Inferring interwell connectivity in a reservoir from bottomhole pressure fluctuations of hydraulically fractured vertical wells, horizontal wells, and mixed wellbore conditions
PETROLEUM EXPLORATION & PRODUCTION
PETROVIETNAM JOURNAL
Volume 10/2020, p. 20 - 40
ISSN 2615-9902
INFERRING INTERWELL CONNECTIVITY IN A RESERVOIR FROM
BOTTOMHOLE PRESSURE FLUCTUATIONS OF HYDRAULICALLY
FRACTURED VERTICAL WELLS, HORIZONTAL WELLS, AND MIXED
WELLBORE CONDITIONS
Dinh Viet Anh1, Djebbar Tiab2
1PetroVietnam Exploration Production Corporation
2University of Oklahoma
Email: anhdv@pvep.com.vn; dtiab@ou.edu
Summary
A technique using interwell connectivity is proposed to characterise complex reservoir systems and provide highly detailed
information about permeability trends, channels, and barriers in a reservoir. The technique, which uses constrained multivariate linear
regression analysis and pseudosteady state solutions of pressure distribution in a closed system, requires a system of signal (or active)
wells and response (or observation) wells. Signal wells and response wells can be either producers or injectors. The response well can
also be either flowing or shut in. In this study, for consistency, waterflood systems are used where the signal wells are injectors, and the
response wells are producers. Different borehole conditions, such as hydraulically fractured vertical wells, horizontal wells, and mixed
borehole conditions, are considered in this paper.
Multivariate linear regression analysis was used to determine interwell connectivity coefficients from bottomhole pressure data.
Pseudosteady state solutions for a vertical well, a well with fully penetrating vertical fractures, and a horizontal well in a closed
rectangular reservoir were used to calculate the relative interwell permeability. The results were then used to obtain information on
reservoir anisotropy, high-permeability channels, and transmissibility barriers. The cases of hydraulically fractured wells with different
fracture half-lengths, horizontal wells with different lateral section lengths, and different lateral directions are also considered. Different
synthetic reservoir simulation models are analysed, including homogeneous reservoirs, anisotropic reservoirs, high-permeability-channel
reservoirs, partially sealing barriers, and sealing barriers.
The main conclusions drawn from this study include: (a) The interwell connectivity determination technique using bottomhole
pressure fluctuations can be applied to waterflooded reservoirs that are being depleted by a combination of wells (e.g. hydraulically
fractured vertical wells and horizontal wells); (b) Wellbore conditions at the observations wells do not affect interwell connectivity
results; and (c) The complex pressure distribution caused by a horizontal well or a hydraulically fractured vertical well can be diagnosed
using the pseudosteady state solution and, thus, its connectivity with other wells can be interpreted.
Key words: Interwell connectivity, bottomhole pressure fluctuations, waterflooding, vertical wells, horizontal wells, hydraulically
fractured wells.
1. Introduction
Numerous studies on inferring interwell connectiv-
ity in a waterflood have been carried out. Some of these
studies used statistical techniques that are very different
from the approach used in this study. Albertoni and Lake
developed a technique that calculates the fraction of flow
caused by each of the injectors in a producer [1, 2]. This
method uses a constrained Multivariate Linear Regression
(MLR) model similar to the model proposed by Refunjol
[3]. The model introduced by Albertoni and Lake, however,
considered only the effect of injectors on producers, not
producers on producers. Albertoni and Lake also intro-
Date of receipt: 12/10/2020. Date of review and editing: 12 - 14/10/2020.
Date of approval: 15/10/2020.
This article was presented at SPE Production and Operations Symposium and licensed by SPE
(License ID: 1068761-1) to republish full paper in Petrovietnam Journal
PETROVIETNAM - JOURNAL VOL 10/2020
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duced the concepts and uses of diffusivity filters to
account for the time lag and attenuation that occur
between the stimulus (injection) and the response
(production). The procedures were proven effective
for synthetic reservoir models, as well as real water
flood fields. Yousef et al. introduced a capacitance
model in which a nonlinear signal processing model
was used [4, 5]. Compared to Albertoni and Lake’s
model which was a steady-state (purely resistive)
one, the capacitance model included both capaci-
tance (compressibility) and resistivity (transmissibil-
ity) effects. The model used flow rate data and could
include shut-in periods and bottom hole pressures
(if available).
connectivity coefficients, the case of different injector well
lengths and unchanged producer well lengths was analysed.
Results for different cases such as all wells are horizontal along
the x-direction, along both x- and y-directions and different
horizontal well lengths are provided.
This study also provides the results for different cases where
mixed wellbore conditions are present. 5 injector and 4 pro-
ducer synthetic reservoirs containing hydraulic fractures and
vertical wells, horizontal and vertical wells or all three types of
wellbore conditions are used in the analysis. The results were
then used to obtain information on reservoir anisotropy, high
permeability channels and transmissibility barriers. Different
synthetic reservoir models were analysed including homoge-
neous, anisotropic reservoirs, reservoirs with high permeability
channel, partially sealing barrier and sealing barrier.
Dinh and Tiab [6 - 9] used a similar approach
as Albertoni and Lake [1, 2], however, bottom hole
pressure data were used instead of flow rate data.
Some constraints were applied to the flow rates
such as constant production rate at every producer
and constant total injection rate. Using bottom hole
pressure data offers several advantages: (a) diffusiv-
ity filters are not needed, (b) minimal data is required
and (c) flexible plan to collect data. All of the stud-
ies above only considered fully penetrating vertical
wells. Dinh and Tiab only considered reservoirs with
vertical wells without any hydraulic fractures or hori-
zontal wells [6 - 9].
2. Analytical model and calculation approach
Previous studies have developed a novel technique to
determine interwell connectivity from bottom hole pressure
fluctuation data. This study extends the application of the tech-
nique to hydraulically fractured, horizontal wells and mixed
wellbore conditions. The technique was described in detail by
Dinh and Tiab [6 - 9]. Key equations and definitions of dimen-
sionless variables below are used throughout this study.
