1
P005 FRACTURING OPTIMIZATION FOR MULTIWELL SYSTEM
R.D.KANEVSKAYA, A.R.ANDRIASOV, A.A.GARIPOVA
4, Kursovoy lane, 119034, Moscow, Russia
Abstract
Hydraulic fracturing has become common and widely used well stimulation technique in oil and
gas industry. It has a long history but still there is a need in commercial simulator taking into
account a reservoir and fracture flow simultaneously for multi-well system. This sort of 3D black
oil simulator is presented. Program deals with numerical finite-difference simulation of flow in
the reservoir and in the fractures, but special in-flow formula for fractured wells was developed.
Fracture orientation and length effects during two-phase flow in low permeable formation were
studied on a basis of five-spot pattern and for multi-well system in heterogeneous reservoir.
Hydraulic fractures in the five-spot pattern model were simulated by individual grid blocks using
non-uniform rectangular grid. The results of Eclipse fine grid calculations were compared with
results obtained by means of in-house code for coarse regular grid blocks. After that, the problem
of fracture characteristic optimization for the real area of low permeable reservoir is under study
using previously described in-house code. 24 producers and 24 injectors are placed in this area
according to triangular well pattern. In these runs such fracture characteristics as length and
orientation were varied. It was shown that the difference in cumulative oil production could
reach up to 15% in relation to fracture orientation.
Introduction
Hydraulic fracturing is one of the most important engineering tools for improving well
productivity either by stimulating performance in low-permeable reservoir or by bypassing nearwellbore damage. The problem of a steady-state flow in an infinite uniform reservoir into finite
conductivity vertical hydraulic fracture with elliptical boundary was considered by Prats [1], who
developed pressure profiles in the reservoir surrounding fracture. The concept of an effective
wellbore radius of a fractured well was presented. That radius is a function of the fracture length
and the dimensionless fracture conductivity, and it defines the negative skin-factor of a fractured
well. Real needs of fracturing engineering are associated with an investigation of multi-phase
flow effects on fractured well behavior. Calculations of this sort are possible only on the basis of
a numerical simulation. However, simultaneous simulation of reservoir and in a fracture flow by
means of finite-difference or finite-element methods presents considerable calculating and
methodical difficulties associated with the need to introduce calculation cells whose dimensions
differ by several orders. The use of local grid refinement near the fracture, flexible grid or hybrid
grid is the tool for multi-phase flow studies [2,3]. But this approach is very time and labour
consuming for multi-well system. Creating such non-uniform simulation grids for each fractured
9th European Conference on the Mathematics of Oil Recovery – Cannes, France, 30 August – 2 September 2004
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well is very difficult in view of full field model. So, detailed simulation on refined grid is usually
conducted on a sector model for several wells or even on individual well basis.
This paper presents the sector model of five-spot pattern with non-uniform grid in the fractured
well vicinity that was used to study fracture orientation effect on two-phase flow of oil and
water.
A complex analysis of multi-phase fluid flow problems for real heterogeneous reservoirs with
varying fractured well spacing requires special efforts in simulation. The use of direct fracture
simulation by fine grid blocks is very difficult. The specification of negative skin-factor for
fractured wells in coarse grid models does not permit the effects of multi-phase flow near wells
to be studied. Some alternate approaches to fractured well simulation were developed in [4-6]. In
the presented paper a universal method of hydraulic fracture simulation in reservoir models is
developed on the basis of the analytical solution of the problem of a steady-state flow into a
finite conductivity hydraulic fracture. This method is an extension of Peaceman result of well
simulation [7,8] to finite conductivity hydraulic fractures with an arbitrary fracture length and
orientation. It is supposed that in a vicinity of the fracture the flow is described by an analytical
solution. Boundary conditions for this solution are determined by grid block parameters. The
flow in the fracture is simulated numerically. One/two dimensional fracture simulator is used in
two/three dimensional reservoir model. The well production rate for the reservoir and for the
fracture is defined on the basis of the analytical solution. This approach is realized in in-house
3D black oil simulator taking into account flow in the reservoir and in the hydraulic fractures
simultaneously for multi-well system without grid refinement. The problem of fracture
characteristic optimization for the real area of low permeable reservoir is under study using
described in-house code.
