Research Journal of Applied Sciences, Engineering and Technology 3(5): 369-376, 2011
ISSN: 2040-7467
© Maxwell Scientific Organization, 2011
Received: September 03, 2010
Accepted: December 25, 2010
Published: May 25, 2011
Theoretical Analysis of Gravity-controlled Waterfloods
1
1
T.N. Ofei and 2R. Amorin
Department of Petroleum Engineering, African University of Science
and Technology, Abuja, Nigeria
2
University of Mines and Technology, Tarkwa, Ghana
Abstract: Gravity waterflooding (also known as “dumpflooding”) is a process in which water flows from a
high pressure aquifer zone to a low pressure oil-producing zone using natural gravity. The water flows through
a well connecting the two zones, enhancing the oil sweep into a producing well. This process is economically
attractive due to the absence of injection surface facilities and injection fluid costs. This research quantifies the
rate of fluid transfer from a high pressure aquifer zone to a low pressure oil producing zone. Detailed theoretical
equations are derived to evaluate the rate of water injection from an aquifer into an oil reservoir with a gas cap
by dumpflooding. Two cases considered are: (1) A finite aquifer injecting into a finite oil reservoir with a gas
cap, and (2) An infinite aquifer injecting into a finite oil reservoir with a gas cap. A single-well model has also
been built using an Excel spreadsheet to solve the ordinary differential equations using the bisection iterative
technique.
Key words: Aquifer, dumpflooding, finite, gas cap, infinite, reservoir
difficulties with flood front control, water breakthrough,
conformance management, and the inability to quantify
the crossflow rate in each well (Rawding et al., 2008).
This study suggests a solution to one of the flaws in
dumpflooding: the inability to quantify the rate of fluid
transfer from a high pressure aquifer zone to a low
pressure oil producing zone.
This finding was significant because it presents
detailed theoretical equations for evaluating the rate of
water injection from an aquifer into an oil reservoir with
gas cap.
Davies (1972), demonstrated that the rate at which
fluid transfers from one zone to another is a constant
value if the reservoir static pressure in both zones is
maintained and presented as:
INTRODUCTION
The technique of injecting water into an oil producing
reservoir for either pressure maintenance or secondary
recovery is a well-established process. However, a
method is required to replace the voidage created by the
volume of oil extracted from the reservoir and to reduce
and reverse the declining pressure and increase oil
recovery (Rawding et al., 2008).
The water sources available for injection purposes
are: produced water, sea water, aquifer water, and fresh
water and economics associated with the injection project
usually dictate which source should be used. Usually the
source of water depends on the quality and quantity of
water available (Davies, 1972). Probably the cheapest
method of injecting water into an oil reservoir is gravity
waterflooding, or dumpflooding, and the two terms will
be used interchangeably in this study.
Dumpflooding refers to the process of allowing a
water-bearing reservoir of high pressure potential to feed
into an oil reservoir of lower pressure potential by placing
the two zones in communication through a casing string
thereby, providing reservoir pressure support and
enhancing oil sweep into a producing well (Davies, 1972;
Quttainah and Al-Hunaif, 2001; Rawding et al., 2008).
This process is often preferred for cost reasons over
conventional waterflooding given the absence of injection
surface facilities and injection fluid costs. However, there
are also challenges with monitoring the dumpflood wells
and controlling the reservoir pressure. These include
⎡1 1
⎤
I w = ⎢ + + ∆ Pfr ⎥ = Pew − Peo
I
J
⎣
⎦
(All terms defined in nomenclature)
Quttainah and Al-Hunaif (2001) analysed the
applicability of long term effects of dumpflood operation
to enhance sweep and maintain reservoir pressure. A
dumpflood pilot project was initiated from Umm Gudair
reservoir to prove the viability of and quantify sweep
benefits of water injection from a source aquifer into a
recipient oil reservoir (injected zone). Their study
recorded that the dumpflood pilot project can be expanded
as a full-field injection to pressure support the falling
Corresponding Author: T.N. Ofei, Department of Petroleum Engineering, African University of Science and Technology, Abuja,
Nigeria
369
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
reservoir pressure and proved an excellent way to analyse
the impact of water injection on the recipient reservoir.
