Download Hydraulic Fracturing Explained Book PDF

Download full Hydraulic Fracturing Explained books PDF, EPUB, Tuebl, Textbook, Mobi or read online Hydraulic Fracturing Explained anytime and anywhere on any device. Get free access to the library by create an account, fast download and ads free. We cannot guarantee that every book is in the library.

Hydraulic Fracturing Explained

Hydraulic Fracturing Explained
  • Author : Erle C. Donaldson,Waqi Alam,Nasrin Begum
  • Publisher :Unknown
  • Release Date :2014-02-25
  • Total pages :224
  • ISBN : 9780127999968
GET BOOK HERE

Summary : Rocks mechanics legend Erle Donaldson, along with colleagues Waqi Alam and Nasrin Begum from the oil and gas consultant company Tetrahedron, have authored this handbook on updated fundamentals and more recent technology used during a common hydraulic fracturing procedure. Meant for technical and non-technical professionals interested in the subject of hydraulic fracturing, the book provides a clear and simple explanation of the technology and related issues to promote the safe development of petroleum reserves leading to energy independence throughout the world.

Environmental Issues Concerning Hydraulic Fracturing

Environmental Issues Concerning Hydraulic Fracturing
  • Author : Anonim
  • Publisher :Unknown
  • Release Date :2017-10-30
  • Total pages :424
  • ISBN : 9780128128039
GET BOOK HERE

Summary : Environmental Issues Concerning Hydraulic Fracturing, Volume One captures the state-of-the-art research currently used to evaluate the potential impact of unconventional gas and oil gas extraction processes. Topics in this comprehensive guide on the topic include chapters on The Human Health Implications of Unconventional Oil and Gas Development, The use of Noble Gas Analysis and other Forensic Techniques in Characterizing Contamination Pathways Associated with Oil and Gas Development, Well Integrity, Contamination Mechanisms and Groundwater Impacts Associated with Unconventional Oil and Gas Development, and Advances in Fracturing and Well Construction: Improving Efficiency and Reducing Risks. This serial explores a wide breadth of emerging and state-of-the-art technologies used to study the potential environmental impact and various processes in the massive industrial process of shale exploration and resource extraction. Covers a wide breadth of emerging and state-of-the-art technologies Includes contributions from an International board of authors Provides a comprehensive set of reviews, covering the potential impact of unconventional gas and oil gas extraction processes

Hydraulic Fracturing Chemicals and Fluids Technology

Hydraulic Fracturing Chemicals and Fluids Technology
  • Author : Johannes Fink
  • Publisher :Unknown
  • Release Date :2013-08-14
  • Total pages :248
  • ISBN : 9780124115217
GET BOOK HERE

Summary : When classifying fracturing fluids and their additives, it is important that production, operation, and completion engineers understand which chemical should be utilized in different well environments. A user's guide to the many chemicals and chemical additives used in hydraulic fracturing operations, Hydraulic Fracturing Chemicals and Fluids Technology provides an easy-to-use manual to create fluid formulations that will meet project-specific needs while protecting the environment and the life of the well. Fink creates a concise and comprehensive reference that enables the engineer to logically select and use the appropriate chemicals on any hydraulic fracturing job. The first book devoted entirely to hydraulic fracturing chemicals, Fink eliminates the guesswork so the engineer can select the best chemicals needed on the job while providing the best protection for the well, workers and environment. Pinpoints the specific compounds used in any given fracturing operation Provides a systematic approach to classifying fracturing fluid technology to meet specific project needs Eliminates guesswork with easy-to-understand language on selection and components of hydraulic fracturing chemicals Addresses environmental aspects of chemicals to safeguard employees and protect the environment

Hydraulic Fracturing in Unconventional Reservoirs

Hydraulic Fracturing in Unconventional Reservoirs
  • Author : Hoss Belyadi,Ebrahim Fathi,Fatemeh Belyadi
  • Publisher :Unknown
  • Release Date :2019-06-18
  • Total pages :632
  • ISBN : 9780128176665
GET BOOK HERE