2.1. Dimensionless variable
Considering a multi-well system with producers or injectors
and initial pressure pi, the solution for pressure distribution due
to a fully penetrated vertical well in a close rectangular reservoir
is as follows [10, 11]:
In this study, bottomhole pressure fluctua-
tions were used to determine the interwell con-
nectivity in a waterflood where horizontal wells,
hydraulically fractured vertical wells or both are
present. MLR model was used to determine the
interwell connectivity coefficients from bottom-
hole pressure data. For the case of hydraulically
fractured vertical wells, a late time solution for a
well with a fully penetrating vertical fracture in a
closed rectangular reservoir was used to calculate
the influence functions and the relative interwell
permeabilities. The case where the fractures are
of different fracture half-lengths is also consid-
ered. Similarly, for the horizontal well cases, the
late time solution for a horizontal well in a closed
rectangular reservoir was used to calculate the in-
fluence functions and the relative interwell perme-
abilities. The cases in which the reservoir contains
horizontal wells of different lengths and different
directions were also considered. In order to quan-
tify the effect of observation wells on the interwell
nwell
p (x , y , t )= q a
(
xD , yD , xwD,i , ywD,i , xeD , yeD ,
[
tDA − tsDA
]
)
(1)
∑
D
D
D
DA
D ,i i
i=1
Where the dimensionless variables are defined in field units
as follows:
x
A
(2)
xD =
y
A
yD =
(3)
kh
(4)
(5)
pD =
(
pini − p
(
x, y, t))
141.2qref Bµ
kt
tDA = 0.0002637
φctµA
ai is the influence function equivalent to the dimensionless
pressure for the case of a single well in a bounded reservoir pro-
duced at a constant rate. Assuming tsDA = 0, the influence func-
tion is given as:
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PETROLEUM EXPLORATION & PRODUCTION
a closed rectangular reservoir provided by Ozkan was used
[13]. The influence function for hydraulically fractured well
becomes:
ai
(
xD , yD , xwD,i , ywD,i , xeD , yeD ,tDA
)
2
F
2 V
W
∞
∞
(
xD+ xwD,i +2nxeD
)
+
4tDA
(
yD +ywD,i +2my
)
1
eD
=
E
1G
∑ ∑
2m=−∞n=−∞
2
yD2 + ywD
G
H
W
X
C
D
D
E
S
T
T
U
yeD
xeD
yD
yeD
1
3
pDf =2πtDA + 2π
−
+
2ye2D
2
F
2 V
(10)
(
xD − xwD,i + 2nxeD
)
)
)
+
(
yD + ywD,i + 2my
)
)
)
eD
+ E
+ E
+ E
1G
W
∞
C
S
C
S
C
2xeD
π
x
xeD
x S
1
1
wDT
D
DT
4tDA
2
G
H
W
X
(6)
D
T
D
+
2 sin kπ
cos kπ
cos kπ
G
(
x , y , y , y ,k
)
∑
eD
eD wD
D
D
T
D
T
D
E
T
xeD
U
xeD
k=1 k
F
2 V
W
E
U
E
U
(
xD + xwD,i + 2nxeD
+
(
yD − ywD,i + 2my
eD
1G
Where the G-function is:
4tDA
2
G
H
W
X
F
2 V
W
G
(xeD, y , ywD, yD ,k)=
eD
(
xD − xwD,i + 2nxeD
+
(
yD − ywD,i + 2my
eD
1G
Cy − yD −ywD
S
T
T
U
C
S
y −
(
yD + ywD
xeD
)
4tDA
eD
eD
G
H
W
X
D
D
D
T
T
coshk
π
+coshk
π
D
xeD
E
U
E
(11)
Equation 6 is valid for pseudosteady state flow and
can be rewritten as below:
C
S
yeD
xeD
D
T
T
sinh k
π
D
E
U
.2Bµ n
p −p
(
x,y
)
=
wellan
[
xD, yD,xwDn, ywDn,x , y ,
t
eD eD
AD
For the case of infinite conductivity fractures, the dimen-
sionless pressure can be obtained by evaluating the above
equation at xD = 0.732 [14].
]
q
n(7)
∑
ini
kh
i=1
Equation 7 is the pressure response at point (xD, yD)
due to a well n at (xwDn, ywDn) in a homogeneous closed
rectangular reservoir. The influence function (an) can
be different for different wellbore conditions as well as
flow regimes (horizontal well, partial penetrating verti-
cal well, fractured vertical well, etc.).
2.3.2. Horizontal wells
The pressure distribution equation for a horizontal well in
a closed rectangular reservoir is [13]:
pDh = ah = pDf + F1
(12)
Where
2.2. Shape factor calculation
∞
2
1
F1 =
cos
(
nπzD
)
cos
(nπzwD
)
Shape factors are used to calculate pressure at
wells at different locations in a reservoir of a certain
shape. Letting CA denote the shape factor, we have the
well known shape factor equation:
∑
xeD LD n=1 n
C
S
T
T
U
yeD − yD − ywD
xeD
C
D
D
E
S
yeD
−
(
yD + ywD
xeD
)
D
D
E
T
T
U
cosh nπ
+ cosh nπ
×
4A
C
S
T
T
U
yeD
xeD
pwD = 2πtDA + 0.5ln
(8)
D
sinh nπ
γ
e C A L2
D
(13)
E
C
S
T
T
C
D
S
C
D
E
S
T
xeD
U
xwD
x
1
xeD
with L = rw, Lxf and Lh/2 for vertical well, vertically
fractured well and horizontal well respectively and γ is
Euler’s constant (γ = 0.5772…)
D
T
D
DT
sin kπ
cos kπ
cos kπ
∞
∞
D
D
T
xeD
U
1
E
U
E
+4 cos
(nπ
z
)
D cos
(
nπzwD
)
∑
∑
k=1 k
b
n=1
Thus, the shape factor can be calculated using
Equation 9 [12]:
cosh b
(
yeD − yD − ywD
)
+ cosh b
(
yeD
−
(yD + ywD ))
sinh
(
byeD
)
4A
F
V
(9)
CA =Exp
(
4πtDA − 2pwD +Log
)
γ
L2e
G
H
W
X
2
2
2
2
2
2
Where
and the L term in the dimen-
b = n π LD + k π /xeD
sionless definition is the horizontal well half-length L = Lh/2,
and zD = z/h and LD = 1/hD = L/2h. xwD and ywD are at the mid-
point of the well length for the uniform flux horizontal well
case. For the infinite conductivity horizontal well case, Ozkan
showed that the point xD = 0.732 used to calculate pressure
distribution for an infinite conductivity fracture can also be
used for an infinite conductivity horizontal well [13]. The term
F1 can be rewritten as follows:
Where the L term in the definitions of dimension-
less quantities is L = Lxf which is the fracture half-length.
2.3. Influence function
2.3.1. Hydraulically fractured well
For a hydraulically fractured well, for simplicity,
the late time solution for a uniform flux fracture in
PETROVIETNAM - JOURNAL VOL 10/2020
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∞
2
1
n
For the case of yD = ywD, if X ≤ a then
F =
cos
(
nπzD
)
cos
(
nπzwD
)
G
(
xeD , yeD , ywD, yD ,nπ
)
1
∑
xeD LD n=1
b
)
b a−X
)
F(a+X
∫
V
+a
(
∞
∞
1
b
2
F
V
α
C
S
T
T
U
C
D
E
S
x
xeD
1
k
1
xeD
K b
(
X−
α
)
W
X
d
=
K0
(
u
)
du+ K0
(
u
)
du
0 G
∫
D
D
wDT (14)
G
W
∫
+4 cos
(
nπzD
)
cos
(
nπzwD
)
sin kπ
cos kπ
H
∑
∑
D
T
G
0
W
X
−a
0
H
E
U
n=1
k=1
(20)
If X ≥ a
then
C
S
T
T
U
xD
xeD
cos D
kπ
GH
(
yeD , ywD , yD , b
)
D
F(
V
b
X
+a
)
b
(
X −a
)
(
+ a
E
1
b
2
F
V
K b
(
X− α
)
dα =
K0
(
u
)
du− K0
u
)
du
G
W
0 G
W
X
∫
∫
∫
Where:
H
G
H
W
−a
0
0
X
k 2π 2
2
2 2
b = n π LD +
(21)
xe2D
then
If X = a
G
(
1, yeD , ywD , yD ,
b
)
+a
1 2ab
K0
GH
(
yeD , ywD , yD ,
b
)
=
2
F
V
b
K
b
(
X − α
)
dα =
(
u du (22)
)
0 G
W
X
∫
∫
H
b
−a
0
To calculate F1 as suggested by Ozkan [13]:
Where a = 1, b = nπLD
F1 = F + Fb1 + Fb2 + Fb3
(15)
(16)
Where
∞
Table 1 presents the dimensionless coor-
dinates for all the vertically fractured wells in
the 5 × 4 synthetic field (5 injectors: I1, I2, I3,
I4 and I5 and 4 producers: P1, P2, P3 and P4
as shown on Figure 1). All wells have the same
fracture half-length of 145 ft. Other data in-
clude xeD = yeD = 21.38 and rwD = 0.0049. Table
2 shows the shape factors for all the wells in
the 5 × 4 synthetic field calculated using PwD
results (influence functions) from the differ-
ent calculation techniques and Equation 9.
