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Unformatted text preview: StreamlineAssisted Ensemble Kalman
Filter for Rapid and Continuous Reservoir
Model Updating
Elkin ArroyoNegrete, Deepak Devegowda, and Akhil DattaGupta, SPE, Texas A&M University, and
J. Choe, SPE, Seoul National University Summary
The use of the ensemble Kalman filter (EnKF) is a promising
approach for data assimilation and assessment of uncertainties during reservoir characterization and performance forecasting. It provides a relatively straightforward approach to incorporating diverse data types, including production and/or timelapse seismic
data. Unlike traditional sensitivitybased history matching methods, the EnKF relies on a crosscovariance matrix computed from
an ensemble of reservoir models to relate reservoir properties to
production data. For practical field applications, we need to keep
the ensemble size small for computational efficiency. However,
this leads to poor approximations of the crosscovariance and,
often, loss of geologic realism through parameter overshoots, in
particular by introducing localized patches of low and high permeabilities. Because the EnKF estimates are “optimal” only for
Gaussian variables and linear dynamics, these difficulties are compounded by the strong nonlinearity of the multiphase history
matching problems and for nonGaussian prior models. Specifically, the updated parameter distribution tends to become multiGaussian with loss of connectivities of extreme values, such as
high permeability channels and low permeability barriers, which
are of special significance during reservoir characterization.
We propose a novel approach to overcome some of these limitations by conditioning the crosscovariance matrix using information gleaned from streamline trajectories. Our streamlineassisted
EnKF is analogous to the conventional assisted history matching,
whereby the streamline trajectories are used to identify gridblocks
contributing to the production response of a specific well. We then
use these gridblocks only to compute the crosscovariance matrix
and eliminate the influence of unrelated or distant observations and
spurious correlations. We show that the streamlineassisted EnKF
is an efficient and robust approach for history matching and continuous reservoir model updating. We illustrate the power and
utility of our approach using both synthetic and field applications.
Introduction
Proper characterization of the reservoir and the assessment of uncertainty are crucial aspects of any optimal reservoir development
plan and management strategy. To achieve this goal, it is necessary
to reconcile geological models to the dynamic response of the
reservoir through history matching. The topic of history matching
has been of great interest and an area of active research in the oil
industry (DattaGupta and King 2007; Emanuel and Milliken
1998; Oliver et al. 2001). The past decade has seen some significant developments in assisted and automatic history matching of
highresolution reservoir models and associated uncertainty quantification. Many of these techniques involve computation of sensitivities that relate changes in production response at a well to a
change in reservoir parameters. Techniques of automatic history
matching that typically do not use parameter sensitivities or gradient of the misfit function are stochastic algorithms such as Copyright © 2008 Society of Petroleum Engineers
This paper (SPE 104255) was accepted for presentation at the 2006 International Oil & Gas
Conference and Exhibition in China, Beijing, 5–7 December, and revised for publication.
Original manuscript received for review 21 August 2006. Revised manuscript received for
review 8 July 2008. Paper peer approved 22 July 2008. 1046 Markov Chain Monte Carlo (MCMC), simulated annealing and
genetic algorithms (Ma et al. 2008; Sen et al. 2005). A relatively
recent and promising addition to this class of techniques is the use
of ensemble Kalman Filters (EnKF) for data assimilation (Gu and
Oliver 2005, 2006; Naevdal et al. 2005; Gao et al. 2006;
Skjervheim et al. 2007; Dong et al. 2006). It is a MonteCarlo
approach that works with an ensemble of reservoir models. Specifically, the method utilizes crosscovariances between measurements and model parameters computed directly from the ensemble
members to sequentially update the reservoir models.
A major advantage of the EnKF is that it can be readily linked
to any existing reservoir simulator. The ability to assimilate diverse data types and the ease of implementation have resulted in
considerable interest in the approach. Moreover, EnKF uses a sequential updating technique; that is, the reservoir data is assimilated as and when it becomes available. The EnKF can assimilate
the latest production data without rerunning the simulator from
the initial conditions. These characteristics make it particularly
wellsuited for continuous model updating. The increased application of downhole monitors, intelligent well systems, and permanent sensors to continuously record pressure, well rates, and temperature has provided a further boost to the sequential model updating through EnKF.
