Characterizing interwell connectivity in waterflooded reservoirs using data-driven and reduced-physics models: a comparative study

Abstract

Waterflooding is a significantly important process in the life of an oil field to sweep previously unrecovered oil between injection and production wells and maintain reservoir pressure at levels above the bubble-point pressure to prevent gas evolution from the oil phase. This is a critical reservoir management practice for optimum recovery from oil reservoirs. Optimizing water injection volumes and optimizing well locations are both critical reservoir engineering problems to address since water injection capacities may be limited depending on the geographic location and facility limits. Characterization of the reservoir connectivity between injection and production wells can greatly contribute to the optimization process. In this study, it is proposed to use computationally efficient methods to have a better understanding of reservoir flow dynamics in a waterflooding operation by characterizing the reservoir connectivity between injection and production wells. First, as an important class of artificial intelligence methods, artificial neural networks are used as a fully data-driven modeling approach. As an additional powerful method that draws analogy between source/sink terms in oil reservoirs and electrical conductors, capacitance–resistance models are also used as a reduced-physics-driven modeling approach. After understanding each method’s applicability to characterize the interwell connectivity, a comparative study is carried out to determine strengths and weaknesses of each approach in terms of accuracy, data requirements, expertise requirements, training algorithm and processing times.

Full text

ORIGINAL ARTICLE
Characterizing interwell connectivity in waterflooded reservoirs
using data-driven and reduced-physics models: a comparative
study
Emre Artun
1
Received: 18 June 2015 / Accepted: 21 December 2015
The Natural Computing Applications Forum 2016
Abstract Waterflooding is a significantly important pro-
cess in the life of an oil field to sweep previously unrecovered
oil between injection and production wells and maintain
reservoir pressure at levels above the bubble-point pressure
to prevent gas evolution from the oil phase. This is a critical
reservoir management practice for optimum recovery from
oil reservoirs. Optimizing water injection volumes and
optimizing well locations are both critical reservoir engi-
neering problems to address since water injection capacities
may be limited depending on the geographic location and
facility limits. Characterization of the reservoir connectivity
between injection and production wells can greatly con-
tribute to the optimization process. In this study, it is pro-
posed to use computationally efficient methods to have a
better understanding of reservoir flow dynamics in a water-
flooding operation by characterizing the reservoir connec-
tivity between injection and production wells. First, as an
important class of artificial intelligence methods, artificial
neural networks are used as a fully data-driven modeling
approach. As an additional powerful method that draws
analogy between source/sink terms in oil reservoirs and
electrical conductors, capacitance–resistance models are
also used as a reduced-physics-driven modeling approach.
After understanding each method’s applicability to charac-
terize the interwell connectivity, a comparative study is
carried out to determine strengths and weaknesses of each
approach in terms of accuracy, data requirements, expertise
requirements, training algorithm and processing times.
Keywords Waterflooding Reservoir characterization 
Interwell connectivity Data-driven modeling Artificial
neural networks Reduced-physics modeling 
Capacitance–resistance models
1 Introduction
Waterflooding is a secondary oil recovery method that is
applied after the oil has been produced from a reservoir
with its natural energy (known as the primary recovery
period). Based on industry experience, it is known that the
recovery factors (ratio of recovered oil volume to the
original oil volume in the reservoir) with primary recovery
ranges between 15–25 %, while a well-managed water-
flood can increase this to 40–50 % depending on the
reservoir characteristics and efficiency of the waterflooding
operation. Although it can be classified as a common
operation for secondary recovery, varying reservoir char-
acteristics and limited water supplies in some areas make it
critical to have a good understanding of the reservoir and
determining the optimum design schemes of the water-
flooding operation under consideration. Poorly managed
waterflooding operations result in underperforming reser-
voirs with reserves that are left behind, and thus, huge
losses in recovered oil and associated monetary income are
realized. Understanding reservoir flow dynamics to apply
proper reservoir management practices is a complex
problem with well data from isolated localities in an oil
field, data distributed spatially across the whole reservoir
and spanned tens of years of history. Current reservoir
management practices highly depend on numerical flow
simulation models that take months to develop and main-
tain, and cost millions of dollars with both significant
manpower and computational power requirements.
&Emre Artun
artun@metu.edu
1
Petroleum and Natural Gas Engineering Program, Middle
East Technical University, Northern Cyprus Campus,
Mersin 10, Kalkanlı, 99738 Gu
¨zelyurt, Turkey
123
Neural Comput & Applic
DOI 10.1007/s00521-015-2152-0

