1
B009 REDUCED-ORDER OPTIMAL CONTROL OF
WATERFLOODING USING POD
*
*
J. VAN DOREN , R. MARKOVINOVIû AND 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, The Netherlands
Abstract
Model-based optimal control of water flooding generally involves multiple reservoir simulations which
makes it into a time-consuming process. Furthermore, if the optimization is combined with inversion, i.e.
with updating of the reservoir model using production data, some form of regularization is required to
cope with the ill-posedness of the inversion problem. A potential way to address these issues is through
the use of Proper Orthogonal Decomposition (POD). POD is a model reduction technique to generate
low-order models using “snapshots” from a forward simulation with the original high-order model. In this
work we addressed the scope to speed up optimization of water flooding a heterogeneous reservoir with
multiple injectors and producers. We used an adjoint-based optimal control algorithm which requires
multiple passes of forward simulation of the reservoir model and backward simulation of an adjoint
system of equations. We developed a nested approach in which POD was first used to reduce the state
space dimensions of both the forward model and the adjoint system. After obtaining an optimized
injection and production strategy using the reduced-order system, we verified the results using the
original, high-order model. If necessary we repeated the optimization cycle using new reduced-order
systems based on snapshots from the verification run. We tested the algorithm on a reservoir model with
4050 states (2025 pressures, 2025 saturations) and an adjoint model of 4050 states (Lagrange multipliers).
We obtained reduced-order models with 20 to 100 states only, which produced almost identical optimized
flooding strategies as compared to those obtained using the high-order models. The resulting reduction in
computing time was between 30% and 60%.
Introduction
Modeling water flooding in a hydrocarbon reservoir is frequently used to determine how to develop and
manage the reservoir. One creates a model of the oil reservoir, based on geological, petrophysical and
seismic data, introduces wells at the desired locations and numerically simulates the response of the
reservoir to the injected water. With increasing computer capacities reservoir models have become more
complex and consist of an increasing number of variables (typically in the order of 104 - 106). Maximizing
hydrocarbon production and minimizing water production of a reservoir can be done in a smart field with
optimal control theory (OCT) [14]. Calculating the optimal valve settings using OCT requires several
passes of forward simulation of the reservoir model and backward simulation of an adjoint system of
equations. The time needed to calculate optimized controls increases with the number of grid blocks and
the complexity of the reservoir model. Reduced-order modelling and reduced-order control may provide
an alternative to this [57].
Parameters and variables of the model can be updated with history matching. In this process output from
the real reservoir is compared to output from the model. With an objective function the model parameters
are updated and the discrepancy between the outputs is minimized. A problem with history matching,
however, is the ill-posedness, even if the correct model is assumed. There are usually more parameters
and variables that can be tuned in order to reach the same result. Normally one tries to overcome this
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
2
problem by constraining the solution space for the model parameters through the addition of
regularization terms to the objective function. Reduced-order models may also provide an alternative to
this process. The present work concentrates, however, on the aspect of using reduced-order models for
increased computational efficiency. We will use proper orthogonal decomposition (POD) for reducedorder optimal control of water flooding. POD is a frequently used tool for model reduction but only
recently it has also been used for control applications [812]. We will design and implement a nested
algorithm, where in the inner iterative loop of the algorithm we use a truncated basis of POD functions to
calculate optimized injection and production rates. After convergence in this loop we will simulate in the
outer loop the original, high-order model with the optimized rates and subsequently adapt the basis and
the truncation of the POD functions. They will be used in the next inner loop to calculate new optimized
injection and production rates. To test the algorithm we will use a 2-dimensional, 2-phase reservoir model
and compare full-order optimal control with reduced-order optimal control.
