 Open Access
 Total Downloads : 281
 Authors : Avram Lazar, Stan Marius
 Paper ID : IJERTV1IS9273
 Volume & Issue : Volume 01, Issue 09 (November 2012)
 Published (First Online): 29112012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Inovative Metohods For Modeling Of Petroleum Mechanical Sistems Using Almost Periodic Functions
Avram Lazar Stan Marius
Universitatea PetrolGaze din Ploiesti Universitatea PetrolGaze din Ploiesti,
Abstract
A method of boring research dynamics is the use of almost periodic functions resorting classical sense and almost periodic probability to study the dy namics of petroleum facilities, taking into account
M r M r (v,bh ,t)
where:
M r ()
l q
rjh (bh ) s jh (t)

1 j 1
(3)
a large number as random factors.

Introduction
Mr( ) are reduced moments;
Rjh(bh) resistant forces rjh (bh) – random reduced to the actuator of the operating system
The dynamics of machinery and plant oilfields equipement operations for extraction requires the substantiation of equivalent mathematical models, generally comprising a number of discrete masses (concentrated), joined by elastic links or elements
J d
r dt
M m ()
m
k 1 i 1
pik (ak ) qik (t)
M r ()
l
h 1 j 1
rjh (bh ) s jh (t)
(4)
with distributed parameters.
Given the complexity there is established a number of simplifying assumptions, considering that the masses are concentrated rigid bodies, elastic con
equivalent mathematical model representing the
operating system generally unconventional for the dynamic study, where:
Mm( ), Mr( ) is deterministic components, and:
necting elements have mass, and the influences of nature are not considered random.
M ik ;
k i
Fjh
h j
(4)
Crucial to solving the corresponding dynamic prob lem, its systems work is equivalent to building mathematical models and simplifying assumptions election.

Problem formulation
Be the equation of motion, written in handling drum shaft in the general case:
random components model
Mik and Fjh as and (4), obtained by generations and reduce random moments drum handling resis tant tree. Solving the mathematical model (4) is not accessible at this point.
From studies, if functions Fjh Mik and certain con ditions, solving the model becomes available, can be obtained the general form of the model solution.
J d
r dt
M m M r
(1)
This phase starts to form premises simulation steps using digital programs of specific applications in
where:
Jr is reduced mass moment of inertia; – angular velocity roads pump;
mechanical drives, the electromechanical field in general and oil in particular.
Definitions:
Mm – engine torque, Mm = Mm( );
Mr – reduced when handling drum resistant tree, Mr = Mr(v) = Mr(k )
Relation (1) is the classic expression of equivalent
P ( t )
P ' ( t )
n
cik


1
q
c' jh
ik ( t )

jh ( t )
n
cik e(
i 1
q (
c' jh e
1 ik t )
1 ' jh t )
(5)
(6)
mathematical model of a working system, in this case: the operating system. Taking into account expressions (4.1.6.6 and 4.1.6.56, [1]), expression engine when taking into account random phenom ena:
j 1 j 1
Mik and Fjh say are random functions almost peri odic in the classical sense FAPC.
These families of functions, the customizations on
M m M m (, ak , t)
M m ()
m n
pik (ak ) qik (t)
(2)
the actual situation will be function that will create the mathematical model study.
k 1 i 1
and of the resistance, taking into account the rela tions (4.1.6.7 and 4.1.6.55, [1]), is:
The relationship (4) becomes:
J r d
dt
M m ()
m n
cik
ik (t)
M r ()
l q
c' jh
' jh (t)
k 1 i 1 h 1 j 1
(7)
If conditions (5) and (6) are not met by approximat
For > 0 (14) and < 0 in (15)
ing the trigonometric polynomials (7) is recom
1 m n
t ( t u )
1 l q
t ( t u )
mended to use random trigonometric polynomials, of the form:
( t )
Jr k 1i 1
e J r
cik
ik ( u )du
Jr h 1 j 1
e J r
c' jh
' jh ( u )du ,
(15)
P ( ak ,t )
n
cik ( ak )
i 1
ik ( t )
n
cik ( ak ) e(
i 1
1 ik t )
(8)
Model solution (15) is similar to relations (12), (13)
( ak ,bh ,t )
P ' ( bh ,t )
q
c' jh ( bh
' jh ( t )
q
c' jh ( bh ) e(
1 ' jh t )
1 m n
( t u )
e Jr
cik ( ak )
ik ( u )du
1 l q
( t u )
e Jr
c' jh ( bh )

