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Copyright 2013, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 30 September–2 October 2013.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been
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Abstract
Reduced-order models (ROM) are considered powerful techniques to address computational challenges associated with reservoir management decision-making. In this sense, they represent perfect alternatives that trade off accuracy for speed in a
controllable manner. In this paper, we describe a model-order reduction technique that entails the use of proper orthogonal
decomposition (POD), truncated balanced realization (TBR) and discrete empirical interpolation (DEIM) to accurately reproduce the full-order model (FOM) input/output behavior. POD allows for a concise representation of the FOM in terms of
relatively small variables while TBR improves the overall stability and accuracy. DEIM improves the shortcomings of POD
and TBR in the case of nonlinear PDEs, i.e.; saturation equation, by retaining nonlinearities at lower dimensional space. In
this work, the use of ROM reduced the computational time by O(100) while providing good overall agreement with FOM.
The use of large reservoir simulation models is expected to add additional speedup factors.
ROMs are potentially perfect alternatives to FOMs in reservoir management intensive studies such as production optimization. However, ROMs presented in this paper and the overall physics-based ROMs have the tendency to perform well
within a restricted zone. This zone is generally dictated by the training simulations used to build the ROM. Therefore, special
care is considered when implementing these training runs. To mitigate the heuristic process of implementing the training
runs, we apply a ROM based trust-region method that provides an adaptive framework to systemically retrain ROM during
the optimization run.
The ROM approach with trust-region methodology is applied to a heterogeneous model containing 13,200 grid blocks
and five wells. The accuracy of the ROM is first demonstrated for several testing simulations in which the injection and
production rates for each well differ from those used to build the ROM. A waterflood optimization case is then considered
to determine the optimum injection and production rates for four producers and one injector at five different times (total
of 25 control variables). Results for optimized net present value using ROM based trust-region is shown to be very close to
those achieved using the full order model with a difference of only 0.2%. The runtime speedup factor for this case was about
31. The ROM approach thus appears to be well suited for use in applications in which many simulations must be performed
such as production optimization and uncertainty assessment.
Introduction
As a result of income and population growth, the global energy demand in 2040 is expected to be approximately 35% higher
than it was in 2010, despite gains in energy efficiency. Oil and natural gas remain to be the prominent source of energy
providing about 55% of global energy demand in 2010, and forecasts indicated that they will continue to be significant
contributors for decades to come, Exxon Mobil (2013). In order to meet the rising demand, E&P industry simultaneously
directs its efforts toward discovering new fields and increasing recovery from existing fields through best practices of reservoir
management.
One of the primary goals of petroleum reservoir modeling and management processes is to enable decisions that determine
the direction and course of billions of dollars every year. Such decisions are made on a long term horizon and may include
valuing a prospective lease, assigning the best drilling locations, and determining development strategy for a given asset. A
decision-making framework requires identifying a methodology to propose possible scenarios and develop an efficient technique
to evaluate them. This can be mathematically translated into an optimization problem where efficient optimization methods
2
SPE 166305-MS
such as gradient-based algorithms are used to propose possible scenarios and reservoir simulation models are used to evaluate
them. The use of finite-difference full-physics reservoir simulation models for reservoir optimization and decision-making is
computationally expensive. A typical reservoir model might require several hours of run time, and the number of simulation
runs required to make a particular reservoir management decision, i.e.; optimum injection/production ratio, can be in the
order of hundreds to even thousands. This high computation cost hinders the popularity of optimization methods in reservoir
management decision-making.
To overcome the associated high-computation cost, we propose to develop a new Reduced-Order Modeling (ROM)
approach to reduce computational demands and enable the practical application of reservoir management decision-making.
This approach incorporates the use of a new model reduction technique based on Discrete Empirical Interpolation (DEIM),
Proper Orthogonal Decomposition (POD) and Truncated Balanced Realization (TBR) to build a reduced version of the
full-order model (FOM) that is much cheaper to evaluate, yet accurately reproduces the full models input/output behavior.
The latter two techniques allow for concise representation of the FOM in terms of a relatively small number of variables while
the former technique enables efficient treatment of nonlinear terms in the low-order space.
Reduction techniques used in this work is physics-based where systematic ROM technique is used to replicate FOM
outputs of interest. The primary motivation for using physics-based ROM comes from the observation that the solution
space of many numerical models is often embedded in a manifold that has much lower dimensions than the dimensions of the
original, spatially descritized, model. A common physics-based ROM approach is based on applying Galerkin projection onto a
low dimensional subspace, which is expected to contain dominant characteristics of the corresponding solution space (pressure
and saturation for example). This subspace can be represented by a set of reduced basis functions with global support. Fine
scale details are encoded in these global basis functions and thus makes it possible to obtain a good approximation with
relatively few basis functions.
Proper Orthogonal Decomposition (POD) constructs reduced basis by running high-order simulation models several
times using different set of forcing input controls, i.e.; production flowrates that covers a particular range of forcing input
controls. These runs are called “training runs” and the accuracy of the ROM is dictated by how well the training runs cover
the solution space. POD has been applied in various applications, i.e.; Bui-Thanh et al. (2004), Zheng et al. (2002), Lim
et al. (2007), including reservoir simulation, Cardoso and Durlofsky (2009), Cardoso and Durlofsky (2010), He et al. (2011),
Suwartadi (2012), van Doren et al. (2006), due to its ability to handle nonlinear systems.