Rawding et al. (2008) described the philosophy and
design of an intelligent well installation in a water
dumpflood well in West Kuwait. They reported on the
application of intelligent well completion for controlled
dumpflood where water from the Zubair aquifer formation
flows to the Minagish Oolite oil formation. The authors
referred to the article by Quttainah and Al-Hunaif, (2001)
who first tried well dumpflood pilot project in the Umm
Gudair field and showed that dumpflood can be expanded
as a full field water dumpflood injection to pressure
support the falling reservoir pressure. In their conclusion,
the use of intelligent completion technology and remotely
controlled hydraulic Interval Control Valve (ICV) proved
reliable and cost effective solution for a controlled
dumpflood.
Fig. 1: Downward flow mechanism
MATERIALS AND METHODS
Dumpflood flow mechanisms: The Fig. 1 describes the
flow mechanism where the aquifer located above the oil
reservoir flows high pressure water by gravity through a
communicating well into the oil reservoir thus, displacing
the oil ahead of it into the producing well.
The upward flow mechanism in Fig. 2 describes an
aquifer located below the oil reservoir and injects high
pressure water into the oil reservoir. In the case where the
aquifer pressure is low, a downhole pump is used to aid
the injection (Yao et al., 1999). The equations associated
with dumpflooding apply equally to both flow
mechanisms in Fig. 1 and 2.
In the aquifer zone, the pressure differential (Pew Pww) results in the movement of water which flows into
the wellbore at a rate qw. As the water flows down the
communicating well between the two zones by gravity,
there exists a pressure drop (i.e., Pww - Pwf) due to friction
and kinetic energy. The water finally injects into the oil
reservoir at a rate Iw and together with the pressure
differential (Pwf - Peo), initiates the displacement of oil into
the producing well. The objective of the above description
is to deduce an equation for the rate of water Iw injecting
into the oil reservoir. The following assumptions are
considered in the derivation:
C
C
C
C
C
C
Fig. 2: Upward flow mechanism
C
C
C
The tubing has no change in diameter
Frictional pressure loss is constant
Aquifer water is compatible with reservoir water
The complete derivation is presented in Appendix A;
thus, the rate of water injection, Iw into the oil reservoir as
adapted from (Davies, 1972) is given as:
Iw[1/I + 1/J] = (Pwf - Peo) + (Pew - Pww)
(1)
Pressure drop in the wellbore: There exists pressure
drop due to friction and kinetic energy as water flows
through the connecting well (tubing string) from the
aquifer zone to the oil reservoir zone. Since tubing string
diameter is constant, the pressure drop due to kinetic
energy is negligible. The pressure drop due to friction is
a function of flow rate and controls the dumpflood rate to
a degree, therefore, there is a relationship between
frictional pressure drop ()Pfr) and rate of water injection,
Iw as presented in Appendix A and given as:
All pressures are datum corrected to the oil reservoir
datum
Single phase (water) flows in the injection well
(tubing)
The fluid (water) is incompressible
No loss of water in the wellbore
(qw = Iw = Contant)
The injectivity I, and productivity J, indices are
constants
There is no water influx
Iw[1/I + 1/J] = Pew - Peo – ) Pfr
370
(2)
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
Material Balance Equation (MBE) for reservoir
boundary pressures: It is noticed from Eq. (2) that the
RHS has two unknown pressures quantities. Since these
pressures Pew and Peo are boundary pressures in the aquifer
zone and oil reservoir zone respectively, they can be
deduced from material balance equation.
