A depth-averaged two-dimensional mathematical model of granular-fluid avalanches is proposed based on the μ(J) rheology. In the model, a depth-averaged deviatoric viscous stress of granular-fluid mixture is formulated and the corresponding depth-integration term is linearly proportional to the flow depth, which is different from the 1.5th-power law for dry granular avalanches. A basal resistance on the granular-fluid mixture, composed of a Coulomb friction term and a viscous shear term, is adopted in the model. Based on the model, a critical Froude number of 1/2 is obtained for the linear instability of steady-uniform granular-fluid mixture flows. The model is capable of capturing the cutoff frequency for the spatial growth rate of perturbation in the mixture flows. The proposed model is applied to simulate the evolution of imposed perturbations in steady granularfluid avalanches on an inclined chute. The intensification of imposed perturbations and the generation of final stable roll waves are captured. A parameter sensitivity analysis is conducted to investigate the effects of the depth-averaged in-plane deviatoric viscous stress and the basal resistance on the final stable roll waves. Further, a preliminary analytical solution for the crest/trough heights of the stable roll waves is proposed to verify the numerical findings.