As shown in Table 2, the shape factors are in
good agreement. These shape factors can be
used to calculate the influence functions us-
ing Equation 8.
+1
2
2
F
V
F= cos
(
nπz
)
cos
(
nπzwD
)
K nπLD
(
xD-xwD -α
)
+
(
yD-ywD dα
)
∑
D
0 G
H
W
X
∫
n=1
−1
∞
2
1
(
n
π
zD
)
cos
(
n
π
z
)
Fb1 =
cos
∑
wD
xeD LD n=1 n
−nπLD
(
2yeD − yD − ywD
)
I
)) Y
−nπLD
(
yD + ywD
(
2yeD
−
(
yD + ywD
) + e
+ e−nπL
]
D
[
e
(17)
L
J
L
Z
∞
∞
F
V
yD − ywD
1+ e−2mnπL
+ e−nπL
e−2mnπL
D yeD
D yeD
D
∑
∑
L
L
G
W
m=1
H
X
K
m=1
[
C
S
C
S
C
S
1
x
xeD
xwD
D
T
T
D
D T
D
T
T
sin k
π
cos k
π
cos k
D
D
T
D
∞
F =4 ∞ cos
(
n
π
zD
)
cos
(
nπ
zwD
)
x
xeD
1
E
eD
U
E
U
E
k2π 2
xe2D
U
∑
∑
k=1 k
b2
n=1
n2 2LD +
π
(18)
k2π 2
xe2D
k2π 2
xe2D
k2π 2
xe2D
I
n2
π
2L2D
+
(
yD + ywD
)
−
n2
π
2L2D
+
(
2y
−
(
yD + ywD
n2
π
2L2D
+
(
2yeD − yD − ywD
)
F −
G
V
))+e−
eD
L
W
Table 3 presents the dimensionless coor-
dinates for all the wells in the 5 × 4 homoge-
neous synthetic field. Other data include xeD =
yeD = 20.67 and rwD = 0.004733. Table 4 shows
the shape factors for the horizontal wells in the
5 × 4 synthetic field calculated using PwD results
(influence functions) from Equations 9 and 12.
× e
+e
J
G
W
LG
W
H
X
K
k2π 2
xe2D
k2π 2
xe2D
k2π 2
n2
π
2L2D
+
yeD
W
−
n2
π
2L2D
+
yD − ywD
n2
π +
2L2D
yeD
F
V
Y
L
Z
× 1+ e−2m
+e
e−2m
∞
∞
xe2D
G
∑
∑
G
G
H
W
W
m=1
m=1
X
L
[
∞
∑
(
π
)
(
π
)
Fb3
=
cos n zD cos n zwD
n=1
I+∫1
L−1
Y
L
2
2
F π
V
X
K0
n
LD
(
xD + xwD
−
α
)
+
(
yD − ywD
)
α
+
Table 1. Dimensionless coordinates of the fractured wells in the 5 × 4
G
Wd
H
synthetic field
L
L
L
L
2
2
IK
Y
F π
V
ꢀꢁꢂꢂꢃ
I01
I02
I03
I04
ꢄꢅꢆꢇ
ꢈꢅꢆꢇ
n
LD
(
xD − xwD− 2kxeD−
α
)
+
(
yD − ywD
)
0
L G
H
W L
X
3.7931
17.5862
10.6897
3.7931
17.5862
10.6897
3.7931
17.5862
10.6897
17.5862
17.5862
10.6897
3.7931
3.7931
17.5862
10.6897
10.6897
3.7931
L
L
L
L
L
L
L
L
2
2
F
V
L
L+ K
π
LD
(
xD + xwD − 2kxeD −
α
)
+
(
yD − ywD
)
Gn
W
J
Z
0
+1
∞
(19)
∫ L
H
XLd
L+ ∑
αL
−1J
Z
L
I05
k=1
2
2
L
L
Fnπ
V
L
α
P01
P02
P03
P04
+ K0
LD
(
xD −xwD +2kxeD−
)
+
(
yD − ywD
)
L
L
L
L
G
H
W
X
L
L
L
L
2
2
Fnπ
V
L
L
+ K
L
(
xD + xwD +2kxeD −
α
)
+
(
yD − ywD
)
L
L
D
0
L
K
G
W
L
[
H
X
K
[
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PETROLEUM EXPLORATION & PRODUCTION
Table 2. Shape factors for the fractured wells in the 5 × 4 synthetic field calculated for
different fracture types
CAf
Wells
Uniform Flux
0.1144
Inꢀnite Conductivity
0.2665
I01
I02
0.1140
0.1606
I03
4.1698
7.5580
I04
0.1144
0.2665
I05
0.1140
0.1606
P01
P02
P03
P04
0.9083
0.9026
0.9003
0.9083
1.6560
1.9678
1.3396
1.6560
Table 3. Dimensionless coordinates of the horizontal wells in the 5 × 4 synthetic field
ꢀꢁꢂꢂꢃ
I01
I02
I03
I04
ꢄꢅꢆꢇ
3.6667
17.0000
10.3333
3.6667
ꢈꢅꢆꢇ
17.0000
17.0000
10.3333
3.6667
I05
17.0000
10.3333
3.6667
17.0000
10.3333
3.6667
Figure 1. Top view of the simulation model showing the LGRs at the fractured wells
in the 5 × 4 homogeneous synthetic field.
P01
P02
P03
P04
17.0000
10.3333
10.3333
3.6667
Table 4. Shape factors for uniform flux and infinite conductivity horizontal wells in 5 × 4
synthetic reservoir
CAh
Wells
Uniform Flux
0.0404
Inꢀnite Conductivity
0.0950
I01
I02
0.0403
0.0563
Figure 2. Cross sectional view showing three wells and the hydraulic fractures
in the 5 × 4 homogeneous synthetic reservoir.
I03
1.4741
2.6713
I04
0.0404
0.0950
No refinement in the vertical direction was applied. Thus,
the number of layers in the LGRs stayed at five layers.
I05
0.0403
0.0563
P01
P02
P03
P04
0.3212
0.5857
Figure 3 presents a zoom-in top view of a LGR con-
taining a high permeability strip representing a hydraulic
fracture. Notice that the permeability of the cell at the tips
of the fracture was set to zero following the assumption
that there was no flow through the tips of the fracture.
The permeability of the fractures was set to 8,000 Dar-
cys. The width of the fractures was 0.8 ft, and the fracture
half-lengths were the same at 145 ft. Thus, the dimen-
sionless fracture conductivity for every fracture, which is
the product of fracture permeability and fracture width
divided by the product of formation permeability and
fracture half-length, is equal to 441. Thus, according to
previous studies [16, 17], the fractures can be considered
as infinite conductivity fractures (dimensionless fracture
conductivity is larger than 300). The porosity of the frac-
ture was input as 0.6 which is higher than the porosity of
the formation of 0.3.
0.3190
0.6997
0.3182
0.4699
0.3212
0.5857
3. Simulation results for hydraulically fractured wells
3.1. Model descriptions for hydraulically fractured wells
The grids in the small areas containing the wells were
refined using the Local Grid Refinement (LGR) options.
