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Mamedov Shakir Ahmad, PhD, Azerbaijan University of Architecture and Construction, Department Test and seismic stability of construction,
Azerbaijan, Baku, E-mail: [email protected] Hasanova Tukezban Jafar, PhD, Azerbaijan University of Architecture and Construction, Department Test and seismic stability of construction,
Azerbaijan, Baku, E-mail: [email protected] Imamalieva Jamila Nusrat, PhD, Azerbaijan University of Architecture and Construction, Department Test and seismic stability of construction,
Azerbaijan, Baku, E-mail: [email protected]
Reserch of movement of the viscous elastic fixed vertically located cylinder in liquid with the free surface under the influence of the seismic waves
Abstract: The problem about movement of the rigid cylinder keeping vertical position under the influence of running superficial waves in a liquid is considered. The indignation of a falling wave caused by presence of the cylinder which moves is thus considered. Special decomposition on a falling harmonious wave is used. The problem dares an operational method. For a finding of the original the decision, considering that the image denominator represents tabular function, Voltaire's integrated equation of the first sort which dares a numerical method is used.
Cylinder movement in the continuous environment under the influence of waves is considered in work [1]. Problems are solved by an operational method, thus originals of required functions are looked for by numerical definition of poles of combinations of transcendental functions and calculation of not own integrals.
Using specificity of a task below, decisions are under construction the numerical solution of the integrated equation of Volter of the first sort that doesn't create computing problems of the complex roots of transcendental functions [2; 3] connected with search. Keywords: cylinder, liquid, wave, movement, surface.
1. Problem definition
It is supposed that the rigid circular cylinder located in liquids with a free surface towers over a surface of liquid and can move in the horizontal direction. Movement of liquids it is considered from the point of view of the theory of long waves [4].
The equation of movement of the cylinder looks like
d2 „ ds , .
m — = P--, (1)
dt dt
where m — the mass of the cylinder, £ — horizontal movement, P — effort on the cylinder from liquid.
Pressure of liquid upon the cylinder [4] is equal in a common ground
p, = pg (Z- z) (2)
where p — density of liquid, g — acceleration of a free fall, Z — a deviation of a surface of liquid from initial situation, z — depth.
The equation of movement of liquid looks like
d x
t LU 27T
q = -pgr0Ç11 J cos2 Odd- z J cosOdO
V 0 0
2n
Jcoscosvd = 0 at v ^ 1,
0
or q = -pgZnnr0
Boundary condition is [4]
dZ =_1 du
dr r=r g dt
r=r0 o
(7)
(8)
where un — a projection of speed of liquid to a normal to a cylinder surface.
Considering (6), a condition (8) it is led to a look:
du
dr
g dt u =u cosd
n x
i
(9)
where ux — cylinder speed
u
V
df dt
a AÇ =
dt2
(3)
where h — liquid depth, A — Laplace's operator. Force having per unit length of a core is equal
2n
q = -r0 J pcosddd (4)
0
where 6 — a polar corner, r — cylinder radius. Having substituted pressure expression from (2) in (4) we will receive
2n
q = -Po gro J(C-z )cos0d6 (5)
0
Size can be presented in a look
$=Jj;vcosv (v = 1,2...) (6)
Having substituted (6) in (5), we will receive
Zp = bH(t)(sinkxcos ct - coskxsinct) =
= bH (t){2[ J1 (kr )cos0- J3 (kr )cos30 + J5 (kr )cos50-...] cos cot
+ [ J0 (kr) - 2 J2 (kr) cos 20 + 2J4 (kr) cos 40 -...] sin cot}
where Jn — Bessel's function. transformed according to Laplace — to Carson in (9),
Having substituted (12) previously having taking into account (6) we will receive:
Change of a surface of liquid consists of two parts: result of a falling harmonious wave and the indignation caused by presence of the cylinder which, thus, moves, i. e.
Z=Zp + Z (10)
zp = bH (t )sin (kx-at) where k — wave number, ffl — frequency.
Indignation of a surface of liquid is defined by the wave equation (3) which decision in Laplace-Carson's images looks like:
Z'=XDK. f—)cosW (11)
i=o v a J
where K — McDonald's function i — about.
Surface deviation on a harmonious wave considering that it is possible to present in a look:
+
(12)
kb {2[ J' (kr0 ) cos0 - J' (kr0 ) cos 30 + J' (kr0 )cos 50 -...