In spite of all its favorable properties, the current implementation of EnKF approach comes with its own share of challenges. A
key requirement in history matching is that the final model should
honor the available geological information and retain geologic
realism. It has been shown that the EnKF works well when the
prior distribution of parameters is Gaussian; however, the estimates are suboptimal for nonGaussian distributions. Over a sequence of many updates, multimodal permeability distributions
tend to transform to Gaussian distribution. During geologic model
updating, this can lead to a loss of structure and connectivity of the
extremes in the permeability field. This has serious implications in
the fluid flow because of the influence of highpermeability channels and lowpermeability barriers. Although there are some variants of the Kalman filter that work with nonGaussian distributions, such as the Gaussian summation approximation, the implementation on an ensemble framework tend to be very expensive
(Anderson and Moore 1979).
In the past few years, we have seen several applications of the
EnKF for fieldscale history matching, including some recent papers that attempt to deal with some of the challenges pertaining to
its use (Gu and Oliver 2005, 2006; Naevdal et al. 2005; Gao et al.
2006; Skjervheim et al. 2007; Dong et al. 2006). In particular,
localized overshooting of permeabilities has been reported, resulting in loss of geologic continuity. This is aggravated by the strong
nonlinearity inherent in multiphase flow simulations.
Another common difficulty experienced when using the EnKF
is filter divergence. The effect of filter divergence is such that the
distribution produced by the filter drifts away from the truth. Filter
divergence normally occurs because the prior probability distribution becomes too narrow (loss of variance) and the observations
have progressively less impact on the model updates.
One common approach to deal with filter divergence is to add
some (white) noise to the prior ensemble to “inflate” its distribution and enhance the impact of new observations. Other problems
and limitations of the EnKF, particularly for nonlinear problems
December 2008 SPE Reservoir Evaluation & Engineering and nonGaussian parameter distributions, can be partly controlled
using a large ensemble. However, for practical field applications,
the ensemble size needs to be kept relatively small for computational efficiency.
This paper describes an approach to address many of the currently reported difficulties in the use of the EnKF applied to reservoir history matching. The unique feature of our proposed approach is that the final models that constitute the ensemble tend to
retain the geological information that went into building them
initially. Over a sequence of many updates, our approach tends to
preserve the shape of the initial permeability distribution and consequently retains key geological features. Our approach greatly
decreases the severity of the overshooting problem reported in
earlier implementations of the EnKF. Moreover, it allows the use
of smaller ensemble size, while providing results comparable or
better than the standard EnKF.
The paper is organized as follows. First, we briefly review the
major steps of the EnKF and the additional streamlinebased conditioning of the crosscovariance proposed here. We also illustrate
these steps using a synthetic example. Next, we discuss the underlying mathematical formulation in detail. We then demonstrate
the power and practical utility of the approach using the benchmark PUNQS3 synthetic example (Gu and Oliver 2005) and a
field example. Finally, an analysis of the scalability and speedup
factor for the parallel implementation of our code is given. blocks that may be far away from the producers, particularly for
highly heterogeneous reservoirs and in the presence of flow channels. The major steps in our proposed approach are outlined next. Approach
A general rule for history matching is to change the parameters
where the uncertainties are large and/or where the changes in the
parameters will have the largest influence on the solution. Thus, it
is vital to identify these regions and then limit the changes to these
areas. Our knowledge of the reservoir drive mechanisms and underlying flow physics can be used to infer these regions. For instance, in primary depletion, the bottomhole pressure is mainly
affected by reservoir parameters within the region defined by the
radius of investigation. Similarly, during waterflooding, the watercut at the producing wells is affected primarily by the rockfluid
properties within the swept zone. A convenient way to identify
these regions of influence during history matching is through examination of the streamline trajectories and the time of flight
(DattaGupta and King 2007). The use of streamlines to decide
regions for changes during reservoir history matching has proved
to be useful in the past. Such streamlineassisted history matching
was first proposed by Emanuel and Milliken (1998). They used
streamlines to identify the gridblocks that affect the production
response in a specific well. When these gridblocks are identified,
it is possible to restrict changes to these gridblocks preferentially.
