The paper examines the problem of the open configuration created when a hydraulic fracture fluid containing a granular proppant is introduced into the fracture. The mathematical modelling examines the problem of an extended cracked region that is wedged open by a granular material present over a finite region of the crack. The combination of the geostatic stress state and the contact stress created between the granular proppant and the elastic rock mass is used to develop a consistency relationship for estimating the dimension of the region of the fracture that will remain open when the pressures applied to create the fracture are released. The interactive mechanics of the fracture and the proppant has an influence on the geometry of the open region that provides the pathway for extraction of the resource.

Revised: 2019-04-01

Accepted: 2019-07-15

Published online: 2019-07-22

Keywords: Hydraulic fracture, proppant-fracture interaction, Griffith fracture models, zones of separation

@article{OGEO_2019__1__A2_0, author = {Selvadurai, A.P.S.}, title = {Proppant-Induced Opening of Hydraulically Created Fractures}, journal = {Open Geomechanics}, publisher = {Alert Geomaterials}, volume = {1}, year = {2019}, doi = {10.5802/ogeo.1}, language = {en}, url={opengeomechanics.centre-mersenne.org/item/OGEO_2019__1__A2_0/} }

Selvadurai, A.P.S. Proppant-Induced Opening of Hydraulically Created Fractures. Open Geomechanics, Volume 1 (2019) , article no. 2, 10 p. doi : 10.5802/ogeo.1. https://opengeomechanics.centre-mersenne.org/item/OGEO_2019__1__A2_0/

[1] Plane-strain propagation of a hydraulic fracture in a permeable rock, Engineering Fracture Mechanics, Volume 75 (2008), pp. 4666-4694 | Article

[2] On certain problems of the theory of elasticity that arise in the investigation of the mechanism of hydraulic rupture of an oil-bearing layer, Prikladnaia Matematika Mehanika Akademii Nauk, Volume 20 (1956), pp. 475-486 | MR 81696

[3] An approximate evaluation of the size of a crack forming in hydraulic fracture of a stratum, Izvestia Akademii Nauk, S.S.S.R., OTN, Volume 3 (1957), pp. 1980-1982

[4] Equilibrium cracks forming during brittle fracture, Dokladi Akademii Nauk, S.S.S.R., Volume 127 (1959), pp. 1980-1982

[5] The mathematical theory of equilibrium cracks in brittle fracture, Advances in applied mechanics, Volume 7, Elsevier, 1962, pp. 55-129 | Article

[6] Rate and structure sensitivity in hydraulic fracturing of fluid-saturated porous formations, Proceedings of the 20th US Symposium on Rock Mechanics (USRMS) (1979)

[7] Comprehensive design formulae for hydraulic fracturing, SPE Annual Technical Conference and Exhibition (1980) | Article

[8] On the computation of the three-dimensional geometry of hydraulic fractures, Symposium on Low Permeability Gas Reservoirs (1979) | Article

[9] Hydraulic fracturing and water resources: separating the frack from the fiction, Pacific Institute, Oakland (2012)

[10] The mechanics and physics of fracturing: applications to thermal aspects of crack propagation and to fracking, Philosophical Transactions of the Royal Society, Volume 373 (2015) | Article

[11] Limit Analysis and Soil Plasticity, Developments in Geotechnical Engineering Volume 7, Elsevier Scientific Publishing Co., Amsterdam, 1975

[12] On the design of vertical hydraulic fractures, Journal of Petroleum Technology, Transactions AIME, Volume 25 (1973), pp. 83-97 | Article

[13] Hydraulic fracture propagation in layered formations, 51st Annual Fall SPE Technical Conference and Exhibition, New Orleans (1976)

[14] The crack tip region in hydraulic fracturing, Proceedings of the Royal Society A, Volume 447 (1994), pp. 39-48 | Article | Zbl 0813.73051

[15] Plasticity and Geomechanics, Cambridge University Press, Cambridge, 2003

[16] A rapid method of predicting width and extent of hydraulically induced fractures, Journal of Petroleum Technology, Volume 21 (1969), pp. 1571-1581 | Article

