The parameters of the seven synthetic surface pairs.
\r\n\tMany tried to define it, and its definition is always related to those who are in power, that being explained by the fact that this power and the abuse of it precisely, gives the access to being corrupted and practicing the acts that fall under corruption.
\r\n\r\n\tWe can find various types of corruption such as bribery, lobbying, extortion, cronyism, nepotism, parochialism, patronage, influence peddling, graft, and embezzlement. Also giving or accepting bribes or inappropriate gifts, double-dealing, under-the-table transactions, manipulating elections, diverting funds, laundering money, and defrauding investors.
\r\n\tNo government is immune to corruption. According to the World Bank, “the causes of corruption are always contextual, rooted in a country's policies, bureaucratic traditions, political development, and social history”.
\r\n\tThis indeed has consequences for increasing inequality, impacts government expenditure and services, shadow economy, and crime.
\r\n\tThis book will be a collection of chapters on Corruption. It welcomes contributions related to the nature of corruption its types and how corruption is undertaken in a certain context and the ways to deal with corruption will be part of this book. We value including materials on Corruption in organizations and ways to solve it. The origins of corruption and the way to deal with corruption, how to provide solutions, and any new insights on corruption will be part of this book.
",isbn:"978-1-80356-696-2",printIsbn:"978-1-80356-695-5",pdfIsbn:"978-1-80356-697-9",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,isSalesforceBook:!1,isNomenclature:!1,hash:"9cda6d2feaa52a6d523da74f2e2d7ffb",bookSignature:"Dr. Josiane Fahed-Sreih",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/11772.jpg",keywords:"Corruption, Origins, Types, Corporate Governance, Organizational Performance, Solutions, Corruption Index, Private Sector, Lebanon, Accountability, Anti-corruption, Public Policy",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"March 23rd 2022",dateEndSecondStepPublish:"April 20th 2022",dateEndThirdStepPublish:"June 19th 2022",dateEndFourthStepPublish:"September 7th 2022",dateEndFifthStepPublish:"November 6th 2022",dateConfirmationOfParticipation:null,remainingDaysToSecondStep:"a month",secondStepPassed:!0,areRegistrationsClosed:!1,currentStepOfPublishingProcess:3,editedByType:null,kuFlag:!1,biosketch:"Dr. Fahed-Sreih is the director of the Institute of Family and Entrepreneurial Business and a chairperson in the Department of Management. She obtained a Ph.D. from Sorbonne University, France, and received the 2007 FFI International Award for outstanding achievement in furthering the understanding of family business issues between two or more countries. She is on the editorial board of the Journal of Family Business Management and a keynote speaker for corporate governance conferences.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"103784",title:"Dr.",name:"Josiane",middleName:null,surname:"Fahed-Sreih",slug:"josiane-fahed-sreih",fullName:"Josiane Fahed-Sreih",profilePictureURL:"https://mts.intechopen.com/storage/users/103784/images/system/103784.jfif",biography:"Dr. Josiane Fahed-Sreih is a full-time associate professor of Management in the School of Business, Lebanese American University. She is the founder and director of the Institute of Family and Entrepreneurial Business and a chairperson in the Department of Management at the same university. She was previously the assistant dean. She obtained a Ph.D. from Sorbonne University, Paris, France. Dr. Fahed-Sreih is the Middle East Coordinator for the Family Firm Institute (FFI), the USA, and a family wealth and family business consultant. She received the 2007 FFI International Award for outstanding achievement in furthering the understanding of family business issues that occur between two or more countries. She has participated in and organized international conferences, workshops, and seminars. She has presented at major conferences locally and internationally and consulted on management issues in many countries, including Saudi Arabia, Dubai, Jordan, Qatar, Kuwait, Syria, Bahrain, Oman, France, Cyprus, and Lebanon. She currently sits on five boards of directors as a shareholder, two as a chairman of the board, and one as an independent director in the private sector. She is also an advisor on boards of community service organizations. \n\nShe speaks regularly to trade and professional groups and presents her research at academic conferences worldwide. She is frequently invited as a keynote speaker to the recognized family business and corporate governance conferences. Her research interests are in management, family business, the functioning of boards of directors, and corporate governance. She has published three books, several book chapters, and academic articles in international journals. 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“Soma” means the body. “Psychosomatic” diseases in psychiatry are mainly due to psychological causes that are not based on an organic etiology or physical disorders.
\nIt is currently believed that conscious or unconscious emotions, thoughts, and behaviors are also effective in psychosomatic disease in biological organisms and should be evaluated together with the dual thoughts related to the body and mind status. The patient may use more atypical and exaggerated expressive words from time to time when defining psychogenic pain. The localization and periodicity of the patient’s pain can also be persistently atypical. The pain of the patient usually begins after some important life events and stresses. The emotional burden of the factors stimulating and triggering pain is reflected in the patient’s voluntary posture and the relevant muscle groups. A psychological dimension, depression and anxiety are usually noted among the accompanying symptoms. It is noteworthy that the characteristics of the pain do not conform to anatomical facts and physiological functioning. Besides, the response to analgesics is also often atypical. Accompanying symptoms such as anger, impatience, helplessness, boredom, and restlessness must be considered in patients with psychological pain.
\nBoth psychosocial and biological factors always play a role in the development and pathophysiological mechanism of psychogenic pain [1, 2, 3, 4].
\nThe
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The following criteria must be present to make a diagnosis of a psychosomatic disorder according to the DSM-IV:
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Stress is one of the major causes of psychosomatic diseases. Stress causes many bodily functions to deteriorate or not work properly.
The most common trigger of psychosomatic diseases is loss and separation.
The American Board of Medical Specialties and the American Psychiatry and Neurology Board have approved specialization in psychosomatic medicine in 2003. This decision has emphasized the importance of this field and also reintroduced the widespread use of the “psychosomatic” term [1, 2, 3].
\nThe Diagnostic Criteria for Psychosomatic Research (DCPR) are considered to be more explanatory than the DSM-IV [1, 2, 3].
\nPersistent somatization subjects with a psychosomatic disorder are believed to have a higher incidence of other nonfunctional system (chronic fatigue) syndromes in the future with this approach, and these are called “multisomatoform disorder,” “pure somatization,” or “chronic somatization.” The functional somatic symptoms secondary to a psychiatric disorder are as follows:
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It is important to have knowledge about the pain and its variety before psychosomatic pain, because pain can cause psychosomatic problems as well as result.
\nPain is thought to be an important clinical and socioeconomic problem all around the world. We investigated the incidence, prevalence, and economic burden of pain conditions in children, adolescents, and adults based on the electronic scanning of databases for articles published between 2000 and 2014 in this review. Differences in methodology and the epidemiological studies make it difficult to give precise predictions of prevalence and incidence; however, the economic burden of psychosomatic pain is clearly high. There is a need to develop concepts and methods to examine pain from a population perspective and to advance the development of pain prevention and management strategies. Family physicians and clinicians have great responsibilities in the diagnosis and treatment of pain and especially psychosomatic pain within this context. The participation of physicians in multidisciplinary training and studies is also a fundamental principle [3, 5].
\nIt is appropriate to explain psychosomatic pain primarily in accordance with general principles. Pain is a defining characteristic in the diagnosis of many diseases. It can serve as an index of the symptoms and activity of the disease or diseases and as a prognostic indicator and a predictor of the use of health care for the underlying etiology [1, 2]. The International Association for the Study of Pain (IASP) and the World Health Organization (WHO) describe pain as “an unpleasant sensory and emotional experience associated with or described in relation to real or potential tissue damage or associated with such damage,” “an unpleasant sensory and emotional experience associated with existing or possible tissue damage or associated with such damage,” and “a protection mechanism” [2].
