We present two types of refractive index sensors by using tunable terahertz (THz) metamaterial (TTM) based on two concentric split-ring resonators (SRRs) with different splits. By modifying the distance between SRRs and substrate, TTM shows tunable single- and dual-resonance characteristic. The maximum tuning range of resonance is 0.432 THz from 0.958 THz to 1.390 THz. To demonstrate a great flexibility of TTM in real application, TTM device is exposed on the surrounding ambient with different refractive index (n). The sensitivity of TTM can be enhanced by increasing SRR height, which is increased from 0.18 THz/RIU to 1.12 THz/RIU under the condition of n = 1.1. These results provide a strategy to improve the sensing performance of the metamaterial-based sensing device by properly arranging the geometric position of meta-atoms. The proposed TTM device can be used for tunable filters, frequency-selective detectors, and tunable high-efficiency sensors in the THz frequency range.