Although being recognized scientifically as the most
powerful approach, full field-scale models do not easily
allow rapid reservoir analysis that results in right reservoir
management decisions.
Over the last two decades, petroleum industry has been
transformed significantly with the advancement of intelli-
gent field technologies that are mostly based on on-site
instrumentation and automation of field operations. Many
technologies have been adapted to collect significant vol-
umes of data in much shorter time frames. However, the
real challenge has been to convert these data into useful
information to make quick and reliable decisions that
generate value. This challenge can only be overcome by
utilizing proper knowledge management, data assimilation
and data analysis practices. The current paradigm in the
evolution of science also requires advanced data analysis to
synthesize all of the earlier empirical, experimental and
computational findings [1].
An efficient way of managing an hydrocarbon reservoir at
any stage of development is the closed-loop reservoir man-
agement approach [2]. As shown in Fig. 1,thisapproach
requires continuous updating of models after collecting recent
data from the high-resolution and high-frequency sensors in
the oil field that record measurements of time-dependent
(dynamic) properties such as pressure, flow rate and temper-
ature. In this workflow, it is very important to have a model
that can respond to the following primary needs:
•A model that can be updated quickly when new data are
available.
•A model that is sufficiently accurate and representative
of the actual system (surface or subsurface) so that it
can be used for decision-making purposes.
There are a wide variety of modeling approaches presented
in the reservoir engineering literature. Each modeling
approach represents various complexities, advantages and
disadvantages. In order to find a model that serves to both
of the aforementioned objectives, it would be a better
approach to have access to different modeling options
readily available and choose the right modeling approach
depending on the problem type and scope.
In this study, two conceptually different modeling
approaches are investigated for the purpose of character-
izing interwell connectivity in a waterflooded reservoir:
1. Fully data-driven modeling: artificial neural networks
(ANN)—no functional relationship presumed.
2. Reduced-physics-driven modeling: capacitance–resis-
tance model (CRM)—physics are incorporated with
certain assumptions that simplify the problem.
Both methods have been proven to be potentially efficient
tools for reservoir engineering problems based on studies
presented in the literature. In this study, it is aimed to
investigate the characteristics of each method including
best practices and challenges associated with them to
characterize interwell connectivity between injection and
production wells. These would allow us to compare these
methods with each other from the practical point of view
and develop guidelines for the practicing engineer or asset
team who is responsible for developing an optimum
waterflooding plan. The primary objectives of the study can
be summarized as the following:
1. Utilizing two different modeling approaches: artificial
neural networks (as a data-driven modeling approach)
and capacitance–resistance models (as a reduced-
physics modeling approach) for quantifying interwell
connectivity between injection and production wells in
a waterflooded petroleum reservoir.
2. Assessing and comparing both methods (ANN and
CRM) from different perspectives to determine
strengths and weaknesses of each approach in terms
of accuracy, data requirements, training algorithm,
processing times and expertise requirements.
This study is the first attempt, to the best of our knowledge,
to compare these two modeling approaches for the purpose
of characterizing reservoir connectivity. This comparison
provides with the necessary insight for the practicing
engineer to implement either of these methods for a
waterflooded petroleum reservoir. These tools have great
advantages over other modeling approaches because of
requiring fewer inputs, being much more computationally
efficient and not being dependent on geological uncer-
tainties. Therefore, having the necessary insights would
help to decide which method is more practical for a par-
ticular problem. Based on the analysis performed during
this study, the decision of choice would be affected by the
expertise of the practicing engineer, availability of the data
and the time frame of the study.
System
(reservoir, wells
and facilities)
Sensors
Data
assimilation
algorithms
System models
Optimization
algorithms
Geology,
seismic,
well logs,
well tests,
PVT,
etc.
Measured
output
Predicted
output
Input
Noise
Controllable
output
Output Noise
Fig. 1 Closed-loop reservoir management workflow [2]
Neural Comput & Applic
123