jh ( u )du ,
j 1 j 1
(9)
Jr k 1i 1 t
Jr h 1 j 1 t
(16)
If polynomials (8) and (9) satisfy the conditions (5) and (6), we say that functions are functions Fjh Mik and almost periodic random probability FAPP.
for < 0, results:
( ak ,bh ,t )
d m n l q
1 m n
t ( t u )
1 l q
t ( t u )
J M ( )
c (a )
(t)
M ( )
c' (b )
' (t)
e Jr
cik ( ak )
ik ( u )du
e Jr
c' jh ( bh )

jh ( u )du
r dt m
ik k ik
k 1 i 1
r jh h
h 1 j 1
jh
(10)
J r k 1i 1
J r h 1 j 1
(17)
Consider a linear variation in the difference deter ministic components:

Analysis of possible cases
M m (
) M r ( )
(11)
If > 0, k = 1 and h = 1, which means that the in
justified by the analysis of possible cases where the DC electric drive.
If, Mm( )=A BÂ· , and Mr( )= A and substituting (11) in (9) or (10), we obtain:
fluence is considered a single random variable from engine to the actuator.
a1) Where almost periodicity in the classical sense. We have the result form:
J r d
dt
m n
cik
ik ( t )
l q
c' jh


jh ( t ) t n
k 1i 1
h 1 j 1
(12)
e 1
r
t J ci1 2
i 1 i 0 2
Jr
Jr sin i 0t
i 0 cos i 0t
Mathematical model solution (12), is: case of al
most periodicity in the classical sense of random phenomena, namely:
e t q ,
sin j 0t
Jr d
dt
m n
cik ( ak )
ik ( t )
l q
c' jh ( bh )

jh ( t )
J c j1 2
r
j 1
j 0 2
j 0 sin 0
cos 0
Jr
k 1i 1
h 1 j 1
Jr
(13)
case of almost periodicity in the probability of phe nomena.
Relations (4) and (13) the general form of general ized mathematical model for system dynamics study of flexibility to both forms of almost perio dicity of random phenomena.
cos j 0t
2
j 0 2
Jr
j 0 cos 0
sin 0 0;
Jr
(18,a)
Mathematical model solution (12), is:
a2) If the probability of almost periodicity
( t )
1 m n
( t u )
e J r
cik
ik ( u )du
1 l q
( t u )
e J r
c' jh ' jh ( u )du ,
Jr k 1i 1 t
Jr h 1 j 1 t
(14)
e t n
t
Jr
ci1 a1 1
Jr sin i 0t

0 cos i 0t
(21)
i 1
Jr
i 0 2
valid for systems with variable yield, where kf is the conversion factor resistance
2
e t1 q , sin j 0t
r
c j1 b1 2
j 1
Jr

0 2
j 0 sin 0
cos 0
Jr
Mr = kF M
(22)
It proposes the following notations:
cos j 0t
j 0 cos 0
sin 0 0; k k
2 2 Jr
A x A
1 F x
B x B
1 F x
Jr j 0
(18,b)
k x k x
(23)