Summary : Hydraulic Fracturing in Unconventional Reservoirs: Theories, Operations, and Economic Analysis, Second Edition, presents the latest operations and applications in all facets of fracturing. Enhanced to include today’s newest technologies, such as machine learning and the monitoring of field performance using pressure and rate transient analysis, this reference gives engineers the full spectrum of information needed to run unconventional field developments. Covering key aspects, including fracture clean-up, expanded material on refracturing, and a discussion on economic analysis in unconventional reservoirs, this book keeps today's petroleum engineers updated on the critical aspects of unconventional activity. Helps readers understand drilling and production technology and operations in shale gas through real-field examples Covers various topics on fractured wells and the exploitation of unconventional hydrocarbons in one complete reference Presents the latest operations and applications in all facets of fracturing

Handbook of Hydraulic Fracturing

Handbook of Hydraulic Fracturing
  • Author : James G. Speight
  • Publisher :Unknown
  • Release Date :2016-03-15
  • Total pages :304
  • ISBN : 9781118673041
GET BOOK HERE

Summary : Presents an up-to-date description of current and new hydraulic fracturing processes Details Emerging Technologies such as Fracture Treatment Design, Open Hole Fracturing, Screenless Completions, Sand Control, Fracturing Completions and Productivity Covers Environmental Impact issues including Geological Disturbance; Chemicals used in Fracturing; General Chemicals; Toxic Chemicals; and Air, Water, Land, and Health impacts Provides many process diagrams as well as tables of feedstocks and their respective products

The Fracking Debate

The Fracking Debate
  • Author : Daniel Raimi
  • Publisher :Unknown
  • Release Date :2017-12-26
  • Total pages :262
  • ISBN : 9780231545716
GET BOOK HERE

Summary : Over roughly the past decade, oil and gas production in the United States has surged dramatically—thanks largely to technological advances such as high-volume hydraulic fracturing, more commonly known as “fracking.” This rapid increase has generated widespread debate, with proponents touting economic and energy-security benefits and opponents highlighting the environmental and social risks of increased oil and gas production. Despite the heated debate, neither side has a monopoly on the facts. In this book, Daniel Raimi gives a balanced and accessible view of oil and gas development, clearly and thoroughly explaining the key issues surrounding the shale revolution. The Fracking Debate directly addresses the most common questions and concerns associated with fracking: What is fracking? Does fracking pollute the water supply? Will fracking make the United States energy independent? Does fracking cause earthquakes? How is fracking regulated? Is fracking good for the economy? Coupling a deep understanding of the scholarly research with lessons from his travels to every major U.S. oil- and gas-producing region, Raimi highlights stories of the people and communities affected by the shale revolution, for better and for worse. The Fracking Debate provides the evidence and context that have so frequently been missing from the national discussion of the future of oil and gas production, offering readers the tools to make sense of this critical issue.

Health Impact Assessment of Shale Gas Extraction

Health Impact Assessment of Shale Gas Extraction
  • Author : Roundtable on Environmental Health Sciences, Research, and Medicine,Board on Population Health and Public Health Practice,Institute of Medicine
  • Publisher :Unknown
  • Release Date :2014-01-13
  • Total pages :139
  • ISBN : 030928791X
GET BOOK HERE

Summary : Natural gas extraction from shale formations, which includes hydraulic fracturing, is increasingly in the news as the use of extraction technologies has expanded, rural communities have been transformed seemingly overnight, public awareness has increased, and regulations have been developed. The governmental public health system, which retains primary responsibility for health, was not an early participant in discussions about shale gas extraction; thus public health is lacking critical information about environmental health impacts of these technologies and is limited in its ability to address concerns raised by regulators at the federal and state levels, communities, and workers employed in the shale gas extraction industry. Health Impact Assessment of Shale Gas Extraction is the summary of a workshop convened in 2012 by the Institute of Medicine (IOM) Roundtable on Environmental Health Sciences, Research, and Medicine to discuss the human health impact of shale gas extraction through the lens of a health impact assessment. Eminent scientists, physicians, public health experts, and representatives from government agencies at federal and state levels, from nongovernment organizations, from the business sector, and from interest groups representing the interests of the citizens met to exchange ideas and to inform on hydraulic fracturing as a means of extraction of natural gas. This report examines the state of the science regarding shale gas extraction, the direct and indirect environmental health impacts of shale gas extraction, and the use of health impact assessment as a tool that can help decision makers identify the public health consequences of shale gas extraction.

