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46
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
UDC 622.276
Study on influence of two-phase filtration transformation on formation of zones of undeveloped oil reserves
Sergey I. GRACHEV, Valentin A. KOROTENKO, Nelly P. KUSHAKOVA»
Tyumen Industrial University, Tyumen, Russia
In order to study the process of fluid filtration during flooding of an oil field, article uses Rapoport - Lis model of non-piston oil displacement by water. During plane-radial filtration in a homogeneous formation, radii of disturbance zones are determined with and without taking into account the end effect. Influence of changes in value of capillary pressure gradient on distribution of water saturation coefficient in the non-piston displacement zone for high and low permeability reservoirs is revealed. Application of an element model for a five-point injection and production well placement system showed that, using traditional flooding technology, flat-radial fluid filtration is transformed into rectilinear-parallel. At solving equation of water saturation, Barenblatt method of integral relations was used, which allows determining the transformation time. By solving the saturation equation for rectilinear-parallel filtration, change in the value of water saturation coefficient at bottomhole of production well for an unlimited and closed deposit is determined. It is shown that an increase in water cut coefficient of a production well is possible only for a closed formation. To determine coefficient of water saturation in a closed deposit, a differential equation with variable coefficients is obtained, an iterative solution method is proposed. In the element of the five-point system, oil-saturated zones not covered by development were identified. For channels of low filtration resistance, conditions for their location in horizontal and vertical planes are established. It is shown that, at maintaining formation pressure, there is an isobar line in formation, corresponding to initial formation pressure, location of which determines direction of fluid crossflow rates. Intensity of crossflows affects application efficiency of hydrodynamic, physical and chemical, thermal and other methods of enhanced oil recovery.
Key words: Rapoport - Lis model; Barenblatt method; water saturation coefficient; transformation of filtration process; residual mobile reserves of oil
How to cite this article: Grachev S.I., Korotenko V.A., Kushakova N.P. Study on influence of two-phase filtration transformation on formation of zones of undeveloped oil reserves. Journal of Mining Institute. 2020. Vol. 241, p. 68-82. DOI: 10.31897/PMI.2020.1.68
Introduction. Model of oil displacement by water has been well studied in classical works on underground hydrodynamics, nevertheless, study of oil recovery process during flooding is one of the most difficult tasks in design and control of oil field development. This is due to fact that when water is injected in bottomhole formation zone (BHFZ) of injection well, an end effect (EE) takes place. Water accumulates in a region of radius rk, in which water saturation coefficient increases to a certain value of sk. When this value is exceeded, water enters the formation, process of two-phase filtration and non-piston oil displacement by water begins. Water flows at internal boundary of EE region, and at external boundary water does not flow out, but oil does. Study of filtration process is complicated by a nonlinear boundary condition on external region of EE [14]. Because of it end effect is neglected, coefficient of water saturation at bottomhole of injection well takes limit (maximum) value s . Porosity and permeability properties (PPP) of the formation, depending on saturation coefficients, geological structure of deposit, location of wells affect change in nature of the fluid flow in formation and, therefore, development of mobile oil reserves.
Statement of the problem. Existing problem of undeveloped residual mobile reserves (RMR) of oil is due to physical properties of the fluids, implemented development system, macroheteroge-neity of the filtration parameters in lateral and thickness of oil reservoir [2, 8, 9, 12, 15, 16, 18]. Surface phenomena occurring at phase boundary, capillary pressure gradient, hydrophilicity or hydrophobicity of reservoir (microheterogeneity) significantly affect deviation of actual development indicators from design technological ones [7, 11]. To eliminate discrepancy between the indicators, technologies (methods) of enhanced oil recovery (EOR) are used. When selecting and evaluating the effectiveness of corresponding method, it is necessary to take into account the change
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
(transformation) of the filtration processes and influence of the fluid saturation degree in reservoir capacitive space.
Methodology. Physical models for multiphase filtration of non-piston oil displacement are considered in [4, 6, 10, 13]. From equations of continuity, motion and rheological equations, a system of partial differential equations is obtained, solution of which is reduced to determining the distribution of saturation and phase pressure coefficients in disturbance region - two-phase filtration. To connect water saturation coefficient with pressure, an additional closing equation is introduced. All considered models can be divided into two groups: taking into account and not taking into account the capillary pressure gradient. Solution of differential equations systems, as a rule, is carried out by approximate numerical methods. In the present work, to solve Rapoport - Lis equations, Barenblatt method of integral relations is used [1]. Transformation of filtration zones in process of developing a production object (PO) is considered. Obtained approximate analytical solutions of water saturation coefficient values allow identifying the zones with existence of two-phase filtration and residual oil.
