Sampling-based stochastic analysis of the PKN model for hydraulic fracturing
Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordg...
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| Vydané v: | Computational geosciences Ročník 23; číslo 1; s. 81 - 105 |
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| Hlavní autori: | , , , , |
| Médium: | Journal Article Publikácia |
| Jazyk: | English |
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Springer International Publishing
01.02.2019
Springer Nature B.V |
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| ISSN: | 1420-0597, 1573-1499 |
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| Abstract | Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture. |
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| AbstractList | Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture. The final publication is available at Springer via http://dx.doi.org/10.1007/s10596-018-9784-y Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture. Peer Reviewed |
| Author | Zlotnik, Sergio van Brummelen, E. Harald Díez, Pedro Verhoosel, Clemens V. Garikapati, Hasini |
| Author_xml | – sequence: 1 givenname: Hasini surname: Garikapati fullname: Garikapati, Hasini email: h.garikapati@tue.nl organization: Eindhoven University of Technology, Universitat Politècnica de Catalunya – sequence: 2 givenname: Clemens V. surname: Verhoosel fullname: Verhoosel, Clemens V. organization: Eindhoven University of Technology – sequence: 3 givenname: E. Harald surname: van Brummelen fullname: van Brummelen, E. Harald organization: Eindhoven University of Technology – sequence: 4 givenname: Sergio surname: Zlotnik fullname: Zlotnik, Sergio organization: Universitat Politècnica de Catalunya – sequence: 5 givenname: Pedro surname: Díez fullname: Díez, Pedro organization: Universitat Politècnica de Catalunya |
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| CitedBy_id | crossref_primary_10_1016_j_compgeo_2023_105709 crossref_primary_10_3390_su162411087 crossref_primary_10_1016_j_engfracmech_2020_107475 crossref_primary_10_1016_j_jmps_2022_105082 crossref_primary_10_3389_feart_2022_906923 crossref_primary_10_3390_en15165811 crossref_primary_10_1016_j_jngse_2021_104158 |
| Cites_doi | 10.1007/s10659-007-9122-4 10.1115/1.321162 10.1108/02644409510799532 10.1016/j.ijrmms.2006.11.006 10.2118/2458-PA 10.1017/S0305004100004175 10.1016/j.petrol.2017.12.048 10.1007/s10596-015-9532-5 10.1016/j.petrol.2011.04.001 10.1002/nag.213 10.1016/j.cma.2015.09.021 10.2118/6089-PA 10.1017/S0013091500011457 10.1016/j.jmps.2016.07.019 10.1007/s10596-017-9702-8 10.1007/s11242-009-9403-4 10.1088/0305-4470/22/9/026 10.2118/10505-PA 10.1098/rspa.1946.0077 10.1016/j.petrol.2010.03.032 10.1016/j.cma.2013.12.005 10.1017/S0305004100037099 10.1016/S0045-7825(96)01087-0 10.1016/j.cma.2014.12.025 10.1061/(ASCE)0733-9399(1989)115:5(1035) 10.1016/j.cma.2012.08.017 10.1007/s10596-013-9396-5 10.2118/8942-PA 10.1061/(ASCE)0733-9399(1986)112:1(85) 10.1098/rsta.1937.0005 10.1139/t99-049 10.1002/cnm.1111 10.1029/GM122p0213 10.2118/3009-PA 10.2172/10176839 10.1017/CBO9780511800955 10.2118/89-PA 10.1146/annurev-fluid-010814-014736 10.2118/8941-PA 10.2118/18959-PA |
| ContentType | Journal Article Publication |
| Contributor | Universitat Politècnica de Catalunya. Departament d'Enginyeria Civil i Ambiental Universitat Politècnica de Catalunya. LACÀN - Mètodes Numèrics en Ciències Aplicades i Enginyeria |
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| Keywords | Finite element method Stochastic analysis Random fields Sensitivity analysis Moving-boundary problem Perkins-Kern-Nordgren model Monte-Carlo method Hydraulic fracturing |
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| Snippet | Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic... The final publication is available at Springer via http://dx.doi.org/10.1007/s10596-018-9784-y Hydraulic fracturing processes are surrounded by uncertainty, as... |
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| SubjectTerms | 49 Calculus of variations and optimal control; optimization 49Q Manifolds 58 Global analysis, analysis on manifolds 58J Partial differential equations on manifolds; differential operators 60 Probability theory and stochastic processes 60H Stochastic analysis Analysis Anàlisi estocàstica Classificació AMS Computation Computer simulation Differential equations, Partial Earth and Environmental Science Earth Sciences Elastic properties Equacions diferencials parcials Equation of continuity Finite element method Geometria Geotechnical Engineering & Applied Earth Sciences Hydraulic fracturing Hydrogeology Investigació operativa Iterative methods Iterative solution Lagrange multiplier Manifolds (Mathematics) Matemàtiques i estadística Mathematical Modeling and Industrial Mathematics Mathematical models Monte Carlo simulation Monte-Carlo method Moving-boundary problem Optimització Original Paper Parameters Perkins-Kern-Nordgren model Pollution Probabilitat Random fields Random variables Sampling Sensitivity analysis Soil Science & Conservation Solution space Solutions Stochastic analysis Stochastic processes Time dependence Tip speed Varietats (Matematiques) Àrees temàtiques de la UPC |
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| Title | Sampling-based stochastic analysis of the PKN model for hydraulic fracturing |
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