+
[ J' (kr0 ) - 2J' (kr )cos20 + 2J' (kr)cos40 -...
(Op
p +O
+
+...
+p
DK '
( PL, Ï
v a J
+ DK '
( Pro I
v a J
cos0 + D2K '
Pro
V a J
P +O
\
cos20 +...
(13)
P2 W 0
= --Ç cos0
Equating coefficients at in (13) it is possible to write If cylinder length , force P = ql. On the cylinder
(for): force of a wave dispersing from it, i. e. acts on length l :
P
-p-4 = 2bk/;(kr0) 2 2 g p
+ — DK 1 a
f PT" ]
V a J
(14)
P = -Pgnro ■ DK,
r k ]
V a J
l
(15)
a
Having substituted (15) in the movement (1) equa- _ m - p
tion in images, we will receive:
- p 2pnr0lK£ +- p 3K£ = 2bkpgnr0lKJ1 r
mp2^ =-pgnr0 Dk
- c^-ßpZ, (16) receive:
V a J
Having excluded from (14) and (16), we will
I =
pkj ;(Kr0 )K , (pr
y a J
Whence: nrl
a
- cpK '¿-H p K
a a
2 2
p
c ■ Pr0 \ ,
PK ' a y a J
+ ^ K' a
aJ
-Pr0lK l
Pr0
\
aJ
m „,( pr +—pKn r a
2bg
2 2 p2
(17)
aJ
Having increased numerator and a denominator in (17) by exp (pro/a), we will define the original ofa denominator of the first factor:
z = -
3l 2a2
}L
2a
Having entered designation S =
(i+e)arch (i+0)->/(i+0)-1+^ [(i+0)-1] j-pnr0iyl (i+0)2 -1 -(1 + 0)(1 + 0)2 -1 + arch(1 + 0)1-i1!^!
expi^}Kl & a J v a J
According to Borel's theorem we have: d
ip)
dt
JS (t-x)z (r)d-
at , — +1
1V ro J
-1
or Js (d-r)z (r)dr = Q, where
(19)
(1 + 0)(l + 0)2 -1 - In (l + 0 + -N/(l + 0}
-1
*=2
For the solution of the integrated equation (19) area of integration breaks into n of small sites of A0 for the purpose of approximation of subintegral functions. Considering that on a piece 0 < 0 < AO the fractional member in (18) bike in comparison with the others, we have:
z'
ß
ß=m
'0 J
On the piece end
z\ = z
ß
"yflÄd From (20) and (21) follows
z = z,
if
(20) (21) (22)
Further we will define S by means of(9), from where considering that at z definition, the numerator and a denominator in (17) were increased by exp (pro/a), let's receive:
ba
S
Or
r ßp
_b_ ß
r ^(1+ 0)2 -1 v
0 = OL -1
T. k. k, (H
\ a ,
na
'0
- P0
1
12pr0 Vt
(18)
(23)
z = ■
ß
z =,
np ; S z = 2a
V20 '
From (23) and (26) follows:
s\
2z2
On the piece end 0 = A0 H3 (23)
S. =-b A0 1 ß
S =_l (2 + ß)
I na '2pr
ß 2z12
=I
(24)
(25)
(26)
For periods the integral in (20) is calculated by linear interpolation of functions S and z on each interval, except the first for z and the last for S. Thus, on the first interval:
0<0<A0, z = zJ—
d
Q
and therefore:
S = (S . - S )-+ S
V n-1 n ; ^q n
l\u
J S (A0-0)z (0)d0
= z
A0
l^J
0
(S , - S )-+ S
V n-l n ) A0 n
d0
100= <27)
2
= 3 z ,A0(2Sn + Sn-i)
Respectively on a piece (n-l)A0 <0 < nA0 will be
2
S
Q
z = (z -z ,)— + nz ,-(n-l)z ,
V n n-1 / ^Q n-1 V / n'
S = S,
d
n--
V Ad
In this case:
nAO nAO f n
J S(nAO-O)z(6)d6 = S, J \n-
(n-i)AO
z -Z , II n-
n n-1 /
_O_
AO
(n-l)AO 6
AO
A6
+
6
6
n--
V A6
62
O
(Zn - Zn-1 )AO + nZn-1 -(n - 11
[nZn-1 -(n - l)Zn ]U =
n
s i l(
(n-l)A6
nA6 6 6 6
S\ - Zn-1 A6~(Zn - Zn-1 )A6 + " ^"Zn-1 -(n - l)Zn H"Zn-1 -(n - l)Zn =
6
z - z 1 62 z - z 1 63
n n-1 n___n n-1
A6
A6 3
+
["zn-1 -(n - 1) zn ] n6
nz
-1 -(n- 1)zn 62
(28)
A6
^1 n (zn - zn-1 )(2n - 1)-3 (zn - zn-1 )(3n 2 - 3n + 1) + n [nzn-1 -(n - 1)zn ]-2 [nzn-1 -(n - 1)zn ]^A6:
= 3 (2zn + zn-1 )A6. On the others i-intervals
S = (S . -S +,) —+ iS .+,-(i-l)S ,
\ n-I n-1 + 1 / AQ n-i + 1 V ' n-l Q
z = (z. -z.,)— + iz., -(i-l)z..
