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Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 191053, 11 pages
http://dx.doi.org/10.1155/2010/191053
Research Article

Asymptotic Method of Solution for a Problem of Construction of Optimal Gas-Lift Process Modes

1Institute of Applied Mathematics, Baku State University, Baku 1148, Azerbaijan
2Department of Mathematics and Statistics, York University, Toronto, ON, Canada M3J 1P3

Received 22 December 2009; Revised 16 April 2010; Accepted 16 April 2010

Academic Editor: Ben T. Nohara

Copyright © 2010 Fikrat A. Aliev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Mathematical model in oil extraction by gas-lift method for the case when the reciprocal value of well's depth represents a small parameter is considered. Problem of optimal mode construction (i.e., construction of optimal program trajectories and controls) is reduced to the linear-quadratic optimal control problem with a small parameter. Analytic formulae for determining the solutions at the first-order approximation with respect to the small parameter are obtained. Comparison of the obtained results with known ones on a specific example is provided, which makes it, in particular, possible to use obtained results in realizations of oil extraction problems by gas-lift method.

1. Introduction

It is known [13] that the gas-lift technique of exploitation of oil wells is widely used when the gushing method does not work for the reason of insufficiency in pressure. The essence of the gas-lift method consists of the fact that by the mean of energy of injected underground gas it is possible to lift fluid to the surface.

While the gas-lift method is widely used in oil extraction for a sufficiently long period of time, construction of an adequate mathematical model is rather an actual problem. Mathematical model describing the oil lifting process in pump-compressor tubes is described in [4].

In [4, 5], using this model, an optimal control problem for gas-lift process is formulated, where the pressure and volume of the injected gas is used as a control parameter. In the same papers, the method of straight lines is used to reduce the optimal control problem to the linear-quadratic optimal control problem (LQOCP). It should be noted that, when the depth of a well is large, in order to obtain sufficiently accurate results it is necessary to divide this distance into large number of relatively small segments. The latter, in its turn increases the dimensions of the system and, consequently, results in higher volume of computations to solve such problems. Thus, generally speaking, solution of these problems yields higher approximation and calculation errors.

Therefore, it seems quite rational to develop other solution methods with lower computational complicatedness [68]. One of the methods applicable to the considered problem for the case when the reciprocal value of well’s depth represents a small parameter is the asymptotic method described in [9].

In the present paper, using the method of straight lines, the mathematical model of the gas-lift process is represented as a system of partial differential equations of hyperbolic type. In a particular case, we arrive to LQOCP for a system of ordinary differential equations containing a small parameter. Applying the algorithm introduced in [10, 11], the solution to LQOCP is obtained as a function of the small parameter. Consequently, as in [6, 9], at the first approximation, analytic formulas for the volume of injected gas (control) and production level (trajectories) are obtained. As it supposed, the presented approach will reduce substantially the amount of required computations.

2. Mathematical Formulation of the Problem

As in [4], mathematical model of gas-fluid mixture flow in pipes is described by the system of hyperbolic-type partial differential equations: 𝜕𝑃𝑐𝜕𝑡=2𝐹𝜕𝑄𝜕𝑥𝜕𝑄𝜕𝑡=𝐹𝜕𝑃[]𝜕𝑥2𝑎𝑄𝑡0,𝑥0,2𝐿,(2.1) which for 𝑥=𝑧/2𝐿and 𝜀=1/2𝐿 can be written as𝜕𝑃𝑐𝜕𝑡=2𝐹𝜕𝑄𝜀𝜕𝑧𝜕𝑄𝜕𝑡=𝐹𝜕𝑃𝜕𝑧𝜀2𝑎𝑄,𝑧(0,1),(2.2) with appropriate boundary and initial conditions𝑃(𝑧,0)=𝑃0(𝑧),𝑄(𝑧,0)=𝑄0(𝑧),𝑃(0,𝑡)=𝑃0(𝑡),𝑄(0,𝑡)=𝑄0(𝑡),𝑃(𝐿+0,𝑡)=𝑃(𝐿0,𝑡)+𝑃𝑝𝑙(𝑡),𝑄(𝐿+0,𝑡)=𝑄(𝐿0,𝑡)+𝑄𝑝𝑙(𝑡).(2.3) It is required to determine a control𝑃𝑈(𝑡)=0(𝑡),𝑄0(𝑡)(2.4) minimizing the functional1𝐽=2𝛼𝑄(2𝐿,𝑇)𝑄deb2+12𝑇0𝑈(𝑡)𝑅𝑈(𝑡)𝑑𝑡,(2.5) where 𝑃 is pressure, 𝑄 is gas-fluid mixture volume and𝑄debis the desired yield.