2 Methodology
2.1 Case study: a synthetic reservoir model
In this study, a synthetic streak case study [3] is selected to
implement the aforementioned methods. It is a synthetic
field that consists of 5 vertical injectors, I1 through I5, and
4 vertical producers, P1 through P4 (Fig. 2). The perme-
ability of the reservoir is 5 md, except two high-perme-
ability streaks:
1. Streak-1: 1000 md between I1 and P1 wells.
2. Streak-2: 500 md between I3 and P4 wells.
The porosity is constant and equal to 0.18. Total mobility
of oil and water (koþkw) is 0.45 and is independent of
saturation. Oil, water and rock compressibilities are
5910
-6
psi
-1
,1910
-6
psi
-1
,1910
-6
psi
-1
, respec-
tively. The model is constructed with 1 layer and 31 grid
blocks in each of the xand ydirections with grid sizes of 80
ft 980 ft (Dxand Dy). The thickness of the reservoir is 12
ft (Dz).
This model has been built and run using a commercial,
numerical reservoir simulator [4] that utilizes black-oil
formulation, which is a common formulation used as a
robust approach for waterflooding problems. A variable
water injection rate scenario is implemented, in which
volumetric injection rates are varied significantly over a
period of 100 months (10 years). It is aimed to charac-
terize the connectivity of the system through these rate
fluctuations [3]. The bottom-hole pressure (BHP) for pro-
ducing wells is fixed at 250 psia, and volumetric liquid
production rates are measured (Fig. 3). The volumetric
water injection rates are varied manually, while setting a
limit for maximum bottom-hole pressure of 5000 psia
(Fig. 4). Both injection and production rates are measured
at reservoir conditions (rbbl/day: reservoir barrels per day).
One hundred months of monthly injection/production his-
tory resulted in a sample size of 100 for each well. Other
descriptive statistics of volumetric injection rates for
injector wells, I1 through I5, and volumetric production
rates for producer wells, P1 through P4, are given in
Table 1.
Although the presented synthetic case has rather simple
permeability contrasts, it is a good example of a real
reservoir with high-permeability streaks that must be con-
sidered during a waterflood optimization study. Existence
of such streaks amplifies the importance of characterizing
the connectivity between wells, since they provide a con-
duit in the reservoir to transport the injected water. The fact
that results from a synthetic model are used should not
raise any concern regarding the validity of the methods
presented since both methods are proven to be successful in
real reservoir cases with a number of examples in the lit-
erature. CRMs were successfully applied to a number of
fields [5], and ANNs were successfully utilized for reser-
voir characterization problems with real-field data [6–9].
Since the main objective in this study is to perform a
comparison of two methods, a synthetic case would be
sufficient. However, it is anticipated that for a more com-
plex reservoir case with more heterogeneities and more
wells, more number of historical observations (more than
10 years of history) of injection and production rates might
be needed for capturing the fluid flow dynamics of the
reservoir.
2.2 Artificial neural networks
Intelligent systems have been applied to many different
types of optimization problems in the petroleum industry.
Most of these problems presented in the literature are based
on development of ANN-based proxy models that can
accurately mimic reservoir models within a reasonable
amount of accuracy and computational efficiency. In some
studies, these models are utilized to construct data-driven
predictive tools and these tools are coupled with evolu-
tionary algorithms to solve the optimization problem effi-
ciently. Several areas of application included reservoir
characterization [6–9], candidate well selection for
hydraulic fracturing treatments [10], field development
[11–13], well placement and trajectory optimization [14–
16], scheduling of cyclic steam injection process [17],
screening and optimization of secondary/enhanced oil
recovery[18–23], history matching [24–26], underground
gas storage management [27], reservoir monitoring and
management [26,28] and modeling of shale-gas reservoirs
[29,30].
1,000
500
5
Permeability (md)
Fig. 2 Synthetic reservoir model used in this study and its
permeability distribution: a reservoir with 2 high-permeability streaks
[3]
Neural Comput & Applic
123

In addition to petroleum engineering and many other
engineering disciplines, artificial neural networks and other
data-driven modeling approaches have been used in many
different kinds of applications such as spatial clustering
[31], cavity-filter optimization [32], electricity load fore-
casting [33], control, pattern recognition, signal processing,
medicine, speech recognition, speech production and
business [34].
The most common training algorithm and also the one
used in this study is the backpropagation algorithm. Also
known as the generalized delta rule, backpropagation
algorithm is a gradient-descent method that minimizes the
total squared error of the output computed by the network.
It played a major role in the re-emergence of neural net-
works in late 1980s. It was introduced as a training method
of multilayer networks to overcome the limitations of
single-layer networks [34]. Backpropagation algorithm is a
supervised training technique (i.e., mapping a given set of
inputs to a specified set of target outputs) and includes
three stages: (1) feedforward of the input training pattern,
(2) calculation and backpropagation of the error and (3)
adjustment of weights. The overall goal is to train the
network such that it can [34]:
•Respond correctly to the input patterns that are used for
training (memorization).
•Give reasonable responses to similar, but not identical,
input patterns (generalization).
Figure 5shows a multilayer, fully connected network
with one hidden layer. There are ninput neurons in the
input layer, phidden neurons in the hidden layer and m
output neurons in the output layer. There are biases also
shown in this figure whose activation value is constant
during the training (1, in this case). While the number of
inputs and outputs is based on the nature of the problem
studied, the number of hidden neurons is a part of the
network design process and must be optimized by the
designer. A rule-of-thumb formula is presented to calculate
the number of the hidden neurons, which is mostly based
on experience [35]:
NHN ¼NIþNO
2þffiffiffiffiffiffiffiffi
NTP
pð1Þ
where NIis the number of inputs, NOis the number of
outputs, and NTP is the number of training patterns. It
should be noted that this is not a theoretical formula, and
this number would not necessarily be the best estimate of
the number of hidden neurons. However, it can be used as a
good start for the optimization process. Algorithm 1 shows
a step-by-step explanation of the backpropagation
algorithm.
0
50
100
150
200
250
300
350
400
450
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 102030405060708090100
Liquid Production Rate (rbbl/d)
P2 and P3
)
d/
lbbr(
e
taR
no
i
t
c
udorP diuqiL
4P dna 1P
Time (months)
P1 P4
P2 P3
Fig. 3 Production history of the
synthetic reservoir model
0
800
1600
2400
3200
4000
0 102030405060708090100
)d/lb
b
r(eta
Rn
oitcejnIret
aW
Time (months)
I1
I2
I3
I4
I5
Fig. 4 Injection history of the synthetic reservoir model
Neural Comput & Applic
123

