CC BY
11
APPLICATION OF A RANK FUZZY REGRESSION MODEL TO PREDICT THE TECHNICAL CONDITION
OF WELL PIPES
Ibrahim Habibov, Oleg Dyshin, Gulnara Feyziyeva, Irada Ahmadova,
Zohra Garayeva
Azerbaijan State Oil and Industry University h.ibo@mail.ru oleg.dyshin@mail.ru gulnara.feyziyeva5@gmail.com ahmadovairada@gmail.com zabiyevaadnsu@gmail.com
Abstract
Oil and gas pipes used in well operations are undergo to aggressive environments. In this case, corrosion wear of the thickness of their walls occurs, which leads to various difficulties. In order to assess the technical condition of well pipes, geophysical methods are used, one of which is the electromagnetic inspection method.
The paper proposes a method for predicting the maximum loss of pipe thickness based on the results of electromagnetic inspection by the step values of the depth of immersion of the lower part of the pipe into the well.
Based on the use of fuzzy regression with fuzzy input/fuzzy output, a method for assessing the level of impact of the main formation parameters on the technical condition of well pipes is proposed.
Keywords: rank transformation, electromagnetic inspection, fuzzy emissions, moving forecasting, membership function
I. Introduction
Despite a fairly wide range of methods for assessing the technical condition of oil and gas pipes, the most widely used method is electromagnetic inspection. (EMI) [1-4].
Experimental studies were carried out for an offshore field, using an electromagnetic inspection of the EMI-43 type, at a depth of 2000-2400 m. The parameters of the pipes under study were Doxdi=127.0x108.6 mm, Doxdi =339.7x 313.6 mm and Doxdi = 473.1 x 446.1 mm (Do and di are the outer and inner string diameter, respectively).
II. The purpose of the work
Is to development of a method for predicting the loss of thickness of the outer pipe of a technical string not accessible to the depth interval for measurements based on previous inspection measurements.
III. Results and discussions
We will demonstrate the application of the RT method to fuzzy regression with a fuzzy input/output the case of measurements using an electromagnetic inspection EMI, which allows us to determine the loss of thickness of the outer pipe of a technical string in a well.
592
Tables 1 and 2 correspondingly show the values of the maximum loss of pipe thickness and the dependence of the formation parameters (density p and viscosity v) on the depth of immersion of the pipe into the well in the period 2019-2022 obtained by EMI methods.
Table 1: Dependence of the maximum loss of pipe thickness on the depth of immersion into the well
Top part Bottom Pipe Nominal Actual Depth of Maximum Classificatio
of the of the length, thickness of minimum maximum loss of pipe n of losses
pipe, (m) pip^ (m) (m) pip^ (mm) thickness of pipe (mm) loss of pipe thickness, (m) thickness, (%)
1 2 3 4 5 6 7 8
2284,10 2292,40 8,29 13,06 12,40 2285,10 5,1 B
2293,70 2302,00 8,29 13,06 12,51 2300.00 4,2 A
2303,30 2311,60 8,34 13,06 12,14 2308,30 7,1 B
2312,90 2321,20 8,29 13,06 12,26 2321,20 6,1 B
2322,40 2330,80 8,34 13,06 12,18 2323,50 6,8 B
2332,10 2340,60 8,50 13,06 12,41 2335,70 5,0 A
2341,90 2350,00 8,11 13,06 12,57 2347,40 3,7 A
2351,30 2359,20 7,98 13,06 8,54 2359,20 34,6 E
2361,20 2369,10 7,86 13,06 8.34 2366,80 36,1 E
2370,40 2381,00 10,61 13,06 8,54 2380,70 34,6 E
Table 2. Dependence of the formation parameters (density p, kg/m3 and viscosity v, spoise) on the depth of _immersion of the pipe into the well in the period 2019-2022_
Years
2019 2020 2021 2022
P v P v P v P v
899 6,25 898 5,73 890 5,55 890 5,15
890 6,20 897 5,68 890 5,50 890 5,05
889 6,17 895 5,66 890 5,48 888 5,00
889 6,14 892 5,60 886 5,46 886 4,98
