Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 2, pp. 771–777 DOI: 10.18514/MMN.2017.1701

ALGORITHM DEFINING THE HYDRAULIC RESISTANCECOEFFICIENT BY LINES METHOD IN GAS-LIFT PROCESS

N.S. HAJIYEVA, N.A. SAFAROVA, AND N.A. ISMAILOV

Received 03 June, 2015

Abstract. A mathematical model of the process of gas-lift in the oil production is considered.In this process the motions of gas and gas-liquid mixture (GLM) in annular space and in thelift are described by the system of partial differential equations of hyperbolic type. Applyingthe straight line method the initial problem is reduced to the system of ordinary differentialequations relatively to the volume of gas, gas-liquid mixture and their pressures. To determine thecoefficient of hydraulic resistance �c statistical data for the fixed well is used (volume of injectedgas at the wellhead of the annular space and GLM at the end of lift) and on the basis of theseresults, constituting the corresponding functional, the functional gradient is derived that allowsone to calculate �c . On the concrete example the statistical value of the coefficient of hydraulicresistance �Qc and the new value of the coefficient of hydraulic resistance �c are compared, it isshown that �c differs from the real �Qc on the order of 10�3.

2010 Mathematics Subject Classification: 49J20; 65K05; 65N40

Keywords: gas-lift, gas-liquid mixture, hydraulic resistance coefficient, gradient of the func-tional

1. INTRODUCTION

As it is known [7,8,12,18,20], the gas-lift method for oil production is used whenthe fountain method impossible due to the lower reservoir pressure. The essence ofthe gas-lift method is that due to the energy of the injected gas, the liquid can belifted to the ground surface. The motions [2, 18, 19] of the injected gas and GLMare described by the partial differential equations of hyperbolic type. To solve thecorresponding optimal control problems (feeding on the estuary such minimum gasvolume, that the debit would be the highest) the method of lines are used, which leadsthe initial problem to the linear - quadratic optimal control problem for a system ofordinary differential equations [1, 3, 6].

However, there are some significant parameters such as the coefficient of hydraulicresistance �c in the lift, included in the equation of fluid motion in the lift (the iden-tifications or the inversions problems [16, 17]), which determines the generation rateof GLM. In this paper we focus on the choice the parameter �c , where choosing thecorresponding quadratic functional, the posed problem is solved that allows one to

c 2017 Miskolc University Press

772 N.S. HAJIYEVA, N.A. SAFAROVA, AND N.A. ISMAILOV

calculate �c [8]. According to the obtained formulas is suggested the algorithm andthe program is made using Matlab. Thus, the offered algorithm allows to find thecoefficient of hydraulic resistance �c and the results coincide with accuracy 10�3

with data from mines.

2. PROBLEM STATEMENT

As it is known [1, 3, 5, 7, 9], the motions of the gas and GLM in the tubes aredescribed by the system of partial differential equations of hyperbolic type:(

@P@tD�

c2i

Fi

@Q@x;

@Q@tD�Fi

@P@x�2aiQ;

i D 1;2; (2.1)

where P DP.x; t/– is pressure,Q– is volume of gas and GLM, correspondingly, ci -

is the sound velocity in the gas and GLM, correspondingly, 2ai Dg

!ic

C�i

c!i

c

2Di.i D

1;2/I g, �ic ; .i D 1;2/– acceleration of gravity and hydraulic resistance in the gas andGLM, correspondingly, !ic ; .i D 1;2/ - averaged over the cross section of the velocityof the gas and mixture in the annular zone and lift, correspondingly. Di ; i D 1;2-are effective internal diameters of lift and annular space, Fi ; i D 1;2– cross-sectionalarea of pump-compressor pipe and is also a constant on axes. Applying the straightline method and denoting l D 1=2n, we obtain [14] from (2.1)(

dPk

dtD�

c2

i

Fi l.Qk �Qk�1/;

dQk

dtD�

Fi

l.Pk �Pk�1/�2aiQk;

(2.2)

where

Fi D

�F1; 0 < k � n;

F2; n < k � 2n;ci D

�c1; 0 < k � n;

c2; n < k � 2n:

Note that for k D nC1 an equation (2.2) has the following form [10](dPnC1

dtD�

c22

F2lQnC1C

c22

F2lQn C

c22

F2lQpl ;

dQnC1

dtD�

F2

lPnC1C

F2

lPn�2a2QnC1C

F2

lPpl ;

where Qpl , Ppl - are volume flow and pressure of the reservoir at the bottom of thewell.

