1
B029 A PRACTICAL, GRADIENT-BASED APPROACH
FOR PARAMETER SELECTION IN HISTORY
MATCHING
Alberto Cominelli, Fabrizio Ferdinandi
ENI Exploration and Production Division
Abstract
Reservoir management is based on the
prediction of reservoir performance by
means of numerical simulation models.
Reliable predictions require that the
numerical model mimics the known
production history of the field. Then, the
numerical model is iteratively modified to
match the production data with the
simulation. This process is termed history
matching (HM).
Mathematically, history matching can be
seen as an optimisation problem where the
target is to minimize an objective function
quantifying the misfit between observed
and simulated production data.
One of the main problems in history
matching is the choice of an effective
parameterization: a set of reservoir
properties that can be plausibly altered to
get an history matched model. This issue is
known as parameter identification problem.
In this paper we propose a practical
implementation of a multi-scale approach
to identify effective parameterizations in
real-life HM problems. The approach is
based on the availability of gradient
simulators, capable of providing the user
with derivatives of the objective function
with respect to the parameters at hand.
Those derivatives can then be used in a
multi-scale setting to define a sequence of
richer and richer parameterisations. At
each step of the sequence the matching of
9th
the production data is improved. The
methodology has been applied to history
match the simulation model of a North-Sea
oil reservoir.
The proposed methodology can be
considered a practical solution for
parameter identification problems in many
real cases. This until sound methodologies,
primarily adaptive multi scale estimation of
parameters, will become available in
commercial software programs.
Introduction
Predictions of the reservoir behaviour
require the definition of subsurface
properties at the scale of the simulation
block. At this scale, a reliable description
of the porous media require to build a
reservoir model by integrating of all the
available source of data. By their nature,
we can categorise the data as prior and
production data. The former type of data
can be defined as the set of information on
the porous media that can be considered as
“direct” measure of the reservoir
properties. The latter type of data includes
dynamic data collected at wells, e.g. water
cut, shut-in pressure and time-lapse data.
Prior data can be incorporated directly in
the set-up of the reservoir model and well
established
reservoir
characterisation
workflows are widely used to capture in
this procedure the inherent heterogeneity.
Production data can be incorporated in the
definition of the model only by HM the
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
2
simulation model. The agreement between
simulation and observations can be
quantified by using a weighted leastsquares type objective function J,
J=
1 T −1
R C R,
2
(1)
where R is the residual vector which i-th
entry is the difference between the
observed (o) and simulated (c) production
datum for the i-th measure,
Ri = oi − ci ,
(2)
while C is the correlation matrix,
accounting for both measurement and
modeling error. The matrix C is symmetric
and positive definite. Notably, in many
practical circumstances the matrix C is
assumed to be diagonal.
HM can then be seen as the minimization
of the function J by perturbing the model
parameters x:
min l ≤ x ≤u J ( x) = min l ≤ x ≤u
1
R( x)T C −1 R( x) (3)
2
where l and u are lower and upper bounds
for the parameters vector x. The problem in
equation (3) is as an inverse problem,
where the target is to recovery some model
parameters by matching the production
data. As inverse problem, HM is ill-posed:
errors affect both the model and
measurement process, the production data
do not include enough information to
condition all the possible model parameters
down to the simulator cell scale.
Ill-posedness can impact the reliability of
the production forecast because models
very similar from an HM point of view
may differ significantly as far as the future
of the reservoir is concerned. Setting-up
the problem in a Bayesian framework, then
adding a quadratic term which accounts for
the prior information on the model, can
provide the required regularization to turn
the problem into a well-posed one.
However in real studies a prior term can
hardly be defined and the choice is to
restrict the set of possible parameters by
means of some kind of zonation, i.e. a
division of reservoir cells into zone of
equal permeability or porosity. Most of the
time that is done with some sensitivity
simulation usually supported by the
physical knowledge of the field at hand.
Then minimization methods for leastsquares problems are often far from being
effective in practical cases. Moreover,
when an acceptable result is achieved, the
trade-off is often an over-parameterization
of the model, with large uncertainty in the
estimated parameters. Hence, there is a
widely recognized need of practical, cost
effective tools and methodology for
parameter identification.
The methodology to identify HM
parameters applied in this work belongs to
the family of multiscale parameter
estimation (MPE) techniques.
