Nordgren PINNs to VQE: Advancing Hydraulic Fracturing Simulations in Shale Reservoirs

Abstract

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 were 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 incorporated Adam, L-BFGS, and Newton-CG optimizers, reaching a final loss of $1.23\times {10}^{-9}$ for a representative benchmark case (e.g., fluid viscosity $\mu$ = 8.0 $\mathrm{Pa}·s$, fracture height $H$ = 2.0 m, and leak-off coefficient $C_L$ = 1.0); these parameters are used for comparison and are not tied to a specific field dataset. However, this approach demands computational resources, with training times exceeding 1454 s and memory usage of 1136 MB. Conversely, the VQE framework leveraged Qiskit on Qbraid to optimize the Hamiltonian representing the fracture system in practice. With qubit-based ansatz circuits and classical optimizers (SPSA, COBYLA, and L-BFGS), VQE achieves energy minimization, converging to -0.583+0j in under 2 s with low memory requirements. Spatiotemporal fracture width predictions from VQE align with trends but exhibit slight oscillations due to quantum noise. This comparative study highlights a trade-off: PINNs show stronger physics-consistent accuracy in the tested cases, while VQE provides lower runtime and memory use. These results suggest that combining PINN accuracy with VQE speed may be useful for decision-support workflows in hydraulic fracturing. Future research will explore integrating these paradigms for scalable, high-fidelity simulations across complex geological settings.