Discussion. Results of the research. To study transformation of filtration processes, a model of flooding a homogeneous reservoir is considered. With commissioning of injection well, a radial non-piston displacement of oil by water occurs. Pressure at the bottomhole of injection well is greater than initial formation and bottomhole pressure of production well. Over time, flat-radial filtration transforms into rectilinear-parallel filtration between injection and production wells [17]. To determine values of water saturation coefficient in the model proposed by Rapoport - Lis, a special case of which is Buckley - Leverett model, formation and fluids are considered incompressible. Task is reduced to solving the differential equation for saturation coefficient taking into account capillary pressure at phase boundary and corresponds to a rigid elastic water drive regime:
m0 * + W (t + ^ div dt dx ( 2
♦ i dp ds
k* (s )f (s)| —-— + Apg sin a + Ago
ds dx
= 0, (1)
where m0 - open porosity coefficient; s - water saturation coefficient in a two-phase filtration zone; W(t) - total phases' filtration rate; x - coordinate, for radial filtration x is replaced by r; fs) - Buckley -Leverett function; k0 - absolute permeability coefficient; k2 (s) - relative phase permeability (RPP) of oil reservoir; pk - capillary pressure; Ap = p2 - p1, p2 and p1 - oil and water densities; g - acceleration of gravity; Ag0 = g2 - g1, g2 and g1 - initial pressure gradients during oil displacement by water in a two-phase filtration zone.
Buckley - Leverett function is determined experimentally:
f (s)= k*( k' (s)k*( ), (o (2)
k1(s)+(0k2 (s) (2
where (1, (2 - dynamic viscosity coefficients of water and oil; k*(s), k2*(s) - reservoir RPP for water and oil.
Equation (1) contains time derivative of the function s(x,t), and operator div corresponds to an equation of parabolic type. Therefore, to solve (1), Barenblatt method of integral relations can be used [1].
Origin of coordinates will be compatible with bottomhole of injection well. Radius of disturbance zone of two-phase filtration is denoted by R(r, t). Since the change in water saturation coefficient corresponds to change in pressure of water injected into reservoir, solution is found in the form [6]
./; (r, R(t )).
f (rw, R(t ))
s(r, t)= s0 +^(t)As J],yL s = s * - s0, (3)
f ( w'
Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
where s0 - coefficient of residual water saturation (bound water); s - water saturation limit value; rw - well radius. Heaviside unit function ^(t) equals 0 at t = 0 and 1 at t > 1:
fi (r, R(t )) = -n ln
Rt) -
R(t I
+1; fi(rw,R(f))=-nln
R(t) -
w
R (t ),
+1.
(4)
Note thatf1(r, R(t)) < 0 for rw < r < R(t).
If the end effect is neglected, then function s(r, t) satisfies following boundary conditions:
s(r,0) = s0, s(rw, t) = s*; s(R(t), t) = ^
dr
= 0.
(5)
r=R(t )
Pressure at the boundary of two-phase filtration zone R(t) is equal to initial pressure in deposit p0.
With a flat radial displacement of oil by water, equation (1) takes form
mn
ds ~dt
df (s ) - k^ 1 d_
r dr \2 r dr
rk* (s)f (s ) % Is
ds dr
+ Apg sin a + Ag0
(6)
where qs = Q^-), Q(t) - flow rate of injection well; h - formation thickness. 2nh
To determine the radius of two-phase filtration zone R(t), (6) is integrated over r in the interval from rw to R(t). Thus
2 ^ { R2 ^
Asn d
4(n + 2) dt V fj(rw, R)
4(t) f (s)
m„
kr.
m0\x2
*t m dpk ds .
+ Apg sina + Ag0
ds dr
(7)
It follows from boundary conditions (5) that, for r = R(t) water saturation coefficient s = s0. RPP of reservoir takes maximum value k2*(s0) = k2*max, and Buckley - Leverett function f (s0) = 0.
At r = rw, s = s*, k 2*( s *) = 0 Buckley - Leverett function f(s*) = 1. As a result, there is an ordinary differential equation:
d_ dt
R2 (t )
v f (rw, R(t ))
- 4(n + 2) n2 m0 As
(t ).
(8)
If qs = const after separation of variables and integration, a transcendental equation for determining R(t) is obtained:
R(t ) =
r2--
^ f (rw, R(t ht
m0 Asn
1
r 2 + 4(^+2) r ln R(t) -1
m0 Asn v rw
n
qst.
(9)
Consequently, under given boundary conditions (5), capillary pressure and RPP do not explicitly affect the radius of two-phase filtration zone.
However, if end effect of radius rk is not neglected, then second boundary condition (5) at bottomhole of injection well will take the form
sk,t)=sk <s*; rk > rw.
At r = rk and s = sk RPP of reservoir for oil k2*(sk) ^ 0, and Buckley - Leverett function f(sk) < 1.
After integration in the interval [rk, R(t)] right part of equation (7) takes following form
n
n
r
r
R
R
r.
r.
w
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
5L_f s )+Mfc) rj ( sk )
m„
m0^ 2
gAp sin a + Ag +
dPk
ds
ds A
s=sk dr rk J
= -^f s )(i+^k );
m
k
A _ k0 k2 ( sk ) Ak - ' qsi 2
gAp sin a + Ag +
dPk
ds
ds A
s II k rk J
At end effect s(r, t) and its derivative are equal to
Kr- ')=so^-flSI; As
k _ sk so ;
ds
«As
1 - k
dr f (rw, R(t R« (t )
'J
For a = 0 and Ag = 0
A =
kok2 (sk ) nAsk dPk
№ fi (rk,R) ds
(10)
sk
After substituting (10) in right part of equation (7), a differential equation is obtained that differs from (8) by its right part:
d_ dt
R2 (t )
fi (rw,R(t))
- 4(« + 2)
«2 m0 As
f (sk k (t )(1 + Ak.
