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На основi розробленог математичног моде-лi двоступеневог сепараци нафти було проведено iдентифiкацiю параметрiв, що дало змогу ство-рити iмiтацiйну модель для першого та другого ступетв сепараци. Дослгджена числовим методом в програмi Ма^аЬ iмiтацiйна модель може бути використана для синтезу ефективних систем керування процесом двоступеневог сепараци та створення математичних моделей у термтах «вхгд-вихгд»
Ключовi слова: синтез систем, двоступенева сепаращя, числовий метод, iдентифiкацiя пара-
метрiв, iмiтацiйна модель
□-□
На основе разработанной математической модели двухступенчатой сепарации нефти была проведена идентификация ее параметров, что позволило создать имитационную модель для первой и второй ступеней сепарации. Исследованная численным методом в программе Ма^аЬ имитационная математическая модель может служить основой для синтеза эффективных систем управления процессом двухступенчатой сепарации и создания математических моделей в терминах «вход-выход»
Ключевые слова: синтез систем, двухступенчатая сепарация, численный метод, идентификация параметров, имитационная модель -□ □-
UDC 681: 519.7
|DOI: 10.15587/1729-4061.2018.121619]
DEVELOPMENT OF THE IMITATION MODEL OF THE TWO-STAGE SEPARATION PROCESS OF OIL
M. Gorbiychuk
Doctor of Technical Sciences, Professor* E-mail: [email protected] D. Povarchuk Postgraduate student* E-mail: [email protected] T. Humeniuk PhD, Associate Professor* E-mail: [email protected] N. Lazoriv* E-mail: [email protected] *Department of computer systems and networks Ivano-Frankivsk National Technical University of
Oil and Gas
Karpatska str., 15, Ivano-Frankovsk, Ukraine, 76019
1. Introduction
Separation systems carry out operations on collection, preparation and storage of oil. One of the main functions is the transport of well products under the action of reservoir pressure or at the expense of the energy of pumps to the point of oil preparation [1]. In these systems, there is a separation of gas from oil and supply of it to consumers, as well as a free water separation from well products (in the case of flooded oil).
The efficiency of the separation process to a large extent is determined by the methods and algorithms of the automatic control systems. The process of separation of oil proceeds under the influence of numerous obstacles and inherent in it complex internal communications [2]. Therefore, the actual task will be the development of effective systems of automatic control, which will be based on adequate mathematical models, which quantitatively and qualitatively will characterize the process of separation in general.
2. Literature review and problem statement
The production economy shows that oil separation processes should be as efficient as possible, since separated oil can contain partially captured water or gas and vice versa. This means that any gas-oil emulsion must be detached to the commercial state, the water content must not exceed 1 % [1]. The reflection of the scientific problem of separation of oil can be found in the papers [2, 3]. However, the greatest
problem of the current state of oil separation processes is the lack of integrated automated control systems, where the parameters of the flow of each component with the enabled function of flow management must be controlled. An integral part of such complexes are two-stage oil separation systems [4]. Typically, single-stage separation systems are used for oil separation at primary oil preparation plants. But the use of such equipment is fair and reasonable for oil with low water and gas content, that is, less than 1/3 of the total mixture. When the total content of water and gas is significantly higher than the oil content or equal, it is expedient to use two-stage separation systems, which will significantly increase the efficiency of the oil separation process [4, 5].
In the first stage of the creation of two-stage separation systems, the issue of mathematical modeling will be put forward [5, 10], which includes such important factors as productivity, gas factor, solubility factor, separation coefficient, geometric dimensions of the separator, etc. The study of oil separation by computer simulation was performed in the paper [6], but the disadvantage was the exclusion of water from the structure of the model, which significantly influences the key parameters: pressure, temperature, level, expense. The development of computer models of the separation process is necessary to provide engineers with valuable tools for obtaining more reliable qualitative and quantitative solutions for the further processing of oil and exploitation of oil fields. In the paper [7], a detailed description of the theory of motion of gas particles in a gravitational field has been made which significantly influences the performance of the separator, but this effect does not investigate the effect on multi-stage separation
©
systems. One of the first mathematical models that describe the process in terms of "input-output" and is suitable for the synthesis of automated control systems was the model suggested in the paper [8]. In the scientific paper [9], the mutual influence of the level of fluid and pressure in the separator was taken into account. But the disadvantage is that the last two papers are not sufficiently substantiated and are intended only for single-stage separation. Two-stage separation systems are considered to be the most effective, since the operation of the first stage significantly affects the efficiency of the second one. In the scientific paper [10], the work of the two-stage separation system is described in details, but according to the authors, it would be expedient to use a system with horizontally placed separators B-1 and B-2, as this can significantly affect the overall separation factor in subsequent calculations [11, 12]. Consequently, the issue of two-stage separation systems will remain inadequate. Therefore, based on the literature review, an imitation model of a two-stage separation system consisting of two horizontally placed separators and a vertical oil storage tank is suggested.
