Studies in the geometry of the discrete plane

This thesis is an attempt to initiate the development of a discrete geometry of the discrete plane H = {(qmxo,qnyo); m,n e Z - the set of integers}, where q s (0,1) is fixed and (xO,yO) is a fixed point in the first quadrant of the complex plane, xo,y0 ¢ 0. The discrete plane was first considered by Harman in 1972, to evolve a discrete analytic function theory for geometric difference functions. We shall mention briefly, through various sections, the principle of discretization, an outline of discrete a alytic function theory, the concept of geometry of space and also summary of work done in this thesis

Effect of Feeding Rate in Polymer Extrusion ”

Starve feeding of single screw extruder was described as an important means of improving the performance characteristics of the extruder. In addition to such improvement with versatility, the starve feeding technique also may affect the mechanical properties of the extrudate since the heat transfer an(l mixing characteristics in the starve fed and Hood fed extruders are not the same. Since the material is more loosely packed in the channels of the starve fed extruder, there may be greater bed mobility and uniformity. Further, the. thermal an(l shear induced degradation are also less since possibilities of developing local high temperatures are less compared to a densely compacted extruder bed. This study has been undertaken mainly to explore the effect of feeding rate on the mechanical properties of rubber and plastic extrudates since the effect of feeding rate has not been analysed from this angle so far.

Design and Programming of Cathodic Protection for SHIPS

In order to minimize the risk of failures or major renewals of hull structures during the ship's expected life span, it is imperative that the precaution must be taken with regard to an adequate margin of safety against any one or combination of failure modes including excessive yielding, buckling, brittle fracture, fatigue and corrosion. The most efficient system for combating underwater corrosion is 'cathodic protection'. The basic principle of this method is that the ship's structure is made cathodic, i.e. the anodic (corrosion) reactions are suppressed by the application of an opposing current and the ship is there by protected. This paper deals with state of art in cathodic protection and its programming in ship structure

Development of Electrochemical and Fluorescent Sensors

Chemical sensors have growing interest in the determination of food additives, which are creating toxicity and may cause serious health concern, drugs and metal ions. A chemical sensor can be defined as a device that transforms chemical information, ranging from the concentration of a specific sample component to total composition analysis, into an analytically useful signal. The chemical information may be generated from a chemical reaction of the analyte or from a physical property of the system investigated. Two main steps involved in the functioning of a chemical sensor are recognition and transduction. Chemical sensors employ specific transduction techniques to yield analyte information. The most widely used techniques employed in chemical sensors are optical absorption, luminescence, redox potential etc. According to the operating principle of the transducer, chemical sensors may be classified as electrochemical sensors, optical sensors, mass sensitive sensors, heat sensitive sensors etc. Electrochemical sensors are devices that transform the effect of the electrochemical interaction between analyte and electrode into a useful signal. They are very widespread as they use simple instrumentation, very good sensitivity with wide linear concentration ranges, rapid analysis time and simultaneous determination of several analytes. These include voltammetric, potentiometric and amperometric sensors. Fluorescence sensing of chemical and biochemical analytes is an active area of research. Any phenomenon that results in a change of fluorescence intensity, anisotropy or lifetime can be used for sensing. The fluorophores are mixed with the analyte solution and excited at its corresponding wavelength. The change in fluorescence intensity (enhancement or quenching) is directly related to the concentration of the analyte. Fluorescence quenching refers to any process that decreases the fluorescence intensity of a sample. A variety of molecular rearrangements, energy transfer, ground-state complex formation and collisional quenching. Generally, fluorescence quenching can occur by two different mechanisms, dynamic quenching and static quenching. The thesis presents the development of voltammetric and fluorescent sensors for the analysis of pharmaceuticals, food additives metal ions. The developed sensors were successfully applied for the determination of analytes in real samples. Chemical sensors have multidisciplinary applications. The development and application of voltammetric and optical sensors continue to be an exciting and expanding area of research in analytical chemistry. The synthesis of biocompatible fluorophores and their use in clinical analysis, and the development of disposable sensors for clinical analysis is still a challenging task. The ability to make sensitive and selective measurements and the requirement of less expensive equipment make electrochemical and fluorescence based sensors attractive.