822 6,10 886 5,51 880 5,40 882 4,90
891 6,05 885 5,51 876 5,38 880 4,87
890 6,03 880 5,51 874 5,33 878 4,82
890 6,01 880 5,48 876 5,28 870 4,80
889 5,95 878 5,45 866 5,25 863 4,76
868 5,93 875 5,40 863 5,06 860 4,72
885 5,90 871 5,33 880 5,15 850 4,69
886 5,87 870 5,30 870 5,10 845 4,67
882 5,81 866 5,27 871 5,07 840 4,65
880 5,78 862 5,25 870 5,02 834 4,61
874 5,73 860 5,22 866 5,00 830 4,58
871 5,73 857 5,20 865 4,95 822 4,55
870 5,71 855 5,20 863 4,90 818 4,52
869 5,73 851 5,18 860 4,86 815 4,48
868 5.70 848 5,16 858 4,81 810 4,40
865 5,68 844 5,16 856 4.76 810 4,35
IV. Numerical implementation of the predicting method and discussion of the
results
1. According to the table 2 in a year 2021 for variable Xfc T°C (T°C is a temperature) build a sequence of points (x^y) (i = 0,1 ,. . ., 5 ), where xj = ^ - the values of variable depth h: xj = 2 000 + i ■ Ax (Ax = 1 0 0 ), i = 0 , 1 ,. . ., 5 ; yj = y(Xj) - the values of variable y = X. By sample
and by method of MNL build a multinomial regression model y = / (X) , / (X) = £4=0 /Sj-xj = /?0 + ftx + /?2 X2 + /?3 X 3 + /?4 X4 ( 1 )
Entering the variables of X0 = 1, X1 = x, X2 = x 2 , A3 = x3 , X4 = x4, we obtain relative variables of A0, X1t . . ., X4 of linear regression model
(2)
or in matrix notation of Y = X ■ p, where X = (.Xy) — ( 6x5 ) - matrix of variable values X j in the ith observation; the Y = (y0,y1( ■ ■ -,ys) is the vector of вектор observations of y variable. According to MNL, coefficient of regression ( - is the matrix transpose sign) calculated
by the formula
p = (АГА)" 1 ■ Y (3)
в coefficients can be calculated using the LIN program in EXCEL with the determination of the error in calculating the predicted value of the output variable
= Д55 = ^ (у—y) 2, (4)
i=o
where у - predicted value of a variable by regression y
= ( 5 )
i=o
Pj - MNL- estimated p coefficient calculated by the formula (41).
If the LIN program is not available for estimating of p, then it calculated by formula (3) using a matrix inversion program to calculate the matrix of (ArX) 1. In our case if n = 6 at number m = 4 independent variables of X^ . . ., X4.
2. Total interval of [x0,x5 ] divide the values of the variable x into intervals of [xj _ 1,xj] with length of 100 m. In each of these intervals select an interpolation node of
10
xj = xj--and construct the interpolation polynomial of Lagrange for xj ^ Xj ( i = 1 ,. . ., n ; n = 5 ) :
■ W=Z/ ^ )П^1 ( 6)
x — x;
^ L-, i -
i=l 1
Since / (n)(x) = 0 (due to the fact that the polynomial degree I = 4, for all x £ [x0,xs ] , satisfying the condition ( )), it is true that and in particular for all
xj = 2 000 + iAx (Ax = 1 0 ; i = 0, 1 ,. . ., 5 0 ). (7)
Therefore, for each variable from table 3, by formula calculated values of
for all , defined by the formula (7).
3. In the example under consideration, can assume (with a sufficiently small error) that
- the maximum depth of immersion into the well of the bottom part of the pipe, which was accessible to inspection measurements in the depth range of [2000; 2500]. Further we will use normalized depth values of hi = 2 000. Then = 1 , 1 9 .
4. Further we will assume that i = 1 ,2 ,. . . - length interval numbers Ax = 0,005 (10/2000) at the depth interval of h j, equal to [1; 1,25] and i 0 - number of subsequent available length interval of A i. For this example i 0 = 3 8.
Starting from the 1st line of table 1, these intervals can be numbered with a series of numbers / = {e|i = 1 , 2,. . ., i0}, every fixed value i correspond to the xj interval with length of Ax = 0,005 .
There is a maximum loss of the pipe thickness (in fractions of a unit to three decimal places) along this interval.