High-order reservoir model
To generate a reduced-order model with POD we first need to run a full-order simulation and produce
snapshots. For the full-order model we use a 2-dimensional, 2-phase reservoir simulator, developed inhouse and written in MATLAB [13]. Spatial discretization of the original partial differential equations
yields the following state-space ordinary differential equation in continuous time t:
x t f x t , u t A c x x t B c x u t ,
Ac
1
ª¬ V x t Ȍ x t º¼ T x t , B c
(1)
1
Ȍ x t ,
(2)
where x is the state vector containing oil pressures po and water saturations Sw for each grid block, V is a
diagonal mass matrix with entries that are a function of grid block volume and fluid densities, Ȍ is a
block diagonal matrix with entries that are primarily a function of compressibility and porosity, T is a
block penta-diagonal matrix containing the transmissibilities for oil and water and u is the input vector
containing oil and water rates qo and qw [14, 15]. We chose not to use a well model, but to use rateconstrained injectors and producers. Ac and Bc are actually not explicit functions of state but functions of
state-dependent parameters. The closure equations for each grid block are So + Sw = 1 and po – pw = pcow,
where So is the oil saturation, pw the water pressure and pcow(Sw) the capillary pressure. The initial
conditions are specified as x 0 x 0 . Semi-implicit Euler discretization by treating the state and input
vectors implicitly, but the matrix coefficients explicitly can be written as,
x (k + 1) x (k )
%t
= A c (x (k )) x (k + 1) + B c (x (k )) u (k + 1) ,
(3)
resulting in
x (k + 1) = f d (x (k ), u (k + 1)) = A d (x (k )) x (k ) + B d ( x (k )) u (k + 1) ,
(4)
where k is discrete time and where the time step-dependent matrices Ad and Bd have now been defined as
1
A d (x (k )) = ¡ I %tA c (x (k ))¯° , B d (x (k )) = %tA d (x (k )) B c (x (k )) .
(5)
¢
±
In our numerical implementation, the matrix inverses in Eq. (5) are not computed, since it is more
efficient to solve the equivalent system of linear equations
I %tA (x (k ))¯ x (k + 1) = x (k ) + %tB (x (k )) u (k + 1)
c
c
¢¡
±°
for the unknown state x(k+1).
(6)
3
Proper orthogonal decomposition
POD has been developed in fluid mechanics and is a mathematical technique to describe ‘coherent
structures’ which represent low-order dynamics in turbulent flow8,9. An approximation of the system
dynamics is obtained by projecting the original n-dimensional state space onto a l-dimensional subspace,
where l << n. First, during simulation of an n-dimensional discrete-time model we record a total of ț
snapshots for the oil pressure state xp and the water saturation state xs. In our case the dimension n will be
equal to twice the number of grid blocks. If we have a reservoir model with one horizontal layer of m u n
grid blocks, the vectors xp and xs will have mn elements each, and therefore the full state vector x will
have length n = 2mn . We keep the pressure and the saturation states segregated because they correspond
to different physical processes and will consequently generate different dominant structures. It is also
useful because we can better control which pressure and saturation POD basis functions should be
truncated respectively. In the following we will use the subscript r { p, s} to indicate the oil pressure po
or the water saturation Sw. After subtracting the means xr
N
1 N ¦ i 1 xr i from the snapshots, we
construct two data matrices:
Xcr = ª¬ xcr 1 , xcr 2 , ..., xcr N º¼
ª¬ x r 1 - xr , x r 2 - xr , ..., x r N - xr º¼ .
(7)
The goal of POD is, given the data matrices, to find orthogonal projections xcr ĭ r z r rr , where ) is a
transformation matrix, zr are reduced state vectors of length lr and rr are residuals, such that the residuals
minimized. This can be achieved through solving for the eigenvectors of Xcr Xcr T , which is equivalent to
determining the singular value decomposition (SVD) of Xcr :
Xcr
Pr Ȉ r QTr ,
(8)
where Pr is an orthogonal matrix of size mn u mn , Qr is an orthogonal matrix of size N u N , and Ȉr is a
matrix of size mn u N given by
Ȉr
0
ªV r ,1
« 0 V
r ,2
«
« #
#
«
0
« 0
« #
#
«
0
0
¬«
"
"
%
"
%
"
0 º
0 »»
# »
» ,
V r ,N »
# »
»
0 ¼»
(9)
where V r ,1 t V r ,2 t ... t V r ,N t 0 are called the singular values of Xcr . The total amount of relative energy
present in the snapshots can be expressed as Er ,tot ¦ N V r2,i . The number of POD basis functions lr we
i 1
want to keep, in order to project the high-order system equations on to them, is then determined by
lr
2
¦ V r ,i
Er
i 1
Er ,tot
dD ,
(10)
where D denotes the fraction of relative energy we want to be captured. The transformation matrix ĭ r is
now defined as the first lr columns of the matrix Pr, where lr << nm . We combine ĭ p and ĭ s to a single
transformation matrix ĭ with a total of l = lp + ls columns and n rows.