\nAn overview of the epidemiology and economic burden of pain conditions in children, adolescents, and adults is summarized below under the relevant headings. The incidence and prevalence of pain conditions as well as the risk factors and the effect of pain on individuals have also been described. The wide range of pain conditions in clinical and research areas include pain in children and adolescents, spinal pain, neuropathic pain, musculoskeletal pain, and fibromyalgia/chronic diffuse pain. In addition to the factors associated with the prevalence of pain, the individual, economic, and social burden of pain conditions should also be considered.
\n\n
According to neurophysiological mechanisms
Nociceptive
Somatic
Visceral
Neuropathic (non-nociceptive) central or peripheral
Psychogenic
According to the duration
Acute
Chronic
According to the etiologic factors
Cancer pain
Postherpetic neuralgia
Pain due to sickle cell anemia
Arthritis pain
According to the pain region
Headache
Facial pain
Low back pain
Pelvic pain
Before making a diagnosis of
Describing the epidemiology of pain is difficult because of the subjective nature of symptoms and the lack of consensus on the definitions of specific diagnoses and conditions. It is problematic to identify pain areas, especially with musculoskeletal pain. Many pain conditions are episodic, and the majority of patients express recurrent symptoms at varying intervals and durations during periods with and without pain. The actual incidence of most pain conditions may therefore remain unknown.
\nSimilarly, study results vary due to differences in the identification of diffuse pain cases and the specific diagnosis. While case definitions may also vary depending on the duration, intensity, or psychological burden of pain in the patient, the diagnoses are based on subjective patient experience, clinical tests, or results of imaging and pathological studies. It may be difficult to compare studies reporting different periods of prevalence (e.g., timepoint, weekly, monthly, lifetime) [5].
\nPain conditions in children and adolescents have gradually become the focus of the scientific literature in recent years. The occurrence of pain in children evidence as indicates childhood or adolescence pain can predict adulthood pain. Children with pain discontinue their education or become withdrawn. Physical inactivity is a possible result. Low back pain, headache, and abdominal pain are the most common types of childhood and adolescence pain.
\nThe reported 1-year incidence of low back pain in children and adolescents varies from 11.8 to 33.0% (median, 22.4%), while the 1-month prevalence varies from 9.8 to 36.0% (median, 22.9%). Since there are a lower number of studies on the prevalence of neck pain (49.0%) and upper back pain (30.0%), some doubt remains about the accuracy of these predictions. A systematic review of chronic pain epidemiology in children and adolescents (pain continuing for more than 3 months) has reported that the 1-month prevalence of chronic back pain was between 18.0 and 24.0% (median, 21.0%) [6, 7, 8, 9]. In addition to these predictions obtained from systematic reviews, the 1-month prevalence of low back pain was reported to be 37.0% in more than 400,000 children and adolescents aged 11–15 years [10, 11, 12, 13, 14].
\nThe predicted 1-month headache prevalence in children and adolescents is 26.0–69.0% (median, 47.5%) in systematic investigations. Swain et al. have reported this figure to be 54.1% in a survey based on 312 schools [15]. Recurrent abdominal pain (at least three episodes that limit the child’s functions for at least 3 months) is the focus of most childhood and adolescence pain studies [16]. Recurrent abdominal pain prevalence has been reported as 0.3–19.0% (median, 8.4%) [17] and 3.8–41.2% (median, 12.0%) [7]. Childhood and adolescence monthly multiple pain prevalence was 12.1–35.7% (median, 23.9%).
\nFurther studies on pain epidemiology in children and adolescents are still required to evaluate the effect of age on the pain prevalence. The effects of the increased pain rates in childhood and adolescence and of the transition to adolescence on the incidence and prevalence of pain conditions are not clear at present [17].
\nSpinal pain, and especially low back pain, is a common problem that most people experience at a certain point in their lives. The lifetime prevalence of low back pain is reported to be between 51.0 and 84.0%. There are many studies on the epidemiology of low back pain compared to other pain conditions.
\nPredictions for the 1-year first low back pain events varied between 6.3 and 15.4% (median, 10.9%) in one review [18] and between 13.5 and 26.2% (median, 19.9%) in others [11, 16, 18, 19, 20, 21, 22]. Predictions of the 1-year incidence of low back pain events (including patients with previous episodes) vary between 1.5 and 38.9% (median, 20.2%). Many people who experience activity-limiting low back pain recover quickly [23], but some have recurrent pains [24]. Predictions for the 1-year recurrence vary between 24.0 and 80.0% (median, 52.0%) [18].
\nImportant information is present on the prevalence of low back pain. The 1-month prevalence is predicted to be between 24.0 and 49.5% (median, 36.8%) [25]. The prevalence of thoracic spine pain varies between 1.4 and 34.8% (median, 18.1%) [26], while the 1-month prevalence of neck pain varies between 15.4 and 45.3% (median, 30.4%) [27].
\nChronic low back pain (CLBP) is usually defined as low back pain continuing for more than 12 weeks [28]. The prevalence of CLBP in the general European population has been predicted as 5.9–18.1% (median, 12%).
\nNeuropathic pain has been defined by the International Association for the Study of Pain as “pain caused by a lesion or disease of the somatosensory nervous system” [29]. It is differentiated from other inflammatory conditions by characteristic signs and symptoms such as “burning” or “freezing,” numbness, tingling, or “pins and needles” sensations [29]. There are only a few studies on the incidence and prevalence of neuropathic pain in the population.
\n\n
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Musculoskeletal system disorders are among the most common causes of disability and incapacity, especially in the elderly [32]. Upper extremity pain is also common; monthly shoulder pain symptoms are present in 18.6–31.0% (median, 24.8%) of adults [33]. Monthly knee pain prevalence in adults is 13.0–28.0% (median, 20.5%) [34]. The prevalence was 28.0, 15.0, and 14.0% for foot ankle and toe pain in a review combining the reported figures [35].
\nChronic diffuse pain is defined as the presence of chronic pain with diffuse localization [36]. The American College of Rheumatology defines chronic diffuse pain as bilateral axial skeleton pain continuing for 3 months or more in its 1990 guidelines [37]. Fibromyalgia diagnostic criteria have been defined using the same definition [38]. Some fibromyalgia prevalence figures are 2.0–5.0% in the USA, 0.7% in Denmark, and 10.5% in Norwegian women. Fibromyalgia is a clinical diagnosis similar to other chronic pain conditions, and the lack of a clear definition therefore limits the comparison of prevalence predictions. The diagnostic criteria for fibromyalgia in the American College of Rheumatology’s 2010 Criteria include the evaluation of diffuse pain together with the other symptoms (such as fatigue and cognitive symptoms) to develop a more specific case definition [38]. Future studies on the epidemiology of fibromyalgia are likely to increase the accuracy of current prevalence predictions by using this definition [39].
\nOnly a few studies have evaluated the prevalence of pain conditions, and most have focused on the prevalence of low back pain. Palmer et al. [40] reported that the 1-year prevalence of low back pain increased by 12.7% over a 10-year period from 1987 to 1997, and this increase was associated with gender, age group, social class, and residence area. A contrasting decrease in the 1-month prevalence from 26.1 to 22.6% has been reported over a 7-year period in another study [41]. The chronic low back pain frequency In the USA has increased from 3.9 in 1992 to 10.2% in 2006 [42].