The general MBE for an oil reservoir is given by
(Dake, 1978; Craft and Hawkins, 1991):
) )
(W
inj
)
9500
Dumping rate ,Iw(BWPD)
( (
N p Bo + R p − Rs B g − We B w −
Iw vs I at (t = 28 days)
− W p Bw = NCT ∆p
(
6500
5500
⎤
)⎥⎥
+
mBoi
∆p
⎦
⎛ Bg
⎞
⎜
⎟
⎜ B − 1⎟
⎝ gi ⎠
(
)
⎧ Bo + R p − R s B g ⎫
⎛ B ⎞
⎪
⎪
Peo = Pio + ⎜ w ⎟ Winj − ⎨
⎬N p
NC
⎝ NCT ⎠
T
⎪⎩
⎪⎭
I w (t ) = I iw e
⎛ Bw ⎞
= piw − ⎜
⎟ Winj
⎝ N w CTW ⎠
45
50
55
⎛ E⎞
−⎜ ⎟ t
⎝ D⎠
+
Fq o
E
⎛ E⎞
⎧
−⎜ ⎟ t ⎫
⎪
⎝ D⎠ ⎪
1
−
e
⎨
⎬
⎪⎩
⎪⎭
(9)
where initial rate of injection is in the form:
(5)
I iw
⎧
⎫
⎪ ∆Pi − TI 1.79 ⎪
⎪
⎪
iw
=⎨
⎬
1
1
⎡
⎤
⎪
⎪
+
⎪⎩ ⎢⎣ I J ⎥⎦ ⎪⎭
(10)
These are nonlinear equations which can be solved
iteratively. All variables are defined in Appendix B.
The bisection method, amongst other iterative
techniques, is much preferred in solving these nonlinear
rate equations due to its guarantee for convergence and
less risky, though it is slow to converge (Kaw, 2009). The
initial dumping rate can be written as a continuous
function in the form:
(6)
Applying the MBE to the aquifer zone at time, t, where
there is no oil (Np=0) gives;
Similarly, the boundary pressure in the aquifer zone
presented in Appendix B is given as:
Winj Bw = N w CTw ( Piw − Pew )
25
deduced from both flow mechanism and MBE are
simplified using Ordinary Differential Equations (ODEs).
The final dumpflood rate equation which is fully
developed in Appendix B can be calculated at any time,
t. Thus:
(4)
The boundary pressure in the oil zone presented in
Appendix B is given as:
Pew
35
40
Injective index, I
7500
Fig. 3: Effects of J & I on in oil reservoir with gas cap (t = 28
days)
when an oil reservoir is with an initial gas cap (i.e., the oil
is initially saturated), there is negligible liquid expansion
energy. It is assumed that with an initial gas cap, water
and pore compressibilities (Cw and Cf) and water influx
(We) are negligible (Dake, 1978). The total
compressibility in the oil zone reduces to:
∆p
30
8500
20
⎞ ⎤ ⎡ Boi (1 + m)
⎛ Bg
⎟
⎜
C w S wc + C f
⎜ B − 1⎟ ⎥ + ⎢ 1 − S
⎝ gi ⎠ ⎥⎦ ⎢⎣
wc
( Bo − Boi ) + ( Rsi − Rs ) B g
J = 200
4500
⎡ ( Bo − Boi ) + ( Rsi − Rs ) B g ⎤
CT = ⎢
⎥+
∆p
⎢⎣
⎥⎦
CT =
J = 100
J = 50
(3)
)p = initial pressure-current pressure (Pi - P), and
⎡ mB
⎢ oi
⎢⎣ ∆p
J = 25
f ( I iw ) = AI iw + TI iw 1.79 − ( Piw − Pio )
(11)
(7)
The continuous function for the final rate of water
injection is given by:
(8)
f ( I w ) = I w − I iw e
RESULTS AND DISCUSSION
⎛ E⎞
−⎜ ⎟ t
⎝ D⎠
−
⎛ E⎞
⎧
−⎜ ⎟ t ⎫
Fqo ⎪
⎝ D⎠ ⎪
e
1
−
⎨
⎬
E ⎪
⎪
⎩
⎭
(12)
The above Fig. 3 indicates how the dumping rate
from a finite aquifer varies as a function of constant
Case 1: Finite aquifer injecting into finite oil reservoir
with gas cap: The rate of water injection equations
371
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
Iw vs I at (t = 28 days)
Dumping rate ,Iw(BWPD)
11500
J = 25
J = 100
J = 50
J = 200
shown in Fig. E1 to E4 in Appendix E. The ‘solid’ curves
record rates from infinite aquifer while ‘dash’ curves
record rates from finite aquifer. It can be deduced that for
constant productivity index in cases 1 and 2, infinite
aquifer injects much water compared to finite aquifer as
time increases from 7 to 28 days.
