@article{LeeMikWheWick2016_Prop,
author = "Sanghyun Lee and Andro Mikeli{\'c}  and Mary F. Wheeler and Thomas Wick",
title = "Phase-field modeling of proppant-filled fractures in a poroelastic medium ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "312",
number = "",
pages = "509 - 541",
year = "2016",
note = "Phase Field Approaches to Fracture ",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2016.02.008",
url = "http://www.sciencedirect.com/science/article/pii/S0045782516300305",
keywords = {Phase-field fracture, Hydraulic fracturing, Proppant transport,Locally conservative,
            Quasi-Newtonian flow, Porous media},
abstract={In this paper we present a phase field model for proppant-filled fractures in a poroelastic medium. The formulation of the coupled system involves four unknowns: displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is then coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton’s method. Our developments are substantiated with several numerical examples in two and three dimensions.}
}

{"_id":"ejgf2aD8acYH56bdk","bibbaseid":"lee-mikeli-wheeler-wick-phasefieldmodelingofproppantfilledfracturesinaporoelasticmedium-2016","author_short":["Lee, S.","Mikelić, A.","Wheeler, M. F.","Wick, T."],"bibdata":{"bibtype":"article","type":"article","author":[{"firstnames":["Sanghyun"],"propositions":[],"lastnames":["Lee"],"suffixes":[]},{"firstnames":["Andro"],"propositions":[],"lastnames":["Mikelić"],"suffixes":[]},{"firstnames":["Mary","F."],"propositions":[],"lastnames":["Wheeler"],"suffixes":[]},{"firstnames":["Thomas"],"propositions":[],"lastnames":["Wick"],"suffixes":[]}],"title":"Phase-field modeling of proppant-filled fractures in a poroelastic medium ","journal":"Computer Methods in Applied Mechanics and Engineering ","volume":"312","number":"","pages":"509 - 541","year":"2016","note":"Phase Field Approaches to Fracture ","issn":"0045-7825","doi":"http://dx.doi.org/10.1016/j.cma.2016.02.008","url":"http://www.sciencedirect.com/science/article/pii/S0045782516300305","keywords":"Phase-field fracture, Hydraulic fracturing, Proppant transport,Locally conservative, Quasi-Newtonian flow, Porous media","abstract":"In this paper we present a phase field model for proppant-filled fractures in a poroelastic medium. The formulation of the coupled system involves four unknowns: displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is then coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton’s method. Our developments are substantiated with several numerical examples in two and three dimensions.","bibtex":"@article{LeeMikWheWick2016_Prop,\nauthor = \"Sanghyun Lee and Andro Mikeli{\\'c} and Mary F. Wheeler and Thomas Wick\",\ntitle = \"Phase-field modeling of proppant-filled fractures in a poroelastic medium \",\njournal = \"Computer Methods in Applied Mechanics and Engineering \",\nvolume = \"312\",\nnumber = \"\",\npages = \"509 - 541\",\nyear = \"2016\",\nnote = \"Phase Field Approaches to Fracture \",\nissn = \"0045-7825\",\ndoi = \"http://dx.doi.org/10.1016/j.cma.2016.02.008\",\nurl = \"http://www.sciencedirect.com/science/article/pii/S0045782516300305\",\nkeywords = {Phase-field fracture, Hydraulic fracturing, Proppant transport,Locally conservative,\n Quasi-Newtonian flow, Porous media},\nabstract={In this paper we present a phase field model for proppant-filled fractures in a poroelastic medium. The formulation of the coupled system involves four unknowns: displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is then coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton’s method. Our developments are substantiated with several numerical examples in two and three dimensions.}\n}\n\n\n\n\n","author_short":["Lee, S.","Mikelić, A.","Wheeler, M. 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