Our proposed streamlineassisted EnKF shares many of the
features discussed previously. Specifically, streamlines and time of
flight are first used to identify areas of influence during history
matching. The crosscovariance calculations that relate reservoir
parameters to production data are then limited to these regions of
influence. Obviously, these regions will change with time as the
flood front progresses or the well conditions change. At each time,
the streamlines are used to localize the crosscovariance calculations. Such conditioning of the crosscovariance has also been used
in other applications such as weather forecasting, where the medium under consideration is more homogeneous (Houtekamer and
Mitchell 2001; Hamill et al. 2001). For example, Houtekamer and
Mitchell (2001) propose the use of an isotropic cutoff radius beyond which the influence of a given observation over the model
parameters is considered to be small. The parameters outside the
cutoff radius are then either damped or excluded during the computation of the crosscovariance matrix. By doing so, they avoided
the estimation of small or erroneous correlation associated with
remote locations. To filter remote observation, they used the Schur
product of the covariance times a correlation function. A similar
idea is used here, but we use streamlines to decide which gridblocks are strongly correlated to an observation. Our approach is
more physically based and intuitive. The use of streamlines is also
bettersuited for reservoir problems compared to the cutoffradius
approach, because production responses are often related to grid The Kalman Update. Update the reservoir model using the Kalman update equation (Eq. 4, discussed later). Repeat all steps again
until all production data are assimilated.
An outline of the procedure for our proposed approach is given
in the flow chart in Fig. 1 and can be easily incorporated in
any existing EnKF code. It only requires an additional step of
streamlinebased covariance localization compared to a standard
EnKF implementation. December 2008 SPE Reservoir Evaluation & Engineering Ensemble Forecast Step. We start with an ensemble of reservoir
models conditioned to static data. For each member of the ensemble, we simulate the production response up to the next available observation time, using either a streamline or a finitedifference simulator. For finitedifference models, we also trace
the streamlines for each member of the ensemble. The streamline
tracing is performed using the total phase fluxes from the simulator
(Jimenez et al. 2007).
Computation and Conditioning of the CrossCovariance Matrix. For each member, we utilize the streamlines to associate
gridblocks or regions that contribute to each producer at a given
time. Next, we stack the selected gridblocks/regions from all ensemble members to define a common region of influence for all
members. Using the production responses at current timestep from
each member, we compute the crosscovariance matrix, including
only the gridblocks within the common region of influence.
Computation of the Kalman Gain. Using the covariance of the
observed data, the model response and the crosscovariance from
the previous step, compute the Kalman gain (Eq. 6 given later). An Illustration of the Procedure. The detailed mathematical formulation behind our proposed approach will be discussed later.
First, we illustrate the procedure using a synthetic 2D example. For
this purpose, we will refer to the approach proposed by Evensen
(1994, 2003) and later introduced to reservoir history matching by
Nævdal et al. (2005) as the “standard EnKF” and the streamlinebased conditioning approach proposed here as the “SL EnKF.”
We set up a 2D reference model with a bimodal distribution of
the permeability. A nonGaussian distribution was chosen deliberately because of the difficulties encountered by standard EnKF
for such distributions. The reservoir consists of a 2D area divided
into 41×41 gridblocks. We simulate waterflood for 2,000 days
with eight producers and one injector as shown in Fig. 2. Both
fluids and the rock are assumed to be incompressible for this
case. The GSLIB (1992) was used to generate the initial ensemble
of 100 models satisfying the prior distribution and spatial continuity (Fig. 3a). Water cuts and bottomhole pressures from the
reference model were treated as observed data for history matching
using the EnKF.