10500
9500
8500
7500
CONCLUSION
6500
C
5500
4500
20
25
35
40
Injective index, I
30
45
50
55
C
Fig. 4: Effects of J & I on Iw in Oil Reservoir with Gas Cap
(t = 28 days)
C
productivity indices J for a time period of 28 days. It is
evident that an increase in dumping rate is proportional to
an increase in constant productivity index.
C
Case 2: Infinite aquifer injecting into finite oil
reservoir with gas cap: The final dumpflood rate
equation which is fully developed in Appendix C can be
calculated at any time, t. Thus;
I w (t ) = I iw e
⎛Y⎞
−⎜ ⎟ t
⎝ X⎠
+
Zq o
Y
⎛Y⎞
⎧
−⎜ ⎟ t ⎫
⎪
⎝ X⎠ ⎪
⎨1 − e
⎬
⎪⎩
⎪⎭
NOMENCLATURE
API symbols are used whenever possible:
Symbol Description
I
Injectivity Index
J
Productivity Index
N
Original Oil in Place
Original Water in Place
Nw
Oil Production, Cumulative
Np
Water Injected, Cumulative
Winj
Oil Producing Rate
qo
Water Producing Rate
qw
from the Aquifer
Iw
Water Producing Rate BWPD
into Oil Reservoir
Initial Water Producing Rate
Iiw
Lower Boundary of the function
Iw(L)
Upper Boundary of the function
Iw(u)
Estimated Root of the function
Iw(m)
n
Number of Iterations
External Boundary Pressure
pe
Pe at initial Conditions
Boundary Pressure in Water Zone
Pew
Pew at initial Conditions
Boundary Pressure in Oil Zone
Peo
Peo at initial Conditions
Pio
Flowing Bottom Hole Pressure
Pwf
)P
Initial Pressure minus
Current pressure
t
Time
Total Compressibility (Oil Zone)
CT
Total Compressibility (Water Zone)
CTW
Current Oil Formation
Bo
Volume Factor
Bo at Initial Condition
Boi
Current Gas Formation
Bg
Volume Factor
Bw at Initial Condition
Bgi
Current Oil Formation
Bw
Volume Factor
Bw at Initial Condition
Bwi
Free Gas
Rsi
Rs at Initial Condition
Rsi
(13)
Where initial rate of injection is in the form;
I iw
⎧
⎫
⎪ ∆P − TI 1.79 ⎪
⎪ i
⎪
iw
=⎨
⎬
1
1
⎡
⎤
⎪
+ ⎥ ⎪
⎢
⎪⎩ ⎣ I J ⎦ ⎪⎭
(14)
The initial dumping rate can be written as a continuous
function in the form:
f ( I iw ) = AI iw + TI iw 1.79 − ( Piw − Pio )
(15)
The continuous function for the final rate of water
injection is given by:
f ( I w ) = I w − I iw e
⎛Y⎞
−⎜ ⎟ t
⎝ X⎠
−
Zq o
Y
⎛Y⎞
⎧
−⎜ ⎟ t ⎫
⎪
⎝ X⎠ ⎪
⎨1 − e
⎬
⎪⎩
⎪⎭
Theoretical dumpflood equations have been
developed to evaluate the rate of water injection into
an oil reservoir with gas cap from an aquifer.
A simple Excel spreadsheet model is built to solve
the rate equations using Bisection iteration technique.
The boundary pressure in the finite aquifer zone
depletes much faster compared to the infinite aquifer
zone.
In both cases, water injection recorded higher rates
for infinite aquifer compared to finite aquifer (Fig. E1
to E4 in Appendix E).
