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6ТЕХНИЧЕСКИЕ НАУКИ
УДК 51-72
ИССЛЕДОВАНИЕ МОДЕЛИ НЕСТАЦИОНАРНОЙ ФИЛЬТРАЦИИ НА ОСНОВЕ ТЕОРЕТИКО-ГРУППОВОГО МЕТОДА ТЕОРИИ ЛИ
ДЖАЛИЛОВА РАХИМА КУРБАНОВНА
Доктор философии по физико-математическим наукам, Доцент Азербайджанского Государственного Педагогического Университета Азербайджанская Республика, г. Баку.
Abstract: On the example of one problem, using the group-theoretic group theory, Ly's theory, the influence of unbalance effects on oil recovery in layered seams is investigated. In this case the first group of factors in traditional flood is unregular:pervisity,porosity,viscosity of oil, configuration,mass expences and porosity of deposit,the first group is concerning to the manager parametres of the production oilfields:geometry and porosity of tuile wells,the system ofplacing of extract and supercharge wells, inflexions ofpressure and ets. Different dynamic of extraction oil and corresponding tempo of the selection set in motion to different tehnico-economics indexes of the production.The dynamic of extracting oil is the unlinear on the stages of the intensive drilling of deposits and increasing of extraction oil, stabilization of extraction oil with influences on the layer, degradation of extraction oil ,emaciation of reserves oil and energy resourses of layer.
Key words: differensial equations, oil recovery, group analysis, unbalance effects, relaxation parameters, unlinear.
Development of methods mathematics modeling for productioning oilfield ,growthing of hydrodynamic models of layer ,so creating of system automation project for working allowed the next step in the direction of determining and growthing of calculate optimal dynamic of extracting oil. For the production of oilfield there are the objective conditions of the creating different dynamic for extracting oil which consisting in being reserves of layer energy for intensification extracting oil in the first period of the production and so for the tehnical extent of resources artificial influence on the oil layers and racing from the layer.The main factors ,which determining the dynamic and the levels of changing oil and liquid can divorce on geological, which characterize the collecor qualiies of layers and filling their fluids and the tehnology conditions of the production oilfields. In this case the first group of factors in traditional flood is unregular:pervisity,porosity,viscosity of oil, configuration,mass expences and porosity of deposit,the first group is concerning to the manager parametres of the production oilfields:geometry and porosity of tuile wells,the system of placing of extract and supercharge wells, inflexions of pressure and ets. Different dynamic of extraction oil and corresponding tempo of the selection set in motion to different tehnico-economics indexes of the production.The dynamic of extracting oil is the unlinear on the stages of the intensive drilling of deposits and increasing of extraction oil, stabilization of extraction oil with influences on the layer, degradation of extraction oil ,emaciation of reserves oil and energy resourses of layer.
Submit for concideration the mathematics model of two-phases in the pit zone:
dp
(1)
(x, x), x e ^
0
7 = (ki,k2), 7 = (fi,f ) f = kt /(Uia)
д -viscosity, k -relative pervious
i=1 apply to oil , i=2 to liquid, 77 -intensive of selection ^-pressures-porosity ,
X, X -co-ordinates of plane ,t-time, J-He$TeoTgana, Vo -primary volume of oil in the layer, £(x, t) -intensive of rolling up oil.
j(^-^)dx = 0 tJ> 0 (2)
T
J = j—jjVid xdt (3)
0 0 Q
T-time of production
Problem consist in definition of manager influences t) , rr(x, t) ,which allow to get maximum H^TeoTgana
T
^rdxdt = Q
0 Q
where Q -definite value
Assign the definite classes of decisions, finding which simpler than finding the general decision, we use theory Ly for it.
