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

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 ρ ₚ ∈ [2500, 3500]\,kg/m³ , fluid viscosity μ ∈ [0.0008, 0.0012]\,Pa· s , and particle diameter dₚ ∈ [0.0005, 0.0010]\,~m . 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 ε = 0.1 . Pressure-gradient analysis showed a 45% reduction in settling depth as Δ P increased from 0.1 to 1.0 bar/m. Model performance was evaluated across symbolic, CFD, and combined datasets. The symbolic model achieved R ² = 0.9934, RMSE = 0.0436; the CFD model yielded R ² = 0.9941, RMSE = 0.2033. The hybrid ensemble outperformed both with R ² = 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.