MPE has become really attracting as way
to estimate parameters in history-matching
type problems. It usually provides with a
connection between the solution of the
inverse problem and the scale lengths
associated to the production data. In this
work we implement the idea proposed by
Chavent et al [6] to use fine-scale
derivatives of the objective function J as an
engine to define a richer and cost-effective
parameterization in a multiscale process.
The paper proceeds as follows: after a
review
of
multi-scale
parameter
estimations,
the
gradient
based
methodology is presented, an application
on a real case is described and, finally,
conclusions are drawn on the basis of the
result obtained.
Multiscale Parameter Estimation
Multiscale estimation aims to avoid overparameterization by adding parameters in a
hierarchical way. A multiscale estimation
process starts from a rather poor coarse
scale parameterization, typically one
parameter for the whole reservoir. Then,
ordinary multiscale estimation (OME)
3
proceeds by adding at a given level of the
hierarchical evaluation all the possible
parameters[1][2]. In this way at the n-th stage
of the estimation process the optimization
problem has 2dn degrees of freedom in a ddimensional case, see Figure 1.
This approach can be effective as regards
to the solution of the minimization problem
but it does not ensure that overparameterization does not occur. Moreover,
in real 3D problems the dimension of the
optimization problems that has to be solved
at each stage of OME increases rapidly.
a)
c)
b)
d)
Figure 1 - Ordinary Multiscale Estimation: in a
32x32 cells model (a) 1-4-16 parameters (b to d)
are added in three steps of the estimation
sequence.
With adaptive multi-scale estimation[4][6]
(AME) the parameters space is not refined
uniformly. Rather, new parameters are
added only if they are warranted by the
production data.
Notably, an analysis of the dependency
between non linearity and scale of
parameterization for similar but simpler
parabolic-type problems has shown that
fine-scale permeability features are
responsible for highly non-linearity in the
model response, while coarse scale updates
are associated with linear behavior of the
model response.
The AME algorithm is terminated when
production data are matched or any richer
parameterization is unreliable.
AME has proven to be a cost-efficient
strategy for parameter estimation in 2D
9th
case, both for simple and for complex
problems[4][6], field-like cases[5]. However
an implementation of this methodology in a
real HM study, including the need to work
with 3D geometry, is not available. This
motivated the application of a similar but
simpler multi-scale parameter estimation,
where, following Chavent et al. [6], the
engine used to define new possible
parameters is based on gradients of the
objective function computed for a
reference fine-scale parameterization.
Gradient Driven Multi-Scale Estimation
In multi-scale parameter estimation the
definition of richer parameterisation at a
given step of the process is based on
geometrical considerations. In OME each
parameter has to be refined (see Figure 1)
by regular cuts of the domain. In AME the
refinement of the parameterisation is
selective, nonetheless also in AME
geometrical considerations play a major
role. Gradient Based Multiscale Estimation
(GBME) uses the gradient of the objective
function to guide the refinement of the
parameterisation.
To introduce the methodology, let us
assume as reference problem to estimate
the permeability of a 2D model consisting
of N=8x8 cells cartesian grid.
In a multiscale setting a uniform
permeability distribution can be estimated
by minimising the objective function J with
respect to on a single parameter x0. An
optimal value J(x0*) can then be found and
the corresponding permeability grid k is a
uniform grid with N=64 entries equal to
x0*.
Next, the gradient ∇J0* of the objective
function with respect to the permeability
values of the 64 model cells can be
computed:
(∇J ) = (∂J / ∂k )
T 0*
i
i
k i = x 0*
, i = 1,..,N , (4)
Following Chavent et al. [6], the gradient
∇J* can be mapped on the computational
grid, see Figure 2-a.
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Based on such map, a new zonation can be
defined by including cells with positive
derivative in one zone (red cells in Figure
2-b) and cells with strictly negative
derivative in another one (light blue cells in
Figure 2-b). Two parameters, x11 and x12,
can then be defined to characterize the
permeability in the two regions.
lumping the cells in each zone according to
the sign of the corresponding gradient. The
cells in the light-blue region in Figure 2-b
are so split (Figure 2-c) in two regions,
light-blue and dark-blue. Similarly, the red
zone (Figure 2-b) can be partitioned in two
zones, red and yellow (Figure 2-c).
By definition, the bound-constrained
minimization of the objective function J
with respect to x0 equals to the
minimization of J under the constraint
α = x − x , α = 0.