5. By sample V = {xt,y0} (i = 1 ,. . . i0) built the best polynominal regressive model of y = F(x) degree I < 1 0 by using of MNL. Let's denote / (x) = F (x) (F (x) - it is derivative function of F (x) ). Let's apply the 4th order Runge-Kutta method with a step of A h = 0,005, assuming io is an even number (otherwise V is considered starting from i=2):
yi+i = y + i (K + 2 fc2 + 2 fc3 + fc4) (8)
ki = A h-/ (x) , /C2=A +ifci),
k 3 = A h-/(x+^2), /C4 = A ft-/ (x + fc 3 ) .
By formula (8), knowing that if , it is possible to calculate the forecasting
value of . The forecast error is estimated according to the following formula
ft=d5ol/(4)(xJl((Ah)5 + 0 (Aft) 6) (9 )
where / (4)) (xto) - is a 4th derivative function of f(x) at a point xto.
If the degree of the polynom F (x) I < 1 , then / (4 ^(x) — F ( 5 ^ (x) = 0 and = 0 ((A h)6), where
- infinitesimal order of ( h) . Shifting the s ample V forward by s tep Ah ~ (taking the obtaining predicted value yt 0+i as the actual value), we will get predicted value of with predicted error
P2=22M-Pi-(A h)5 + 0(A h)6
Calculated by fomula (8) with replacement on the right side of by . If we also obtain that ( h) . In this way, forecasts can be calculated by , where
h imaX, ima^ = 5 0 (limiting value in the table 3) in this case the forecast error x„0+1/ will equal to ( h) .
6. By sample of last 20 values y : yto_ i9,. . .,yio fuzzify the variable 7. Denote that y ( 1 ' = yt
- i 9,
y (2 ) = yi0- i8 , y (w)) = yi0 (^ = 2 0) and calculate for indexes i = i0 — 9 , i0 — 8, ..., i0 of value
a0 = — ( max y (t)) + min y (t))),
2 Vl<i<n 1 <i<n /
(maxy (t)) + min y (t))) ( 1 0)
£0 _ v i < t < n_i < t <n / _
D — , £.a — Z.
2ea
Then with probability , will satisfy the following ratio
A'y(y) = = y) = exp
—m21
(ii)
in order, L Y (y) = fl y (y) = exp [ — (*jf) ].
If a , then subsequent shifts to right by 1 and replaced by
subsequent of ,
where y (n+i) at n > n0 then has a forecast value calculated by formula (8).
By formula (41) is calculated new values of , values of and etc. until, at a certain step of where the condition is satisfied of a with a certain error (for example,
1 0 _2). Then, with the approximation error eo, can accept that the mode of the fuzzy number of y (n)) is equal to a .
Truth, for fixed depth of h the value of of each fuzzy parameter of at a depth of
v° v° v° v°
h = h t find number of a and ¿¿ kk (denote their as a and b t kk), where fc represented by LR-
form
k = ( ^к'^ъкfc'^ifcfc) with the membership function fiXnk(y) = exp
V?,,
g,. Ik
if
У-« £
<k
Ak i , f
>s£.
i,к
<bi,lk and ^ (У) = 0, if
7. Via Xk denote the output fuzzy variable in linear fuzzy regression with close-cut input
h:
*k( k)+Pi( k (12)
I'M
Which the coefficients p( ; and p( ; calculated by sample of 7 = {x^yj ( i = 1 ' ■ . . ,i0), where xi = i - the depth interval length Д h = 0. 005 with number of n, i.e. xi = 1 + ¿Д h and Yi - the value of variable Y = Xk at depth interval xk, obtained by calculation according to the equation (12).
Denote xk = 0, x1 = h (xk0 = 1, xk' 1 = ^), via X (10x2) - matrix of . = (xy) ( i = i0 —
8.....i 0 + 1 ; j = 0 ' 1 ). Y = ( Yi0_8.....Yi0+i)T, Yi = i) , i = i 0 — 8.....i о + 1 ; P = (p( k) 'P( k ) / (T - is
matrix transpose sign). Then the value of p determined by the formula
P = (.r.) _ 1 .rY (13)
or calculated by the program of или LIN in EXCEL for simple linear regression with single input variable of .
8. Will differ the fuzzy number of in the record, obtained by fuzzification using the formula (11), fron fuzzy number of Y , obtained by calculation with regression (12).
Due to the symmetry of y the left span of (a) ' a 6 [ 0 ' 1 ] of the formula (49), will be equal
to the right span , and Y .