(11)
From formula (10) it can be seen that parameter Ak depends on technological parameter qs, phase permeability of oil, derivative of capillary pressure with respect to s at point sk, radius of a two-phase filtration zone R. According to the data given in [3], capillary pressure derivative with respect to s is equal to tangent of inclination angle to axis s (tga) in interval s0 < s < s . The highest values of capillary pressure derivative are observed in vicinity of water saturation coefficient s0 corresponding to front of oil displacement by water, distanced from bottomhole of injection well (Fig. 1). Derivative values vary in range -10 < tga < -8. At the bottomhole of injection well, a change in capillary pressure for a high permeable reservoir (curve 1) has practically no effect on oil displacement: tga « -0.1. For a low permeable reservoir (curve 2) tga « -1.
Indeed, let us consider variation interval of perturbation zone radius 4 < R < 200 m. For a high permeable reservoir at following parameters: Q = 300 m3/day; h = 200 mD; k2* = 0.6; | = 2 mPa-s;
tga = -0.1 MPa; Ask = 0.72 values of parameter Ak change in the interval 0.01< Ak < 0.06. For a low permeable reservoir at set parameters: Q = 8 m3/day; k0 = 4 mD; k2* = 0.2; |2 = 2 mPa-s; tga = -1 MPa; Ask = 0.27 values of parameter Ak change in the interval 0.03 < Ak < 0.16. With increasing radius R(t) parameter Ak decreases. As Ak < 1, then to determine the radius of perturbation zone, formula (9) can be used.
Example 1. Given: rw = 0.1 m; h = 10 m; s0 = 0.2; s* = 0.8; m0 = 0.2; Q = 300 m3/day.
Ph MPa
0.2
0.15
0.1
0.05
0.2
0.4
0.6
0.8
Fig. 1. Capillary pressure dependence on coefficient of water saturation
1 - for a high permeable reservoir;
2 - for a low permeable reservoir
0
k
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
Determination of values for disturbance zone radius by formula (9) for different values n
Table 1 Calculation results are shown in Table 1.
Let us consider an element of operational object consisting of injection and production wells (Fig.2). While inequality R(t) + R2(t) < L holds, where L - distance between wells; R2(t) - radius of disturbance zone 2 for production well, in which only oil is filtered, there is flat radial filtration. Since injection pressure is greater than initial formation pressure and bottomhole pressure of production well, bulk of injected water will move to perforation zone of production well. Filtration becomes rectilinear-parallel. Pressure at interface of zones 1 and 2 (points Ci) is equal to initial formation pressure p0. Another part of water will flow at a speed V1 to zone 1. In disturbance region 2 of production well, oil will flow into linear filtration channel at a speed V2. Moreover, flow speeds will decrease over time. Note that, under given boundary conditions, points Ci are motionless. Transformation time t is determined from condition [6, 7]
t, days R(t), m
n = 1 n = 2 n = 3 n = œ
0.01 3 3 3 2.2
0.1 14 12 11 8
1 50 42 39 30
2 73 61 56 44
10 176 147 135 105
100 607 506 464 361
365 1210 1006 923 718
R(t*)+ R2(t*)= L.
(12)
According to well-known formulas
R2 (t)=VCx7; x 2 = k°k2
\ 2P2
where C = const.
Substituting R2 and expression (9) into (12), transformation time from transcendental equation is obtained:
t * =
Cx 2
L2
1 + 4(n + 2)qs rln R(t*) -1
1 + . 1 +
Cx 2m0 nAs
n
(13)
R22 D\ R22 B1 Q-,-O B2
R2
D4
B4
D2
R2
B3
D3
Fig.2. Transformation scheme of the filtration process for element of inverse five-point flooding system: a: R - radius of zone 1 of two-phase filtration of an injection well A; R2, R22 - radii of disturbance zones 2 for production wells B; 3 - zone (region), not involved in flooding; b: Vy, V2 - fluid flow speeds in zones 1 and 2; 1 - injection well area A!
2
r
w
b
a
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
Table 2
Determination of transformation start time by formula (13) for different values n
t*, days R, m
n = 1 n = 2 n = 3
3 92
3.3 80
3.4 75
Example 2. Given: rw = 0.1 m; h = 10 m; s0 = 0.2; s* = 0.8; m0 = 0.2; Q = 300 m3/day; L = 400 m; k) = 100 mD; k2* = 0.6; | = 2 mPa-s; p2= 10-3 1/MPa; L = 400 m; C = 12
Calculation results t according to formula (13) are given in Table 2.
