Peter von Schultzendorff

arXiv2026

Efficient design of continuation methods for hyperbolic transport problems in porous media

Full-physics modeling of multiphase flow in porous media, e.g., for carbon storage and groundwater management, requires the nonlinear coupling of various physical processes. Industry standard nonlinear solvers, typically of Newton-type, are not unconditionally convergent and computationally expensive. Homotopy continuation solvers have recently been studied as a robust and versatile alternative. They tackle challenging nonlinear problems by first solving a simple auxiliary problem and then tracing a solution curve towards the more complex target problem. Robustness and efficiency of the method depends on the iterative numerical curve tracing algorithm as well as on careful design of the auxiliary problem. We assess the traceability of the solution curve for different choices of the auxiliary problem. For the Buckley-Leverett equation, modeling two-phase flow in one dimension, we exemplarily compare the previously introduced vanishing-diffusion and linear constitutive laws homotopy continuation, and a new approach based on the entropy solution of the problem. This provides insight toward systematically and robustly designing homotopy continuation methods for solving complex multiphase flow in porous media.

math.NA↗

arXiv2026

A Machine-Learned Near-Well Model in OPM Flow

Recent advances in reservoir simulation increasingly utilize hybrid approaches that couple physics-based simulators with machine-learning (ML) components. ML components offer high fidelity to training data and fast inference, enabling efficient and accurate modeling of complex multi-scale or multi-physics phenomena. Modern reservoir simulators rely on automatic differentiation (AD) to support efficient and flexible strategies for nonlinear solvers, inverse problems, and optimization problems. Efficient hybrid modeling therefore requires tight integration of the ML components with the simulator's AD framework. We present the first integration of neural networks into the high-performance reservoir simulator OPM Flow. Networks are trained in TensorFlow and imported into OPM, where they are accessed as native AD functions. This presents an efficient framework for hybrid modeling and enables seamless integration in existing simulator workflows. As an application, we introduce a novel, data-driven near-well model. Near-well models are essential in reservoir simulation for accurately representing singular pressure gradients around wells. Commonly used are the Peaceman near-well model and its extensions, or local grid refinement around the wells. Peaceman-type models are limited to simplified flow regimes, whereas local grid refinement is computationally expensive. We address these limitations by training a neural network to infer a Peaceman-like well index from fine-scale ensemble simulations of the near-well region. It is then integrated into OPM Flow with the new framework. Tested on relevant examples for CO$_2$ storage, the method offers high fidelity to fine-scale results at low computational cost, demonstrating the potential of the OPM Flow-Neural Network framework for hybrid modeling.

math.NA↗