After some transformations the system (2.2) can be reduced to

Px D A.�c/xCBuCV; (2.3)

with initial conditionX0 D

�P 01 ;Q

01; :::;P

02n; Q

02n

�0

; (2.4)

where X0 is the initial position of the well, u is controlling influence and AD�aij�;

ALGORITHM DEFINING THE HYDRAULIC RESISTANCE COEFFICIENT 773

B D�bij�; V D

�vij�;

aij D

8̂̂̂̂<̂ˆ̂̂:a11; if i D j and 1� i � n;

a12; if i D j C1 and 1� i < n�1;

a22; if i D j and n < i < 2n;

a21; if i D j C1 and n�1 < j � 2n�1;

0; in other cases;

a11 D

�0 �c21

ıF1l

�F1=l �2a2

�; a12 D

�0 c21

ıF1l

F1=l 0

�;

a21 D

�0 c22

ıF2l

F2=l 0

�; a22 D

�0 �c22

ıF2l

�F2=l �2a2

�;

bij D

8<: c21ıF1l ; if i D 1; j D 2;

F1=l; if i D 2; j D 1;0; in other cases;

vij D

8<: c22ıF2l ; if i D nC1; j D 2;

F2=l; if i D nC2; j D 1;0; in other cases:

x D ŒP1;Q1;P2;Q2; : : : ;Pn;Qn;PnC1;QnC1; : : : ;P2n;Q2n�0

Let’s have some statistics, which at the given initial volumes of gas the debit QQi2n

is measured at the output, i.e. QQi0; and QQi

2nare known (i D 1; N /, where N is the

number of measurements [4, 8].It is required to find such values of the coefficient of hydraulic resistance �c

[4,8,11,13], which system (2.3) will describe the motion of GLM in the lift, closer topractice (adequate mathematical model ). To solve this problem, the objective func-tion based on the lowest square deviation from the real initial data QQi

2nis constructed,

i.e. it is required to minimize the functional

f .a2.�c//D

NXiD1

hQQi2n�Q

i2n

i2: (2.5)

Thus, the problem of defining the coefficient of hydraulic resistance �c is reducedto the finding the gradient of functional (2.5).

774 N.S. HAJIYEVA, N.A. SAFAROVA, AND N.A. ISMAILOV

3. CALCULATING THE GRADIENT OF THE FUNCTIONAL (2.5)

The general solution of equation (2.2) can be represented as follows

x.t/D eAtx0CtR0

eA.t��/Bud�CtR0

eA.t��/Vd� D

D eAtx0CA�1.eAt �E/BuCA�1.eAt �E/V;

(3.1)

where from the equation (3.1) Qi2n has the form

Qi2n D J �x.T /; (3.2)

where J D Œ0; 0; 0; 0; :::; 1�, E– is the identity matrix.In fact, u is a function of the variable t . To get any value of the volume of gas or

the corresponding value of the pressure, we can take them in the mouth of the well inthe form of sin.bt/u0; that formula (3.1) turns into the following

x.t/D eAtx0CtR0

eA.t��/Bud�CtR0

eA.t��/Vd� DeAtx0C

C

tR0

eA.t��/B sin.b�/u0d�CtR0

eA.t��/Vd� D

D .eAtx0CE.eAt �u0 cos t �Asin t /Bu0/.E�A2/�1CA�1.eAt �E/V:

Then, substituting (3.1) and (3.2) in (2.5) we have

f .a2.�c//DNPiD1

�QQi2n�J �x.T /

�2D

D

NPiD1

Œ QQi2n�JeAxi

0�JA�1.eAT �E/Bu�JA�1.eAT �E/V �2:

(3.3)

For the solution of the optimization problem (2.3) - (2.5) we find the gradient of thefunctional f .a2.�c// and equating it to zero, we have