Results From the Standard EnKF. Fig. 3b shows the updated
ensemble mean permeability and some ensemble members after
the application of the standard EnKF. Notice that much of the
geologic continuity in the initial model is absent from the updated
models. This is because of the tendency of the standard EnKF to
transform the logpermeability statistics to a Gaussian distribution
after many updates. The logpermeability histogram for one of the
updated ensemble members is shown in Fig. 4 (center), which
clearly shows the lack of the bimodal character. The fact that the
prior model statistics is not preserved is also indicated by the
nonlinear behavior of the QQ plot. Multivariate model parameter
distributions derived from the updated realizations can also be
assessed for normality (Vasco et al. 1996). The scaled and squared
deviations of each ensemble member from the ensemble mean,
represented by the respective Mahalanobis distances, tend to follow a chisquare distribution, which indicates posterior parameter
distributions that are multiGaussian (Devegowda 2008). This can
1047 Fig. 2—Ninespot waterflooding example; figure shows the porosity of the reference model and the position of the producer
and injectors. vicinity of the producer shows a stronger and more linear relationship compared to gridblock 1,523. In fact, there is very little impact of the permeability for gridblock 1,523 on the watercut response as is evident from the scatter in the crossplot. For most
practical applications with modest ensemble size, we do not have
enough samples to adequately estimate this crosscovariance, resulting in noisy or spurious estimates. Hamill and Whitaker (2001)
have shown that for such gridblocks where the correlation coefficient is small, assimilation of information using the inaccurate
values of the crosscovariance might have a detrimental outcome
in terms of poor state/parameter estimates and potential filter divergence. We intend to avoid this situation using our proposed
streamlinebased covariance localization approach. Fig. 1—Streamlineassisted ensemble Kalman filter flow chart. have serious ramifications in geologic modeling as the multiGaussian posterior models are incapable of reproducing the connectivities of the extremes, specifically flow channels and barriers.
It is interesting to note that in spite of the loss in the geologic
continuity, the final members are able to closely match the production data as shown in Fig. 5. Overall, the EnKF greatly reduces
the spread of the model responses around the observations. This
also shows the inherent nonuniqueness in history matching and
that not all models reproducing history can be considered valid for
prediction purposes. Next, we repeat the same history matching,
but this time we condition the crosscovariance matrix using the
streamline path information.
The Need for Covariance Localization. An important aspect of
the EnKF is to quantify the relationship between the state variables
using the appropriate crosscovariance measure. For our particular
example, the crosscovariance matrix relates the water cut and the
permeability at each gridblock. The computation of this cross covariance can be explained as follows. The EnKF uses an ensemble
of realizations of the permeability. For each realization, flow simulation is carried out to calculate the watercut response. Fig. 6
shows a cross plot of the different values of the permeability for
two selected gridblocks (gridblock 172 and 1,523) from the different ensemble members vs. the computed water cut at a particular assimilation time. Clearly, the gridblock 172, which is in close
1048 Conditioning the CrossCovariance and the SL EnKF. In practical applications of the EnKF, we need to keep the ensemble size
reasonable to minimize the computation cost. To accomplish this,
we condition the crosscovariance matrix by selecting those gridblocks for which the correlation with production response is expected to be significant. Fig. 7 shows all the gridblocks crossed by
streamlines arriving at the producer P8 at a particular time. The
results for two ensemble members, 15 and 73, are shown here. We
have also shown the result after stacking all the ensemble members. The highlighted gridblocks in the stack indicate those gridblocks that have a streamline passing through them and arriving at
producer P8 at least for one member of the ensemble. This stacked
representation now defines our region of influence for the entire
ensemble and only the gridblocks within this region are included in
the crosscovariance calculations.
Fig. 3 shows a comparison of the results from the standard and
the SL EnKF. It is clear that while the standard EnKF produces
over/undershooting of the permeability, the SL EnKF does not.
Also, a visual examination of the permeability fields indicate that
the geologic continuity is maintained. Fig. 4 shows a comparison
of the histograms of the prior and the updated permeabilities for
one of the ensemble members. Unlike the standard EnKF, the
bimodal distribution is preserved here. This is further reinforced by
the QQ plot, which now shows a linear behavior. Overall, the
results indicate that the SL EnKF appears to outperform standard
EnKF in terms of preserving the geologic realism during history
matching. Fig. 5 shows the initial and final spread of the water cut
at wells P1, P2, and P4 using the SL EnKF. The results show that
our proposed approach is able to assimilate the production data
without difficulty.