Truncated Balanced Realizations (TBR) is one of the few projection techniques that possesses global error bounds on
the accuracy of the resulting reduced model. The basic idea behind TBR is to perform a change of coordinates where states
such as pressure and saturation are ordered from most important to least important. The “important” states are defined as
those that are either very controllable or very observable. Conversely, the least important states are those that are neither
controllable nor observable. The controllability and observability of states can be quantified via the controllability and
observability Grammians, respectively. TBR has been mainly applied in control system theory where systems are mostly
static and therefore suitable for liner time invariant systems (LTI), Gugercin and Antoulas (2004), Dones et al. (2011),
Vasilyev et al. (2003). Vasilyev et al. (2003) recently proposed an approach where TBR has been used with Trajectory
Piecewise Linearization approach to model nonlinear systems, but we are not aware of prior work on the use of TBR for
reservoir simulation.
Since reservoir simulation equations are nonlinear, reduction techniques, although reduce dimensions in the sense that
far fewer variables are present, still depend on the dimension of the original full-order model through nonlinear terms. The
complexity of evaluating the nonlinear terms remains dependent on the full-order model hindering the performance and speed
of the constructed ROM. This limitation was seen in reservoir simulation by Cardoso and Durlofsky (2009), where POD was
effective at the linear solver level while full residual and Jacobian equations were still constructed at every iteration of every
time step before being reduced. Detailed description of computational complexity of POD with nonlinear model can be found
in Chaturantabut and Sorensen (2010).
In order to use the above mentioned reduction techniques to construct ROM for reservoir simulation, an efficient treatment
for nonlinearities must be devised. One approach, Trajectory Piecewise Linearization (TPWL), uses first-order taylor series
expansion to approximate nonlinearities around saved “closest” equilibrium states, Rewienski (2003), Cardoso and Durlofsky
(2009), He et al. (2011), Vasilyev et al. (2003). Reasonable accuracy and substantial speedups were reported by Cardoso
and Durlofsky (2010) and He et al. (2011) when TPWL was used for reservoir simulation. The approach however requires
storing Jacobian matrices for both states and controls, in addition to solution states, which occupies substantial disk space
as these matrices can be quite large and difficult to manipulate. In addition, it requires output of partial Jacobians such as
flux Jacobian and accumulation Jacobian.
The use of high fidelity reservoir simulation models for decision-making is presently infeasible for large models due to
the high computation cost. In response to this shortfall, ROMs are perfect alternatives that trade off accuracy for speed
in a controllable manner. However, for decision-making and optimization, ROMs are reliable in a restricted zone around
the control values used to construct them “root-point” and generally reliability decreases as we deviate away from the root
SPE 166305-MS
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point. Cardoso and Durlofsky (2010) applied a heuristic procedure in which they varied control variables, i.e.; BHPs of
production wells, randomly during ROM construction to cover the optimization study range and ensure that deviation from
the root-point during the optimization process is minimal. He et al. (2011) applied a retraining strategy where initial ROM
is used for optimization until a certain number of function evaluations is reached. The optimization run is then stopped and
the ROM is retrained using the last control variables. Both strategies are sub-optimal in the sense that there is no indication
to how many training runs are required initially to cover the control variable space nor error quantification is available to
determine suitably when to retrain the ROM. Trust-region methods offer an effective way to manage ROM retraining over
the course of the optimization, Dennis and Schnabel (1983), Conn et al. (2000). It ensures that the optimizer for a certain
iteration stays close to the root-point. The implementation of trust-region with ROM was adapted by Arian et al. (2000)
where POD was used as ROM construction approach for flow control of viscous fluids using Navier-Stokes equations. Agarwal
and Biegler (2010) implemented trust-region strategy for ROM using POD to optimize a two-bed isothermal pressure swing
adsorption system for CO2 capture. Alexandrov et al. (1997) applied trust-region methods for general approximate models
while Suwartadi (2012) used trust-region to maximize recovery from a five-spot well pattern using POD and DEIM as ROM
construction approach.
The approach we will adopt for model order reduction in this work is Discrete Empirical Interpolation Method (DEIM).
It enables the representation of nonlinear terms through evaluation of the full-order nonlinear terms only at few selected grid
blocks (interpolation points). DEIM can be thought of as a clever extension to reduction techniques such as POD to retain
nonlinearities at lower dimensional space. DEIM is the discrete variant of EIM, developed by Barrault et al. (2004) , and was
first introduced by Chaturantabut and Sorensen (2010), where it was applied for miscible viscous fingering in porous media,
Chaturantabut and Sorensen (2011). Subsequently, Suwartadi (2012) applied DEIM for 2D reservoir simulation models. High
accuracy along with high speedups has been reported by the above mentioned investigators.
In this paper, we further develop a modified version of the original DEIM for reservoir simulation that is more stable
to incorporate fluid flow in porous media. We also incorporate trust-region methodology for ROM accuracy measure and
update during the optimization and decision-making process.
Governing Equations for Reservoir Simulation
In this section, we briefly discuss the governing equations for the reservoir simulator, including the formulation of the problem.
The discretization procedure and solution strategy will then be discussed. Additional description can be found in, i.e.; Lie
et al. (2011).
Oil-Water Flow Equations We consider incompressible oil-water flow and neglect capillary pressure and gravity effects.
The continuity equation for each phase, designated j (where j = o for oil and w for water), is given by:
φ
∂Sj
+ ∇ · (ρj vj ) = qj ,
∂t
(1)
where φ is porosity, Sj is the saturation of phase j, vj is the phase Darcy velocity, and qj is the source term. Eq. 1 is rewritten
into a more tractable system of equations consisting of a pressure equation and a saturation or fluid-transport equation for
easier dimensionality reduction. The pressure equation is given by:
vj = −
krj
K∇p,
µj
∇ · vj = qj ,
(2)
where µj is the phase viscosity, K is the absolute permeability, krj (Sj ) is the relative permeability of phase j, and p is
pressure. The derived water saturation equation is given by:
∂Sw
qw
φ
+ ∇ · fw (Sw )v =
,
(3)
∂t
ρw
where fw (Sw ) = λw /(λw + λo ) is the fractional-flow, λj = krj /µj is the mobility of phase j, and v is total velocity. The term
fw (Sw )v represent viscous forces.