Reduced-order reservoir model
After determining transformation matrix ĭ the normalized state vector xc x x of the original problem
is projected on the axes in the reduced space,
z k 1 ĭT f ĭz k , u k 1 .
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
4
As described in reference [6] it may appear at first sight as if the reduced-order model does not lead to a
reduction in simulation time. If we compute z(k+1) explicitly we even obtain a slight increase in
simulation time, because every time step we have to perform two additional transformations (from z to x
and back) because we need the original state vector x to compute the functions f. However, if we consider
implicit or semi-implicit computation of z(k+1), and if f(x) can be written in the form of matrices with
coefficients that are functions of x, a considerable efficiency gain may be achieved. Applying the
transformation ĭ to Eq. (6), we obtain
I %tA (ĭz (k ))¯ ĭz (k + 1) = ĭz (k ) + %tB (ĭz (k )) u (k + 1) .
c
c
¢¡
±°
In our implementation we successfully used
ĭT ¡ I %tA c (ĭz (k ))°¯ ĭ z (k + 1) = z (k ) + ĭT %tB c (ĭz (k )) u (k + 1) ,
¢
± (12)
(13)
lxl
which is obtained from equation (12) by premultiplying with )T. The number of state variables is thus
reduced from originally n 2mn to l. The matrix dimensions are consequently reduced from n u n to
l u l. The simulation time of the reduced-order model using semi-implicit discretization is decreased
because we have to solve l equations instead of n equations in the full-order model, where l << n. For
fully-implicit simulation where more than one systems of equations have to be solved during every
timestep the decrease in simulation time is expected to be even higher. Unfortunately, the original pentadiagonal matrix structure (hepta-diagonal for 3-dimensional systems) is changed to a full matrix, because
we multiply the penta-diagonal matrix with a full matrix ĭ from the left side and from the right side.
This counteracts the computational advantage obtained by reducing the size of the state vector. In our
numerical implementation we solved Eq. (13) with LU-decomposition.
It is known from literature that when we simulate reduced-order reservoir models with the same controls
as the original full-order models we can obtain almost identical states, as long as a sufficient fraction of
the relative energy of the full-order model is preserved. However, if we alter the controls, and therefore
the structures of the states, we experience difficulties reproducing the states of the full-order model with
the reduced-order model. Apparently we cannot reproduce structures that are not represented in the data,
and it is difficult to extrapolate to flows of a different nature [12]. We will use this information when
designing the optimization reduced-order algorithm.
Reduced-order optimal control
To compute the optimized injection and production rates for an oil reservoir, optimal control theory
(OCT) can be applied [14]. This is a gradient-based optimization technique, where the gradients are
obtained with the aid of an adjoint equation in terms of Lagrange multipliers O. The multipliers represent
the objective functions’s sensitivities to changes in the state variables and originate from adding the
dynamic system as a constraint to the objective function. We use an objective function
$ k $ x k , u k which represents NPV. Following the derivation in reference [1], the adjoint
equation can be written as in discrete time as
T § wg k 1 ·
Ȝ k ¨¨
¸¸
© wx k ¹
where g x k 1 , x k , u k 1 0
ª
w$ k w$ k 1 º
T wg k « Ȝ k 1
» ,
wx k wx k wx k ¼»
¬«
(14)
is the implicit form of system equations (4). For our
implementation, instead of the full-order model we added the reduced-order model as a constraint to the
objective function $ k with the aid of a set of low-order Lagrange multipliers µ:
K 1
$ red
¦ ª¬$ ĭz k , u k µ k 1
k 0
T
ĭT g ĭz k 1 , ĭz k , u k º .