Table 1 Descriptive statistics
of volumetric injection and
production rates that are output
from the model and used in the
study
Injection well I1 I2 I3 I4 I5
Volumetric water injection rates (rbbl/day)
Mean 1,683.1 999.5 860.4 779.0 931.8
Standard error 55.5 20.9 28.3 21.5 28.1
Median 1,595.2 1,018.6 854.7 792.0 1,020.1
Standard deviation 554.7 208.8 283.4 215.3 280.5
Sample variance 307,638.1 43,595.6 80,313.9 46,345.7 78,686.5
Kurtosis 1.24 0.60 0.37 -0.07 -0.05
Skewness 0.71 -0.74 -0.27 0.04 -0.86
Range 3,258.7 1,053.0 1,288.9 1,034.2 1,128.5
Minimum 305.9 227.0 172.7 272.0 145.0
Maximum 3,564.6 1,280.0 1,461.6 1,306.3 1,273.5
Sum 168,313.9 99,945.2 86,043.1 77,896.8 93,179.6
Count 100 100 100 100 100
Production well P1 P2 P3 P4
Volumetric liquid production rates (rbbl/day)
Mean 2,494.9 157.1 274.3 2,386.1
Standard error 56.8 4.0 3.3 43.8
Median 2,458.5 158.1 272.3 2,354.8
Standard deviation 568.4 39.7 33.2 438.4
Sample variance 323,095.6 1,577.6 1,105.2 192,167.8
Kurtosis 0.34 0.23 0.30 0.41
Skewness 0.27 0.47 -0.33 0.52
Range 3,141.9 193.4 174.5 1,899.1
Minimum 917.5 85.0 182.5 1,509.9
Maximum 4,059.4 278.4 357.0 3,408.9
Sum 249,486.5 15,712.0 27,431.2 238,606.3
Count 100 100 100 100
Neural Comput & Applic
123

The iterative procedure for each training pair shown in
Algorithm 1 is repeated for all training patterns, until a pre-
specified stopping condition is achieved. Processing of
each training data is known as a training event or iteration.
When all training data are processed once, one epoch
(training cycle) is completed. Each training data can be
processed either by random selection, or by rotation. After
each training event, average mean-squared error is calcu-
lated. Achieving minimum mean-squared error of outputs
and maximum number of epochs is among most common
stopping conditions. Once the defined stopping criteria are
satisfied, weights on connection links achieve their opti-
mum states. Provided that the training performance is
satisfactory, the trained network with optimum weights can
be used as a predictive model.
In this study, mapping input–output relationships is
achieved with the inputs of injection rates from the water
injectors and the output of the liquid production rate of a
given producer. By analyzing the weights on connection
links of the trained neural network, interwell connectivity
is quantified. The optimized value of the weight on a given
connection link indicates the degree of influence of the
given input parameter on the output parameter. Therefore,
we propose that once the training is completed, the relative
values of connection links that connect each injector signal
to the producer can be used to quantify the connectivity.
There are individual neural networks for each producer
well in the field. A neural network for a given producer
would provide the connectivity of each injector to that
producer. Once all neural network models are trained, all
connectivities between all injector–producer pairs would
be quantified. This would provide insights about the
waterflood dynamics in the reservoir and help to under-
stand the overall reservoir connectivity to be used for
further optimization studies.
A feedforward artificial neural network is constructed
for each producing well in the reservoir. The training
algorithm used is the Levenberg–Marquardt backpropaga-
tion algorithm [36,37], and due to the low number of total
input/output parameters (5 injectors and 1 producer: 6
parameters), only 1 hidden layer is used with 12 neurons.
The schematic of the architecture of the neural network is
shown in Fig. 6. Eighty percentage of the historical pro-
duction/injection are used for training, 10 % are used for
validation during the training to prevent over-training, and
10 % are used for blind-case testing.
2.3 Capacitance–resistance models
It was suggested that the development of an electrical
model offers the promise of rapid evaluation for non-
mathematical analysis of complex reservoir problems
including understanding of the waterflood performance [3,
38]. An analogy between the flow behaviors of electricity
in electric units and fluid in reservoir units was made
through an experimental study [38]. This analogy implies
that the electrical unit acts as a device which stores the
electrical charge just as reservoir rock is acting as the
storage of reservoir fluids [38]. After considering that the
current may be equivalent to fluid flow, and the pressure is
Fig. 5 Architecture of a
multilayer network
Neural Comput & Applic
123