\nPain, and especially chronic pain, creates a significant burden for patients and their families. It adversely affects the general health perception, significantly inhibits daily activities, is associated with depressive symptoms, and significantly and negatively affects the relationships and interactions with others. The World Health Organization Global Burden of Disease uses the term “disability” to assess the potential for non-mortality-related disease. They define disability as any short- or long-term loss of health [43]. Disability-adjusted life years (DALYs) and years lived with disability (YLDs) are needed to measure and compare the limitations of a wide range of disorders associated with pain. Pain-related disorders that are characterized or defined by the presence of pain (low back pain, neck pain, other musculoskeletal disorders, migraine, and falls) constitute 5 of the top 10 conditions responsible for YLDs in the world. Acute low back pain has caused 83 million DALYs, and according to the effects of the chronic types of back pain, this constitutes 10.7% of all YLDs [43]. Neck pain and migraine/headache each account for about 24 million DALYs. Other musculoskeletal disorders are responsible for 28 million DALYs and traumas due to falls for 19 million DALYs. Other important contributions include osteoarthritis (17 million DALYs) and accident-related injuries (13 million DALYs) [44]. These results for 1990 and 2010 supersede all previously published Global Burden of Disease results.
\nIt is difficult to determine the factors initiating pain episodes in the population as studies only specify an estimate due to the differences in the methodology and reporting. It is necessary to focus on the main risk categories such as age, gender, social group, and individual factors. Generally, there is no evidence for the risk factors of pain. Future studies that evaluate all aspects of the pain experience from both the individual and the population point of view are needed. These studies must employ multidimensional methods in the case of psychosomatic pain.
\nStudies on pain in children and adolescents have shown that females generally suffer more pain than males. The relationship between pain and gender is clear in adults. Females report more severe pain, more frequent pain, and longer-lasting pain than males in most studies. However, it is not known whether this gender difference is due to underlying biological pain mechanisms or the effects of psychological and social factors.
\nAs regards age, the prevalence of some pain disorders such as back pain increases from childhood to adolescence. The effect of age is not relevant on the pain prevalence in the elderly as some studies report that it increases, while others report it decreases with age. The effect also varies by gender and the pain location [41]. It is believed that musculoskeletal pain is most common in adults in the employment age and the prevalence therefore decreases from the middle of the sixth decade [41]. However, recent studies have shown that pain continues to be a widespread and serious problem in the elderly. The prevalence of chronic pain in the active elderly (>65 years) varies between 25.0 and 76.0%, while it is much higher in the sedentary elderly at between 83.0 and 93.0% [41].
\nThe role of social factors continuously increases throughout life [45]. The socioeconomic status is usually measured by the complex created by education, income, and occupation. Many studies in children and adolescents have evaluated the relationship between pain and socioeconomic status, but there is some evidence of a conflict between these studies [7]. An inverse relationship is present between the socioeconomic status and pain prevalence in adults. The data show that lower educational status, low income, and being unemployed are related to increased pain prevalence [5].
\nMore recently, pain prevalence studies have been conducted in populations of various cultural, ethnic, and socioeconomic statuses. Native Americans, Alaska Natives, and Aboriginal Canadians have been found to have a higher prevalence of pain than the general population of the United States [45]. Studies in Africa have found the 1-year prevalence of pain to be 33.0% in adolescents and 50.0% in adults [46]. This value is higher than reported in studies conducted in most Western countries (mean prevalence of 38.1%) [25]. However, it is difficult to make a definite comparison due to the differences in methodology. Another study based on the World Bank Human Development Index has reported the prevalence of chronic pain to be 24.8% in less developed countries and 28.1% in more developed countries [47].
\nVarious individual risk factors have been associated with the development of pain disorders. The demands of employment, lack of job security, an immobile job position, dissatisfaction with work, low levels of social support at work, and vibrating bodily work conditions have been associated with various occupational factors that lead to musculoskeletal pain. Individual lifestyle factors that create health problems such as smoking and obesity can also play a role in the development of pain disorders [34]. The psychosocial variables believed to influence the pain prevalence include stress, anxiety, lack of sleep, depression, low self-confidence, and the presence of chronic health problems (irregular heartbeat, dizziness, pain, cardiovascular problems, gastrointestinal discomfort, erectile dysfunction, feeling of lump or pressure in the throat, chest problems, hallucinations, and double vision).
\nPain and disorders that are not clearly attributable to an organic disease are called somatoform disorders. These disorders can be an expression of untreated mental pain and life experiences resulting from serious loss, profound personal injury, and disrespect. These symptoms occur in almost all humans, but they can become a serious problem in 4–20% of the population [1, 2, 3, 5, 6, 7, 8, 9, 48].
\nGenetic research on pain is increasing, and chronic pain is being seen as a classic example of the gene–environment interaction [49]. It is generally believed that the first trigger of chronic pain syndromes are inflammatory processes or nerve trauma. Once chronic pain develops, the pain intensity and response to analgesics are also quite variable. However, evidence is lacking regarding the influence of genetic effects and the interaction with psychosocial environmental factors as regards the development of chronic pain. Patients with chronic pain feel that the cause is life challenges, but the disease also makes life more difficult. This contradictory approach is related to the importance attributed to the pain. A person with intrapsychic conflict may not have to express the problem verbally and may have to use the organs to do so (alexithymia).
\nThe presence of enhancers in the environment: one example is caring for one’s wife when she has pain but not caring when she does not. Other examples are as follows: finding comfort by suffering pain for a bad action; excessive interest in pain and then relaxing when the test results are normal (somatization); and feeling sure an illness is present and changing physicians frequently (hypochondriasis).
\nType A persons (hasty, impatient, hyperactive) perceive pain more easily than type B persons (calm, cool), probably due to autonomic hyperactivity. The risk of hypertension and coronary heart disease is 3–5 times higher in type A persons [50, 51].
\nThe cumulative cost of chronic pain to the patient, the health-care system, and the economy is huge. In Australia, with a population of 22.7 million, the total annual cost of chronic pain was estimated as 34.3 billion dollars in 2007 or 10.847 dollars per person [52]. The total cost in Europe is estimated as 1.5–3.0% of the European GDP [53]. In the USA, approximately 100 million adults have been affected by chronic pain in 2008, including joint pain and arthritis [54]. The total cost in 2010 was 560–635 billion dollars. The annual cost of pain is higher than that of heart disease (309 billion dollars), cancer (243 billion dollars), and diabetes (188 billion dollars) [53, 55, 56, 57, 58, 59, 60].
\nThe term psychosomatic means the person. It combines two basic components, including the mind and the body. The reason is that physical complaints are at the forefront. However the research will investigate if there are any physical symptoms to explain such bodily complaints.
\nThere is no medical illness, and this is a definable psychiatric disorder.
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Pain experts still do not know whether pain is one of the senses or an experience. The fact that pain can be learned and that it can be affected by beliefs, expectations, and emotional states is quite important in its diagnosis and treatment. It is known that psychological disorders increase pain, while extreme fear, stress, and shock decrease pain. Many studies have reported that cultural norms and expectations play a major role in feeling pain and the related behavior.
\nThe gate control system that monitors pain is especially influenced by neurotransmitter modulation that is associated with cortical stimulants in anxious subjects. The lack of an adequate 5-hydroxyl tryptamine (5HT) level in the synapse disturbs pain perception and decreases the pain threshold and pain tolerance [50, 51].
\nThe physical and psychological problems of the person described under the following titles also play a role in psychosomatic pain.
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Emotional crises and chronic distress can lead to various psychosomatic complaints. The whole organism can be affected and the effects are therefore not listed here.
