This document discusses numerical methods and their application in reservoir simulation. It defines numerical methods as procedures that can solve mathematical problems that are difficult to solve using traditional analytical methods. It also discusses numerical approximation, significant figures, error, Taylor series, numerical simulation, and the history and recent advances in reservoir simulation.

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NUMERICAL METHODS
DUBAN CASTRO FLOREZ
PETROLEUM ENGINEERING
6th SEMESTER
2010

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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 the
number of digits t, which can be used
with confidence to measure a variable,
for example, three significant figures on
the speedometer and 7 significant
figures 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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1.1.4 ALTERNATIVES SELECTION
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 COMPUTER TOOLS ARE
-PC MACHINES "IDIOTS" THAT JUST DO IT
TO BE ORDERED, HOWEVER, THE
-Graphing calculator THE FIGURES DO TEDIOUS CALCULATIONS
VERY FAST AND VERY GOOD, NO HASSLE.
-Scientific pocket calculator

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1.1.4 ALTERNATIVES SELECTION
SOFTWARE :
Program Development
"C" language
-Basic
-Fortran
Using mathematical software:
-Maple
-MatLab
-Mathcad
-Mathematica.
Managing spreadsheets on PC:
-Excel
-Lotus
Expedited handling of a graphing calculator

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1.1.4 ALTERNATIVES SELECTION
• 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

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4.EQUIPMENT
• Computer
• Calculator

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5.SOFTWARE
-Software development
-Mathematical software
-Spreadsheet
-Graphing calculator

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2. ROUNDING ERROR
-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 failure
e= -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 only
e = 2.118281828 = -0.000000002 -2.11828183 ..

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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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3. TOTAL NUMÉRIC ERROR
• The total numerical error is defined as the sum of
the rounding and truncation errors introduced in the
calculation.

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4. TAYLOR`S SERIE
Colin maclaurin

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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 function
f (x) is called analytic. To check whether the series
converges to f (x), is often used an estimate of the
remainder of Taylor's theorem. A function is analytic if
and only if it can be represented by a power series, the
coefficients of this series are necessarily determined in
the formula for the Taylor series.

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5. NUMERIC SIMULATION
• A numerical simulation is a mathematical recreation of a
natural process. Using numerical simulations we study
the physical, engineering, economic and even biological.
http://es.wikipedia.org/wiki/Simulaci%C3%B3n_num%C3%A9rica

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6. RESERVOIR SIMULATION NUMERIC
• 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.
http://modelaje-de-pozos.blogspot.com/2009/05/simulacion-numerica-de-yacimientos_140.html

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6. RESERVOIR SIMULATION NUMERIC
• During the 80s, the range of simulation applications for
deposits continued to expand. The description of sites
moved toward the use of geostatistics for describing
heterogeneities and provide a better definition of the field.
http://modelaje-de-pozos.blogspot.com/2009/05/simulacion-numerica-de-yacimientos_140.html

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6. RESERVOIR SIMULATION NUMERIC
Recent advances have focused mainly on the
following points:
-Description of reservoir.
- Naturally fractured reservoirs.
- Hydraulic Fracturing.
- Horizontal Wells.
http://modelaje-de-pozos.blogspot.com/2009/05/simulacion-numerica-de-yacimientos_140.html

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BIBLIOGRAPHY
http://es.wikipedia.org/wiki/Simulaci%C3%B3n_num%C3%A9rica
http://modelaje-de-pozos.blogspot.com/2009/05/simulacion-numerica-de-
yacimientos_140.html
http://www.google.com/