equivalent to the electrical potential, oil reservoir, as a
porous continuum, is divided into small blocks so that the
material balance can be used assuming the reservoir fluid is
flowing in at one face of the block and out at the opposite
face [38]. Role of such models for rapid estimation of
waterflood performance and optimization was investigated
by calling them capacitance–resistance models (CRMs)
[3]. The base data necessary to run this model are pro-
duction/injection data and well bottom-hole pressure
(BHP) to calibrate the model against a specific reservoir.
CRMs were primarily used for the characterization of
interwell connectivity between injection and production
wells rapidly without needing a geological model [39,40].
After characterizing the connectivity, they are then used to
optimize injection allocation and well locations in water-
flooded reservoirs. An integrated capacitance–resistance
model (ICRM) was presented that uses cumulative water
injection and cumulative total production instead of water
injection rate and total production rate while investigating
the advantages of a linear reservoir model over the non-
linear capacitance–resistance model [41,42].
The main advantage of CRM is that it requires very few
inputs as little as the production/injection history. It is based
on the main assumption that reservoir properties can be
drawn only from production/injection history of wells. Also
it requires that no significant changes in the field are observed
during the analysis period. The primary application area is
for the fields that are observing a secondary recovery period
with water or gas injection. Its applications for primary and
tertiary recovery periods are still being developed.
In these applications, the method enables to quantify
connectivity between injector–producer pairs and aquifer
strength, through history matching the production history
by adjusting model parameters. After the capacitance
models were introduced to understand interwell connec-
tivity [43], CRMs for dynamic evaluation of waterfloods
were presented [5]. Being a simple and user-friendly tool,
the methodology proved to be very powerful in field
applications, especially by quantifying interactions
between injector and producer wells [44]. CRMs for
three different control volumes are presented with semi-
analytical formulations, with each of them having differ-
ent level of complexities [3]:
1. One producer’s control volume,
2. An injector–producer pair’s control volume,
3. A field’s control volume.
In this study, a producer-based control volume is consid-
ered to focus on production wells and how they are con-
nected to different injection wells. Considering Ninumber
of injectors and Npnumber of producers, an in situ volu-
metric balance over the effective pore volume of the pro-
ducer is defined by the following differential equation [45]:
dqjðtÞ
dtþ1
sj
qjðtÞ¼1
sjX
Ni
k¼1
fijiiðtÞJj
dpwf ;j
dtð2Þ
where sjis the time constant for producer jand defined as a
function of total compressibility, ct, pore volume, Vp, and
productivity index, J, of the producer for its effective area:
sj¼ctVp
J

jð3Þ
and, fij is defined as the fraction injection rate of injector, i,
toward producer, j:
fij ¼qijðtÞ
iiðtÞð4Þ
The solution of this equation, including a variation in the
producer’sbottom-hole pressure (BHP), is the following [43]:
qjðtnÞ¼qjðt0Þetnt0
sj

!
zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{
Primary depletion
þet
sj
Zt
to
e
t
sj
1
sjX
Ni
i¼1
fijiiðnÞdt
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Injection input signal
et
sj
Zt
to
e
n
sj
Jj
dpwf;j
dndn
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
BHP variation
ð5Þ
I1
I2
I3
I4
I5
Injection
Rates
Liquid
Production
Rate
12 hidden
neurons
1 output
neuron
5 input
neurons
P
Fig. 6 Schematic of the
architecture of the neural
network constructed with 5
inputs, 1 output and 12 hidden
neurons
Neural Comput & Applic
123

which includes three components for representing the
production rate signal q(t) at any given time on the right-
hand side of the equation:
1. Primary depletion
2. Injection input signal
3. Variation in the producer’s bottom-hole pressure
(BHP)
By applying numerical integration, the integrals in the
above solution were evaluated by proposing two approaches
[45], which includes linear variation of BHP during the
consecutive time intervals, and either stepwise variation in
the injection rate (constant injection rate during a timestep),
or linearly varying injection rate during a timestep. In this
study, injection rates are kept constant during a timestep;
therefore, the former solution is utilized. For the case of
fixed injection rate of iðDtkÞ¼IðkÞ
i, and a linear BHP vari-
ation during time intervals Dtk,ðk¼1;2;...;nÞ,by
assuming a constant productivity index at any given time,
tn, total production rate of producer jcan be written as:
qjðtnÞ¼qjðt0Þetnt0
sj

