The design of a hydraulic fracturing treatment typically requires using a computational model that provides rapid results. One such possibility is to use the so-called classical pseudo-3D (P3D) model with symmetric stress barriers. Unfortunately, the original P3D model is unable to capture effects associated with fracture toughness in the lateral direction due to the fact that the assumption of plane-strain (or local) elasticity is used. On the other hand, a recently developed enhanced P3D model utilizes full elastic interactions and is capable of incorporating either toughness or viscous regimes of propagation by using the corresponding asymptotic solution at the tip element. Since either the viscous or toughness asymptote is used, the intermediate regime is not described accurately. To deal with this problem, this study aims to implement the intermediate asymptotic solution into the enhanced P3D model. To assess the level of accuracy, the results are compared to a reference solution. The latter reference solution is calculated numerically using a fully planar hydraulic fracturing simulator (Implicit Level Set Algorithm (ILSA)), which also incorporates the asymptotic solution for tip elements that captures the transition from viscous to toughness regime.


Hydraulic fracturing (HF) plays a crucial role in the petroleum industry, as it allows one to perform reservoir stimulation and intensify hydrocarbon production [1]. To design a HF treatment, an appropriate HF model needs to be utilized. The simplest model is the onedimensional Khristianovich-Zheltov-Geertsma-De Klerk (KGD) model [2], in which the fracture propagates in a plane, the elastic interactions are modelled assuming that plane strain conditions prevail, and the coupling between viscous fluid flow and elasticity is included. To represent the fracture geometry more realistically, the Perkins-Kern- Nordgren (PKN) model [3, 4] was developed to predict fracture propagation in a horizontally layered medium. The PKN model assumes that the fracture height is always equal to the thickness of the reservoir layer, the fracture opening in each vertical cross-section is taken to be elliptic, while the fluid pressure is calculated assuming that a plane strain condition holds in each cross-section. Given the fact that the PKN model does not allow for the height growth, the pseudo-3D (P3D) model, which permits height growth, has been developed [5]. Later, with the increase of the computational power, more accurate planar 3D models (PL3D) were developed [7, 8]. As follows from the name, the fracture is contained in one plane, where the fracture geometry within this plane is discretized using a two-dimensional grid. Since the KGD, PKN and P3D are essentially one-dimensional models, while all varieties of PL3D are two-dimensional, the CPU time increases dramatically. The PL3D models improve accuracy and open the possibility of capturing different fracture geometries. Recently, researchers have shifted their effort to investigate the interaction between multiple hydraulic fractures that are growing simultaneously [9], and to describe non-planar fracture propagation [10].

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