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Somatoform pain disorder is characterized by intense and agonizing pain that is subjectively felt in a part of the body for at least 6 months and that cannot be reasonably explained by a physical disorder or physiological event. The onset of the pain is related to a significant problem that has created serious emotional and/or psychosocial stress, conflict, or trauma. The increased interest in the person and the medical care received are the possible gains from the disorder. When compared with somatization disorders, these pains are long-lasting, and the patient focuses on them. The differential diagnosis of pain syndromes requires differentiation of organic physical pain from histrionic processing.
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The absence or modification of physical functions without a physical cause is usually the result of an intrapsychic conflict and can lead to psychogenic paralysis, coordination disorders, tremors, and myoclonus (muscle twitching).
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Major depression affects the entire body including the metabolism and the musculoskeletal system. Inactivity or pain syndromes may be present. The pain, lack of exercise, social withdrawal, smoking habits, and malnutrition lead to significant difficulties in the patient’s life.
\nThe patient comes to the doctor because of the somatic complaints as he/she has usually not noticed the depression: it is therefore not an independent disease. It is possible to determine the real psychic etiology during the examination if retrospective evaluations are also performed. Neurobiological studies have demonstrated that somatic symptoms are associated with brain dysfunction that is also responsible for depression. Psychological pain and emotional pain have been shown to cause activation of the same sites as physical pain stimuli on MR investigations.
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The documentation of neuroendocrine abnormalities in cases of depression and pain have revealed the parallel course of the functional changes in depression and pain and the hypothalamus-hypophysis-adrenal axis with excess production of corticotropin-releasing hormone (CRH). It is also known that a deficiency in the serotonin and norepinephrine monoamines can play an important role in the decreased inhibition of pain pathways and the development of somatic symptoms in depression [50, 51].
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Masked depression can have various symptoms. Urogenital system symptoms include dysuria, painful urination and defecation, signs of urinary and fecal incontinence, functional prostate problems (prostatitis), and bladder dysfunction in women without additional genital muscle weakness. There may also be upper abdominal discomfort, bloating, colic-like abdominal discomfort, stomach pain, and constipation [51].
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Negative emotions such as fear and anger permanently weaken our immune system and defense. The risk of catching infections such as influenza increases many times, and wounds heal slower and in a worse manner.
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Sleep problems are common when pain is present. Lack of sleep affects both the social life and performance of the patient. Fatigue can lead to depression and accidents. Sleep disorders have various signs such as difficulty falling asleep, waking up frequently and quickly, long period of staying awake during the night, being irritable, superficial sleep, loud and irregular snoring, leg restlessness, waking up early in the morning, and disturbing thoughts. Anger and hopelessness can also have a strong effect on sleep disorders.
\nIt has recently been found that our brain is active in a very special manner during sleep. The brain sends impulses, produces active substances, and is involved in coding and storing data 24 h a day with its 100 billion nerve cells, and it is the organ that benefits most from a good night’s sleep. This has been demonstrated with decreased brain capacity when we get little sleep. The first sign of cerebral fatigue is difficulty with concentrating and performing coordinated tasks such as driving or tasks that require a great deal of attention. We then become irritated and feel pain because of the related fatigue.
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Snoring during sleep is present in 10–30% of adults and it is usually not dangerous. However, pauses in respiration during sleep are an indicator of sleep apnea syndrome, which affects approximately 3 million people in Germany alone. The most common type of the disorder is “obstructive sleep apnea.” The pharyngeal muscles relax excessively and do not let air pass, leading to a pause in the respiration during sleep. This breathing problem goes on for about 2 min, usually with explosive snoring, and the subject then starts to breath normally again. In severe cases, these periods of paused respiration can recur hundreds of times every night. These patients are usually prone to falling asleep during the day, and the muscles can be weak and painful.
\nThe most important diagnostic step in the diagnosis of sleep apnea syndrome is talking to the patient and family. In case of increased sleep apnea suspicion, the next step is a sleep laboratory investigation. Electrodes record the ECG, blood pressure, and brain waves; observe movements of the eyes and legs; measure the oxygen content of the blood; and record each snoring and breathing sound during this test.
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Our breathing becomes quite shallow in case of stress, depression, or sadness. The lungs receive less oxygen and can provide less oxygen to the blood, increasing the risk of infection. Pneumonia is five times more common in the elderly than in healthy subjects.
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Somatization disorder and pain syndromes develop after a heart attack in approximately 30% of all patients. A heart attack is experienced as a “spontaneous infarction.” Physician appointments are frequently avoided and the recommended medication is not used. This increases the risk of new infarction development two to four times when added to the biological changes in the metabolism.
\nThe infarction risk is increased several times (deaths due to a heart attack are four to five times more common in depression patients). The more severe the somatization disorders, the worse the prognosis of a heart disorder. Factors such as emotional stress, dissatisfaction with work and the partner, anxiety, and long-term stress increase the heart attack risk more than classical risk factors such as smoking and high blood pressure.
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The reason for white hair is mineral deficiency in the hair and scalp, and there can be several causes: decreased nutrition due to age, acidity or nutritional disorder, and psychological reasons. Mineral intake is decreased with fear or stress, resulting in hair loss or white hair.
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The metabolism slows down and the body functions deteriorate during stress. Free radicals that attack the skin cells and slow down the regeneration of the natural protective layer are created. The skin ages faster and spots develop. The face appears stressed.
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Gastric absorption becomes difficult in patients with repressed emotions, anger, or anxiety. The stomach becomes tense with stress and anger, leading to increased gastric acid secretion. This in turn causes heartburn and can result in gastric ulcer, bloating, nausea, and cramps. Many subjects suffer from irritable stomach or irritable colon. Psychological components also play a role. Excitement and anxiety increase irritable stomach or irritable colon symptoms.
\nIt is very important to investigate the relationship of psychosomatic pain in many patients admitted between orthopedics and traumatology clinics. In this context;
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Depression leads to a long period of internal stress and increased muscle tension, causing a predisposition to motility disorders, immobility, and arthritis. The patient usually sees an orthopedist before going to a neurologist.
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Patients suffering from depression for a long time can be exposed to fractures more commonly as the mineral content of bone is decreased. Heavy psychological burdens can significantly decrease the blood oxygen content in the elderly as the breathing becomes superficial. The cells cannot receive adequate nutrition and renewal deteriorates. Inflammation and arthritis can develop in the joints.
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Orthopedic surgeons believe that back pain is the result of emotional problems, not organic ones, in most cases. If job dissatisfaction is high, the person feels overwhelmed and does not seek solutions to change the circumstances, leading to a high risk of pain as the spinal system reacts very strongly to mental stress.
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Approximately 6% of Germans are affected by neuropathic pain. At least 20% of the patients at pain centers suffer from neuropathic pain syndromes. Peripheral neuropathies can commonly develop after postherpetic neuralgia or trauma. Generalized neuropathies include those due to chemotherapy and diabetic neuropathies.
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Nociception is the perception of pain. The responsible receptors are called nociceptors. Nociceptors are present in all the pain-sensitive tissues of the body as the free nerve ends of sensitive neurons of the spinal cord. Nociceptors trigger various types of pain according to their localizations:
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Myofascial pain has local causes. Individual muscles are affected more intensely by chronic pain than muscle groups. Individual muscles contain trigger points that signify very sensitive areas. Overloading of the muscle can prevent excess calcium intake and the related muscle relaxation and therefore create localized oxygen deprivation.
\nLong-term contraction treatment becomes difficult in the case of myofascial pain. The sensitivity leads to the corresponding areas creating a perception of pain even with mild contact, and these areas are therefore called trigger points.
\nExcessive stress on such muscles can be the result of muscle damage, inadequate nutrition, hormonal imbalances, immobility, muscle weakness, hypothermia, contractions, and neurological damage.