!
þX
n
k¼1
etntk
sj
1eDtk
sj

!(
X
Ni
i¼1
fijIðkÞ
i
hi
Jjsj
DpðkÞ
wf ;j
Dtk
"#)
ð6Þ
where IðkÞ
iand DpðkÞ
wf ;jrepresent injection rate of injector, i,
and changes in the BHP of the producer, j, during time
interval, tk1to tk, respectively. The stepwise variation in
injection rates is consistent with the discrete nature of field
data that are typically reported in monthly averages [3]. If
the bottom-hole pressure for producing wells does not
change with time, the equation becomes:
qjðtnÞ¼qjðt0Þetnt0
sj

!
þX
n
k¼1
etntk
sj
1eDtk
sj

!
X
Ni
i¼1
fijIðkÞ
i
hi
"#()
ð7Þ
The history-matching process is achieved by inputting
observed production and injection rates for liquids and by
changing the unknown parameters:
•Initial production rates, qjðt0Þ,
•Time constant for each producer, j,sj,
•Fraction injection rate of injector, i, toward producer, j,
fij (i.e., the connectivity between injector, i, and
producer, j.
Through an optimization routine, these parameters are
changed until the average error between observed and
calculated production rates is minimized. This error is
defined as:
MSE ¼PNdata
n¼1ðqobs qestÞ2
Ndata ð8Þ
where Ndata is the number of observations (sample size),
qobs is the observed flow rates, and qest is the flow rate
estimated by the model. The routine is initialized by
assuming values for the time constant and initial flow rates
and calculating the initial fractional flow parameter using
the inverse-distance method [46]:
fij ¼
1
dij
PNpro
j¼1
1
dij
ð9Þ
where dij represents the distance between each injector and
producer. After this initialization, the trust-region reflective
search algorithm [47–51] is used to search for the combi-
nation of parameters that provides the lowest range of error
between the observed and estimated rates. After a pre-
specified stopping criteria are met, then the solution is
accepted as the optimum solution. An error tolerance of
1e-07 and a maximum number of function evaluations of
10,000 are used as the convergence criteria. Then, the
system parameters (e.g., fractional flow) are used to char-
acterize the reservoir.
3 Results and discussion
Proposed methods, namely ANNs and CRMs, are applied
to the case study presented in the previous section. The
primary objective was to quantify the connectivity between
injector/producer pairs using both methods. This is
achieved by a contribution parameter derived from the
trained neural network weights, w, in the case of ANNs and
by the fractional flow parameter, f, in the case of CRMs. In
the following subsections, results obtained using these two
methods are presented and discussed.
3.1 Artificial neural networks
The history-matching results are shown in Fig. 7. These
figures show that the training was successful in matching the
historical rates observed. Once the training is completed, it is
expected that the neural network would capture the dynamics
of the reservoir system from observed data. Since no pre-
sumed physical laws are introduced, we call such models
data-driven models. The model, during the training, captures
the physics of the process through the neural network train-
ing, which is an iterative procedure. After the training is
completed by satisfying certain stopping criteria, weights
remain in their optimum state, at which the neural network
can predict the output (production rate) within high levels of
accuracy. In that case, the optimum set of weights would
Neural Comput & Applic
123

represent the contribution of each injection well to the pro-
ducing well’s production, which can be used as a proxy to the
connectivity between injection and production wells. Higher
quantity of weights indicate stronger contribution and thus
stronger connectivity, and lower quantity weights indicate
weaker contributions and weaker connectivity. Contribution
factors calculated in this case are given in Table 2. After
calculating these contribution factors, a connectivity map is
drawn which represents the strength of connection between
each producing and injector well (Fig. 8). This figure is very
similar to the actual reservoir grid in which the permeabili-
ties are shown (Fig. 2). Since the neural network was able to
capture the high-permeability streaks in the reservoir system,
this gives us the confidence that artificial neural networks can
be used to characterize the connectivity of an oil reservoir
system which goes through water injection.
3.2 Capacitance–resistance models
The history-matching results are shown in Fig. 9. These
figures show that the training was successful in matching
the historical rates observed. Based on the CRM formula-
tion, the search process includes searching for the optimum
combination of initial production rates, fraction of flow and
0500 1000 1500 2000 2500 3000
1000
1500
2000
2500
3000
3500
4000
P1
Time, days
q, bbl/d
Actual
ANN
(a) Producer-1
0500 1000 1500 2000 2500 3000
100
150
200
250
300
P2
Time, days
q, bbl/d
Actual
ANN
(b) Producer-2
0500 1000 1500 2000 2500 3000
180
200
220
240
260
280
300
320
340
360
380
P3
Time, days
q, bbl/d
Actual
ANN
(c) Producer-3
0500 1000 1500 2000 2500 3000
1500
2000
2500
3000
3500
P4
Time, days
q, bbl/d
Actual
ANN
(d) Producer-4
Fig. 7 History matching of the producing wells during ANN training
Neural Comput & Applic
123

