Nordgren Physics-Informed Neural Networks to Variational Quantum Eigensolver: Advancing Hydraulic Fracturing Simulations in Shale Reservoirs

This study advances hydraulic fracturing simulations in shale reservoirs using two computational paradigms, Physics-Informed Neural Networks (PINNs) and the Variational Quantum Eigensolver (VQE). PINNs are employed to solve Nordgren’s equation, which governs fracture width evolution, by embedding physical laws into the neural network architecture. Using TensorFlow on Google Colab, the PINN training process incorporates Adam, L-BFGS, and Newton-CG optimizers, achieving an accuracy of 1.23x10 ^-9 for fluid viscosity \mu = 8.0 Pa.s, fracture height H = 2.0m, and leak-off coefficient C _L = 1.0. However, this approach demands significant computational resources, with training times exceeding 1454 seconds and memory usage of 1136 MB in some cases. Conversely, the VQE framework leverages Qiskit on Qbraid to optimize the Hamiltonian representing the fracture system. With qubit-based ansatz circuits and classical optimizers (SPSA, COBYLA, and L-BFGS), VQE achieves rapid energy minimization, converging to approximately -0.583+0j in under 2 seconds with negligible memory requirements. Spatiotemporal fracture width predictions from VQE align well with expected trends but exhibit slight oscillations due to quantum noise. This comparative study highlights the trade-offs where PINNs excel in accuracy and physical adherence, while VQE offers unparalleled efficiency for rapid computations. The hybrid application of PINN precision and VQE speed provides a robust toolkit for optimizing hydrocarbon recovery. Future research will explore integrating these paradigms for scalable, high-fidelity simulations across complex geological settings.