Praxisrelevante Auswirkungen des Vorschlages für eine EU-Verordnung KOM/2000/7/final, 2000/0212(COD) des europäischen Parlaments und des Rates von der Kommission der Europäischen Gemeinschaften auf die Verkehrsplanung des öffentlichen Personennahverkehrs

Die Arbeit behandelt den Vorschlag für eine EU-Verordnung KOM/2000/7/final, 2000/0212(COD) des europäischen Parlaments und des Rates von der Kommission der Europäischen Gemeinschaften als Grundlage einer Marktöffnungsverordnung und welche Veränderungen sich dadurch in Deutschland einstellen werden. Ausschreibungen von Verkehrsleistungen werden zunehmen. Die Ausschreibungsarten werden sich in ländlichen Regionen von denen in Verkehrszentren unterscheiden. In der Region werden sich Bedarfslösungen stärker durchsetzen. Kürzungen von Verkehrsleistungen werden hier stärker ausfallen als in den Zentren und damit zu einem kleineren Leistungsvolumen führen. Aufgrund des geringen Leistungsumfangs gibt es weniger Interessenten. Bei Standardausschreibungen werden deshalb auch häufig die Varianten der beschränkten oder die freihändige Vergabe gewählt. Funktionale Ausschreibungen haben nur eine untergeordnete Bedeutung. In den Verkehrszentren sind die Lose größer und damit für viele Anbieter interessant. Die Verkehrszusatzleistungen sind zudem komplexer. Standardausschreibungen in öffentlicher Vergabeart werden sich hier vermutlich als Norm durchsetzen. Die VOL/A wird sicherlich ihre Bedeutung und ihren dafür notwendigen Regelungsumfang in Deutschland als deutsches oder als europäisches Recht behalten. Ob der empfehlende Charakter der DIN EN 13816 Norm „ÖPNV: Definition, Festlegung von Leistungszielen und Messung der Servicequalität“ erhalten werden kann und nicht als Steuerungselement zur Standardisierung im ÖPNV beitragen wird, ist dabei zu bezweifeln. Durch diese Wettbewerbspflicht wird der Aufgabenträger zum Besteller von Verkehrsleistungen. Damit geht die Verkehrsplanung in die Verantwortung des Aufgabenträgers über und gerät stärker in den Einflussbereich der Politik. Die strategisch abstrakte und die konkrete Verkehrsplanung wachsen für den Normfall der Standardausschreibung zusammen. Die Hoffnung auf eine bessere Netzintegration und eine Standardisierung des ÖPNV Angebots und der ÖPNV Qualität entsteht. Es entwickelt sich dadurch aber auch die Gefahr der Abhängigkeit des Nahverkehrsangebots von der derzeitigen Haushaltslage oder der Interessenlage der Politik. Kontinuität in Angebot und Qualität werden zu erklärten Planungszielen. Der Verkehrsplaner auf der Bestellerseite muss die Planung in Ausschreibungsunterlagen umsetzen. Dies erfordert erweiterte Kompetenzen in den Bereichen Betriebswirtschaft, Logistik, Jura, Informatik und Führungskompetenzen. Ausbildende Institutionen müssen darauf bereits im Vorfeld der Umsetzung reagieren. Durch die zeitliche Verzögerung der Umsetzung der Planung durch die Ausschreibungsschritte sind in der Verkehrsplanung längere Planungsvorlaufzeiten einzukalkulieren. Vorausschauender zu planen, wird dabei wichtiger. Auch eventuelle Fehler in der Planung sind nicht mehr so einfach zu korrigieren. Durch den gestiegenen Einsatz von Technologien in den Fahrzeugen besteht für den Verkehrsplaner dafür häufiger die Möglichkeit, Planungsänderungen auf ihre Wirksamkeit im Hinblick auf Attraktivität für den Fahrgast anhand von den ermittelten Fahrgastzahlen zu kontrollieren. Dasselbe gilt auch für Marketing- und Vertriebsmaßnahmen, wie für die Tarifpolitik. Die Zahlen stehen nicht nur für diese Rückkopplung zur Verfügung, sondern dienen auch als Planungsgrundlage für zukünftige Maßnahmen. Dem Planer stehen konkretere Zahlen für die Planung zur Verfügung. Ein Aspekt, der aufgrund der Sanktionsmaßnahmen bei Ausschreibungen an Bedeutung gewinnen wird, ist die Möglichkeit, Qualität von Verkehrsleistungen möglichst objektiv beurteilen zu können. Praxisrelevante Auswirkungen auf die Verkehrsplanung des öffentlichen Personennahverkehrs ergeben sich hauptsächlich durch die gestiegene Komplexität in der Planung selbst und den dadurch unverzichtbaren gewordenen Einsatz von Computerunterstützung. Die Umsetzung in Ausschreibungsunterlagen der Planung und die Kontrolle stellen neue Elemente im Aufgabenbereich des Verkehrsplaners dar und erfordern damit breiter ausgelegte Kernkompetenzen. Es werden mehr Verkehrsplaner mit breiterer Ausbildung benötigt werden. Diese Arbeit hat aufgezeigt, dass sich mit der Integration des Ausschreibungsgedankens in den Ablauf der Verkehrsplanung eine sprunghafte Entwicklung in der Planungstätigkeit ergeben wird. Aufgrund der in Zukunft steigenden Qualität und Quantität der Planungsgrundlagen und der ebenfalls gestiegenen Ansprüche an die Bewertungsparameter ergeben sich Veränderungen und neue Anforderungen auf diesem Gebiet, die in erster Linie für die Hochschulen und andere ausbildende Einrichtungen, aber auch für die Verkehrsplanung unterstützende Industrie.