Surface Chemistry and Geochemistry of Hydraulic Fracturing

Surface Chemistry and Geochemistry of Hydraulic Fracturing
  • Author : K. S. Birdi
  • Publisher :Unknown
  • Release Date :2016-10-14
  • Total pages :223
  • ISBN : 9781482257199
GET BOOK HERE

Summary : Unique in focus, Surface Chemistry and Geochemistry of Hydraulic Fracturing examines the surface chemistry and phenomena in the hydrofracking process. Under great scrutiny as of late, the physico-chemical properties of hydrofracking are fully detailed and explained. Topics include the adsorption-desorption of gas on the shale reservoir surface and relevant waste-water treatment dependent on various surface chemistry principles. The aim of this book is to help engineers and research scientists recognize the basic surface chemistry principles related to this subject. Written by a long-time expert in the field, this book presents an unbiased account of the hard science and engineering involved in a resource that is gaining growing attention within the community.

Water-Based Chemicals and Technology for Drilling, Completion, and Workover Fluids

Water-Based Chemicals and Technology for Drilling, Completion, and Workover Fluids
  • Author : Johannes Fink
  • Publisher :Unknown
  • Release Date :2015-02-06
  • Total pages :288
  • ISBN : 9780128026434
GET BOOK HERE

Summary : Oil and gas engineers today use three main factors in deciding drilling fluids: cost, performance, and environmental impact, making water-based products a much more attractive option. Water-Based Chemicals and Technology for Drilling, Completion, and Workover Fluids effectively delivers all the background and infrastructure needed for an oil and gas engineer to utilize more water-based products that benefit the whole spectrum of the well’s life cycle. Helping to mitigate critical well issues such as formation damage, fluid loss control, and borehole repair, more operators demand to know the full selection of water-based products available to consistently keep a peak well performance. This must-have training guide provides the necessary coverage in the area, broken down by type and use, along with an extensive list of supportive materials such as a chemical index of structural formulas and helpful list of references for further reading. In addition to understanding the types, special additives, and chemical compatibilities of the products available, the reader will also learn proper waste disposal techniques, including management of produced water, a component mandatory to hydraulic fracturing operations. Concise and comprehensive, Water-Based Chemicals and Technology for Drilling, Completion, and Workover Fluids details all the necessary educational content and handy references to elevate your well’s performance while lowering your environmental impact. Understand the basics and functions on all water-based fluids for drilling, completion, cementing, and enhanced oil recovery operations Get up to date with the growing need for water-based fluids in hydraulic fracturing operations including supportive materials such as an index of trade names, acronyms, and chemicals Stay responsible and know the environmental aspects and current regulations, including disposal and discharge

Environmental Considerations Associated with Hydraulic Fracturing Operations

Environmental Considerations Associated with Hydraulic Fracturing Operations
  • Author : James A. Jacobs,Stephen M. Testa
  • Publisher :Unknown
  • Release Date :2019-04-23
  • Total pages :576
  • ISBN : 9781119336099
GET BOOK HERE

Summary : A guide to environmental and communication issues related to fracking and the best approach to protect communities Environmental Considerations Associated with Hydraulic Fracturing Operations offers a much-needed resource that explores the complex challenges of fracking by providing an understanding of the environmental and communication issues that are inherent with hydraulic fracturing. The book balances the current scientific knowledge with the uncertainty and risks associated with hydraulic fracking. In addition, the authors offer targeted approaches for helping to keep communities safe. The authors include an overview of the historical development of hydraulic fracturing and the technology currently employed. The book also explores the risk, prevention, and mitigation factors that are associated with fracturing. The authors also include legal cases, regulatory issues, and data on the cost of recovery. The volume presents audit checklists for gathering critical information and documentation to support the reliability of the current environmental conditions related to fracking operations and the impact fracking can have on a community. This vital resource: Contains the technical information and mitigation recommendations for safety and environmental issues related to hydraulic fracturing Offers an historical overview of conventional and unconventional oil and gas drilling Explains the geologic and technical issues associated with fracking of tight sand and shale formulations Presents numerous case studies from the United States EPA and other agencies Discusses issues of co-produced waste water and induced seismicity from the injection of wastewater Written for environmental scientists, geologists, engineers, regulators, city planners, attorneys, foresters, wildlife biologists, and others, Environmental Considerations Associated with Hydraulic Fracturing Operations offers a comprehensive resource to the complex environmental and communication issues related to fracking.

