1
B033 CLOSED-LOOP WATERFLOODING
‡
†
K.M. OVERBEEK*, D.R. BROUWER* , G. NAEVDAL , C.P.J.W. VAN KRUIJSDIJK* , J.D. JANSEN*
*Delft University of Technology, Dept of Geotechnology,
PO box 5028, 2600 GA Delft, The Netherlands
†
Shell International Exploration and Production, Exploratory Research,
PO box 60, 2280 AB Rijswijk
‡
RF – Rogaland Research,
Thormøhlensgate 55, N-5008 Bergen, Norway
Abstract
Closed-loop waterflooding involves the combined use of model-based optimization techniques and
data assimilating techniques, which are used to maximize ultimate recovery or net-present value
(NPV) and to update the reservoir model during the producing life of the reservoir. In particular we
used Optimal Control Theory (OCT) in combination with an Ensemble Kalman Filter (EnKF) to
perform closed-loop water flooding. Testing the effectiveness of the closed-loop approach was done
on a synthetic reservoir model (SRM), mainly because of repeatability and time aspects. We used the
SRM to generate synthetic data such as production measurements and time-lapse seismics. As a first
step we used simple, two-dimensional models to demonstrate the feasibility of this workflow, i.e. the
use of closed-loop reservoir management techniques combined with a SRM. We used a commercial
reservoir simulator as the SRM, and an in-house Matlab reservoir model for the OCT and EnKF. The
EnKF uses an ensemble of model representations to update the permeability field and the states
(pressures and saturations) of the reservoir from ‘measured’ pressures from the SRM. OCT is then
applied, using the most recently estimated permeability field, to optimize injection and production
rates in a fixed configuration of wells. We obtained an increase in NPV of 29% after injecting one
pore volume, as a result of a higher oil production (increase of 20%) and a lower water production
(decrease of 34%). The estimated permeability field revealed the main heterogeneities of the SRM.
The use of the SRM allowed for rapid testing of different varieties of OCT and EnKF techniques.
1
Introduction
In order to establish the most favorable development and management of a hydrocarbon reservoir, we
usually use a numerical reservoir model. These models are a crude approximation of reality and
frequently contain a large amount of uncertainties. History matching is a way to adapt the parameters
of the model, but it is usually done on a campaign basis, for example, once every five years. With data
assimilation techniques we can shift from a campaign-based ad-hoc history matching procedure to a
near-continuous systematic updating of the reservoir parameters based on data from different sources
during the producing life of the reservoir. In our study we have only used pressure and flowrate
measurements as data, but the scope for expansion in data sources is wide, e.g. the use of time-lapse
seismics or passive seismics.
In a numerical model for water flooding we can systematically optimize reservoir production
strategies by using smart field technology, in particular when we use model- based optimization
techniques to steer the waterfront in the reservoir and in this way optimize the ultimate recovery or
net-present value (NPV). In this study we will combine model-based optimization techniques and data
assimilating techniques. In particular we will use Optimal Control Theory (OCT) as described by
Brouwer et al. [1], in combination with an Ensemble Kalman Filter (EnKF) to perform closed-loop
water flooding. The EnKF uses an ensemble of model representations to update the reservoir states
(pressures and saturations) and parameters (permeabilities), and the associated model uncertainty.
With the most recent updated states and parameters, we will then apply OCT to optimize injection and
production controls in a fixed configuration of wells. The pressure measurements from downhole
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
2
gauges are used as data for the EnKF to estimate the states and the permeability field, the OCT
assigns the control settings to the surface chokes. Earlier studies on estimating permeability fields
with EnKF by Naevdal et al. [2] and maximizing recovery with OCT by Brouwer et al. [1] have
shown promising results, and a first step of combining the two techniques was described by Brouwer
et al. [3].
Testing the effectiveness of closed-loop reservoir management was done on a synthetic reservoir
model (SRM), an environment that represents the real asset, because of repeatability and time aspects.