Applying the so-called method of straight lines to (2.2) and taking 𝑙=1/𝑁, we obtain𝑑𝑃𝑘𝑐𝑑𝑡=2𝜀𝑄𝐹𝑙𝑘𝑄𝑘1,𝑑𝑄𝑘𝑑𝑡=𝐹𝜀𝑙𝑃𝑘𝑃𝑘12𝑎𝑄𝑘.(2.6) Note that for 𝑘=𝑁+1 (2.6) can be written as ̇𝑃𝑁+1𝑐=22𝜀𝐹2𝑙𝑄𝑁+1+𝑐22𝜀𝐹2𝑙𝑄𝑁+𝑐2𝜀𝐹2𝑙𝑄𝑝𝑙,̇𝑄𝑁+1𝐹=2𝜀𝑙𝑃𝑁+1+𝐹2𝜀𝑙𝑃𝑁2𝑎2𝑄𝑁+1+𝐹2𝜀𝑙𝑃𝑝𝑙,(2.7) where 𝑄𝑝𝑙,𝑃𝑝𝑙 denote gas-fluid outlay (yield) and pressure at the bottom of a well, respectively. As in [10], a linear-quadratic optimal control problem is formulated for this system. It is required to find 𝑥,𝑢 satisfying the equation𝐴̇𝑥=0+𝐴1𝜀𝑥+𝐵𝜀𝑢+𝐶𝜀(2.8) with initial condition𝑥(0)=𝑥0(2.9) such that the value of the functional1𝐽=2𝑥(𝑇)𝑥𝐾𝑥(𝑇)𝑥+12𝑇0𝑢𝑅𝑢𝑑𝑡(2.10) is minimized.

Here, 𝐴1=𝐴11𝐴0000000000012𝐴1100000000000𝐴12𝐴1100000000000𝐴12𝐴11000000000000𝐴1100000000000𝐴12𝐴1100000000000𝐴21𝐴2200000000000𝐴21𝐴22000000000000𝐴2200000000000𝐴21𝐴2200000000000𝐴21𝐴2200000000000𝐴21𝐴22𝐴,(2)0=00000000000002𝑎100000000000000000000000002𝑎100000000000000000000000002𝑎100000000000000000000000002𝑎200000000000000000000000002𝑎200000000000000000000000002𝑎2,𝐴11=𝑐021𝐹1𝑙𝐹1𝑙0,𝐴12=0𝑐21𝐹1𝑙𝐹1𝑙0,𝐴22=𝑐022𝐹2𝑙𝐹2𝑙0,𝐴21=0𝑐22𝐹2𝑙𝐹2𝑙0,𝑃𝑥=1,𝑄1,𝑃2,𝑄2,,𝑃𝑁,𝑄𝑁,,𝑃2𝑁,𝑄2𝑁,𝑥0=𝑃01,𝑄01,𝑃02,𝑄02,,𝑃0𝑁,𝑄0𝑁,,𝑃02𝑁,𝑄02𝑁,𝑃𝑢=0𝑄0,0𝑐𝐵=21𝐹1𝑙𝐹1𝑙00𝑐00000000,𝐶=000022𝐹2𝑙𝐹2𝑙0𝑃0000𝑝𝑙𝑄𝑝𝑙.(1) Let 𝑅=𝑅𝜀. Then the corresponding Euler-Lagrange control problem can be written aṡ𝜆=𝐴̇𝑥0+𝐴1𝜀𝜀𝐵𝑅1𝐵0𝐴10𝐴11𝜀𝑥𝜆+0𝐶𝜀,(2.12)𝑥(0)=𝑥0,𝜆(𝑇)=𝑁𝑥(𝑇)𝑥.(2.13) Thus, initial problem (2.2)–(2.5) is reduced to finding the solution of problem (2.12), where 𝜀 is a small parameter. Therefore, the asymptotic method (see [9]) allowing to expand the solution of system (2.12) with respect to small parameter𝜀 can be applied.

