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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Bibliographic Details
Published in:Computational geosciences Vol. 23; no. 1; pp. 81 - 105
Main Authors: Garikapati, Hasini, Verhoosel, Clemens V., van Brummelen, E. Harald, Zlotnik, Sergio, Díez, Pedro
Format: Journal Article Publication
Language:English
Published: Cham Springer International Publishing 01.02.2019
Springer Nature B.V
Subjects:
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
Finite element method
Stochastic analysis
Random fields
Sensitivity analysis
Moving-boundary problem
Perkins-Kern-Nordgren model
Monte-Carlo method
Hydraulic fracturing
ISSN:1420-0597, 1573-1499
Online Access:Get full text
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Summary: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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ISSN:1420-0597
1573-1499
DOI:10.1007/s10596-018-9784-y