In contrast to a related study by Brouwer et al. [3], which used the same (low-order) model for the
SRM as the reservoir model, we are using a high-order, commercial reservoir model for the SRM and
a low-order, in-house Matlab reservoir model with which we perform the EnKF and the OCT.
2
Role of the synthetic reservoir model in closed-loop reservoir management
The SRM is a numerical model that, as closely as feasible, represents the real reservoir, and has all the
relevant parameters and characteristics of the real reservoir. The benefits of using a SRM instead of a
real reservoir are mainly repeatability and time aspects. Of course, the use of a numerical model to
test another numerical model (the reservoir simulator) can be questioned. The SRM is a numerical
approximation of the truth and not reality. If the number of states in the SRM is large with respect to
the number of states of the reservoir simulator, which is used for optimization, we feel that this
approach can be justified as a first step in this research. Similar SRM approaches are also used in e.g.
Saputelli et al. [4].
Just as for a reservoir in reality, the parameters of the SRM before the start of production are known
to a limited extent to the user of the reservoir simulator that is used for the optimization. When we
start producing, we obtain production data from the wells. The wells in the SRM are controlled by
surface chokes and are monitored by downhole gauges and multiphase flow meters. The flow and
pressure measurements are the main data sources for data assimilation in this project.
The SRM, after running for a certain period, gives us ‘measured’ output, which is used with the EnKF
to obtain updated states of the reservoir model. These updated states are used in the OCT to control
the rates and optimize the NPV of the reservoir simulator and thus the SRM. The closed-loop
procedure can be illustrated by figure 1 and 2.
1.
asset
Synthetic Virtual
reservoir
model
Noise
Input
Controllable
input
Output
Outpu
High-order
System
reservoir model
Output
Noise
3.
Optimization
Control
algorithms
Sensor
models
Low-order
reservoir model
4.
Measured
output
Parameter
updating
5.
Fig. 1– Schematic overview of closed-loop
reservoir management.
3
2.
Run the simulation for the synthetic reservoir model for
a certain period. Obtain measurement data.
Create an ensemble of model representations, and run
the forward simulation until the time of measurement.
Perform the EnKF with the measurement data and the
ensembles. Obtain ensemble of analyzed states.
(Parameter updating in figure 1).
Perform optimal control with most recent analyzed
states. Obtain new controls for injection and production
wells. (Optimization in figure 1).
Constrain the SRM with the new controls, run it for
another certain period, obtain new measurement data
and continue with step 3.
Fig. 2 – Closed-loop procedure.
Ensemble Kalman filter
We use an EnKF for continuous updating of the reservoir model by using measured data. The EnKF is
a variation of the original Kalman filter, a set of mathematical equations that provides an efficient
computational (recursive) solution of the least-squares method [5]. The EnKF computations are based
on an ensemble of realizations of the reservoir model. The filter consists of sequentially running a
forecast step, followed by an analysis step. During the forecast step the reservoir simulator is run for
each of the model realizations up to the time where new measurements become available. With these
measurements, all realizations in the model are updated by combining the new real measurements
with forecasts from the ensemble, which happens during the analysis step. In the Kalman filter the
CLOSED-LOOP WATERFLOODING
3
ensemble members are propagated in time by the non-linear dynamic system eq. (5) which represents
the reservoir model and which is defined in terms of the state vector x containing oil pressures and
water saturations. In our implementation of the Kalman filter we also treat the permabilities as
unknowns, which leads to the use of an extented state vector x , containing oil pressures, water
saturations and permeabilities. The extended state vector after running the N th forecast step is denoted
f
a
x N 1 , after the analysis step x N . To initialize the filter we generate a random initial
ensemble, with a specified mean and covariance matrix. The mean value of the initial ensemble
should preferably be a good estimate of the initial true state, and therefore the mean is based on well
data. The members of the ensemble are drawn randomly assuming a Gaussian distribution, with a predefined spatial correlation length. A more detailed explanation of the generation of the initial
ensemble can be found in Naevdal et al. [2]. The analyzed state can be calculated as the mean of the
analyzed ensemble, because the true state can be approximated by the mean of the ensemble
1 N ª
a
f
obs
f
(1)
x N 1
¦ x N 1i K N 1 y N 1 Cx N 1i º¼ ,
N i 1¬
obs
where N is the sample size of the ensemble, y N 1i
is the observed measurement including
measurement noise, C is a matrix that selects from the extended state vector only the pressures in the
wells and K is the Kalman gain
f
f
K N 1 P N 1 CT CP N 1 CT R N 1
1
.