Decentralized model reference control of flexible cable-stayed beam structures

A decentralized model reference controller is designed to reduce the magnitude of the transversal vibration of a flexible cable-stayed beam structure induced by a seismic excitation. The controller design is made based on the principle of sliding mode such that a priori knowledge

A comparative study of two Casas de la Cultura in Bogota: Tunjuelito and Usaquen

Es un estudio descriptivo/exploratorio de dos Casas de la Cultura en Bogotá (Tunjuelito/Usaquén), basado en la revisión de documentos y recolección de datos a través de herramientas cualitativas, con aspectos cuantitativos en la presentación de datos por medio de cuadros de comparación. Presenta resultados de las siguientes categorías: grupos de interés, aspectos administrativos y productos.

Les opinions séparées et leurs répercussions sur l’indépendance du juge international

Différents points de vue pour déterminer la portée du secret des délibérations dans certains tribunaux internationaux ont débouché sur le fait que les juges aient le droit de présenter des opinions séparées; alors que d’autres n’ont pas ce droit. En tenant compte du rôle et des objectifs des missions internationales, les juges devraient avoir le droit de présenter des opinions séparées, de la même façon que dans le système de common law et dans un grand nombre de tribunaux constitutionnels.Cependant, ces analogies ont joué un rôle marginal dans les travaux préparatoires du Statut de la Cour Permanente de Justice Internationale en 1920. D’autant que les Etats ne trouvaient pas  orrect qu’une opinión juridique d’un juge international soit condamnée a l’anonymat comme consequence du principe du secret des délibérations, ceci comme conséquenced’un «technicisme» relatif au fait que ladite opinion était contraire à la position majoritaire de la Cour au moment de voter le projet de la décision.Les règles générales de droit international public garantissent un pouvoir autonome au pouvoir judiciaire international. Selon les règles de procédure des tribunaux internationaux, les juges ont le droit de se prononcer avec une opinion séparée, même si ce droit ne se trouve pas typifié de façon expresse dans le Statut ou dans le traité constitutif de l’organisation. Cette règle est présumée à moins qu’il y ait eu une claire volonté des Etats dans le sens contraire.Le droit relatif aux opinions séparées peut être analysé sous la perspective des juges en tenant compte de leur droit à la liberté d’expression. En ce sens, un juge international peut avoir la liberté pour démontrer, de façon systématique, par le biais d’opinions séparées, les vides argumentatifs de la majorité, en évitant un style qui puissent être offensif envers ses collègues. Cette façon de s’exprimer est considéré inoffensive envers l’autorité judiciaire.Les effets positifs par l’absence, ou l’interdiction, d’opinions séparées, en relation avec l’indépendance des juges internationaux ne sont pas faciles à mettre de côté. Cependant, ce genre des mesures restrictives à la liberté d’expression n’est pas suffisamment effectif ni proportionné pour légitimer l’objectif du juge. Il y a des instruments bien plus effectifs y moins restrictifs qui mènent au même résultat (par exemple, un seul mandat, non renouvelable, des juges nternationaux).

The project of land restitution in Colombia: an illustration of the civilizing force of hypocrisy?

