Interpretation of interwell connectivity tests in a waterflood system
PETROLEUM EXPLORATION & PRODUCTION
PETROVIETNAM JOURNAL
Volume 6/2021, pp. 18 - 36
ISSN 2615-9902
INTERPRETATION OF INTERWELL CONNECTIVITY TESTS
IN A WATERFLOOD SYSTEM
Dinh Viet Anh1, Djebbar Tiab2
1Petrovietnam Exploration Production Corporation
2University of Oklahoma
Email: anhdv@pvep.com.vn; dtiab@ou.edu
Summary
This study is an extension of a novel technique to determine interwell connectivity in a reservoir based on fluctuations of bottom
hole pressure of both injectors and producers in a waterflood system. The technique uses a constrained multivariate linear regression
analysis to obtain information about permeability trends, channels, and barriers. Some of the advantages of this new technique are
simplified one-step calculation of interwell connectivity coefficients, small number of data points and flexible testing plan. However, the
previous study did not provide either in-depth understanding or any relationship between the interwell connectivity coefficients and
other reservoir parameters.
This paper presents a mathematical model for bottom hole pressure responses of injectors and producers in a waterflood system.
The model is based on available solutions for fully penetrating vertical wells in a closed rectangular reservoir. It is then used to calculate
interwell relative permeability, average reservoir pressure change and total reservoir pore volume using data from the interwell
connectivity test described in the previous study. Reservoir compartmentalisation can be inferred from the results. Cases where producers
as signal wells, injectors as response wells and shut-in wells as response wells are also presented. Summary of results for these cases are
provided. Reservoir behaviours and effects of skin factors are also discussed in this study.
Some of the conclusions drawn from this study are: (1) The mathematical model works well with interwell connectivity coefficients
to quantify reservoir parameters; (2) The procedure provides in-depth understanding of the multi-well system with water injection in
the presence of heterogeneity; (3) Injectors and producers have the same effect in terms of calculating interwell connectivity and thus,
their roles can be interchanged. This study provides flexibility and understanding to the method of inferring interwell connectivity from
bottom-hole pressure fluctuations. Interwell connectivity tests allow us to quantify accurately various reservoir properties in order to
optimise reservoir performance.
Different synthetic reservoir models were analysed including homogeneous, anisotropic reservoirs, reservoirs with high permeability
channel, partially sealing fault and sealing fault. The results are presented in details in the paper. A step-by-step procedure, charts, tables,
and derivations are included in the paper.
Key words: Interwell connectivity, multi-well testing, waterflood system, well test analysis, reservoir characterisation.
1. Introduction
The previous study carried out by Dinh and Tiab has
introduced a new technique to infer interwell connectivity from
bottom-hole pressure fluctuations in a waterflood system. The
technique was proven to yield good results based
on numerical simulation models of various cases of
heterogeneity [1].
In this study, an analytical model for multi-
well system with water injection was derived for
the technique. The model is based on an available
solution for a fully penetrating vertical well in a
closed rectangular multi-well system and uses
the principle of superposition in space. Based on
Date of receipt: 5/4/2021. Date of review and editing: 5 - 13/4/2021.
Date of approval: 11/6/2021.
This article was presented at SPE Annual Technical Conference and Exhibition and licensed
by SPE (License ID: 1109380) to the republish full paper in Petrovietnam Journal.
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROVIETNAM
analytical analysis, a new technique to analyse data of
interwell connectivity test was developed. This technique
utilises the least squares regression method to calculate
the average pressure change. Thus, reservoir pore volume,
average reservoir pressure and total average porosity can
be estimated from available input data. The results were
verified using a commercial black oil numerical simulator.
2. Literature review
In 2002, Albertoni and Lake developed a technique
calculating the fraction of flow caused by each of the
injectors in a producer [2]. This method uses a constrained
Multivariate Linear Regression (MLR) model. The model
introduced by Albertoni and Lake, however, considers
only the effect of injectors on producers, not producers on
producers.AlbertoniandLakealsointroducedtheconcepts
and uses of diffusivity filters to account for the time lag and
attenuation occuring between the stimulus (injection) and
the response (production) [2]. Yousef et al. introduced the
capacitance model in which a nonlinear signal processing
model was used [3]. Compared to Albertoni and Lake’s
model which was a steady-state (purely resistive),
the capacitance model included both capacitance
(compressibility) and resistivity (transmissibility) effects.
The model used flow rate data and could include shut-in
periods and bottom hole pressures (if available). However,
the technique is somewhat complicated and requires
subjective judgement.
The practical value of interwell coefficients was
investigated. In order to derive the relationship between
interwell connectivity coefficients and other reservoir
parameters, a pseudo-steady state solution of the
previously mentioned model was used. The wells were
fully penetrating vertical wells flowing at constant rates.
The investigation proves that the interwell coefficients
between signal (active) and response (observation) wells
are not only associated with the properties between the
two wells but also the properties at the signal wells. To
calculate Relative interwell permeabilities, we assumed the
properties at the signal wells are constant. Thus, by varying
permeability between well pairs to match the Relative
interwell connectivity coefficient calculated from analytical
model and simulation results, the interwell permeabilities
can be found. Different cases of heterogeneous synthetic
fields were considered including anisotropic reservoir,
reservoir with high permeability channel, partially sealing
fault and sealing fault. In the sealing fault case, the results
indicated 2 groups of average reservoir pressure change
corresponding to 2 reservoir compartments. Thus,
reservoir compartmentalisation can be detected.
Recently, Dinh and Tiab [1] used a similar approach
as Albertoni and Lake [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.
Some advantages of using bottom-hole pressure data
are: (a) Diffusivity filters are not needed, (b) Only minimal
number of data points are required and (c)The programme
for collecting data is flexible.
The technique presented in the previous paper
requires several constraints including constant production
rates and constant total injection rates. These constraints
make it difficult to apply the technique in a real field
situation where production rates are hardly kept constant.
In this study, the systems with constant injection rates
and changing production rates were investigated. The
obtained interwell connectivity coefficients were almost
the same as the results from the case with constant
production rates and changing injection rates. The
technique is also applicable for fields with only producers;
where some producers are used as signal wells and others
as response wells provided that all assumptions are valid.
This suggests the technique is applicable to depletion
fields as well. Also, response wells can act as shut-in wells.
This study is to extend the work by Dinh and Tiab [1]
on interwell connectivity calculation from bottom-hole
pressure in a multi-well system. The purpose of this paper
is to incorporate a pseudo-steady state analytical solution
for closed system to the problem. Thus, other reservoir
parameters such as relative interwell permeability, and
reservoir pore volume can be quantified. This paper also
provides in-depth understanding of the method and its
applications.
3. Analytical approach
Numerousstudiesconcerningmulti-wellsystemshave
been carried out. Bourgeois and Couillens [4] provided a
technique to predict production from well test analytical
solution of multi-well system. Umnuayponwiwat et al.
investigated the pressure behaviour of individual well
in a multi-well closed system [5]. Both vertical well and
horizontal well pressure behaviours were considered.
This new study provides a tool to analyse reservoir
heterogeneity and to have a better understanding of
multi-well systems with the presence of both injectors
and producers.
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
Valko et al. developed a solution for productivity index
for multi-well system flowing at constant bottom-hole
pressure and under pseudo-steady state condition [6].
Marhaendrajana et al. introduced the solution for well
flowing at constant rate in a multi-well system [7, 8].
The solution was used to analyse pressure build-up test
and to calculate the average reservoir pressure using
decline curve analysis. Lin et al. [9] proposed an analytical
solution for pressure behaviours in a multi-well system
with both injectors and producers based on the work by
Marhaendrajana et al. [7].