Discretization and Solution Strategy Eq. 2 and Eq. 3 are nonlinearly coupled through the saturation-dependent mobility
λj in the pressure equation and through the pressure-dependent total velocity v in the saturation equation, in addition to
other terms that depends on pressure, i.e.; viscosities. A sequential method is applied to obtain solution states where
saturation from previous step (or initial condition) is used to compute saturation-dependent coefficient, i.e.; λj in Eq. 2
before it is solved for pressure and subsequently total velocity. Then, total velocity is kept constant while saturation from
Eq. 3 is solved and advanced in time. Next, the new saturation is used to update saturation-dependent term in Eq. 2 and
pressure is solved again, and so on. Eq. 2 is descritized explicitly while Eq. 3 is discretized implicitly in time as follow:
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SPE 166305-MS
T(Sn )pn+1 = Bun+1 ,
(4)
Here T is diagonal transmissibility matrix, n represents time step, u is controls, i.e.; well rates or BHP, and B is the
arrangement matrix for the controls. The descritized pressure equation, i.e.; Eq. 4, is a linear equation as the transmissibility
matrix T(Sn ) does not depend on pressure (since viscosity is constant for incompressible flow). Eq. 3 is desecrated as:
4t
n+1
n+1
n
Afw (S
) + Qw ,
(5)
S
=S +
φ
where A is upwinded flux matrix, fw (Sn+1 ) is a vector of fractional flow, and Qw is a vector of source/sink terms.
Model Order Reduction Representation and Procedure
This section will describe how to reduce the pressure equation, i.e.; Eq. 4, using an approach that merges POD and TBR to
formulate stable reduced pressure basis. DEIM will then be applied to the nonlinear saturation equation, i.e.; Eq. 5 providing
cost-efficient representation of nonlinearities. Explanation of the shortcomings when applying POD-TBR solely to nonlinear
equations and how this is treated using DEIM will also be discussed.
Reduced-Order Pressure Equation We provide a succinct description of the Proper Orthogonal Decomposition (POD)
and Truncated Balanced Realization (TBR). Interested readers can refer to Cardoso and Durlofsky (2009) and van Doren
et al. (2006) for POD and Gugercin and Antoulas (2004) and Vasilyev et al. (2003) for further discussion on TBR. Both POD
and TBR are projection-based techniques that enable the representation of pressure vector p in terms of a reduced pressure
coefficients p̃ using a right basis matrix ΦR ; i.e.;
p = ΦR p̃,
(6)
The basis matrix ΦR is the key to the accuracy and stability of the ROM and can be constructed either using POD or
TBR with computational complexity associated to each approach. POD constructs the basis matrix from singular value
decomposition (SVD) of snapshots, which are discrete samples of trajectories associated with a particular set of inputs, i.e.;
each snapshot corresponds to a vector of pressure values for each grid block at a given time step generated using a certain
flowrate or BHP. We denote P ∈ RN ×ns as the pressure snapshot matrix where N is the number of grid blocks and ns is the
number of snapshots such that:
P = p1 p1 · · · pns ,
(7)
Performing singular value decomposition (SVD) to P, the left singular vectors provide the columns of the basis matrix U
such that each column corresponds to a singular value. The total number of columns retained in U is q, where q << N . The
number of columns (q) is determined based on an ‘omitted energy’ criterion where vectors yielding low energy are discarded.
Truncated Balanced Realization (TBR) constructs the basis matrix by exploiting the structure of the system of equations.
It analyzes the response of the states of a system for a given control input and measures its behavior through the so-called
controllability and observability concepts. A system with state vector x1 is called controllable if and only if the system states
can be changed by changing the control inputs. On the other hand, a particular state x1 is called observable if there exists a
control input that transfers the state of the system from the initial state x0 to x1 in some finite time interval. TBR combines
these two concepts to determine a left basis matrix WL and a right basis matrix WR using Algorithm 1 where T and B are
the transmissibility and input control arrangement matrices in Eq. 4 respectively. The matrix C is the observation matrix
with ones at the well locations and zero elsewhere.
TBR is applicable to linear systems and hence suitable to the pressure equation, Eq. 4. It provides guaranteed stability
and improved accuracy over POD with a-priori error bound. However, it is limited to models with few hundred unknowns
as the computational complexity is O(N 3 ). In order to allow effective and efficient reduction for large models, we apply a
two-stage reduction strategy where we perform an intermediate reduction using POD method to reduce the model from N
to q, and then apply a TBR-based projection around initial stable point (p0 , s0 ) to further reduce the model from q to `.
The two stage reduction is summarized in Algorithm 2. The dimension of ΦL and ΦR is ` × N and N × ` respectively. The
projection matrices therefore reduce the number of unknown from N to `, with ` << N .
