Fracture growth in layered formations with depth-dependent properties has been a topic of interest amongst researchers because of its critical influence on well performance. This paper revisits some of the existing height-growth models and discusses the evaluation process of a new and modified model developed after incorporating additional constraints.The net-pressure is the primary driver behind fracture propagation and the pressure distribution in the fracture plays an important role in vertical propagation, as it supplies the necessary energy for fracture advancement in the presence of opposing forces. The workflow adopted for this study included developing a preliminary model that solves a system of non-linear equations iteratively to arrive at fracture height versus net pressure mapping. The theoretical results were then compared to those available in the literature. The solution set was then extended to a 100-layer model after incorporating additional constraints using superposition techniques.The predicted outcomes were finally compared to the fracture height observations made in the field on several treatments.
A reasonable agreement between model-predicted and observed height was observed when a comparison between the two was made, for most cases.The majority of these treatments were pumped in vertical wells, at low injection rates of up to 8.0 bbl/min (0.021 m3/s) where net pressures were intentionally restricted to 250 psi (1.72 MPa) in order to prevent fracture rotation to the horizontal plane.The leak-off was minimal given the low permeability formations. In some cases, however, the pumping parameters and fluid imparted pressure distribution appeared to dominate. Overall, it was apparent that for a slowly advancing fracture front, which is the case in low injection rate treatments, the fracture height could be predicted with reasonable accuracy. This condition could also be met in high rate treatments pumped down multiple perforation clusters such as in horizontal wells, though fracture-height measurement may not be as straightforward as in vertical wells.
The model developed under the current study is suitable for vertical wells where fracture treatments are pumped at low injection rates. The solid-mechanics solution that is presented here is independent of pumping parameters and can be readily implemented to assist in selection of critical design parameters prior to the job, with a wide range of applicability worldwide.