This article offers a theoretical interpretation of the dispositions on land restitution contained in the famous “Victims’ Bill”, which was debated in the Colombian Congress during the year 2008. The bill included specific mechanisms aimed at guaranteeing the restitution of land to victims of the Colombian armed conflict. At the time, the bill was endorsed by all the main political actors in the country –notably the government and the elites that support it, on the one hand, and victims’ and human rights organizations and other opposition groups, on the other–. The fact that the restitution of land to victims of the Colombian armed conflict was being considered as a serious possibility by all political actors in the country seemed to indicate the existence of a consensus among actors whose positions are ordinarily opposed, on an issue that has traditionally led to high levels of polarization. This consensus is quite puzzling, because it seems to be at odds with the interests and/or the conceptions of justice advocated by these political actors, and because the restitution of land faces enormous difficulties both from a factual and a normative point of view, which indicates that it may not necessarily be the best alternative for dealing with the issue of land distribution in Colombia. This article offers an interpretation of said consensus, arguing that it is only an apparent consensus in which the actors are actually misrepresenting their interests and conceptions of justice, while at the same time adopting divergent strategies of implementation aimed at fulfilling their true interests. Nevertheless, the article concludes that the common adherence by all actors to the principle of restorative justice might bring about its actual realization, and thus produce an outcome that, in spite (and perhaps even because) of being unintended, might substantively contribute to solving the problem of unequal land distribution in Colombia. Even though the article focuses in some detail on the specificities of the 2008 Bill, it attempts to make a general argument about the state of the discussion on how to deal with the issue of land distribution in the country. Consequently, it may still be relevant today, especially considering that a new Bill on land restitution is currently being discussed in Congress, which includes the same restitution goals as the Victims’ Bill and many of its procedural and substantive details, and which therefore seems to reflect a similar consensus to the one analyzed in the article.

Equilibrium Principle in Colombian Public Contracting

Public contracting in Colombia is conflicting and inefficient. It frequently leads to damage to State property. The Colombian legal system cannot assure efficient and transparent public contracting. The cause is the institutional environment characterized by high transaction costs. Colombian law worsens the process by recognizing the principle of economic equilibrium in public contracts. This principle increasese contract incompleteness and renders impossible the use of economic incentives to control the opportunism of the economic agents. The authors present the hypothesis that the economic equilibrium principle increases the conflictive nature of public contracting. They test the hypothesis empirically. The first section of the paper presents a summary of the literature on transaction costs economics, as well as the legal literature on the historical origin and the content of the economic equilibrium principle. The second section describes the methodology of the empirical study. The third section shows the empirical evidence of the effects that the economic equilibrium principle exerts over the public contracting. The last section presents the conclusions.