Thus, there are nine LGRs in this model [15]. Figure 1
shows the top view of the permeability distribution for
this case. The LGRs can be seen at each well. Figure 2 is a
permeability distribution plot showing the cross-sectional
view through three wells. The hydraulic fractures are rep-
resented in red indicating high permeability. The LGR ar-
eas are 300 ft × 20 ft each with a global grid configuration
of 13 × 1 which is refined to a grid configuration of 65 × 25.
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I01
I02
P01
P02
P03
I03
Figure 3. A zoom-in view of a LGR showing a high permeability strip representing
a hydraulic fracture - 5 × 4 homogeneous system.
I01
I02
P01
P04
I05
I04
Figure 4. Representation of the interwell connectivity coefficients for the 5 × 4 homoge-
neous system with hydraulically fractured wells.
I01
I02
P01
P02
P03
I03
P03
P02
I03
P04
I05
I04
Figure 5. Representation of the relative interwell permeability for the 5 × 4 homoge-
neous reservoir with hydraulically fractured wells.
3.2. Homogeneous reservoir with hydraulic fractures
P04
I05
I04
Table 5 and Figure 4 show the results for the interwell
connectivity coefficients. Similar to previous cases, the re-
sults are as good as the results obtained in the case of ho-
mogeneous reservoir with vertical wells only with asym-
metry coefficient of 0.0048. Table 6 and Figure 5 present
the corresponding relative interwell permeabilities with
the equivalent time of 5.66 days, and the reference per-
meability of 100 mD. The difference between the high and
low interwell connectivity coefficients is more significant
than in the case of vertical wells suggesting an observa-
tion well is less affected by a far away active fractured well
than by a vertical unfractured well of the same distance
away. This is reasonable because with the same flow rate,
the pressure drop in a fractured well is less than its unfrac-
tured counterpart.
Figure 6. Representation of the connectivity coefficients for the case of 5 × 4 anisotropic
reservoir - hydraulically fractured wells.
3.3. Anisotropic reservoir with hydraulic fractures
Similar to the anisotropic case in the previous chapter,
the effective permeability in the x direction is tenfold the
fracture permeability in the y direction. Table 7 and Fig-
ure 6 show the results for the interwell connectivity coef-
ficients. As expected, the results are good indications of
the anisotropy with large coefficients for well pairs in the
direction of high permeability. Table 8 and Figure 7 pres-
ent the corresponding relative interwell permeabilities
with the equivalent time of 5.66 days, and the reference
permeability of 316 mD.
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Table 5. Interwell connectivity coefficient results from simulation data for the 5 × 4
Table 6. Relative interwell permeability results for the 5 × 4 homogeneous synthetic
homogeneous synthetic field with hydraulic fractured wells (As = 0.0048)
field with hydraulic fractured wells (kref = 100 mD, Δteq = 5.66 days)
ꢀ1
114
114
92
92
91
ꢀꢁ
112
91
96
111
92
ꢀꢂ
90
111
99
91
113
101
ꢀꢃ
91
91
ꢄꢅꢆrꢇꢈꢆ
102
102
95
102
ꢀ1
-223.6
0.32
0.32
0.24
0.06
0.06
1.00
ꢀꢁ
-226.1
0.31
0.06
0.25
0.31
0.06
1.00
ꢀꢂ
-225.7
0.06
0.31
0.26
0.06
0.31
1.00
ꢀꢃ
-223.6
0.06
0.06
0.25
0.32
0.32
1.00
ꢄuꢅ
-899
0.75
0.75
1.01
0.75
0.75
I1
I2
I3
I4
I5
β
I1
I2
I3
I4
I5
Sum
0j (psia)
93
114
117
101
103
Average
101
101
Table 8. Relative interwell permeability results for the 5 × 4 anisotropic synthetic field -
Table 7. Interwell connectivity coefficient results from simulation data for the 5 × 4
hydraulically fractured wells (kref = 316 mD, Δteq = 5.66 days)
anisotropic synthetic field - hydraulically fractured wells
ꢀ1
353
351
90
80
77
ꢀꢁ
75
152
444
75
ꢀꢂ
152
76
444
151
77
ꢀꢃ
78
80
ꢄꢅꢆrꢇꢈꢆ
164
164
267
164
ꢀ1
-69.6
0.43
0.43
0.11
0.02
0.02
1.00
ꢀꢁ
ꢀꢂ
ꢀꢃ
-69.6
0.02
0.02
0.11
0.42
0.43
1.00
ꢄuꢅ
-332
0.67
0.67
1.32
0.67
0.67
I1
I2
I3
I4
I5
β0j (psia)
-96.5
0.13
0.10
0.55
0.13
0.10
1.00
-96.5
0.10
0.13
0.55
0.10
0.13
1.00
90
I1
I2
350
357
191
153
180
166
I3
Average
190
180
I4
I5
Sum
I01
I02
P01
P02
P03
I03
I04
I05
P04
Figure 7. Representation of relative interwell permeability for the case of 5 × 4 synthetic
reservoir - hydraulically fractured wells.
Figure 8. Top view of the simulation model showing the permeability in x direction for
the high permeability channel case of the 5 × 4 synthetic field with fractured wells.
3.4. Reservoir with a high permeability channel
Table 9 and Figure 9 show the results for the interwell
connectivity coefficients. Similar to previous cases of high
permeability channels, the results reflect well the pres-
ence of the channel. Different from the previous cases,
well I03 has much higher connectivity with producers P02
and P04. The reason for this is that in the previous cases,
well I03 was not connected to the high permeability chan-
nel while in this case, due to the extension provided by
the hydraulic fracture, it is directly connected to the chan-
nel and has better connectivity with the producers.
Figure 8 shows the top view of the permeability dis-
tribution for this case. The cells in yellow color have high
permeability in both x and y direction. Similar to the high
permeability channel cases in the previous chapters, the
permeability of the channel was ten-fold (1,000 mD) of
that in the other areas of the reservoir (100 mD). There
are nine vertically fractured wells with the same fracture
half-length of 145 ft.
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Table 9. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic reservoir with a high permeability channel - hydraulically fractured wells
ꢀ1
-153.5
0.46
0.23
0.26
0.02
0.03
1.00
ꢀꢁ
-54.1
0.42
0.02
0.45
0.07
0.04
1.00
ꢀꢂ
-194.2
0.10
0.28
0.33
0.03
0.25
1.00
ꢀꢃ
-65.4
0.16
0.02
0.53
0.13
0.16
1.00
ꢄuꢅ
β0j (psia)
-467
1.14
0.55
1.57
0.25
0.48
I1
I2
I3
I4
I5
Sum
Table 10. Relative interwell permeability results for the 5 × 4 synthetic reservoir with high permeability channel - hydraulically fractured wells. (kref = 300 mD, Δteq = 5.66 days)
ꢀ1
369
162
202
79
ꢀꢁ
337
77
ꢀꢂ
153
210
256
92
ꢀꢃ
200
84
ꢄꢅꢆrꢇꢈꢆ
265
I1
I2
133
I3
I4
347
24
412
69
304
66
I5
90
94
184
179
104
174
118
Average
180
176
P01
I01
I02
I01
P01
I02
P03
P02
P02
P03
I03
I03
P04
P04
I05
I05
I04
I04
Figure 10. Representation of relative interwell permeability for the 5 × 4 synthetic
reservoir with a high permeability channel - hydraulically fractured wells.
