Improved pseudo three-dimensional model for hydraulic fractures under stress contrast

The conventional pseudo three-dimensional (P3D) model is improved to consistently account for arbitrary stress contrast when simulating hydraulic fractures in a rock stratum. The improvement is based on proper combining the model by Khristianovich-Geertsma-de Klerk (KGD), generalized to a medium with stress contrast, and the original P3D model. We suggest the correspondence principle, which establishes a correspondence between the two models in terms of physical quantities present in each of them. These are: (i) the positions of fracture tips, and (ii) the fluid volumes above and below the injection point. When these quantities are same for the both models, it may be expected that the speeds of height growth, not present in the original P3D model, may be equated to those of the KGD model, which includes these speeds in its formulation. This approach is applicable in all the cases when the solution to a KGD problem uniquely defines the speeds of fracture tips via their positions and fluid volumes below (above) a tip. The comparison of fracture footprints, obtained by the method suggested, with published truly 3D solutions for cases with strong stress contrast, has shown good agreement. This confirms that the P3D model, complemented with the correspondence principle, provides acceptable results for the height growth. A simple approach to meet the correspondence principle is suggested and successfully tested. Examples demonstrate abilities of the improved model for simplified modeling of HF in a layered stratum with stress contrast. The model developed serves also to establish conditions, under which the P3D modeling is relevant.