For n = 2, distance from injection well to transformation boundary is R = 80 m, pressure at the boundary is p0. In zone 1, two-phase filtration takes
place - oil displacement by water according to Rapoport - Lis model. Disturbance zone radius of production well, in which single-phase oil filtration exists, is 320 m. Since commissioning of injection well, time of changing the nature of the filtration process for given PPP and specified technological parameters does not exceed 4 days. Since pressure at bottomhole of production wells is less than pressure at other points in deposit, flat-radial filtration is transformed into rectilinear-parallel. Flooding of production wells will begin through channels A1Bi.
To describe linear displacement of oil by water, equation (1) is used. Origin of coordinates coincides with bottomhole of injection well. Boundary conditions in this case change, since on interval [xw, R] a change in coefficient of water saturation has already occurred in formation, and at bottomhole of injection well for x = xw = rw coefficient of water saturation s = s . Over time at border x = R coefficient of water saturation will increase: s > s0. Therefore consider s(x, t) on two intervals: s1(x, t) in interval xw < x < l(t) < R and s2(x,t) in interval R < x < l(t).
For interval xw < x < l(t) < R solution is found in the form
,(x - xw )2 .
Sj (x, t)= sR (x, R)+
x ~
l (t )
R ( x
- + b-
l2 (t )
(x, R )= s0 + As
fj (x, R).
fj (x w, R ) '
R ( x
(14)
fj (x, R)=-n -1^1 +1; fj (xw, R )= -n I +1 .
x IR
x
R
For R < x < l(t)
( , x - R (x - R )n
s2 (x, t ) = a0 + aj—rr- + a„ _ / ; ; n > 2.
l (t ) n ln (t )
(j5)
Here R - constant, corresponding to n (Table 2); c, b, a0, a1, an - unknown parameters determined from boundary conditions and conditions for merging of functions s1 and s2. Boundary conditions for function s2 are set at the boundary of displacement front:
Ss,,
,(l(t),t) = So.
Sx
= 0.
x=l (t )
Conditions for merging of functions s1 and s2 for x = R will be
\ \ Ss, Ss2 S2 s, S2s2
Sj (R, t) = s 2 (R, t ); -1 = j- 2
dx dx ' dx2 dx2 From a joint solution of system (16), (17) for n = 2 parameters ai, b, c are
(j6)
(17)
L RY „ L R^ 2As
a0 = s0 + a21 j--I ; aj = -2a2l j--I; a2 =---,-T ;
0 0 2 1 l J I l J 2 fj (xw,R)
2As
l
b = a + / m 2 fj (xw,R)lR
— I ; c = -2a -
4 As l fj (x w, R ) R '
(18)
n
n
s
2
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
After substitution of (18) into (15) we get
M X
fat ) = s0 + a 2
(l (t )-x )2 l2 (t) .
(19)
For practical testing of determining the coefficients of water saturation and watering at bot-tomhole of production well, let us consider the case R < x < l(t), s = s2.
After substitution of s2 into equation (1) and subsequent integration over x in range from R to l(t), we obtain a differential equation for determining linear size of two-phase filtration zone l(t):
d_ dt
(l - R )3 l2
3W (t \ f s )
a2 m0
1 + A
(l - R )
m l2
A = 4Ask0k2* (s ) dPk kl \ 2W (t )f (x w, R ) ds
(20)
Here s is value of water saturation coefficient corresponding to x = R. It should be noted, that s > s0 is variable value, which will increase over time. Buckley - Leverett function varies in the interval 0 < f (s ) < 1 and also increases.
Verification calculation of parameter Akl shows that for various values included in formula, its order does not exceed 10-4, and from (20) with increasing l(t) the second term in square bracket tends to zero. Therefore, equation for determining l(t) has the form
d_ dt
(l (t )-R )3
l (t )2
m f f (s >
(l (t )-R )3 lit)1
1 (t )
3W (t ) f ( s )dt = ®(i ).
(21)
R a 2m0
From where cubic equation is obtained
l3 -(3R + 0)/2 + 3R2l - R3 = 0.
(22)
Using the Cardano method, l(t) is found:
l (t )=R+3 ®(t )
1+3
1+
R
+3
,®(t)J 1
1 -
R '
(23)
Value s is determined from formula (19), which for x = R will be
- (l (t )-R )2
s = s + a -—
0 a l (t )2
l > R.
(24)
To obtain an approximate analytical solution, we estimate the function O(t), included in right part of (21). Let total phase filtration rate W(t) be constant. Then, using mean value theorem, we can write
) = ^ J. f (s d . -^f (s )t;
n TTI J n m
3W
s )t; ^ < s < s < s .
a2m0 0
a 2 m0
(25)
It follows from (24) that value s depends on time, consequently, s also increases over time t. Limit value
- (26)
s = s0 + a2 = s0 + r
ln--0.5
x,„
a2 m0
1,5
1,5
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
For so = 0.2; As = 0.6 and R = 80 m value of s = 0.2485. Thus, Buckley -Leverett function f(s) in the integrand expression changes insignificantly (a weakly changing function) and its removal beyond the sign of integral is justified. In the future we will take s = s .