The voters of Colombia are free?

A fundamental principle of democracy is citizenship freedom. We suggest that a fair electoral contest is possible if a relationship between free will, electoral preferences and respect to the public institutions (constitutionalism) (section 1) exists. We focus on three illiberal practices that perturb the voter's decision: political clientelism and political markets (sections 2 to 4), media influence (which feeds on the voter's limited rationality and limited information) (section 5), and the suppression of opposition options (section 6). Later (section 8), we provide a brief balance and, additionally, we show how in Colombia the political system has missed opportunities to expand the voters' freedom. Our interpretation of the electoral process in Colombia is an appeal, supported on theoretical arguments and empirical evidence, to doubt about the voters' freedom. Also we make a call for more etudies.

Obligación política en el caso del genocidio armenio

La autoridad legítima del soberano y la obligación política del ciudadano constituyen el fundamento del poder del Estado, no obstante, una y otra se ven seriamente cuestionadas ante situaciones de extrema injusticia como lo es el Genocidio. En casos como este la obligación de obedecer la ley por parte del súbdito puede e, incluso, debe ser exceptuada y sustituida por los mandatos de la razón moral. El caso de un tribunal alemán que decide absolver al asesino del principal perpetrador del Genocidio Armenio, desobedeciendo de esta manera la ley penal de su país, apoya la tesis enunciada apoyándose en argumentos de la teoría contractualista de Thomas Hobbes y John Locke. La misma tesis es analizada desde la perspectiva del Juego Limpio, en este caso la conclusión no es tan clara, sin embargo, plantea un nuevo problema sobre el Derecho de los Pueblos que deberá ser abordado en otra instancia.

La ejecución de las sentencias urbanísticas

El Estado español tiene más de 100.000 construcciones ilegales. En la actualidad, la ejecución de las sentencias en materia urbanística no es eficaz. Existe, por tanto, una situación de inejecución de las sentencias urbanísticas. Un ciudadano puede construir una casa al amparo de una licencia. Sin embargo, a veces, esta licencia es ilegal porque infringe el plan urbanístico aplicable y incluso la propia ley de urbanismo. La consecuencia jurídica de la ilegal es la obligatoriedad del derribo de lo ilegalmente construido. Lo mismo sucede ante la declaración de ilegalidad de un plan urbanístico. En este caso, el juez o tribunal estimará la ilegalidad del plan y el fallo deberá ser ejecutado. La solución no es un fácil porque la demolición o derribo de la construcción ilegal trae consigo el pago de una indemnización a los propietarios que han edificado al amparo de una licencia ilegal.El derecho a la ejecución de sentencias es prescrito por el artículo 24 de la Constitución española. Este artículo 24 garantiza el derecho de todos ciudadanos a una tutela judicial efectiva. El Tribunal Constitucional español en sus sentencias número 67/1984 y 28/1989, reconocieron que el derecho a la ejecución de las resoluciones judiciales deriva del propio derecho a la tutela judicial efectiva. La justicia administrativa está regulada por la Ley 29/1998, del 13 de julio, reguladora de la jurisdicción contencioso-administrativa. En su artículo 103.1 de la Ley 29/1998, establece que los jueces y los tribunales de la jurisdicción administrativa son los competentes para hacer ejecutar las sentencias. Por otro lado, el artículo 117.3 de la Constitución Española otorga la competencia en exclusiva a los jueces y tribunales para hacer ejecutar las sentencias, como manifestación del principio de división de poderes.En último lugar, los valores que la legislación urbanística española incorpora, tales como, el desarrollo sostenible, urbanismo sostenible y la cohesión social, quedan inaplicados si las sentencias urbanísticas no se ejecutan.

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.

The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/ porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic - viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown-Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown-Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz-clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks. (C) 2003 Elsevier B.V. All rights reserved.