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NEWTONS DIVIDED DIFFRENCE POLYNOMIAL INTERPOLATION..

As mentioned earlier, there is variety of alternative forms for expressing an interpolating

polynomial. Newtons divided difference interpolating polynomial is among the most popular

and useful forms. Before presenting the general equation, we will introduce the linear and

quadratic because of their simple visual interpretation.

Linear interpolation:the simplest form of interpolation is to connect two data points with a

straight line. This technique is called interpolation.

Applying similar triangles

!"f#"x$ % f"x$$&"x % x'$( ) !"f"x#$ % f"x'$$&"x#% x'$(, then multiplying both sides of this equation by

x % x'and then re % arranging, the equation becomes

f"x$ ) f"x$ * !f"x#$ % f"x'$&"x#% x'$("x % x'$.

Newton !i"i!e! !i##eren$e interpolation:the preceding analysis can be generali+ed to fit an

nth % order polynomial to n*# data points, fn"x$ ) b'* b#"xx'$ -- bn"xx'$"xx#$ , as done

earlier the linear and quadratic interpolations data points can be used to evaluate the coefficients

b', b#,-..bn. for an nth % order polynomial, n*# data points are required x', x#, x---. /n,

using these data points, the following equations are used to compute the coefficients

b') f"x'$

b#) f !x#x'(

b) f !xx#x'(

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bn) f !xnxn#-.. x#x'(, where the brac0eted function evaluations are finite differences for

example, the first finite divided difference is represented generally as

f !xix1( ) !"f"xi$ % f"x1$$&"xi% x1$( xi* x1, where the second finite divided difference which

represents the difference of the first divided differences. After close evaluation the general

analysis yields the interpolating polynomial. This is given as thus

f"x$ ) f"x'$ * "x % x'$ f!x#x'* --.xn$ which is the Newton divided difference interpolating

polynomial, which implies that the data points ta0en should be equally spaced.

Error o# Newton interpolatin% pol&no'ial:observed that the structures of the equations in

Newton divided method is similar to the tailors series expansion in the sense that terms are

added sequentially in order to capture the higher order behavior of the underlying function.

These terms are finite divided differences and thus represent approximations of the higher order

derivatives. 2onsequently as with the Taylor series, if the true underlying function is an nth %

order polynomials, the nth % order interpolating polynomial based on n*# data points will yield

exact results. 3ecall from the previous equation that the truncation error for the Taylor series

could be expressed generally as 3n) !f"n*#$"4$& "n*#$( "xi#% xi$

n)#. 5or an nth % order

interpolating polynomial, an analogous relationship for the error is 3n) !f"n*#$"4$& "n*#$( "x % x'$

"x % x#$-..

6here 4 is somewhere in the interval containing the un0nown and the data. 5or the formula to be

use, the function in question must be 0nown and differentiable. This is not usually the case.

5ortunately, an alternative formulation is available that does not require prior 0nowledge of the

function. 3ather it uses a finite divided difference to approximate the n*# derivative.

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APPLICATION OF POLYNOMIAL INTERPOLATION TO (AS EN(INEERIN(

7olynomial interpolation is applied in the area of gas engineering, drilling and production

engineering. But in this study we will be loo0ing at the failure criteria analysis of both vertical

and deviated gas wells with emphasis on insitu principal stresses.

PRO)LEM:

#. 8 gas wells are drilled and the corresponding insitu stresses down hole when 9ohr

coulomb criterion is followed, determine the intermediate insitu stress using the Newton

interpolation method. :f the following data is given

6ell number /i"psi$ ) stresses ;ogx

#'"psi$

#

= #'' >.'?'' >.##>$&"#>''#''$( ) '.''''''''''''#8

Then to get the intermediate stress for wellbore stability, since the intermediate stress is

considered to be the strengthening effect of our wellbores from roc0 mechanics part of

drilling engineering view.

5) .

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6ell

numbe

r

Ktresse

s x

"psi$

;og of

stresse

s

5irst Kecond Third 5ourth

# '.'''=#>

'.''''''

8

'.'''''''''#

@

'.''''''''''''

#8

= #'' >.'??@

@

'.''''''#?

#

'.'''''''''#

>

8 #>'' >.##>< '.'''>=?

'.''''''#8

@

. The solution is based on the fact that, the octahedral shear stress is computed for each in

situ principal stress and then a polynomial plot is made to show the data spread and data

space for each failure criterion.

# > m,

>

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=#> ' = #@'

=I= ># ' =# #@#

== =' =' =@ #= ># =' @? '

5irst polynomial interpolation plot for the well. To show stability based on octahedral shear

stress, it can be seen that the pea0 of the polynomial plot shows the wellbore stability stress

which is the F2K.

f(x) = 0x^3 - 0.01x^2 + 1.53x + 380.87

R = 0.69

plot of 1 vs 3 for polynomial interpolation

1

Polynomial (1)

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Kecond plot is also to show the shear stress stability based on the tensile nature of the well to

avoid fracture while drilling in that well. Then a plot of mean effective stress is polt againt the

octahedral shear stress to show the spread of result. Lbeying the Newton polynomial

interpolation.

f(x) = - 0x^3 + 0.27x^2 - 67.37x + 5647.02

R = 0.81

plot of m,2 vs to show stability for polynomial interpolation

Polynomial ()

5rom the graph above you can see that the stability of the well based on octahedral and mean

effective stress also depends on the pea0 stress, showing that the higher the stresses the more

stable and the more the wellbore is prone to failure criteria, inview of the fact that elastic moduli

also contributes to the strenght of the roc0.