Reduced-Order Saturation Equation Nevertheless, reduction techniques reduce dimensions of models in the sense that
far fewer variables are present, they still depend on the dimension of the original full-order model through nonlinear terms
and thus marginal computational gain is achieved. To see this, lets consider the residual form and Jacobian of Eq. 5:
4t
n+1
n+1
n
Afw (S
) + Qw ,
(8)
g=S
−S −
φ
SPE 166305-MS
5
Algorithm 1: TBR algorithm
Input : System matrices T, B, C
Output: Projection bases WL and WR
1
2
3
4
5
Find observability Grammian P :
T P + P AT = −BB T ;
Find controllability Grammian Q:
T T Q + QT = −C T C;
Using SVD compute Cholesky factors of P and Q:
P = Zc ZcT ,
Q = Zo ZoT ;
Compute SVD of Cholesky product:
U ΣV = Zo ZoT
Compute WL and WR :
WL = Zc V Tbal Σ−1/2 , WR = Zo U Tbal Σ−1/2 ;
where Tbal is an N × q truncation matrix
Algorithm 2: Two-stage reduction algorithm
input : Pressure snapshot matrix P
output: Projection bases ΦL and ΦR
1
2
3
Compute first-stage reduction basis U using POD:
UΣV = P;
where U = N × q;
Compute second-stage basis WL , WR using TBR:
[WL , WR ] = TBR(T̂ , B̂, Ĉ)
where T̂ = UT T U, B̂ = UT B,
C = I;
Construct final bases
T
ΦL = WL
U,
ΦR = UT WR ;
J=I−
4t 0 n+1
Afw (S
),
φ
(9)
where g is the residual vector we seek to drive to zero, and J is the Jacobian. Note that fractional flow terms fw (Sn+1 ) and
0
(Sn+1 ) are evaluated component wise, i.e.; at each grid block. Representing the saturation state vector S in terms of a
fw
reduced state vector Sr using basis matrix Υ obtained from saturation snapshot matrix S:
S = ΥSr ,
(10)
Substituting Eq. 10 into Eq. 8 and Eq. 9 and premultiplying by ΥT , the reduced form of the residual and Jacobian become:
4t
n
T
n+1
ĝ = ΥT g = Sn+1
−
S
−
Υ
Af
(ΥS
)
+
Q
,
(11)
w
w
r
r
r
φ
Ĵ = ΥT JΥ = I −
4t T 0
Υ Afw (ΥSn+1
)Υ,
r
φ
(12)
The evaluation of the nonlinear terms still requires O(N ) nonlinear evaluations for the N entries of the vector fw (ΥSn+1
)
r
0
and the matrix fw
(ΥSn+1
)
which
is
proportional
to
the
dimension
of
the
full-order
model.
r
To avoid full evaluations of the nonlinear term and the Jacobian matrix, the Empirical Interpolation Method (EIM) was
proposed to approximate these terms via interpolation over a subset of points that are independent of of the full-order model
dimension N . In this context, DEIM partially evaluates the nonlinear term at “interpolation indices” which represent in
this problem the grid blocks that are important to the saturation equation while preserving the continuity properties of the
saturation equation. The first step of the DEIM is to approximate the nonlinear term fw (ΥSn+1
) using a separate set of
r
basis vectors Ψ that are different from those used for the states:
fw (S) ≈ Ψc,
(13)
6
SPE 166305-MS
where Ψ = [ψ1 , ψ2 , · · · , ψM ] ∈ RN ×M , M << N are the basis vectors for the nonlinear term and c ∈ RM ×1 is the vector
containing expansion coefficients. The basis vectors Ψ = {ψi }M
i=1 are derived from the snapshots of the nonlinear term
s
F = {fwi }ni=1
, using the same POD method described earlier. The collection of these nonlinear snapshots does not incur
additional computational cost in sampling because the nonlinear terms are already evaluated during the sampling of the
state snapshots P and S. Note that Eq. 13 represents an overdetermined system, so to compute c at a computational cost
independent of N , we select M interpolation points of fw (S) through a rectangular permutation matrix:
P = [e℘1 , e℘2 , · · · , e℘m ] ∈ RN ×M ,
(14)
where e℘i = [0, · · · , 0, 1, · · · , 0]T ∈ RN is the ℘th
i column of the rectangular permutation matrix. The selection of the
interpolation indices ℘i and permutation matrix P is determined inductively via Algorith 3 and this results in a system from
which the coefficient vector c can be uniquely determined:
PT fw (S) = (PT Ψ)c,
(15)
fw (S) ≈ Ψc = Ψ(PT Ψ)−1 PT fw (S),
(16)
and the final approximation from Eq. 13 becomes:
Applying the approximation in Eq. 16 in the reduced residual, Eq. 11:
4t
T
T
−1 T
n+1
n+1
n
Υ AΨ(P Ψ) P fw (ΥSr ) + Qw ,
ĝ = Sr − Sr −
φ
(17)
And similarly for the Jacobian matrix in Eq. 12:
Ĵ = I −
4t T
Υ AΨ(PT Ψ)−1 PT fw (ΥSn+1
)PT Υ,
r
φ
(18)
The computation of the reduced residual and Jacobian in Eq. 17 and Eq. 18 is now efficient because firstly, the term
ΥT AΨ(PT Ψ)−1 , which is ` × M , can be precomputed offline meaning that it is computed only once and thus does not need
to be computed within the Newton iteration and secondly, the term PT fw (ΥSn+1
) does not entail the full-order dimension
r
)
corresponding
to ℘1 , · · · , ℘m and it is therefore O(M ).
N anymore as it can be obtained by extracting rows of fw (ΥSn+1
r
Algorithm 3: DEIM Interpolation Point Selection Algorithm
Input : Ψ = {ψi }M
i=1
Output: ℘
~ = [℘1 , ℘2 , · · · , ℘m ] ∈ RM ×1
1
[|ρ|, ℘1 ] = max{|ψ1 |}
2
Ψ = [ψ1 ], P = [e℘1 ], ℘
~ = [℘1 ]
3
for i ← 2 to M do
Solve (PT Ψ)c = PT ψi for c
r = ψi − Ψc
[|ρ|, ℘i ] = max{|r|}
Ψ ← [Ψ, ψi ] , P ← [P e℘i ], ℘
~ = [℘
~ T , ℘i ]
end
ROM Based Trust-Region for Decision Making
A target application of the ROM is to be able to make a rational informed decision in a reasonable time. Within the
context of reservoir management, this includes the ability to apply changes in production strategy that maximize recovery
factor or net present value for instance. ROMs are suitable replacement to full-order models and are sufficiently accurate only
in a restricted zone around the point in decision variable space, i.e.; BHP for production wells, where they are constructed.
