Physics-informed multi-task learning for permeability prediction and probabilistic HFU modeling: a case study from the Lower Bahariya Reservoir, Shahd SE field Egypt

Abstract

Accurate permeability prediction is essential for reliable reservoir characterization and simulation, yet remains challenging due to complex nonlinear relationships and subsurface heterogeneity. Conventional hydraulic flow unit (HFU) methods rely on discrete rock typing and fixed porosity–permeability relationships, limiting their ability to capture continuous variations. Physics-informed neural networks (PINNs) offer a data-driven alternative with embedded physical constraints, but their effectiveness is often limited by weak enforcement of physics during inference. In this study, a physics-guided multi-task neural network (MT-PINN) is proposed to simultaneously predict permeability and hydraulic flow units within a unified framework. The model integrates data-driven learning with physics-based relationships and probabilistic rock typing, enabling permeability to be estimated as a weighted combination of multiple flow units and allowing smoother transitions between facies. The proposed approach was evaluated using core and well log data and compared against conventional HFU and standard PINN methods. Within the studied dataset, the MT-PINN demonstrated improved predictive performance, with a higher correlation coefficient (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:R\:=\:0.936$$\end{document}) compared to HFU (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:R\:=\:0.89$$\end{document}) and PINN (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\:R\:=\:0.90$$\end{document}), along with a reduction in log-scale error. The model also provides more continuous and stable permeability predictions along depth. In addition to permeability estimation, the MT-PINN outputs both discrete HFU classifications and associated class probabilities, which can be incorporated into 3D stochastic reservoir modeling workflows for uncertainty-aware multi-realization analysis. The proposed framework demonstrates the potential to integrate traditional petrophysical methods with modern machine learning techniques, providing a promising workflow for permeability prediction and reservoir characterization within the studied reservoir.