\nMyofascial pain mainly develops in the
Psychotherapy in the treatment of pain can be explained as follows:
\nPatients with psychosomatic pain may have had a very disturbing experience in the past. This event can create links with the memory and senses, and the traumatic disorder may occur at any time when the present experience is once again dominant. Psychological stress may also cause physical illnesses. Within the soul-soma-soul sequence, an ever-growing chain of causes can be present. Life-threatening diseases such as cancer or myocardial infarction and the relevant medical interventions can also lead to mental trauma.
\nTraumas, undesirable social experiences, accidents, or stressful experiences have been scientifically proven to be the most common triggers and the causes of many physical disorders, pain, and other illnesses without a physical cause. Changes related to psychological trauma in the brain can now be scientifically demonstrated by imaging (such as fMRI, a magnetic resonance method) and other diagnostic methods. Brain structure and brain metabolism are altered by the corresponding changes in the autonomic nervous system during the stress process. The entire spectrum of mental trauma should be considered. Psychotherapy for psychosomatic pain due to trauma is performed in three stages and with multiple sessions.
\nThere are some fundamental differences between type I and type II trauma.
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The results of type II trauma include the consequences of childhood and long-term adult violence. This violence can also be mental as seen with coldness, extreme violence and indifference, emotional wounds, and frequently repeated trauma, especially in childhood. Type II traumas can lead to complex symptoms as in borderline disorder with comorbidities.
\nA three-stage model is used for the treatment of disorders that are psychosomatic or accompanied by pain. These stages consist of
It is not mandatory to include each stage in every treatment process. Stabilization is the foundation of all treatment steps. It can be integrated into other therapeutic methods such as specific interventions for trauma, behavioral therapy, systemic treatment, or deep psychology-associated therapies.
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Sufficient time is allocated to get to know the patient and to create the basis of trauma therapy while building a mutually trusting relationship.
\nThe stabilization stage creates a foundation for a common understanding of the clinical picture. An objective viewpoint is obtained about the emotions as much as possible. Anxiety and depression are responses to psychologically disturbing experiences and the emotional dissociations within the patient or those around him/her. An emergency state plan is developed together with the patient at this stage. Relaxation techniques such as meditation or Jacobson’s progressive muscle relaxation are then used as guides for self-passivation techniques, and the patient thus learns the relevant methods to use the powers of healing within.
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Behavioral therapy for the confrontation state has been specifically designed to treat phobias and anxiety. Cognitive behavioral therapy and especially systematic desensitization techniques are used. The aim is to re-evaluate the traumatizing event.
\nSpecial therapeutic procedures for trauma are also used such as eye movement desensitization and reprocessing (EMDR), imagery rescripting and reprocessing therapy (IRRT), and psychodynamic imaginative trauma therapy (PITT).
\nEye movement desensitization and reprocessing has been developed by the American psychologist Francine Shapiro (* 1948). The literary translation indicates “eye movement, desensitization and reprocessing.” This method is not hypnosis. When the patient focuses on an especially stressful stage of the emotionally traumatic experience, the therapist slowly asks the patient to perform rhythmic eye movements by slowly moving his fingers and gives the patient confidence. This stimulates cerebral processes. The aim is to decrease and even eliminate the fears produced by the memories. More than 20 controlled studies have shown the long-term effect of EMDR. EMDR is also included in guidelines as a preferred procedure.
\nImagery rescripting and reprocessing therapy has been developed by the American clinical psychologist Mervin Smucker (* 1949). A traumatizing experience is created together with the therapies as though it had happened today. The patient imagines how he dealt with it in the past and how he is dealing with it now. The patients no longer view themselves as helpless victims and feel they are the designers of the condition who can act and maintain control even in the most difficult situations. We can think about it as deleting an old text and then writing over it.
\nPsychodynamic imaginative trauma therapy has been developed by psychoanalyst Luise Reddemann (* 1943). PITT is based on the idea that people have self-regulating powers to cope with disturbing events even after terrible experiences. Establishing a supportive therapeutic relationship is very important for such self-understanding, and it is also important for helping oneself. At the heart of PITT’s therapeutic approach is the “internal phase” that the person is currently acting on.
\nIn this mental “imaginary” game, the patient confronts the previous ego states with the therapist’s support. Understanding the multiple egos in the consciousness comes from a scientific and philosophical tradition that has been present in all cultures for a thousand years, and the treatment relies on a systemic approach to therapy. With PITT, you can experience the injustice you have experienced and the area where you feel helpless from a safe distance. The patient learns to accept this part of his personality and to relax and make others relax at the same time. The patient also learns to heal his/her emotional wounds and therefore regains his/her confidence.
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This is the moment the joy of living and the relevant control are regained and understood.
\nThe aim of this stage is to gradually integrate the traumatic experiences of the patient into his/her consciousness. These events are parts of the person’s life, and control over life and social integration can be achieved once again by confronting these events [4, 61, 62, 63, 64].
\nOur understanding of the epidemiology of psychosomatic pain is limited to a small number of studies that provide estimates of the prevalence in the general population. These studies are usually difficult and costly to conduct and require very large samples. The way data is collected and reported may also have an impact on the estimates with various results obtained from studies that depend on surveys, interviews, or clinical investigations. Large-scale population-based studies can provide richer data related to the age and gender distribution of pain, and assessments over extended periods of time can provide comprehensive information about the incidence and risk factors. Epidemiological studies in various cultural, social, and ethnic groups can clarify the effects and also the interactions between the individual and population-based risk factors. Physicians should be able to understand the information related to psychosomatic pain, search the relevant information available, and perform research on the subject themselves.
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A lot of natural and engineering materials can be categorized as visco-elastic materials, such as rock, elastomers, and rubbers. In engineering applications, it is very important to understand and simulate the visco-elastic deformation of rough fractures. For example, in hydrocarbon extraction, we need to accurately simulate the visco-elastic deformation of rock fractures to predict production rates. In biomedical devices, we need to simulate the visco-elastic deformation of artificial joints to evaluate safety and effectiveness. Due to the geometrical complexity of rough fractures and the time-dependent properties of engineering materials, it is extremely difficult to obtain closed-form mathematical solutions. Thus, numerical models are required to simulate the time-dependent behavior of rough fractures.
The boundary element method (BEM) has been extensively used in solving rough surface contacting problems for distinct advantages compared with the traditional finite element method (FEM). First, it only requires discretization and calculation on the boundaries of the calculation domain, which is two-dimensional. On the contrary, FEM requires discretization and calculation for the whole calculation domain. As a result, to achieve the same stress calculation resolution, BEM requires much fewer numbers of elements and therefore, much less calculation time. In addition, since all the approximations are limited to the boundary, BEM has better stress calculation accuracy compared with FEM.
In recent years, researchers have been combining the BEM and fast numerical algorithms to achieve more efficient numerical simulations for contact problems. Stanley and Kato [1] published the first paper using the fast Fourier Transform (FFT) method to calculate the elastic deformation of rough surfaces under normal stresses. The FFT method makes the BEM simulation more efficient because FFT turns complicated convolution into simple matrix multiplication. Later, Polonsky and Keer [2] proposed the conjugate gradient (CG) method and combined it with the FFT method to further improve the efficiency. Liu et al. [3] improved the drawbacks of the FFT method proposed by Stanley and Kato [1]. Then, the CG and FFT methods have been applied to simulate plastic and visco-elastic deformations of rough fractures. Jacq et al. [4] and Sahlin et al. [5] considered perfect plasticity to simulate deformations of rough metal surfaces; and Wang et al. [6] considered strain-hardening plasticity.
