A Semi-Analytical Solution to Three-Dimensional Hydraulic Fracture Propagation in Multistress Media

Abstract

abstract Hydraulic fracturing plays a very important role in the enhancement of gas and oil production rates from low permeability reservoirs. To mathematically model a hydraulically induced fracture, one must couple equations describing formation elasticity, fracture fluid flow, and fracture mechanics. Also, the effects of fluid leakoff into the poro-elastic formation must be considered in association with the fracture fluid mass conservation equation. Fracture propagation models in multilayer systems are needed to better understand hydraulic fracturing and to optimize a fracture treatment design. Currently, the determination of the optimum fracture treatment in formations of three or more layers are being carried out using pseudo three-dimensional or full three-dimensional fracture propagation models. Pseudo three-dimensional models usually run rapidly on personal computers. However, pseudo three-dimensional models are mainly applicable to fractures where the total fracture length is several times greater than the fracture height. If fracture height growth is a problem, full 3-D models are required to solve the fracture propagation problem rigorously. However, these sophisticated 3-D models require considerable computer memory, run time, and expertise by the user. Saving CPU time is very important in the optimization of the fracture design process because numerous fracture predictions and sensitivity analyses are needed to determine the optimum fracture design. The objectives of this paper are (1) to develop an efficient three-dimensional, semi-analytical fracture propagation model that can be used to predict the created fracture dimensions and pressure inside the fracture for a given injected volume in a multilayer reservoir, and achieve results more compatible with a full 3-D model, and (2) to develop an accurate 3-D model that is easy to run and can execute rapidly on a personal computer. In this research, We have developed a full 3-D fracture propagation model that allows us to compute efficiently the fluid pressures and fracture dimensions in a multilayer medium with asymmetric characteristics in each layer of rock. This model considers two-dimensional fluid flow inside the fracture and computes pressure drop in both directions of the fracture height and fracture length. We have used cubic polynomial approximations to represent both the fluid pressure and the in-situ stress profiles. Thus, we could use the analytical solution to determine displacements of an elliptical crack under arbitrary normal loading presented by Shah and Kobayashi1 (SK). We use the first variational technique to approximate certain partial differential equations. These equations are coupled with the analytical solution presented by SK1. As such, asymmetric fracture growth can be solved for a multilayer system, where each layer can have unique mechanical properties such Poisson's ratio, in-situ stress, Young's modulus, and fracture toughness.