Analysis of fracturing effect on water flooding dynamics using direct fracture simulation.
2D two-phase flow in five-spot pattern sector model of thin homogeneous reservoir is
considered. All wells are hydraulically fractured; individual grid blocks simulate fractures
explicitly. Grid block size is increasing according to the logarithmic law from 0.005 m in the
fracture to 5 m away from the fracture. Well spacing is 850 m. Porosity is constant and equals
20%; reservoir permeability is 0.001 or 0.005 D in different cases. Layer thickness is 9 m.
Fracture parameters are the following: permeability is 200 D, width is 0.01 m, half-length differs
from 100 to 200 m. Producer’s bottom hole pressure is 100 atm, injection pressure is 350 atm.
Initial reservoir pressure is 250 atm. Fluid properties can be found in Table 1.
In order to estimate fracture orientation influence on well performance in low permeable
formation two extreme cases are studied. For the first one, fractures in both wells are oriented
along the rows of wells. For the second case, fractures are arranged at the angle of 45o to the
rows of wells, i.e. producer and injector fractures are pointed towards each other, fig. 1. For the
first case the distance between tips of 100 m half-length fractures is 721 m, and the distance
between tips of 200 m half-length fractures is 632 m. For the second case the distance between
tips of fractures is 650 и 450 m correspondingly.
All the cases are simulated by means of Eclipse. Fig. 2 represents recovery efficiency factor and
water-cut versus dimensionless injected volume for the model with permeability 0.005 Darcy.
These plots show that the worst case is when 200 m half-length fractures are situated opposite
each other. Recovery coefficient is the lowest and waterfront is coming much earlier. This issue
is correct for shorter fractures also. For the reservoir of lower permeability these effects are more
intensive. On the other hand, subject to time total oil production for the 200 m half-length
fracture case is higher up to 10% in comparison with 100 m half-length fractures, but liquid
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production is increasing tremendously, fig.3. Water cut is also higher for long fractures
independently of their orientation. Effect of fracture orientation can be estimated by comparison
of water breakthrough moments. It occurs two times earlier if 200 m half-length fractures are
pointed towards each other. Nevertheless for short fractures difference in water breakthrough
moments is not so noticeable. So, the fracture orientation is very important parameter in well
placement planning for water flooded low-permeable reservoir where the majority of wells are to
be completed by massive hydraulic fracturing. That’s why stress distribution is essential to be
measured when fracture direction is estimated.
Approach to simulate reservoir and fracture flow simultaneously in large systems.
Simultaneous simulation of the reservoir and fracture flow by means of finite-difference methods
presents considerable calculating and methodical difficulties associated with the need to
introduce calculation cells whose dimensions differ by several orders. The use of local grid
refinement near the fracture, flexible grid or hybrid grid is the tool for multi-phase flow studies
and comprehensive analysis of fracture performance [2,3]. But this approach is very time and
labour consuming for multi-well system. Other approaches neglect the volume of the fracture.
For example, Settari et al. [4] proposed to combine the fracture transmissibility with the
transmissibility of the reservoir blocks in the fracture plane, without having the fracture
represented by separate grid blocks. Nghiem published another simulation method based on
representing the fracture as a set of point sources or sinks and using special production or
injection formulae, the fractures being considered to be ideal [5,6]. The flow into or out of the
fracture is computed from the fracture pressure and the pressures of the blocks surrounding those
containing the fracture. In both methods substantial constraints were imposed on the fracture
orientation and position with respect to the finite-difference grid.
