A New Method to Determine the Injection Allocation of Single... Well in Low Permeability Reservoirs

EASTERN ACADEMIC FORUM
A New Method to Determine the Injection Allocation of Single Water
Well in Low Permeability Reservoirs
ZHANG Junting1, YIN Hongjun1, CHEN Xiaoqi2
1. Key Laboratory of Ministry of Education. PRC, Northeast Petroleum University, China, 163318
2. Tianjin Subsidiary Company of China Offshore Oil Company, China, 300452
[email protected]
Abstract: The theory of non-Darcy flow is analyzed in low permeability reservoirs, and the reservoir
average pressure equation is established, which considers the impact of threshold pressure gradient and
injection-production ratio, according to material balance equation and injection-production ratio
equation. A new method is advanced to determine the injection-production ratio and injection allocation
flux in waterflooding reservoir. This method takes into account the impact of pressure gradient, while
the amount injection allocation does not ensure oil well production, but also recover the formation of the
average pressure. The computation process adopt the multilayer iteration method, consider the impact of
the injection well and production well pressure in areal well pattern, the target liquid producing capacity,
the average formation pressure and injection-production ratio. According to this method and combining
with the real data of the Chinese daqing oilfield S1 block, the water injection of single well is 8.26m3/d.
The result is advisable and accurate, which provides theoretical guide for practical application.
Keywords: Water injection, Iteration, Low permeable reservoir
1 Introduction
In recent years, with the development of low permeability and more low permeability reservoir, people
pay more and more attention to the rule which the fluid move in the low permeable and more low
permeable reservoir[1]. The fluid does not follow the classical Darcy equation, but non-Darcy. At the
same time, through a large number of studies, it has been realized that there is threshold pressure
gradient when the permeability is very low in the low permeable reservoir. As the reservoir’s formation
is dense, the ability of transmission pressure is bad, and the natural energy is small. The first recovery
efficiency is very low. In order to achieve better development effect and economic benefits,
development mode with artificial methods has been adopted, at the moment using artificial water
flooding.
In the process of waterflooding development, when the waterflooding time and water injection mode are
defined, reservoir engineer mainly research the problem that where the formation should be injected
with water [2]. A new method defined the water injection is proposed, which considers the impact of
threshold pressure gradient. The low permeability non-Darcy mathematical model is established, which
is through using the material balance equation and the injection-production ratio formula [3].To use the
iterative method to define the water injection. It is directive and referred for low permeable flood
development fields [4][5].
Many research data show that, due to solid and liquid interfaces, there is a boundary layer of oil in the
inner surface of the porous rock. In the boundary layer, the oil component and quality are very different,
the composition changes orderly, structural viscosity and yield value will be generated. The thickness of
boundary layer is not with itself nature, but with the size of pore path and the drive pressure on the
gradient, and so on [6].
Due to the small porous channel radius in low permeable reservoirs, the channel less than one
micrometer cover a great proportion, and the crude oil boundary layer has a significant influence on the
flow, which is demonstrated by the threshold pressure gradient in the process of flowing. Lots of
research data show that threshold pressure gradient is inversely proportional to the permeability.
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EASTERN ACADEMIC FORUM
2 The Research of Single Well Injection Allocation Method in Low Permeability
Reservoirs
2.1 Basic theoretical basis
In different waterflooding oilfields, there are many production wells and injection wells. But the
production and injection rate between the wells can be different. When producing, every well has a
certain size of supply area. In developed reservoir, there are three possible trends when a well supply
pressure or the entire border reservoir pressure changes with time: the first, pressure reduces with time;
the second, pressure increases with time; the third, pressure is constant with time.
It is assumed that the pressure changes linearly, that is:
p(t ) = at + c
(2-1)
When a is smaller than zero, the pressure reduces with time; when a is more than zero, the pressure
increases with time; when a equals zero, the pressure is constant with time. The equation (2-1) is to
describe the pressure variation in pseudo steady-state flow (including the steady fluid). Compared with
the conventional steady fluid which formed in a closed reservoir to be non-invasive fluid or injection,
this reflects a more general case. So it is called generalized pseudo steady-state fluid. When a equals
zero, it indicates steady fluid. Here premises the generalized pseudo steady-state fluid, and derives the
velocity equations in the change of linear pressure.