For visco-elasticity, Chen et al. [7] first used the CG and FFT method to simulate visco-elastic deformations of rough fracture surfaces. They simulated three load-driven scenarios: rigid sphere indenting into PMMA surface, contact area evolution under constant load, and contact area evolution under harmonic cyclic load. Spinu and Cerlinca [8] applied different cut-off values for contact pressure to account for the plastic deformation of contacting asperities.
However, it appears that there is not much work that systematically simulates the visco-elastic deformation of rock fracture surfaces. Kang et al. [9] reported that for Musandam limestone fractures, the effect of mechanical compression on rock fracture time-dependent deformation is non-negligible, and should be systematically investigated. In addition, previous articles suggest that the fracture surface geometry has a significant effect on fracture time-dependent deformation. Therefore, we should systematically study the effect of surface geometry on rock fracture visco-elastic deformations.
Brown [10] proposed a simple probabilistic model to describe rock fracture surface geometry. In his model, the rock fracture surface geometry can be completely described by three key parameters: the Hurst exponent, the root mean square (RMS) roughness, and the mismatch length scale. In this research, his model will be used to generate synthetic fracture surface pairs, and the three key parameters will be changed systematically. The numerical method proposed by Chen et al. [7] will be used to simulate the visco-elastic deformation of synthetic fracture surfaces.
This chapter is organized as follows. Section 2 introduces and explains the principles and procedures of the numerical method. Section 3 provides a detailed example. The method for generating synthetic rough surfaces is introduced, and the effect of surface geometry parameters on the creep deformation is shown and discussed. Section 4 mentions the limitations of this method. Section 5 summarizes the findings.
Before explaining the method for visco-elastic deformation calculation, it is essential to introduce the method for elastic deformation calculation. The author has developed an in-house numerical code, which is similar to the algorithm proposed by Polonsky and Keer [2]. In this section, only the key mathematical concepts will be shown; the details can be found in their work [2]. It is worth noting that only the compressive stress (stress normal to the fracture surface) is considered; the shear stress (stress parallel to the fracture surface) is not considered.
First, the aperture (surface gap between two rough surfaces) distribution h (x,y) needs to be defined:
where h0(x,y) is the initial aperture distribution, ue(x,y) is the elastic deformation of fracture surfaces, and δ is the rigid body displacement between two surfaces under applied normal stress. Here, compressive stress and fracture closure are defined as positive.
The boundary conditions are expressed as:
where p(x,y) is the contacting stress (normal to the surface) acting on location (x,y). Eqs. (2) and (3) indicate that the contacting stress is larger than zero at contacting regions, and is zero at non-contacting regions.
Boussinesq and Cerrutti [11] stated that the vertical displacement ue (x,y) can be calculated as:
where p(x′, y′) is the normal pressure acting on location (x′, y′), K is the influence matrix, which represents the normal displacement at location (x, y) caused by unit normal pressure acting on location (x′, y′), and ue (x,y) is the elastic displacement at location (x, y). The influence matrix K can be expressed as:
where G is the shear modulus, and v is the Poisson’s ratio.
As mentioned in the introduction section, it is difficult to obtain the analytical solution for rough surface deformation under normal stress. However, the numerical solution can be obtained. To obtain the numerical solution, the fracture surface area needs to be discretized into rectangular grids:
where xi, yj are x and y coordinates, respectively; N and M are total number of grids in x- and y-direction, respectively; and Δx and Δy are the grid dimensions in x- and y-direction, respectively. After discretization, the aperture distribution function and boundary conditions can be expressed as:
Love [12] first discretized Eqs. (4) and (5) as:
where
As mentioned before, Stanley and Kato [1] first the FFT method to solve Eq. (11) to make the calculation more efficient. The FFT method turns complicated convolution into simple matrix multiplication. By using the FFT method, Eq. (11) becomes:
where IFFT represents the inverse of Fourier transform. The FFT method reduces the number of operations from N2 * M2 to N*M*log(N*M) [1]. Therefore, when N and M are large, the FFT method can greatly reduce the calculation time.
The force balance over the entire fracture surface needs to be satisfied:
Eqs. (8)–(10), (14), and (15) are solved iteratively using the CG method proposed by Polonsky and Keer [2].
As described before, Chen et al. [7] first combined the FFT and CG method to simulate visco-elastic deformations of rough fractures. The author has developed an in-house numerical code, which is similar to the algorithm described by Chen et al. [7]. In this section, only the key mathematical aspects will be introduced; the rest can be found in their work [7].
In this simulation, the rock materials are assumed to be linear viscoelastic. Therefore, is it essential to introduce the concept of linear viscoelasticity first. For linear viscoelastic materials, the stress/strain response scales linearly with the strain/stress input, and the behavior follows the rule of linear superposition. Mathematically, the stress/strain at time t can be expressed as:
where J(t) and E(t) are the creep compliance function and the relaxation modulus function, respectively. J(t) represents the time-dependent strain change with a step change in stress, and E(t) represents the time-dependent stress change with a step change in strain. Based on Eq. (17), the Boussinesq and Cerrutti equation can be modified to represent linear viscoelasticity by adding the creep compliance function:
In Eq. (18), the creep compliance function J(t-τ) replaces the term 1/2G. Rearranging Eq. (18) gives:
Eq. (19) cannot be solved analytically for rough fracture surfaces. However, if the time integration term can be de-coupled with the pressure integration term, Eq. (19) will become a linear equation system, and can therefore be solved numerically. To de-couple the time integration term, the time duration t is discretized into Nt time steps. The time interval is uniform, and is termed as Δt. The time interval is assumed to be sufficiently small that the pressure distribution field within each time interval does not change. In addition, based on the fundamental theorem of calculus, the term ∂p(x′, y′, τ) dτ/ ∂τ can be substituted by a finite difference p(x′, y′, τ + dτ) – p(x′, y′, τ). Therefore, Eq. (19) becomes:
where α = 1, 2, …, Nt.
In addition, within each time interval, the pressure distribution field does not change. Therefore, the pressure distribution field can be removed from the integration term:
Eq. (21) indicates that the time integration term is de-coupled with the pressure integration term. The pressure integration term can then be discretized, similar to Eq. (11). From Eqs. (4), (5), and (11), the Boussinesq equation can be discretized as:
Based on Eq. (22), the integration part of Eq. (21) can then be discretized:
Therefore, Eq. (21) can be discretized as:
To implement FFT, Eq. (24) can be decoupled into two equations:
and
Eq. (26) can be solved by the FFT method, similar to Eqs. (13) and (14):
Within each time step, Eqs. (8)–(10), (15), (25), and (27) are solved using the CG method. The pressure distribution field is obtained and stored. Then, a new time step will be added (α will be increased by one), and the new deformation and pressure fields will be solved based on the historical pressure fields. Figure 1 summarizes the main calculation algorithm based on the above mathematical concepts.
Summary of the calculation algorithm (Kang et al. [
Before simulating visco-elastic deformations of rough rock fractures, it is essential to validate the numerical code against analytical solutions. In this research, the analytical solutions provided by Radok and Lee [14] will be used for validation. In their solutions, a rigid spherical indenter is indented into a flat visco-elastic surface; and the visco-elastic models for the flat surface are the Maxwell and Standard Linear Solid (SLS) model. Figure 2 illustrates the geometry setup for the analytical solution, and Figure 3 shows the concepts of the Maxwell and SLS model.