In modeling vertical wells it is usual to assume that in a vicinity of the well the flow is nearly
radial and the production rate Q0 can be described by the formula
Q0 =
2π k r h pc − p w
µ ln Rc rw
(1)
Here, pc is the pressure in the block where the well is located and the radius R c is determined
by the grid block size [7,8]; p w is the bottomhole pressure and rw is the wellbore radius; k r and
h correspond to the permeability and thickness of the reservoir; µ is the fluid viscosity.
An analogous method of finite conductivity hydraulic fracture simulation in reservoir models is
developed on the basis of the analytical solution of the steady-state flow problem [1,9]. This
method considers the hydraulic fractures with an arbitrary length and orientation and may
include the fracture face damage. The analytical solution describes a plane flow of a
homogeneous incompressible fluid into vertical fracture. The well is simulated by the point
source (sink) Q located at the center of the reservoir. This point sink is surrounded by elliptical
fracture characterized by permeability k f and semiaxes l and w corresponding to the halflength and half-width. The focal distance is f: l 2 − w2 = f 2 . The pressure distribution for the
reservoir and fracture is determined by Laplace’s equation, and there exists the complex
potential of the flow into a point source located at the center of elliptical inclusion in
9th European Conference on the Mathematics of Oil Recovery – Cannes, France, 30 August – 2 September 2004
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a homogeneous reservoir:
Φm =
km
µ
p m + iψ m .
(2)
Here, pm is the pressure, ψ m is the stream function, the subscript m=r,f corresponds to the
reservoir and fracture.
The approach to hydraulic fracture account in reservoir model is based on the assumption that in
a vicinity of the well the flow structure is described by the analytical expressions of Φ m
functions, which define the fractured well production rate. Boundary conditions for this solution
are determined by grid block parameters. The flow along the fracture and the flow interchange
with the reservoir are calculated by means of finite-difference methods, the flow rate formula is
introduced only for the block containing the central segment of the fracture. One/two
dimensional fracture simulator is used in two/three dimensional reservoir model.
Let pr is the pressure in the point z. Then the analytical expression of potential (2) gives [9]
2πkr h pr − pw
µ
Ρ(z)
2
∞
⎛
⎞
⎡
2
2
⎡z
⎤
⎞ ⎤
z
z
z
4m ⎛ z
4m ⎟
m
⎜
[
]
(
)
λ
λ
P( z) = Re (1 − λ) ln + λ ln⎢ +
1
1
ln
1
q
1
1
−
q
−
+
−
+
−
−
⎜
⎟
⎥
⎢
∑
2
⎜
⎟ (3)
w
f
f
f 2 ⎥⎦
f
m=1
⎣
⎝
⎠
⎥
⎢
⎦
⎣
⎝
⎠
k f − kr
λ=
,
q= l−w
k f + kr
l+w
Q=
Here, z = x + iy = re iϕ is a complex variable, r is the distance from the source, ϕ is the polar
angle reckoned from the direction defined by the major axis of the fracture.
In general, the fracture passes through several grid blocks and can be arbitrarily oriented with
respect to the grid. If r1,2 are the distances of the block boundaries from the center of the
fracture, reckoned along the fracture axis, the curves z1,2 = r1,2 e iϕ bound the segment of the
fracture inside the block. The flow q from the reservoir into the fracture through the sections of
the boundary inside the block is determined by the expression [9]
r2
r2
r1
r1
q = 2 ∫ vn ds = −2 ∫
∂ψ
ds = 2(ψ (r1 ) − ψ (r2 )) ,
∂s
w
Ψ(ri ) = (1 − λ ) + λ ⋅ arctan
ri
f 2 − ri 2
ri
ψ (ri ) = Im(Φ r ( z i )) =
∞
− (1 − λ ) ∑ λ ⋅ arctan
m
m=1
4m
(4)
f 2 − ri 2
q 4 m ri
(1 − q ) f
Q
Ψ (ri )
2π
2
/ 2 + q 4 m ri 2
Ψ( rw ) = Ψ( w) = π 2 , Ψ( l) = 0
Here, ν n is the flow velocity component normal to the fracture boundary, and s is the tangent to
the boundary. ri >> w , so that ϕ ≈ sin ϕ = w ri .