Equation 1 is valid for pseudo-steady state flow and
can be rewritten as below:
nwell
141.2B
µ
p −p
(
x,y
)
=
a
[
xD , yD, xwDn, ywDn, xeD , y ,tAD qn
]
(7)
∑
n
eD
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
vertical well, fractured vertical well, etc.). This study only
considered the case of fully penetrating vertical well in a
closed rectangular reservoir under pseudo-steady state
condition.
3.1. Analytical model application
Considering a multi-well system with producers or
injectors and initial pressure pi, the solution for pressure
distribution due to a fully penetrating vertical well in a
close rectangular reservoir is as follows [8, 10]:
Equation 7 is applicable to a field where all the wells
are either producing or injecting. Lin and Yang [9] have
extended the model to a field with both injectors and
producers based on the model suggested by Equation 7
as shown below:
nwell
(1)
p (x , y ,t )= q a
(
x y ,xwD,i , ywD,i ,x , y ,
[tDA −tsDA
]
)
,
∑
D
D
D
DA
D,i
i
D
D
eD eD
i=1
In
pr
141.2B
kh
µ
L
pini −p
(
x, y
)
=
a
[
xD , yD , xwDj , ywDj xeD , yeD ,tAD
]
q
J
∑
j
j
where the dimensionless variables are defined in field
units as follows:
L j=1
K
(8)
ninj
Y
L
x
A
−
a
[
xD , yD , xwDi , ywDi , xeD , yeD ,tAD q
]
i Z
∑
i
xD =
(2)
(3)
(4)
L
[
i=1
y
A
where i and j denote injectors and producers,
respectively. Equation 8 is for a homogeneous reservoir
with initial reservoir pressure (pini) equal everywhere.
Applying Equation 8 to each time interval of an interwell
connectivity test, since the total injection and production
are kept constant, the average reservoir pressure change
is assumed to be constant for every time interval. The first
term in the bracket on the right-hand side of Equation
8 is constant due to constant rates at every producer
throughout the test. Applying to each time interval in the
interwell connectivity test, assuming the initial pressure
at the beginning of each interval increases at the same
rate as the average reservoir pressure (Δpave), Equation 8
can be rewritten as:
yD =
kh
141.2qref Bµ
pD =
(
pini − p
(
x, y, t))
kt
φctµA
(5)
tDA = 0.0002637
ai is the influence function equivalent to the
dimensionless pressure for the case of a single well in
bounded reservoir produced at a constant rate. Assuming
tsDA= 0, the influence function is given as:
∞
∞
1
2
ai
(
xD , yD , xwD,i , ywD,i , xeD , yeD ,tDA
)
=
E
∑ ∑
1
m=−∞ n=−∞
2
F
2 V
(
xD + xwD,i + 2nxeD
)
+
(
yD + ywD,i + 2my
)
eD
G
W
141.2Bµ
4tDA
G
H
W
X
p −p
(
x,y =
)
ave
kh
xD, yD,xwDi , ywDi x ,y ,t
(9)
2
2 V
n
F
−
inj a
[
]
q +∆p
Z
(
xD − xwD,i + 2nxeD
)
)
)
+
(
yD + ywD,i + 2my
)
)
)
I
Y
eD
(6)
+ E
+ E
+ E
1G
W
,
J
K
∑
i
eD eD AD
i
pr
4tDA
2
G
H
W
X
i=1
[
F
2 V
(
xD + xwD,i + 2nxeD
+
(
yD − ywD,i + 2my
where
eD
1G
W
npr
µ
4tDA
2
G
H
W
X
141.2B
∆p =
a
[
xD , yD, xwDj , ywDj xeD, y ,tAD
]
qj +∆p
ave (10)
∑
pr
j
eD
2 V
kh
F
j=1
(
xD − xwD,i + 2nxeD
+
(
yD − ywD,i + 2my
eD
1G
W
4tDA
pave = pini − ∆pave
G
W
H
X
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROVIETNAM
∆pave
Both ∆ppr and
are assumed to be constant.
pressure at injector i (pi) on producer j. Δt is the length
of the time interval as the injection rates were changed
after each time interval. Including the average reservoir
pressure, pave to Equation 15, we have:
Applying Equation 9 for a point at the circumference of
the well bore of producer j’ and taking into account the
skin factor, we obtain:
I
141.2Bµ
pave − pwf , j'
(
∆t
)
= β0 j' + pave
−
β p
(16)
pave − pwf , j'
(
xwDj', ywDj'
)
=
∑
ij' wf ,i
kh
i=1
(11)
ninj
I
L
L
K
Y
L
One of the properties of Equation15 is:
−
a
[
xwDj', ywDj' + rwDj', xwDi , ywDi , yeD
]
qi + sj'q + ∆p
J
j'Z
∑
ij'
pr
I
L
[
i=1
β
= 1
(17)
∑
ij'
i=1
where the third term in the bracket accounts for the
skin at well j’. For injector i’, we have:
Thus Equation 16 becomes:
141.2Bµ
I
pave − pwf ,i'
(
x
wDi', ywDi' =
)
β
+
β
(
pave − pwf ,i
)
pave − pwf , j'
=
(18)
∑
0 j'
ij'
kh
(12)
i=1
n
inj
I
J
K
Y
−
a
[
x
wDi', ywDi' + rwDj', xwDi , ywDi , yeD
]
q + s q + ∆p
i' Z
∑
ii'
i
i'
pr
Marhaendrajana et al. introduced the concept of
interference effect as a regional pressure decline to
analyse pressure build-up data at a production well [8].
Lin and Yang extended the work to a field with both
injectors and producers [9]. Their solutions basically state
that the pressure response of a well (injector or producer)
in a multiwell system is affected by the flow rate at the
well plus an interference effect due to other wells in the
field flowing under the pseudo-steady state. The solution
for a producer (j’) can be written as:
i=1
[
To simplify the problem, we assume all skin factors are
equal to zeros. Equations 11 & 12 can be rewritten for each
time interval as:
I
141.2Bµ C
S
p
ave − pwf , j' = −
q a + ∆p for j’ = 1...J (13)
D
ij' T
∑
ij'
pr
kh
E
i=1
U
I
141.2Bµ C
S
(14)
pave − pwf ,i' = −
q a + ∆p for i’ = 1...I
D
ii' T
∑
ii'
pr
kh
E
i=1
U
where qij’ = qii’ = qi are the flow rates at injectors (signal
wells).
141.2Bµ
p −pwf, j'
(
xwDj', ywDj',t
)
=
[
q
(
a −2πtDA
)
+2π∆qtottDA
]
(19)
ini
j' j'j'
kh
For injector i’, we have
3.2. Interpretation of interwell connectivity coefficients
using bottom-hole pressure data
p −pwf i'
(
xwDi'
,
y
,
wDi' t
)
ini
,
(20)
141.2B
µ
=
[q
(
i' ai'i'+2
π
tDA
)
+2 ∆qtottDA
π
]
Now, let us consider the interwell connectivity test.
In order to obtain better results, the reservoir should
reach pseudo-steady state before the test begins.