9. Let's put a* = 0' 5 .
By the sample of ,й(Й1)'Д ( (a* ) ) : i = i0 — 8' ■ . .' i0 + 1 - will construct following regressive model
R ( ^(a*) ) = P0 (a* ) + Pi (a*) й(Йi), (14)
where R(fti) - rang of number in the subsequent of {fti} ( i = i0 — 8' ■ . .' i0 + 1 ), first number in ascending number order h i, R = 1 rank is assigned, to the second R = 2 and so on; in the case of two identical numbers, equal for example r, in an unordered row, the first one is given rank of
R = [ (r — 1 ) + r]/2, but the second rang represented by the following R = [ (r + 1 ) + r]/2.
For simple regression of (14), assum that Yi = ( a*) and R (.) = R( h i) rank transformation (14) reduces to the following equation
(n + 1)1
0+1) R(Yi) = ^-L + ß
(15)
where - is the size of selected subsequent of (in this case
).
ffl-l 1 "i (tl-\-1 ")
Assume that yj = R (Yj)---—, xj = R (JQ---—, in this case we will get following
regression equation yj = p ■ xj, in which the MNL- is the value of p cofficient of p determined from the equation of ^ = 0, Z = £f= 1 (y — pxj) 2 , the solution of which is determined by the formula:
£ = £?= iyjXj/sr_iXt2 (16) Then from the equation of (15) find the predicted value
)=^+[R«)—(17)
10. Let's calculate the values ^ (a* ) by the following rules
Zo. (a* ) =
'iy( 1 )( a*) , i/ R ( Zy. (a* ) ) < R ( Zy( 1 )( a*) ), < ^w(a* ) - ^ R( wt (a* ) ) >R ( ^ ( a*) ), (18)
>a)(a* ) - i/ Д ( Zy. (a* ) ) < R ( гУ(Л (a*) ) .
If R ( Zyy) (a*) ) < R ( Zy. (a* ) ) < R ( Zy(.+i}(a*) ), then
( ) ( ) iti(a*;> = Z^^o + ( zyo+i, - zy( (a* ) — v
/r( W a+1 )(a*) )-R( W a)( a*:) )
where Z? (a*) - is a value of the left span Zy. (a*) of the fuzzy number % (a) = Y (JQ(a), constructed by the rang regression (14); Zyy) (a*) - jth is the ascending value of / of subsequent , Zy (a*) -, i = i 0 - 8,. . ,,i 0 + 1 . Since the left-side value of the а-level set of a fuzzy number must
be no more than its mode, then the estimation of the left-side range of the fuzzy number Yi (ал*) will be written as
Ffi (a*) = min(ryi (a*) ,yj, (19)
where у t - a priori estimation of the mode of a fuzzy number %, for which the assessment can be taken as a(^o)) mode of моды fuzzy number Y, obtained due to the fuzzifi cation according to the formula (11). This estimate is subsequently corrected using a parametric estimator of span.
Similarly, according to the sample of (R( h¿),R (ту. (a* ) : i = i 0 — 8,. . ., i 0 + 1 } with replacement in the formulas (17)-(19) of l by r will get value of right span of the fuzzy number as
following form
f?i (a*) = тах(ту. (a*),yj|. (20)
Under the numbers of Zy. (a) and ту. (a) understood the projection to the y axes of intersection points with line correspondingly of left branch and rigth branch membership
functions of of fuzzy number .
In the formulas (17) and (19) mode of of the fuzzy number satisfies, respectively, the equalities
aY. = lY.(a) + I ■ L_1(a),
У , ( V (21)
ay. = rY.(a)-r-L 4«).
where .
In the case, when the membership functions of of fuzzy number is represented by the
Gaussian
—Ш
that get
. (22)
11. Parametric assessment of spans and modes of the fuzzy number (X^) (a) with a ^ a* at a close-cut input Xt = ftj is constructed as follows.
Based on the obtained estimations of f?. (a* ) and l? (a* ) are built following estimations
(m ax(max{ a< s<a*}{ fy.00 ,yj], i/ a < a*, fy. (a) = { , .