¼
(15)
5
Taking the first variation of Eq. 15, and reworking the results, we obtain a reduced-order equation in
terms of reduced-order Lagrange multipliers:
ª
º
«
»
·
§
·
w
w
k
k
g
$
T §
(16)
« µ k 1 ¨¨ ĭT
ĭ ¸¸ ¨¨ ĭT
µ k ¸¸ » ,
wx k ¹ ©
wx k ¹ »
«
©
»
«¬
lxl
lxl
1 xl
¼
T
Starting from the final condition µ(K) = 0 it can be integrated backward in time. Because the derivatives
in Eq. (16) consist of state-dependent parameters we first calculate the full-order derivatives. They are
then transformed and reduced by projecting them on the axes of the low-order model. After calculating µ
every timestep we can calculate:
T
§ T wg k 1 ·
ĭ ¸¸
¨¨ ĭ
wx k ©
¹
w
k wu k red
w$ k T
ĭT red
µ k 1
wu k lx1
­° T wg k ½°
®ĭ
¾.
wu k ¿°
¯°
(17)
lx1
We
u new
compute improved controls using a steepest ascend method according to
u old H ªw
¬ red k wu k º¼ where İ is a weighting factor determined according to reference [1].
The computational advantage of using reduced-order models in OCT is that the system of equations
involves only l unknowns, whereas the original system involved n 2mn unknowns. This decreases the
simulation time considerably, especially for large systems where l << n. Unfortunately, the original block
wg k 1
wg k and the block-diagonal matrix structure of
are
penta-diagonal matrix structure of
wx k wx k changed to full matrices, because we multiply them with a full matrices ĭ and ĭT . This counteracts the
computational advantage obtained by using reduced-order optimal control.
Algorithm description
The implementation of the full-order OCT algorithm for
water flooding was described in reference [1]. In reducedorder optimal control based on POD (see Fig. 1) we first
simulate the dynamical behavior of the system over time
interval 0 to K with an initial choice of u and compute the
NPV. We store ț snapshots of pressures and saturations
and calculate POD transformation matrix ). Now instead
of using the full-order derivatives of the system we use the
reduced-order derivatives for the backward calculation
and calculate P with Eq. (16). Based on the derivatives
computed with Eq. (17) we compute new controls and use
them for the next reduced-order forward simulation. For
this simulation we use the same transformation matrix ).
This means that the computational ‘overhead’ of
calculating ) is shared by multiple runs of the reducedorder model. To determine convergence of the inner loop
we use a convergence criterion c. The inner loop has
converged when the NPV of the last reduced-order
forward simulation is less than the c times the NPV of the
previous reduced-order simulation. Entering the outer loop
again we use the improved controls in a full-order forward
simulation and calculate the NPV. The previous
transformation matrix ) is replaced with a new ) that
reflects the altered dynamics. The algorithm has
9th
START
Apply initial input u
Simulate full forward model
Calculate NPV full model
Yes
DONE
Full NPV
converged?
No
Calculate (truncated) transformation matrix )
Substitute x = )z into the Hamiltonian and
run the reduced adjoint
Produce optimized input
Simulate reduced forward model
Calculate NPV reduced model
No
Reduced NPV
converged?
Yes
Fig. 1 – Algorithm outline for reducedorder OCT for water flooding.
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
6
converged when the NPV of the last forward simulation is less than the NPV of the previous forward
simulation. The advantage of the algorithm is that we use reduced-order forward simulations and reducedorder optimal control, which have a shorter simulation time. A disadvantage of the algorithm is that an
improved control of the reduced-order model is not necessarily an improved control for the full-order
model. In the numerical example below we will see that in our example this is however the case.
