Spontaneous imbibition of a liquid into glass-fiber wicks is modeled using the single-phase Darcy's law after assuming a sharp flow-front marked by full saturation behind the front occurring in a transversely isotropic porous medium. An analytical expression for the height of the wicking flow-front as a function of time is tested through comprehensive experiments involving using eight different wicks and one oil as the wicking liquid. A good fit with experimental data is obtained without using any fitting parameter. The contact-angle is observed to be important for the success of the modellower contact angle cases marked by higher capillary pressures were predicted the best. The proposed model provides a nice upper bound for all the wicks, thereby establishing its potential as a good tool to predict liquid absorption in glass-fiber wicks. However, the sharp-front model is unable to explain region of partial saturation, thereby necessitating the development of part II of this article series (Zarandi and Pillai, Spontaneous Imbibition of Liquid in Glass fiber wicks. Part II: Validation of a Diffuse-Front Model. AIChE J, 64: 306-315, 2018) using Richard's equation. (c) 2017 American Institute of Chemical Engineers AIChE J, 63: 294-305, 2018