1 max|min{a«<s<a}{ fY 00i/ a* < a,
f ( C- ^ (23)
mi n,max{ a«<s <«}{ lY.00,&}], i/ a* < a, Zy. (a) = { , _
1 (min jmin{a<s< a.}{ ly. (X) ,yj], i/ a < a*,
Since Zy. (a) , fy. (a) increase by decreasing of a, then the mi n a Zy. (a) = Zy. ( 0 ) and the maxa Zy. (a) = Zy. (1 ) ; minify, (a) = fy. (1 ) = $ and maxafy. (a) = fy. (0) , in this case Ly. ( Zy. (1 ) ) = R y. (fy. ( 1 ) ) = 1 and y i - is the value of the fuzzy number Yi (X) (a) where a = 0, to be confirmed.
For each fixed i, intend sample data of { Zy. (afc) , afc:Zc = 0,1 ,. . . ,Zc0} and {f> (afc) , afc:Zc = 0,1 ,. . ,,Zc 0} with increasing subsequent of {av} the value parameters a (for instance, afc = Zc/ 1 0, Zc = 0, 1 ,. . ., 1 0) and by the values of Zy. (av) and fy. (av) , determined by the formula (23), the measure of the fuzzy number Yi = Yi (X) is adjusted and the membership function py. (y) is approximated. However, if a = av Ly. ( Zy. (afc) ) = av ( Zc = 0 , 1 ,. . .,Zc0) (Ly; (y) - that is the left branch of function py. (y) ), then by the regression
1 L?t (y)=^)+/?ii)-y + /?2W-y 2 (24)
Based on the sample of { Zy. (afc) , afc: Zc = 0,1 ,. . ., Zc0 } it is possible to obtain of predicted value Ly. (y) for all of y £ suppY = [-ft^0 ^ ^+. Similarly, taking into account the equalities of iy (ffj (afc) ) = afc,Zc = 0,1 ,. . ,,Zc0, the right branch is being restored Ry. (y) by the following regression
?yi(y)=^0r)+^r-y + ^2r)-y2. (25)
(7) ~ (r)
Further, denoting through y>; = Zy (1) and y> ; = fy(1) obvious estimates of the mode of a fuzzy number, for evaluation of y i the mode of the fuzzy number Yi(Xl) can be accept as following
y i =-2-■ ( 2 6 )
The spans Zy. and fy. of the fuzzy number Yi = Yi (X) estimate as following
1 1 Zy. =9i-L? 1 (0 ) , fY=? ( 0 )-yi, (27)
where L^ 1 ( 0 ) and R ( 0 ) are solutions, respectively, of the equations L y; (y) = 0 and R y; (y) = 0. However, the fuzzy number Y represented by the LR-form as following
Y = (yi, yi - L^1 ( 0 ) , R f1 ( 0 ) - yi) (28)
v 1 1 'LR
Tuth, for the fuzzy variables Y = X fc(h i), i = i 0 - 8 ,. . ,,i 0 + 1 is obtained as the form (28) represented by the LR-form.
12. Suppose that the - is the variable, characterized by the maximum losses of the pipe thickness at the of hi, i = i 0 - 8 ,. . ., i 0 + 1 , the value of which i = i 0 - 8 ,. . ., i 0 determined in the 7th column of the table 1 and expressed in fractions of a unit (so if i = i0 Y = 0,346, i.e. 34,6%), and the value calculated by the formula of predicting (8).
Let's make fuzzification of number Yi ( i = i0 - 8,. . ,,i0 + 1 ) by the formula (10) with appropriate modes dy0 and spans by.o).
To describe the main characteristics of a fuzzy number of , - are vector
of parametrs of the formation with the values from the table 3, integrated using the formula (6) whole of 10-meter depth scale by step A h = 0,005 along all depth interval [1, 1.25].
Consider the last 11 values of the vector X at a depth hi with indexes / = {i|i = i0 - 8,. . ., i0 + 1 }. By sample { Z* (a) , Z*2 (a) , Z*3 (a) , Zy. ( a) }, i £ /, let's construct a rank regression instead of (52)
3 3 3 . .
R(Zy. (a)) =/?®(a) + ^/?®R(Z*.p(a))+ £ £/?$, (a) R (z*tp (a)) ■ R ( Z*ip, (a) / (2 9)
p=i p'=ip=i ^ '
however by selected data {f* (a) , f*2 (a) , f* (a) , fy. (a) }, i £ / - a rank regression.
3 3 3 . .