Different testing schemes were also considered including
(a) injectors as response wells, (b) producers as both
response and signal wells and (c) shut-in wells as response
wells. The response wells need to be directly affected by
the signal wells. The case where total injection equals to
total production is not considered for the test due to the
reason stated in the previous publication [1].
kh
npr
ninj
∆q = q −
q
i. Equations 19 and 20 state
where
∑ ∑
tot
j
that the pressure change i=a1t a producer or injector is a
j =1
combination of two terms as shown on the right-hand
sides of the two equations. The first term is proportional
to the flow rate of the well itself and the second term
accounts for the regional effect of other wells. In our case,
the second term in the brackets is constant for each time
interval. Using the material balance, we have:
In the previous study, Dinh and Tiab [1] defined the
interwell connectivity coefficients using the bottom-hole
pressure data that satisfy the equation:
∆pave 0.23394B
∆t
(21)
=
∆qtot
ctVp
I
where the constant 0.23394 is the conversion factor
for field units and Vp is the reservoir pore volume in
reservoir barrels. Applying the definition of tDA (Equation
5) and Equation 21 to the second term in the right-hand
side bracket, Equation 20 becomes:
ˆ
pj
(
∆t
)
=
β0j
+
β
ij pi
(
∆t for j = 1...J
)
(15)
∑
i=1
ˆ
pj
∆t
β
where
is the bottom-hole flowing pressure
is a constant and
coefficient accounting for the effect of bottom-hole
β
at producer j,
is the weighting
ij
0j
∑
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
(29)
β0 j' = ∆ppr (∆teq )
pini − pwf ,i'
(
xwDi', ywDi',t
)
(22)
141.2B
kh
µ
Equation 28 indicates that the interwell connectivity
coefficient βij reflects the effect of both the flow rates at
the signal wells and the influence of other wells on the
=
[qi'
(
ai'i' + 2
π
tDA + ∆pave(t)
)
]
Moving Δpave to the left-hand side, Equation 22
can be rewritten for each time interval of the interwell
connectivity test as:
I
aij'
aii +2πtDA
= 1
, pave on both sides is
signal wells. Since
∑
(
)
i=1
cancelled out and Equation 27 can also be written as:
141.2Bµ
pave(t) − pwf ,i'
(
xwDi', ywDi', t
)
=
[
qi'
(
ai'i' + 2πtDA
)
]
I
(23)
(24)
aij'
aii + 2πtDA
kh
pwf , j'
=
p
(
xwDi , ywDi
)
+ ∆ppr (∆teq )
(30)
∑
wf ,i
(
)
i=1
p (t)− pwf ,i'
(
x
wDi', ywDi', t
)
ave
I
I
q =
or
aij'
π
i'
141.2B
kh
µ
β
Eventhough
and
arebothequal
∑
∑
ij'
[
(
ai'i' + 2 tDA
π
)
]
(
aii +2
tDA
)
I
i=1
to 1, the meaninig=s1 are different for each case.
β =1
∑
ij'
Substitute qi’ defined in Equation 24 into Equation 13,
i=1
indicates the pressure fluctuation at the response wells
we have:
I
aij'
I
aij'
aii +2
=1
due to signal wells only while
indicates
∑
p − pwf,j'
=
[
p − p
(
xwDi , ywDi
)
]
+ ∆ppr
(25)
∑
ave
ave
wf,i
aii + 2πtDA
i=1
(
π
tDA
)
i=1
a state of pressure distribution due to pseudo-steady
state flow after the period Δteq.
Equation 25 can only be applied to the pseudo-steady
state flow and equivalent to Equation 18 if the following
condition satisfied:
Since the interwell connectivity coefficients were
calculated without the knowledge of pressure history
during each time interval, it is reasonable to apply the
pseudo-steady state equation (Equation 25) with the
flow duration of Δteq to each pressure data. Thus, the
original test system is now set to an equivalent pseudo-
steady state system with the time interval of Δteq. The
model works with the assumption that the bottom-hole
pressures at the response wells reach pseudo-steady state
before the rates at the signal wells are changed.
I
I
aij'
aii + 2πtDA
β =
= 1
(26)
∑
∑
ij '
(
)
i=1
i=1
NoticethatEquation25doesnotdependonproduction
history and holds true for any time interval assuming the
I
aij'
∑
pseudo-steady state flow. The sum
can be
(
aii +2πtDA
)
i=1
set to 1 by adjusting the time duration (Δt). The equivalent
time duration (Δteq) obtained indicates the time of the
pseudo-steady state required so that Equation 26 is satisfied
3.3. Model verification
at the response well. Thus, Equation 25 can be written as:
I
In order to verify the analytical model,
2
p
ave − pwf , j'
=
[
p
ave − pwf ,i
(
x
wDi , ywDi
)
]
∑
i=1
homogeneous synthetic fields were used. One field has
5 injectors and 4 producers (the 5×4 synthetic field) and
the other has 25 injectors and 16 producers (the 25×16
synthetic field). The used reservoir simulator was ECLIPSE
100 Black Oil Simulator. Figures 1 and 2 show the grid
systems for the 2 models and the well locations with I
and J indicating injector and producer respectively.
The grid configuration for the 5×4 synthetic field was
73×73×5 and for the 25×16 synthetic field was 59×59×5.
The dimensions for the 5×4 synthetic field were 3100
ft × 3100 ft × 60 ft and for the 25×16 synthetic field
were 5900 ft × 5900 ft × 60 ft. The initial static reservoir
pressure was 650 psia. Other reservoir properties for the
homogeneous case are shown in Table 1. One-phase
flow of water was assumed. The 5×4 synthetic field was
run for 50 months representing 50 data points (time
(27)
aij'
+ ∆p (∆teq )
pr
(
aii + 2πtDA
)
I
aij'
where
and ∆ppr (∆teq ) is the
= 1
∑
aii +2πtDA
i=1
pressure change defined by Equation 10 corresponding
to ∆teq ∆ppr (∆teq ) depends on the pseudo-steady state
.
initial pressure, the total field flow rate and the influence of
producers, but not on the actual time interval. Thus, with
the same total field flow rate (Δqtot), assuming the pseudo-
∆p (∆t )
steady state has been reached,
any test time interval (Δt). Equation 27 is true for any pave.
Since Equations 27 and 18 are now equivalent, we should
is constant with
pr
eq
have:
aij'
with i = 1…I and j’=1…J
(28)
β =
ij'
aii + 2πtDA
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROVIETNAM
Table 1. Input data for homogeneous simulation models
Horizontal permeability
Vertical permeability
Porosity
Viscosity
Initial reservoir pressure
Water saturation
kh = 100 mD
kv = 10 mD
φ = 0.3
μ = 2 cp
pi = 650 psi
Sw = 0.8
Water compressibility
Oil compressibility
Rock compressibility
Total compressibility
Formation volume factor
Wellbore radius
cw = 1E-6 psi-1
co = 5E-6 psi-1
cr = 1E-6 psi-1
ct = 2.8E-6 psi-1
B = 1.03 bbl/STB
rw = 0.355 ft
Figure 1. Grid system for the 5×4 synthetic field (73×73×5).
Figure 2. Grid system for the 25×16 synthetic field (59×59×5).
Table 2. Interwell connectivity coefficient results from MLR for the 5×4 synthetic field
P1
-740.6
0.25
0.25
0.22
0.14
0.14
1.00
P2
-740.3
0.26
0.14
0.21
0.25
0.14
1.00
Pꢀ
-741.3
0.13
0.26
0.22
0.14
0.25
1.00
Pꢁ
-741.0
0.14
0.14
0.22
0.25
0.25
1.00
Sum
β
I1
I2
I3
I4
I5
Sum
0j (psia)
-2963
0.78
0.78
0.87
0.78
0.78
Table 3. Interwell connectivity coefficient results from analytical solution with Δteq = 12.63 days for the 5×4 synthetic field
P1
P2
Pꢀ
Pꢁ
Sum
0.77
0.77
0.91
0.77
0.77
I1
I2
I3
I4
I5
Sum
0.24
0.24
0.23
0.15
0.15
1.00
0.24
0.15
0.23
0.24
0.15
1.00
0.15
0.24
0.23
0.15
0.24
1.00
0.15
0.15
0.23
0.24
0.24
1.00
interval, Δt = 30 days), while the 25×16 synthetic field
was run for 130 months. However, only data after the 2nd
month were used to better satisfy the condition of over
all pseudo-steady states.
5×4 Synthetic field
Both Equations 27 and 30 were used to verify the
analytical model. The bottom-hole pressure calculated
from Equations 15 and 30 were compared.The coefficients
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
calculated from the influence function were
also compared to those obtained from
simulation data. Investigation on the effect
of different teq on the interwell connectivity
coefficients was also carried out.
