Drag-reducing nanofluids are complex non-Newtonian fluids. Their constitutive characteristics are the basis of flow mechanism analysis in porous media. However, the rheological effects of drag-reducing nanofluids have not been thoroughly studied. In the present work, rheological properties of several nanofluids were measured, and the shear thickening mechanism was investigated experimentally. The results show that all the nanofluids examined have complex characteristics and critical shear rates. The viscosity exhibits a slow linear increase with the shear rate below the critical shear rate, while the shear thickening power-law fluid behaviour appears above the critical shear rate. The critical shear rate increases with the increase of particle concentration, which indicates the injection rate needs to be controlled to avoid significant increase of nanofluids viscosity. The rheological curve of increasing shear rate nearly coincides with that of decreasing shear rate, which indicates that the shear thickening of nanofluids studied in this work is transient and reversible. A constant index constitutive equation with an exponent of 0.5 is obtained from test results by the fixed index method, and its coefficient $k(c)$ is a linear function of the concentration, which can replace a set of conventional constitutive equations with different concentrations. The constant index constitutive equation also clarifies the coefficient dimension. Similar results have been obtained by analysing several other nanofluids using the fixed index method, which validates the new effective method for constructing the constitutive equations of non-Newtonian nanofluids.