(2)
f
Here, P N 1 is the forecasted error covariance matrix for the states of the system and R N 1 is
the covariance matrix for the measurement errors. The error covariance matrix that we need for
computing the Kalman gain can be calculated as
P N 1
where L N 1
f
f
L N 1
f
T
L N 1 f
,
(3)
ª x N 1 f x N 1 f , ! , x N 1 f x N 1 f º and
N
1
¼
N 1 ¬
1
a
x
1 N ¦ xi .
N i1
f
The analyzed covariance error matrix P N 1 can be computed similarly as P N 1 . For a more
detailed description of EnKF see Evensen [6] and Naevdal et al. [2].
4
Optimal control theory
This short introduction to OCT is largely based on Brouwer et al. [1]. Optimal control is a gradientbased optimization technique that allows us to find those values of the input variables u that minimize
or maximize a certain objective function $ . The gradients are obtained with the aid of an adjoint or
co-state equation in terms of Lagrange multipliers which represent the objective functions’s
sensitivities to changes in the state variables. In our study, the objective function $ represents the
net-present value (NPV)
K 1
K 1 ª N prod r q
º
o o k n rw qw k n
k ¦« ¦
$¦ $ (4)
» 't k ,
a
k 0
k 0« n 1
1 b /100 ¬
¼»
where r is the price per unit volume, q is the flow rate, b is the annual interest rate, a is the number of
years since the start of production, ǻt is the time interval corresponding to timestep k, K is the total
number of time steps, and Nprod is the number of production wells. To derive the influence of changes
in u on the magnitude of $ , we need to work out the changes in x first, because $ is a function of
x, while x is a function of the input variable u. The problem we have stated here can be interpreted as
a constrained optimization problem, where the constraint is the dynamic system
g x k 1 , x k , u k 0 ,
(5)
which represents the reservoir model and specifies the relation between x and u. In short, the solution
to the constrained optimization problem is as follows. First, the dynamic system (eq. (5)) is included
as a constraint with the aid of a vector of Lagrange multipliers Ȝ in the objective function $ . We
derive the modified objective function $ :
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
4
K 1
$
ª$ x k , u k Ȝ k 1T g x k 1 , x k , u k º
¦
¬
¼
k 0
K 1
¦ k ,
(6)
k 0
is an auxiliary variable known as the Hamiltonian. It is then possible to express the effect
where
of a small change in u on the value of
(and therefore the value of $ ) provided the Lagrange
multipliers obey a system of ‘adjoint’ differential equations which can be derived through taking the
first variation of eq. (6) and reorganising the results leading to
1
ª
w$ k º § wg k 1 ·
T
T
T wg k (7)
Ȝ k Ȝ
k
1
¸ , with final condition Ȝ K 0 .
« »¨
wx k ¬«
wx k ¼» ©¨ wx k ¹¸
After a forward integration of the dynamic system (5) we can integrate equations (7) backward in
time, and when all the Lagrange multipliers have been calculated, we can calculate G $
T
§ w k ·
G uk ¦
¨¨
¸¸
k 0 © wu k ¹
K 1
G$
T
ª w$ k T wg k º
Ȝ k 1
«
» G uk .
¦
wu k ¼»
k 0¬
« wu k K 1
(8)
Eq. (8) shows the first order change in the objective function resulting from a change in the input
vector u. It can be shown that in a constrained optimum
reaches a maximum, and we used the
steepest ascent method to find the corresponding (local) optimal value of u:
w k ,
(9)
u k new u k old H
wu k where H is a factor determining the optimal change in u along the search direction w
Brouwer et al. [1].