R (fy. (a) ) = /?(r) (a) + £ R (f*tp (a) ) + £ £ (a) R (f*tp (a) ) ■ R L, (a) / ( 3 0 )
p=i p'=ip=i ^ '
From the equations (29) and (30) based on the MNL-estimations of their coefficient the predicted values r(Zy. (a) ) and R (fy. (a) ) are found, of which using the formulas (18)-(23) the values of ff. (a) and if. (a) are calculated, and using formulas (24)-(28) the left and right branches of the membership function and the LR-form of the fuzzy number % = Y; (X;) are calculated. 13. Discrepancy between fuzzy numbers of Y; u % is calculated by the formula (11):
ifti) = CM,)3^tti(Yi(i)ii(i))i (3i)
J _ „My 00 dx v y
where Y; ( 1 ) = sup pY;, % ( 1 ) = sup p - carriers of fuzzy numbers Y; and %, represented by intervals
y (i) = [4? } - ¿y1 (i) - R_1 (i) - <o) ]
Y ( 1 ) = [yj-if t ( 1 ) - f? t ( 1 )-?;]
Let's denote for intervals of Y1 ( 1 ) uand Y2( 1 ) respectively via [ a 1 - b 1] and [a 2 -b 2 ] . The first term on the right side (31) characterizes the relative error of approximation of the membership function of the fuzzy number Yi by the membership function of the fuzzy number Y . In order for the second term to characterize the relative error in approximating the interval to the interval of Y will wrote as a following form
h, (Y; (1) -Y (1) ) = g[(a - kl (1) ) + (fy (1) -&)] -
-1(4Vo) - ¿y 1 (1) ) + (R_> (1) - 4V° ))1--1-}—r- ( 3 3 )
2Vy y )) R_i( 1 )+iyi( 1 )-2a(v°) ^ ^
where is the first multiplier in (33) is the distance between the centers of the intervals Y and , and the second multiplier - is an inverse number to the length of the interval Yi (0) Simpson's compound rule. Assume that the points of a = t0 < < t2 < • • • < ts _ 1 « b split the cut [a-b ] to the s-subcutes (elementary cuts). Having put the h = (b - a) /2s, t; = a + 2 th will obtain the following
I. 1 4
r" i rH+i
/ = J f (x) dx = ^ J f (x) dx.
Simpson's compound formula is written as following ft
I = - [/(a) + 4/(a + ft) + 2/(a + 2ft) + 4/(a + 3ft) + 2/(a + 4ft) + 8
ft4
+4/(a + 5ft) + /(a + 6ft)] - (ft - a) ■ -— ■ /(4) (a + 5ft), ft =
ft — a
with remaining members
180
R 2 s+1 = - (b - a) ^ ■ f (4 ) ( a +1 (b - a) ).
With increasing number of 5 elementary cuts R 2 s+1 — 1.
The integral in the denominator of the fraction on the right-side of (69) is written as
r oo b1
J My (x) dx = J f (x) dx- f (x) =
oo
x - a
In this case, the 4th derivative is written as following
f<4> W=|-2(^)i
exp
exp
x - a.
;(v0)
(v0r
Y
By choosing s, any degree of smallness of the remainder term R 2 s+1 is achieved in (35). To calculate the integral in the numerator on the right side of (68), calculate ,
and calculate of
(34)
(35)
(36)
pCO pb
I |my; 00—My 00|dx=I lfiY.00—My 00|dx, ( 3 7)
J—oo J a
where My (x) is the / (x) function from (36), but My (x) has a left Ly (x) and right Ry (x) branches, modelling by the regression of (24) and (25), respectively. Therefore, the derivatives My (x) = 0 and remaining members of (35) for the function / (x) = |mY; (x) — My (x)| is completely determined by the 4th derivative of the function of
r x — ) V
My(x) = exp
Tuth, the relative approximation accuracy (RAA) of the fuzzy number Yi with a fuzzy number y expressed as the following
RAA( Y = y) = l — d (Y,y ) , (38)
moreover, equality (38) is true with accuracy (38) and confidence probability
Y=y) = ( «) , a = e" 4 , (39)
where v0 - iteration step, on which the assessment a^ 0 ^ i s a chi eve d for the mode of the number fuzzified by formula (11).