Geometry setup for the analytical solution (Kang et al. [
Concepts of the Maxwell and SLS model (Kang et al. [
The Maxwell model consists of a dashpot and a spring. The dashpot represents viscosity, with a viscosity of η; the spring represents elasticity, with a shear modulus of G. Under constant stress σ0, the strain can be obtained:
Eq. (28) implies that under constant stress, the strain rate does not change with time. The creep compliance can be expressed as:
Another parameter, the relaxation time T, is defined as:
In the numerical simulation, Eq. (29) will be implemented into Eq. (27), and the displacement and pressure field will be solved as described in Sections 2.1 and 2.2. For the geometry setup shown in Figure 2, the analytical solution for the contacting region radius and pressure field can be obtained:
and
where p is the pressure field, t is the total time duration, υ is the Poisson’s ratio, and r is the distance from the center of the contacting region.
The SLS model consists of one dashpot and two springs. The dashpot represents viscosity, with a viscosity η; the two springs represent elasticity, with a shear modulus of G1 and G2, respectively. Under constant stress σ0, the strain can be obtained:
The creep compliance J(t) is expressed as:
The relaxation time T is defined as:
In the numerical simulation, Eq. (34) will be implemented into Eq. (27), and the displacement and pressure field will be solved as described in Sections 2.1 and 2.2. For the geometry setup shown in Figure 2, the analytical solution for the contacting region radius and pressure field can be obtained:
and
where p is the pressure field, t is the total time duration, υ is the Poisson’s ratio, and r is the distance from the center of the contacting region.
Johnson [15] solved Eqs. (31), (32), (36), and (37), Figures 4 and 5 compare the numerical and analytical solutions for the SLS and Maxwell models, respectively. The solid lines are the numerical solutions obtained by the author, and the dashed lines are the analytical solutions solved by Johnson [15]. In Figures 4 and 5, rh is the contacting region at time zero, ph is the maximum contacting pressure at time zero, and T is the relaxation time.
Numerical and analytical solutions for the SLS model (Kang et al. [
Numerical and analytical solutions for the Maxwell model (Kang et al. [
Figures 4 and 5 indicate the deviation between the numerical and analytical results is less than 10%. Therefore, the numerical code can be used to simulate the visco-elastic deformations of rough fractures. For the two validation cases, the numerical simulation accuracy is not strongly dependent on the total number of elements, but on the time interval Δt. The deviation between numerical and analytical solutions will be smaller if the time interval Δt is reduced.
In this chapter, synthetic fracture surface pairs are generated based on Brown’s model [10]. Brown’s probabilistic model assumes that the surface is self-affine, and the surface height distribution follows Gaussian distribution [10]. The surface geometry can be completely described by three parameters: the Hurst exponent H, the mismatch length λc, and the root mean square roughness RMS.
Mathematically, a self-affine surface is defined as:
where H is the Hurst exponent, z is the surface height, and ε is a constant for scaling at the x-direction. The H value is between 0 and 1, and it describes local roughness. A smaller H value corresponds to a rougher local surface profile.
The H value can be obtained from the power spectrum of surface height. The power spectrum of a surface can be obtained by decomposing the surface profile into a series of sinusoidal waves via Fourier transform, and each sinusoidal wave has its own amplitude A, wavelength λ, and phase. Figure 6 shows the schematic of the decomposition process. The power (A2) is defined as the square of the amplitude A; and the plot of power versus the wavelength number (the inverse of wavelength, which is 2π/λ) is defined as the power spectrum. Figure 7 shows the schematic of power spectrum.
Schematic of wave decomposition via Fourier transform (Kang et al. [
Schematic of a power spectrum (Kang et al. [
For a self-affine fracture surface, the power C (=A2) can be related to the wavelength number q (=2π/λ) as:
In Figure 7, the q has an upper bound and a lower bound. For the lower bound, qmin = 2π / λL, where λL is the surface dimension; for the upper bound, qmax = 2π/λ1, where λ1 is the surface measurement resolution.
The second parameter is the mismatch length, λc. As illustrated in Figure 6, each wave component has its own wavelength λ. Glover et al. [16] and Brown [10, 17, 18] stated that for most natural rock joints, the two surfaces have relative shear displacements. At long wavelengths, the wave components match well; at short wavelengths, the wave components are not identical. Based on the above observation, Brown [10] proposed a parameter: critical wavelength λc, which is also called the mismatch length scale. Brown [10] assumed that above the mismatch wavelength, the decomposed wave components of two surfaces match perfectly; they have the same amplitudes, wavelengths, and phases. On the contrary, below the mismatch wavelength, the decomposed wave components of two surfaces do not match; they have the same amplitudes and wavelengths, but the phases are independent. Figure 8 illustrates the concept of the mismatch wavelength.
Illustration of the mismatch wavelength (Kang et al. [
The third parameter is the root mean square roughness, RMS. It represents the absolute scale of surface asperity elevation. Mathematically, the RMS is defined as:
where C is the power, q is the wavelength number, and σ is the RMS value. When generating the synthetic surface, the surface heights are normalized by its own RMS value, σini, and then multiplied by the designated RMS value, σdes:
where zini is the initial surface height and zdes is the surface height after linear scaling. In this chapter, only the key mathematical concepts of Brown’s [10] model is introduced; other details can be found in [10].
Brown [10] measured the Hurst exponent H, mismatch length λc, and RMS for 23 natural rock joints. His measurement results imply that the H value is normally between 0.5 and 1.0; the normalized λc value (λc/fracture profile length) is normally between 0.02 and 0.2, and the normalized RMS value (RMS/fracture profile length) is normally between 0.005 and 0.015. Based on the above conclusion, seven synthetic fracture surface pairs are generated, with different H, λc, and RMS values. Table 1 summarizes the parameters of the seven synthetic surface pairs. It is worth noting that surface pair No. 2 is the reference surface pair.
Surface Pair No. | Profile length L (mm) | H | λc | RMS | ||
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λc/L | Absolute value (μm) | RMS/L | Absolute value (μm) | |||
1 | 10 | 0.6 | 0.1 | 1000 | 0.005 | 50 |
2 | 10 | 0.8 | 0.1 | 1000 | 0.005 | 50 |
3 | 10 | 1.0 | 0.1 | 1000 | 0.005 | 50 |
4 | 10 | 0.8 | 0.2 | 2000 | 0.005 | 50 |
5 | 10 | 0.8 | 0.3 | 3000 | 0.005 | 50 |
6 | 10 | 0.8 | 0.1 | 1000 | 0.010 | 100 |
7 | 10 | 0.8 | 0.1 | 1000 | 0.015 | 150 |
The parameters of the seven synthetic surface pairs.
Table 1 shows that between surface pairs 1, 2, and 3, the H value is varied; between surface pairs 2, 4, and 5, the λc value is varied; between surface pairs 2, 6, and 7, the RMS value is varied. For each surface pair, the aperture distribution field can be plotted. Figure 9 plots the aperture fields for surface pairs 1, 2, and 3; Figure 10 plots the aperture fields for surface pairs 2, 4, and 5, and Figure 11 plots the aperture fields for surface pairs 2, 6, and 7.
Aperture fields for different H values (Kang et al. [
Aperture fields for different λc values (Kang et al. [
Aperture fields for different RMS values (Kang et al. [
Based on Figures 9–11, we have the following observations:
According to Figure 9, when H increases, the average and standard deviation of the aperture decreases;
According to Figure 10, when λc deceases, the average and standard deviation of aperture decreases;
According to Figure 11, the average and standard deviation of aperture scales linearly with the RMS value.
Table 2 summarizes the mean and standard deviation of aperture for each surface pair. In the numerical code, each calculated aperture field (shown in Figures 9–11) is considered as the initial aperture field.
