Why Fracking Works

2014 ◽  
Vol 81 (10) ◽  
Author(s):  
Zdeněk P. Bažant ◽  
Marco Salviato ◽  
Viet T. Chau ◽  
Hari Viswanathan ◽  
Aleksander Zubelewicz
Keyword(s):  
Fracture Mechanics ◽  
Numerical Solutions ◽  
Pressure Profile ◽  
Crack Spacing ◽  
Classical Solutions ◽  
Hydraulic Pressure ◽  
Pumping Rate ◽  
Time To Peak ◽  
Fracture Localization ◽  
Localization Instability

Although spectacular advances in hydraulic fracturing, also known as fracking, have taken place and many aspects are well understood by now, the topology, geometry, and evolution of the crack system remain an enigma and mechanicians wonder: Why fracking works? Fracture mechanics of individual fluid-pressurized cracks has been clarified but the vital problem of stability of interacting hydraulic cracks escaped attention. First, based on the known shale permeability, on the known percentage of gas extraction from shale stratum, and on two key features of the measured gas outflow which are (1) the time to peak flux and (2) the halftime of flux decay, it is shown that the crack spacing must be only about 0.1 m. Attainment of such a small crack spacing requires preventing localization in parallel crack systems. Therefore, attention is subsequently focused on the classical solutions of the critical states of localization instability in a system of cooling or shrinkage cracks. Formulated is a hydrothermal analogy which makes it possible to transfer these solutions to a system of hydraulic cracks. It is concluded that if the hydraulic pressure profile along the cracks can be made almost uniform, with a steep enough pressure drop at the front, the localization instability can be avoided. To achieve this kind of profile, which is essential for obtaining crack systems dense enough to allow gas escape from a significant portion of kerogen-filled nanopores, the pumping rate (corrected for the leak rate) must not be too high and must not be increased too fast. Furthermore, numerical solutions are presented to show that an idealized system of circular equidistant vertical cracks propagating from a horizontal borehole behaves similarly. It is pointed out that one useful role of the proppants, as well as the acids that promote creation of debris in the new cracks, is to partially help to limit crack closings and thus localization. To attain the crack spacing of only 0.1 m, one must imagine formation of hierarchical progressively refined crack systems. Compared to new cracks, the system of pre-existing uncemented natural cracks or joints is shown to be slightly more prone to localization and thus of little help in producing the fine crack spacing required. So, from fracture mechanics viewpoint, what makes fracking work?–the mitigation of fracture localization instabilities. This can also improve efficiency by fracturing more shale. Besides, it is environmentally beneficial, by reducing flowback per m3 of gas. So is the reduction of seismicity caused by dynamic fracture instabilities (which are more severe in underground CO2 sequestration).

Cathodic Disbondment of Rubber/Steel Adhesive Bonds Modeled as Liquid-Solid Reactions

2011 ◽  
Author(s):  
Ali A. Tarhini ◽  
Ramsey F. Hamade
Keyword(s):  
Fracture Mechanics ◽  
Failure Criterion ◽  
Numerical Solutions ◽  
Strain Energy Release ◽  
Chemical Reactor ◽  
Bonded Joints ◽  
Chemical Failure ◽  
Adhesive Bonded Joints ◽  
Semi Empirical ◽  
Reaction Processes

Under cathodic conditions, rubber/steel adhesive bonded joints have been documented to ‘weaken’ due to attack by the generated alkali. If this were to occur under the action of cleavage mechanical loads, the bonds are likely to completely ‘delaminate’ causing the bonded constituents to physically separate. These two modes of disbondment are referred to as ‘weakening’ and ‘delamination’, respectively. Previously, Hamade and coworkers have implemented empirical and semi-empirical approaches to modeling cathodic disbondment of adhesive joints. Here, a method is presented to simulate bond weakening progress via numerical solutions. Bond degradation is modeled as a liquid-solid chemical reactor due to the attack by the alkaline medium. Specifically, the diffusion and chemical reaction processes involved in weakening are mathematically represented via a simplified, 2 partial differential equations (p.d.e.) boundary value problem (BVP). This is a reduced version of the more complex electrochemical formulation needed to fully describe the chemistry at the bondline under cathodic conditions. The weakening model is capable of simulating weakened bond lengths vs. time as function of electrolyte type (artificial sweater, ASW, or 1N NaOH), cathodic potential, and temperature. Furthermore and to model bond delamination, a mechano-chemical failure criterion is incorporated into the weakening formulation effectively coupling fracture mechanics principles with those of cathodic degradation. A fracture mechanics parameter, applied strain energy release rate, G, is used to represent the effect of externally applied loads. The failure criterion stipulates that the bond will delaminate if the applied G exceeds that of the degraded bond’s residual resistance. Both, the weakening and delamination formulations are validated against experimental data of bond weakening and delamination under a variety of conditions. As such, the numerical simulations developed in this work may be used to provide first order estimates of the life of rubber/steel bonded joints (weakened or delaminated lengths vs. time) as function of cathodic parameters and applied G (if the joint is loaded in the case of delamination).