Note, that at t ^ ro formula (23) will take a simple form:
l (t ) = R + 0(t ) = R + (s) t. (27)
Table 3
Dependence of change in the size of linear filtering zone l(t) on time in an infinite formation
t, days l(t), m (23) l(t), m (27) x > R s2(x,t) (19)
0.1 80.1 80.03 81 0.2924
1 80.9 80.3 100 0.2913
2 82 81 300 0.2803
10 89 83 361 0.2771
100 169 117 500 0.2701
365 405 394 800 0.2560
3650 3333 3332 3333 0.2000
From (23) and (27) it follows that after transformation, size of two-phase filtration zone l(t) linearly depends on time. Formula (23) is valid for short times, and for R c 0(t) (27) can be used.
Values of Buckley - Leverett function f(s) are determined if the formulas for dependence of RPP on coefficient of water saturation are known. According to research of Chen Jung-xiang [16]
k* (s ) =
s - 0.2 0.8
>3,5
; 0.2 < s < 1; k2* (s) =
0.85 - s 0.85
>2,8
(1 + 2.4s); 0 < s < 0.85.
(28)
In this case for s = 0.2485 Buckley - Leverett function equals f (s) = 1.81-10 4.
Table 3 shows the results of calculating l(t) for different values of time.
Second and third columns show calculated values of l(t) in time interval from 0.1 days to 10 years, calculated by the formulas (23) and (27). In fourth column, the x coordinate values are from 81 to 3333 m, in fifth column there are calculated values of water saturation coefficient in this interval. Note, that if x = L - distance between wells, then from formula (19) it follows that for l(t) ^ ro, s(L,t) = s0 + a2, x - constant value. This is contrary to practice, since water cut increases, therefore, at bottomhole of production well, proportion of water in fluid flow should increase. Accordingly, size of filtration zone should be limited to lk = l(tk), which is determined from geometric structure of the deposit.
Pressure distribution in confined deposits was considered in [1, 16]. By analogy, change in coefficient of water saturation in a closed limited deposit is found in the form
s3 (x, t ) = D(t K (x, lk ) = D(t )
s0 + a2
v lk J
R < x < lk ; D(0)= 1.
(29)
Unknown function D(t) is defined after substitution (29) in equation (1) and after integration in the interval [R,lk] we get differential equation:
dD = G - BD, dt
(30)
where
G = ^/Èf! ; h = (, _ R +1 a
m0 H | 3
b = 1 2a2 k0k2 (sr )f (sr ) (/m - R) dpk
m0H
\ 2
l
ds
lk - R
V lk J
; B < 0.
(31)
Here sR corresponds to value of water saturation coefficient at x = R, sk corresponds to x = lk, s > sR > sk > s0. For the Buckley - Leverett functions, relation f(sk) < f(sR) ^ 1 is true.
It follows from (29) that function D(t) is limited above, its maximum value Dmax=s*/s0.
2
2
s
R
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
For a constant total filtration rate W(t) = W and limited time-dependent functions G, B result of (30) will be
D(t) = B[G-(G-B)e Bt].
Expanding the exponent in a row, we obtain an approximate solution:
D(t ) = 1 + (G - B)t.
For calculation, following iterative scheme is proposed:
1. By known values lk and R from first formula (31) parameter H is found.
2. Set time values t and parameter D(t) = 1,01.
3. From (29) we define
(32)
(33)
«3, (L, t ) — D(t K (L, lk ) — D(t )
S0 ^ a2
flt - 2
V lk y
; R < L < lk ;
S d — S
— «3, (R, t ) — D«2 (r, h ) — D(t )
S0 ^ a2
- ^2
V lk y
Sl, — S31 (lk , t) — Ds2 (lk , lk ) — d(t)s0 ,
where L - distance between wells.
4. Calculate values of RPP coefficients and Buckley - Leverett function by the formulas (28) and (2) for sR and slk.
5. From Fig. 1 or functional dependence we find
dp,
ÔS
, by formulas (31) we define pa-
rameters B, G.
6. We calculate new value of coefficient s32. If [s31 - s32 ]> 10-3, then repeat the process from point 3) iterative scheme.
Setting a new value of time, we determine increase in coefficient of water saturation s3(L,t) at bottomhole of production well.
To determine water cut ratio a on channel A1B1 we use the formula
a = — ; W = v + v2, W
where v - water filtration rate; v2 - oil filtration rate; W - total fluid flow rate at bottomhole of production well. If W is constant, then terms change, oil filtration rate decreases, and water filtration rate increases.
Water filtration rate is expressed through total rate and depends on capillary pressure gradient [15]:
— f (s )
W+k
k2* (s) dp, ds ^ 2 ds dx
Water cut coefficient
. — f (s )(1 + A) A — k k2* (s)dpk
W|J, 2 ds
ds dx
— - D(t )k o
k2 (S) 2a2 (lk- - L) dpk
w^ lk ds
- ' ds
< 0.
(34)
s
R
v
L
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
For small values s, close to s0, RPP k2*(s) takes values, close to maximum, and Buckley -Leverett function is small. At s ^ s* function f(s) ^ 1, and k2*(s ) ^ 0. Therefore, for large values of water saturation coefficient at bottomhole of production well, Buckley - Leverett function corresponds to water cut coefficient of well fluid. Thus, experimental work on determination of RPP values plays an important role in modeling of technological indicators and choice of well operation mode.