Surface pair No. | H | λc (μm) | RMS (μm) | Average aperture (μm) | Standard deviation of aperture (μm) |
---|---|---|---|---|---|
1 | 0.6 | 1000 | 50 | 63.41 | 14.29 |
2 | 0.8 | 1000 | 50 | 37.30 | 8.57 |
3 | 1.0 | 1000 | 50 | 21.89 | 5.14 |
4 | 0.8 | 2000 | 50 | 55.94 | 15.01 |
5 | 0.8 | 3000 | 50 | 66.10 | 20.12 |
6 | 0.8 | 1000 | 100 | 74.59 | 17.15 |
7 | 0.8 | 1000 | 150 | 111.89 | 25.72 |
The average and standard deviation of seven synthetic surface pairs.
The author uses the Maxwell model to calculate the visco-elastic deformation of seven synthetic surface pairs. The mechanical properties of Vaca Muerta Shale measured by Mighani et al. [19] are used as the input parameters, and those properties are summarized in Table 3.
Parameters | Value |
---|---|
Shear modulus, G (GPa) | 7.0 |
Poisson’s ratio, υ | 0.25 |
Viscosity, η (GPa*s) | 2.0 |
Relaxation time, τ = η / G (s) | 2.857 |
Input parameters for the Maxwell model.
Before showing the results, two parameters are introduced: macroscopic stress σ and contact ratio:
The macroscopic stress σ = total force applied to the fracture/fracture surface area;
Contact ratio = 100 * (the number of grids in contact/total number of grids).
Figures 12 and 13 show the mean aperture and contact ratio evolving with time for seven synthetic surface pairs, respectively. The total time duration is 2τ, and the macroscopic stress σ = 10 MPa. The initial changes of the mean aperture and contact ratio correspond to fracture elastic deformation.
Mean aperture changing with time (Kang et al. [
Contact ratio changing with time (Kang et al. [
Based on Figures 12 and 13, several conclusions can be drawn:
As H decreases, the mean aperture increases, and the contact ratio increases slower with time;
As RMS increases, the mean aperture increases, and the contact ratio increases slower with time;
As λc increases, the mean aperture increases, and the contact ratio increases slower with time.
Under current macroscopic stress, time duration, and surface parameters, the contact ratio is generally less than 9.5%.
Table 4 summarizes the effect of surface parameters on the mean aperture and contact ratio.
Parameters | Average aperture | Contact ratio | ||
---|---|---|---|---|
Initial value | Decrease rate | Initial value | Increase rate | |
H↓ | ↑ | ↑ | ↓ | ↓ |
λc↑ | ↑ | ↑ | ↓ | ↓ |
RMS↑ | ↑ | ↑ | ↓ | ↓ |
Effect of surface parameters on the mean aperture and contact ratio.
Figure 14 shows the contact region and local contacting stress evolution of surface pair
Contact region and local contacting stress evolution before and after the creep stage (Kang et al. [
The author also uses the SLS model to calculate the visco-elastic deformation of seven synthetic surface pairs. The mechanical properties of Vaca Muerta Shale measured by Mighani et al. [19] are used as the input parameters, and those properties are summarized in Table 5.
Parameters | Value |
---|---|
Shear modulus, G1 (GPa) | 7.0 |
Shear modulus, G2 (GPa) | 7.0 |
Poisson’s ratio, υ | 0.25 |
Viscosity, η (GPa*s) | 2.0 |
Relaxation time, τ = η / G2 (s) | 2.857 |
Input parameters for the SLS model.
Figures 15 and 16 show the mean aperture and contact ratio evolving with time for seven synthetic surface pairs, respectively. The total time duration is 5τ, and the macroscopic stress σ = 10 MPa. The total time duration is extended from 2τ to 5τ to show the time-decaying creep rate. The initial changes of the mean aperture and contact ratio correspond to fracture elastic deformation.
Mean aperture changing with time (Kang et al. [
Contact ratio changing with time (Kang et al. [
Based on Figures 15 and 16, several conclusions can be drawn:
As H decreases, the mean aperture increases, and the contact ratio increases slower with time;
As RMS increases, the mean aperture increases, and the contact ratio increases slower with time;
As λc decreases, the mean aperture increases, and the contact ratio increases slower with time.
Under current macroscopic stress, time duration, and surface parameters, the contact ratio is generally less than 7.0%.
Under current macroscopic stress, time duration, and surface parameters, the creep rate decreases significantly with time. This is mainly because the SLS model assumes an exponentially decaying creep rate.
Table 6 summarizes the effect of surface parameters on the mean aperture and contact ratio.
Parameters | Average aperture | Contact ratio | ||
---|---|---|---|---|
Initial value | Decrease rate | Initial value | Increase rate | |
H↓ | ↑ | ↑ | ↓ | ↓ |
λc↑ | ↑ | ↑ | ↓ | ↓ |
RMS↑ | ↑ | ↑ | ↓ | ↓ |
Effect of surface parameters on the mean aperture and contact ratio.
Figure 17 shows the contact region and local contacting stress evolution of surface pair
Contact region and local contacting stress evolution before and after the creep stage (Kang et al. [
In this numerical method, the contacting asperities deform visco-elastically, and there is no upper limit on the local contacting stress. For some synthetic surfaces, the contacting stress in a few cells exceed 1.3 GPa. In reality, under such high contacting stresses, the asperities may deform plastically. Ignoring the plastic deformation will underestimate the contact ratio and overestimate the local contacting stress. In addition, asperity breakage is ignored in this numerical method. Under high contacting stresses, asperities may break, which will further change the contacting regions and the contacting stress distribution [20]. Furthermore, the effect of shear stress on fracture visco-elastic deformations is also not considered. In engineering applications (especially in oil and gas production), fractures may subject to shear stress, which may significantly change the visco-elastic deformations.
This chapter explains how to use the boundary element method to calculate visco-elastic deformations of rough fractures. Fast numerical algorithms (CG and FFT) are implemented to further improve the efficiency. In addition, one example, which investigates the effect of surface geometry on visco-elastic deformations of rough rock fractures, is given. In this example, the author builds two in-house numerical codes: one code generates synthetic fracture surface pairs using Brown’s probabilistic model [10], and the other simulates the visco-elastic deformations of the synthetic surface pairs. Seven synthetic surface pairs are generated by systematically changing the values of the root mean square roughness RMS (50 μm, 100 μm, and 150 μm), mismatch length λc (1000 μm, 2000 μm, and 3000 μm), and Hurst exponent H (0.6, 0.8, and 1.0). Then, the author simulates the visco-elastic deformation of the seven surface pairs by using the Standard Linear Solid (SLS) and the Maxwell model. The following key conclusions can be drawn:
As RMS increases, the average aperture increases, and the contact ratio increases slower with time;
As λc increases, the average aperture increases, and the contact ratio increases slower with time;
As H decreases, the average aperture increases, and the contact ratio increases slower with time;
For the macroscopic stress (10 MPa), time durations (5τ for the SLS model and 2τ for the Maxwell model), and the surface roughness parameters (RMS between 50 and 150 μm, λc between 1000 and 3000 μm, H between 0.6 and 1.0) used in the examples, the contact ratio is less than 9.5%.
While the results are useful, future work would be helpful. First, more surface roughness parameter values can be used so a quantitative relationship between surface parameters and contact ratio or average aperture can be obtained. In addition, other visco-elastic models, such as the Burgers model and the Power Law model, can be implemented. Furthermore, in this simulation, the plastic deformation of contacting asperities is not considered. As a result, the local contacting stress may be overestimated. The plastic deformation of contacting asperities can be considered so the results can be more realistic. Last but not least, the effect of shear stress can be simulated to make the results more applicable.
The authors gratefully acknowledge Farrokh Sheibani, Stephen Brown, Herbert Einstein, and John Germaine for their useful suggestions.
The authors declare no conflict of interest.
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