Numerical example
To test the algorithm described earlier we used a 2-dimensional
model with 2025 (45x45) grid blocks, which is the same model as
used in reference [1]. The dimensions of the reservoir were
450x450x10m and the permeability field is shown in Fig. 2.
Initially the reservoir is completely saturated with oil. We
assigned liquid compressibilities of 1*10-10 Pa-1 to both water and
oil. At the left side of the reservoir we introduced one horizontal
water injector divided in 45 segments by interval control valves
(ICV). At the right side we introduced one horizontal producer,
also divided in 45 segments. The wells were operated without
well model under rate-constraint: the liquid rate in an injection
segment was equal to the water rate, and in a production segment
Fig. 2 – Permeability field (m2).
the rate was equal to the sum of water rate and oil rate. The total
production and injection rates were equal to each other during the
WATER
entire simulation time. During optimization the flow rates were
redistributed maintaining a constant total sum of the rates. In the
NPV calculation we used for the oil price ro $80 per m3 and the
OIL
produced water cost rw $20 per m3. We compared the NPV
obtained with the reduced-order and full-order optimal control
algorithms with the NPV of a reference case. In the reference case
the injection and production rates were constant over time and a
function of water and oil mobility, reflecting a conventional water
flood where the wells are operated on constant well bore pressure.
Fig. 3 –Final water saturation after 1
We simulated the reservoir model for 949 days with variable time
PV production for the reference
step size and in this period we injected and produced one pore
case.
volume of liquid. We obtained a saturation profile as depicted in
Fig. 3. The total NPV for the reference case is $ 10.1 million.
Full-order optimal control example
Starting from the reference case we ran the full-order control algorithm. With a 2.4 Gh Pentium 4
processor and 1 Gb RAM memory it took 8701 s (148 minutes) to run the full-order algorithm. We
reached convergence after 19 full-order forward simulations and 18 full-order backward simulations.
5
0.8
10
0.7
15
0.6
20
0.5
25
30
35
0.3
40
0.2
45
5
injection rates (m3/d) vs time for all wells
35
5
100
60
25
40
well number
well number
20
30
35
40
45
0
0.7
WATER
0.6
20
20
0.5
20
25
25
15
30
40
40
0.8
15
45
cum. time (d)
0.4
30
10
35
0.3
5
40
0.2
0
45
20
cum. time (d)
25
OIL
5
25
35
35
45
20
30
15
80
30
15
10
10
10
15
10
production rates (m3/d) vs time for all wells
5
5
10
15
0.4
20
25
30
35
40
45
0.1
Fig. 4 – Optimized production rates (left), injection rates (middle) in m3/d vs. the simulation time, calculated
with full-order control algorithm. The picture in the right depicts the f inal oil/water saturation distribution
for full-order optimal control.
7
The average simulation time for the full-order forward simulation was 144 s and for the full-order
backward simulation 266 s. The resulting optimized rates are given in the two most left pictures of Fig. 4,
which correspond to a final saturation profile as depicted in the right picture of Fig. 4.
Reduced-order optimal control example
For reduced-order optimal control we used the algorithm as depicted in Fig. 1. We choose to use an
energy level of 0.999 as cut-off criterion for POD. With an energy level of 0.999 we obtained a final NPV
of $19.8 million. This is an increase of 96% with respect to the reference case. The maximum NPV
obtained with reduced-order control approached the NPV obtained with full-order optimal control to
within 97%. We reached convergence in 4949 s, which is 43% faster than the full-order optimal control.