scheme of the simulation model of the two-stage separation unit is shown in Fig. 1.
Production of wells of an oil and gas deposit at a pressure of P0=4.0 MPa enters the horizontal separator B-1. Oil from the separator B-1 at a pressure of Pi=1.6 MPa enters the second degree of separation into a horizontal separator B-2 through the actuator (executive device) EDh. Gas from the separator B-1 enters the compressive compressor station "CCS" through the actuator EDp (Fig. 1).
Fig. 1. Technological scheme of the simulation model of the two-stage separation unit
3. The aim and objectives of the study
The aim of the work is to develop a numerical simulation model of the two-stage oil separation model in the MatLab program, taking into account the features of the technological process for the synthesis of efficient systems for automatic control of the separation process and the further development of mathematical models in terms of "input-output". In other words, it is about determining the mutual influence between the technological parameters of each of the separators of the two-stage separation system.
To achieve this aim, it was necessary to fulfill the following objectives:
- the creation of a mathematical model of the material balance of a two-stage separation for the first B-1 and the second B-2 horizontally placed separators;
- the identification of parameters of the mathematical model of two-stage separation for further simulation in MatLab;
- the development of a simulation model of two-stage separation and examine it numerically in the MatLab program to change the key parameters of the level and pressure in each separator.
4. The object and methods of research of the separation unit
The object of research is the technological process of oil separation. The two-stage separation system (Fig. 1), which contains the first and second degree of separation [11], is taken into consideration.
Working pressure at the first stage of separation is created by reservoir pressure at the outlet of the well. Such pressures range from 4.1 to 8.3 MPa [11]. The technological
Oil from the separator B-2 at a pressure of P2=0.6 MPa enters the reservoirs of the GSR. Further oil through the pipeline is provided at the entrance to the oil pumping station and to the oil pipeline, where it is shipped to the tanker. Gas from B-2 at a pressure of P2=0.6 MPa enters the "CCS" through the actuator EDh1. At the same time, gas from B-2 is supplied also for industrial and technological needs (ITN).
The main technical and technological parameters of the separation unit are given in Table 1.
Table 1
The main technical and technological parameters
Unit of measurement Parameters
No. Device Parameter Maximum Installed
name name permissible technologically
pressure MPa 1.6 0.6+1.5
1 Separator B-1 temperature °C -40++100 -20++40
volume m3 100 30
pressure MPa 0.6 0.1+0.6
2 Separator B-2 temperature °C -40++100 -20++40
volume m3 100 30
3 High-pressure gas diameter length mm m - 89 1,700
line pressure MPa 16.0 8.0+15.0
4 Gas line CCS diameter length pressure mm m MPa 6.4 159 930 3.5+4.5
5 Gas line ITN diameter length mm m - 219 5,740
pressure MPa 0.6 0.52+0.57
Some characteristics of oil and gas mixture together with oil and gas parameters after separation are shown in Table 2.
Table 2
Characteristics of oil and gas mixture together with oil and gas parameters after separation
Parameter name Designation Value Unit of measurement
Physical properties of crude oil
Oil density Po 843 kg/m3
Mass fraction of water xw 0.5 0/ %
The gas separated from the equipment
Gas density at t=20 °C Pg 0.819 kg/m3
the separator B-1; M0=V0po, where po is the density of oil; so is the oil separation factor;
% f =
1
Af =Xf
f D,
f 1
2F/'
The necessary efficiency of the separation process is achieved by stabilizing such mode parameters as the level of fluid and pressure in the separator [12]. Therefore, material flows in a two-stage separation system are described using the equations of the material balance, respectively, for each separator individually.
where X is the coefficient of friction resistance, Df is the diameter of the inlet pipe, lf is the total length of the section, which includes the equivalent lengths of the local resistances, Ff is the cross-sectional area of the pipeline; pf, pf1 are the density of oil and gas mixtures entering the separation system and the second stage of separation, pg stands for gas density; P0 is the pressure at the inlet of the separator B-1; ag(U1)=KL1Kv(U1)^nc, where KL1 is the shutter release ratio, which is proportional to the command signal of the controller U1, KV(U1) is the throughput of the EDp actuator
(Fig. 1);
=
PcX
P„,
5. Mathematical model of operation of a two-stage separation unit
The two-stage separation unit has two horizontal separators B-1 and B-2 (Fig. 1).