[17] Beit zum Vollstandigen ebenen Olastizitats problem, Proceedings of the 3rd International Congress of Applied Mechanics, Stockholm, Volume 2 (1930), pp. 185-190

[18] Comparison of the theories for predicting width and extent of vertical hydraulically induced fractures, Transactions ASME, Volume 101 (1979), pp. 8-19

[19] The contact problem for a rigid obstacle pressed between two dissimilar elastic half-spaces, Quarterly Journal of Mechanics and Applied Mathematics, Volume 34 (1981), pp. 25-263 | Zbl 0462.73093

[20] On contact problems for a medium with rigid flat inclusions of arbitrary shape, International Journal of Solids and Structures, Volume 32 (1995), pp. 383-389 | Article | Zbl 0865.73052

[21] Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics, Volume 1461, Springer-Verlag, Berlin, 1991 | Zbl 0717.45002

[22] Über die Integration der Differentialgleichungen des ebenen Gleichgewichtszustandes für allgemeinen-plastischen Körper, 1925 (Thesis, Göttingen)

[23] Hydraulic Fracturing, Monograph Series, Society of Petroleum Engineers of AIME, Richardson, TX. 2, 34, 1970

[24] The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950 | Zbl 0041.10802

[25] Fracking: The Operations and Environmental Consequences of Hydraulic Fracturing, John Wiley and Sons, New York, 2013

[26] Effect of residual opening on the inflow performance of a hydraulic fracture, International Journal of Engineering Science, Volume 74 (2014), pp. 80-90 | Article | Zbl 06976546

[27] On a rigid inclusion pressed between two elastic halfspaces, Mechanics of Materials, Volume 68 (2014), pp. 38-44 | Article

[28] Formation of vertical fractures by means of highly viscous liquid, Proceedings of the 4th World Petroleum Congress, Rome, Sec. II (1955), pp. 579-586

[29] Elasticity, Prentice-Hall, Upper Saddle River, New Jersey, 1975

[30] A comprehensive review on proppant technologie, Petroleum, Volume 2 (2016), pp. 26-39 | Article

[31] Dimensions of the Fractures Obtained by Hydraulic Fracturing of Oil Bearing Formations, Revue de I’lnstitut Francois du Petrole (1967), pp. 44-98

[32] Hydraulic Proppant Fracturing and Gravel Packing, Elsevier, Amsterdam, 1989

[33] Sur le lignes de glissement et le calcul des déplacements dans la déformation plastique, Comptes Rendus, Volume 225 (1947), p. 1272-1273 | Zbl 0029.32903

[34] The compression of a slab of ideal soil between rough plates, Acta Mechanica, Volume 3 (1967), pp. 82-92 | Article

[35] A review of hydraulic fracture modeling, Part II: 3D modeling and vertical growth in layered rock, ASME Journal of Energy Resources Technology, Volume 106 (1984), pp. 543-553 | Article

[36] A review of hydraulic fracturing modeling, Part I: General concepts, 2D Models, Motivation for 3D Modeling, ASME Journal of Energy Resources Technology, Volume 106 (1984), pp. 369-376 | Article

[37] Unconventional Oil and Gas Resources Handbook: Evaluation and Development, Elsevier, Amsterdam, 2016

[38] Residual opening of hydraulic fractures filled with compressible proppant, International Journal of Rock Mechanics and Mining Sciences, Volume 61 (2013), pp. 223-230 | Article

[39] Hydraulic Fracturing and Geothermal Energy, Proceedings 1st Japan-USA Joint Seminar on Hydraulic Fracturing and Geothermal Energy, Martinus-Nijhoff Publ (1983) | Article

[40] Propagation of a vertical hydraulic fracture, Society of Petroleum Engineers Journal, Volume 12 (1972) no. 04, pp. 306-314 | Article