500
R 2 = 0.95
480
Δp (Δt ) = -760 psi
pr eq
460
440
420
400
380
360
340
320
Tables 2 and 3 show the interwell
connectivity coefficients obtained from
simulation data using MLR technique [1]
and calculated from analytical solution
with equivalent time Δteq = 12.63 days. The
coefficients for each well pair from both
tables are close with the difference less than
10%.
Simulated
Calculated
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Time (month)
Figures 3 and 4 show the results obtained
from Equations 27 and 30 with the simulation
results, respectively. The average pressures
for analytical solution (Equation 27) were
calculated using material balance equation
(Equation 21). The constant term ∆ppr (∆teq)
was calculated using trial-and-error method
by matching 2 representative equivalent
points on both graphs. The coefficient of
determination (R2) does not depend on this
constant term. Good match is observed on
Figure 3 with R2 = 0.95. The error could be
because the average reservoir pressure is not
exactly constant due to the change in total
compressibility. However, excellent match is
observed in Figure 4. The constant terms Δppr
(Δteq) for both cases are close to β0j calculated
from simulation data using MLR technique
(Table 2).
Figure 3. Absolute values of (pave- pwf) from Equation 28 and from simulation results for well P-1, the 5×4
homogeneous field.
16320
R2 = 1.00
Δppr (Δteq) = -735 psi
14320
12320
10320
8320
6320
4320
Simulated
Calculated
2320
320
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Time (month)
Figure 4. pwf results from Equation 30 and from simulation for well P-1, the 5×4 homogeneous field.
Similar results were obtained for other
producers. Thus, the analytical approach
works well for the 5×4 homogeneous
reservoir. Figure5showsaplotoftheconstant
β0j' calculated from simulation results versus
different length of the test time interval (Δt).
β0j' for different Δt are almost the same with
less than 1% difference. Hence, the results
agree with the analytical model that the term
Δppr (Δteq) = β0j' does not depend on the test
time interval.
-720
P1
P2
P3
P4
Average
-725
-730
-735
-740
-745
-750
-755
-760
25×16 Synthetic field
5
10
15
20
25
30
Similar procedure was used to verify
the application of an analytical model to
Time interval, dt (days)
Figure 5. Plot of the term βoj' = Δppr (Δteq) versus different time interval (Δt), the 5×4 homogeneous field.
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROVIETNAM
the 25×16 synthetic field. The equivalent
time was found to be 5.87 days (Δteq = 5.87
days). Again, Figure 6 shows the results
obtained from Equation 30 for Well P1.
Again, a perfect match was obtained for
bottom-hole pressures calculated using
Equation 30 and from simulation results.
However, the pressure difference plots
display a good match only at early time. The
poor match at late time resulted in a low
value of R2 (0.42). At a later time, as more
water was pumped in, the change of water
saturation became more significant. Since
water and oil compressibility were different,
the change in water saturation would lead
to a change in total compressibility. Thus,
the constant average reservoir pressure
change assumption was violated. Pave used
in Equation 27, which was calculated from
material balance, was no longer accurate
with changing total compressibility. When
the actual average field pressure from
simulation results was used for Equation
27, we obtained a much better match as
shown in Figure 7 (R2 = 0.92). Since excellent
match was again obtained for bottom-hole
pressure results even at late time (Figure 6),
it was confirmed that once the well reaches
pseudo-steady state, the bottom-hole
pressure is independent from production
history [6].
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
R2 = 1.00
Δppr(Δteq ) = -799 psi
Simulated
Calculated
0
1
9 17 25 33 41 49 57 65 73 81 89 97 105 113 121
Time (month)
Figure 6. pwf results from Equation 30 and from simulation for well P-1, the 25×16 homogeneous synthetic
field (Δteq = 5.87 days).
700
R2 = 0.92
Δppr (Δteq) = -798 psi
Simulated
Calculated
600
500
400
300
200
100
1
8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127
Time (month)
Figure 7. Absolute values of (pave- pwf) calculated and simulated with pave taken from simulation results for
well P-1, the 25×16 homogeneous synthetic field (Δteq = 5.87 days).
Different values of permeability were
applied to the same reservoirs (the 5×4 and
25×16 synthetic fields) to investigate the
behaviour of the equivalent time (Δteq). Plots
of permeability of both the 5×4 and 25×16
synthetic fields vs. the equivalent time are
shown on Figure 8. It is clear that as the
permeability increases, Δteq decreases. The
fact that Δteq of the 25×16 field was higher
than that of the 5×4 field indicated that with
the designed flow rates, the 25×16 field
reached the pseudo-steady state quicker
than the 5×4 field.
45
5×4 ꢀeld
40
25×16 ꢀeld
35
30
25
20
15
10
5
0
0
20
40
60
80
100
120
140
160
180
Formation permeability (mD)
Figure 8. Equivalent time (Δteq) as a function of permeability, the homogeneous 5× 4 and 25×16 synthetic fields.
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
4. Calculation techniques for interwell connectivity tests
Average pressure change calculation
4.1. Least squares linear regression (LSLR) and
multivariate linear regression (MLR) techniques
Now, assuming constant B, µ, ct and Δpave, we can
subtract the previous equation in the system of Equation
31 from the next equation taking into account that Δppr
stays constant. Thus, we have:
Albertoni and Lake [2] introduced the Multivariate
Linear Regression (MLR) technique to solve a system of
linear equations for interwell connectivity coefficients
using flow rate data. Dinh and Tiab [1] used the same
technique to calculate interwell connectivity coefficients
from bottom-hole pressure data. Least squares linear
regression is another technique to solve a system of linear
equations by least square fitting [11, 12]. According to
Yousef et al., MLR technique is equivalent to least squares
linear Regression (LSLR) [13]. Thus, using either MLR or
LSLR is an option based on convenience. In this study,
both MLR and LSLR were used. More details about LSLR
technique are provided below.
I
I
(2)
(1)
(
q(2) − q(1)
)
M ij' − ∆pave = −
(
p
p
− p
− p
)
)
∑
ij'
ij'
wf , j'
wf , j'
L
L
i=1
I
(3)
wf , j'
(2)
wf , j'
q(3) − q(2)
)
M ij' − ∆pave = −
(
L
L
(
∑
(32)
ij'
ij'
J
L
L
L
i=1
M
I
(L)
wf , j'
(L−1)
wf , j'
(
q(L) − q(L−1)
)
M ij' − ∆pave = −
(
p
− p
)
∑
ij'
ij'
L i=1
K
where Mij’ are coefficients account for the state of
the well regardless of production history. Since the total
injection rate was kept constant, when one equation was
subtracted from the other, the sum of the rate differences
was equal to zero. The sum of the resulting coefficients
(Mij’) was also equal to zeros indicating that if the flow
rates are kept constant and equal, the change of bottom-
hole pressure is equal to the change of the average
pressure. However, since Mij’ were calculated without the
information of production rates, they do not reflect the
actual state and are not used in the analysis.
4.2. Calculation approaches
Consider a system of J producers and I injectors where
injectors are signal wells and producers are response
wells. All wells are fully penetrating vertical wells. The
reservoir is assumed to be homogeneous with constant
rock properties. The fluid saturations are assumed to be
constant. Single phase flow of a slightly compressible
fluid of constant viscosity is also assumed. In an interwell
connectivity test as described by Dinh and Tiab [1], the
injection rates were changed after a constant time interval
(Δt) while the production rates were kept constant
and equal throughout the test. The total injection and
production rates were also kept constant. The reservoir
was assumed to have reached the pseudo-steady state at
the end of each time interval.
Equation 32 can be solved using either LSLR or MLR
technique. In this study, LSLR was used to calculate Δpave.
