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Study on the Productivity of Multi-fractured Horizontal Wells in
EASTERN ACADEMIC FORUM
Study on the Productivity of Multi-fractured Horizontal Wells in
Ultra-Low Permeability Reservoirs
YANG Chuncheng 1, XU Ziyi 2, YIN Chao 3, ZHONG Huiying 1, YIN Hongjun 1
1. Key Laboratory of Ministry of Education. PRC, Northeast Petroleum University, Daqing, China,
163318
2. College of Petroleum Engineering, China University of Petroleum, Beijing, China, 102249
3. Oil Recovery Plant No.5, Daqing Oilfield Corp. Ltd., Daqing, Heilongjiang, 163513
[email protected]
Abstract: Based on the equivalent radius model, a productivity model for ultra-low permeability
reservoirs was developed, which considered the influence of the threshold pressure gradient and the
variable mass flow in fractures, and the effect of threshold pressure gradient, fracture number, fracture
half-length, fracture angel, and fracture distribution on the productivity were analyzed. The results show
that the greater the threshold pressure gradient, the greater the effect of it on multi-fractured horizontal
well productivity, so the threshold pressure gradient must be accounted to evaluate the productivity in
ultra-low permeability reservoirs. With the increase of fracture number, fracture half-length and fracture
angel, the productivity increases. In order to obtain higher productivity, the angle between the fracture
and horizontal well should be between 75 and 90 degree. When the total half-length of the fracture is
certain, reducing the fracture interference, the length of the outer fractures should be bigger than that
among the inner fractures. The research results provide a scientific basis for the design of multifractured horizontal wells in ultra-low permeability reservoirs.
Keywords: Multi-fractured horizontal wells, Ultra-low permeability, Threshold pressure gradient,
Productivity
1 Introduction
In recent years, development of multi-fractured horizontal well technologies has become one of the
important technologies for tight reservoir development. This has caused tremendous interest in oil and
gas productivity from unconventional reservoirs all over the world, especially in ultra-low permeability
reservoirs [1-2]. With the increase in oil and gas demands and the limited amounts of conventional
reserves, ultra-low permeability reservoirs are expected to play a key role in meeting the energy needs in
decades to come.
It is highly desirable to have access to methods capable of predicting productivity of multi-fractured
horizontal wells. Many researchers have made important efforts and laid foundation contributions for
the productivity prediction of fractured horizontal well. Lang Zhaoxin, et al. [3-4] suggested an analytical
model of horizontal well with multiple transverse hydraulic fractures by using potential energy theory
and superposition principle, they assumed linear flow from reservoir to fracture, and linear and
converging radial flow inside the fracture. In their approach, flow convergence is modeled using the
choke skin factor. Zeng Fanhui, et al. [5-7] studied an unsteady state model for fractured horizontal well
with finite conductivity fracture. By dividing the fracture into N segments, a series of equations had
been solved to calculate the pressure distribution and contribution of each segment to the total flow by
assuming each segment as a uniform flux source. Wan and Aziz [8] presented a semi- analytical well
model for horizontal wells with multiple hydraulic fractures. Guo Boyun and Yu Xiance [9] proposed
productivity index models for oil and gas horizontal wells with multiple hydraulic fracturesproducing
under pseudo-steady state conditions. In their model, radial flow in the non-fractured area of reservoir is
combined with linear flow towards fractures in the fractured region, and linear and radial flow in the
fractures toward the wellbore. Lian Peiqing, et al [10] developed an unsteady state model to predict the
performance of multilateral wells, which coupled a reservoir inflow model with a wellbore flow model
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EASTERN ACADEMIC FORUM
to calculate the production rate from each lateral and pressure drop along lateral wellbore was
considered in the model.
In previous studies productivity prediction of fractured horizontal well, it has considered a more
comprehensive in the flow characteristics and fracture parameters, but largely have ignored the problem
of the threshold pressure gradient in the ultra-low permeability reservoirs. The results often have larger
error, and mostly the whole fracture is assumed to be a point sink or line sink, that the fluid is uniform
inflow along the fracture plane. For this reason, on the basis of previous studies, considering the variable
mass flow in fractures and non-darcy flow in reservoir, a percolation model of fractured horizontal well
in accord with the actual oilfield was developed in ultra-low permeability. The effect of fracture number,
fracture half-length, fracture spacing and other factors on the productivity of fractured horizontal well
under different threshold pressure gradient has been analyzed, and got some useful conclusions.
2 Physical Models
There is a horizontal well in the reservoir which the effective thickness is h, the number of fracture are N,
the length of upper wing is xfu, the length of lower wing is xfd, the width of fracture is w, the
permeability of fracture is Kf, the productivity all comes from fractures, the upper and lower wing of
each fracture can be divided into some segments, the number of segment are M, the length of each small
segment is Δxf. The basic assumptions of the model as follows:
(1) The reservoir is homogeneous, with the coordinates aligned with the principal permeability
directions.
(2) Reservoir and fracture fluid are single-phase and slightly compressible with a constant
compressibility.
(3) Formation properties are independent of pressure.
(4) Fractures completely go through the formation, fracture height is equal to the thickness of the
reservoir.
(5) The fracture plane may have an angle to the wellbore.
3 Mathematical Models
3.1 Reservoir-inflow
For the fracture has been divided into many small segments, according to the equivalent radius
principles, each segment can be equivalent to a vertical well [11].
The equivalent radius of each segment can be expressed as:
rwef , kl  0.5xf , kl exp  ln 2  1.5 
(1)
Where rwef,kl is the equivalent radius of fracture k in segment l, m; Δxf, kl is the length of fracture k in
segment l, m.
According to the principle of superposition, the potential at any point in the formation can be expressed
as:
 M=
1 N 2M
 xf ,kl qf , k l ln rM, kl  C
2πh k 1 l 1
(2)
Where ΦM is any point in the formation, m2·
Pa/Pa·
s; qf, kl is the unit length oil productivity of fracture k
in segment l , m3/s/m; rM, kl is the distance from fracture k in segment l to any point in the formation, m;
C is the constant.
To take a special point in the supply boundary and each equivalent vertical well borehole wall, the
relationship between the each equivalent vertical well bottom hole flowing pressure and flow rate which
considering the threshold pressure gradient can be expressed as:
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EASTERN ACADEMIC FORUM
pe  pwfff ,ij 
rij , kl