Hydraulic Fracturing for Oil and Gas

Hydraulic Fracturing for Oil and Gas
  • Author : U.s. Environmental Protection Agency
  • Publisher :Unknown
  • Release Date :2017-06-09
  • Total pages :664
  • ISBN : 1547257636
GET BOOK HERE

Summary : This final report provides a review and synthesis of available scientific information concerning the relationship between hydraulic fracturing activities and drinking water resources in the United States. The report is organized around activities in the hydraulic fracturing water cycle and their potential to impact drinking water resources. The stages include: (1) acquiring water to be used for hydraulic fracturing (Water Acquisition), (2) mixing the water with chemical additives to prepare hydraulic fracturing fluids (Chemical Mixing), (3) injecting the hydraulic fracturing fluids into the production well to create fractures in the targeted production zone (Well Injection), (4) collecting the wastewater that returns through the well after injection (Produced Water Handling), and (5) managing the wastewater via disposal or reuse methods (Wastewater Disposal and Reuse). EPA found scientific evidence that hydraulic fracturing activities can impact drinking water resources under some circumstances. The report identifies certain conditions under which impacts from hydraulic fracturing activities can be more frequent or severe.

Rock Fractures and Fluid Flow

Rock Fractures and Fluid Flow
  • Author : National Research Council,Division on Earth and Life Studies,Commission on Geosciences, Environment and Resources,Committee on Fracture Characterization and Fluid Flow
  • Publisher :Unknown
  • Release Date :1996-08-27
  • Total pages :568
  • ISBN : 9780309049962
GET BOOK HERE

Summary : Scientific understanding of fluid flow in rock fractures--a process underlying contemporary earth science problems from the search for petroleum to the controversy over nuclear waste storage--has grown significantly in the past 20 years. This volume presents a comprehensive report on the state of the field, with an interdisciplinary viewpoint, case studies of fracture sites, illustrations, conclusions, and research recommendations. The book addresses these questions: How can fractures that are significant hydraulic conductors be identified, located, and characterized? How do flow and transport occur in fracture systems? How can changes in fracture systems be predicted and controlled? Among other topics, the committee provides a geomechanical understanding of fracture formation, reviews methods for detecting subsurface fractures, and looks at the use of hydraulic and tracer tests to investigate fluid flow. The volume examines the state of conceptual and mathematical modeling, and it provides a useful framework for understanding the complexity of fracture changes that occur during fluid pumping and other engineering practices. With a practical and multidisciplinary outlook, this volume will be welcomed by geologists, petroleum geologists, geoengineers, geophysicists, hydrologists, researchers, educators and students in these fields, and public officials involved in geological projects.

Shale Gas

Shale Gas
  • Author : Ali Al-Juboury
  • Publisher :Unknown
  • Release Date :2018-08-29
  • Total pages :110
  • ISBN : 9781789236187
GET BOOK HERE

Summary : Natural gas, particularly shale gas, is one of the main sustainable energy sources in the current century. It is an abundant energy resource, playing an active role in future energy demand and enabling nations to transition to higher support on renewable energy sources. The book aims to add some contributions and new advances in technologies and prospects on shale gas reserves in selected regions of the world, in terms of new technologies of extraction, new discoveries of promising reserves, synthesis and applications to get high quality of this cleanest consuming non-renewable energy source.

Oil Field Chemicals

Oil Field Chemicals
  • Author : Johannes Fink
  • Publisher :Unknown
  • Release Date :2003-08-19
  • Total pages :495
  • ISBN : 0080497578
GET BOOK HERE

Summary : Oil field chemicals are gaining increasing importance, as the resources of crude oil are decreasing. An increasing demand of more sophisticated methods in the exploitation of the natural resources emerges for this reason. This book reviews the progress in the area of oil field chemicals and additives of the last decade from a rather chemical view. The material presented is a compilation from the literature by screening critically approximately 20,000 references. The text is ordered according to applications, just in the way how the jobs are emerging in practice. It starts with drilling, goes to productions and ends with oil spill. Several chemicals are used in multiple disciplines, and to those separate chapters are devoted. Two index registers are available, an index of chemical substances and a general index. * Gives an introduction to the chemically orientated petroleum engineer. * Provides the petroleum engineer involved with research and development with a quick reference tool. * Covers interdisciplinary matter, i.e. connects petroleum recovery and handling with chemical aspects.

