The document discusses the importance of mathematics in petroleum engineering. It outlines how mathematical modeling allows for virtual experimentation and analysis of complex phenomena. Areas like differential equations, numerical analysis, statistics, and harmonic analysis play key roles in modeling processes for applications such as image processing, telecommunications, data transmission, and oil prospecting. Developing strong links between mathematics and petroleum engineering is important for addressing challenges through interdisciplinary collaboration.

1. ESCUELA DE INGENIERÍA DE PETROLEOS

2. ESCUELA DE INGENIERÍA DE PETROLEOS CONTENT 4. BIBLIOGRAPHY 3. APPLICATIONS 2. Mathematical Model Benefits 1. INTRODUCTION

3. ESCUELA DE INGENIERÍA DE PETROLEOS 1- INTRODUCTION The challenges facing today's science and engineering are so complex that can only be solved by interdisciplinary relationship in which mathematics plays a very important role. The math, science and engineering have a long and close relationship that is crucial and increasingly important for them. The math is now required very significantly by the technology of communications, finance, and business. Scientific progress, in all its branches, requires close and strong relationship with mathematics.

4. ESCUELA DE INGENIERÍA DE PETROLEOS The main themes that emerge consistently in the relationship of mathematics to science has been the following: • Mathematical modeling: adequate description of a scientific phenomenon in a mathematical framework allows the use of powerful tools for the construction of effective algorithms for the characterization, analysis and prediction of the phenomenon. Mathematical models allow virtual experiments whose real analogues would be expensive, dangerous or impossible, make unnecessary the actual destruction of an airplane, spread a deadly virus or witness the origin of the universe.

5. ESCUELA DE INGENIERÍA DE PETROLEOS • Complexity and dimension as the simple fact is almost never requires complex models. However more complex models eventually lead to fundamentally different problems, not just bigger and more complicated. It is impossible to characterize disordered systems with the same tools that are suitable for systems of good behavior. • Uncertainty: Uncertainty is inevitable though, ignore it can be justified when studying physical processes isolated, small-scale and well understood. This is not true for large-scale systems with many components, such as the atmosphere and oceans, chemical process where there is no way to determine exactly the sequence of reactions and of course in biological and medical applications, or systems that rely on human involvement.

6. ESCUELA DE INGENIERÍA DE PETROLEOS • Multiple scales: the need to model or compute on multiple scales arises when disparate scales (space, time or both) contribute simultaneously to an observable result. For example, in turbulent combustion, the shape of the chamber is as important as are the small fluctuations of temperature controlled chemical reactions. • Large data sets: the enormous data sets that are generated today in many scientific areas to be exhibited, analyzed and scrutinized to discover the hidden order and patterns. Not all large data sets have the same characteristics, their quality varies from very specific to those consistently noisy, often with variations on the same set. Also, large sets of data to be analyzed in real time, such as guided surgery or control of an aircraft, pose important challenges in mathematics.

7. ESCUELA DE INGENIERÍA DE PETROLEOS Predictive modeling of complex relationships. Predictive Modeling financial behavior of the markets for the oil industry. Computer and statistics applied to petrophysics, for example in the Nuclear Magnetic Resonance Logs. Hybrid systems for controlling oil production.  Design and operation of processes assisted by mathematical models.

8. ESCUELA DE INGENIERÍA DE PETROLEOS The importance of strong links between mathematics and Petroleum Engineering should be evident from the examples presented, which are but a small sample of a much larger. Unfortunately, in the country a clear shortage of people able to bridge the gap between mathematics and science. This is recognized in developed countries where active policies are established to overcome this deficiency, due to its high economic impact. In our country, this shortage reaches alarming levels and frankly one of the challenges we face is how to overcome the underlying educational problem. It is clear that mathematicians and math students should be able to understand scientific problems, and that researchers and science students should understand the strength and scope of mathematics.

9. ESCUELA DE INGENIERÍA DE PETROLEOS The following fields of mathematics, with interdisciplinary activities that involve applications and are areas of vacancy in our country under development achieved by some research groups in related topics, should have good potential for development. The support is divided equally among the five areas of mathematics described here.

10. ESCUELA DE INGENIERÍA DE PETROLEOS Possible fields of application are:  Analysis of variance: a statistical technique to measure the importance of various sources of variability. It applies, for example, heterogeneous oil reservoirs.  Time series: models used in production studios for predicting volatility. Spatial statistics: applied to the exploration of materials and hydrocarbons. Statistical processing of images. Stochastic processes: between applications of this area of probability theory can be mentioned the following: Stochastic Differential Equations: Physical apply (eg. In statistical mechanics) and are of great importance in Finance. Markov Fields: used in image processing. Seismic trace analysis: applied in the exploration of oil reservoirs.  Design and control of networks of reservoir fluid flow.

11. ESCUELA DE INGENIERÍA DE PETROLEOS The differential equations play an essential role in the modeling of physical and chemical processes. They are also used in industry to control production processes, for computer simulation of processes, etc.. The effective resolution of differential equations requires, in almost all cases, the use of numerical methods. The design and analysis of its effectiveness is one of the central issues of Numerical Analysis. Note that a type of numerical methods widely used, especially in engineering, is the finite element. This kind of methods requires the use of advanced mathematical techniques. The aim is to encourage participation both in developing mathematical models of processes such as those mentioned above as in the numerical solution of the same.

12. ESCUELA DE INGENIERÍA DE PETROLEOS Important developments in Harmonic Analysis, Functional and Approximation Theory, such as Atomic Decomposition Signal Spaces, Theory of Wavelets, Theory of Marcos, Transform Analysis, Boundedness of Operators, Theory of Sampling, Characterization of Function Spaces as Models in Engineering, Approximation and Multiresolution Spaces, Time Frequency Analysis, are the appropriate theoretical framework for treatment, among others, the following technology applications:  Image Processing.  Celular and satellite phones - Telecommunications  Data Transmission - Transmission of Images over the Internet  Oil Prospecting

13. ESCUELA DE INGENIERÍA DE PETROLEOS www.conicet.gov.ar/becas/archivos_gral/.../matematica.doc Stewart, James. "Calculus, Early Transcendent." 4 ed. Tr. Andrew Sesti. Mexico, Ed Thomson, 2002. p. 1151