Where A and B according to and considerations point to remain valid. The general solution of the model (14) becomes
The general solution of the model (19), in this case, becomes:
B x
t 2 A t
B B
– B t
A 0 e Jr
B
(19)
t 2A x x t A x
B x B x B x
t
0 e J r
(24)
The function expression (19) involved constants A, B, whose expressions are known.
For explicit expression (t) is presented as result ing from rationalmentul above. Significance of
where :
x t
B x n
J r i 1
ci1
B x t u
e Jr
t
sin i0u du
measurements involved in relationship (19) is known.
Where:
B x q
J r j 1
B x t u
c
e
, Jr
j1
t
sin i0u
0 du ;
(25)
B n
t
J r i 1
ci1
e
B 2
J r
t
i 0 2
B sin i 0 t J r
i 0 cos i 0 t
case of almost periodicity in the classical sense.
c
j1
Case of almost periodicity in probability under the conditions agreed, the general solution is (24) or
B e t q ,
sin j t B
(25), but coefficients
ci1 i
' will take values
J r
c j1
0 j 0 sin 0
cos 0 '
J r j 1
2
B j 0 2
J
ci1 a1
and
c j1 b1
. This is a general case with
r
cos j 0 t j
cos
B sin .
real phenomena which are more specifically stu
died. This results in the use and usefulness of the proposed approach to modeling mechanical sys
2 0
B j 0 2
J r
0 J r 0
0
(19)
tems of these functions.

Analysis of the structure for the oil field
Figures at the end paper are simulated using com puter applications.


Cases studied previously, are cases of work specific systems has consistently yield. Using rela tions, the functionality of the system actuator con sidering
equipment
Plant fluids from wells with pumping progressive cavity pumps, submersible (Figure 1) have the fol lowing main components: Operating system (SM ST SCA) in the rotating pump surface mounted ;
Pipes (P) that are screwed to the stator to be placed
M m A B
is given by expression (11) becomes:
(20)
in column operation..Pumping rods (PR )which transmit the rotation of the rotor drive system with progressive cavity pump, pipes are placed in the probe, with the rotor screw to the bottom of the ram pump road.
M m
M r
A B
1 kF
x
k x
Progressive cavity pump (PCP), submersible known as other names that screw pump, eccentric screw pump, rotary pump thaw, or thaw pump ro tating twister. The following figures are numerical applications of these results.
11
( t )
F( t )
M( t )
SCA
PR
Mr(k )
M ik ;
k i
SC/TA
0
0 t 60
,
i1
50.8883
c
,
c
j1
ci1 a1
PC
j1 1
c, b
( t )
M( t )
Figure 1 Installation of pumps with progressive cavity pump
11.8624
0 t 30
27.8034
1( x1 , t )
2( x2 , t )
3( x3 , t )
7.65267
0 t 30
77.9023
M1 ( x1 , t )
M2 ( x2 , t )
M3 ( x3 , t )
21.5927
0 t 30
Figure 2 Variation of angular velocity and momen tum with variable yield case of almost periodicity in the classical sense
Figure 3 Variation of angular speed and engine torque case of almost periodicity in the probability
4. Conclusions
In this article it presented the mathematical model of the whole drive system composed of equipment from the top of the gasket seal Pomar and pump assembly.
This was due to the fact that in establishing the functional caracteristics of the pumping sistem, the deterministic trearment of the mathematical model has shown that a series of influiences linked to the action of some random factors are not found in the analysis and/or the synthesis of the system and of the functional characteristic.
An element of novelty innovation in the authors view is the introduction of the almost periodic functions in the classical sense and of the probabili ty almost periodic functions allow the identification of the influence of random factors.
Using these test functions transfers the proble matic of the study from the domain of the mathe matic statistics to that of mathematic analisysis, which is more accessible to the requirements.
The results are materialized by developing a com puterized calculation program whose results are presented in graphical simulation system operation after at various depth for pumps.
A great importance in the dynamic study of a work ing system in technological pumping operation is given to the way in which the system structure is set.
A first concern in this regard was to set the struc ture of the working system in order to design the dynamic simulation as a continuous system.
By comparison with mathematical models estab lished by already existing energy method proposed by researchers, some original contributions have been made by including the mathematical equa tions of the model.
They are represented by the effect given by the gel resistance of the drilling mud and of the "plunger", by additional hydrodynamic pressure occurrence during bit run back operation. It has also been con sidered that energy losses of the drill string result from viscous and dry amortization.
Produced mechanical waves can become dangerous for both the road string (area of threaded joints) and the surface guidance structure on which the waves have effect.

Acknowledgements
I wish to express our homage to Octav Onicescut, a Romanian mathematician, whose work made it posible to the work of a to achieve this research paper.
We also thank our colleagues manufacturing engi neers without whose input we could not have veri fyed in practice the results of research and software developed

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