Comparison With DistanceDependent Localization. To illustrate the benefits of using the SL EnKF, we perform a comparison
between SL EnKF and EnKF using distancedependant covariance
December 2008 SPE Reservoir Evaluation & Engineering Fig. 3—(a) Initial permeability map for randomly selected members in the ensemble; (b) updated permeabilty after application of
the standard EnKF; and (c) updated permeability after application of the SL EnKF. localization (Houtekamer and Mitchell 2001). For distancedependant localization, we consider a localizing function centered
at the well and monotonically decreasing from a value of one at the
well location to a value of zero at some predefined cutoff radius
(Gaspari and Cohn 1999). Outside of this cutoff radius, it is assumed that the model parameters have no influence on the performance of the well. The choice of a cutoff radius is quite subjective and is a
potential weakness of the approach. We consider the synthetic
example shown in Fig. 2, which is 410 feet (41×10 feet) on each
side and use a cutoff radius of 100 feet for each well. Fig. 8b
shows the final spread of the watercut from wells P1, P2, and P4
in comparison with the initial model predictions in Fig. 8a. The
poor match underscores the importance of the proper choice of Fig. 4—Comparison of the logpermeability statistics for member 23. Left: Histogram of the reference logpermeability values and
the QQ plot comparing the true and initial histograms. Center: Histogram of the updated logpermeability values and the QQ plot
comparing the initial and updated histograms after using the standard EnKF. Right: The QQ plot comparing the initial and updated
histograms after using the SL EnKF and the corresponding histogram of the updated logpermeability values.
December 2008 SPE Reservoir Evaluation & Engineering 1049 Fig. 5—Comparison of the watercut spread for initial members (first row), the standard EnKF (second row), and the SL EnKF (third row). Fig. 6—Cross plot of ln permeability vs. water cut at Well P1 for different realizations at two gridblocks. Notice how the correlation
coefficient is much higher for the gridblock closer to Well P1.
1050 December 2008 SPE Reservoir Evaluation & Engineering Fig. 7—Gridblocks (left and center) selected by the streamlines arriving producer P8 at time 750 days for members 15 and 73.
Conditioning criteria for the covariance matrix for Well P8: the region of influence (right) is generated by stacking the gridblocks
from all the members at time 750 days. localization. Next, we chose a larger cutoff radius of 250 feet.
This choice seems to be more appropriate, as seen in Fig. 8c.
However, the benefits of using the streamlinebased localization
are evident when we compare the permeability distribution for an
arbitrarily chosen ensemble member shown in Fig. 9. Clearly, the
streamline based covariance localization keeps the changes minimal and is able to better preserve the characteristics of the prior
permeability field.
In general, the selection of the cutoff radius for distancedependent localization will require trial and error. Furthermore, the localization may not be consistent with the underlying heterogeneity that might include highpermeability channels with contributions from large distances. Using streamlines, we can effectively
define regions of influence with a sound physical basis that is tied
to the underlying flow field and geology.
Mathematical Background
This section briefly reviews the standard EnKF and discusses the
implementation of our proposed enhancements using streamlines.
A more detailed description including the underlying theoretical Fig. 8—(a) Initial watercut predictions. The reference model is shown in red. (b) Watercut predictions from the updated members
using a cutoff radius of 100 feet. The matches to the observed data in red are not satisfactory. (c) Watercut predictions from the
updated members using a cutoff radius of 250 feet.
December 2008 SPE Reservoir Evaluation & Engineering 1051 Fig. 9—Updated permeability for one realization. (a) The initial model; (b) the updated model using distancedependent localization,
and (c) the updated model using the SL EnKF. basis and derivations of EnKF can be found in several earlier
publications (Naevdal et al. 2005; Evensen 1994, 2003).