5
5.1
wu ; see
Examples and results
Project setup
To represent the SRM, we`used a two-dimensional, horizontal, two-phase, high-order reservoir
model, implemented in commercial reservoir simulation software. The reservoir parameters have been
displayed in table 1.
Porosity
Permeability
Oil density
Water density
Initial oil pressure
Initial water saturation
Connate water saturation
Residual oil saturation
ij
k
ȡo
ȡw
po,ini
Sw,ini
Swc
Sor
[-]
[mD]
[kg/m3]
[kg/m3]
[MPa]
[-]
[-]
[-]
0.2
200 – 10000
1000
1000
80
0.1
0.1
0.1
Table 1 – Reservoir parameters.
We assumed the compressibility of the fluids to be constant at a value of 10-10 1/Pa. The formation
volume factors were also kept constant at a value of 1. The viscosity of the water and the oil were set
to 10-3 Pa s. The permeability field, depicted in figure 3, ranged from 200 mD to 10000 mD. There
were 10 vertical injection and 10 vertical production wells, located at opposite sides at the borders of
the reservoir. The wells were operated under voidage replacement at constant field rates of 2600
m3/day, and we assumed that they were rate-controlled by surface chokes, and monitored by
downhole pressure gauges and surface multiphase flow meters. The accuracy of well-callibrated
gauges according to various manufacturer is around 0.01 MPa (see table 2). However we used a value
of 0.5 MPa to account for additional operational error sources. The accuracy of the flow meters was
not quantified explicitly, but was accounted for in the model error. The SRM was 1100 by 1100
meters wide, with a depth of 10 meters, discretized using 110 by 110 by 1 grid blocks such that the
amount of states (pressures and saturations) in the SRM was 24200.
CLOSED-LOOP WATERFLOODING
5
10000
Injection well
Production well
200
Fig. 3 – Top-view of the SRM.
Permeability field [200 – 10000 mD].
Fig. 4 – Top-view of the low-order, in-house
Matlab reservoir model.
Manufacturer
Range
Accuracy
0 – 68.9 MPa
0.1 % full scale
0.07 – 140 MPa
21 kPa full scale
10.3 – 138 MPa
0.05 % full scale
0 – 138 MPa
0.02 % full scale
Tubel Technologies [7]
Roxar [8]
Senz-it [9]
Calsan [10]
Product
In-Line Real Time Downhole
Wireless Gauge
Roxar Quartz Pressure and
Temperature Gauges (RQPG)
Real Time Pressure
Temperature Module (RTPT)
Wolvirine Quartz Gauge
Table 2 – Accuracy of various downhole pressure gauges.
The low-order, in-house Matlab reservoir model, represented by eq. (5) that we used for EnKF and
OCT was a two-dimensional, horizontal, two phase reservoir model. We assumed all the reservoir
properties, except the permeability, to be known. The PVT data and relative permeabilities were also
known. The model had 10 injectors and 10 producers at opposite sides, as shown in figure 4. The rates
in the wells were set equal to the rates in the corresponding SRM wells, i.e. to a total field rate of
2600 m3/day. Because the wells were assumed to be rate-constrained, no well models were used. The
model had a dimension of 1100 by 1100 by 10 meters, discretized in 10 by 10 by 1 grid blocks. The
ensemble for the EnKF consisted of 100 members. Because we updated the permeability field of the
reservoir model with the EnKF, we had to use an initial ensemble of (ln-)permeability fields. These
fields were based on the permeability values from the wells in the SRM and were randomly generated
using a pre-defined spatial correlation length of 2 grid blocks with a standard deviation of 0.1 grid
blocks. The standard deviation for the generation of the ln-permeability field was 0.5 ln-mD.