Heat Conduction in Printed Circuit Boards: Part II — Small PCBs Connected to Large Thermal Mass at Their Edge

2007 ◽  
Author(s):  
Wataru Nakayama ◽  
Tatsuya Nakajima ◽  
Hiroko Koike ◽  
Ryuichi Matsuki
Keyword(s):  
Heat Conduction ◽  
Heat Sink ◽  
Continuum Model ◽  
Printed Circuit Boards ◽  
Numerical Solutions ◽  
System Level ◽  
Classical Solutions ◽  
Conduction Path ◽  
Two Samples ◽  
Thermal Conductivities

The PCB in high density packaging environment often provides a critical heat conduction path from the package to the electrical connectors or the system enclosure that serves as a solid heat sink at the edge of the PCB. Where convective heat transfer on the PCB surface is negligible, heat flow is almost unidirectional from the area under the package footprint to the solid heat sink. However, the heat conduction analysis requires a large computational resource, if the full details of PCB’s internal organization are taken into account. To perform the board and system level thermal analysis with available computational resources, it is imperative to replace the actual PCB’s internal structure by a simplified model. In the most popular modeling the PCB is replaced by a medium having orthotropic thermal conductivities. The assumption of equivalent thermal conductivities, however, is not a straightforward practice, as illustrated by the present study. Two samples are subject to heat conduction analysis; they are rectangular bar specimens cut out from the PCB, one measuring 6mm (L) × 0.5mm (W) × 1.27mm (HPCB), and the other 12.35mm (L) × 1.27mm (W) × 1.27mm (HPCB). The specimen contains two layers of continuous copper, and one row of through-vias in the zone under the package footprint. Numerical solutions are obtained taking into account the full details of the via zone structures. Classical solutions are also obtained for a two-zone continuum model having different sets of orthotropic thermal conductivities for the via zone and the remaining zone. The thermal conductivities in the continuum model are varied to bring the key temperatures to those obtained by the numerical analysis. Presented in this paper is a methodology to estimate the equivalent thermal conductivities for a given set of dimensional data and thermal boundary condition.

Influence of hydraulic pressure in fracture mechanics modelling of crack propagation in concrete

1998 ◽  
Vol 31 (3) ◽  
pp. 203-208 ◽  
Author(s):  
Ulf Ohlsson ◽  
Mikael Nyström ◽  
Thomas Olofsson ◽  
Knut Waagaard
Keyword(s):  
Fracture Mechanics ◽  
Crack Propagation ◽  
Hydraulic Pressure

scholarly journals AN INVESTIGATION OF IRREGULAR CRACK PATH EFFECTS ON FRACTURE MECHANICS PARAMETERS USING A GRAIN MICROSTRUCTURE MESHING TECHNIQUE

2012 ◽  
Vol 04 (01) ◽  
pp. 1250001 ◽  
Author(s):  
A. MEHMANPARAST ◽  
F. BIGLARI ◽  
C. M. DAVIES ◽  
K. M. NIKBIN
Keyword(s):  
Finite Element ◽  
Fracture Mechanics ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Crack Path ◽  
Crack Tip ◽  
Grain Structure ◽  
Element Mesh ◽  
Mesh Structure ◽  
Crack Paths

