A novel computational framework that synergistically integrates capillary force modeling with the phase-field lattice Boltzmann method (LBM) is developed. The proposed approach simplifies capillary force computation at solid boundaries through three fundamental parameters: order parameter magnitude, its spatial gradient, and solid surface normal orientation. Through rigorous theoretical derivations and systematic numerical benchmarking tests, this study demonstrates both the methodological robustness and enhanced accuracy of the proposed formulation. Theoretical analysis establishes well-defined error bounds for the force calculation scheme, while computational validation employing two benchmark cases - a liquid bridge between two static solid particles and a particle-adhering bubble system - reveals significant improvements in accuracy over conventional methodologies in three-dimensional simulations. Notably, our model achieves superior three-dimensional precision compared to existing approaches, with the particle-bubble adhesion case study demonstrating that LBM-discrete element method (DEM) coupling, when applying the accurate capillary force modeling, forms an effective computational platform for investigating three-phase gas-liquid-solid interactions. These advancements offer enhanced research capacity for complex interfacial phenomena.
 

Phase Field (PF) method,DEM-LBM; Capillary force evolution; Grain Size Gradation; Liquid bridge morphology; unsaturated porous medium