1
1
1
2
(5)
Then, the proposed parameterization can be
seen as a relaxation of the constraint (5)
and the corresponding Lagrange multiplier
λ,
λ = (∂J 0* / ∂x11 ) = −(∂J 0* / ∂x12 ) ,
(6)
can be interpreted as a measure of the
impact on the objective function of the
relaxation of constraint (5),
λ = (dJ 0* / dα )α =0 ,
(7)
b)
thus Ben-Ameur et al.[3] named λ a
refinement indicator.
By construction the refinement of the
coarse parameterization on the basis of the
gradients sign is likely to give the best
cost-effective parameterization in terms of
reduction of J. The new parameterisation
can be used in the second step of the
GBME to improve the match, then getting
two
calibrated
parameters,
1* T
1*
1*
x
= ( x1 , x 2 ) , and a value for the
1*
objective function J 1* = J ( x ) . Notably,
by definition J 1* ≤ J 0* . Although the new
value J 1* represents and improvement of
the HM, it is quite likely that a further
improvement is required.
Then, at this point the calibrated
parameters x1* can be used to compute the
gradient of the objective function ∇J1*
with respect to the 64 permeability values
on the fine scale. The parameterization
shown in Figure 2-b can be refined by
c)
Figure 2 –Gradient map after the first step of the
GBME (a), the related 2 parameter zonation (b)
and the 4 parameter zonation (c).
By associating a uniform permeability
value to each of the four sub-zones, the
new parameterisation consisting of 4
degree of freedom, x2T=(x12, x22, x32, x42),
can be defined. The couple (x12, x22) comes
from the refinement of the parameter x11,
while the couple (x32, x42) comes from the
refinement of the parameter x21. This
parameterization can be calibrated in the
next step of the GBME to get an objective
2*
J 2* = J ( x ) ,
with
function
J 2* ≤ J 1* ≤ J 0* .
5
More generally, at the n-th step of the
GDME 2n calibrated parameters can be
refined by means of the corresponding fine
scale gradient ∇Jn*. The refined
parameterisation consists of 2n+1 degree of
freedom to be calibrated in the n+1 step.
The n-th step parameterization can be
recovered from the n+1 by adding n
equality constraints:
α = x − x , α = 0, i = 1,.., n .
i
n+1
N +1
2 i −1
2i
i
(8)
Before the n+1 step regression. the
effectiveness of the proposed richer
parameterization can be evaluated by
computing the n refinement indicators
associated to the corresponding equality
constraints:
λi = (∂J *n / ∂x2ni+−11 ) = −(∂J *n / ∂x2ni+1 ) ,
(9)
With respect to OME, GBME is more
efficient because the number of parameters
increase as a power of 2 at each step, while
in OME this happens only for 1D
problems. AME is more efficient because
the parameters are added selectively.
The selection criterion is based on the
estimation of the potential uncertainty
associated
with
a
proposed
parameterization. This usually prevents the
risk of over-parameterization. In the
current GBME the refinement indicators λi
can provide with a guess on the
improvement in the parameterization
associated with the refinement of the i-th
set of parameters. However, the refinement
indicators do not represent an estimate of
the reliability of the n new couples of
parameters. In this work we have dealt with
the over-parameterization issue only a
posteriori by means of a L-curve type
analysis[4][10], where the uncertainty in the
n*
calibrated parameterization x is based on
the maximum eigenvalue of the posterior
parameters
covariance
matrix
~
T
−1
C = (∇R C∇R ) .
From a computational viewpoint the
bottleneck in GDME is the computation of
the fine scale array of derivatives, which
9th
entails the computation of a large number
of HM parameters. The ideal choice, one
parameter for each cell, can be selected
only if the gradient ∇J is computed by
means of Adjoint formulation. In this work
we used a commercial simulator[8] which
computes HM parameters by means of the
so-called forward gradient methods.
This approach is quite effective, because it
directly provides with the sensitivity matrix
∇R. This allows to compute the GaussNewton approximation to the Hessian
matrix, ∇RT C −1∇R , hence cost-efficient
methods like Levenberg-Marquardt can be
implemented. However, this way of
computing
gradients
is
time
consuming[8][9], because the cpu-time
scales linearly with the number of
parameters. Hence, in real or even realistic
cases it is prohibitive to define a parameter
for each cell. In this work we introduced
the
concept
of
finer
allowed
parameterization (FAP), which means the
finer parameterization that can be used for
a detailed gradient computation at each
step of GBME. The FAP constraints are
case dependent and their definition is
tightly related to the cpu-time required to
simulate the model.