The well production rate Q is divisible into two parts
Q = Q f + Qr
(5)
Here, Q f is the flow along the fracture into the grid block 0, where the well is located, and Qr is
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the inflow from the neighbouring cells into the grid block 0. For the fracture segment where the
center of the fracture (sink) is located, the expression (4) gives
Qr = Q(π − Ψ (r1 ) − Ψ (r2 )) π
(6)
The five-point finite-difference of the continuity equation for the block 0 is as follows
kr h
µ
4
∑a (p
i =1
i
i
− p0 ) − Qr = 0,
a1,3 =
∆y
,
∆x1,3
a 2, 4 =
∆x
∆y 2, 4
(7)
Here, ∆x , ∆y are the dimensions of the block 0; ∆xi , ∆yi are the distances from the node 0 to
neighbouring nodes, i=1,...,4; pi are the nodal pressures, p 0 is the pressure in the grid block 0.
As a result the flow Qr is given by the following expression obtained from (3), (6), (7)
Qr =
2(π − Ψ (r1 ) − Ψ (r2 ) )k r h ( p 0 − p w )
µ
Ρ( z 0 )
⎤⎛ 4 ⎞
⎡ 4
Ρ( z 0 ) = ⎢∑ ai Ρ( z i ) − 2(π − Ψ (r1 ) − Ψ (r2 ) )⎥⎜ ∑ ai ⎟
⎦⎝ i =1 ⎠
⎣ i =1
−1
(8)
Here, zi is the complex coordinate of the ith node in the coordinate system associated with the
fracture, r1, 2 are the distances between fracture center and the points of intersection of the
fracture and the 0th block boundaries. If there is no fracture, then Ρ( z i ) = ln z i rw and
Ψ(r1,2 ) = 0 , and the formula (8) coincides with the Peaceman formula [8].
The Q f value is defined by (3), (5), (8) as follows
Qf =
l0 =
2(Ψ (r1 ) + Ψ (r2 ) )k r h ( p 0 − p w ) 4k f wh ( p 0 − p w )
=
µ
µ
l0
Ρ( z 0 )
C f lΡ ( z 0 )
(Ψ (r1 ) + Ψ (r2 ) )
, Cf =
(9)
2k f w
kr l
Here C f is the dimensionless fracture conductivity; 2w is the fracture width taken to be constant
within the grid block. In the calculation, the flow along the fracture is assumed to be linear. And
the expression (9) shows that the p 0 value can be interpreted as the pressure in the fracture at the
distance l 0 from the center. Because of this, when calculating the fracture flow into the grid
block 0 from neighbouring blocks, the internodal distances due to be reduced by l 0 .
The presented approach is used in in-house 3D black oil simulator. This approach gives a good
agreement with the exact solution (2) [9]. Some calculations were carried out to test the
presented method for simulation of multiphase flow into fractured wells. The above-mentioned
model of five spot pattern with fractured wells was analyzed, fracture half-length is 100 m,
reservoir permeability is 1 mDarcy. And the described results of direct fracture simulation by
refined grid were compared with results obtained by means of presented program for coarse
regular grids. The fig. 4 shows the good agreement of oil and liquid production as well as the
water injection for different sizes of grid blocks. In case of 25 m regular grid the number of grid
blocks is lowered by a factor of 5, for 100 m block size this factor is equal to 80.
9th European Conference on the Mathematics of Oil Recovery – Cannes, France, 30 August – 2 September 2004
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Effect of fracture orientation on performance of multi-well system.
