Sampling-based stochastic analysis of the PKN model for hydraulic fracturing

descriptionPublicationkeyboard_double_arrow_right Article 20 Nov 2018 Netherlands English Publisher:Springer Science and Business Media LLCJournal:Computational Geosciences, volume 23, pages 81-105 (issn: 1420-0597, eissn: 1573-1499,

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Authors: Garikapati, Hasini; Verhoosel, Clemens V.; van Brummelen, Harald; Zlotnik, Sergio; Díez, Pedro;

doi: 10.1007/s10596-018-9784-y

handle: 2117/125346

Sampling-based stochastic analysis of the PKN model for hydraulic fracturing

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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.

Country

Netherlands

Related Organizations

Eindhoven University of Technology
Netherlands
Universitat Politècnica de Catalunya
Spain
Technical University Eindhoven
Netherlands
Universitat Polite`cnica de Catalunya
Spain

Keywords

Àrees temàtiques de la UPC::Matemàtiques i estadística::Probabilitat, moving-boundary problem, Manifolds (Mathematics), sensitivity analysis, Classificació AMS::58 Global analysis, Classificació AMS::49 Calculus of variations and optimal control; optimization::49Q Manifolds, Stochastic analysis applied to problems in fluid mechanics, :60 Probability theory and stochastic processes::60H Stochastic analysis [Classificació AMS], :Matemàtiques i estadística::Probabilitat [Àrees temàtiques de la UPC], :58 Global analysis, analysis on manifolds::58J Partial differential equations on manifolds [Classificació AMS], Moving-boundary problem, Classificació AMS::58 Global analysis, analysis on manifolds::58J Partial differential equations on manifolds; differential operators, Flows in porous media; filtration; seepage, Àrees temàtiques de la UPC::Matemàtiques i estadística::Investigació operativa::Optimització, Monte Carlo methods, Stochastic analysis, Differential equations, Partial, Equacions diferencials parcials, Classificació AMS::58 Global analysis, analysis on manifolds::58J Partial differential equations on manifolds, :49 Calculus of variations and optimal control [Classificació AMS], Engineering, Mechanical, optimization::49Q Manifolds, Classificació AMS::49 Calculus of variations and optimal control, random fields, Random fields, Sensitivity analysis, :49Q Manifolds [optimization], Partial, Differential equations, Finite element method, Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria, Engineering, Civil, Anàlisi estocàstica, Varietats (Matematiques), finite element method, stochastic analysis, Engineering, Multidisciplinary, Classificació AMS::60 Probability theory and stochastic processes::60H Stochastic analysis, hydraulic fracturing, Perkins-Kern-Nordgren model, Engineering, Ocean, analysis on manifolds::58J Partial differential equations on manifolds, Engineering, Aerospace, Engineering, Biomedical, Hydraulic fracturing, Computer Science, Software Engineering, Monte-Carlo method, Engineering, Marine, Engineering, Manufacturing, differential operators, Engineering, Industrial, :Matemàtiques i estadística::Geometria [Àrees temàtiques de la UPC], :Matemàtiques i estadística::Investigació operativa::Optimització [Àrees temàtiques de la UPC]

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