Since the first stage B-1 separator is horizontal and cylindrical, the degree of its filling will be given by the equations [14]:
where pcv, Tcv, Pcv stand for gas density, temperature and pressure under normal conditions (Tnc=273 K and Pnc= =0.1013 MPa); Pg2 is pressure after shutting down of the regulating body (RB); z is the gas compressibility factor, which is calculated at temperature T1 and pressure
- P,+ Pg 2
p _ _J_g2 ;
g 2 ;
1
v (h)=n
h-i
h-i
i -
h-i
\2
(1)
(2)
1
dP1 _
dt ~ 1 -vp(h)
+_P_ 0s m0
VV g 0
cf 1
%/VP/ (p0 -p1 ) -
a s (U ) ¡(p12 - P% )
zT1
M a (U2 )Jpf 1 (p1+P/1 gh 10-6 -p2;
(3)
dh
dt qp (h) Mo
xL fP/ (Po-P1 ) -a0 (U2^P/1 (P1+ P/ 1gh 10-6-P2
(4)
ao(U2)=KL2Kv(U2), where KL2 is the proportionality factor between the transfer of the stem RB of the executive device EDh and the command signal U2, Kv(U2) is the throughput of the actuator EDh.
The mathematical model of the material balance for the separator B-2 will be determined by the equations [14]:
where h is the oil level in the separator B-1; rs is the radius of the separator B-1.
Consequently, the mathematical model for the first degree of separation will be as follows [14]:
dP
1
0g1
VV g1
dt 1 -vp1 (h1 ) X/Pf 1 (P1 + Pf 1gh 10-6 -P
M
ao (U2
A
M„.
a„1 (US2 ^P0 (P2 +Pogh1 10-6 -P3 ) -
a ,U ^
dh1
dt q^K(U2Up/ 1 (p+Pf 1gh10-6^
-a„1 (Us 2 ^ (P2 +P„gh1 10-6 - P3 ) ),
V (h1 ) = -n
'A.-1I+Îh-1I h - h -1
/ 1 \
2
h K,
V's1 /
(5)
(6)
,(7)
where vp(h) and qp(h) are calculated by formulas (1) and (2).
Other notations in equations (3) and (4) will be as follows: eg stands for the gas separation factor; Q=V0/zRgT1, where Vo is the total volume of the separator B-1, z is the gas compressibility factor, Rg is the gas constant; T1 stands for gas temperature in the separator B-1; Pi is gas pressure in
qp1 (h1 J1 -
2
h -1
nr.,
(8)
where h1 is the oil level in the separator B-2; rs1 is the diameter of the separator B-2.
g
As the initial conditions for differential equations (3)-(6), we take the fixed values of the corresponding quantities - p (0) = P1(0), h (0) = h(0), P2 (0) = P2(0) and h (0) = h1(0).
In equations (5) and (6), the following designations are taken: P2 is gas pressure in the separator B-2; eg1=(1-rg1) G1 stands for the gas separation factor, where rg1 is the efficiency indicator of the second stage of separation, G1 is the gas factor of the second stage of separation; 0g1 = =V01/RgT2, where V01 is the total volume of the separator B-2, T2 is the gas temperature in the separator B-2; M01=V01po is the mass of oil in the separator B-2 at its full filling; ao1(Us2)=KsLKsv(Us2), where KsL is the proportionality factor between the transfer of the stem RB of the operating device EDh1 and the command signal Us2, Ksv(Us2) is the throughput of the actuator EDh1 (Fig. 1); P3 is an average pressure in reservoirs (Fig. 1); Qgi=V01/RgT2, where T2 is the gas temperature in the separator B-2; ag1('Us1)=Ks1Kg1(Us1) £,cv, where Ks1 is the gate transfer coefficient of the EDp1 actuator, which is proportional to the command signal of the controller Us1, Kgi(Usi) is the throughput of the EDp1 actuator (Fig. 1); Pg3 is the gas pressure after the RB of the EDp1 actuator of the second stage of separation.