The area of real field was simulated in order to estimate fracture orientation effect on production
performance. Priobskoye is a large West-Siberian oil field. Productive reservoirs can be
characterized as isolated oil-saturated low permeable lenses. That’s why hydraulic fracturing is
supposed to be the main completion technique for all wells. Geological model of the area under
consideration was built using Geoframe (Schlumberger). Fig. 5 demonstrates the fence grid of
permeability distribution. Average permeability is 0.01 Darcy; average porosity equals 16%,
fluid properties are presented in the table 1. After upscaling the simulation grid was reduced to
159 x 216 x 5 сells. Grid block size is 25 m in XY plane. 24 producers and 24 injectors are
placed according to regular triangular pattern with 600 m well distance. Consideration is being
given to some cases where all wells are fractured. Fracture width is 0.01 m; its permeability is
200 D. Two alternatives for fracture half-length was chosen: 100 and 200 m. Wells operate under
bottom hole pressure control: 100 atm for producers and 350 atm for injectors. Three options of
fracture orientation were simulated: parallel to the rows of wells (i.e. along X axis),
perpendicular to the rows of wells (i.e. along Y axis) and at the angle of 60o to the rows of wells.
Case’s nomenclature and description can be found on the figure 6. One should note that the
distance between tips of fractures in injector and producer is different depending on fracture
orientation: the largest being in the case of fractures parallel to the rows of wells and the least
being in the case of fractures arranged at the angle of 60o to the rows of wells, i.e. producer and
injector fractures are pointed towards each other.
Runs were performed using previously described simulator. Fig. 7, 8 represent recovery factor
and water cut versus time for fracture half-length of 100 and 200 m correspondingly. For
fractures with half-length of 100 m the water breakthrough occurs 1.5 times earlier when
fractures in injector and producer are oriented towards each other, but total liquid production
differs only by 2% with approximately equal total oil production, fig. 7. The same graph for the
cases with 200 m half-length fractures, fig. 8, shows more drastic picture. The water
breakthrough occurs almost at the beginning of production in case of fractures arranged at the
angle to the rows of wells, whereas this happens two years later in other cases. Thus, the
cumulative oil production for this case is less by 4% in comparison with the case with fractures
parallel to the rows of wells. And the maximum difference in oil rate between cases amounts to
1.5 times. For the reservoir with lower permeability the difference in total oil production
between cases distinguishing by fracture orientation grows to 15%.
Conclusions
The method of hydraulic fracture simulation in reservoir numerical models has been developed
on the basis of the analytical solution obtained. This method has given satisfactory results even
when a finite-difference grid with large cells has been used. 3D black oil simulator taking into
account a flow in the reservoir and fractures simultaneously for multi-well system has been
created.
Calculations for five spot pattern and real field area have shown that for water flooded lowpermeable reservoir where the majority of wells are to be fractured the fracture direction is very
important parameter in well placement planning. If rows of producers are parallel to the fracture
direction cumulative oil production grows and the water breakthrough occurs much later. For
low permeable reservoir the difference in total oil production between cases distinguishing by
fracture direction can reach 15%.
Authors would like to thank dr. R.Kats for helpful advices given during this research.
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References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Prats M. (1961). Effect of vertical fractures on reservoir behavior - incompressible fluid
case. Soc. Petrol. Eng. Journal,1: 105-118.
Aziz H. (1993). Reservoir simulation grids: opportunities and problems. Paper SPE 25233.
Mlacnik M.G., Heinemann Z.E. (2003). Using well windows in full-field reservoir
simulation. SPE Res Eval.&Eng.: 275-285.
Settari A., Puchyr P.J., Bachman R.C. (1990) Partially decoupled modeling of hydraulic
fracturing processes. SPE Prod. Eng., 5: 37-44.
Nghiem L.X. (1983) Modeling infinite-conductivity vertical fractures with source and sink
terms. Soc. Petrol. Eng. Journal, 23: 633-644.