Figure 9. Representation of the connectivity coefficients for the case of 5 × 4 synthetic
reservoir with a high permeability channel - hydraulically fractured wells.
Table 10 and Figure 10 present the corresponding
relative interwell permeabilities with the equivalent time
of 5.66 days, and the reference permeability of 300 mD.
Table 11 and Figure 12 show the results for the inter-
well connectivity coefficients. The presence of the partial-
ly sealing barrier is well established by the results. Table
12 and Figure 13 present the corresponding relative inter-
well permeabilities with the equivalent time of 5.66 days,
and the reference permeability of 100 mD. The relative
interwell permeability for well pair I01-P01 was negative
because the influence function for the pair was calculated
using the late time solution. When the interwell connec-
tivity coefficients are small, they are translated to early
3.5. Reservoir with a partially sealing barrier
Figure 11 shows the top view of the x-direction per-
meability distribution for this case. The permeability for
the cells in grey color were set to zero and thus, those cells
served as a partially sealing barrier. The formation perme-
ability was 100 mD.
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I01
I02
P01
P02
P03
I03
P04
I05
I04
Figure 11. Top view of the simulation model showing the permeability distribution in x
direction for the case of 5 × 4 synthetic field with a partially sealing barrier - hydraulically
fractured wells.
Figure 12. Representation of the connectivity coefficients for the case of 5 × 4 dual-
porosity reservoir with a partially sealing barrier - hydraulically fractured wells.
Table 11. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic field with partially sealing barrier - hydraulically fractured wells
ꢀ1
-440.1
0.01
0.79
0.06
0.04
0.11
1.00
ꢀꢁ
-204.0
0.34
0.02
0.25
0.32
0.07
1.00
ꢀꢂ
-306.9
0.01
0.49
0.08
0.05
0.37
1.00
ꢀꢃ
-226.1
0.06
0.06
0.22
0.33
0.33
1.00
ꢄuꢅ
-1177
0.42
1.36
0.61
0.73
0.87
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 12. Relative interwell permeability results for the 5 × 4 synthetic field with partially sealing barrier - hydraulically fractured wells (kref = 100 mD, Δteq = 5.66 days)
ꢀ1
-40
347
23
80
115
105
ꢀꢁ
127
71
95
114
95
ꢀꢂ
68
199
29
88
141
105
ꢀꢃ
90
92
ꢄꢅꢆrꢇꢈꢆ
62
177
58
100
I1
I2
I3
I4
I5
83
119
125
102
119
Average
101
time periods and thus the late time solution becomes in-
accurate. Solutions that are good for both early time and
late time should be used for better results.
two compartments. Based on the change in average res-
ervoir pressure calculated from each producer, this com-
partmentalisation can be inferred.
Table 13 and Figure 15 show the results for the inter-
well connectivity coefficients. Similar to previous cases,
the results clearly reflect the presence of the sealing bar-
rier. Some connectivity coefficients are very small and
even negative. They indicate poor connectivity or no con-
nectivity at all. Small connectivities were still observed for
some pairs of wells on different sides of the sealing barrier.
3.6. Reservoir with a sealing barrier
Figure 14 shows the top view of the x-direction per-
meability distribution with a sealing barrier case. The
permeability of the cells in grey color was set to zero and
thus, those cells served as a sealing barrier. As seen in the
figure, the barrier completely divides the reservoir into
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I01
I02
P01
P03
P02
I03
I05
I04
P04
Figure 13. Representation of relative interwell permeability for the case of 5 × 4 dual-
porosity reservoir with a partially sealing barrier - hydraulically fractured wells.
Figure 14. Top view of the simulation model showing the permeability in x direction for
the case of 5 × 4 synthetic field with a sealing barrier - hydraulically fractured wells.
Table 13. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic field with a sealing barrier - hydraulically fractured wells
ꢀ1
-336.6
0.00
0.87
0.05
-0.02
0.07
0.97
ꢀꢁ
-266.0
0.35
-0.01
0.27
0.36
0.04
1.01
ꢀꢂ
-225.4
0.00
0.60
0.05
-0.02
0.35
0.97
ꢀꢃ
-365.7
0.10
-0.01
0.35
0.53
0.05
1.02
ꢄuꢅ
-1194
0.45
1.44
0.73
0.84
0.51
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 14. Relative interwell permeability results for the 5 × 4 synthetic field with a sealing barrier - hydraulically fractured wells (kref = 100 mD, Δteq = 5.66 days)
ꢀ1
0.00
385.6
0.00
0.00
98.2
97
ꢀꢁ
131.5
0.00
101.7
137.4
0.00
74
ꢀꢂ
0.00
253.1
0.00
0.00
132.6
77
ꢀꢃ
112.5
0.00
132.6
216.6
0.00
92
ꢄꢅꢆrꢇꢈꢆ
61.01
159.7
58.6
88.5
57.7
I1
I2
I3
I4
I5
Average
As explained before, these non-zero connectivity coeffi-
cients are due to the noises in the data as the injection
rates were generated randomly. This problem can be re-
solved by increasing the number of data points. For this
case, the interwell connectivity coefficients should be
analysed with the average reservoir pressure change re-
sults. If the pressure changes indicate reservoir compart-
mentalisation, then the small interwell connectivity coef-
ficients can be evaluated to decide whether the injectors
and producers are on different side of the barrier.
Table 14 and Figure 16 present the corresponding
relative interwell permeabilities with the equivalent time
of 5.66 days, and the reference permeability of 100 mD.
A cut-off coefficient of 0.06 was applied to eliminate the
low connectivity coefficients. Thus, the relative interwell
permeability corresponding to the coefficients lower than
0.06 were set to zeros. The resulting relative interwell per-
meabilities show a clear presence of the sealing barrier.
Table 15 shows the results for the average reservoir
pressure change for all producers in each case described
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I01
I02
I01
I02
P01
P02
P03
P02
P03
I03
I03
P04
I05
I04
I04
P04
Figure 16. Representation of relative interwell permeability for the 5 × 4 synthetic field
with a sealing barrier - hydraulically fractured wells.
Figure 15. Representation of the connectivity coefficients for the 5 × 4 synthetic field
with a sealing barrier - hydraulically fractured wells.
Table 15. Average pressure change (ΔPave) after each time interval for different cases of 5 × 4 synthetic field - hydraulically fractured wells
ꢀꢁꢂꢃꢂ ꢄ1 ꢄꢅ ꢄꢆ
285.93 285.93 285.74
ꢄꢇ
Homogeneous eservoir
Anisotropic reservoir
Channel
285.74
285.77
285.82
298.84
390.18
285.83
285.82
295.33
180.93
285.82
285.82
300.01
390.14
285.82
285.81
296.38
180.77
Partially sealing barrier
Sealing barrier
above. Similar to the results obtained from the previous
systems, except for the case of sealing barrier, the changes
in average reservoir pressure for all the cases are consis-
tent and close to the pressure changes obtained from the
simulation results. For the case with the presence of seal-
ing barrier, the calculated pressure changes for wells P01
and P03 (about 181 psi) are different from those for wells
P02 and P04 (about 390 psi) indicating two different pore
volumes and thus, two different reservoir compartments.