(16)
All variables are defined in Appendix C.
In the above Fig. 4, the dumping rate from an infinite
aquifer also shows an increase in proportion to the
increase in constant productivity indices J. Comparisons
of dumping rates from both finite and infinite aquifer are
372
Units
BWPD/psi
BWPD/psi
MMSTB
MMBW
MMSTB
MMBW
BOPD
BWPD
BWPD
BWPD
BWPD
BWPD
Dimension less
PSIG
PSIG
PSIG
PSIG
PSIG
PSIG
PSIG
PSIG
DAYS
1/PSI
1/PSI
RB/STB
RB/STB
RB/SCF
RB/SCF
RB/STB
RB/STB
SCF/STB
SCF/STB
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
Rp
Swc
)Pfr
D
:
d
dc
dt
h
Cumulative GOR
Connate Water Saturation
Friction Pressure Drop
Density of Dumping Fluid
Viscosity of Dumping Fluid
Diameter or Equivalent*
diameter of Pipe, Internal
Internal Diameter of Casing
Internal Diameter of Tubing
Distance between Mid-point of
Source Zone Producing Interval
to Mid-point of Injected Zone
Producing Interval
SCF/STB
FRACTION
PSI
GM/CC
CP
INCHES
⎛ B ⎞
Peo = Pio + ⎜ w ⎟ Winj
⎝ NCT ⎠
(
INCHES
INCHES
FEET
d
= ( dc − d t )
2.79
× ( dc + dt )
(B2)
Total compressibility in oil zone is reduced to:
CT =
The equivalent* diameter of the annular space between the casing and
tubing is computed using the formula:
4.79
)
⎡ Bo + R p − Rs Bg ⎤
⎥Np
−⎢
NCT
⎢
⎥
⎣
⎦
Appendix A:
Dumpflood flow mechanism: Reference to Fig. 2, the rate of water
injection into oil zone, Iw from Darcy is given by:
Winj Bw = Nw CTw (Piw - Pew)
Pew = Piw - (Bw / Nw CTw) Winj
⎡1 1 ⎤
I w ⎢ + ⎥ + TI w1.79 = ( Piw − Pio )
⎣I J⎦
(A3)
⎡ 1
1 ⎤
− Bw ⎢
+
⎥Winj
⎣ N w CTw NCT ⎦
(
(A4)
(A5)
Letting
(A6)
)Pfr = [518D0.79 :0.207 h / d4.79 × 1000 × 14401.79] qw1.79 = Tqw1.79 (A7)
⎡1 1 ⎤
D=⎢ + ⎥
⎣I J⎦
(
)
⎛ Bo + R p − Rs Bg ⎞
⎟
F =⎜
⎜
⎟
NC
T
⎝
⎠
Since qw = Iw
Rate of water injection from flow mechanism gives:
(A8)
∆Pi = ( Piw − Pio )
Appendix B:
Case 1: Finite aquifer injecting into finite oil reservoir with gas cap:
Assumptions of oil reservoir with gas cap are, Cw = 0 and Cf = 0, thus
MBE in the oil zone at time, t, is deduced as:
Winj Bw - Np (Bo + (Rp -Rs) Bg) = NCT (Peo - Pio)
)
⎛ 1
1 ⎞
E = Bw ⎜
+
⎟
⎝ N w CTw NCT ⎠
For non-Newtonian fluids (Darley and Gray, 1988):
Iw [1/I + 1/J] + TIw1.79 = Pew - Peo
(B6)
⎡ Bo + R p − R s B g ⎤
⎥N p
+⎢
⎥
⎢
NCT
⎦
⎣
For constant tubing cross-section, pressure drop due to kinetic energy
)PKE is negligible. Thus:
Iw [1/I + 1/J] = Pew - Peo - )Pfr
(B5)
Substituting (B2) and (B5) into (A8) and simplifying gives:
(A2)
Pressure drop in the wellbore is given by:
Pww - Pwf = )PKE + )Pfr
(B4)
Boundary pressure in aquifer zone from gives:
Adding equations (A-1) and (A-3) gives:
Iw [1/I + 1/J] = (Pwf - Peo) + (Pew - Pww)
(B3)
(A1)
Productivity index J is a measured constant
Assuming qw = Iw (no loss of fluid in wellbore)
Iw / J = Pew - Pww
⎞
⎛ Bg
⎜⎜
− 1⎟⎟
⎝ Bgi ⎠
Likewise, MBE in aquifer zone at time, t, gives:
Injectivity index I is a measured constant
For aquifer zone, the rate of water flow into the wellbore is given by:
qw = J(Pew - Pww) or Iw / J = Pew - Pww
∆p
mBoi
+
∆p
2
Iw = I(Pwf - Peo) or Iw / I = Pwf - Peo
( Bo − Boi ) + ( Rsi − Rs ) Bg
Equation (B6) becomes:
DIw + TIw1.79 = )Pi - EWinj + FNp
(B1)
(B7)
For constant frictional pressure drop, i.e., TIw1.79 and differentiating Eq.