N 2M
 x
2πKh
k 1 l 1
q
f , kl f , k l
ln
Re
   Re  rij ,c 
rij , kl
(3)
2
2

  xij  xkl    yij  ykl   i  kor  l 


rwef , kl  i  kand j  l 
(4)
Where K is the reservoir permeability, m2; μ is oil viscosity, Pa·
s; pe is the boundary pressure, Pa; pwfff, ij
is the borehole wall pressure of fracture i in segment j, Pa; Re is the supply radius, m; λ is the threshold
pressure gradient, Pa/m; rij, kl the distance from fracture k in segment l to fracture k in segment l, m; rij, c
is the distance from fracture i in segment j to the center of horizontal wellbore, m.
3.2 Fracture-inflow
As shown in the Figure 1, due to the flow to the wellbore after fluid to flow into the fractures is
one-dimensional linear flow, the relationship for fracture i can be expressed as:
For upper wing of fracture i:
pf ,ij  pf , i  j 1 
pf , ij  pwf,i 
qfc, i j xf ,ij
K fi wh
qfc, i j xf , ij
2 K fi wh
 j  M 
(5)
 j  M 
(6)
For lower wing of fracture i:
pf , ij  pwf,i 
qfc, i j xf , ij
2 K fi wh
pf ,ij  pf , i  j 1 
 j  M  1
qfc, i j xf , ij
K fi wh
(7)
 j  M  1
(8)
Where the cross section rate of fracture i in segment j is:
qfc,i j
 j
 xf , ij qf , im
 m 1
  2M
 x q
f , ij f , im