4. Simulation results for horizontal wells
4.1. Model description for horizontal wells
Figure 17 shows the top view of the permeability
distribution of the 5 × 4 homogeneous synthetic field
with horizontal wells. All the wells were horizontal wells
with their centres at the cell where the vertical wells were
completed as described in the previous section (Table
3). Figure 18 shows the permeability distribution cross
section cutting through three representative horizontal
wells. Thus, all the wells were completed in the centre
Figure 17. Top view of the simulation model showing the horizontal wells of the 5 × 4
homogeneous synthetic field.
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layer of the reservoir so that their distances to the top and
bottom boundaries of the reservoir were equal. The for-
mation permeability was set to 100 mD in the x, y and z
directions. All wells are at the same length of 300 ft and
completed along the x-direction. The wells were assumed
to be infinite conductivity horizontal wells. Thus, the influ-
ence functions were calculated using the pressure distri-
bution equation (Equation 12) evaluated at the point xD =
0.732 and yD = ywD.
I01
I02
P01
P02
P03
I03
4.2. Homogeneous reservoir
Table 16 and Figure 19 show the results for the inter-
well connectivity coefficients obtained from the simula-
tion data for this case. Similar to the same cases in the
previous section, the results are very close to the results
obtained for the homogeneous reservoir with vertical
wells. Small value of the asymmetry coefficient for this
case (As = 0.00445) indicates good results for the interwell
connectivity coefficients. Table 17 and Figure 20 present
the corresponding relative interwell permeabilities with
the equivalent time of 6.59 days, and the reference per-
P04
I05
I04
Figure 19. Representation of the connectivity coefficients for the case of 5 × 4 homoge-
neous reservoir with horizontal wells.
I01
I02
P01
P02
P03
I03
P04
I05
I04
Figure 18. Cross sectional view showing three horizontal wells and their completions in
the 5 × 4 homogeneous synthetic reservoir.
Figure 20. Representation of the relative interwell permeability for the case of 5 × 4
homogeneous reservoir with horizontal wells.
Table 16. Interwell connectivity coefficient results from simulation data for the 5 × 4 homogeneous synthetic field with horizontal wells (A = 0.00445)
ꢀ1
-291.9
0.29
0.29
0.24
0.09
0.09
1.00
ꢀꢁ
-293.7
0.30
0.08
0.24
0.29
0.09
1.00
ꢀꢂ
-294.0
0.08
0.30
0.25
0.09
0.29
1.00
ꢀꢃ
-292.1
0.09
0.09
0.23
0.29
0.30
1.00
ꢄuꢅ
-1172
0.76
0.76
0.96
0.76
0.76
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 17. Relative interwell permeability results for the 5 × 4 homogeneous synthetic field with horizontal wells (kref = 100 mD, Δteq = 6.59 days)
ꢀ1
108
107
93
98
96
ꢀꢁ
112
94
93
107
97
ꢀꢂ
92
109
98
96
106
100
ꢀꢃ
97
97
ꢄꢅꢆrꢇꢈꢆ
102
102
94
102
I1
I2
I3
I4
I5
93
106
109
100
102
Average
100
101
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Table 18. Interwell connectivity coefficient results from simulation data for the 5 × 4 anisotropic synthetic field - horizontal wells
ꢀ1
-131.3
0.38
0.38
0.14
0.05
0.04
1.00
ꢀꢁ
-165.7
0.15
0.13
0.43
0.15
0.13
1.00
ꢀꢂ
-165.7
0.13
0.15
0.43
0.13
0.15
1.00
ꢀꢃ
-131.5
0.05
0.05
0.14
0.38
0.39
1.00
ꢄuꢅ
-594
0.71
0.72
1.14
0.71
0.72
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 19. Relative interwell permeability results for the 5 × 4 anisotropic synthetic field - horizontal wells (kref = 316 mD, Δteq = 6.59 days)
ꢀ1
319
317
117
100
91
ꢀꢁ
ꢀꢂ
ꢀꢃ
95
96
117
314
321
189
ꢄꢅꢆrꢇꢈꢆ
173
174
236
173
I1
I2
I3
I4
I5
104
177
354
103
177
183
175
105
355
174
105
183
174
Average
189
I01
I02
P01
I01
I02
P01
P02
P03
P02
P03
I03
I03
P04
I05
I04
P04
I05
I04
Figure 21. Representation of the interwell connectivity coefficients for the case of 5 × 4
Figure 22. Representation of relative interwell permeability for the case of 5 × 4
anisotropic reservoir - horizontal wells.
synthetic reservoir - horizontal wells.
meability of 100 mD. Notice that the differences between
the high and low interwell connectivity coefficients are
less significant than in the case of vertically fractured wells
of similar half-length suggesting the observation wells are
less affected by the nearby active horizontal wells than as
in the vertically fractured well case. This is reasonable be-
cause for the same flow rate, the pressure drop in a frac-
tured well is less than in a horizontal well considering the
fracture half-length is approximately equal to the horizon-
tal well half-length.
tion (100 mD). Similar to the homogeneous base case, all
wells have the same horizontal half-lengths. Table 18 and
Figure 21 show the results for the interwell connectivity
coefficients. As expected, the results are good indications
of the reservoir anisotropy with large coefficients for well
pairs in the direction of high permeability. Table 19 and
Figure 22 present the corresponding relative interwell
permeabilities with the equivalent time of 6.59 days, and
the reference permeability of 316 mD.
4.4. Reservoir with high permeability channel
4.3. Anisotropic reservoir with horizontal wells
Figure 23 shows the top view of the permeability dis-
tribution for this case. The cells in red color indicate high
permeability in both x and y directions. Similar to the high
In this case, the effective permeability in the x-direc-
tion (1,000 mD) is tenfold the permeability in the y-direc-
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I01
I02
P01
P02
P03
I03
P04
I05
I04
Figure 23. Top view of the simulation model showing the permeability in x-direction for
the high permeability channel case of the 5 × 4 synthetic field - horizontal wells.
Figure 24. Representation of the connectivity coefficients for the high permeability
channel case of the 5 × 4 synthetic field - horizontal wells.
Table 20. Interwell connectivity coefficient results from simulation data for the high permeability channel case of the 5 × 4 synthetic field - horizontal wells
ꢀ1
-197.5
0.46
0.20
0.26
0.03
0.04
1.00
ꢀꢁ
-73.2
0.45
0.03
0.41
0.07
0.04
1.00
ꢀꢂ
-241.7
0.14
0.25
0.34
0.05
0.21
1.00
ꢀꢃ
-83.5
0.22
0.04
0.48
0.12
0.15
1.00
ꢄuꢅ
-596
1.27
0.51
1.50
0.27
0.45
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 21. Relative interwell permeability results for the high permeability channel case of the 5 × 4 synthetic field - horizontal wells (kref = 300 mD, Δteq = 6.59 days)
ꢀ1
374
142
209
83
ꢀꢁ
368
76
321
29
ꢀꢂ
179
188
271
97
ꢀꢃ
245
86
384
66
ꢄꢅꢆrꢇꢈꢆ
292
123
296
69
I1
I2
I3
I4
I5
91
180
92
177
155
178
95
175
108
Average
permeability channel cases in the previous chapters, the
permeability of the channel was ten-fold (1,000 mD) of
that in the other areas of the reservoir (100 mD). There are
nine horizontal wells with the same horizontal well half-
length of 150 ft.