Shale Gas and Fracking

Shale Gas and Fracking
  • Author : Michael Stephenson
  • Publisher :Unknown
  • Release Date :2015-02-02
  • Total pages :170
  • ISBN : 9780128017623
GET BOOK HERE

Summary : Shale Gas and Fracking: The Science Behind the Controversy explains the relevant geological principles before examining the peer-reviewed evidence and presenting it through a simple and compelling illustrated narrative. Each chapter focuses on a particular controversy, such contamination of well water with gas from fracking, and follows a similar format: starting with the principles; then detailing peer-reviewed case studies for earthquakes, radioactivity, and climate change; and concluding with a judgment of the general risks involved. Shale Gas and Fracking: The Science Behind the Controversy provides readers with the unbiased information they need to make informed decisions on the controversial issue of fracking. Presents a clear and unbiased view of the pros and cons of fracking in Europe and the US, through a simple and compelling narrative from an informed publicly-funded scientist Includes full-colour diagrams, photographs, and maps to present information clearly and simply Focuses on peer-reviewed, documented examples, particularly of earthquakes and groundwater contamination due to fracking

Fracking 101

Fracking 101
  • Author : Eric George
  • Publisher :Unknown
  • Release Date :2016-04-24
  • Total pages :110
  • ISBN : 9780994585004
GET BOOK HERE

Summary : Fracking, or hydraulic fracturing to give its proper name, has become part of our lives recently, due to the massive exploitation of America’s shale oil and gas fields. Along the way it has stirred up controversy, with passionate opponents fighting against the oil companies. The fight has generated a lot of heat, but not much understanding. This guide, written by some-one who knows what he is talking about takes a detached, neutral view of the subject. Without pushing a view for or against, it provides the factual background you need to form an opinion of your own. An Informed and Neutral Introduction Like most people I have heard of fracking, but did I really understand what it was? To answer honestly, no. I knew it had something to do with mining and was perhaps destructive to the land. To me, it was just one of those words of the moment. This guide has given me a real sense and understanding of what fracking is. It allowed me, someone who has no experience in this field, to learn about the pros and cons of fracking, without having the good and bad of it forced down my throat. If you want an informed and neutral introduction into fracking, then this is the guide for you. ~ Debbie Prewer

Petroleum Rock Mechanics

Petroleum Rock Mechanics
  • Author : Bernt Aadnoy,Reza Looyeh
  • Publisher :Unknown
  • Release Date :2019-06-15
  • Total pages :460
  • ISBN : 9780128159040
GET BOOK HERE

Summary : Petroleum Rock Mechanics: Drilling Operations and Well Design, Second Edition, keeps petroleum and drilling engineers centrally focused on the basic fundamentals surrounding geomechanics, while also keeping them up-to-speed on the latest issues and practical problems. Updated with new chapters on operations surrounding shale oil, shale gas, and hydraulic fracturing, and with new sections on in-situ stress, drilling design of optimal mud weight, and wellbore instability analysis, this book is an ideal resource. By creating a link between theory with practical problems, this updated edition continues to provide the most recent research and fundamentals critical to today’s drilling operations. Helps readers grasp the techniques needed to analyze and solve drilling challenges, in particular wellbore instability analysis Teaches rock mechanic fundamentals and presents new concepts surrounding sand production and hydraulic fracturing operations Includes new case studies and sample problems to practice

Unconventional Oil and Gas Resources Handbook

Unconventional Oil and Gas Resources Handbook
  • Author : Y Zee Ma,Stephen Holditch
  • Publisher :Unknown
  • Release Date :2015-10-06
  • Total pages :550
  • ISBN : 9780128025369
GET BOOK HERE

Summary : Unconventional Oil and Gas Resources Handbook: Evaluation and Development is a must-have, helpful handbook that brings a wealth of information to engineers and geoscientists. Bridging between subsurface and production, the handbook provides engineers and geoscientists with effective methodology to better define resources and reservoirs. Better reservoir knowledge and innovative technologies are making unconventional resources economically possible, and multidisciplinary approaches in evaluating these resources are critical to successful development. Unconventional Oil and Gas Resources Handbook takes this approach, covering a wide range of topics for developing these resources including exploration, evaluation, drilling, completion, and production. Topics include theory, methodology, and case histories and will help to improve the understanding,integrated evaluation, and effective development of unconventional resources. Presents methods for a full development cycle of unconventional resources, from exploration through production Explores multidisciplinary integrations for evaluation and development of unconventional resources and covers a broad range of reservoir characterization methods and development scenarios Delivers balanced information with multiple contributors from both academia and industry Provides case histories involving geological analysis, geomechanical analysis, reservoir modeling, hydraulic fracturing treatment, microseismic monitoring, well performance and refracturing for development of unconventional reservoirs