m j
( 9)
Figure 1 Schematic diagram of reservoir-fracture seepage
Where pf, ij is the center pressure of fracture i in segment j, Pa; pwf,i is the intersection of pressure
between fracture i and wellbore, Pa; Kfi is the permeability of fracture i, m2; qf, im is the unit length oil
productivity of fracture i in segment m, m3/s/m; qfc, ij is cross section rate of fracture i in segment j, m3/s.
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EASTERN ACADEMIC FORUM
3.3 Wellbore flow
According to the hydrodynamic theory, due to the horizontal wellbore wall friction and the fracture fluid
convergence, there is a certain pressure in the process of horizontal wellbore flow. As showed in the
Figure 2, to take a short between fracture i and fracture i+1 on the wellbore do a research, the outlet
pressure of fracture i left end is p1i, inlet pressure of right end is p2i. According to the conservation of
mass and momentum conservation, if considering the frictional and acceleration pressure drop in the
wellbore flow model, the wellbore flow equation can be expressed as:
(10)
p1(i 1)  p1i  pfrici  pacci
Where the acceleration pressure drop is:
pacci   v12i   v1(2 i 1)  p2i  p1i   v12i   v22i
(11)
The pressure drop from the left end of fracture i+1 to the right end of fracture i is:
p1( i 1)  p2i  f i
Qof2 i 
Li
4π 2 rw5
3 ,N 
 i  2,,
(12)
And the cross section rate of fracture i is:
Q of ( i 1)  qofi
Qofi = 
qofi
(i  N)
(13)
(i  N)
Where fi is the wellbore wall surface friction coefficient of fracture i; ρ is the crude oil density, kg/m3, rw
is the wellbore radius, m; Qofi is the cross section rate of fracture i, m3/s; qofi is the oil productivity of
fracture i, m3/s; ΔLi is the distance between fracture i to fracture i+1, m.
fracture i
p1i
fracture i+1
p1(i+1)
p2i
p2(i+1)
wellbore
Figure 2 Schematic diagram of wellbore flow
3.4 Coupling model
The following coupling relationships are made for the couple model.
1) The coupling relationship of reservoir-fracture: the pressure of the reservoir inflow in the fracture
wall is equal to the pressure of fracture inflow in the fracture wall.
(14)
pwfff,ij  pf ,ij
2) The coupling relationship of fracture-wellbore: the pressure of fracture inflow in the intersection
between the horizontal wellbore and fracture is equal to the pressure of wellbore flow in the intersection.
pwf,i 
p1i  p2i
2
(15)
4 Solution of Mathematical Model
By using the reservoir–fracture inflow equation (3), (5)-(8), the wellbore flow equation (10)-(12), and
the coupling relationship, the coupling model is obtained.
The model is composed of 2N+4NM equations consisting of nonlinear equations. Therefore, an iterative
method should be used to solve the equations. Iterative scheme is as follows:
q n 1  A1 p n , p n 1  Fq n 1
(16)
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EASTERN ACADEMIC FORUM
Where pwf is pressure at the heel of the horizontal wellbore, which is assumed to be known. From the
initial value p0=[pwf, pwf, …, pwf] being started, the new pressure vector and flow rate vector are obtained
by the iterative forms until the changes of p, q were less than a certain value, that is
 max  pf,nij1  pf,nij  1
1 i  N ,1 j  2 M