ALGORITHM DEFINING THE HYDRAULIC RESISTANCE COEFFICIENT 775

@f .a2.�c//@a2.�c/

D 2NPiD1

. QQi2n�JeAT x0�

�A�1.eAT �E/Bu�JA�1.eAT �E/V /�

�

NPiD1

.�J.1PkD1

kPjD1

T k

kŠ.A.a2.�c///

j�1�

�@A.a2.�c//@a2.�c/

.A.a2.�c///j�1/xi0�J

�1 @A.a2.�c//@a2.�c/

�1Bu�

�.1PkD1

kPjD1

J Tk

kŠ.A.a2.�c///

j�1 @A.a2.�c//@a2.�c/

�

�.A.a2.�c///j�1/BuCJ�1 @A.a2.�c//

@.a2.�c//

�1VC

CJ.��1 @A.a2.�c//@a2.�c/

�1C .

1PkD1

kPjD1

J T kC1

.kC1/Š�

�.A.a2.�c///j�1 @A.a2.�c//

@a2.�c/.A.a2.�c///

k�j /V /D 0:

(3.4)

Using the method [4] for the solution of the nonlinear equation we define �c in theform

�c D 4a2D=!cC2Dgı!2c :

Thus, we formulate the following algorithm for finding �c .

Algorithm:

1. The initial data and parameters A;B;V;u are introduced from (2.3);2. The statistical (observations) data

x0 D ŒP0i1 ;Q

0i1 ;P

0i2 ;Q

0i2 ; � � � ;P

0in ;Q

0in ;P

0inC1;Q

0inC1; � � � ;P

0i2n;Q

0i2n�0

and the debit QQi2n from the practice for the same well (i D 1;N , N -is the number ofmeasurements) are introduced;

3. The matrix A;B are founded from (3.1)

x.t/D ŒP1;Q1; : : : ;PnC1;QnC1;P2n;Q2n�I

4. The functional f .a2.�c// is formulated from (3.3);5. The solution of equation (3.4) is founded using the steepest descent method and

univariate method (golden section search method) [15].6. The condition

ˇ̌̌@f .a2.�c//@a2.�c/

ˇ̌̌< " is checked out for sufficiently small number ".

If this condition is not satisfied we go to the step 2, else calculating procedure ends.

Example. Let’s consider the realization of the proposed algorithm on the example[10]. After applying of the proposed algorithm we obtain that �c D 0:2357157. HereQ�c D 0:23, where Q�c is the hydraulic resistance value from practice [4, 8]. Note that

776 N.S. HAJIYEVA, N.A. SAFAROVA, AND N.A. ISMAILOV

�c differs from Q�c to the order 10�3, and it shows the efficiency of the proposedalgorithm.

4. CONCLUSION

The quadratic functional is formed for the solution of the indicated problem and itallows to find the coefficient of hydraulic resistance. The offered algorithm confirmthe adequacy of the constructed mathematical model with practice.

REFERENCES

[1] A. P. Afanas’ev, S. M. Dzyuba, I. . Emelyanova, and A. Ramazanov, “Optimal control with feed-back of some class of nonlinear systems via quadratic criteria,” Appl. Comput. Math., vol. 15,no. 1, pp. 78–87, 2016.

[2] F. A. Aliev, M. K. Ilyasov, and M. A. Jamalbekov, “Modeling of work of gas lift well,” Reports ofANAS, vol. 22, no. 4, pp. 107–116, 2008, in Russian.

[3] F. A. Aliev, M. K. Ilyasov, and N. B. Nuriev, “Problems of modeling and optimal stabilization ofthe gas lift process,” International Applied Mechanics, vol. 46, no. 6, pp. 709–717, 2010.

[4] F. A. Aliev and N. A. Ismailov, “Inverse problem to determine the hydraulic resistance coeffficientin the gaslift process,” Appl. Comput. Math., vol. 12, no. 3, pp. 306–313, 2013.

[5] F. A. Aliev, N. A. Ismailov, H. Haciyev, and M. F. Guliyev, “A method to determine the coefficientof hydraulic resistance in different areas of pump-compressor pipes,” TWMS J. Pure Appl. Math,vol. 7, no. 2, pp. 211–217, 2016.