A 2D problem to highlight the concept of
FAP is shown in Figure 3-a: a 2D reservoir
with an highly permeable channel up to
1000 mD, embedded in a low permeable
matrix. Synthetic history data have been
created by simulating a water flooding with
one producer, located in the channel, and
two injectors. Then, using as baseline
permeability a constant 80mD value, a
GBME was carried out, using as FAP the
8x8 parameterization shown in Figure 3-b.
The parameterizations calibrated in the
second and third step of the sequence are
shown in Figure 3-c and in Figure 3-d,
respectively.
Although the FAP is relatively coarse (2x2
boxes), nonetheless the first FAP gradient
evaluation gives a rough but reasonable
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
6
estimate of the permeability trend (see the
green area in Figure 3-c).
WCT measurement error of 0.05 was
chosen.
From a practical point of view we assumed
that a single GDME would run in a day,
with the cpu time-consuming FAP gradient
simulation running overnight.
Figure 3 – 16x16 cells permeability distribution
with an high permeable channel along the main
diagonal(a), a FAP consisting of 64 2x2 cells
boxes (b), stage 2 parameter zonation (c) and the
stage 3 parameter zonation (d).
Application to Camelot History Match
The Camelot field is a Paleocene
undersaturated oil reservoir located in the
UK Continental Shelf of the North Sea.
The field has brought onstream for
production since 11/1986 by means of 14
production wells. For pressure maintenance
purposes water was injected till 1998. After
1998 the surrounding aquifer still provided
with enough energy to support reservoir
pressure largely above the bubble point.
Since 11/1986, only water cut (WCT) data
have been systematically collected. To
predict field production the reservoir has
been simulated by means of a dead oil
model. Grid model consists of 46x150x39
cells, with only 55394 active. Layer 1 to 9
correspond to the main oil bearing sand – F
– while the sequences 11-25 and 26-35
correspond to E and D sands.
In this work we implemented the GDME to
reconcile the baseline model with the WCT
data collected in a history time-window
going from 1986 to 2002. Field production
was simulated by running the model with
liquid rate boundary conditions. GDME
has been implemented to calibrate the
model by selecting as HM parameters
horizontal transmissibility multipliers
(HMTRNXY). Constant, uncorrelated
Figure 4 – FAP defined for the Camelot field:
view from the top
In addition, to preserve the main geological
layering, it was required that the fine scale
zones associated to FAP did not mix cells
belonging to different oil bearing sands.
To fulfill both aforequoted constraints a
FAP consisting of 645 regions were been
defined. The regions are columns of cells
spanning vertically an entire oil-bearing
sand, with base box of 3x3 cells. In Figure
4, a view from the top of the FAP regions
can be seen.
On a Pc with Xeon 3.2 GHtz processors the
cpu time to compute the FAP gradients was
about 8 hours, with 11 minutes required to
simulate the production time-window.
GBME has been run until the value of the
root means square (RMS) production
misfit, defined as
RMS = R T C −1 R ,
is lower than 1. This is the default stopping
criterion in a commercial gradient based
HM package[9].
7
At each step of GBME, the parameters
were optimized by using a commercial
of
Levenbergimplementation[11]
Marquardt method in the framework of a
research HM package.
8000
Objective Function
7000
Numerical Results
6000
8 Parameters
5000
4000
32 Parameters
3000
2000
1000
A sequence of 6 GDME steps were run
until the stopping criterion was fulfilled
with a final parameterization consisting of
32 regions. The sequence of the optimal
values J* is plotted in Figure 5 vs the
number of parameters, alongside with the
values of the RMS.
10000
0
0
2.5
2
6000
5000
1.5
4000
RMS
Objective Function
7000
8
10
12
14
By means of this analysis, see Figure 6, we
noticed that the uncertainty in the
parameterization is fairly constant up to the
4th stage of the GBME sequence (8
parameters space), while a rapid increase in
the uncertainty could be seen in the two
subsequent step
1
3000
2000
1
0.5
Well PA2
0.9
1000
0.8
0
0
10
15
20
25
30
0.7
35
Number of HM parameters
Figure 5 - objective function and RMS values
versus the number of parameters.
The behavior of the objective function J*
shows some typical feature of multiscale
type history matching: larger improvement
are achieved in the early steps of the
process, where the linearity associated to
large scale features in the –unknown –
property
trend
dominates.
