# Inovative Metohods For Modeling Of Petroleum Mechanical Sistems Using Almost Periodic Functions

DOI : 10.17577/IJERTV1IS9273

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#### Inovative Metohods For Modeling Of Petroleum Mechanical Sistems Using Almost Periodic Functions

Avram Lazar Stan Marius

Universitatea Petrol-Gaze din Ploiesti Universitatea Petrol-Gaze 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. 1 j 1

(3)

a large number as random factors.

1. 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 oil-fields 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.

2. 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

2. 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:

1. 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

1. 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

2. 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)

1. 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 ra-tionalmentul 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.

1. Analysis of the structure for the oil field

Figures at the end paper are simulated using com- puter applications.

2. 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 )

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 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.

1. 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

2. References

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