Thus, the mathematical model of a two-stage separation system, which has two cylindrical horizontal separators B-1 and B-2 in its composition, is given by systems of differential nonlinear equations (3)-(6). The peculiarity of the two-stage separation system is that the output (regulated) quantities Pj and h of the first degree of separation, as well as the control action U2, act as a perturbation for the second degree of separation. Such interference with the separation stages worsens the efficiency of the separation unit and requires systemic solutions to eliminate or reduce the effect of the first stage on the efficiency of the second stage. Such solutions can be found by developing structural schemes that will make it possible to compensate or weaken the effect of the first degree of separation on the second stage.
In formulas (3)-(6), it is taken into account that the pressure is measured in MPa, and the hydrostatic pressure pgh in Pa. The linearization was carried out to solve the problem of mathematical models of the two-stage separation system [15].
6. Identification of the parameters of the mathematical model of the separation system
The initial data for calculating the parameters of the mathematical models of the separation system are given in Table 3.
The calculation of the compressibility factor 2 of natural gas is carried out according to the modified Benedict-Web-ba-Rabin equation [8]:
23-22-a2-b=0, (9)
where
a=n((0.1237/x)-(0.3468/T2)-(0.1188/x4));
b=n2((0.0291/x2)-(0.0273/T3)+(0.0390/x5));
rc=(P+1.3340-4Pa)/Pkp is the given pressure; T=(tg+273)/Tkp is the given temperature; Pkp=4.67-0.1A is the critical pressure measured in MPa; Tkp=99.8+162.8A is the criti-
cal temperature measured in K; Pa is atmospheric pressure measured in mm Hg.; P is excess pressure of natural gas measured in MPa; tg is temperature of natural gas measured in °C.
Table 3
Initial data for calculating the parameters of the mathematical models of the separation system (the established mode of operation of the separation system)
Parameter name Designation Value Unit of measurement
Pressure at the inlet of the separator B-1 Po 4.0 MPa
Gas pressure in the separator B-1 P1 1.6 MPa
Gas pressure in the separator B-2 P2 0.6 MPa
Gas temperature at the entrance to the B-1 To 286 K
Oil density Po 843 kg/m3
Gas density at standard conditions Pcv 0.820 kg/m3
Gas pressure in GSR tanks P3 0.1 MPa
Gas pressure after EDp Pg2 0.85 MPa
Gas pressure after EDp1 Pg3 0.32 MPa
The volume of the separator B-1 Vo 100 m3
The volume of the separator B-2 V01 100 m3
Diameter of the separator B-1 rs 3.0 m
Diameter of the separator B-2 Ts1 3.0 m
Oil level in the separator B-1 h 1.51 m
Oil level in the separator B-2 hi 1.47 m
Diameter of the inlet pipeline Df 0.219 m
The volume of oil supplied to the separation unit Qf 3.71 m3/hour
Atmospheric pressure Pa 762 mm. Hg.
Mass fraction of water w 0.036 -
Productivity of the oil plant Gol 100 tons/day
Productivity of the gas plant Qgn 105 nm3/day
The relative gas density by air is calculated by the formula [8] A=pcv/1.205, where pcv is the density of natural gas under standard conditions.
The real root of the Benedict-Webba-Rabin equation determines the value of the gas compressibility factor.
The calculation of the parameters of the mathematical model of the separation system, given by the equations (3)-(6), was carried out according to a program written in the algorithmic language of the MatLab environment. The results of the calculations are summarized in Table 4.
The productivity of the separation unit for oil is Go/= =100 tons/day, for gas Qgn=105 nm3/day (Table 3), which we shall express in units of mass per unit of time according to the following formula
Gg=Qgnpcv/k,
where k is the conversion factor, which makes it possible to express Gg in kg/sec (k=86.400).
Then we will determine the mass fraction of gas in the oil and gas mixture as follows
x=Gg/G0ki+Gg, where k1=103/k.