Δpave is positive when the average pressure increases and
negative when it decreases. Assuming constant total
compressibility and porosity, the reservoir pore volume
(Vp) can be estimated using Equation 21. Knowing the
initial static pressure, the average pressure after each time
interval can be estimated by adding the total pressure
change (Δpave), With the known total reservoir volume
(Vb), the total porosity can also be calculated:
Equations 18 and 19 were used as models for the
interwell connectivity test. Thus, the equations were
appliedtoeachtimeintervalduringthetest. Sincethetotal
field-wise flow rate and the time interval are constant, the
average reservoir pressure change is constant for every
time interval. Let the superscript l be the order of the data
points used for the test, we obtain a system of equations
for L data points for producer j’as follows:
Vp
Vb
(33)
φ
=
tot
Least squares linear regression (LSLR)
Considering the following model representing each
data point:
I
(34)
Y = A0 + C1A + C 2 A2 +K+ C I AI +
ε
I
L
L
141.2Bµ C
S
(1)
1
(1)
D
q(1)a + ∆p = p
ij ' T
− p
∑
i=1
I
ij '
pr
ave
ave
wf , j'
kh
141.2Bµ C
E
U
where the response is Y. The regression model
parameters are A0 and Ai, the explanatory variables are Ci
and ε is random error (11, 12]. With (L-1) data sets, (I+1)
estimated model parameters, we have the following
equation:
S
L
L
(2)
D
q(2)
a
ij ' T
+ ∆p = p
− p
wf , j'
(2)
(31)
∑
ij'
pr
J
L
L
L
kh
E
i=1
U
M
I
141.2Bµ C
L
S
(L)
D
q(L)a T + ∆p pr = pave − p(L)
∑
E
i=1
ij ' U
ij '
wf , j'
kh
K
PETROVIETNAM - JOURNAL VOL 6/2021
26
PETROVIETNAM
coefficients from analytical model and simulation results
for each response well equal zero. Different from the
interwell connectivity coefficients, the relative interwell
permeabilities do not depend on the distance between
wells and the position of the wells.
C1(1)
C1(2)
M
C 2(1) M M C I(1)
F
G
G
G
G
H
0 V
W
Y
Y2
F
1
1
V
W
W
W
W
A
F
G
G
G
G
V
W
W
W
W
X
1
G
C 2(2) M M C I(2)
M M
A
1 W
(35)
(36)
G
G
G
=
×
W
W
X
M
M
M
M
M
1 C (L−1) C 2(L−1) M M C I(L−1)
YL−1
AI
G
H
W
X
H
1
4.3. Calculation procedures
The short form of Equation 35 is:
Step 1: Obtain both flow rate and pressure data from
the interwell connectivity test. The number of data points
should be more than I+1 to get good results [1]. The time
interval should be long enough for every well to reach the
pseudo-steady state. However, if the reservoir is already in
the pseudo-steady state, the time required for each well
to reach the pseudo-steady state after each rate change
will be much shorter than the time required for the
reservoir to reach the pseudo-steady state from a static
initial pressure [5]. The interwell connectivity coefficients
can then be calculated using MLR method as described by
Dinh and Tiab [1].
Y = C × A
By minimising the sum of the squared differences
between the observed responses and the predicted
responses for each set of Ci(l) , the least squares estimation
of the parameter vector A is (11, 12]:
C TC
−1C TY
(37)
A =
where CT is the transpose of C. For example, to solve
Equation 32 for well j’, we consider
,
A0 = −∆pave
,
Ai = M ij'
(l +1)
wf , j'
(l)
wf , j'
(l+1)
ij'
(l)
ij'
C i(l)
=
q
− q
, and
.
Y = −
p
− p
l
Relative interwell permeability calculation from interwell
Step 2: Calculate the average reservoir pressure
change corresponding to each producer, Δpave using
Equation 32. Δpave for every producer should be close
if all producers are connected to the same reservoir
pore volume. The bulk volume (Vb) of the reservoir can
also be calculated knowing the reservoir geometry. The
pore volume and the total average porosity can then be
calculated using Equations 21 and 33.
connectivity coefficients using bottom hole pressures
A direct relationship between interwell connectivity
coefficients and the influence functions (aij) is presented
in Equation 28, in which aij’ represents the connectivity
between the 2 wells i and j’ and the term (aii + 2πtDA) is
associated with the injector i. Thus, the permeability
value in aij’ reflects the permeability between wells i
and j’ relative to the permeability given to the injector i
in the term (aii + 2πtDA). If permeability values given for
every injector are equal, then the permeabilities in aij’ are
relative to one another among injector - producer pairs
and the permeability at the injectors. The equivalent
time Δteq was calculated using trial-and-error technique
with an assigned homogeneous permeability system
Step 3: Define a homogeneous pseudo-steady state
reference reservoir by assuming a reference permeability
(kref). The kref should be representative of the entire
reservoir. Further details about the characteristics of kref
will be discussed later. The equivalent time interval (Δteq)
corresponding to the reference reservoir can be calculated
using trial-and-error method as described before.
I
aij'
aii + 2πtDA
= 1
. Thus, by
to the injectors so that
∑
Step 4: Using kref and Δteq from Step 3, match the
interwell connectivity coefficients from analytical
equation (Equation 28) with those calculated from the
bottom hole pressure data. The denominator in Equation
28, (aii + 2πtDA), is associated with the injector i and is
calculated using kref. The nominator is calculated using
the relative interwell permeability (kir). Thus, kir is varied
to obtain the match while kref is kept constant. The match
is obtained when the percent error between interwell
connectivity coefficients calculated from the analytical
equation and simulation is 0%. The results include a value
of kir for each injector - producer pair. These kir are relative
interwell permeability corresponding to the assumed
reference permeability.
(
)
i=1
varying permeability between each well pair so that
aij'
aii + 2πtDA
= β
ij' , the relative permeability among wells
can be estimated.
The reference reservoir is homogeneous with
permeability equal to the one given to the signal wells
(injectors). We call the permeability assumed for the signal
wells reference permeability (kref) and the permeability
accounting for the flow property between signal and
response relative wells interwell permeability (kir). The
matching process can be carried out using trial-and-
error method by varying relative interwell permeabilities
until the total difference between interwell connectivity
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
Table 4. Relative interwell permeability results for the 5×4 homogeneous synthetic field (kref = 100 mD, Δteq = 12.63 days)
P1
105
104
95
P2
109
95
Pꢀ
93
108
97
Pꢁ
98
98
Aꢂeꢃ
101
101
95
I1
I2
I3
94
95
I4
I5
Ave.
99
97
100
106
97
100
97
105
100
104
106
100
101
101
Table 5. Relative interwell permeability results from the pseudo-steady state equation for the 5×4 anisotropic synthetic field (kref = 316 mD, Δteq = 4.0 days)
P1
P2
Pꢀ
Pꢁ
Aꢂeꢃ
302
303
363
302
303
I1
I2
I3
I4
I5
494
490
220
197
182
317
203
326
505
201
327
313
324
204
506
323
205
312
188
191
220
486
498
317
Ave.
Step 5: The obtained results are used to
I01
I02
P01
analyse the reservoir properties including high
permeability channel, permeability barrier and
reservoir compartmentalisation. More details are
discussed in the next section.
5. Simulation results
P03
P02
The calculation approaches presented in the
last section were applied to data from 2 synthetic
fields, one with 5 injectors and 4 producers
(the 5×4 synthetic field) and the other with 25
injectors and 16 producers (the 25×16 synthetic
field). These synthetic fields are already described
in the previous sections. Both homogeneous
reservoirs and reservoirs with heterogeneity were
considered.
I03
P04
I05
I04
Figure 9. Representation of relative interwell permeability for the case of the 5×4
homogeneous reservoir.
I02
I01
P01
5x4 Synthetic field
Consider a waterflood system of 5 injectors
and 4 producers as shown in Figure 1, where
production and injection rates were kept constant
during constant time intervals. Injection rates were
changed after each time interval but production
rates and total injection rate stayed constant (qtot
= constant) as described by Dinh and Tiab [1]. The
system was assumed to be in the pseudo-steady
state so Equations 18 and 19 apply.
