Sensitivity analysis of hydraulic fracturing Using an extended finite element method for the PKN model

Abstract

Hydraulic fracturing is a process that is surrounded by uncertainty, as available data on e.g. rock formations is scant and available models are still rudimentary. In this contribution sensitivity analysis is carried out as first step in studying the uncertainties in the model. This is done to assess which of the parameters are key drivers to the model's results. As a baseline model for hydraulic fracturing, the classical PKN model of hydraulic fracturing by Perkins and Kern (J. Pet. Tech. Trans. AIME, 222:937949 (1961)) and Nordgren (J. Pet. Tech. 253:306314 (1972)) which is widely used in the Oil and Gas industry to assist in the design of the hydraulic fracturing treatment is considered. The problem under consideration is characterized by a moving boundary, strong non-linearities and a singularity at the moving tip due to vanishing of the fracture aperture. Sufficient accuracy for such problems is required to obtain a high quality parametric sensitivity analysis and uncertainty quantification. In order to achieve this, various advanced numerical methods are studied. An XFEM approach is adopted to simulate the fracture initiation, propagation and opening of the fracture profile. Using this method, the sensitivity analysis of parameters such as fracturing fluid viscosity, fracture geometry characteristics, soil or rock formations properties via plain strain modulus and leak off coefficient (of the soil) are presented. The impact of these input parameters with respect to the output parameters such as the fracture length and width are discussed.