Consequently, the ROM needs to be updated in a systematic manner over the course of the optimization and decision making
process. In this section, we describe an adaptive framework using ROM based trust-region methodology that allows for an
automatic update and validation of the ROM throughout the optimization process.
SPE 166305-MS
7
ROM Based Trust-Region Trust-region methods provide an excellent adaptive framework for ROM based optimization.
They not only restrict the optimization step within the ROM’s validity, but also synchronize ROM updates with information
obtained during the course of the optimization, thus providing a robust and globally convergent framework. To see this, let’s
start at the k th iteration of an optimization cycle. ROM is constructed at a particular point, xk = {pk , Sk }, which is used to
build the model function for the trust-region subproblem. We define the ROM based trust-region subproblem at iteration k
as:
max JkR (uk + δ ),
δ ∈Rn
subject to:
FR
δ) = 0
k (x̂, u + δ
gkR (uk + δ ) = 0
hR
k (uk + δ ) ≥ 0
L B ≤ uk + δ ≤ U B
k δ k∞ ≤ 4k
(19)
where JkR (uk +δδ ) is the reduced objective function computed using ROM, FR
δ) is the adjoint constraint, gkR (uk +δδ )
k (x̂, u + δ
R
and hk (uk + δ ) are equality and inequality constraints computed from the reduced set of state variables of the ROM. L B
and U B are upper and lower control boundaries and 4k is the control trust-region at iteration k. Finally, δ is the increment
in controls variable we want to optimize.
ROM based trust-region procedure starts by constructing a ROM using POD, TBR, and DEIM from the original FOM
using an initial control u0 . The basis functions ΦL , ΦR , ΥandΨ are computed using snapshots P, S, and F obtained from
the FOM. Then Eq. 19 is used to compute the optimum δ that maximizes a quadratic reduced objective function where δ is
bounded by a trust region 4k defined by the user. 4k should be initialized in such a way that the ROM’s results are very
close to that obtained using the FOM inside the trust-region. The optimum step δ is used to run the FOM and compute a
calibration parameter to investigate the success of the iteration. The calibration parameter is defined as:
ρk =
Jk (uk + δ ) − Jk (uk )
aredk
= R
predk
Jk (uk + δ ) − JkR (uk )
(20)
where aredk is the actual reduction using the FOM and predk is the predicted reduction using the ROM. The calibration
parameter determines how well the ROM approximates the FOM. If it is larger than one, it means the ROM approximates
the FOM very well and thus the trust-region is enlarged for subsequent steps. If it is between 0.3 and 1, it means the
approximation is good but not sufficient to enlarge the trust-region and therefore the trust-region remains similar to previous
step. Finally, if the calibration parameter is small, then the step is rejected and the trust-region is shrunk. Details of the
procedure is outlined in Algorithm 4. In addition, the overall optimization algorithm is depicted in Fig. 1.
Figure 1: Optimization flowchart including trust-region step
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SPE 166305-MS
Algorithm 4: ROM Based Trust-region Algorithm
(a) Initialization:
Chooese 0 < η1 < η2 < 1 ≤ η3 , 0 < γ1 ≤ γ2 < 1 ≤ γ3 ,
Specify initial trust-region radius 40 , minimum radius 4min , and maximum radius 4max .
Compute state snapshot matrix X0 , based on initial control u0 and compute J0 (u0 )
Set k = 0
(b) Construct ROM:
Compute basis based on snapshot Xk and construct ROM
(c) Step Calculation:
Compute step δk by solving trust-region subploblem, Eq. 19
(d) Compute Calibration ratio:
Compute new snapshot X+ based on uk + δk , and evaluate full objective function Jk (uk + δk )
Define calibration ration as:
ρk =
Jk (uk + δk ) − Jk (uk )
aredk
= R
predk
Jk (uk + δk ) − JkR (uk )
(e) Trust-region Evaluation and Update:
• If ρk ≥ η3 :
set uk+1 = uk + δk ,
Jk (uk+1 ) = Jk (uk + δk ),
Xk+1 = X+ ,
update trust-region radius 4k+1 = min(4max , γ2 4k ),
set k = k + 1 and go to (b).
• If η2 ≤ ρk < η3 :
set uk+1 = uk + δk ,
Jk (uk+1 ) = Jk (uk + δk ),
Xk+1 = X+ ,
update trust-region radius 4k+1 = 4k ,
set k = k + 1 and go to (b).
• If η1 ≤ ρk < η2 :
set uk+1 = uk + δk ,
Jk (uk+1 ) = Jk (uk + δk ),
Xk+1 = X+ ,
update trust-region radius 4k+1 = γ2 4k ,
set k = k + 1 and go to (b).
• If ρk < η1 or ρk = ∞:
set uk+1 = uk ,
update trust-region radius 4k+1 = γ1 4k ,
set k = k + 1 and go to (b).
SPE 166305-MS
9
ROM Construction and Verification
In this section, The ROM construction procedure using POD, TBR, and DEIM is applied to a synthetic reservoir model
containing 13,200 grid blocks. The performance of the ROM is tested and verified for a variety of test controls.
