ABSTRACT: This paper presents a numerical model for the simultaneous growth of multiple parallel hydraulic fractures with a constant height. The model uses an idealized formulation based on the Elliptic Displacement Discontinuity Method (EDDM). The EDDM assumes each fracture element to have displacement discontinuities of an elliptical shape and solves the one-dimensional elasticity problem. In addition to the EDDM, the model employs the multi-scale tip asymptotic solution that allows a coarser mesh near the fracture tip, compared to the Linear Elastic Fracture Mechanics solution. To show the capabilities of the developed model, the paper presents the comparison between the computed numerical solution and a reference solution. The latter is calculated using a fully 3D hydraulic fracturing simulator for multiple parallel hydraulic fractures. We investigate the effect of perforation friction and spacing on the results. The comparison shows that the EDDM model agrees with the reference solution when spacing between fractures is greater than the fracture height. However, a discrepancy appears in the zero perforation friction case once the fracture spacing becomes comparable or smaller than the fracture height.
Hydraulic fracturing is a method used to crack rock formations using high-pressure fluid. The technology is often applied in oil and gas well stimulation (Economides and Nolte, 2000), waste disposal (Abou-Sayed et al., 1989), rock mining (Jeffrey and Mills, 2000), and geothermal energy extraction (Brown, 2000). Typically, multiple fractures are simultaneously induced to reduce the operational costs. Therefore, the ability to simulate multiple interacting hydraulic fractures can improve the design of hydraulic fracture treatment. Hydraulic fracturing simulators often use various approximations that affect the accuracy and the computational time of the numerical procedures (Adachi et al., 2007, Olson, 2008, Kresse et al., 2013, McClure and Zoback, 2013, Wu et al., 2015, Peirce and Bunger, 2014, Peirce, 2015, Donlsov and Peirce, 2015a). Constant height hydraulic fractures, considered in this paper, resemble classical Perkins-Kern-Nordgren (PKN) fracture geometry (Perkins and Kern, 1961, Nordgren, 1972). The classical PKN model uses a local elasticity assumption and ignore. an essential part of constructing a simulator for multiple growing fractures - the interactions between the fracture elements. This issue is addressed by the enhanced PKN (EPKN) method for a single fracture (Dontsov and Peirce, 2016, Protasov and Donstov, 2017), in which the elastic interactions between cross-sectional elements are based on the elasticity equation for a planar fracture (Adachi and Peirce, 2008). The elliptic fracture opening profile from the classical PKN model is taken as an assumption for the EPKN method making it possible to reduce the planar elasticity equation to a one-dimensional relation. In addition, the EPKN method employs the multiscale tip asymptotic solution (Garagash et al., 2011, Dontsov and Peirce, 2015b) to make it possible to use a relatively coarse mesh near the fracture tip without losing accuracy, compared to the Linear Elastic Fracture Mechanics (LEFM). (As shown in (Dontsov and Peirce, 2016, Protasov and Donstov, 2017), the EPKN method possesses a high computational efficiency, compared to the fully planar simulators, while being able to accurately predict fracture size for a wide set of parameters.