time constant parameters. The fraction of flow parameter is
used as a proxy to the interwell connectivity between each
injector and producer. The values obtained after the opti-
mization routine is completed are given in Table 3. Using
these values, similar to the ANN case, a connectivity map is
drawn which represents the strength of connection between
each producing and injector well (Fig. 10). As in the case
with ANN, CRM was able to capture both high-perme-
ability streaks in the reservoir model which are shown in
the reservoir grid (Fig. 2). Therefore, we also conclude that
in a similar fashion with artificial neural networks, capac-
itance–resistance models can also be used to characterize
the connectivity of an oil reservoir system which goes
through water injection.
3.3 Comparison of data-driven and reduced-physics
modeling approaches
One of the main objectives of this study is to provide a
comparison of these two methods for the practicing
reservoir engineer or asset team with respect to a number of
aspects. Table 4shows a comparison of these two methods
in different aspects, and a discussion of each aspect is
presented as the following:
Accuracy Prediction capabilities of each method can be
analyzed by comparing each method’s ability to inden-
tify high-connectivity zones in the reservoir. When all of
the 20 interwell connectivities between each of the five
injectors and four producers are ranked, upper 10 values
can be classified in the high-connectivity category, while
lower 10 values can be classified in the low-connectivity
category. One can approximate the connectivity values
for the numerical simulation model, by utilizing the
average permeability between two wells and normaliz-
ing the permeability by the distance between two wells
(if two wells are close to each other and have a high-
permeability streak between them, their connectivity
would be the highest). These values are given in Table 5.
After sorting the connectivity values in the numerical
model together with the predictions of data-driven and
the reduced-physics models, it is seen that the data-
driven and the reduced-physics models were able to
correctly estimate 80 and 70 % of each connectivity
category, respectively. These acceptable accuracy levels
indicate that both methods have similar prediction
capabilities, while the data-driven model has a slightly
better performance.
Data requirements Data-driven modeling approach (ar-
tificial neural networks) is better since it does not have
any limitation regarding the types of input/output data
set used. In this study, only injection and production
rates are used. This can be considered as a minimum
required set of data, since a set of signals would be
needed to be able to relate the connectivity between
wells, and production/injection rates are the most
commonly available data set that can be incorporated.
Meanwhile, the data included can be expanded, by
including well locations, known reservoir properties and/
or pressure data. Artificial neural networks are very
advantageous in terms of flexibility and can be modified
or restructured depending on the reservoir, wells, or any
other known aspects of the field that are studied. On the
other hand, capacitance–resistance models have a certain
formulation and require the data in that formulation
(production/injection rates, bottom-hole flowing pres-
sures, well locations). These are common type of data
available in any oil field; therefore, this does not create a
significant problem in implementing the method. How-
ever, it does not have the flexibility to incorporate more
data, if needed. In that case, the formulation needs to be
modified and redeveloped, which is not practical to do
for a quick application of the method but can be done in
Table 2 Contribution values obtained from weights, w, after ANN
training which is a proxy to the interwell connectivity
fP1 P2 P3 P4
I1 1.000 0.250 0.293 0.204
I2 0.117 0.177 0.521 0.165
I3 0.206 0.205 0.184 0.561
I4 0.239 0.903 0.373 0.438
I5 0.213 0.239 0.402 0.405
0500 1000 1500 2000 2500
−2200
−2000
−1800
−1600
−1400
−1200
−1000
−800
−600
−400
−200
X (m)
Y (m)
Connectivity
Producer
Injector
Fig. 8 Connectivity map of injectors/producers using the contribu-
tion values derived from the trained neural network
Neural Comput & Applic
123

the long term to have a suite of CRMs. This also can be
explained with the modeling approach utilized in each
method. ANNs are purely data-driven, and CRMs are
reduced-physics-driven as explained earlier.
Training algorithm ANNs offer a number of different
training algorithms, and all of them are purely data-
driven algorithms. The choice of the algorithm requires
subject-matter expertise in ANNs. ANNs do not limit the
use of any training algorithm and provide the flexibility
in choosing from the available options. With CRMs, the
0 500 1000 1500 2000 2500 3000
1000
1500
2000
2500
3000
3500
4000
P1
Time, days
q, bbl/d
Actual
CRM
(a) Producer-1
0 500 1000 1500 2000 2500 3000
100
150
200
250
300
P2
Time, days
q, bbl/d
Actual
CRM
(b) Producer-2
0 500 1000 1500 2000 2500 3000
180
200
220
240
260
280
300
320
340
360
380
P3
Time, days
q, bbl/d
Actual
CRM
(c) Producer-3
0500 1000 1500 2000 2500 3000
1500
2000
2500
3000
3500
P4
Time, days
q, bbl/d
Actual
CRM
(d) Producer-4
Fig. 9 History matching of the producing wells during CRM training
Table 3 Fraction of flow, f, values obtained after CRM training
which is a proxy to the interwell connectivity
fP1P2P3P4
I1 1.000 0.002 0.003 0.005
I2 0.548 0.003 0.137 0.303
I3 0.057 0.010 0.033 0.974
I4 0.117 0.178 0.001 0.662
I5 0.126 0.001 0.106 0.766
Neural Comput & Applic
123