Rock Failure Mechanisms

Rock Failure Mechanisms
  • Author : Chun'An Tang,John A. Hudson
  • Publisher :Unknown
  • Release Date :2010-08-06
  • Total pages :364
  • ISBN : 9780415498517
GET BOOK HERE

Summary : When dealing with rock in civil engineering, mining engineering and other engineering, the process by which the rock fails under load should be understood, so that safe structures can be built on and in the rock. However, there are many ways for loading rock and rock can have a variety of idiosyncracies. This reference book provides engineers and researchers with the essential knowledge for a clear understanding of the process of rock failure under different conditions. It contains an introductory chapter explaining the role of rock failure in engineering projects plus a summary of the theories governing rock failure and an explanation of the computer simulation method. It subsquently deals in detail with explaining, simulating and illustrating rock failure in laboratory and field. The concluding chapter discusses coupled modelling and the anticipated future directions for this type of computer simulation. An appendix describing the RFPA numerical model (Rock Failure Process Analysis program) is also included. About the Authors Chun'an Tang has a PhD in Mining Engineering and is a Professor at the School of Civil & Hydraulic Engineering at Dalian University of Technology in China. He is an advisor for design and stablity problem modelling in mining and civil rock engineeringand and Chairman of the China National Group of the International Society for Rock Mechanics. John Hudson is emeritus professor at Imperial College, London and is active as an independant consultant for Rock Engineering Consultants. He has a PhD in Rock Mechanics and completed over a 130 rock engineering consulting assignments in mining and civil engineering. He is a fellow at the Royal Academy of Engineering in the UK and President of the International Society for Rock Mechanics.

Understanding Trust in Government

Understanding Trust in Government
  • Author : Scott E. Robinson,James W. Stoutenborough,Arnold Vedlitz
  • Publisher :Unknown
  • Release Date :2017-04-28
  • Total pages :182
  • ISBN : 9781315519517
GET BOOK HERE

Summary : Growing disenfranchisement with political institutions and policy processes has generated interest in trust in government. For the most part, research has focused on trust in government as a general attitude covering all political institutions. In this book, Scott E. Robinson, James W. Stoutenborough, and Arnold Vedlitz argue that individual agencies develop specific reputations that may contrast with the more general attitudes towards government as a whole. Grounded in a treatment of trust as a relationship between two actors and taking the Environmental Protection Agency as their subject, the authors illustrate that the agency’s reputation is explained through general demographic and ideological factors – as well as policy domain factors like environmentalism. The book presents results from two approaches to assessing trust: (1) a traditional attitudinal survey approach, and (2) an experimental approach using the context of hydraulic fracturing. While the traditional attitudinal survey approach provides traditional answers to what drives trust in the EPA, the experimental results reveal that there is little specific trust in the EPA across the United States. Robinson, Stoutenborough, and Vedlitz expertly point the way forward for more reliable assessments of trust, while demonstrating the importance of assessing trust at the agency level. This book represents a much-needed resource for those studying both theory and methods in Public Administration and Public Policy.

Hydraulic Fracture Mechanics

Hydraulic Fracture Mechanics
  • Author : Peter Valkó,Michael J. Economides
  • Publisher :Unknown
  • Release Date :1995
  • Total pages :298
  • ISBN : UOM:39015032516489
GET BOOK HERE

Summary : The book explores the theoretical background of one of the most widespread activities in hydrocarbon wells, that of hydraulic fracturing. A comprehensive treatment of the basic phenomena includes: linear elasticity, stresses, fracture geometry and rheology. The diverse concepts of mechanics are integrated into a coherent description of hydraulic fracture propagation. The chapters in the book are cross-referenced throughout and the connections between the various phenomena are emphasized. The book offers readers a unique approach to the subject with the use of many numerical examples.