A PKN Hydraulic Fracture Model Study and Formation Permeability Determination

Hydraulic fracturing is an important method used to enhance the recovery of oil and gas from reservoirs, especially for low permeability formations. The distribution of pressure in fractures and fracture geometry are needed to design conventional and unconventional hydraulic fracturing operations, fracturing during water-flooding of petroleum reservoirs, shale gas, and injection/extraction operation in a geothermal reservoir. Designing a hydraulic fracturing job requires an understanding of fracture growth as a function of treatment parameters.

There are various models used to approximately define the development of fracture geometry, which can be broadly classified into 2D and 3D categories. 2D models include, the Perkins-Kern-Nordgren (PKN) fracture model, and the Khristianovic-Geertsma-de. Klerk (KGD) fracture model, and the radial model. 3D models include fully 3D models and pseudo-three-dimensional (P-3D) models. The P-3D model is used in the oil industry due to its simplification of height growth at the wellbore and along the fracture length in multi-layered formations.

In this research, the Perkins-Kern-Nordgren (PKN) fracture model is adopted to simulate hydraulic fracture propagation and recession, and the pressure changing history. Two different approaches to fluid leak-off are considered, which are the classical Carter's leak-off theory with a constant leak-off coefficient, and Pressure-dependent leak-off theory. Existence of poroelastic effect in the reservoir is also considered.

By examining the impact of leak-off models and poroelastic effects on fracture geometry, the influence of fracturing fluid and rock properties, and the leak-off rate on the fracture geometry and fracturing pressure are described. A short and wide fracture will be created when we use the high viscosity fracturing fluid or the formation has low shear modulus. While, the fracture length, width, fracturing pressure, and the fracture closure time increase as the fluid leak-off coefficient is decreased.

In addition, an algorithm is developed for the post-fracture pressure-transient analysis to calculate formation permeability. The impulse fracture pressure transient model is applied to calculate the formation permeability both for the radial flow and linear fracture flow assumption. Results show a good agreement between this study and published work.