Numerical Methods In Petroleum Engineering: 2010

NUMERICAL METHODS are techniques by which mathematical problems can be formulated in such a way that can be solved using arithmetic. There are many types of numerical methods, and share a common characteristic: they invariably must make a number of tedious arithmetic.

The numerical methods are very powerful tools to solve problems. They can handle large systems of equations, nonlinearities and complex geometries common in engineering. It is also possible to use commercially available software containing numerical methods. The intelligent use of these programs depends on knowledge of the basic theory of these methods, plus there are many problems that can arise when using customized programs, knowing well the numerical methods can design their own programs and so do not buy expensive software. At the same time learn to know and control the rounding errors that are inseparable from the large-scale numerical calculations.

Numerical methods are a means to enhance understanding of mathematics, because delve into the issues that would otherwise be obscure, it increases your ability to understand and understanding in this matter.

Edited from: www.norte.uni.edu.ni/estudiantes/aproximaciones.doc

MODELING

a perception of the world can be described as a succession of phenomena. From the beginning of time man has sought to discover whether they understand them or not.

It is apparent that an interpretation of the world is necessary, which must be sufficiently abstract to avoid being affected by the dynamics of the world (small changes) and should be robust enough to represent the data and the world are related. A tool like this is called data model, which can represent more or less reasonable than any reality. The data model allows for abstractions of the world, allowing focus on the macros, without worrying about the specific, so our concern is focused on generating a representation scheme, and not the values of the data.

Edited from: http://carmuz.tripod.com/modelamiento/modelamiento.htm

COMPONENTS OF A MATHEMATICAL MODEL

Independent variable:

It is the one variable q q constantly changes its value alters the result in the same way the whole equation

Dependent variable:

It is the one variable q depends on the independent variable and change dramatically with changes made to q is the same.

DARCY´S LAW

Darcy's Law describes, based on laboratory experiments, the characteristics of the movement of water through a porous medium.

Darcy's Law is one of the cornerstones of soil mechanics. From the initial work of Darcy, a monumental work for the time, many other researchers have analyzed and tested this law. Through these later works has been determined and is applicable for most types of fluid flow in soils. In order to seepage of liquids at very high speeds and gas at very low speeds, Darcy's law becomes invalid.

In the case of water flowing in soils, there is overwhelming evidence for the purposes of verifying the validity of Darcy's law for soils ranging from silts to medium sands. It is also perfectly applicable in the clays, for steady flows.

For soils of higher permeability than the sand media, determined experimentally in the actual relationship between the gradient and velocity and porosity for each soil studied.

The mathematical expression of Darcy's law is:

Q=k*A*(h3-h4)/L=k*i*A

Where:

Q=expense, discharge or flow. [m3/s]

k=a constant known as Darcy permeability coefficient. Variable depending on the sample material.[m/s]

A=cross-sectional area of the sample[m2]

L=length in meters of the sample.[m]

i=(h3-h4)/L

h3=height as the water reaches a tube placed at the entrance of the filter bed.

h4=water height reaches a tube placed at the exit of the filter bed.

Edited from: http://es.wikipedia.org/wiki/Ley_de_Darcy

MATHEMATICAL APPROACH

The significant figures (or significant digits) represent the use of a level of uncertainty under certain approximations.

The use of these considers the last digit of approach is uncertain, for example, to determine the volume of a liquid using a graduated cylinder with a precision of 1 ml, implies an uncertainty range of 0.5 ml. It may be said that the volume of 6ml of 5.5 ml will be really to 6.5 ml. The previous volume is represented as (6.0 ± 0.5) ml. For specific values closer would have to use other instruments of greater precision, for example, a specimen finest divisions and thus obtain (6.0 ± 0.1) ml or something more satisfying as the required accuracy.

edited from: http://es.wikipedia.org/wiki/Cifras_significativas

SIGNIFICANT FIGURES

When using a number in a calculation, there must be assurance that can be used with confidence. The concept of significant figures has two important implications in the study of numerical methods.

1 .- The numerical methods obtain approximate results. Therefore, criteria must be developed to specify how accurate are the results obtained.

2 .- Although certain numbers represent specific number can not be expressed exactly with a finite number of digits.

ACCURACY AND PRECISION

Accuracy refers to how close it is measured or calculated value of the true value. Accuracy refers to how close an individual value is measured or calculated with respect to the others.

The inaccuracy is defined as a systematic departure from the truth. Imprecision, on the other hand, refers to the magnitude of the spread of values.

The numerical methods should be sufficiently accurate or no bias to meet the requirements of a particular engineering problem.

ERROR

In general, for any error, the relationship between the exact number and obtained by approximation is defined as:

Error = Actual value-estimated value

Sometimes they know exactly the value of the error, denoted as Ev, or we estimate an approximate error.

Now, to define the magnitude of error, or incidence on calculating the error detected, we can normalize its value:

Ea = relative error (fraction) estimated error = true value I

Since the value of Ea can be either positive or negative, in many cases we want to know more the magnitude of the error, in which case we will use the absolute value of this.

An interesting case is an investigation conducted by Scarborough, which determined the number of significant figures contained in the error as:

If we replace it in Eq. We will get the number of significant figures is the approximate value obtained reliable.

Thus, if our calculation has an error less than the criterion for two significant figures, we get numbers that correspond to a minimum:

Es = (0.5x 102-2)% = 0.5%

TAYLOR SERIES

Special attention is the approximation of functions using Taylor series expansion. Thus, if a function is continuous and differentiable in the range of interest can be written as a finite power series, or series of Taylor.

This method can not, however, be used to fit experimental data [Xi, f (x) J], but to transform known and differentiable functions to a more user-friendly.

There are certain observations that should be known by applying this formula. For example, to get a better approximation of the function to an interval [a, b], the value of Xo must be chosen as close as possible to the center of this range. This will minimize the maximum contribution of the term (X-Xo) n + l of the residue in the calculation of R (x) between

a <= x <= b. Another way to minimize the value of the waste is to raise the degree of the polynomial adjustment, or include more terms in the series and reduce the exponent of R (x). Edited from: www.norte.uni.edu.ni/estudiantes/aproximaciones.doc

CALCULATION OF ROOTS OF EQUATIONS

The purpose of calculating the roots of an equation to determine the values of x for which holds:

f(x) = 0

The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum eigenvalues of matrices, solving systems of linear differential equations, etc ...

The determination of the solutions of the equation can be a very difficult problem. If f (x) is a polynomial function of grade 1 or 2, know simple expressions that allow us to determine its roots. For polynomials of degree 3 or 4 is necessary to use complex and laborious methods. However, if f (x) is of degree greater than four is either not polynomial, there is no formula known to help identify the zeros of the equation (except in very special cases).

Edited from: http://www.uv.es/~diaz/mn/node17.html

Numerical Methods In Petroleum Engineering

martes, 27 de julio de 2010

lunes, 26 de julio de 2010

LU DESCOMPOSITION

Matrices and determinants

Systems of equations - Gauss

GAUSS-JORDAN METHOD

Calculations of roots

martes, 20 de julio de 2010

Roots equations

Darcy´s law

viernes, 7 de mayo de 2010

Numerical Methods.