Results of calculations of the parameters of the model (3)-(6)
Table 4 mathematical
Parameter name Designation Value Unit of measurement
Gas compressibility factor z 0.9979 -
Degree of filling of the separator B-1 Vp(h) 0.504 -
Degree of filling of the separator B-2 Vp1(h0 0.487 -
Density of the oil and gas mixture py 70.47 kg/m3
- 0g 830.25 kg/MPa
The intensity of filling of the separator B-1 qp(h) 0.424 m-1
The intensity of filling of the separator B-2 qp1(hù 0.423 m-1
Mass fraction of gas in oil and gas mixture x 0.45 -
Natural gas solubility factor «1 0.3593 MPa-1
Productivity of the separator B-1 (of gas) Gg1 0.6453 kg/sec
Productivity of the separator B-2 (of gas) Cg2 0.3037 kg/sec
Gas density under normal conditions pK 0.8801 kg/m3
Productivity of the separator B-1 (of gas) Qg1 6.3356-104 nm3/day
Productivity of the separator B-2 (of gas) Qg2 2.9818-104 nm3/day
- ag(U1) 0.4761 kg/sec (K/MPa)1/2
Gas separation factor % 0.3064 -
Productivity of the separator B-1 on the liquid Gy1 1.4433 kg/sec
Separation factor of the separator B-1 Ey1 0.6852 -
Productivity of the separator B-2 for oil Go2 1.1574 kg/sec
Mass fraction of gas (separator B-2) X1 0.1981 -
- a0(U2) 0.1881 kg/sec (K/MPa)1/2
The liquid density in the separator B-1 py1 60.30 kg/m3
Mass of liquid at h=D where D is the diameter of the separator B-1 Mo 6.0301-103 kg
- 1.9672 kg/sec (K/MPa)1/2
Gas separation factor (second stage) Eg1 0.2079 -
- 0g1 829.05 kg/MPa
Mass of liquid at h1 =D1, where Dy is the diameter of the separator B-2 M01 84300 kg
- a01(Us2) 0.0703 kg/sec-(m3/ kg-MPa)1/2
- ag1(Us1) 10.1204 kg/sec (K/MPa)1/2
We will determine the density of the oil and gas mixture py as far as we know the proportion of gas in the oil and gas mixture x. The results of the calculations by the formula (10) are recorded in Table 4.
Since gas is supplied to the first stage of separation in the amount of Gg, and Gy=Go;k1+Gg, then by comparing with the formula (10), we arrive at the conclusion that G0=x.
The efficiency of the first stage of gas separation Qg1 can be calculated using the following formula
Qgi = Qy (1-w)(G '0- a^), (11)
where a1 is the solubility factor of natural gas in oil. Formula (11) can be represented as follows:
Qg1pg = Qf Py (Pg / Py M1- W >(G 'o -ap),
(12)
where py is the density of the gas is reduced to the conditions of separation of the first degree.
Since Ggi = Qgipg and Gy=Qypy and taking into account that G0 = G0•py/pg, the equation (12) will take the form:
Ggi = Gf(1-wy(Go-pg/pf(aipi)).
(13)
The density of the gas for the conditions of separation in B-1 is calculated by a formula that is similar to the formula:
Pg=(P/zTi)lc;
(14)
where Ycv=(pcv'Tcv)/Pcv.
The gas compressibility factor 2 is calculated from the Benedict-Webb-Rabin equation at a pressure P1 and temperature T1, while T1=T0. The solubility factor of natural gas a1 was determined from the graphical dependence, which is given in [2] (Table 4).
The productivity of the B-2 separator for gas Gg2 will be determined as the difference between the overall performance of the separation unit and the performance of the separator B-1
Gg2 = Gg- G.
g-Gg1.
(15)
The calculated values Gg1 and Gg2 are recorded in Table. 4. Table 4 contains the data on the performance (volumetric) of Qg1 and Qg2 of B-1 and B-2 separators, which are brought to normal conditions under the following formulas
Qg,i =k(Gg,i/pn), i=1, 2,
(16)
where Qgi has a dimension of nm3/day.
Recalculation of gas density pcv to normal conditions (Pcv=0.1013 MPa, Tcv=273 K) was carried out according to the following formula:
pcv pcv(PcvTcv/PcvTcv)•
(17)
Taking into account that the established mode of the separation system is considered, we find that:
%f = f Vpf (p0 - P1 ).
(18)
Oil and gas mixture is supplied to the second stage of separation, which we calculate as the difference between the mass amount of oil and gas mixture Gy, supplied to the
separation system and the amount of gas Gg1 that has been isolated at the first stage of separation,
Gf1 = Gf-Gg1.
