One of the most conspicuous geological features of the Niger Delta is its growth fault pattern. Growth faults are a particular type of normal fault that develops during ongoing sedimentation, so the strata on the hanging wall side of the fault tend to be thicker than those on the footwall side. In this study, a detailed treatment is given of the transient pressure behaviour of a well located near a leaky boundary with unequal formation thickness on opposite sides of the barrier. These types of barriers are typical features of growth fault. The results of this study generalize previously published works which assumed the thickness index for media on different sides of the fault was zero, one, or infinity. These assumptions are not always valid for growth fault reservoirs commonly found in the Niger Delta. One application of this work would be to facilitate detection of formation thickness on the other side of the fault without drilling a well in that location, this means reduced capital development cost thereby maximizing NPV. The range of variables examined includes thickness index from zero to one and from one to infinity, also, the effect of transmissibility factor ranging from zero to infinity was included. Based on this study, it was observed that for high values of the thickness index, the response is similar to that of a constant pressure boundary, while the behaviour resembles the sealing boundary characteristics for low values of the index.