Time of partial watering of fluid tL for well B1 through channel A1B1 corresponds to value of water saturation coefficient, defined by formula (29), at t = tL and x = L. After substitution in (33) we get
,L = ^ ; Dfc)=-f—2. (35)
G r lk - ^
s0 + a 2
lk
Parameter G is defined from formula (31) at f (sR) = 1, k2(s ) = 0.
Example 3. Given: Q = 200 m3/day; h = 10 m; R = 80 m; L = 400 m; lk = 1000 m; for value of function f(sk) = 0.5 time of fluid watering, calculated with formula (35) is tL = 6.3 days. If value of Buckley - Leverett function will be f(sk) = 0.9, then tL = 31.7 days. If filtration zone size is lk = 3000 m, then for value of Buckley - Leverett function f(sk) = 0.5 and f(sk) = 0.9 values of fluid watering time tL = 16.1 days and tL = 90 days respectively.
Filtration zone size lk depends on geological structure of deposit and significantly affects time of watering. In Rapoport - Lis model, porosity coefficient is constant, fluids and reservoir are incompressible, and therefore, linear filtration channels can be neglected. Consequently, in areas 3 (Fig.2, a) there will remain oil reserves not involved in development.
To determine oil reserves not involved in development in zones 3 we define t22 - contact time of disturbance zones of production wells B1B2 in point D1.
Distance between production wells B1B2 = L2 Radius of oil filtration zone R22 for ho-
mogeneous reservoir
R22 = ^ = ^L. (36)
22 2 2
Denote by t22 time corresponding to reaching of point D1, - interface between zones of filtration of production wells:
R22 = VC%2-22 , (37)
where x22 - coefficient of oil piezoconductivity; C - numerical coefficient.
Radius R2 and transformation start time t satisfy the relation
R = L - R = V^/. (38)
From formulas (36) and (37) we get
R 2 L2
t = t = t *_L_ (39)
t22 Rl 2(L - R )2. ( )
We use data of example 2 and calculation results of Table 2. For L = 400 m, R = 80 m, t = 3.3 days and R2 = 320 m from (38) and (39) we get t22 = 2.6 days, R22 = 283 m. Consequently, disturbance zones of two production wells are in contact at a point D1 earlier, than they connect at point C1 of disturbance zone of injection and production wells.
We make an essential remark. Formation pressure in point D1 at t22 = 2.6 days is p0, which will further decrease. Therefore, in time interval t22 < t < t in zone 3 oil filtration will begin in direction of point D1. If due to "proximity" of values time t22 and t is neglected, it is possible to determine oil reserves in areas not covered by flooding.
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
B
n
D
B2
Fig.3. Fragment of an operational object element for determining the area of zone 3. Arbitrary point Da e curve DA. Angle 0 < a < 45°.
So, at time t three zones with different saturation patterns form in the PO element of the five-point system (Fig.2).
1. In vicinity of injection well first zone has the shape of a circle with radius R. Saturation pattern - water and oil.
2. Four zones 2, each of which has the shape of a closed region bounded by curved lines CD;. Saturation pattern - oil.
3. Four zones 3, bounded by curved lines CjCjDi, not covered by development, saturated with oil, pressure in which is equal to initial pressure p0.
Denote the area of zone 3 by S3. Consider an isosceles right triangle B1B2A1, whose legs are equal L. From Fig.2 determine the area of zone 3:
1 nR 2
S3 = SA S A1C1C2 -2Sa = 2 L 4--2Sa
(40)
where Sa - area of a triangle AiBiB2; SA1c1c2 - area of a sector A1C1C2; Sa - area of a figure B\D\C\
with center in point B1, limited by arc D1C1 with central angle a0 = 45°. Curvature radius B1C1 changes from R22 to R2. Area of zone S3 depends on the accuracy of determining (calculating) area Sa.
Formula for determining Sa for a variable radius of curvature has the form
1 "-0
Sa= 2 J r2da .
(41)
Consider several options for calculating the area of a figure B1D1C1 (Fig.3). As can be seen from the figure, arc D1C1 is convex with respect to the center B1. At changing the angle in the interval 0 < a < tc/4 radius Ra increases from R22 = 283 m to R2 = 320 m. This condition is satisfied by function
BiDa= Ra= L-
sin I 45 -a
+
cos a
(L - R )tga.
(42)
For interval 0 < a < tc/4 distance from injection well A1 to point Da is determined by ratio
sin
ADa= L-
n
-a
cos a
+ Rtg a.
(43)
It follows from formulas (42) and (43), that curvature radius Ra and distance ADa to injection well A1 depends on angle a. Calculation results are presented in Table 4.
Example 4. Given: R = 80 m; L = 400 m; R2 = 320 m; R22 = 283 m; 0 < a < rc/4.