Model Description The reservoir model represents the first layer of the SPE 10th comparative study, Christie and Blunt
(2001). The grid is 60 × 220 × 1 (Nx × Ny × Nz , where Nk is the number of grid blocks in direction k). The physical dimension
of each grid block is 20 ft × 10 ft × 2 ft. There are four production wells located at the corners of the model and one injection
well located at the middle of the model. All of the wells are under flowrate control.
P3
P4
I1
P1
P2
0.01
0.1
1
10
100 1000
Figure 2: Permeability in the x-direction in mD with well locations
Fig. 2 shows the permeability in the x-direction. Permeability is taken to be a diagonal tensor, with kx = ky . The mean kx
and kz are 74.3 mD and 17.1 mD respectively. The porosity is constant in this model and set equal to 0.25. The initial water
and residual oil saturations are zero. For oil, we set ρo = 45 lb/ft3 , µo = 5 cp; for water, we set ρw = 65 lb/ft3 , µw = 1 cp.
The system is incompressible and capillary pressure and gravity effects are neglected. The relative permeabilities for the oil
and water phases are specified as:
a
1 − Sw − Sor
0
kro (Sw ) = kro
,
(21)
1 − Swr − Sor
b
Sw − Sor
0
krw (Sw ) = krw
,
(22)
1 − Swr − Sor
0
0
0
0
where kro
and krw
are the endpoint relative permeabilities. Here we set kro
= krw
= 1 and a = b = 2.
ROM Construction We perform a training simulation using the FOM to construct the ROM. The flowrate control schedule
is shown in Fig. 3a. We specify the injection well to inject water at a constant rate of 211 STB/day, equivalent to 0.6 of the
pore volume (0.6 PVI), while the production wells produce also constantly at 53 STB/day. The training case was simulated
R
for 1000 days using MATLAB
Reservoir Simulation Toolbox, Lie et al. (2011), and a total of 200 snapshots were collected
and used to build the basis matrices. The pressure basis was first constructed using POD with 70 columns and then reduced
further using TBR to 25 columns. The saturation and fractional flow basis matrices were reduced using POD and contained
16 and 15 columns respectively. We thus reduced the dimension of the problem from 13,200 variables to only 41 variables.
The nonlinear term, fractional flow, basis matrix is used to determine the interpolation indices for DEIM. The spatial location
of each DEIM point is depicted in Fig. 7. The DEIM points are concentrated in locations where the rate of change in water
cut is high, i.e.; the path toward well P2. Although, DEIM points corresponding to all snapshots are shown in Fig. 7, only
the red DEIM points (15 points) are used to recover the fractional flow term. Thus, the whole saturation field, evaluated
using the saturation equation, can be recovered accurately by evaluating the fractional term at only those locations.
10
SPE 166305-MS
ROM Testing Prior to using the ROM for optimization and decision-making, it is important to assess the accuracy of
the ROM. We follow He et al. (2011) testing methodology where we define a ‘target’ flowrate schedule for each well. The
flowrate for each well in the target flowrate schedule, depicted in Fig. 3b, is randomly set within an interval between 43
STB/day and 74 STB/day under the constraint that injected fluid (water) has to be equal to produced fluid (oil and water)
for incompressible flow model. The target flowrate schedule for each well is perturbed every 100 days. Therefore, the total
number of control variables is 10 × 5 = 50. It is clear that the training flowrate schedule, taken to be constant for all wells,
is vastly different from the target flowrate schedule. We then interpolate between the training and target flowrates to enable
systematic perturbation away from the training run. Specifically, we specify the test case flowrates as follow:
utest = (1 − α)utraining + αutarget ,
(23)
where α is taken to be between 0 and 1. The error between the FOM and ROM is expected to increase with increasing
α as the test case is entirely made from the target flowrate schedule when α is 1. The ROM is assessed using three test
cases. Fig. 4 shows the the oil production rate and water cut for the four production wells for Test 1. In this and subsequent
figures, the dashed lines show the training simulation solution which controls were used to build the ROM, the solid lines
display the reference test case solution simulated using the full-order model, and the circles depict the test case solution
simulated using the ROM. Test 1 is simulated using α = 0 indicating that testing rate schedule is similar to that used for
training. The results show full agreement between the FOM and the ROM when similar controls are used. For Test 2, we
specify α = 0.5. The oil production and water cut comparisons are shown in Fig. 5. ROM results are in close agreement
with the FOM for all quantities. We see that the test simulation results are not drastically different from the training results,
though there are clear difference between the two cases particularly for the oil production in wells P3 and P4. In Test 3,
we set α = 1, which means we apply the flowrate schedule shown in Fig. 3b. Results for this case are shown in Fig. 6. The
ROM results are slightly less accurate than Test 2 which is expected as we deviate away from the ROM training root-point.
They are however still in close overall agreement with the full-order model. We notice less agreement for wells P1 and P2
toward the end of the simulation run as water cut increases. This might be due to inadequate DEIM points and the error
can be rectified by increasing the DEIM points. Overall, the results for all test cases demonstrate that the ROM is able
to provide simulation results in reasonable agreement with those from the full-order model. For this reservoir model, the
full-order simulation runtime is about 280 seconds, while the runtime for the ROM is about 3 seconds. We thus achieve a
reduction of computational time by a factor of 93. Although this reduction in computation is considered modest as compared
to computation time reduction of O(1000) attained by Chaturantabut and Sorensen (2011) when using DEIM for nonlinear
miscible viscous fingering problem, we anticipate larger speedup factors when larger models are used (O(105 ) grid blocks). It
is firstly because TBR reduction will be more significant (current reduction factor for this model is 528 for pressure equation).
Secondly, DEIM eliminates full evaluation of nonlinear terms and therefore less time will be spent during Newton-Raphson
iterations which will be very advantageous for large models with large number of variables.