3. Application of Asymptotic Method

Let us apply the asymptotic method to the Euler-Lagrange equation (2.12) with boundary conditions for 𝑥(0)and𝜆(𝑇). According to [12], 𝑥(𝑇)𝜆(𝑇)=𝑒𝐴0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴1𝑇𝑥0+𝑒𝜆(0)𝐴0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴1𝑇𝐴𝐸00𝐸0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴110.𝐶𝜀(3.1) Further, let us expand the expression𝑒𝐴0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴1𝑇=𝑒𝐴000𝐴0𝑇+𝜀𝐴1𝐵𝑅1𝐵0𝐴1𝑇(3.2) from (3.1) with respect to 𝜀. If we denote 𝐻1i.e=𝐴000𝐴0,𝐻2i.e=𝐴1𝐵𝑅1𝐵0𝐴1,(3.3) then according to [12], the expansion of expression (3.2) with respect to 𝜀 can be represented as: 𝑒𝐻1𝑇+𝜀𝐻2𝑇𝑒𝐻1𝑇+𝜀10𝑒𝐻1𝑇(1𝑠)𝐻2𝑇𝑒𝐻1𝑇𝑠𝑑𝑠(3.4) Denote the integral in (3.4) by 𝐿0. Then it is not difficult to show that matrix 𝐿0 is a solution of the following Sylvester's equation:𝐻1𝐿0𝐿0𝐻1=𝑒𝐻1𝑇𝐻2𝐻2𝑒𝐻1𝑇.(3.5) Therefore, for expression (3.2) we obtain the expansion𝑒𝐻1𝑇+𝜀𝐻2𝑇𝑒𝐻1𝑇+𝜀𝐿0.(3.6) Introducing notations 𝐿0=𝐿1𝐿2𝐿3𝐿4,𝐴0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴11=𝑆1𝑆20𝑆4,(3.7) expression (3.1) can be simplified to 𝜆=𝑒𝑥(𝑇)(𝑇)𝐴0𝑇𝑒𝑥(0)𝐴0𝑇𝐿𝜆(0)+𝜀1𝑥(0)+𝐿2𝜆(0)+𝑒𝐴0𝑇𝑆1𝐶𝑆1𝐶𝐿3𝑥(0)+𝐿4𝜆(0).(3.8) Hence, adding boundary conditions from (2.12), we arrive to the following system of algebraic equations: 𝑥(𝑇)𝜀𝐿2𝜆(0)=𝑒𝐴0𝑇𝑒𝑥(0)+𝜀𝐴0𝑇𝑆1𝐶𝑆1𝐶+𝐿1,𝑒𝑥(0)𝜆(𝑇)𝐴0𝑇+𝜀𝐿4𝜆(0)=𝜀𝐿3𝑥(0),𝜆(𝑇)𝑁𝑥(𝑇)=𝑁𝑥(3.9) which in the matrix form can be written as𝐸𝜀𝐿200𝑒𝐴0𝑇𝜀𝐿4𝐸𝑥𝜆=𝑒𝑁0𝐸(𝑇)𝜆(0)(𝑇)𝐴0𝑇𝑥0𝑒+𝜀𝐴0𝑇𝑆1𝐶𝑆1𝐶+𝐿1𝑥0𝜀𝐿3𝑥0𝑁𝑥.