Construction the infinitesimal operator for the mission in the following view : (4)
x = a>1 fa x2,t, p + fa x2,t, p + fa X,t, P )-° + cXj ox2 ot
и \ д д д д д
а fo,x2,t,p)— + vx — + + -3 — + -11~-+
дР дРх, дРх2 дР> дРхЛ
д д д д д
■ + -+ + -23 ~-+ -33
12 ^ 22 ^ 13 ^ 23 ^ 33 /-s
°Px,x2 °Px2x2 °Pxlt °Px2t 0Ptt In result of the action of the operator we get the next differencial equation:(5)
afa, X2 )(v„ + V22) + PXi fa X2 >1 + px2X2 ax2 (xi, X2V
+ P (X1, x2V + Px2 ax2x2 fa x2 V + «x! (X1, x2 Vl + , . ^ „ ^
where vn u v22 express in full differencial D
ax2 (xi, x2 )v2 - C(X1, x2 )v3 - PtCx, (xi, x2 )® -PtCx2 (X1, x2 )®2 = Q
with formulas:
V11 = DXi (Vi) - PxnDXi (c3)-PXiXiDXi fa )-PX2XiDXi (c2) V22 = DXi (V2) - PDXi (c3)- PXiXDXi (c1)- PXiXDXi (c2)
In order to define a view function , c1 fa, x2, t, p ), c2 fa, x2, t, P), c4 fa, X, t, P) it is necessary with reckoning expressions for vu,v22,v2,v3 to split the definite equation ,which is differencial
concerning of unknown co-ordinates c:,c2,c3,c4 .It is necessary to mark if do not use with the infinetizimal create of invariancy differencial equation and the formula substitute into the equation , in this case receiving system will be unlinear,if the first equation is unlinear,and ,mission of finding group will seem very complex and unwieldy . The definition equations always unlinear it means the
application of the infinitesimal creates of invariancy actually do unlinear the mission of finding group of the transformation with permissible system of the differensial equations. Splitting the equation (5) concerning "free" co-ordinate P and its derivatives,we get:
aa4 -2aa>\ + a C = Oa1 = 0,c3 = 0,c3 = 0
p Xl Xl ? p ? p ? x2
xi
am 4 - 2amX + ax m2 = 0,m2 = 0, mX = 0, m4 = 0
p Xi x2 ' p ' xi ' pp
cm4 + cm3 - <mlcx - <m2cx = 0
p t Xi Xi
4 4 4 4 4
-cm>t + am>xx + am:c x + ax 0)4 + ax cox -m Q +
t xjxj X2X2 Xj Xj X2 X2
(6)
+%f+LJ = 0
cm1 + 2aa4p - amlXXi - aml2^ + aXXi m1 - aXim^ - a^m1^ +
aX mp. = 0 X1 p
2 4 2 2 2 2 2
cmt + 2am^ - am„ - am_ + a „ m - a m„ - a m„ +
x2 p
aX m4 = 0
X2 p
It is possible to decide the mission of the group classification concerning element j = j{x) .Proposition about the arbitrary j = j{x) and its derivative is made in receiving definition equations, that assists to additional splitting system (6) and give us the definition equations of nucleus of main groups.
m + + m1 X m1 = 0
tjm
4
pXi 4
pX2
2 2 + x, + cmt -1m
2 X2 2 X1X1
-1 ml = 0
After unheavy transformations we get the following treatments:
m4" = m],m\
= 0;m2_ = 0
The following equations appear in view of definite receiving system ,which keep only ?](x) .These equations are classifications:
f \ f \
m2 =
1
V 1X2 J
(m2!-m? ),m1 =
1
v 1xi j
(2miXi -mf )
(7)
We get some expanse of vector fields of tangent to groups for each decision of classification equations. We see the addition base of the main expanse till the base of expanse L( (7)). The main group is generated with transformations, which belong to 5 groups:
д д д д д д
Х^ = x--^ X —^ t—, X2 = —, Х^ =-, X4 = —,
дх дх2 дt дх дх2 дt
х 5=д. дp
It is possible to get the invariancy decision for each of these .It is interesting the chance of submiting the operators X2 u X3 .For X2 corresponds the invariancy decision p=p(x2 t),putting up in (1) we get:
_д_
дх.
1(X2
дх-,
- c f=Q(t )
дt
>
22
22
i -n
22
>
1-4
2-v2
The receiving equation represents the single-measure equation of heat-conducting with variable coefficient and source ,so the appling of theory Ly gave us the splitting of the first equation in directions xi and X2 . When we use operator X3 ,we get:
_d_
dx1
l(xi )
dp dx1
- c dP = Q(t ) dt
We can begin the classification.The main base is found ,the group of equivalent of equation is looked for in the class of equations joining with condition,so as arbitrary r depended on from X2 (or from xi).
When we difference the first equation in (7) on X2, but the second on xi ,we get the following classification equation:
f \ r
f \ Л
\1x2
= 0
1 j
= 0
71
— = —f + r2 ,где r, Г, Г', r2 -const
x,
The decisions is written in that view
r x2
— = ^ + r2 ; VXl ri rxi '1
The linear combination of invariant decisions of the main base of the group transformations are the invariancy decision too .Then the general decision of the classification system is noted in following view:
l(x ) = сгх" +
,где
^ 1
ri ri
In that case we see the widening of the main group on vectors: (axl ,0,2 p) The cultivate method of synthesis and approach of the group analysis of the differensial equations and the counting analysis reveals the good possibilities and allows to reseach numerous unlinear appearances, and the management influence ,which influence on the productivity of layer.
xx
x,x
2^2
1-4
p
c2 x2
LITERATURE
[1].G.A.Virnovski ,E.I.Levitan, About identification of two-measure model of current of homogeneous liquid in porosity space. J.V.M. And M.F.T.XXX,№5,M.1990.
[2].L.V.Ovsannikov.The group analysis of differencial equations,M.1978.
[3].R.G.Jalilova, S.Yu.Kasumov, K.F.Shirinov. The application of theory Ly to reseaching of single class for prosesses of filtration. News AN Azerb. Seriya f.t..and m.n.,№5-6,B.1995
