Magnetorotational instability (MRI) is considered as the most likely mechanism driving angular momentum transport in astrophysical disks. Because of its importance in astrophysics, there has been great interest and efforts to detect MRI in laboratory over the last two decades with varied success. In preparation for upcoming liquid sodium DRESDYN-MRI experiments, recently we have performed global linear analysis of the standard MRI (SMRI) in an infinitely long cylindrical Taylor-Couette (TC) flow with an imposed axial magnetic field, showing that SMRI can in principle be well detected in those ranges of Lundquist (Lu), Reynolds (Re) and magnetic Raynolds (Rm) numbers accessible in these experiments (Mishra, Mamatsashvili, and Stefani (2022)). In this follow-up study also related to DRESDYN-MRI experiments, we focused on the nonlinear evolution and saturation properties of SMRI and analyzed its scaling behavior with respect to various parameters of the basic TC flow. We conducted a detailed analysis over the extensive ranges of Lu ∈ [1.5, 15.5], Rm ∈ [8.5, 37.1] and Re ∈ [103, 105]. For fixed Rm, we investigated the nonlinear dynamics of SMRI for small magnetic Prandtl numbers P m down to P m ∼ 10−4, aiming ultimately for those very small values of P m typical of liquid sodium used in the experiments. In the saturated state, the magnetic energy of SMRI and associated torque exerted on the cylinders, characterising angular momentum transport, both increase with Rm for fixed (Lu, Re), while for fixed (Lu, Rm), the magnetic energy decreases and torque increases with increasing Re. We also studied the scaling of the magnetic energy and torque in the saturated state as a function of Re and found a power-law dependence of the form Re−0.6...−0.5 for the magnetic energy and Re 0.4...0.5 for the torque at all sets of (Lu, Rm) and sufficiently high Re ≥ 4000 (see 1). We also characterized the dependence on Lundquist number and the ratio of outer to inner cylinder angular velocities. Extrapolating the scaling laws to very high, experimental values Re ≳ 106, we estimated the rms of velocity and magnetic field perturbations, 0.5−0.94 ms−1 and 0.45 − 0.97 mT , respectively, which can be expected in the experiments. These scaling laws will be instrumental in the subsequent analysis for a finite length TC flow with endcaps and comparison of numerical results with those obtained from the DRESDYN-MRI experiments in order to conclusively and unambiguously identify SMRI in Lab.