Numerical Simulation of Hydraulic Fracturing in the Phase Field from Dynamic Adaptive and Fully Unstructured Gridding

Summary Hydraulic fracturing remains a dynamic and evolving research field, with laboratory investigations continually revealing phenomena that challenge conventional modeling tools. Despite advances in simulation techniques, key limitations persist—particularly in representing fracture branching and the influence of distinct fracturing fluids. These complexities are well-suited to phase-field approaches, which replace sharp crack discontinuities with diffusive damage zones, enabling the modeling of intricate crack geometries. By incorporating surface energy—known to vary in the presence of different fluids—phase-field models also capture fluid-specific effects. Although the method introduces additional mathematical constructs, its foundations remain deeply rooted in Griffith’s classical energetic framework for crack propagation. Resolving narrow diffusive zones demands fine mesh resolution, resulting in significant computational overhead. The present work introduces a computational framework that accelerates phase-field simulations via dynamic adaptive gridding (DAG), concentrating refinement in regions of active fracture propagation. A low-pressure condition is imposed on newly formed cracks to reproduce pressure responses consistent with experimental studies. DAG is implemented on fully unstructured triangular grids, recursively refining elements while preserving child-triangle geometry. Hanging nodes generated during refinement are systematically removed to ensure compatibility with the numerical method. Newly formed cracks are identified via the phase-field variable, and affected cells are assigned low-pressure values in accordance with experimental observations. The mixed hybrid finite element method (MHFEM) is used to solve for pressure, guaranteeing local mass conservation. Phase-field evolution is also addressed with MHFEM, while mechanical deformation is solved using conventional finite element method (FEM). Fluid properties are characterized using equations sourced from National Institute of Standards and Technology databases, and the coupled system of three equations is solved using a sequentially iterative scheme. Speedups are computed across two laboratory-scale domains and one intermediate-scale example, providing insight into future field-scale developments. The DAG approach shows pronounced benefits at larger scales, yielding orders-of-magnitude improvements in computational efficiency. The characteristic pressure history—buildup, rapid decay, and slow propagation—is recovered with the imposed low-pressure condition. Few implementations of DAG exist for phase-field hydraulic fracturing models, and prior efforts are generally restricted to structured meshes and open-source libraries. This work also avoids the negative pressures at the fracture tip reported in other approaches.