The material balance equation which is of no native water injection in the waterflooding reservoir is:
N p Bo + Wp B w − Wi B w = NBoi C e ∆p
(2-2)
Where Bo—oil volume ratio,m3/m3;Boi—initial oil volume ratio,m3/m3;Bw—oil volume ratio,
m3/m3;Ce—available fluid total compressibility,MPa-1; N—initial geologic reserve,m3;Np—cumulative
oil production,m3; Wi—cumulative water injection,m3; Wp—cumulative water injection,m3;
∆p —average pressure difference.
When the equation (2-2) is used to describe single well, these parameters above-mentioned become the
case of single well system.
The equation (2-2) can be differentiated and written as:
d∆ p
(2-3)
(1 − RIP ) qB = 24 A1hϕ Ct
dt
d(Wi Bw )
1 d ( N p Bo + Wp B w ) RIP =
q=
d
(
N
p Bo + W p B w )
B
dt
;
Where q—daily oil production, m3/d; RIP—injection-production ratio; t—production time, h.
In the term of linear pressure, that is t ≥ t pss (the time pseudo steady-state flow comes), giving:
dp
d∆p
dp
=−
= − wf
dt
dt
dt
The equation (2-3) can be written as:
dpwf
qB
=−
(1 − RIP )
dt
24 Ahϕ Ct
(2-4)
Using the dimensionless form is:
dpwD
= 2π(1 − RIP )
dtAD
Where the parameter is:
pwD =
Kh
3.6K
( pi − pwf ) tAD =
t
1.842 ×10−3 qBµ
ϕµ
Ct A
;
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(2-5)
EASTERN ACADEMIC FORUM
Equation (2-5) is the velocity equation of the linear pressure change. This formula is applicable to any
reservoir, the injection-production ratio reflected reservoir conditions is: for the closed reservoir, RIP
equals zero; for the part of the fluid is injected or invasive, RIP is more than zero and smaller than one;
for adequate supply of fluid injection or constant pressure boundary, RIP equals one; for the fluid
over-injection, RIP is more than one.
2.2 The determination of single water injection formula for Non-Darcy flow in low permeability
reservoir
In the context of linear pressure, the law of conservation of matter and formula (2-4) are used to
establish the mathematical model, which considers the impact of threshold pressure gradient and
injection-production ratio.
The control equation is:
2 ×1.842 ×10−3 ql µ B
∂ 2 p 1 ∂p
(
)
(1 − RIP )
λ
+
−
=
−
B
( re2 − rw2 ) Kh
∂r 2 r ∂r
p( rw , t ) = pwf (t )
(2-6)
(2-7)
−3
r(
∂p
1.842 ×10 ql µ B
− λB ) |r = re =
RIP
∂r
Kh
(2-8)
The formulas (2-6) to (2-8) are solved, the formula of pressure distribution can be determined, which
considers threshold pressure gradient and the oil well of controlling band and injection-production ratio,
that is:
p(r,t ) = pwf (t ) +
1.842 ×10−3 ql µ B
r
r2
[ln − (1 − RIP ) 2 ] + λB (r − rw )
Kh
rw
2re
(2-9)
The formula of pressure distribution which considers threshold pressure gradient and the water well of
controlling band and injection-production ratio, that is:
p(r,t ) = piwf (t ) −
1.842 ×10−3 qw µw Bw
r
r2
[ln − (1− RPI ) 2 ] − λB (r − rw ) (2-10)
Kw h
rw
2re
The average pressure of using area weighting method is:
∫
p=
re
rw
p ( r , t )2π rdr
π(re2 -rw2 )
(2-11)
By substituting (2-9) to (2-11), the average pressure formula controlled by oil well band is:
p = pwf (t ) +
r
1.842 × 10 −3 ql µ B
3 − RIP
2
(ln eo −
) + λB ( reo − rw )
Kh
rw
4
3
(2-12)
The average pressure formula controlled by water well band is:
p = piwf (t ) −
r
1.842 ×10−3 qw µ w Bw
3 − RPI
2
(ln ew −
) − λB ( rew − rw ) (2-13)
Kwh
rw
4
3
When the distance s is given the boundary distance re, the reservoir pressure of boundary will be
obtained. When the formula (2-9) is changed, the radius formula controlled by oil well is obtained. And
the threshold pressure gradient and injection-production ratio are considered in the formula. That is:
reo =

1 
1.842 ×10−3 ql µ B reo 1
[ln − (1 − RIP )] + rw
 pe − pwf (t ) −
Kh
rw 2
λB 

The formula (2-14) is also the iteration formula of the radius controlled by oil well.