training is basically an optimization process, in which
the optimization algorithm and parameters can be
modified. Therefore, CRMs are also can be trained with
a different number of optimization-related options.
Training speed Both methods are promising as the
training of both models took \30 s of CPU time. Even
for more complex reservoir systems, it is fair to expect
training times not more than a few minutes, which is a
reasonable amount of time for decision-making
purposes.
Expertise requirements Both methods are considered to
be moderate in terms of expertise requirements. In
developing a tool that utilizes either of the method, both
methods require significant expertise in certain subjects.
ANNs require to be familiar with the ANN theory and
terms; CRMs require the knowledge of CRM formula-
tion and reservoir engineering concepts. To train a
readily available tool using available data, ANNs still
require to be familiar with the related theory to
determine the neural network architecture and training
parameters. For training of CRMs, although not neces-
sary, knowledge of optimization algorithms and param-
eters would help to find the best optimization approach.
4 Summary and conclusions
In this study, two methods of different modeling approa-
ches, ANNs and CRMs, are studied to quantify the inter-
well connectivity between water-injecting wells and oil-
producing wells for a petroleum reservoir that has gone
through a reasonably long period of waterflooding. The
methods were tested on a synthetic reservoir case, in which
there are two high-permeability streaks (500 and 1000 md)
in a reservoir having a permeability of 5 md, elsewhere.
After calculating the connectivities from the model
parameters, two methods are compared with each other
considering various aspects from the practical points of
view.
Only liquid production and injection rates as well as
well locations are used as the data input to the two meth-
ods. Among these methods, artificial neural networks are
purely data-driven, with no assumption regarding the
governing laws of physics made. The other method,
capacitance–resistance models, can be defined as reduced-
physics-driven. The reason is that the physical laws
included in the formulation require a large number of
assumptions regarding the fluid flow. Both methods require
a training process, in which the developed model learns
from observed data, to capture the actual dynamics in the
reservoir. Therefore, success of both methods depends on
data quality and quantity. It was observed that for a known
reservoir scenario with 2 high-permeability streaks, both
methods were able to capture these streaks within reason-
able ranges of accuracy. Since both methods are practical
and easy to implement, a recommendation can be made
that both methods are applicable for such applications.
The key conclusions obtained from this study can be
summarized as follows:
0500 1000 1500 2000 2500
−2200
−2000
−1800
−1600
−1400
−1200
−1000
−800
−600
−400
−200
X (m)
Y (m)
Connectivity
Producer
Injector
Fig. 10 Connectivity map of injectors/producers using the contribu-
tion values derived from the trained capacitance–resistance model
Table 4 Characteristics of data-driven and reduced-physics modeling approaches in various aspects
Data-driven modeling (ANN) Reduced-physics modeling (CRM)
Accuracy (based on the specific example presented in this study) 80 % 70 %
Data requirements Flexible Fixed
Modeling approach Flexible data-driven model Physics-based, include assumptions
Training algorithm Flexible Flexible
Training speed Fast Fast
Expertise requirements Moderate Moderate
To develop the tool ANN background needed Reservoir eng. background needed
To train ANN background needed Optimization knowledge (for fine-tuning)
Neural Comput & Applic
123

1. Both ANNs and CRMs can be used to quickly estimate
the interwell connectivity between injection wells and
production wells in a reservoir. These tools have great
advantages over numerical modeling because of
requiring fewer inputs and being much more compu-
tationally efficient, while also providing the ability to
utilize available historical data and not being depen-
dent on geological uncertainties.
2. In the example presented here, 10 years of production/
injection history was sufficient for training and
achieving an accurate history matching. For more
complicated reservoir cases (with more number of
wells), higher duration of production and injection
histories might be needed.
3. Both methods are efficient in terms of CPU time
requirements with training times\30 s reported for the
example used in this study. Even for more complex
reservoir systems, this time is not expected to be
greater than a few minutes.
4. In terms of data requirements and modeling approach,
ANNs are more flexible than CRMs, since ANNs are
purely data-driven and do not require any presumed
functional relationship between process variables.
Instead, it derives the relationships through training
of observed data.
5. In both methods, some degree of knowledge is needed
for fine-tuning of results during the training process
which involves optimization of weights or the frac-
tional flow parameter.
Acknowledgments This work is supported by the Middle East
Technical University Northern Cyprus Campus (METU-NCC)—
Campus Research Fund; Project No. BAP-FEN-13-YG-2. The sup-
port is gratefully acknowledged.
Compliance with ethical standards
Conflict of interest We declare that there is no any conflict of
interest to report.
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