Finer
parameterization generated at later stage is
associated with small scale feature and
leads to more and more moderated
improvements as the sequential estimation
proceeds. We expected then to notice some
trade-off between accuracy in the matching
and reliability of the parameter estimate.
Following [4] we have measured the
uncertainty in the parameterization by
means of the maximum eigenvalue of the
~
parameter covariance matrix C . Then, the
tradeoff between accuracy and reliability
can be visualized by means of a L-curve
plot, where the objective function J* is
represented vs the maximum eigenvalue of
~
the corresponding C matrix.
0.6
WCT
5
0.5
0.4
0.3
0.2
0.1
0
0
1000
2000
3000
4000
5000
6000
2000
2500
3000
time (days)
1
Well PA-B2
0.9
0.8
0.7
0.6
WCT
0
9th
6
Figure 6 – L-type curve: optimal objective
function vs. the maximum eigenvalue of the
posterior covariance matrix.
3
Objective Function
RMS
4
Maximum eigenvalue
9000
8000
2
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
time (days)
Figure 7 – WCT vs time for well PA2(top) and
PA-B2: history (black), baseline model (blue), 8
(green) and 32 (red) parameters GBME models
results.
Thus, the price for improvement in the
RMS value from stage 4 (RMS=1.17) to
stage 6 (RMS=0.87) is a fairly large
increase of the uncertainty in the
parameters. In most of the wells the
difference between 8-parameters and 32 –
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
8
parameters simulation models is fairly
small, see for instance well PA-2 results in
Figure 7. Nonetheless, in the case of well
PA-B2 (see Figure 7) the less uncertain 8parameters model falls short of matching
the history data .
References
[1]
Yoon, S., Malallah, A. H., Datta-Gupta,
A.,Vasco, D. W, Beherens, R. A., A
Multiscale approach to Production Data
Integration Using Streamline Models, SPE
56653, SPE ATCE, 1999.
[2]
Lee, S.H., Malallah, A., Datta-Gupta, A.,
Higdon, D.: “Multiscale Data Integration
Using Markov Random Fields,” paper SPE
63066, SPE ATCE, 2000.
[3]
Ben Ameur, H., Chavent, G., Jaffré, J.:
“Refinement
and
Coarsening
of
Parameterization for the Estimation of
Hydraulic Transmissivity,” in Proceedings of
the 3rd Int. Conference on Inverse Problems
in Engineering: Theory and Practice, Port
Ludlow, WA, USA, June 13-18 1999.
[4]
Grimstad, A.A., Mannseth, T., Nævdal, G.,
Urkedal, H., “Scale Splitting Approach to
Reservoir Characterization”, SPE 66394,
SPE Reservoir Simulation , 2001.
[5]
Grimstad A, Mannseth , T., Aanonsen, S. A.,
Aavatsmark, I., Cominelli, A., Mantica, S.,
“Identification of Unknown Permeability
Trends from History Matching of Production
Data”, SPE 77485, SPE ATCE 2002.
[6]
A.-A. Grimstad, T. Mannseth, G. Nævdal,
and H. Urkedal. “Adaptive multiscale
permeability estimation”, Computational
Geosciences, 7: 1-25,
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2003.Chavent, G., Bissel, R., “Indicator for
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Eclipse 2003A Technical Description, 2003,
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Conclusions
In this work an implementation of the
methodology to select parameterization in
a multiscale setting by means of
gradients[6] has been presented.
The application was performed on a real
case by exploiting the gradient capabilities
of a commercial simulator, together with a
HM gradient-based package. In terms of
reduction of the objective function J, the
implementation has achieved the goal.
However, the GBME methodology is still
subject to the risk of over-parameterization,
as shown by means of a L-curve type
analysis.
The methodology then has to be improved
by means of the addition of selective
parameterization criterion. One possibility
is to use the refinement indicators, see
equation (9), to select new possible
Although
of
easy
parameters[3].
implementation, this solution does not
provide with any guess on the reliability of
the proposed parameterization. On the
other side, the use of sensitivity matrices[6]
to guess the potential improvement in the
objective function and associated standard
deviation is much more promising.
Besides the risk of over-parameterization, a
real limitation of this approach is the
computation of FAP to define new
parameters.
Currently
the
forward
sensitivity simulation is time-consuming
and in many real cases the dimension of the
FAP space may become unrealistically
small because of computational constraints.
In this case, the extension of AME type
methodologies [4][5][6] to 3D problems could
provide with a sound and cost effective
alternative.
Guide,
2003,
Libraries,