(3.10) Multiplying (3.10) by [𝐸000𝐸𝐸00𝐸] on the left, we obtain𝐸𝜀𝐿20𝑁𝑒𝐴0𝑇𝜀𝐿40𝑥𝜆=𝑒𝑁0𝐸(𝑇)𝜆(0)(𝑇)𝐴0𝑇𝑥0𝑒+𝜀𝐴0𝑇𝑆1𝐶𝑆1𝐶+𝐿1𝑥0𝑁𝑥+𝜀𝐿3𝑥0𝑁𝑥.(3.11) Hence, if we denote the coefficient matrix in (3.11) by 𝑀, then it can be written as𝑀=𝐸𝜀𝐿20𝑁𝑒𝐴0𝑇𝜀𝐿40=𝐾𝑁0𝐸𝐾(𝜀)01𝐸,(3.12) where𝐾(𝜀)=𝐸𝜀𝐿2𝑁𝑒𝐴0𝑇𝜀𝐿4,𝐾1=𝑁0.(3.13) It is not difficult to show that𝐾𝐾(𝜀)01𝐸1=𝐾1𝐾(𝜀)01𝐾1(𝜀)𝐸.(3.14) Hence, using the fact that𝐾1(𝜀)𝐾1(0)𝐾1̇(0)𝐾(0)𝐾1(0)𝜀(3.15) we obtain the inverse matrix 𝑀1𝐸+𝐿2𝑒𝐴0𝑇𝑁𝜀𝐿2𝑒𝐴0𝑇𝑒𝜀0𝐴0𝑇𝑁𝑒𝐴0𝑇𝐿4𝑒𝐴0𝑇𝑁𝜀+𝑒𝐴0𝑇𝑁𝐿2𝑒𝐴0𝑇𝑁𝜀𝑒𝐴0𝑇𝑒𝐴0𝑇𝑁𝐿2𝑒𝐴0𝑇𝜀+𝑒𝐴0𝑇𝐿4𝜀0𝑁+𝑁𝐿2𝑒𝐴0𝑇𝑁𝜀𝑁𝐿2𝑒𝐴0𝑇𝜀𝐸(3.16) and, consequently, multiplying the both sides of (3.11) by 𝑀1 on the left, we obtain the following analytic formulae to determine values of 𝑥(𝑇),𝜆(0),𝜆(𝑇): =𝑒𝑥(𝑇)𝜆(0)𝜆(𝑇)𝐴0𝑇𝑥0𝑒𝐴0𝑇𝑁𝑒𝐴0𝑇𝑥0+𝑒𝐴0𝑇𝑁𝑥𝑁𝑒𝐴0𝑇𝑥0𝑁𝑥𝑒+𝜀𝐴0𝑇𝑆1𝐶𝑆1𝐶+𝐿1𝑥0+𝐿2𝑒𝐴0𝑇𝑁𝑒𝐴0𝑇𝑥0𝐿2𝑒𝐴0𝑇𝑁𝑥𝑒𝐴0𝑇𝑁𝑒𝐴0𝑇𝑆1𝐶𝑒𝐴0𝑇𝑁𝑆1𝐶+𝑒𝐴0𝑇𝑁𝐿1𝑥0+𝑒𝐴0𝑇𝐿3𝑥0𝑒𝐴0𝑇𝐿4𝑒𝐴0𝑇𝑁𝑒𝐴0𝑇𝑥0+𝑒𝐴0𝑇𝑁𝐿2𝑒𝐴0𝑇𝑁𝑒𝐴0𝑇𝑥0𝑒𝐴0𝑇𝑁𝐿2𝑒𝐴0𝑇𝑁𝑥+𝑒𝐴0𝑇𝐿4𝑒𝐴0𝑇𝑁𝑥𝑁𝐿2𝑒𝐴0𝑇𝑁𝑒𝐴0𝑇𝑥0𝑁𝐿2𝑒𝐴0𝑇𝑁𝑥+𝑁𝐿1𝑥0+𝑁𝑒𝐴0𝑇𝑆1𝐶𝑁𝑆1𝐶.(3.17) Further, as in (3.1), using determined in (3.17) value 𝜆(0), it is possible to find 𝑥(𝑡𝑖),𝜆(𝑡𝑖) in the form𝑥𝑡𝑖𝜆𝑡𝑖=𝑒𝐴0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴1𝑡𝑖𝑥0+𝑒𝜆(0)𝐴0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴1𝐴𝐸00𝐸0+𝜀𝐴1𝜀𝐵𝑅1𝐵0𝐴0𝜀𝐴110𝐶𝜀(3.18) for every 𝑡𝑖[0𝑇].

