This paper reports the study on spontaneous and fast motion for a microliter water droplet on nanotextured glass capillary surfaces with a wide range of curvature gradients. The surface is highly related to the surface tension gradient that is mainly formed by three distinct driving forces, including surface hydrophilicity gradents, chemically patterned nanotexture, and curvature gradient capillaries. In the experiments, the droplet velocity shows a dependency to the droplet position on the conical capillary curvature surface and moves toward the more wettable part of the gradient. The speed of the droplet on the oxygen plasma treated nanotextured glass capillary is up to 238.5 mm/s with more than two times of that, 101.7 mm/s, on the untreated surface. Therefore, we can conclude that a gradual variation of wettability property governs the droplet motion.