4.5. Reservoir with a partially sealing barrier
Figure 26 shows the top view of the x-direction per-
meability distribution for this case. The cells in white color
were inactive and thus, served as a partially sealing bar-
rier. The formation permeability was 100 mD. Table 22 and
Figure 26 show the results for the interwell connectivity
coefficients. The presence of the partially sealing barrier
is well established based on the results. Table 23 and Fig-
ure 28 present the corresponding relative interwell per-
meabilities with the equivalent time of 6.59 days, and the
reference permeability of 100 mD. Similar to the same
case for fractured wells, the relative interwell permeability
Table 20 and Figure 24 show the results for the inter-
well connectivity coefficients. Similar to the fractured well
case of a reservoir with high permeability channel, the re-
sults reflect accurately the presence of the channel. Table
21 and Figure 25 present the corresponding relative inter-
well permeabilities with the equivalent time of 6.59 days,
and the reference permeability of 300 mD.
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I01
I02
P01
P02
P03
I03
P04
I05
I04
Figure 25. Representation of relative interwell permeability for the high permeability
channel case of the 5 × 4 synthetic field - horizontal wells.
Figure 26. Top view of the simulation model showing the permeability distribution in x
direction for the 5 × 4 synthetic field with partially sealing barrier - horizontal wells.
Table 22. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic field with partially sealing barrier - horizontal wells
ꢀ1
-540.6
0.01
0.73
0.07
0.05
0.13
1.00
ꢀꢁ
-260.1
0.34
0.03
0.24
0.30
0.09
1.00
ꢀꢂ
-391.4
0.02
0.47
0.10
0.07
0.34
1.00
ꢀꢃ
-291.3
0.09
0.09
0.22
0.30
0.31
1.00
ꢄuꢅ
-1483
0.46
1.31
0.63
0.73
0.87
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 23. Relative interwell permeability results for the 5 × 4 synthetic field with partially sealing barrier - horizontal wells (kref = 100 mD, Δteq = 6.59 days)
ꢀ1
-32
321
30
80
119
104
ꢀꢁ
130
67
93
114
98
ꢀꢂ
64
195
38
89
130
103
ꢀꢃ
97
96
ꢄꢅꢆrꢇꢈꢆ
65
170
62
98
116
I1
I2
I3
I4
I5
85
111
115
101
Average
101
for well pair I01-P01 was negative because the influence
function for the pair was calculated using the late time
solution. When the interwell connectivity coefficients
are small, they are translated to early time-periods and,
thus, the late time solution becomes inaccurate. Thus, the
negative value was set to zero due to small connectivity
coefficient.
ing barrier. As seen on the figure, the barrier completely
divides the reservoir into two compartments. Based on
the change in average reservoir pressure calculated from
each producer, the compartmentalisation can be inferred.
Table 24 and Figure 30 show the results for the in-
terwell connectivity coefficients. Similar to the previous
cases, the results clearly reflect the presence of the seal-
ing barrier. Some connectivity coefficients are very small
and even negative. They indicate poor connectivity or no
connectivity at all.
4.6. Reservoir with a sealing barrier
Figure 29 shows the top view of the x-direction per-
meability distribution for the sealing barrier case. The cells
in white colour were inactive and thus, served as a seal-
Table 25 and Figure 31 present the corresponding
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I01
I02
I01
P01
I02
P01
P02
P03
P03
P02
I03
I03
P04
I05
I04
P04
I05
I04
Figure 28. Representation of relative interwell permeability for the case of 5 × 4 dual-
Figure 27. Representation of the connectivity coefficients for the case of 5 × 4 dual-
porosity reservoir with a partially sealing barrier - horizontal wells.
porosity reservoir with a partially sealing barrier - horizontal wells.
I01
I02
P01
P02
P03
I03
P04
I05
I04
Figure 29. Top view of the simulation model showing the permeability in x direction for
the case of 5 × 4 synthetic field with a sealing barrier - horizontal wells.
Figure 30. Representation of the connectivity coefficients for the 5 × 4 synthetic field
with a sealing barrier - horizontal wells.
relative interwell permeabilities with the equivalent time
of 6.59 days, and the reference permeability of 100 mD.
A cut-off coefficient of 0.06 was applied to eliminate the
low connectivity coefficients. Thus, the relative interwell
permeability corresponding to the coefficients lower than
0.06 were set to zeros. The resulting relative interwell per-
meabilities show a clear presence of the sealing barrier
(Figure 31).
Table 26 shows the results for the average reservoir
pressure change for all producers in each representative
case described in this section. Similar to the previous sec-
tion, the changes in average reservoir pressure for all the
cases are about the same and close to the simulated pres-
sure changes. For the case with the presence of a sealing
barrier, the resulting pressure changes for wells P01 and
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Table 24. Interwell connectivity coefficient results from simulation data for the 5 × 4 synthetic field with a sealing barrier - horizontal wells
ꢀ1
-336.6
0.00
0.87
0.05
-0.02
0.07
0.97
ꢀꢁ
-266.0
0.35
-0.01
0.27
0.36
0.04
1.01
ꢀꢂ
-225.4
0.00
0.60
0.05
-0.02
0.35
0.97
ꢀꢃ
-365.7
0.10
-0.01
0.35
0.53
0.05
1.02
ꢄuꢅ
-1194
0.45
1.44
0.73
0.84
0.51
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 25. Relative interwell permeability results for the 5 × 4 synthetic field with a sealing barrier - horizontal wells (kref = 100 mD, Δteq = 6.59 days)
ꢀ1
0
391
0
0
89
96
ꢀꢁ
137
0
106
143
0
ꢀꢂ
0
259
0
0
135
79
ꢀꢃ
104
0
138
222
0
ꢄꢅꢆrꢇꢈꢆ
60
163
61
I1
I2
I3
I4
I5
91
56
Average
77
93
I01
I02
P01
P02
P03
I03
P04
I05
I04
Figure 31. Representation of relative interwell permeability for the 5 × 4 synthetic field
with a sealing barrier - horizontal wells.
Figure 32. Top view of the simulation model showing the x-direction permeability for
the 5 × 4 homogeneous synthetic field - mixed hydraulically fractured and vertical wells.
Table 26. Average pressure change (ΔPave) after each time interval for different cases of 5 × 4 synthetic field - horizontal wells
ꢀꢁꢂꢃꢂ ꢄ1 ꢄꢅ ꢄꢆ
285.98 286.04 285.79
ꢄꢇ
Homogeneous reservoir
Anisotropic reservoir
Channel
285.84
285.79
285.91
298.96
390.18
285.93
285.90
294.99
180.93
285.92
285.94
300.45
390.14
285.92
285.82
296.10
180.77
Partially sealing barrier
Sealing barrier
P03 (about 181 psi) are different from those for wells P02
and P04 (390 psi) indicating two different reservoir com-
partments. Thus, the reservoir pressure change results are
consistent.
5. Results for mixed wellbore conditions
5.1. Mixed case of fully penetrating vertical wells and
fully penetrating hydraulic fractures
Figure 32 shows the top view of the permeability dis-
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I01
I02
I01
I02
P01
P01
P02
P03
P03
P02
I03
I03
P04
I05
I04
P04
I05
I04
Figure 33. Representation of the connectivity coefficients for the 5 × 4 homogeneous
Figure 34. Representation of relative interwell permeability for the 5 × 4 homogeneous
synthetic field - mixed hydraulically fractured and vertical wells.
synthetic field - mixed hydraulically fractured and vertical wells.