V . i-1> AQ .-1 V ) .
Then:
iA6
iA6
J S(nAd-d)z(6)d6 = J
(¡-1)A6 (¡-1)A6
iAe
(S -S .+1 ) —+ iS + -(i- 1)S
\ n-i n-i +1 / AQ n-i+1 \ / n-
6
(z. -z. 1)— + iz. 1 -((-l)z. V . A6 1-1 V ' '
d6 =
- Sn-M )(z. - z-1 +( - )) .z,-i -((-1) ~]{e+[iSn_M-((- i)Sn-, ]( - z-1
+
[¿S-.+i -((-l)Sn-.]|Xi-((-l)z.]}d6 = (( -Sn-.+i)(z. -z.-i)[i3 -((-1)
A6
+
(29)
A6
+
+ {(-. - Sn-+! )((- -(( - l)z. )+[.Sn-.+! -(( - l)Sn-. ]( - z,-! )}[. 2 -(( - 1)2
+ \[.S„-.+,-(-l)Sn-.~\[tzi!-(-l)z.]A6.
Putting integrals (27), (28) and (29) we will lead the integrated equation (20) to a look:
2 1 Q
2 (2S„+S„ )z,3 (+z„ )S, = Q
(30)
Here:
z=]-:3 -+1-z.))3-((-1)3
i=2 3 L
+
+2
2 {(- - Sn-,+.) [iz,-. - ( -1) Zi ] + [i'S„-i+, - ( -1) S„-i ]} X
i2 - (i -1)2 ] + [iSB-i+, - ((-1) S„ ] [iz,- (i -1) Zi ]
(31)
Expression (31) can be transformed as follows
(Sn -1
(Sn -1 - Sn - , +1 - zi -1
i2 - i + V3 1 +
iz. - (i - 1)z. i -1 i
(i
-1
+
+
iS . - (i - 1)S 1 n -i +1 n -1.
z. - z. , m -i i -1
12)
+
iz. - (i - 1)z. i -1 i
=(S .-S . V n -i n -i +1
i1zt - izt + Y3 z, - i2zt_l + izw -((3\z,-i + i2z,_l -((2 )iz'-i -
-i (i -1) z, + ()/)(i -1) z, ] + [iSn-,+, - (i -1) S, = (S . - S
\ n-i n-i
p - i + 1/ - i2
^ )i - >2
iz,-((2)z, -z-1 + (X
-i2 - i - 1/ + i2 - 1
z.t-1 + iz i-1 -(i -1) z, ] =
i z
1 ]+
z. - iz. , +
z,-1+z-1- iz,+z,]=[((.- >6 )z,+((.- >3 )
V i- 1/ |z +
+ [+1 - (i - 1)Sn-1]
x(s,-, - Sn-+1 ) + (( zf +12 zf-1 )[iSn-,+i -(( - l)Sn-, ] = (Sn-, - s, +(( i - % )z,-1 ]+[iSn-i+i-(i - l)Sn-i ] 12 (z, + z,-1 ) = Sn-i [(( i -16 )z, +(( i - % )z,-1 -(i -0>2 (z, + zf-1 ) + Sn-,+1 [ 12. (z, + zf-1 )-((.- % )z,-((.- >3 )z,-1
=Sn-, ((3z,+16 z,-1 )+Sn-,+1 ((z,+% z,-1 )=>3 Sn-, (z,+Xz,-1 )+13 Sn-,+1 ((2z,+z,-1 )•
Finally:
4 = 2 J J S (0)sin (O-r)d
1 n-1
1=3 z
3 ¡=2
' 1 ^
z. +—z
S, +
21 -1 n-1
J V2
—z. + z.