(19)
Then we determine the performance of the separator B-2 for oil by the formula, which is similar to the formula (19):
Go2 = Gf1-Gg2.
(20)
The mass fraction of gas in the oil and gas mixture entering the second stage of separation will be as follows:
X1 = Gg2/Gf1.
(21)
Since the established mode of the separation system is considered, we obtain the following system of algebraic equations from the equations (3)-(6), which describe the dynamics of the separation system:
« g (U1 )t
1 P - Pg22)
e„ V ~zTx
+«„ (U2) Mpryj Pf 1 (P1+P/ 1gh 10-6 - P2) =
'Mo
fg +P
eg M0
v g 0
4fVPf (p0 - p1),
ao (U2 ^Pf 1 (P1 + Pf 1gh 10-6-P2)
= f f Pf (P0 -P1),
(22)
(23)
ao (U2
fg!+P
Veg1 M01 y
Jpf1 (P1 + Pf 1 gh 10-6 -P2]
-a g 1 Ui
P
«01 (Us 2) ^ (P2 + Poght 10-6 - P3) = 0, (24)
ao (U2 ^Pf 1 (P1 + Pf 1gh 10-6-P2) --a„1 (Us 2 (P2 + P0gh[ 10-6 - P3) = 0.
1 /(P12 - Pg22)
11"e J zT1 '
«12=-M-JPf 1 (P1+Pf 1gh P2),
«22 = yjpf 1 (P1 + Pf 1gh P2 ), a13=0, a14=0, a21=0, a23=0, a24=0,
a31=0, a41=0, a43=0,
eg1
V g1
P
Mr],
Vp/ 1 (P1 + Pf 1gh P2 ),
= VPo (P2 + Pogh1 -P3),
1 IP2 - P2
1 M7? rc
g 3
e gj T2
«42 =a/P/ 1 (P1 +Pf 1gh P2 ), «44 = ^Po (P2 +Pogh1 -P3 ),
b =
e g
g
m„
1
4 fV p f (p0 - p1),
b2 = Ef fPf (P0 -P1), b3=0, b4=0,
X1=ag(U1), X2=a0(U2),
X3=ag1(US1), X4=ao1(Us2). a11X1+a12X2+a13X3+a14 X4=b1, a21X1 + a22X2 + a23X3+a24X4=b2, a31X1+a32X2+a33X3+a34X4=b3,
a41x1+a42x2+a43x3+a44x4=b4.
(26)
(25)
The system of equations (25) was solved by the Gauss's reverse method with the choice of the maximal element [13].
When recording equations (22)-(25), it was taken into account that the pressures P1, P2, P3, Pg2 i Pg3 are measured in MPa, and the hydrostatic pressure P=pgh is measured in Pa.
When calculating the values of 0g i 0g1 (Table 4), we assume that the temperatures in the separators B-1 and B-2 are the same. The gas constant Rg for natural gas, which is included in the expressions 0g i 0g1 (Table 4), is calculated by the following formula [8]:
The system of equations (22)-(25) is linear according to the unknown values ag(U1), a0(U2), ag1(Us1), and ao1(Us2). We introduce the following designations:
Rg=(288.15/A>10-6, where Rg is measured in MJ/kg-K.
(27)
7. Results of studies of a two-stage separation unit
The mathematical model (3)-(6) of the separation system will be solved with the following assumptions:
- pressures P0, Pg2 and Pg3 are accepted as stable;
- the temperature of the gas stream entering the separation system and the temperature of the products in the separators B-1 and B-2 are the same and unchanged in time for the period of the transition time (T0 =T2 =T1);
- we neglect the influence of temperature on the density of oil;
- the gas compressibility factor is computed at pressure
a™ =
32
a
34
Industry control systems
Pg =
P + P
g 2
2
and temperature T1;
- gas in the separator B-2 is considered ideal;
- we consider mass particles of gas x and x1 in the separators B-1 and B-2 unchanged.
The input values that cause the change in the output values P1 and P3 and levels h1 and h2 in the separators B-1 and B-2 will be considered as changes in the values of ag (U1), ao (U2), ag1 (Us;) and ao1 (US2).