P03
P02
I03
Homogeneous reservoir
P04
I05
I04
The interwell connectivity coefficients
calculated from simulation data and analytical
Figure 10. Representation of relative interwell permeability for the case of the 5×4
anisotreopic reservoir.
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROVIETNAM
Table 6. Relative interwell permeability results from pseudo-steady state equation for the 5×4 synthetic field/reservoir with high permeability channel (kref = 300 mD,
Δteq = 4.21 days)
P1
P2
924
161
158
97
Pꢀ
Pꢁ
Aꢂeꢃ
859
179
163
148
175
I1
I2
I3
I4
I5
873
156
136
173
181
304
749
219
166
199
189
304
888
182
192
125
144
306
187
305
Ave.
model were presented in the previous section.
LSLRtechniquewasusedtocalculatetheaverage
pressure change as described before. ΔPave is in
perfect match with the results obtained from
material balance and the resulting porosity was
0.301. By keeping the permeabilities associated
with injectors constant at 100 mD, the interwell
coefficient in Table 3 can be matched with those
inTable 2 by adjusting the permeability between
injector/producer pairs or the influence function
aij. The resulting relative interwell permeabilities
are shown in Table 4. Figure 9 shows the
representation of the permeabilities in Table
4 in the form of inverse arrows. The lengths of
the arrows are proportional to the permeability
between injectors and producers. The relative
interwell permeabilities are very close to each
other and to the input formation permeability.
P01
I01
I02
P03
P02
I03
P04
I05
I04
Figure 11. Representation of relative interwell permeability for the case of the 5×4 synthetic field with
high permeability channel.
Anisotropic reservoir
connectivity for some well pairs such as I1P2 and I2P3 is larger
than the others such as I2P2 and I1P3. However, the permeabilities
between I2P2 and I1P3 are larger than those of I1P2 and I2P3 even
though the distance between the former pairs is less than the latter.
Thus, the relative interwell permeabilities are independent of the
distance between wells or the position of the wells. Results for the
change of average reservoir pressure for this case are almost the
same as the previous case, thus, the average pressure change does
not depend on permeability.
In this case, the permeability in x direction
(1,000 mD) is 10-fold the permeability in y
direction (100 mD). The results for relative
interwell permeability are shown in Table 5.
The permeability at the injectors was set to
the geometric average of the maximum and
minimum permeability which equals 316 mD.
The equivalent time (Δteq) was found to be 4.00
days.
Reservoir with high permeability channel
Figure 10 shows the representation of the
relative interwell relative permeabilities. The
results agree with the actual permeability of
the field with high permeability in x direction
and low permeability in y direction. The results
indicate that the relative permeability is not
directional permeability between well pairs
but rather be the average permeability of the
effective area between the 2 wells. The interwell
In this case, a high permeability channel was present as shown
on Figure 11. The shaded area is the high permeability channels
with permeability of 1000 mD which is 10-fold the permeability
in other area of the reservoir (100 mD). For this case, permeability
at the injectors was set to 100 mD. Again, the relative interwell
permeability between the well pairs was calculated by matching
the values of interwell connectivity coefficients calculated from
the analytical model with the values obtained from MLR technique
using simulation results. Some resulting permeabilities were lower
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
Table 7. Relative interwell permeability results from the pseudo-steady state equation for the 5×4 synthetic field/reservoir with partially sealing fault (kref = 100 mD, Δteq = 12.63 days)
P1
20
249
52
P2
129
65
Pꢀ
62
174
60
Pꢁ
98
95
Aꢂeꢃ
77
146
76
I1
I2
I3
99
94
I4
I5
Ave.
79
116
103
111
95
100
87
120
101
106
108
100
96
110
than the reservoir permeability, which was
unreasonable. It was because well I1 was
actually located in the high permeability
zone and thus, assuming the permeability
of well I1 (kref) was the same as the
formation permeability would lead to
unrealistic results. Thus, in order to address
this problem, an approximate average
reservoir permeability of 300 mD was
assumed for well I1. The same permeability
was applied to other injectors to guarantee
comparable relative permeability. A new
set of relative interwell permeabilities were
found as shown in Table 6.
I01
I02
P01
P03
P02
I03
Representation of the relative interwell
permeabilities is shown in Figure 11.
A clear trend of the high permeability
channel can be observed by looking at
the relative interwell permeabilities on
Figure 11. The flow in the channel seems
to affect the relative interwell permeability
between wells on each side of the channel.
For example, kir for the pair I03-P02 is lower
than kir for the pair I03-P03 even though
the permeability between I03-P02 is
higher. Thus, flow interference may affect
the relative interwell permeability.
P04
I05
I04
Figure 12. Representation of relative interwell permeability for the case of the 5×4 synthetic field with partially
sealing fault.
I01
P01
I02
Reservoir with partially sealing fault
P03
I05
P02
I03
In this case, a reservoir with partially
sealing fault similar to the case discussed
by Dinh and Tiab [1] was investigated.
The partially sealing fault is indicated by
the shaded strip as shown on Figure 12.
The fault was set to zero porosity and
permeability. Permeability at injectors was
equal to formation permeability of 100 mD.
P04
I04
Figure 13. Representation of relative interwell permeability for the case of the 5×4 synthetic field with sealing
fault.
The relative interwell permeability
results are shown in Table 7. Figure 12
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROVIETNAM
Table 8. Change of average reservoir pressure results for the 5×4 synthetic field/reservoir with sealing fault (kref = 100 mD, Δteq = 12.63 days)
P1
P2
Pꢀ
Pꢁ
Aꢂe
p (psia)
I1
I2
I3
I4
I5
Sum
181.0
-0.13
0.42
-0.21
-0.09
0.00
390.3
0.14
-0.24
0.16
0.07
-0.13
0.00
180.8
-0.18
0.30
-0.23
-0.12
0.23
390.2
-0.01
-0.20
0.19
0.12
-0.11
0.00
285.6
-0.18
0.28
-0.08
-0.02
0.00
∆
ave
0.00
0.00
shows the representation of the relative
interwell permeabilities in the form
of reverse arrows. It is clear that the
permeabilities of well pairs with wells
on different side of the fault are small.
Unlike the homogeneous case, the
constant β0j calculated for each producer
were different indicating each producer
was under different influence by other
producers.
120
100
80
60
40
20
0
The average pressure change for this
case is higher than that of the previous
case indicating a decrease in pore
volume. This is because the fault was set
to zero porosity causing a decrease in
overall pore volume. The calculated total
porosity was 0.29, which is slightly lower
than assigned formation porosity (0.30).
0
20
40
60
80
100
120
140
Injector-producer pairs
Figure 14. Plot of relative interwell permeability (kir) after cut-off (βij-cut-off = 0.04) for the 25×16 homogeneous
synthetic field (kref = 100 mD, Δteq = 5.87 days).
Reservoir with sealing fault
This case is similar to the partially
sealing fault; however, the fault seals
completely as shown in Figure 13.
Thus, the reservoir is divided into two
compartments. The results for interwell
connectivity coefficients were similar
to those presented in the previous
publication [1]. Some coefficients are
significantly small compared to others
for the same producers. To simplify the
calculation, a cut-off value was set at 0.1.
Thus, any coefficients less than 0.1 were
set to zeros. Since the relative interwell
permeabilities do not exist at zero
interwell coefficients, they were also set
to zero.
The representation of relative
interwell permeability results is presented
Figure 15. Representation of relative interwell permeability after cut-off (βij-cut-off = 0.04) for the case of the
25×16 homogeneous synthetic field (kref = 100 mD, Δteq = 5.87 days).
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PETROLEUM EXPLORATION & PRODUCTION
in Figure 13. The resulted average pressure change along
with coefficient Mij are shown in Table 8. It is obvious that
there are 2 sets of average pressure changes (181 psi and
390 psi) corresponding to 2 groups of producers (P1, P3)
and (P2, P4) suggesting 2 different reservoir pore volumes.