In this example we needed 10 full-order forward simulations, which took on average 156 s per simulation,
and we needed 9 reduced-order forward simulations, which took on average 114 s per simulation. The
reduction in simulation time for the reduced-order forward simulation was 30%. Furthermore we needed
9 reduced-order backward simulations, which took on average 187 s per simulation. The reduction in
simulation time for the reduced-order backward model was 30% (where the full-order backward
simulation took on average 266 s per simulation). In order to maintain an energy level of 0.999 we would
need for the reference case in total 30 POD basis functions. For the optimal case we used 49 POD basis
functions. The number of POD basis functions increased when we used improved controls. This speaks in
favor of our nested approach where we adapt the transformation matrix after a full-order forward
simulation. The resulting rates for this case are given in the two most left pictures of Fig. 5, which
correspond to a final saturation profile as in the right picture of Fig. 5. The resulting rates and the final
saturation profile obtained with reduced-order optimal control differed from the resulting rates and final
saturation profile obtained with full-order optimal control. Apparently we ended up in two different local
optima.
production rates (m3/d) vs time for all wells
injection rates (m3/d) vs time for all wells
70
5
OIL
5
5
60
0.8
60
10
10
10
0.7
50
50
15
40
20
25
30
20
40
0.5
30
10
45
40
45
cum. time (d)
0.4
20
35
0.3
10
40
0.2
0
45
35
40
WATER
30
20
35
20
25
25
30
30
15
0.6
well number
well number
15
cum. time (d)
5
10
15
20
25
30
35
40
45
0.1
Fig. 5 – Optimized production rates (left), injection rates (middle) in m3/d vs. simulation time calculated with
reduced-order control algorithm. Energy level is 0.999. The picture in the right depicts the final oil/water
saturation distribution.
Conclusion
In the example discussed, and in several others not discussed, we found that reduced-order optimal
control of water flooding using POD improved the NPV with respect to an uncontrolled reference case.
Within shorter simulation time the NPV obtained by the full-order optimal control algorithm was
approached closely by the NPV obtained by the reduced-order algorithm. The increase in computational
efficiency was achieved by reducing the number of states in the forward and backward simulations
considerably and consequently the number of equations that needed to be solved every time step.
Considering a reservoir model with 4050 states (2025 pressures, 2025 saturations) and an adjoint model
of 4050 states (Lagrange multipliers) we obtained reduced-order models with 20 to 100 states only. The
NPV obtained by reduced-order optimal control was approached to within 97% of the NPV obtained by
full-order optimal control. In the various examples investigated, the resulting reduction in computing time
was between 30% and 60%. In general, the number of POD basis functions preserving a certain fixed
level of relative energy increases during optimization, which speaks in favor of our nested reduced-order
optimal control algorithm where we adapt the transformation matrix after simulating the full-order
reservoir model with improved controls. The potential benefit of reduced-order models in history
matching, where they may form an alternative to regularization, is part of ongoing research.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
8
Nomenclature
A
B
c
E
f
g
k
K
l
$
=
=
=
=
=
=
=
=
=
=
=
system matrix
input matrix
convergence criterion in inner loop
relative energy present in snapshots
nonlinear system function vector, explicit form
nonlinear system function vector, implicit form
Hamiltonian, $
discrete time step counter, or permeability, m2
total number of time steps
total number of preserved POD basis functions
objective function, $
m = number of grid blocks in x direction
n =
=
N =
p =
p =
P =
q =
q =
Q =
r =
R =
t =
S =
T =
u =
V =
n
system order
number of grid blocks in y direction
number of wells
pressure, m/(L t2), Pa
pressure vector
orthogonal matrix in SVD
flow rate, L3/t, m3/s
flow rate vector
orthogonal matrix in SVD
price per unit volume, $/m3
covariance matrix
time, s
saturation
transmissibility matrix
input vector
mass matrix
x
X
z
z
Į
İ
ț
Ȝ
µ
ı
Ȉ
ij
ĭ
Ȍ
=
=
=
=
=
=
=
=
=
=
=
=
=
=
state vector
data matrix, containing state vectors
transformed state variable
transformed state vector
fraction of relative energy
weighting factor
number of snapshots in POD
vector of Lagrange multipliers
vector of reduced-order Lagrange multipliers
singular value
diagonal matrix of singular values
eigen function
transformation matrix
compressibility/porosity matrix
Subscripts
c
d
i
o
p
r
red
s
tot
w
=
=
=
=
=
=
=
=
=
=
continuous
discrete
counter
oil
pressure
state variable specification, r  { p , s}
reduced
saturation
total
water
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
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