(17)
 qf,nij1  qf,nij   2 
1 i  max
N
,1

j

2
M

Here the value of ε1 and ε2 depends on the degree of accuracy needed in prediction of well productivity.
5 Mathematics Simulation
Taking a ultra-low permeability reservoir a fractured horizontal well as the research object, using the
productivity predicting model simulation. The Reservoir and fluid data are given in Table 1.
Table 1 Parameters values for the horizontal well
Parameter
Value
Reservoir permeability, 10-3 μm2
1.1
Supply radius, m
500
Threshold pressure gradient, MPa/m
0.002
Reservoir thickness, m
8
Horizontal wellbore length, m
400
Fracture half-length, m
100
Fracture number
4
Fracture conductivity, μm2·
cm
25
Fracture angle with wellbore, °
90
Wellbore radius, m
0.07
Oil viscosity, MPa·
s
5.5
Oil density, kg/m3
864
Wellbore roughness, m
0.000 15
Volume factor
1.12
5.1 Fracture number
Figure 3 shows the effect of the fracture number on the productivity of fractured horizontal well under
different threshold pressure gradient. As showed in the Figure 3, as the fracture number increases, the
productivity also increases, and because of the interference of the number of fractures, the productivity
increased amplitude decreases. Figure 3 also shows, as the threshold pressure gradient increases, the
productivity decreases. So when the threshold pressure gradient is greater, it suggests selecting larger
number of fractures of the same productivity.
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EASTERN ACADEMIC FORUM
9.0
3
Q (m /d)
7.5
6.0
0 M Pa/m
0.002 M Pa/m
0.004 M Pa/m
0.006 M Pa/m
4.5
3.0
0
2
4
6
8
10
N
Figure 3 Effect of the fracture number on the productivity of fractured horizontal well
5.2 Fracture half-length
Figure 4 shows the effect of the fracture half-length on the productivity of fractured horizontal well
under different threshold pressure gradient. As showed in the Figure 4, as the half-length increases, but
the productivity increased amplitude decreases. Figure 4 also shows, as the threshold pressure gradient
increases, the productivity decreases, the contribution of the per unit length of fracture to productivity
decreases.
9.0
Q (m3/d)
7.5
6.0
0 M Pa/m
0.002 M Pa/m
0.004 M Pa/m
0.006 M Pa/m
4.5
3.0
0
30
60
90 120
x f(m)
150
180
Figure 4 Effect of the fracture half-length on the productivity of fractured horizontal well
5.3 Fracture spacing
Keeping the other parameters constant, fixed the toe and heel end of two fractures, changing the internal
two inner fractures spacing, and the productivity under different inner spacing is calculated , as showed
in the Figure 5. When the inner spacing is between 40-100m, as the inner spacing increases, the
productivity increased; when the inner spacing is between 100-140m, as the inner spacing increases, but
the productivity increased amplitude decreases; when the inner spacing is larger than 140m, as the inner
spacing increases, the productivity decreases. When the inner spacing is very large when the total
spacing is certain, the fracture spacing of toe and heel end is small, the productivity significantly reduces
as for fracture interference. This indicates that it may be appropriate to increase the spacing of inner
fractures for greater productivity.
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EASTERN ACADEMIC FORUM
7.26
Q (m3/d)
7.24
7.22
7.20
7.18
7.16
0
40
80
120 160
d (m)
200
240
Figure 5 Effect of inner fractures spacing on the productivity of fractured horizontal well
5.4 Fracture angel
Figure 6 shows the effect of the fracture angel on the productivity of fractured horizontal well. As shown
in the Figure 6, as the fracture angel increases, the productivity increases, when the fracture angel
increases from 15 degree to 45 degree, the productivity increased significantly, when the fracture angel
increases from 60 degree to 75 degree, the productivity increases slow, when the fracture angel increases
from 75 degree to 90 degree, the productivity is almost equal, so the optimal fracture angel is 75 degree
to 90 degree.
7.4
3
Q (m /d)
7.2
7.0
6.8
6.6
6.4
15
30
45
60
θ (°)
75
90
Figure 6 Effect of the fracture angel on the productivity of fractured horizontal well
5.5 Fracture distribution
Taking four fractures of the horizontal well for example, the fracture spacing is equal, the total fracture
half-length is certain, and the fracture half-length close to the toe and heel end gradually increase from
0m to 200m, and the productivity under different fracture half-length distribution is calculated. As
showed in Figure 7, as the fracture half-length increases, the productivity decreases and then increases,
when the inner fracture half-length is 0m, the productivity is the most. Therefore, from the perspective
of single well productivity optimization, in order to improve the fractured horizontal well productivity,
the fracture half-length distribution should be fully accounted during the fracture parameter optimization
analysis.
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EASTERN ACADEMIC FORUM
8.5
3
Q (m /d)
8.0
7.5
7.0
6.5
0
40
80
120
x f (m)
160
200
Figure 7 Effect of different fracture half-length distribution on the productivity of fractured horizontal well
6 Conclusions
(1) The coupling model of variable mass flow in fractures and non-darcy flow in reservoir was
developed, which considered the effect of threshold pressure gradient. The model can simulate on any
fracture spacing distribution, any angle between wellbore and fracture, and any conductivity fracture
conductivity.
(2) Threshold pressure gradient is an important factor which affects the ultra-low permeability reservoirs
productivity; it suggests selecting a larger number of fractures when the threshold pressure gradient is
greater.
(3) From the perspective of the single well productivity, during the fracture parameter optimization, it
should take full account of the effect of fracture distribution and fracture spacing, reducing the sutural
interference for higher productivity.
Acknowledgements:
This research was supported by the Scientific Research Project of the Heilongjiang Education
Department (Grant No: 12521044).
Corresponding Author:
Yin Hongjun:
Tel: 0459-6504066
E-mail: [email protected]
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