2 Capitulo Metodos Numericos

NUMERICAL METHODS

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DUBAN CASTRO FLOREZ

PETROLEUM ENGINEERING

6th SEMESTER

2010

DEFINITION

• The numerical methods are useful alternative procedures to solve

math problems for which complicates the use of traditional

analytical methods and, occasionally, are the only possible solution.

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1.1 NUMERICAL APPROXIMATIÓN

• Numerical approximation is defined as X * a figure that represents a

number whose exact value is X. To the extent that the number X * is

closer to the exact value X, is a better approximation of that number.

Examples:

▫ 3.1416 is a numerical approximation of ,

▫ 2.7183 is a numerical approximation of e,

▫ 1.4142 is a numerical approximation of 2, and

▫ 0.333333 is a numerical approximation of 1/3.

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1.1 APPROXIMATIONS

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1.1.2 SIGNIFICATIVES FIGURES

The number of significant figures is thenumber of digits t, which can be usedwith confidence to measure a variable,for example, three significant figures onthe speedometer and 7 significantfigures on the odometer.

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1.1.3 EXACTITUDE AND PRECISION

Exactitude = refers to the number of significant figures

represents a quantity.

Precision = refers to the approach of a number or measure

the numerical value is supposed to represent.

The numerical methods should provide sufficiently accurate

and precise solutions. The error term is used to represent

both the inaccuracy and to measure the uncertainty in the

predictions

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1.1.4 ALTERNATIVES SELECTION

The use of numerical methods in engineering is not trivial, because

it requires choosing between:

-Several alternative numerical methods for each type of problem

-Several technological tools

There are different ways to approach problems from one person to

another, depending on:

-The level of participation in the mathematical modeling of the

problem

-Ingenuity and creativity to confront and resolve

-The ability to choose, according to criteria and experience

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Type of problem to solve:

-Roots of equations

-Systems of simultaneous linear equations

-Interpolation, differentiation and integration

-Ordinary Differential Equations

-Partial Differential Equations

-Other (not covered in this course, seen in other subjects)

Team:

-Supercomputer

-PC

-Graphing calculator

-Scientific pocket calculator

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COMPUTER TOOLS ARE

MACHINES "IDIOTS" THAT JUST DO IT

TO BE ORDERED, HOWEVER, THE

THE FIGURES DO TEDIOUS CALCULATIONS

VERY FAST AND VERY GOOD, NO HASSLE.


SOFTWARE :Program Development"C" language-Basic-Fortran

Using mathematical software:-Maple-MatLab-Mathcad-Mathematica.

Managing spreadsheets on PC:-Excel-LotusExpedited handling of a graphing calculator

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• Numerical method: there is no better, but if the favorites

-Extent of application

-Friendliness

-Stability

-Fast convergence

-Required number of initial values

Be taken into account, besides

-Model complexity

-Turbulence data

-Ingenuity and creativity

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1.TYPE OF PROBLEM

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2. MATHEMATIC MODEL

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3. NUMERIC METHOD

4.EQUIPMENT

• Computer

• Calculator

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5.SOFTWARE

-Software development

-Mathematical software

-Spreadsheet

-Graphing calculator

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-Many times, computers cut decimal numbers between e17 and 12th decimal thus introducing a rounding error.-For example, the value of "e" is known as 2.718281828 ... to infinity.-If we cut the number 2.71828182 (8 significant digits after the decimal point) we are obtaining or failuree= -2.71828182 2.718281828 = 0.000000008 ...-However, as we do not consider the number that was cut was greater than 5, then we should have let the number as 2.71828183, in which case the error would onlye = 2.118281828 = -0.000000002 -2.11828183 ..

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2. ROUNDING ERROR

ROUNDING RULES

-If the digit to round greater than 5 increases by one who is left: 8236 = 8.24-If the digit to round is less than 5 increases do not make changes which is: 8231 = 8,23-If the digit is 5 to remove a number other than 0 which is increasing: 8.2353 = 8.24-If the digit to be deleted is 5 followed by 0 you look at the number below, if odd couple and if you increase left: 8.23503 = 8.24; 8.23502 = 8,23

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• The total numerical error is defined as the sum of

the rounding and truncation errors introduced in the

calculation.

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3. TOTAL NUMÉRIC ERROR

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Colin maclaurin

4. TAYLOR`S SERIE

• Here, n! is the factorial n and f(n)(a) indicates the n-esima- derivative of f in a.If this series converges for all x belonging to the interval(a-r, a + r) and the sum is equal to f (x), then the functionf (x) is called analytic. To check whether the seriesconverges to f (x), is often used an estimate of theremainder of Taylor's theorem. A function is analytic ifand only if it can be represented by a power series, thecoefficients of this series are necessarily determined inthe formula for the Taylor series.

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4. TAYLOR`S SERIE

• A numerical simulation is a mathematical recreation of anatural process. Using numerical simulations we studythe physical, engineering, economic and even biological.

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5. NUMERIC SIMULATION

http://es.wikipedia.org/wiki/Simulaci%C3%B3n_num%C3%A9rica






















• In the '60s, the development of reservoir simulation, was

aimed at solving problems of oil fields in three phases. The

recovery methods were simulated depletación included

various forms of pressure and pressure maintenance.

Developed programs operating on large computers

(Mainframe) and used cards for data entry.

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6. RESERVOIR SIMULATION NUMERIC

http://modelaje-de-pozos.blogspot.com/2009/05/simulacion-numerica-de-yacimientos_140.html

























• During the 80s, the range of simulation applications fordeposits continued to expand. The description of sitesmoved toward the use of geostatistics for describingheterogeneities and provide a better definition of the field.

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Recent advances have focused mainly on the following points:

-Description of reservoir.- Naturally fractured reservoirs.- Hydraulic Fracturing.- Horizontal Wells.

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BIBLIOGRAPHY


http://modelaje-de-pozos.blogspot.com/2009/05/simulacion-numerica-de-

yacimientos_140.html

http://www.google.com/














































http://www.google.com/

2 Capitulo Metodos Numericos

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