The transient behavior of a vertically fractured pressure response due to the presence of an infinite-conductivity vertical fracture is determined by solving the diffusivity equation in elliptical coordinates. The solution is then extended to a composite elliptical system to provide for the different fluid banks present during water injection. The validity of the analytical solutions presented is demonstrated by comparing limiting forms with those available elsewhere in the literature. Computational issues which became evident during the verification stage of our work are also discussed. The solutions have been developed in the Laplace domain to facilitate the addition of fissures and variable rate production (i.e. wellbore storage). A pressure transient test for tracking the advancement of a water front during the early stages of waterflooding is described. We utilize the composite elliptical model developed herein to provide for two distinct regions in which the flow behavior resulting from an induced fracture is elliptical rather than radial. A relationship between the increasing elliptical distance to the waterfront and the resulting change in the apparent (total) skin factor is obtained. Through the analysis of successive falloff tests, this relationship may be used to monitor the advancement of the front provided the cumulative volume of injected water is known. The fluid saturations and the mobilities of the swept and unswept regions are assumed unknown and are obtained from the test analysis. Finally, we present methods for computing the Mathieu functions necessary in solving the diffusivity equation in elliptical coordinates. Mathieu functions are utilized in many applications involving elliptical geometry and we feel the efficient evaluation of these functions is an important contribution of this work.