14. According by the formula (30) the relative approximation of y to Yi is calculated, in the case, when the is the parametr of the formation and is the depth of the well and and Xi = hi (1 et's denote their as d(^ k,Xik)) for each parametrs of the formation Xk (Cc = 1 ,2 , 3 ) and in the case when Y is the maxumum losses of the pipe thickness and the X = (X1(X2 ,X3 ) is the vector parametrs of the formation and Yi = Yi (X^, Xi = (Xi 1(Xi2 ,Xi3 ) (let's denote their as d(Yi( y). Taking into account the error depth predictions valuves of macsimum losses of the thickness, determined by the formula (9), total (general) error of predictions y0+1 are calculated as following
£l = P + (fjd (Xio+i,k,Xyio+i,k)) ■ d( Yi o+! ,?io+1 ), (4 0)
where x?1o+1k -predicted parameter value Xk at a depth hi by regression (12) and Yio+1 - predicted value of the maxumum losses of thickness, determined by the rank regressions (24) and (25). To error of (40) corresponds to relative approximation accuracy y0+1 to the true (unobserved) value Yio+1 (i.e. the accuracy of predict Yio+1), defined by equality
RAA( y o+i ,Y^+0 = 1 — £1 , (41)
which is valid with confidence probability
po, 1 =
j
n
Lfc=i
, k (Xi o + 1, k = % io + 1, k)
' Yi o + 1, Yo +1 )'
(42)
where Pa , k(Xio+^ k = x?1o+^ k) = (e" 4) v o , 1 o+1 k (v0,i 0 + 1 , Zc - is iteration step, on which achieved by the way of fuzzification mode estimation of a fuzzy number £io+ 1k) and ia(Yio+1 , y0+1) = ( - is iteration step, on which achieved by the way of fuzzification mode
estimation of a fuzzy number Yio+1). Printing the forecast yio+1.
According to the rolling forecasting scheme, shifting a set of indices / = { i|i = i 0 — 8,. . ., i 0 + forward one step , we will get a set of indices | .
Then the forecasting error Yio+2 will be equal
£ 2 = e 1 ^2 ■ (rid(Xi 0+2 , k,*i0+2 , fc)) ■ d(Y 0+2 , Y 0+2 ), ( 4 3 )
where p2 - is forecasting To error of value yio+2 maxumum losses of thickness, determined by the formula (19) and following
RAA( Y )
with confidence probability
, 2 — I , fc (X 0+2 , 0+2 , fc) I Y o+2 'Y o+2 )-
(44)
. k = 1
At step m of this algorithm, a forecast of values will be obtained with the error
3
£m = £m - 1 ■ Pm ■ ( ^^ ^ (Xi 0+m, fc^i 0+m, fc) I ■ Yi 0+m, Y 0 + m), ( 4 5 )
\k = 1 /
If ( - is the permissible forecast error), then and go to point 6.).
Otherwise STOP. We print.
Using this algorithm, the following predictions were obtained:
i = i0 + 1 = 39 (2390 m), y3 9 = 0, 3 5 7; i = i0 + 1 = 40 (2400 m), ), y40 = 0,3 62 .
Thus, based on the above studies, the following main conclusions can be drawn.
V. Conclusion
1. Inspection measurements carried out on the basis of studying the electromagnetic field created by a sounde located inside casing or pumping and compression pipe (tubing) require significant costs, especially at large well depths and an increased degree of aggressiveness of their environment. In this regard, the problem of predicting the results of the inspection at greater depth of immersion of the pipe into the well becomes fundamentally important.
2. For this purpose, we have proposed a new methodology for predicting the maximum loss of pipe thickness by step of 10 m after any certain depth of immersion of the bottom part of the pipe into the well, for which the maximum loss of thickness (in percentage) is considered given.
3. Forecasting is carried out by sliding in the direction of increasing losses with a forward shift of depth by 10 m. The error of the resulting forecast is estimated using a numerical procedure for calculating the measure of difference between fuzzy numbers.
References
[1] Koskov, V.N. Comprehensive assessment of the condition and operation of oil wells using field geophysical methods: textbook. allowance /V.N. Koskov, B.V. Koskov, I.R. Yushkov. Perm: Perm State Technical University Publishing House, 2010, - 226.
[2] Huseyinov, G.G. Accelerated technology for eliminating accidents in wells and the mechanisms used in this case. Oilfield business, 2012, 8, 36-38.
[3] Technical guidance on the application of geophysical surveys and operations in oil and gas wells. SOCAR guidance document. Baku, 2019, 320.
[4] Feyziyeva, G.E Results of using geophysical methods in assessing the operating condition of oil and gas wells. Equipment, Technologies, Materials, 2022, 38-42.