65
P1
P2
P3
75
P4
P1
P2
P3
P4
Oil Flowrate (STB/DAY)
Oil Flowrate (STB/DAY)
70
60
55
50
65
60
55
50
45
45
0
200
400
600
Time (DAYS)
(a) Training flowrates
800
1000
40
0
200
400
600
Time (DAYS)
(b) Target flowrates
Figure 3: Time varying flowrate schedules for training and testing
800
1000
SPE 166305-MS
11
P1
P2
0.8
60
80
1
0.8
60
0.6
40
0.6
40
0.4
20
0
0
0.2
200
400
600
800
0
1000
0.4
20
0
0
0.2
200
400
P3
Oil Flowrate (STB/DAY)
600
800
1
0.8
60
80
1
0.8
60
0.6
40
0.6
40
0.4
20
Oil (Test−Full)
0
1000
P4
80
0
0
0.2
200
Water Cut (%)
1
400
600
Time (DAYS)
Oil (Test−ROM)
800
0
1000
Oil (Train−Full)
0.4
20
0
0
Water Cut (%)
Oil Flowrate (STB/DAY)
80
0.2
200
WC (Test−Full)
400
600
Time (DAYS)
800
WC (Test−ROM)
0
1000
WC (Train−Full)
Figure 4: Comparison for Test1
P1
P2
0.8
60
80
1
0.8
60
0.6
40
0.6
40
0.4
20
0
0
0.2
200
400
600
800
0
1000
0.4
20
0
0
0.2
200
P3
Oil Flowrate (STB/DAY)
400
600
800
1
0.8
60
80
1
0.8
60
0.6
40
0.6
40
0.4
20
Oil (Test−Full)
0
1000
P4
80
0
0
0.2
200
400
600
Time (DAYS)
Oil (Test−ROM)
Water Cut (%)
1
800
0
1000
Oil (Train−Full)
0.4
20
0
0
Water Cut (%)
Oil Flowrate (STB/DAY)
80
0.2
200
WC (Test−Full)
Figure 5: Comparison for Test2
400
600
Time (DAYS)
WC (Test−ROM)
800
0
1000
WC (Train−Full)
12
SPE 166305-MS
P1
P2
80
0.8
60
1
0.8
60
0.6
0.6
40
40
0.4
20
0
0
0.4
20
0.2
200
400
600
800
0
1000
0
0
0.2
200
400
P3
Oil Flowrate (STB/DAY)
600
800
1
80
0.8
60
1
0.8
60
0.6
0.6
40
40
0.4
20
Oil (Test−Full)
0
1000
P4
80
0
0
0.4
20
0.2
200
Water Cut (%)
1
400
600
Time (DAYS)
Oil (Test−ROM)
800
0
1000
0
0
Oil (Train−Full)
Water Cut (%)
Oil Flowrate (STB/DAY)
80
0.2
200
400
600
Time (DAYS)
WC (Test−Full)
WC (Test−ROM)
800
0
1000
WC (Train−Full)
Figure 6: Comparison for Test3
P3
P4
I1
P1
P2
0.2
0.4
0.6
0.8
1
Figure 7: Oil saturation map at final time step with discrete points for fractional flow term
SPE 166305-MS
13
ROM Based Trust-Region Optimization
In this section, we demonstrate the use of ROM based trust-region optimization and decision-making for reservoir
management. The problem involves the maximization of undiscounted net present value (NPV) by adjusting the injection
and production rates. Specifically, we seek to maximize the objective function J(u), where
J(u) = ro Qo (u) − cwp Qwp (u) − cwi Qwi (u),
(24)
Here ro is the price of oil ($/STB), cwp and cwi are the cost of handling produced water and the cost of water injection
($/STB) respectively. Qo , Qwp and Qwi are the cumulative oil production, water production and water injection in (STB)
respectively. We use the previously tested reservoir model in the preceding section with injection and production rates
perturbed every 200 days (total of 25 control variables). The optimization and trust region parameters are summarized in
Table 1 and Table 2. As highlighted in Algorithm 4, we start the ROM based trust-region by constructing ROM using initial
injection and production controls (root-point). The constructed ROM has 10 basis for pressure, 7 basis for saturation and 5
basis for fractional flow term. We then establish a trust-region in which the ROM is robust enough and can reproduce the
FOM input/output behavior. The trust-region used in this work is a box trust-region. With a current-trust-region radius
4k , the shape of the box is defined as:
|u0 − uk,i | ≤ 4k ,
(25)
Where u0 is the initial control at the beginning of the trust-region step, and uk,i is the control setting suggested at each
trust region iteration. This indicates that we can only perturb the initial controls for every trust-region step by ±4k .