Table 27. Interwell connectivity coefficient results from simulation data for the 5 × 4 homogeneous synthetic field - mixed hydraulically fractured and vertical wells
ꢀ1
-281.1
0.37
0.16
0.32
0.04
0.11
1.00
ꢀꢁ
-502.1
0.36
0.04
0.33
0.16
0.11
1.00
ꢀꢂ
-282.1
0.10
0.16
0.34
0.04
0.35
1.00
ꢀꢃ
-501.9
0.11
0.04
0.33
0.16
0.36
1.00
ꢄuꢅ
-1567
0.94
0.41
1.32
0.41
0.93
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 28. Relative interwell permeability results for the 5 × 4 homogeneous synthetic field - mixed hydraulically fractured and vertical wells (kref = 100 mD, Δteq = 7.33 days)
ꢀ1
123
81
ꢀꢁ
122
74
ꢀꢂ
92
80
ꢀꢃ
93
74
ꢄꢅꢆrꢇꢈꢆ
108
I1
I2
77
I3
I4
109
75
110
81
114
74
110
80
111
78
I5
93
96
94
96
121
96
125
97
108
Average
tribution for this case. As shown on the figure, wells I01,
P01, I03, P03 and I05 are hydraulically fractured wells and
all the other wells are fully penetrating vertical wells. Table
27 and Figure 33 present the interwell connectivity coef-
ficient results for this case. It is obvious that hydraulically
fractured injectors have better connectivity with the pro-
ducers than the vertical injectors.
is in good agreement with the input permeability for the
model of 100 mD.
Figure 35 shows the comparison of the interwell con-
nectivity coefficients results obtained from simulation
data and calculations using influence functions. The coef-
ficients are in good agreement with R2 = 0.9875.
5.2. Mixed case of fully penetrating vertical wells and
horizontal wells
Table 28 and Figure 34 show the corresponding rela-
tive interwell permeability results for this reservoir. The
relative permeabilities for the well pairs of vertical injec-
tors are slightly lower than those of hydraulic fractures.
However, the calculated relative interwell permeability
Figure 36 shows the top view of the permeability dis-
tribution for this case. As shown on the figure, wells I01,
P01, I03, P03 and I05 are horizontal wells and all the other
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PETROLEUM EXPLORATION & PRODUCTION
wells are vertical wells. Figure 37 shows the cross section
through wells I04, P04 and I05. Wells I04 and P04 are fully
penetrating vertical wells and similar to other horizontal
wells, and horizontal well I05 is completed in the middle
layer.
0.45
R2 = 0.9875
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Figure 36. Top view of the simulation model showing the x direction permeability for the
5 × 4 homogeneous synthetic field - mixed horizontal and vertical wells .
0
2
4
6
8
10
12
14
16
18
20
We ll Pairs
I01
I02
P01
Simulated
Calculated
Figure 35. Comparison of the interwell connectivity coefficient results for the 5 × 4
homogeneous synthetic field - mixed hydraulically fractured and vertical wells.
P02
P03
I03
P04
I05
I04
Figure 37. Cross sectional view showing three wells of the 5 × 4 homogeneous synthetic
Figure 38. Representation of the connectivity coefficients for the 5 × 4 homogeneous
field - mixed horizontal and vertical wells.
synthetic field - mixed horizontal and vertical wells.
Table 29. Interwell connectivity coefficient results from simulation data for the 5 × 4 homogeneous synthetic field - mixed horizontal and vertical wells.
ꢀ1
-349.3
0.34
0.17
0.30
0.06
0.13
1.00
ꢀꢁ
-540.1
0.35
0.06
0.29
0.17
0.13
1.00
ꢀꢂ
-350.7
0.12
0.17
0.31
0.06
0.33
1.00
ꢀꢃ
-540.5
0.14
0.06
0.30
0.17
0.34
1.00
ꢄuꢅ
-1781
0.94
0.47
1.20
0.46
0.93
β0j (psia)
I1
I2
I3
I4
I5
Sum
Table 30. Relative interwell permeability results for the 5 × 4 homogeneous synthetic field - mixed horizontal and vertical wells (kref = 100 mD, Δteq = 7.33 days)
ꢀ1
117
80
ꢀꢁ
122
80
ꢀꢂ
96
82
ꢀꢃ
102
83
ꢄꢅꢆrꢇꢈꢆ
109
I1
I2
81
I3
I4
107
83
104
79
111
82
106
78
107
81
I5
100
97
100
97
116
97
118
97
108
Average
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I01
I02
P01
0.45
0.4
R2 = 0.9681
0.35
0.3
0.25
0.2
P03
0.15
0.1
P02
I03
0.05
0
0
2
4
6
8
10
We ll Pairs
12
14 16
18
20
Simulated
Calculated
Figure 40. Comparison of the simulated and calculated interwell connectivity coefficient
results for the 5 × 4 homogeneous synthetic field - mixed horizontal and vertical wells.
P04
I05
I04
Figure 39. Representation of relative interwell permeability for the 5 × 4 homogeneous
synthetic field - mixed horizontal and vertical wells.
- The complication of pressure distribution caused
by a horizontal well can be captured using the analytical
model and thus its connectivities with other wells can be
interpreted and quantified.
Table 29 and Figure 38 present the interwell con-
nectivity coefficient results for this case. It is obvious that
horizontal injectors have better connectivity with the
producers than the vertical injector. Table 30 and Figure
39 show the corresponding relative interwell permeabil-
ity results for this reservoir. The relative permeabilities for
the pairs of vertical injectors are slightly lower than those
of horizontal injectors. This could be due to numerical er-
rors and analytical assumptions. However, the calculated
relative interwell permeability is in good agreement with
the input permeability for the model of 100 mD. Figure 40
shows the comparison of the interwell connectivity coef-
ficients results obtained from simulation data and by cal-
culation using influence functions. The coefficient results
are in good agreement with R2 = 0.9681.
- The results obtained from the mixed wellbore
condition cases showed that connectivities between wells
with different and complicated wellbore conditions in a
reservoir can be inferred using the bottomhole pressure
fluctuation technique knowing the shape factors of the
wells.
References
[1] Alejandro Albertoni and Larry W. Lake, “Inferring
interwell connectivity only from well-rate fluctuations
in waterfloods”, SPE Reservoir Evaluation and Engineering
Journal, Vol. 6, No. 1, pp. 6 - 16, 2003. DOI: 10.2118/83381-
PA.
6. Conclusions and recommendations
[2] AlejandroAlbertoni,“Inferringinterwellconnectivity
from well-rate fluctuations in waterfloods”, The University
of Texas at Austin, 2002.
- The
interwell
connectivity
determination
technique can be applied to reservoirs even when the
wells are hydraulically fractured;
[3] Belkis Teresa Refunjol, “Reservoir characterization
of North Buck Draw field based on tracer response and
production/injection analysis”, M.S. Thesis, The University
of Texas at Austin, 1996.
- The effect of a vertically fractured well on other
wells at far distance is very close to the effect of its vertical
well counterpart given the same flow rate. Thus, only the
pressure drops at the wells themselves are different;
[4] Ali Al-Yousef, “Investigating statistical techniques
to infer interwell connectivity from production and injection
rate fluctuations”, PhD Dissertation. University of Texas at
Austin, 2006.
- Theinterwellconnectivitydeterminationtechnique
can be applied to reservoirs containing horizontal wells;
- The well length at the observations wells or the
well directions do not affect the interwell connectivity
results;
[5] Ali A. Yousef, Pablo Hugo Gentil, Jerry L. Jensen,
and Larry W.Lake, “A capacitance model to infer interwell
PETROVIETNAM - JOURNAL VOL 10/2020
39
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