Toward a unified continuum framework for dense granular flows: $\boldsymbol{\mu} \boldsymbol{(I)}$ rheology extended with granular temperature

Dense granular flows exhibit pronounced non-local behaviours, particularly in creeping regions and shear-localised zones, which challenges classical local inertial rheologies. In this work, we develop a continuum framework for dense granular flows by extending the $\mu (I)$ rheology through the inclusion of granular temperature as an explicit state variable, thereby establishing a direct link between grain-scale velocity fluctuations and macroscopic stresses, and enabling the representation of non-local effects. The model is implemented within a finite-volume computational framework, and systematically validated against three canonical configurations spanning steady and transient regimes: heap flows, split-bottom Couette flows, and granular column collapse. Across these benchmarks, the formulation captures key non-local features observed experimentally and numerically, including sustained creeping below yield, shear-band broadening and migration, and the transient evolution of free surfaces and runout dynamics. Overall, the granular-temperature-extended $\mu (I)$ rheology provides a unified continuum description that reconciles local and non-local behaviour in dense granular flows, retains the predictive capability of inertial rheology in rapid regimes, and extends its applicability to creeping and shear-localised flows. The proposed framework offers a physically interpretable and scalable basis for modelling granular processes in both geophysical and industrial contexts.