Table 4
Calculated values of curvature radii of the figure BjDjCj and distance m
Parameters Calculation number
1 2 3 4 5 6 7
a, rad 0 0.087 0.175 0.349 0.524 0.698 0.785
Ra, (42) 283 286 289 296 304 314 320
AiDa, (43) 283 265 247 209 166 113 80
4
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
After substitution of (42) in (41) area B1D1C1 is
S »= 2
*L+42l
4 2
R -
'S
+1
L
-In 2 + 2
R-
'S
+1
L
1 -*
. 4,
= 28682 m2. (44)
If, to determine area Sa, we take average radius rav, equal to half-sum of radii R2 = 320 m and
R22 = 283 m, then
S„
8
= 35661m2.
Thus, approximation of section BDiCi to sector of constant average radius gives an overestimated value of area, and, as follows from formula (40), a smaller value of area of zone 3, which will lead to an incorrect estimate of reserves.
Element of PO B1B2B3B4 (see Fig.2) contains four zones 3, each with area S3. Volume of oil (geological reserves)
V = 4S3hm0 (1 - ). (45)
To determine volume of recoverable oil reserves in zone 3 we take coefficient of oil displacement by water
I =
l.oil
S - Sn
1 - sn
where sinoil = 1 - s0 - coefficient of initial oil saturation; sresoil = 1 - s* - coefficient of residual oil saturation; s0 - coefficient of residual water saturation; s - limiting coefficient of water saturation at which oil filtration stops.
Volume of recoverable oil reserves
V3lz = IlK = 4|S3hm0 (s^ - S0 )
(46)
where - coefficient of formation coverage by flooding.
Example 5. Take m0 = 0.2; h = 10 m; s0 = 0.2; s = 0.8; = 1. Remaining data will be taken from example 3.
From formulas (40), (45), (46) we get S3 = 17613 m2, V3 = 12721 m3, V3iz = 84540 m3. If input flow rate of well that penetrated zone 3 is 50 m3/day, then time spent on extraction of mobile reserves V3iz, is 4 years, if boundaries of zone 3 do not change. Volume of geological oil reserves in PO element is 512000 m3. Share of unextracted reserves will be 0.22.
However, over time, there will be an intrusion of water in zone 3, in which two-phase filtration will occur, the oil flow rate will fall.
Let us consider how filtration processes change over time with simultaneous operation of wells. Following remark is appropriate. Due to given boundary conditions in PO element, following inequality is true:
PB1 < PD1 < PDa < Pc1 = Pg1 = P0 < Pa, , (47)
where arbitrary point Ga e zone 3.
Condition (47) is satisfied from the moment the zone formation begins. Consequently, flooding of zones 2 and 3 will occur unevenly. With growth of time t > t > t22 in zone 3, two-phase filtration begins, boundary C,Ga of which is limited by an isobar with initial pressure p0. Single-phase filtration zone 2 is deformed, water injected into formation will begin to flow through the boundary along channel CDa.
Thus, in a reservoir, which is permeability homogeneous in Oxy plane, linear filtration will occur in zones 3 and 2. Values of saturation coefficients and pressures in filtration channels will differ from initial values at beginning of operation.
2
2
2
S — S
in.oil res.oil
S
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
Consider the filtering process in element of PO in form of a channels system connecting injection A1 and production B1 well.
Article [5] shows that watering of well fluid happens in presence of low filtration resistance (LFR) channels in formation. Cases of LFR channels' location along the thickness of reservoir and in the planes Oxy are considered. To determine channel length, parameters ray, were introduced, depending on concentration of indicator peaks and fixation time, j is indicator fixation number. For the element of the five-point development system considered above, we have
L1 = < L = ad B =
L
cos a
2sinI —-a 1 + sin a
14 J
(48)
From which
© i =
2sin I —-a | + sin a 14
cosa
(49)
Angle a = 45° corresponds to a straight line A1B1 = L - the shortest distance between injection and production wells, parameter < = 1. Angle a = 0° c corresponds to the greatest distance AiDiBi = lv2, parameter < = 42 . Therefore, LFR channels that satisfy the conditions
L < L} < lA/2; 1 < © j <42, (50)
lie in horizontal Oxy planes. If conditions (50) are not satisfied, then LFR channels are located in a vertical plane.
In Tables 3-5 of article [5] cases of horizontal arrangement of LFR channels Lj for three given parameters < are considered.
Table 5 shows values < and Lj from article [5, see Table 3] and dependencies < and A1DaB1 on angle a.
Table 5
Comparison of parameters m-, Lj for L = 505 m [5] with calculated values (48) (49)
Peaks j < [5] Lj = ©jL, m [5] a, deg. <(49) A1D0B1 (48)
1 1 505 45 1 505
2 1.44 707 30 1.17 593
3 2.6 1313 20 1.26 638
4 3.8 1919 10 1.34 677
5 5.4 2828 0 1.41 714
Obviously, for a set parameter ra j = tj /11 there are two channels in horizontal plane Lf. j = 1
and j = 2. Remaining LFR channels are located in a vertical plane. Note that methodology for calculating the volume of pore channels, necessary to apply alignment of injectivity profile is given in the article [5].
Each model has its own limitations, its own field of application, including the Buckley -Leverett model. Confirmation of reliability is compliance of calculated and actual field indicators [7]. It is known that watering of a production well does not occur "abruptly", but gradually increases with increasing water saturation coefficient.