Table 1: Trust-region parameters
40
4min
4max
17.4 STB/day
0.3 STB/day
43.4 STB/day
η1
η2
η3
0.03
0.3
1
γ1
γ2
γ3
0.35
0.7
1.2
Table 2: Optimization parameters
Initial injection rate
Initial production rate
ro
cwp
cwi
Injection rate range
Production rate range
21.7 STB/day
5.4 STB/day
$70/STB
$25/STB
$10/STB
[0 - 217] STB/day
[0 - 217] STB/day
Table 3: NPV ($106 )
Method
FOM
ROM
NPV(initial)
3.516
3.516
NPV(final)
10.994
10.970
Time (seconds)
51376
1636
Fig. 8 depicts the evolution of the NPV using ROM at the end of each trust-region step against that of the FOM. We
first see reasonable agreement between NPV obtained using ROM and FOM as injection rate was relatively small. Toward
the end of the optimization run, we can see how the trust-region is adjusted (shrunk) to minimize the difference and achieve
better accuracy by remaining close to the ROM’s root-point. The changes in trust-region radius is shown in Fig. 9. Here
() indicates accepted trust-region steps where η1 ≤ ρ, while (×) represents rejected steps. We can see declining trend
in trust-region radius as water injection and production rates increase. This indicates smaller calibration ratio values due
to larger differences between the results of the ROM and FOM at the end of the trust-region step. The algorithm tries
to restore a high calibration ratio value by reducing the trust-region radius. Eventually, the algorithm terminates after 14
trust-region steps due to shrinking trust-region. Table 3 presents the NPV for FOM and ROM. It is evident that the NPV
using FOM is only 0.2% higher than that computed suing ROM while the runtime speedup factor is about 31. The optimum
injection and production flowrates are shown in Fig. 10 for both FOM and ROM. The results from FOM and ROM appears
to be different for both injection and production rates. Similar results were also indicated by Agarwal and Biegler (2010)
14
SPE 166305-MS
and Suwartadi (2012). This is due to the difference in the sequence of trust-region radius employed by the ROM based
trust-region and the FOM trust-region algorithm. Agarwal and Biegler (2010) successfully implemented a procedure allowing
FOM and ROM to globally converge by enforcing equal gradients at the end of each trust-region step. This however resulted
in larger computational costs.
6
14
x 10
12
NPV [$]
10
8
6
4
Reduced−order
Full−order
2
0
2
4
6
Iterations
8
10
12
Figure 8: NPV progress during optimization
Trust−region Radius [STB/DAY]
30
step accepted
step rejected
25
20
15
10
5
0
0
5
10
15
Iterations
Figure 9: Trust-region radius changes during optimization
120
190
P1
P2
P3
P4
180
Flowrate [STB/DAY]
Flowrate [STB/DAY]
100
80
60
40
20
0
0
170
160
150
140
200
400
600
Time [DAYS]
(a) Production flowrate
800
1000
130
0
200
400
600
Time [DAYS]
800
1000
(b) Injection flowrate
Figure 10: Optimum controls using ROM and FOM (solid line is ROM, dashed line is FOM)
SPE 166305-MS
15
Conclusions
In this work, we developed a reduced-order modeling technique for reservoir management decision-making and optimization. The technique entails combining proper orthogonal decomposition (POD) and truncated balanced realization (TBR)
for model-order reduction and using discrete empirical interpolation (DEIM) for nonlinear terms approximation. A high
degree of efficiency is achieved as far fewer variables are required to reproduced the full-order model input/output behavior.
DEIM improves on the shortcomings of the original projectoin-based model-order reduction techniques by intelligently approximating the nonlinear parameters depicted in the saturation equation. Computational speedup factor is O(100) and is
expected to be higher for large models.
Decision-making based on multiple scenarios and optimization are potential application for reduced-order models (ROMs).
Physic-based ROMs are restricted to a validity zone and special care is considered in designing training runs to build ROMs.
Since, the training process is entirely heuristic, there is no guarantee that the ROMs stay close to the root-point (within
the restricted zone). To mitigate this shortcoming, we applied a ROM based trust-region method that provides an adaptive
framework to systemically retrain ROM within the optimization run. The ROM approach with trust-region methodology was
applied to a heterogenous model containing 13,200 grid blocks and five wells. The accuracy of the ROM was demonstrated
for a sequence of testing runs. A waterflood optimization case was then considered to determine the optimum injection and
production rates. Results for optimized net present value using ROM based trust-region were shown to be very close to those
achieved using the full-order model but with runtime speedups of a factor of 31.
In future work, we plan to develop similar ROM techniques for black-oil models. In addition, we plan to implement the
ROM approach on a realistic field model with larger number of grid blocks including 3D cases.
Nomenclature
A=
upwinded flux matrix
B=
arrangement matrix for the controls
F
c=
DEIM coefficients vector
c=
cost
F=
fractional flow snapshots matrix
R
adjoint constraint
=
fw =
fractional flow vector
fw =
fractional flow
g=
residual vector
R
equality constraint
g
=
ĝ =
reduced residual vector
R
inequality constraint
h
=
I=
J
R
=
identity matrix
objective function of reduced-order model
J=
J=
objective function of full-order model
Jacobian matrix
Ĵ =
reduced Jacobian matrix
K=
permeability tensor
kr =
LB =
relative permeability
lower bound
p=
Pressure vector
P=
pressure snapshots matrix
P=
permutation matrix
p=
pressure
p̃ =
pressure coefficients vector
q=
flowrate
16
SPE 166305-MS
Qw =
source/term vector
r=
price
S=
saturation snapshots matrix
Sr =
saturation coefficient vector
S=
Saturation
S=
Saturation vector
t=
time
T=
u=
UB =
transmissibility matrix
controls vector
upper bound
v=
velocity
x̂ =
reduced state
X=
state snapshot
Greek
α = distance variable
φ=
porosity
ρ=
density
µ=
viscosity
λ=
δ=
mobility
trust-region optimum incement/decrement
ρk =
calibration ratio at step k
η1 =
trust-region parameter
η2 =
trust-region parameter
η3 =
trust-region parameter
γ1 =
trust-region parameter
γ2 =
trust-region parameter
γ3 =
trust-region parameter
ΦR =
right pressure basis matrix
ΦL =
left pressure basis matrix
Υ=
saturation basis matrix
Ψ=
fractional flow basis matrix
℘=
interpolation point
4=
trust-region radius
Superscripts
n = time level
Subscripts
k = trust-region step
j=
phase
wp =
water production
wi =
water injection
SPE 166305-MS
17
Acknowledgements
The authors would like to thank Saudi Aramco and Geonumeric research group for supporting this work.
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