We use the Runne-Kutta method [13] to solve the system of differential equations that describe the dynamics of the separation unit, which in the vector form implements the following iterative procedure:
t;w=f (% uw=f (y+f ,
f(k) = f f y + h^72(k)^, f4(k) = f (ft + hjf),
y*+1=F*+| (F+2 7f+27^+
where
y =
Pi h P2
A.
is the vector of output quantities;
7 (y )=
1
1 -v„
£g P1 + £ f <
0g M° 71 VV g °
4fVP7 (P° - Pi) -
a. (U) ¡(Pi2 -P2)
571
-MM- «0 (U 2 ^71 (P1+ P71 gh 1°-6-P2,
1
»(h) M° 1
£77 VP7 (P° - P1) -
-a0 (U 1 (P1+ P71gh 10-6 - P2)
+A_
Vv9g1 M°1
1 -VP1 (A) X/P71 (P+ P71.h 1°-6-P2
a o (U2 )X
M
-ao1 (U 2 UPo (P2 +Pogh 1°-6 -P3) -
-t a g 1 u )j
(ao(u 2
-ao1 U 2 ^Po (P2 + Pogh1 1°-6 - P3))
is a vector-function, the components of which are right parts of the system of differential equations (3)-(6).
The step of discreteness ht is calculated by the formula ht=(tf-t0)/n, where t0, tf is the start and end time; n is the number of iterations in the computational process.
The program for solving the system of differential equations (3)-(6) is written in the algorithmic language of the MatLab package.
As an example of solving the system of differential equations (3)-(6), consider the change in the output values P;, h, P2 and h; in time, depending on the change in the input value ao (U2), which is defined as follows:
ao(U2)=ao(U 2°))+Aao(U2).
(29)
The increment of the value Aao (U2) will be calculated as Aao (U2)=xao (Uf).
Taking into account the value Aao (U2), the formula (3°) will take the form of:
ao(U2)=ao(U2°))(1+x).
(3°)
(28)
The values of ag (U;), ao (U2), ag; (Usi) and aoi (Us2), in essence, are the throughput characteristics of the regulatory bodies of the corresponding actuators (Fig. 1).
Table 5 establishes the correspondence between the parameters of the mathematical model (3)-(6) of the separation system and machine variables. In the same table, the numerical values of the values included in the model (3)-(6) are given.
In the process of solving the mathematical model (3)-(6), the following values of variables selected by the user were selected:
- the number of iterations n=500;
- the duration of the transition process tf=800 sec;
- the change value of ao ( U20) ) - X=1.0.
The result of a numerical solution of a mathematical model is shown in Fig. 2-4.
£ 4r
PQ 3-5 3
* 2.5 . S
1.5
/ /
/ /
/ /
/
0 100 200 300 400 500 600 700 800 Time t, s
Fig. 2. The change of pressure P1(t) in the separator B-1 as a function of time t
E 1.65 ® 1.625
Q
Ü 1.6
£1.575 . S
* 1.55 !
<«1.525
1.5
________ JL /
/
/
^ *"0 100 200 300 400 500 600 700 800 Time t, s
Fig. 3. The change of the fluid level h(t) in the separator B-1
Table 5
Correspondence between the parameters of the mathematical model and the machine variables
Parameter name Designation Machine variable Value
The degree of filling of the separator B-1 Vp(h) Vp 0.5042
The degree of filling of the separator B-2 Vpi(hi) Vp1 0.4823
Density of oil and gas mixture Pf ro f 70.47
- 0« teta g 830.2487
The intensity of change in the degree of filling of the separator B-1 qp(h) Q 0.424i
The intensity of change in the degree of filling of the separator B-2 qpi(hi) qi 0.4243
Gas solubility factor ai alphal 0.44
Productivity of the separator B-1 by gas (mass) G«l Ggl 0.6453
Productivity of the separator B-2 by gas (mass) g«2 Gg2 0.3037
- a«(Ul) alphag U1 0.476i
Gas separation factor ePs_g 0.3064
Productivity of the separator B-1 by liquid Gfi Gfl i.46ii
Separation factor of the separator B-1 Efi eps_f1 0.6936
Productivity of the separator B-2 by oil Go2 Go2 i.i574
Density of liquid in the separator B-1 Pfi ro fl 60.30
Mass of liquid at h=D, where D is the diameter of the separator B-1 Mo M0 6.030M03
- a0(U2) alphaN_U2 0.i88i
- f ksi f 6.9298
Gas separation factor (second separation segment) E«i ePs_g1 0.2079
- 0«i teta gl 829.05
Mass of liquid at h1=D1, where D1 is the diameter of the separator B-2 Moi M01 84.300
- aoi(Us2) alphaNl US2 0.0703
- a«i(Usi) alphaG1_US1 i0.i204
Oil density Po ro ol 843
=3
<N
03
0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35
0
100 200
300 400 Time t, s
500 600 700 800
Fig. 4. The change of pressure P2 in the separator B-2
m1.475
<N
m
O
1.47
1.465
1.46
■ S3 1.455
.a -a
1.445
>
o J
1.45
1.44
100 200
300 400 Time t,
500 600 700 800
Fig. 5. The change of the fluid level hi (t) in the separator B-2
The above graphs (Fig. 2-5) for the steady-state operation of the two-stage separation system are the result of a program that is written in the Matlab algorithmic language.