If formation is layered-inhomogeneous, consists of interlayers (seams) of different permeability, then for each ith seam, its own zones 1, 2, 3 are formed. Filtration transformation time for each inter-layer will be different, determined by formulas (9), (12), (13). In absence of crossflows between seams (within framework of accepted Rapoport - Lis model), following boundary conditions are possible. flow rate of water injected into formation should be distributed in proportion to conductivity coefficients kihi of seams [8, 15] or it should be assumed that bottomhole pressure is constant in all
ê Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
seams [11]. Note that each case requires a separate study. LFR channels correspond to highly permeable seams (HPS). For stationary filtration, presence of water "tongues" in the most permeable sections of formation, which is carried out through HPS, has been established. Change in water saturation coefficient is determined by formulas (29), (31), (33).
Original method for studying the distribution of residual mobile oil reserves in a laterally heterogeneous formation was proposed by M.I.Maksimov in a monograph [8]. It is shown that, depending on location of zones with permeability of 40, 200, and 1000 mD, a "development self-regulation" takes place in an area heterogeneous formation. In our opinion, this is a consequence of fluid flows (see Fig.2, b), individual for each zone.
In the 50s of the twentieth century. V.N.Shchelkachev [18] formulated conditions (criteria) for the use of forced fluid extraction (FFE) recommended for PO at a late stage of development: fluid water cut of at least 90 %, high productivity coefficients, collector stability, etc. We believe that large value of water cut coefficient is not a decisive criterion for determining the late fourth stage of PO.
As it is known, FFE technology consists in a phased change in operating modes of production and injection wells, in creation of high-pressure gradients in formation, allowing low-permeability interlayers, dead end and stagnant zones to be involved in development. This leads to a change in directions of the filtration flows, and presence of a hydrodynamic connection between differently permeable seams involves developing low-permeability differences, i.e. cyclic flooding. Therefore, there is a combined unstationary flooding. Existence of regions 3, formed as a result of filtration flows transformation and containing mobile unextracted oil reserves, indicates the need for oil recovery enhancement methods (EOR) that were not previously used in this PO. There are techniques that determine density of oil reserves in areas not covered by development, in particular the Voronoi method. In numerous scientific works of recent years [15, 16], commissioning of second wellbores with a shallow and horizontal end and multi-stage hydraulic fracturing are recommended. Length of the wellbore in reservoir, as follows from above calculations, should not exceed the radius of drainage zone R2 of production well (Fig.2, b). Methods for calculating the horizontal drain in formation are quite fully considered in literature [15, 16].
Thus, a change in direction of the filtration flows leads to creation of linear filtration channels between production wells and formation of zones not covered by development (Fig.3). Obviously, this must be taken into account at designing and arranging second wellbores for efficient development of oil reserves.
Since bottomhole pressure of injection wells exceeds initial formation pressure p0, and bottom-hole pressure of production wells, on the contrary, is less than p0, then isobar line corresponding to it - G0 will exist in the formation. Moreover, existence of G0 line does not depend on chosen physical model. Location of this line in space depends on boundary conditions and affects formation of filtration flows. On the other hand, isobar line G0 limits efficient effect of all existing EOR: hydro-dynamic, physical and chemical, gas, heat, fracturing, zones of efficient operation of horizontal wells endings, etc. Presence of fluid crossflows in horizontal plane (see Fig.2, b), as well as flows in hydrodynamically interconnected seams in layered-heterogeneous formations affect efficiency of the technologies used.
Conclusion
1. Study of traditional flooding technology using Rapoport - Lis model of non-piston oil displacement by water for plane-radial filtration taking and taking not into account the end effect found that changing the capillary pressure gradient practically does not affect distribution of saturation coefficients in two-phase filtration zone.
^ Sergey I. Grachev, Valentin A. Korotenko, Nelly P. Kushakova
Study on influence of two-phase filtration...
2. Simulation of simultaneous operation of injection and production wells revealed that planeradial filtration is transformed into a straight-parallel flow. Time of this process and distribution of water saturation coefficients for rectilinear-parallel filtration in an infinite and final (closed) formation are determined.
3. On example of considered model, formation zones containing 22 % of unextracted oil reserves are identified. Under real conditions of development of an operational object, results that are more negative can be obtained.
4. For a layered-heterogeneous formation without taking into account crossflows between seams, a scheme for determining water saturation coefficients in each interlayer is proposed.
5. Criteria for possible location of low filtration resistance channels in lateral and formation thickness are established.
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Authors: Sergey I Grachev, Doctor of Engineering Sciences, Professor, grachevsi@tyuiu.ru (Tyumen Industrial University, Tyumen, Russia), Valentin A. Korotenko, Candidate of Engineering Sciences, Associate Professor, korotenkova@tyuiu.ru (Tyumen Industrial University, Tyumen, Russia), Nelly P. Kushakova, Candidate of Engineering Sciences, Associate Professor, kushakovanp@tyuiu.ru (Tyumen Industrial University, Tyumen, Russia). The paper was received on 30 May, 2019. The paper was accepted for publication on 12 September, 2019.