5.2
Results
We applied the filter to take into account the information gained from 20 measurements at each
assimilation, one for each production well and one for each injection well. The measurements were
the bottom hole pressures from the 20 wells and were generated after 2, 4, 7, 9, 11, 23, 46, 69, 92,
116, 231, 463 and 694 days. We finished the simulation after 750 days. The time span of producing
for 750 days was based on the reference case, in which we flooded the SRM with one pore volume of
water assuming constant pressure in the wells. (In the reference case, we choose the rates in the wells
depending on the permeability in the neighbouring gridblocks to achieve an almost constant pressure
situation. The corresponding rates were equal to the rate at t 0 days depicted in figure 5d.) The
water and oil saturations resulting from the conventional flooding strategy are shown in figure 5a. In
figure 5b we present the outcome of the optimized case after 0, 46, 116, 463 and 750 days of
optimized flooding of the SRM. In figure 5c the estimated permeability field of the reservoir model is
shown. The estimated permeability ranges from 20 to 7000 mD. The initial permeability field (figure
5c at t = 0 days) ranges from 640 to 800 mD. Figure 5d shows the injection rates for the SRM at the
different measurement times. The initial injection for the optimized case is also based on a constant
pressure in the wells, like in the reference case. In figure 6a and 6b the oil and water rates and the
cumulative production of the reference case versus the optimized case are shown.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
6
a. Saturations for
conventional water
flooding
(white = oil, black
= water).
b. Saturations for
optimized water
flooding with
estimated
permeability field
(white = oil, black
= water).
c. Estimated
permeability field
[20 (white) – 7000
(black) mD].
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
8
8
8
8
9
9
9
9
9
10
10
10
10
10
2
3
4
5
6
7
8
9
10
d. Injection rates. The
total injection rate
is constant and
2600 m3/day.
8
Fig. 5 – Saturations, permeability field and injection rates at times t = 0, 46, 116, 463 and 750 days.
3000
1600000
1400000
2500
1200000
cumulative production [m3]
rate [m3/day]
2000
1500
1000
1000000
800000
600000
400000
opt oil
opt wat
ref oil
ref wat
500
opt oil
opt wat
ref oil
ref wat
200000
0
0
0
100
200
300
400
500
600
700
800
0
100
200
time [days]
300
400
500
600
700
800
time [days]
Fig. 6a – Oil and water rates as a function of time.
Fig. 6b – Cumulative oil and water production
as a function of time.
The NPV (eq. 4) was calculated with a revenue of produced oil, ro, of $80 per m3, the cost of the
produced water, rw, was $20 per m3. The costs of water injection were disregarded. We used an annual
interest rate of zero, b 0 , thus the results are non-discounted NPVs. The economic results are
shown in table 3.
CLOSED-LOOP WATERFLOODING
7
Results after 750 days
reference case
NPV
cum oil
cum waterrecovery tot PV
[$]
[m3]
[-]
[%]
[-]
8.44E+07 1.23E+06 7.16E+05 64%
1.01
Normalised results after 750 days
reference case
NPV
cum oil
cum waterrecovery tot PV
[-]
[-]
[-]
[-]
[-]
1.00
1.00
1.00
1.00
1.00
optimized case
NPV
cum oil
cum waterrecovery tot PV
[$]
[m3]
[-]
[%]
[-]
1.09E+08 1.48E+06 4.71E+05 76%
1.01
optimized case
NPV
cum oil
[-]
[-]
1.29
1.20
cum waterrecovery tot PV
[-]
[-]
[-]
0.66
1.20
1.00
Table 3 – Results after 750 days for the reference case and the optimized case.
Conclusions
x
x
x
x
x
x
Combining model-based optimization and data assimilation to perform closed-loop water flooding
gave promising results for both optimizing NPV and estimating heterogeneities in the reservoir.
The dynamic optimization led to a distinct increase in NPV compared to the reference case. This
was due to a significant increase in oil production and an even more significant decrease in water
production.
With EnKF we were able to tune the state variables and the permeability of the reservoir model.