A sub-grain size finite element modelling approach is presented in this paper to investigate variations in fracture mechanics parameters for irregular crack paths. The results can be used when modelling intergranular and transgranular crack growth where creep and fatigue are the dominant failure mechanisms and their crack paths are irregular. A novel method for sub-grain scale finite element mesh consisting of multiple elements encased in ~50–150 μm-sized grains has been developed and implemented in a compact tension, C(T), mesh structure. The replicated shapes and dimensions were derived from an isotropic metallic grain structure using representative random sized grain shapes repeated in sequence ahead of the crack tip. In this way the effects of crack tip angle ahead of the main crack path can be considered in a more realistic manner. A comprehensive sensitivity analysis has been performed for elastic and elastic-plastic materials using ABAQUS and the stress distributions, the stress intensity factor and the J-integral have been evaluated for irregular crack paths and compared to those of obtained from analytical solutions. To examine the local and macroscopic graph path effects on fracture mechanics parameters, a few extreme cases with various crack-tip angles have been modelled by keeping the macroscopic crack path parallel to the axis of symmetry. The numerical solutions from these granular mesh structures have been found in relatively good agreement with analytical solutions.

scholarly journals Flood Hydrograph Prediction Using Machine Learning Methods

Water ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 968 ◽  
Author(s):  
Gokmen Tayfur ◽  
Vijay Singh ◽  
Tommaso Moramarco ◽  
Silvia Barbetta
Keyword(s):  
Machine Learning ◽  
Numerical Solutions ◽  
Central Italy ◽  
Computing Methods ◽  
Flood Routing ◽  
Flow Equations ◽  
Time To Peak ◽  
Wide Range ◽  
Flood Hydrograph ◽  
Artificial Neural Network Ann

Machine learning (soft) methods have a wide range of applications in many disciplines, including hydrology. The first application of these methods in hydrology started in the 1990s and have since been extensively employed. Flood hydrograph prediction is important in hydrology and is generally done using linear or nonlinear Muskingum (NLM) methods or the numerical solutions of St. Venant (SV) flow equations or their simplified forms. However, soft computing methods are also utilized. This study discusses the application of the artificial neural network (ANN), the genetic algorithm (GA), the ant colony optimization (ACO), and the particle swarm optimization (PSO) methods for flood hydrograph predictions. Flow field data recorded on an equipped reach of Tiber River, central Italy, are used for training the ANN and to find the optimal values of the parameters of the rating curve method (RCM) by the GA, ACO, and PSO methods. Real hydrographs are satisfactorily predicted by the methods with an error in peak discharge and time to peak not exceeding, on average, 4% and 1%, respectively. In addition, the parameters of the Nonlinear Muskingum Model (NMM) are optimized by the same methods for flood routing in an artificial channel. Flood hydrographs generated by the NMM are compared against those obtained by the numerical solutions of the St. Venant equations. Results reveal that the machine learning models (ANN, GA, ACO, and PSO) are powerful tools and can be gainfully employed for flood hydrograph prediction. They use less and easily measurable data and have no significant parameter estimation problem.

Importance of convective heat transfer in flow of non-Newtonian nanofluid featuring Brownian and thermophoretic diffusions

2019 ◽  
Vol 29 (12) ◽  
pp. 4624-4641 ◽  
Author(s):  
Waqar Azeem Khan ◽  
Mehboob Ali ◽  
Muhammad Waqas ◽  
M. Shahzad ◽  
F. Sultan ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Source ◽  
Convective Heat Transfer ◽  
Design Methodology ◽  
Numerical Solutions ◽  
Pressure Profile ◽  
Boundary Region ◽  
Heat Transfer Analysis ◽  
Content Type ◽  
Source Sink

Purpose This paper aims to address the flow of Sisko nanofluid by an unsteady curved surface. Non-uniform heat source/sink is considered for heat transfer analysis. Design/methodology/approach Numerical solutions are constructed using bvp4c procedure. Findings Pressure profile inside boundary region is increased when A and K are enhanced. Originality/value No such analysis is yet presented.