[41] Polymers for proppants used in hydraulic fracturing, Journal of Petroleum Science and Engineering, Volume 145 (2016), pp. 154-160 | Article

[42] Widths of hydraulic fractures, Journal of Petroleum Technology, Volume 13 (1961) no. 09, pp. 937-949 | Article

[43] Anwendungsbeispiele zu einem Henckyschen Satz über das plastiche Gleichgewicht, Zeitschrift für Angewandte Mathematik und Mechanik, Volume 3 (1923), pp. 401-406 | Article | Zbl 49.0596.02

[44] Zweidimensional problem der elastizitätstheorie, Zeitschrift für Angewandte Mathematik und Mechanik, Volume 8 (1928), pp. 107-121 | Article

[45] Fracture evolution during indentation of a brittle elastic solid, Mechanics of Cohesive-Frictional Materials, Volume 5 (2000), pp. 325-339 | Article

[46] Partial Differential Equations in Mechanics Vol. 2, Springer-Verlag, Berlin, 2000 | Article | Zbl 0967.35002

[47] On an invariance principle for unilateral contact at a bi-material elastic interface, International Journal of Engineering Science, Volume 41 (2003), pp. 721-739 | Article | MR 1959202 | Zbl 1211.74192

[48] Stationary damage modelling of poroelastic contact, International Journal of Solids and Structures, Volume 41 (2004), pp. 2043-2064 | Article | Zbl 1106.74404

[49] On an integral equation governing an internally indented penny shaped crack, Mechanics Research Communications, Volume 12 (1985), pp. 347-351 | Article

[50] Separation at a pre-fractured bi-material geological interface, Mechanics Research Communications, Volume 21 (1994), pp. 83-88 | Article | Zbl 0795.73064

[51] Some moving boundary problems associated with poroelastic media, Moving Boundaries IV, Computational Modelling of Free and Moving Boundary Problems, Proceedings of the International Symposium on Moving Boundary Problems, Ghent, Belgium, A.A. Balkema, The Netherlands, 1997, pp. 251-261

[52] Computational modelling of steady crack extension in poroelastic media, International Journal of Solids and Structures, Volume 35 (1998), pp. 4869-4885 | Article | Zbl 0967.74583

[53] Hydraulic Fracturing, CRC Press, Boca Raton, FL, 2015

[54] Compression and shear of a layer of granular material, Journal of Engineering Mathematics, Volume 52 (2005), pp. 251-264 | Article | MR 2159405 | Zbl 1079.74018

[55] Handbook of Hydraulic Fracturing, John Wiley & Sons, Inc., New York, 2016 | Article

[56] A theory of the kinematics of ideal soils under plane strain conditions, Journal of the Mechanics and Physics of Solids, Volume 13 (1964), pp. 337-351 | Article | MR 171447 | Zbl 0125.15208

[57] Mandel-Cryer effects in fluid inclusions in damage susceptible poroelastic media, Computers and Geotechnics, Volume 37 (2004), pp. 285-300 | Article

[58] On the expansion of a penny shaped crack by a rigid circular disc inclusion, International Journal of Fracture, Volume 25 (1984), pp. 69-77 | Article

[59] Some annular inclusion problems in elasticity, International Journal of Solids and Structures, Volume 20 (1984), pp. 129-139 | Article | Zbl 0526.73018

[60] The annular crack problem for an isotropic elastic solid, Quarterly Journal of Mechanics and Applied Mathematics, Volume 38 (1985), pp. 223-243 | Zbl 0559.73090

[61] Contact mechanics of a dilatant region located at a compressed interface, International Journal of Engineering Science (2018), (in press) pages | Article | MR 3869729 | Zbl 06986255

[62] Hydraulic Fracture Mechanics, John Wiley, New York, 1996

[63] Fluid loss from hydraulically induced fractures, Journal of Petroleum Technology, Volume 249 (1970), pp. 882-888 | Article

[64] Hydraulic fracture of an oil-bearing bed, Izvestia Akademii Nau, Volume 5 (1955), pp. 3-41