(B7) with respect to time, t gives:
Boundary pressure in oil zone is given as:
373
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
∂Winj
∂Np
∂I
D w +E
−F
=0
∂t
∂t
∂t
Final dumpflood rate equation gives:
[
(B8)
Fq
− E D t
− E D t
I w ( t ) = I iw e ( ) + o 1 − e ( )
E
But
∂Winj
= Iw
∂t
Thus:
and
∂Np
= qo
∂t
∂I
D w + EI w − Fqo = 0
∂t
⎡ ∆P − TI 1.79 ⎤
I iw = ⎢ i 1 1iw ⎥
+J
⎢⎣
⎥⎦
I
(
(B9)
(B10)
dI w ⎛ E ⎞
Fqo
+ ⎜ ⎟ Iw =
dt ⎝ D ⎠
D
(B11)
Pew = Piw
Iw
Let
1
J
] + TI
1.79
w
X=
[
1
I
+
1
J
= ( Piw − Pio )
(C2)
]
( )
1
NCT
(
)
∆Pi = ( Piw − Pio )
Equation (C2) becomes:
(B14)
XIw + TIw1.79 = )Pi - YWinj + ZNp
Substituting (B14) into (B12) at t = 0
D
+
⎛ Bo + R p − Rs Bg ⎞
⎟
Z=⎜
⎜
⎟
NC
T
⎝
⎠
(From A6)
(B13)
∆Pi − ∆Pfr
D
∆Pi − ∆Pfr
1
I
Y = Bw
Substituting the above conditions into (A6):
K=
[
⎛ B ⎞
⎡ Bo +( R p − Rs ) Bg ⎤
− ⎜ w ⎟ Winj + ⎢
⎥Np
NCT
⎝ NCT ⎠
⎣
⎦
(B12)
Iw = Iiw, Pew = Piw and Peo = Pio
I iw =
(C1)
Substituting (B2) and (C1) into (A8) gives:
where K, is a constant.
At initial condition:
DI iw = ∆Pi − ∆Pfr
(B18)
Appendix C:
Case 2: Infinite aquifer injecting into finite oil reservoir with gas
cap: For an infinite aquifer, Nw = 4, therefore the boundary pressure
(B5) in the aquifer zone reduces to:
Solution to this ordinary differential equation (ODE) (B11) with respect
to time t, is given as:
⎡1 1 ⎤
I iw ⎢ + ⎥ = Piw − Pio − ∆Pfr
⎣I J⎦
)
These nonlinear equations (B17) and (B18) can be solved by an iterative
technique.