From the relative interwell permeability results, we can
identify the wells connected to the same pore volumes
by analysing both relative interwell permeabilities and
average pressure changes. The results indicate 2 groups
of wells. One group of wells connected to the same pore
volume includes well P1, P3, I2 and I5. The other group
includes P2, P4, I1, I2 and I4. This agrees with the actual
reservoir model setup. Thus, the new technique can
be used to detect reservoir compartmentalisation and
identify the wells that are in the same compartment.
The relative interwell permeability results are close to
one another. However, the average value for kir is slightly
lower than the input permeability of 100 mD as shown in
Figure 14. This could be due to cross flow effects among
wells. As shown in Figure 15, only kir between well pairs
that did not have any other well between them could be
determined. The relative interwell permeabilities of the
well pairs with farther distance were slightly higher than
those with closer distance. This agreed with a conclusion
drawn by Umnuayponwiwat et al. [5] that“the interference
effects are not always dominated by the nearby wells.
Under certain conditions, farther wells may play more
important roles on the well performance.”
6. Different flowing conditions at the response and
signal wells
The 25x16 synthetic field
The previous study [1] considered injectors as signal
wells (changing rates) and producers as response wells
(constant rates). However, in a real field situation, it is not
always possible to keep the production rates constant.
Thus, different test designs should be considered. The
characteristics of the analytical model discussed in the
previous section indicate that either injector or producer
can be used as response wells or signal wells. Hence,
the technique should not be restricted to the case
where injectors serve as signal wells and producers as
response wells. In this section, we obtained simulation
results from several scenarios to verify this theory.
Resulting interwell connectivity and discussion on any
necessary modification to the analytical solutions are
also presented.
Only the homogeneous case was considered for this
field. As mentioned before, 128 data points were obtained
to calculate interwell connectivity coefficients using MLR
technique. Similar results to the results presented by Dinh
and Tiab [1] were obtained. The interwell connectivity
coefficients are very low for the well pairs that are too
far apart. Since the percentage errors as mentioned in
Step 4 were magnified for low interwell coefficients, a
cut-off value of 0.04 was applied. Thus, the percentage
errors of any coefficients lower than the cut-off value
were set to zero, and the corresponding relative interwell
permeability was considered as undetermined. Only
relative interwell permeability corresponding to the
connectivity coefficients higher than or equal to the cut-
off values were calculated. The results are shown in Figures
14 and 15.
Tables 9 and 10 summarise the results for all the
cases discussed in this section. The second column
Table 9. Interwell connectivity result summary for different test schemes for the 5×4 homogeneous synthetic field (kref = 100 mD, Δteq = 12.63 days)
Cases
Base case
Constant injection
All producers
Shut-in producers
Shut-in injectors
Ave. % Error for βij
0.00%
A
Δqtot (STB/day)
-800
Ave. Δpave (psi)
286.0
% Error for Δpave
0.01%
Porosity
0.301
0.301
0.277
0.302
0.0035
0.0045
0.0044
0.0035
0.0431
2.28%
2.27%
0.04%
2.35%
-800
2,400
-2,000
2,000
285.6
-930.3
711.5
-683.2
0.12%
8.30%
0.63%
4.58%
0.315
Table 10. Interwell connectivity result summary for different test schemes for the 25×16 homogeneous synthetic field (kref = 100 mD, Δteq = 5.87 days)
Cases
Base case
Constant injection
Shut-in producers
Shut-in injectors
Ave. % Error for βij
0.00%
A
Δqtot (STB/day)
-3600
Ave. Δpave (psi)
353.0
% Error for Δpave
0.58%
Porosity
0.0059
0.0059
0.0072
0.1307
0.303
0.303
0.308
0.306
0.70%
1.37%
420.85%
-3600
-10000
10000
352.5
964.6
-970.6
0.70%
2.45%
1.74%
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shows the average percent error of interwell connectivity
coefficients compared to the base case (constant production
rate and changing injection rate in homogeneous reservoir).
The 3rd column presents the asymmetric coefficients (A). The
4th column is the total field flow rates. The 5th and 6th columns
show the Δpave results and their percent error compared to the
material balance solution, respectively. The last column is the
calculated porosities with input porosity of 0.3 for all the cases.
of 8.3% (Table 9). This was because as all wells were
producing, the water saturation decreased leading
to changing total compressibility or deviation
from original assumption. Thus, Δpave was actually
different for each time interval.
Similar approach was applied to the 25×16
homogeneous synthetic field. However, with the
original flow rates, when all wells are producing, it
was impossible to maintain the production rates as
scheduled due to quick depletion of the reservoir.
Thus, noresultswereobtainedforthe25×16synthetic
field in this case. Therefore, the challenge to carry
out the interwell connectivity test when all wells are
producing is to maintain the scheduled production
rates and make adjustments to the change in total
compressibility.
6.1. Constant injection rates and changing production rates
For this case (constant injection), the injectors of the
5×4 homogeneous synthetic field described before were
converted to producers and the producers were converted to
injectors. Thus, the 5×4 synthetic field now has 5 producers
and 4 injectors. Flow rates of the new producers are the same
as of the original injectors except they are now producing flow
rates. The new injectors were maintained at constant rates (850
STB/day) so that the difference between total injection and
total production was the same as the base case. The results are
shown in Table 9.
6.3. Shut-in wells as response wells
In this case, all response wells in the previous
cases were shut-in (shut-in producers and shut-in
injectors). The results obtained were also similar
for both changing injection rates and changing
production rates. Both cases of shut-in producers for
the constant production rate and changing injection
rate case and shut-in injectors for constant injection
rates and changing production rates case for the
5×4 homogeneous synthetic field were investigated.
Results for the shut-in injector case (A = 0.0431) were
not as good as the results for the shut-in producers
(A = 0.0035) as shown in Table 9. The reason could be
a more significant change in total compressibility in
the case of shut-in injectors.
Determination coefficients of R2=1 and the low asymmetric
coefficient A = 0.004482 indicate good results. The coefficients
and average pressure change are almost the same as for the
case of constant production rates and changing injection rates
(Table 9).
Similar results were obtained for the 25×16 synthetic field
with asymmetric coefficient A = 0.0059. Almost the same Δpave
was also obtained. Table 10 summarises the results.
A few changes are required for the analytical model in this
case. The negative sign in front of the first terms on the right-
hand side of both Equations 13 and 14 become positive and
the Δppr becomes:
The same approach was applied to the 25×16
homogeneous synthetic field. The case of all
producers with shut-in wells as response wells
could be simulated for this field. Good results
were obtained for the case of shut-in producer
and changing injection rates (Table 10). However,
poor results with an average percent error of βij =
420.85% were obtained for shut-in injectors and
active producers even after a cut-off value of 0.04
was applied to the interwell connectivity coefficients
as shown in Table 10. Again, these errors were due
to the significant change in total compressibility
as water was drawn from the reservoir and the
decreasing reservoir pressure leading to weak signals
from active producers.
n
inj
141.2B
kh
µ
∆p = −
a
[
xD , yD , xwDj , ywDj xeD , yeD ,tAD
]
qj + ∆p
ave (38)
∑
pr
j
j=1
j and i are now standing for injectors and producers,
respectively. Equation 19 should be used instead of Equation
20 to derive the flow rates for active wells (producers).
6.2. All production wells with constant rates at response wells
In this case (all producers), for the 5×4 homogeneous
field, the injectors in the base case were converted to
producers and acted as signal wells. Thus, all wells in the
system were producers. The response wells were set to
constant production rate of 100 barrels/day. The results are
shown in Table 9. Poorer result was obtained for Δpave with
the percentage error compared to the material balance result
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PETROLEUM EXPLORATION & PRODUCTION
As seen in Tables 9 and 10, with a negative total field
flow rate (total injection is higher than total production),
the calculated Δpave are positive indicating an increase
in reservoir pressure and vice versa. The results for the
base case and the constant injection case are very close
indicating the roles of injectors and producers can be
switched without significantly affecting the interwell
connectivity results.
