Summary
A high minimum in-situ stress contrast between pay and bounding zones can be very effective in restricting vertical fracture- height growth. For a highly elongated fracture, the fracturing-fluid flow can be approximated by one-dimensional (1D) flow along the pay-zone direction. If the stress contrasts are smaller (i.e., weak barrier), however, the height growth is greater and two-dimensional (2D) fluid flow needs to be incorporated into the modeling. This major difficulty has been solved only recently by a fully numerical solution. The goal of this research was to develop a fracture model for length/height ratio ≤ 4 that includes 2D flow (and a line source corresponding to the perforated interval) but makes approximations that allow a semianalytical solution, with large computer-time savings over the fully numerical model. The height, maximum width, and pressure at the wellbore in this semianalytical model are calculated and compared with the results of the fully three-dimensional (3D) model. There is reasonable agreement in all parameters, the maximum discrepancy being 24%. Comparisons of fracture volume and leakoff volume also show reasonable agreement in volumes and fluid efficiencies. The values of length/height ratio, in the four cases in which agreement is found, vary from 1.5 to 3.7. The model offers a useful first-order (or screening) calculation of fracture-height growth through weak barriers (e.g., low stress contrasts). When coupled with the model developed for highly elongated fractures of length/height ratio ≥ 4, which are also found to be in basic agreement with the fully numerical model, this new model provides the capability for approximating fracture-height growth through barriers for vertical fracture shapes that vary from penny to highly elongated. The computer time required is estimated to be less than the time required for the fully numerical model by a factor of 10 or more.