dI w
+ EI w = constant
dt
Fq
− E D t
I w ( t ) = Ke ( ) + o
E
(B17)
Where initial rate of injection is in the form:
If oil production rate qo is a specified constant, then:
D
]
−
Fqo
E
By similar approach in Appendix B, final dumpflood rate equation
gives:
[
Zq
−Y X t
−Y X t
I w ( t ) = Iiw e ( ) + o 1 − e ( )
Y
(B15)
Substituting (B15) into (B12) and simplifying gives:
⎡ ∆ Pi − ∆ Pfr ⎤ − ( E D )t
I w (t ) = ⎢
⎥e
D
⎣
⎦
Fq o ⎡
− E D t
1 − e ( ) ⎤⎥
+
⎢
⎦
E ⎣
(C3)
]
(C4)
where initial rate of injection is in the form:
⎡ ∆P − TI 1.79 ⎤
I iw = ⎢ i 1 1iw ⎥
+J
⎢⎣
⎥⎦
I
(
(B16)
)
(C5)
These nonlinear equations (C4) and (C5) can be solved by an iterative
technique.
374
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
[ I w ( u) − I w ( L) ]
Appendix D:
Solving the rate of water equation using bisection iterative
technique: The root of the function is estimated as (Kaw, 2009):
Iw (m) = Iw (L) + Iw (u)
(D5)
The number of iterations, n, of the bisection method needed to determine
the root within an error of at most 5×10G8 is given by:
The absolute error is estimated as (Kaw, 2009):
| Iw (m)new - Iw (m)old |
I w ( u) − I w ( L )
(D2)
Similarly, the relative percentage error is given as (Kaw, 2009):
I w ( m)
new
− I w ( m)
I w ( m)
(D4)
2 n +1
2 n +1
(D5)
where
old
new
≤ 5 × 10−8
× 100
⎡ log⎛⎜ I w ( u ) − I w ( L ) ⎞⎟ + 8 ⎤
5
⎠
⎥
⎢ ⎝
n≥⎢
⎥ −1
log( 2)
⎥
⎢
⎦
⎣
(D3)
After n steps of iterations, the approximate root is computed with an
absolute error of at most (Kaw, 2009):
(D6)
Appendix E:
Case 1 & 2 (t = 7 days)
Case 1.J=25
Case 1.J=50
Case 1.J=50
12500
Case 1 & 2 (t = 21 days)
Case 1.J=100
Case 1.J=200
Case 1.J=100
Case 1.J=200
13500
12500
11500
Dumping rate ,Iw(BWPD)
Dumping rate ,Iw(BWPD)
Case 1.J=25
10500
Case 1.J=25
Case 1.J=50
Case 1.J=50
Case 1.J=100
Case 1.J=200
Case 1.J=100
Case 1.J=200
11500
10500
9500
8500
7500
6500
5500
9500
8500
7500
6500
5500
4500
4500
20
25
30
40
35
Injective index, I
45
50
20
55
Fig. E1: Dumping rate comparison: Oil reservoirs with gas cap for Case
1 & 2 (t = 7 days)
13500
25
30
35
40
Injective index, I
45
50
55
Fig. E3: Dumping rate comparison: Oil reservoirs with gas cap for Case
1 & 2 (t = 21 days)
Case 1 & 2 (t = 28 days)
Case 1 & 2 (t = 14 days)
Case 1.J=25
Case 1.J=25
Case 1.J=25
Case 1.J=25
Case 1.J=50
Case 1.J=50
Case 1.J=50
Case 1.J=50
Case 1.J=100
Case 1.J=200
Case 1.J=100
Case 1.J=200
Case 1.J=100
Case 1.J=200
Case 1.J=100
Case 1.J=200
11500
Dumping rate ,Iw(BWPD)
12500
Dumping rate ,Iw(BWPD)
Case 1.J=25
10500
11500
10500
9500
8500
7500
6500
5500
9500
8500
7500
6500
5500
4500
4500
20
25
30
35
40
Injective index, I
45
50
20
55
Fig. E2: Dumping rate comparison: Oil reservoirs with gas cap for Case
1 & 2 (t = 14 days)
25
30
35
40
Injective index, I
45
50
55
Fig. E4: Dumping rate comparison: Oil reservoirs with gas cap for Case
1 & 2 (t = 28 days)
375
Res. J. Appl. Sci. Eng. Technol., 3(5): 369-376, 2011
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