Therefore, expansions for 𝑥(𝑡𝑖)and 𝑢(𝑡𝑖) with respect to 𝜀 can be obtained in the form𝑥𝑡𝑖=𝑒𝐴0𝑡𝑖𝑒+𝜀𝐴0𝑡𝑖𝑆1𝐶𝑆1𝐶+𝐿11𝑥0𝐿22𝑒𝐴0𝑡𝑖𝑁𝑒𝐴0𝑡𝑖𝑥0+𝐿22𝑒𝐴0𝑡𝑖𝑁𝑥,𝜆𝑡𝑖=𝑁𝑒𝐴0𝑇𝑥0+𝑁𝑥+𝜀𝑁𝐿22𝑒𝐴0𝑡𝑖𝑁𝑒𝐴0𝑡𝑖𝑥0𝑁𝐿22𝑒𝐴0𝑡𝑖𝑁𝑥𝑁𝐿11𝑥0𝑁𝑒𝐴0𝑡𝑖𝑆1𝐶𝑁𝑆1𝐶,𝑢𝑡𝑖=𝑅1𝑡𝐵𝜆𝑖.(3.19) Here 𝐿11,𝐿22 are block matrices of matrix-solutions 𝐿𝑖 of Sylvester's equation [12, 13]𝐻1𝐿𝑖𝐿𝑖𝐻1=𝑒𝐻1𝑡𝑖𝐻2𝐻2𝑒𝐻1𝑡𝑖(3.20) for every 𝑡𝑖.

4. Computational Experiments

On the basis of the obtained formulae, we have developed an algorithm and computer program using MATLAB system. In [5] an algorithm for solving problem (2.8)–(2.10) for the case when 𝐶=0,𝑁=2 is developed. It is clear that for 𝑁>2 the algorithm will yield more accurate solutions. However, for computational comparison of the asymptotic method with the method described in [5], authors have considered only the case when𝐶=0,𝑁=2. It is possible to prove that in this case Sylvester's equation has infinitely many solutions. While using a special program in MATLAB one particular solution is determined.

In Figures 1, 2, 3, and 4 the graphs of the functions 𝑃1,𝑄1,𝑃2 and𝑄2 for both algorithms are given. Namely, the graphs for the functions obtained using the algorithm introduced in [5, 14, 15] are shown in solid lines and by the algorithm elaborated using the asymptotic method are given in dashed lines. A comparative analysis shows that the results for the functions𝑃1,𝑄1,𝑃2, and 𝑄2 for both algorithms coincide with a sufficiently high accuracy.

fig1
Figure 1: Dependence of 𝑃1 on t.
fig2
Figure 2: Dependence of 𝑃2 on t.
fig3
Figure 3: Dependence of 𝑄1 on t.
fig4
Figure 4: Dependence of 𝑄2 on 𝑡.

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