The main heterogeneities in the estimated permeability fields, corresponding to the
heterogeneities in the SRM, were rapidly recovered.
It is important that the generation of the initial permeability fields correctly reflects the
uncertainty. It is not straightforward to obtain an adequate initial ensemble, but it is essential for
realistic results. Therefore, more research has to be done on this part in the future.
Accounting for modeling errors is crucial, especially since we are working with a reservoir model
to influence the flow in a real asset or synthetic reservoir model. Adding modeling errors
influences the uncertainty in the states. The nature and magnitude of the model errors require
further investigation.
The use of the SRM allowed for rapid testing of different varieties of OCT and EnKF techniques.
Nomenclature
a
b
C
g
k
k
N
K
K
$
L
p
P
9th
= time, t, a
= annual interest rate
= measurement matrix
= non-linear vector function
= Hamiltonian, M, $
= simulation time step
= permeability, L2, m2, mD
= measurement time step
= final time step
= Kalman gain matrix
= objective function (cost function),
M, $
= left factor of covariance matrix P
= pressure, m/(L t2), Pa, bar
= covariance matrix of model
uncertainty
= flow rate, 1/t, L3/t, 1/s, m3/day
= price per unit volume M/L3, $/m3
= covariance matrix of measurement
error
= saturation
= input (control) vector
= state vector
= measurement vector
= time interval, t, s
q
r
R
S
u
x
y
ǻt
H
ȡ
ij
Ȝ
0
= weighting factor
= density, m/L3, kg/m3
= porosity
= vector of Lagrange multipliers
= null vector
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
8
Subscripts
Superscripts
i
ini
inj
n
o
or
prod
w
wc
a
f
obs
t
= index (ensemble members)
= initial
= injector
= index (number of wells)
= oil
= residual oil
= producer
= water
= connate water
= analyzed (aposteriori)
= forecasted (apriori)
= observed
= true
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Brouwer, D.R. and Jansen, J.D.: “Dynamic optimization of water flooding with smart wells
using optimal control theory”, paper SPE 78278 presented at the SPE European Petroleum
Conference, Aberdeen, U.K., 29-31 October 2002.
Naevdal, G., Johnsen, L.M., Aanonsen, S.I., Vefring, E.H.: “Reservoir monitoring and
continuous model updating using Ensemble Kalman filter”, paper SPE 84372 presented at the
SPE Annual Technical Conference and Exhibition, Denver, USA, 5-8 October 2003.
Brouwer, D.R., Naevdal, G., Jansen, J.D., Vefring, E.H., van Kruijsdijk, C.P.J.W.: “Improved
reservoir management through optimal control and continuous model updating”, paper SPE
90149 accepted for presentation at the SPE Annual Technical Conference and Exhibition in
Houston, USA, 26-29 September 2004.
Saputelli, L., Nikolaou, M., Economides, M.J.: “Self-learning reservoir management,” paper
SPE 84064 presented at the SPE Annual Technical Conference and Exhibition, Denver, USA,
5-8 October 2003.
Welch, G. and Bishop, G.: “An introduction to the Kalman Filter”, UNC-Chapel Hill, TR 95041, March 2002.
Evensen, G.: “Sequential data assimilation with a nonlinear quasi-geostrophic model using
Monte Carlo methods to forecast error statistics”, J. Geophys. Res., 99: 10, 143-10, 162,
1994.
Tubel Technologisch, Inc.: Wireless gauges, Inline Real Time Downhole Wireless Gauge.
http://www.tubeltechnologies.com
Roxar: Production Management, Proven Permanent Downhole Monitoring System (PDMS),
Roxar Quartz Pressure and Temperature Gauges (RQPG). http://www.roxar.com.
Senz-it: Real Time Pressure Temperature Module (RTPT). http://www.senz-it.com.
Calsan: Products, Quartz Downhole Gauges, Wolvirine. http://www.calsan.net.
CLOSED-LOOP WATERFLOODING