The hemiwicking state has attracted much interest because of numerous important potential applications in inking, printing, boiling heat transfer, and condensation. However, the mechanism of the emergence of hemiwicking has not been well understood, especially the effects of geometry of patterned surfaces on the hemiwicking state has not been systematically investigated. Here, we presented a new method to study the critical conditions for hemiwicking on patterned surfaces. By minimizing the variation of the free energy, we obtain the corresponding stable height of the hemiwicking film and find that it is easier for a droplet to be in the hemiwicking state if the pillar surface has small spacing, large radius and height, and a small intrinsic contact angle. Our established model is applied to a flat-topped cylindrical pillar-patterned surface, and the modeling results are in well agreement with experiments and other existing theories. Besides, our model is also applied to other kinds of patterned surfaces including hemispherical-topped cylindrical and conical pillars, about which the other existing theories are deficient. Our theoretical results not only are in well agreement with the experimental observations but also provide some important predictions, which implies that the established model could be applicable to understanding the basic physical mechanism of the hemiwicking state and be useful in guiding the design and fabrication of hemiwicking surfaces.