- Further investigation on the characteristics of
relative interwell permeability and the effect of interwell
flow on the interwell permeability should be conducted.
- Interwell connectivity tests with varied test time
intervals and multi-phase flow should be investigated.
- Extension of the study to include wells with
different well bore conditions such as horizontal wells and
hydraulic fractured wells is recommended.
7. Conclusions and recommendations
- Extension of the study to infinite reservoirs
and closed reservoirs with different shapes is also
recommended.
The previous study by Dinh & Tiab [1] has been
extended in this study. A pseudo-steady state flow
solution for a well in a multi-well system was used to
model the interwell connectivity test. The model was
verified using 2 synthetic reservoir models, one with 5
injectors and 4 producers and the other with 25 injectors
and 16 producers. Results from the model fit well with
the simulation results. Average reservoir pressure change
can be calculated, and the total reservoir porosity can
be estimated. By defining a reference permeability, the
interwell connectivity can be presented in terms of the
relative interwell permeability. Some of the conclusions
and recommendations drawn from this study are:
Nomenclature
= modelled pressure change (psia)
φ = porosity, fraction
φtot = total field porosity, fraction
a = influence function
A = asymmetric coefficient or area (ft2)
B = formation volume factor (rbbl/STB)
co = oil compressibility (psi-1)
- The analytical model presented in this study
works well with the interwell connectivity test with
the assumption that the pseudo-steady state has been
reached at the end of each time interval.
cr = rock compressibility (psi-1)
ct = total compressibility (psi-1)
cw = water compressibility (psi-1)
E1 = exponential integral function one
h = formation thickness (ft)
- Tests that are longer than required (more data
points) may create errors because of deviation from the
constant total compressibility assumption due to the
change of total reservoir saturation. Thus, an adequate
number of data points should give better results.
I = total number of signal or active wells (injectors) or
injector indicator in well names
- The relative interwell permeability does not
depend on the position and the distance between wells.
Thus, it provides an additional parameter to evaluate
interwell connectivity.
J = total number of response wells (producers),
producer indicator or productivity index, (STB per day/psi)
k = permeability (mD)
kir = interwell relative permeability (mD)
kref = reference permeability (mD)
LSLR = least square linear regression
M = coefficients in average pressure change calculation
m,n = numbers of calculation terms
MLR = multivariate linear regression
ninj = total number of injectors
- The average reservoir pressure change with the
interwell connectivity information can be used to identify
reservoir compartmentalisation as well as the wells
connected to each compartment.
- Results from this study have shown that the signal
wells could be either producers or injectors, and so are
the response wells. The response well could also be
either flowing or shut-in. Thus, this study provided more
flexibility in design of interwell connectivity tests to fit a
field situation.
npr = total number of producers
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nwell = total number of wells
e = boundary value
p = pressure (psia)
eq = equivalent
pave = average pressure (psia)
i' = investigated signal/active well (injector)
i = signal or active well (injector) index
ini = initial value
pini = initial pressure (psia)
pj = pressure at the observation well (psia)
pwf = bottom-hole flowing pressure (psia)
q = flow rate (STB/day)
j = response/observation well (producer) index
j’ = investigated response/observation well (producer)
tot = total
qref = reference flow rate (STB/day)
R2 = coefficient of determination
rw = wellbore radius (ft)
w = well
wf = flowing conditions
s = skin factor, dimensionless
Superscripts
t = time (hours)
l = order of data point
L = total number of data points
T = transposed
ts = starting time (hours)
Vb = reservoir bulk volume (ft3)
Vp = pore volume (ft3)
References
x = coordinate or dimension in x-direction (ft)
xe = dimension of study area in the x-direction (ft)
xw = individual well x-coordinate (ft)
y = coordinate or dimension in y direction (ft)
ye = dimension of study area in the y direction (ft)
yw = individual well y-coordinate (ft)
β0j = additive constant term in MLR
βij = weighting coefficient in MLR
Δp = pressure change/difference (psi)
Δpave = average pressure change (psi)
[1] Djebbar Tiab and Dinh Viet Anh, “Inferring
interwell connectivity from well bottom hole pressure
fluctuations in waterfloods”, SPE Reservoir Evaluation
& Engineering, Vol. 11, No. 5, pp. 874 - 881, 2008. DOI:
10.2118/106881-PA.
[2] 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.
[3] A.A.Yousef, P.Gentil, J.L.Jensen, and Larry W.Lake,
“A capacitance model to infer interwell connectivity from
production and injection rate fluctuations”, SPE Annual
Technical Conference and Exhibition, Dallas, Texas, 9 - 12
October 2005.
Δppr = pressure change corresponding to influence of
response wells and change in average pressure (psi)
[4] M. Bourgeois and P. Couillens, “Use of well test
analytical solutions for production prediction”, European
Petroleum Conference, London, United Kingdom, 25 - 27
October 1994. DOI: 10.2118/28899-MS.
Δqtot = field total flow rate (STB/day)
Δt = time interval (hours)
Δteq = equivalent pseudo-steady state time interval
μ = fluid viscosity (cp)
[5] Suwan Umnuayponwiwat, Erdal Ozkan, and
R.Raghavan, “Pressure transient behavior and inflow
performance of multiple wells in closed systems”, SPE
Annual Technical Conference and Exhibition, Dallas, Texas,
1 - 4 October 2000. DOI: 10.2118/62988-MS.
Subscripts
ave = average
D = dimensionless quantity
DA = dimensionless corresponding to area
[6] P.P.Valko, L.E.Doublet, and T.A. Blasingame,
“Development and application of the multiwell
PETROVIETNAM - JOURNAL VOL 6/2021
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PETROLEUM EXPLORATION & PRODUCTION
productivity index (MPI)”, SPE Journal, Vol. 5, No. 1, pp. 21 -
31, 2000. DOI: 10.2118/51793-PA
pressure fluctuations of hydraulically fractured vertical
wells, horizontal wells, and mixed wellbore conditions”,
Petrovietnam Journal, Vol. 10, pp. 20 - 40, 2020. DOI:
10.47800/PVJ.2020.10-03.
[7] Taufan Marhaendrajana, “Modeling and analysis
of flow behavior in single and multiwell bounded reservoirs”,
PhD dissertation, Texas A&M University, Texas, May 2000.
[11] Jerry Lee Jensen, L.W.Lake, Patrick William
Michael Corbett and D.J.Goggin, Statistics for petroleum
engineers and geoscientists. New Jersey: Prentice Hall,
1997.
[8] T.Marhaendrajana,
N.J.Kaczorowski,
and
T.A.Blasingame, “Analysis and interpretation of well test
performance at Arun field, Indonesia”, SPE Annual Technical
Conference and Exhibition, Houston, Texas, 3 - 7 October
1999. DOI: 10.2118/56487-MS.
[12] Ali Abdallah Al-Yousef, “Investigating statistical
techniques to infer interwell connectivity from production
and injection rate fluctuations”, PhD dissertation, University
of Texas at Austin, Austin, Texas, May 2006.
[9] Jia En Lin and Hui-Zhu Yang, “Analysis of well-
test data from a well in a multiwell reservoir with water
injection”, SPE Annual Technical Conference and Exhibition,
Anaheim, CA, 11 - 14 November 2007. DOI: 10.2118/110349-
MS.
[13] Steven C.Chapra and Raymond P.Canale,
Numerical methods for engineers, 2nd edition. McGraw-Hill,
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[10] Dinh Viet Anh and Djebbar Tiab, “Inferring
interwell connectivity in a reservoir from bottomhole
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