@article{scholars12451,
           title = {A review of PKN-type modeling of hydraulic fractures},
       publisher = {Elsevier B.V.},
         journal = {Journal of Petroleum Science and Engineering},
             doi = {10.1016/j.petrol.2020.107607},
            year = {2020},
          volume = {195},
            note = {cited By 15},
          author = {Nguyen, H. T. and Lee, J. H. and Elraies, K. A.},
            issn = {09204105},
             url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85088117447&doi=10.1016\%2fj.petrol.2020.107607&partnerID=40&md5=e9d6d80f9a61a3bce4f0f18368eeac22},
        keywords = {Fracture; Hydraulic fracturing; Numerical models; Velocity control, Chronological order; Dependent variables; Modeling and computation; Modeling technique; Particle velocities; Robust algorithm; The standard model; Three-dimensional model, Three dimensional computer graphics, computer simulation; flow field; future prospect; hydraulic fracturing; model test; nonlinearity; numerical model; petroleum engineering; three-dimensional modeling},
        abstract = {Many numerical modeling techniques have been employed to understand the phenomena occurring during hydraulic fracturing. Despite the various complex models developed, such as the planar three-dimensional and pseudo three-dimensional models, the one-dimensional Perkins{\^a}??Kern{\^a}??Nordgren (PKN) model is still of great interest because it serves as a starting point for the development of more efficient and robust algorithms. This work reviews all PKN-type models from the first published model in 1961 to recent advancements in the model design. The underlying assumptions of the fundamental physical processes of the PKN model are first analyzed. Developments of and notable contributions to PKN models are then summarized into four main groups and discussed in a relatively chronological order. The differences between the standard modeling approach that employs the fluid flux as a dependent variable and the other that utilizes the particle velocity are highlighted. The trend of accounting for the poroelastic effect in the leak-off process and the effect of turbulence in the flow regime is also discussed. A number of challenges in modeling and computations caused by the strong nonlinear and degenerate governing system and how they have been tackled are then discussed in detail. Finally, some open suggestions for future work are mentioned. {\^A}{\copyright} 2020 Elsevier B.V.}
}