8. Discussion of the results of the study of the simulation model of the process of two-stage oil separation
The mathematical model of the process of two-stage separation is obtained, which, in contrast to the known models [10, 13], takes into account the interaction of the first and second stages of separation. The simulation model of the two-stage separation system is investigated by a numerical method in the MatLab's mathematical laboratory to change the key parameters of the level and pressure in each separator. The results obtained in the form of graphs (Fig. 2-5) describe the dynamics of transients for each separator individually, which collectively reflects the work of the system in general.
Due to the combination of two horizontally placed B-1 and B-2 separators and an oil storage tank, functional dependencies were obtained that allow establishing the interconnection of the technological parameters, in other words, the change in the value of the parameters at the second stage of separation will depend on the first one and vice versa. The advantage of such use of technological equipment is the possibility of improving the one-stage technological process of oil separation, in particular, increasing the overall oil separation factor. The study was conducted with the following assumptions:
- pressures P0, Pg2 and Pg3 are accepted as stable;
- the temperature of the gas stream entering the separation system and the temperature of the products in the
0
s
separators B-1 and B-2 are the same and unchanged in time for the period of the transition time (T0=T2=T;);
- the gas compressibility factor is computed at pressure and temperature T1;
- we considered gas in the separator B-2 ideal;
- we considered mass particles of gas x and x; in the separators B-1 and B-2 unchanged. The disadvantage is the neglect of the influence of temperature on the density of oil.
This simulation model is investigated by a numerical method in MatLab and can be used to synthesize effective control systems for the process of two-stage separation and to create mathematical models in terms of "input-output". Difficulties in further research may be low informing about the processes associated with oil production and not taking into account new perturbations that can influence the system in general.
9. Conclusions
1. A mathematical model of the material balance of the two-stage separation for the first B-1 and the second B-2 horizontally placed separators is created, which takes into account the interaction and mutual influence of the first and second stages of separation. Automated control systems that operate on the principle of negative feedback were used to stabilize the main technological parameters of oil separation. Since the deviations of the regulated quantities from their prescribed values are small, this allowed linearizing the nonlinear model of the two-stage process of oil separation.
2. The parameters of the mathematical system of two-stage separation for simulation modeling are identified. All values of physical quantities are obtained for the steady mode of operation of the two-stage separation system. During the identification of the parameters, it was assumed that the pressures P;, P2, P3, Pg2 and Pg3 are measured in MPa, and the hydrostatic pressure P=pgh is measured in Pa, and it is assumed that the temperatures in the separators B-1 and B-2 are the same when calculating the values of 0g and 0g;.
3. As a result of the research of the newly created simulation model in the MatLab program, it was determined that the increase in pressure P; is the increase in the fluid level h in the separator B-1 from 1.51 m to 1.6319 m. It was also established that an increase in the hydraulic resistance due to the closure of the regulatory body of the executive mechanism of the EDh, which is mounted on the initial line of the separator B-1, causes an increase in pressure P; from 1.6 MPa to 3.9970 MPa. On the basis of changes of the technological parameters P; and h in the separator B-2, it is obvious that the liquid level h1 is from 1.47 m to 1.4448 m due to the decrease in the flow of liquid from the separator B-1. Such a decrease in the liquid level in the separator B-2 entails a reduction in the pressure P2 from 0.6 MPa to 0.3210 MPa. This means that the received scientific result in the form of functional dependencies allows us to establish the relationship of technological parameters, in other words, the change of the value of the parameters at the second stage of separation will depend on the first one and vice versa. Thus, the applied aspect of using the obtained scientific result is the possibility of improving the typical technological process of oil separation.
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