD, and combined datasets. The symbolic model achieved R = 0.9934, RMSE = 0.0436; the CFD model yielded R = 0.9941, RMSE = 0.2033. The hybrid ensemble outperformed both with R = 0.9970, RMSE = 0.1801. This framework enables interpretable, accurate, and computationally efficient prediction of PSR, eliminating the need for full-scale CFD–DEM simulations. It is well-suited for decision support in multiscale fracture design and proppant transport analysis.

Computational geomechanics , Fluid–solid , Hydraulic fracturing , Machine learning , Proppant

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Faculty of Chemical and Process Engineering Technology, Universiti Malaysia Pahang Al-Sultan Abdullah, Kuantan, 26300, Malaysia
Department of Petroleum Engineering, School of Mining and Geosciences, Nazarbayev University, Astana, 010000, Kazakhstan
State Key Laboratory of Oil and Gas Reservoir Geology and Exploration & College of Energy, Chengdu University of Technology, Chengdu, 610059, China
Department of Computational and Applied Mechanics, Federal University of Juiz de Fora, Juiz de Fora, 36036-900, Brazil

Faculty of Chemical and Process Engineering Technology
Department of Petroleum Engineering
State Key Laboratory of Oil and Gas Reservoir Geology and Exploration & College of Energy
Department of Computational and Applied Mechanics

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