[6] F. A. Aliev, N. A. Ismailov, E. V. Mamedova, and N. S. Mukhtarova, “Computational algorithm forsolving problem of optimal boundary-control with nonseparated boundary conditions,” J. Comput.Syst. Sci.Int., vol. 55, no. 2, pp. 700–711, 2016, doi: 10.1134/S1064230716030035.

[7] F. A. Aliev, N. A. Ismailov, and N. S. Mukhtarova, “Algorithm to determine the optimal solutionof a boundary control problem,” Automation and Remote Control, vol. 76, no. 4, pp. 627–633,2015, doi: 10.1134/S0005117915040074.

[8] F. A. Aliev, N. A. Ismailov, and A. A. namazov, “Asymptotic method for finding the coefficientof hydraulic resistance in lifting of fluid on tubing,” Journal of Inverse and Ill-posed Problems,vol. 23, no. 5, pp. 511–518, 2015.

[9] F. A. Aliev, N. A. Ismailov, A. A. Namazov, and M. F. Rajabov, “Algorithm for calculating theparameters of formation of gas-liquid mixture in the shoe of gas lift,” Appl. Comput. Math., vol. 15,no. 3, pp. 370–376, 2016.

[10] F. A. Aliev, M. M. Mutallimov, I. M. Askerov, and I. S. Raguimov, “Asymptotic method of solu-tion for a problem of construction of optimal gas-lift process modes Mathematical Problems inEngineering, vol. 2010, article id 191053, 11 pages,” Mathematical Problems in Engineering, vol.2010, 2010, article ID 191053, doi: 10.1155/2010/191053.

[11] A. D. Altshul, Hydraulic resistance. Moscow: Nedra, 1970, in Russian.[12] R. L. Barashkin, “Modeling of gas liquid mixture motion in tubing of gas lift well,” Automation,

telemechanization and connection in the oil industry, vol. 5, pp. 41–46, 2011, in Russian.[13] X. H., Y. H. Wang, H. Wang, and S. N. Balakrishnan, “Variable time impulse system optimization

with continuous control and impulse control,” Asian J. Control, vol. 16, no. 1, pp. 107–116, 2014,doi: 10.1002/asjc.769.

[14] K. K. Hasanov and T. S. Tanriverdiyev, “Optimal control problem describing by the cauchy prob-lem for the first order linear hyperbolic system with two independent variables,” TWMS J. PureAppl. Math, vol. 6, no. 1, pp. 100–110, 2015.

ALGORITHM DEFINING THE HYDRAULIC RESISTANCE COEFFICIENT 777

[15] D. M. Himmebblau, Applied nonlinear programming. New-York: Craw-Hill Book Company,1972.

[16] N. Huang and C. F. Ma, “Two inversion-free iterative algorithms for computing the maximalpositive definite solution of the nonlinear matrix equation,” Appl. Comput. Math, vol. 14, no. 2,pp. 158–167, 2015.

[17] M. A. Mabrok, M. A. Haggag, and I. R. Petersen, “System identification algorithm for negativeimaginary systems,” Appl. Comput. Math, vol. 14, no. 3, pp. 336–348, 2015.

[18] A. K. Mirzadjanzadeh, I. M. Akhmetov, A. M. Khasaev, and V. I. Gusev, Technology and techniqueof oil production. Moscow: Nedra, 1986, in Russian.

[19] D. V. Shternlikht, Hydraulics. Moscow: Kolos, 2005, in Russian.[20] V. I. Shurov, Technology and technique of oil production. Moscow: Nedra, 1983, in Russian.

Authors’ addresses

N.S. HajiyevaInstitute of Applied Mathematics, BSU, Z. Khalilov str.23, AZ1148 Baku, AzerbaijanE-mail address: nazile.m@mail.ru

N.A. SafarovaInstitute of Applied Mathematics, BSU, Z. Khalilov str.23, AZ1148 Baku, AzerbaijanE-mail address: narchis2003@yahoo.com

N.A. IsmailovInstitute of Applied Mathematics, BSU, Z. Khalilov str.23, AZ1148 Baku, Azerbaijan

Institute of Information Technology of ANAS, B.Vakhabzade str.9, AZ1141 Baku, AzerbaijanE-mail address: inao212@rambler.ru