Accurate modeling of dense granular flows is critical for achieving two-phase sediment transport simulations. This study proposes an improved μ(I)-rheology model enhanced by a novel L1-regularization technique and integrates it into the Material Point Method (MPM) framework. We first identify numerical instabilities inherent to the standard μ(I)-rheology under vanishing shear rates and derive a regularization strategy to resolve ill-conditioned viscosity divergence. The reformulated model is then coupled with MPM to simulate dense granular flows problem with the large deformation characteristic, leveraging its coupled Eulerian-Lagrangian descriptions to avoid the mesh distortion. Numerical benchmarks demonstrate that the L1-regularized μ(I)-rheology achieves superior numerical conditioning while retaining fidelity to the original rheological behavior. Comparison with experimental data validate the framework’s robustness in capturing key flow dynamics.