Capsular flow in pipelines

1972 ◽  
Vol 56 (1) ◽  
pp. 49-59 ◽  
Author(s):  
A. E. Vardy ◽  
M. I. G. Bloor ◽  
J. A. Fox
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Steady Motion ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Hydraulic Pressure ◽  
Central Difference ◽  
Navier Stokes Equations

The problem considered is that of the steady motion of a series of neutrally buoyant, flat-faced, rigid, cylindrical capsules along the axis of a pipeline under the influence of a hydraulic pressure gradient. The Navier-Stokes equations are non-dimensionalized and expressed in central-difference form. Numerical solutions are found by the method of relaxation for Reynolds numbers up to 20 000 and a close agreement is obtained with readings from a laboratory apparatus for Reynolds numbers up to 2200.The flow is examined in detail and the existence of toroidal vortices between successive capsules is demonstrated. Their shape is shown to be increasingly influenced by inertial forces as the Reynolds number increases, but the overall pressure gradient is not greatly dependent on the Reynolds number.

Stress—Strain Fields in Compressed Elastomeric Seals and Their Extension to Fracture Mechanics

1986 ◽  
Vol 59 (5) ◽  
pp. 709-721 ◽  
Author(s):  
Gianluca Medri ◽  
Antonio Strozzi
Keyword(s):  
Fracture Mechanics ◽  
Pressure Profile ◽  
Mechanical Analysis ◽  
Integral Approach ◽  
J Integral ◽  
Elastomeric Materials ◽  
Viscoelastic Effects ◽  
Numerical Tools ◽  
Experimental Findings ◽  
Elastomeric Seals

Abstract The modeling of an elastomer in terms of hyperelastic material has been discussed. The peculiarities of the finite element method when applied to nearly incompressible materials have been underlined. The mechanical analysis of various uncracked elastomeric seals has been treated with particular regard to possible crack initiation. The theoretical aspects of fracture mechanics applied to elastomers have been discussed, and the validity of the J-integral approach has been checked against experiments. The J-integral has been computed numerically in cracked O-ring seals for various fractional compressions and crack lengths. The numerical stress field has been compared to the experimental findings. The influence of the crack length on the contact pressure profile has been considered. Various difficulties still exist in modeling efficiently the elastomeric materials and in developing suitable numerical tools. Nevertheless, it is believed that statistical predictions on the debasement in the sealing characteristics due to viscoelastic effects and to crack propagation will become shortly feasible.

Limitations of the Short Bearing Approximation in Dynamically Loaded Narrow Hydrodynamic Bearings

1993 ◽  
Vol 115 (3) ◽  
pp. 544-549 ◽  
Author(s):  
M. A. Rezvani ◽  
E. J. Hahn
Keyword(s):  
Numerical Solutions ◽  
Parametric Design ◽  
Pressure Profile ◽  
Squeeze Film ◽  
Aspect Ratios ◽  
Transient Solutions ◽  
Dynamically Loaded ◽  
Axial Profile ◽  
Dynamic Loading Conditions ◽  
Improved Accuracy

Transient solutions are still widely used for evaluating the vibrational behavior of rotor bearing systems containing dynamically loaded journal bearings with large unbalance, or noncircular orbit type squeeze film dampers, such as dampers without centralizing springs. For parametric design studies, such transient analyses need rapid means for evaluating the motion dependent fluid film forces and for narrow bearings or dampers (aspect ratios less than 0.5) the short bearing approximation (SBA) to the Reynolds equation has generally been assumed. Comparisons with exact numerical solutions under conditions of static loading and pure squeezing show that the SBA pressure profile predictions are significantly in error for aspect ratios as low as 0.25 at eccentricities around 0.9, whereas the optimal parabolic axial profile approximation (MSBA), while retaining all the rapid calculation features of the SBA, is accurate to within 1 percent under the same conditions and to within 3 percent for aspect ratios around 1.0. Using the MSBA as a yardstick under transient solution conditions, the SBA, while satisfactory for aspect ratios of 0.05, was found to be inadequate in predicting transient and steady state orbits and transmitted forces at aspect ratios of 0.5. At these aspect ratios, jump speeds and instability threshold speeds were also found to be erroneously predicted, with speed overestimates of 30 percent possible for practical unbalance situations. In view of the vastly improved accuracy obtainable by the MSBA, its use is to be preferred to the SBA under dynamic loading conditions for aspect ratios around 0.5, and probably around 0.25 or lower.

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