(32)
(33)
The formula (30) taking into account (32) represents a recurrent formula for definition Sn.
From (17), considering that the original: p sinat
In dimensionless sizes: ϔ; 9
X =
2kbgJ[
7 = J S (9) sin (O-r)dr
Replacing integral with approximately final sum it is
2 2 p + m
m
had:
according to Borel's theorem it is possible to define movement:
X = A6XS, sin [(n - i )A0]
(34)
Figure 1
\
\ z
vr n-i
nAQ
Figure 2
)
n=5000; k=0.01
Figure 3 n=5000; k=0,02
Figure 4 n=5000; k=0.02
0.4
-0.3
Figure 5
In figure 3 schedule S (0) is submitted at ^ = 4, and in fig. 4 the schedule of dependence of movement from time is submitted: oscillations are imposed on cylinder oscillations with the big period answering to elastic fixing with the smaller period, caused by the frequency of waves in liquid.
References:
1. Forrestol M. Zh., Alzkheymer B. E. Unsteady movement of the rigid cylinder under the influence of elastic and acoustic waves. Applied mechanics. Series E, ASME, 1968, P. 278-283.
2. Kubenko V. D., Panasyuk N. N. Action of non-stationary waves on cylindrical bodies in compressed liquid. Applied mechanics, 1973, t. 9, century 12. P. 77-82.
3. Agalarova T.J. Interaction of an acoustic wave with an oscillator. The collection of scientific works on mechanics, No. 7, - Baku, 1997. P. 181-184.
4. Kochin N. E., Kibel I. A., Rose N. C. Theoretical hydromechanics. Prod. technic. - a teor. lit., - Moscow. 1955. P. 509-513.
5. Ditkin V. A., Prudnikov A. P. Directory on operational calculation. The higher school, - M. 1965.
Murvatov Faxraddin Tadji, Scientific research institute "Geotechnological Problems of Oil Gas and Chemistry"
Republic of Azerbaijan, Baku Leading research fellow E-mail: [email protected]
Application of the influence of nanostructural coordination polymers based composite solution on well-bottom zone (WBZ)
Abstract: In the article study and application results of influence measure carried out in WBZ of well N111 of the South-East Sadan oil-gas area of Siyazan monoclinal oil field by 3% layer water composition of1% mixture of BF-1 and BF reagents have been analyzed and use of the reagents in heavily extracted oils fields has been recommended to increase the production efficiency.
Keywords: rubber, permeability, nanostructural coordination polymer based composite fluid, WBZ influence.
Nowadays negative tendencies as worsening of hydraulic condition in the old oil fields, deteriorating of oil reserves structure, formation of great number of heavily extracted reserves, complication of exploitation condition of the wells, increase of interval between repairs and wells with little production, increase of geological production, technological and etc. risks are observed. The main part of hydrocarbon reserves of oil deposits consists of oils having high viscosity, anomal property, and asphalten-resin-paraffin (ARP) containing compositions. For speeding up the flow of such oils into the well, the well bottom zone is influenced by chemical, thermochemical, thermal, microfoam system and other methods [1-4]. But in many cases these methods are not efficient and create problems with unknown results. For adopting of such hydrocarbons use of more complicated technologies brings to the increase of various
risks. By the carried out analyses it has been determined using traditional oil extracting methods in the development 66-58% of oil reserves can remain in the earth [5-7].
Let's mention that in Siyazan monoclinal oil deposits (SMOD) simultaneous exploitation of several reservoirs with various regimes, pressure, productivity and volume filtration characteristics is followed by incompatibilities. In such case as a result of strong flow from various characteristic formations complex to the well, artificial opportunities for the movement of water with oil and gas appear. Natural isolation of reservoirs is gradually disturbed, their content and properties worsen in the contact with layer fluids having various temperature, pressure, lithological — hydrogeological, oil-gas properties, reservoir oils become heavier, the structure changes and overturns to heavily extracted one.