Nghiem L.X., Forsyth P.A.Jr., Behie A. (1984) A fully implicit hydraulic fracture model.
J.Petrol.Technol., 36: 1191-1198.
Peaceman D.W. (1978) Interpretation of well-block pressures in numerical reservoir
simulation. Soc. Petrol. Eng. Journal, 18: 183-194.
Peaceman D.W. (1983) Interpretation of well-block pressures in numerical reservoir
simulation with nonsquare grid blocks and anisotropic permeability. Soc. Petrol. Eng.
Journal, 23: 531-543.
Kanevskaya R.D. (1999) Simulation of oil and gas reservoir engineering using hydraulic
fracturing, Moscow, “Nedra” (in Russian).
Table 1
Multiwell
case
247
10
1.14
2.22
0.35
879
80
0.35
879
35
Water viscosity, cP
3
Oil density, kg/m
3
3
Gas oil ratio, m /m
850 m
600 m
850 m
Recovery efficiency, %
40
100
parallel, Lf = 100 m
parallel, Lf = 200 m
at the 45 angle, Lf = 100 m
at the 45 angle, Lf = 200 m
30
75
20
50
10
25
Vinj/Vp
0
0
Fig.1 Sketch plot of fracture orientation for 5spot
pattern element
0.1
0.2
0.3
Water cut, %
Initial pressure, MPa
Bubble point pressure, MPa
Oil formation volume factor
Oil viscosity, cP
5spot
pattern
240
6.6
1.14
2.2
0
0.4
Fig.2 Recovery efficiency and water cut vs.
Fig.2
Recovery efficiency
water cut vs.
dimensionless
injectedand
volume
dimensionless injected volume
9th European Conference on the Mathematics of Oil Recovery – Cannes, France, 30 August – 2 September 2004
8
2
6
in-house code, 6 m cells
in-house code, 25 m cells
5
at the 45 angle, Lf = 100 m
Oil, liquid rate, injection rate,
3
thous.m
Total oil and loquid production, Mm3
parallel, Lf = 200 m
at the 45 angle, Lf = 200 m
1.5
Injection
Eclipse, nonuniform fine grid
parallel, Lf = 100 m
Liquid
1
Oil
0.5
in-house code, 100 m cells
Liquid
4
3
Oil
2
1
Years
0
01
15
30
45
16
31
46
Fig.3 T otal oil and liquid production vs. time
Y ears
0
60
61
0
20
40
60
80
100
Fig.4 Comparison of simulation results obtained
Fig.4 Comparison of simulation results
with Eclipse and in-house code
Fig.3 Total oil and liquid production vs. time
obtained with Eclipse and in-house code
p100y,
p200y
p100x,
p200x
r100x,
r200x
500 m
600 m
5
10
15
20
r100y
r200y
Fig.5 Permeability distribution in real field model,
mDarcy
100
50
100
p100x
p200x
p100y
40
20
10
20
0
0
40
10
49
4
73
6
97
8
Fig. 7 Recovery efficiency and water cut versus time
Fig.7
Recovery
efficiency
and water
for cases
with fracture
half-length
100cut
m vs.
time for cases with fracture half-length 100 m
W ater cut, %
Recovery efficiency, %
Recovery efficiency, %
20
20
25
2
r200y
60
60
Years
Months
80
30
30
0
p200y
40
80
r100y
01
100
200
Fig.6 Sketch plot of fracture orientation for real field model
50
40
at 60o angle
121
10
Years
Months
01
25
2
49
4
73
6
97
8
Fig. 8 Recovery efficiency and water cut versus time
for cases
with fracture
half-length
200cut
m vs.
Fig.8
Recovery
efficiency
and water
time for cases with fracture half-length 200 m
0
121
10
Water cut, %
0
Fracture
Fracture
orientation with
Case
halfnomenclature respect to the
length, m
rows of wells
p100x
100
parallel
p200x
200
p100y
100
perpendicular
p200y
200