When the formula (2-10) is changed, the water well injection pressure formula is:
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(2-14)
EASTERN ACADEMIC FORUM
piwf (t ) = pe +
r
1.842 ×10−3 qiw µ w Bw
1
[ln ew − (1 − RPI )] + λB (rew − rw )
Kwh
rw 2
When the formula (2-15) is changed, the iteration formula of the injection rate is:
K w h [ piwf ( t ) − pe − λB ( rew − rw ) ]
qiw =
r
1
1.842 × 10 −3 µ w Bw [ln ew − (1 − RPI )]
rw 2
(2-15)
(2-16)
2.3 To determine the single well injection through using the iteration method in low permeability
reservoir
For areal well pattern, if only research the relation between injection pressure and influx, it is not disable
to reflect the block’s comprehensive exploitation situation, but also the data which is for computing is
not justifiability. So, a method is proposed to determine the injection-production ratio by using
multilayer iteration method, considers the impact of pressure in areal well pattern.
This method applies the well fluid and bottom hole flowing pressure of oil well as the known parameters.
Using the formula (2-9) and formula (2-10) to determine the location of oil/water well boundary
pressure, and then determine water well pressure controlling area. And according to the
injection-production ratio, water injection can be computed. Then substituting them to the formula
(2-15), the injection pressure can be computed. The concrete flow graph is graph 2-1.
Known parameters: pwfo ,ql ,µ ,B ,K ,h ,rw
Process variables: pe , RIP
×
Iterate reo
qi=ql RIP
reo
rew=L-reo
qi
rew
Known parameters:µ, B, K, h ,rw, qi,
Process variables:pe, RIP , rew
pe
pwfi
Known parameter:pwfo
variables:pe, pwfi
Solve pwfi
pwfi
Known parameters:pwfo, µ, B, K, h, rw
Process variables:pe, RIP , rew, pwfi
Solve pe
Interate RIP
pe
RIP
Figure 2-1 Iteration flow graph
Define the single well fluid ql, oil well bottom flowing pressure pwfo as known quantity, and oil/water
well boundary pressure’s starting value pe1, substituting them to the formula (2-14), can obtain the first
operating distance reo1 which oil/water well boundary is from oil well.
So the distance rew1 which oil/water well boundary is from water well will be known.
For iterating to compute pressure, first to compute the injection pressure piwf, using the initial value
injection-production ratio RIP and the objective well fluid of oil well to substitute the formula (2-15), the
injection pressure piwf will be obtained.
So far, substituting the injection pressure piwf and well boundary pressure pe1 and the initial value
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EASTERN ACADEMIC FORUM
injection-production ratio R to the formula (2-16), to iterate and modify the injection-production ratio.
When completing the variable parameters iteration of initial value, using the computed initial value to
iterate cyclically like foregoing steps. Then the constriction’s influx will be acquired.
3 Application Example
The S1 block of Daqing oil field in China, is a typical low permeable reservoir, how to confirm the
water injection reasonably is an important issue. So, the method introduced above is used to compute the
water injection.
The basic parameters for iteration list as follows K is 3×10-3µm2,h is 8.7m, µ is 4.2 mpa·s, Bo is 1.103, ql
is 1.5m3/d, pwfo is 1.99 MPa, λ is 0.0146 MPa/m, rw is 0.1m.
According to the graph 2-1, the single well fluid ql=1.5m3/d and oil well bottom flowing pressure pwfo=1.99MPa are used as known quantity, and oil/water well boundary pressure’s starting value pe1=5MPa,
substituting them to the formula (2-15), the first operating distance reo1=131.56m can be obtained. The
single well injection is 8.26m3/d for iterating cyclically. A conclusion is summarized for calculating,
which oil/water well boundary pressure’s starting value is greater than oil well bottom flowing pressure
is appropriate.
4 Conclusion
(1) The mathematical model of pressure distribution is established, which considers the impact of
threshold pressure gradient and injection-production ratio, according to material balance equation and
injection-production ratio equation. Then the formula to determine the pressure distribution and the
average pressure in the area controlled by oil/water well has been obtained.
(2) It is proposed that derivates the iteration formula about the oil/water well controlling radius and
water injection, and the flow graph of iteration method is described.
(3) According to the basic data of the S1 block of Daqing oil field, the S1 block water injection of single
well which equals 8.26m3/d is determined. The water injection obtained by iterative method guarantees
target liquid producing capacity and buildups formation pressure.
Acknowledgments:
This author gratefully acknowledges financial support from Heilongjiang Provincial Science and
Technology Plan Project (Grant No: GZ09A407) and Research Program of Innovation Team of Science
and Technology in Enhanced Oil and Gas Recovery (Grant No: 2009td08)
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