Reservoir State Vector. In EnKF, the joint model parameterdata
state vector y includes three types of elements: static variables ms
k
(e.g., permeabilities, porosity), dynamic variables obtained from
flow simulation md (e.g., pressure, phase saturation), and the obk
served production data dk (e.g., bottomhole pressure, well production rates, water cuts measured at the wells). Ά·
ms
k yp =
k md
k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) dk The state vector at time k is defined by Eq. 1, where superscript
p denotes prior, s stands for static and, d stands for dynamic. One
characteristic of the EnKF is that it uses an ensemble of state
vectors to estimate the mean and the covariance. The ensemble of
state vectors is represented by Eq. 2.
p
⌿p = ͕yp yp . . . yk,Ne͖, . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
k
k,1
k,2 where Ne represents the number of ensemble members. Each state
vector represents an individual member of an ensemble of possible
states that are consistent with the initial measurements from core,
well logs, and seismic data. The static parameters of the ensemble
members can be generated using appropriate geostatistical techniques such as sequential Gaussian simulation, indicator simulation, or by other means (Deutsch and Journel 1992). In general,
each ensemble member will result in a different forecast and we do
not know a priori which one is closer to the truth, if any. The best
we can do is to provide an uncertainty analysis from the ensemble.
A general way to describe a model state invokes a probability
distribution over the model space. Our goal is to explore the model
space by suitable stochastic simulation techniques to sample an
ensemble of models that have enough “broadness” to provides a
good estimation of the central tendency (mean) and dispersion
(covariance matrix).
The Forecast and the Update Steps. Ensemble Kalman filters
have two main steps: a forecast step and an update step. In this
work, the forecast step is carried out by a commercial streamline
simulator (Frontsim 2005). This action can be represented in our
equation notation as follows: ͭͮ
md
k
dk = f ͑ms , md ͒, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
k−1
k−1 where f represents a numerical solution of the porous media fluid
flow equations moving forward from time step k–1 to timestep k.
The forecast step is followed by the update step, whereby the state
variables are updated using the Kalman update equation as follows
(Evensen 2003):
1052 ⌿u = ⌿p + ⌲͑Dk − H⌿p͒. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
k
k
k
The superscript u denotes updated and p denotes prior. Here, matrix K is the Kalman gain and the matrix D represents an ensemble
of sampled observations, both defined later. The measurement matrix H is given below, where I is simply the identity matrix.
H = ͓0 I͔. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) The basic function of the matrix H in Eq. 4 is to select rows in ⌿p
k
corresponding to the calculated production data dk. The Kalman
gain matrix is given as follows (Evensen 2003):
K = Cp HT͑HCp HT + CD͒−1, . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)
⌿
⌿
where Cp represent the state vector covariance matrix; and CD
⌿
represent observation covariance matrix. The ensemble of sampled
observations Dk can be represented as follows:
Dk = ͕dk,1 dk,2 . . . dk,Ne͖, . . . . . . . . . . . . . . . . . . . . . . . . . . (7) dk,i = dk + i, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)
where dk represents a vector of any type of production data measured at time k, and i represent the noise in the observation for
each member i. The noise is assumed to be normally distributed
with mean zero a covariance given by CD.
Because the true state vector is not known, we approximate it
with the mean of the ensemble. Then the covariance matrix Cp can
⌿
be computed at any point in time.
yp = 1
Ne Cp =
⌿ Ne ͚y,
p
i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9) i=1 1
Ne − 1 Ne ͚ ͑y − y ͒͑y − y ͒ .
p
i p p
j p T . . . . . . . . . . . . . . . . . . . . (10) i,j=1 It is important to point out that the Kalman update in Eq. 4 is a
minimum variance estimate that is optimal only for Gaussian variables and linear dynamics. These conditions are typically violated
for reservoir history matching, rendering the estimates suboptimal.
Nevertheless, the results can be useful from practical point of view.
In the Kalman gain equation the covariance appears always
multiplied by the matrix H. Thus, in practice, there is no need to
compute the whole covariance matrix as we require only a small
portion of it. In our proposed approach here, we will further condition the matrix Cp to include only the regions that are crossed by
⌿
the datarelevant streamline trajectories. We call this streamlinebased covariance localization. To account for the conditioning
using streamlines, the covariance matrix is redefined as
Cp HT = ؠ
⌿ ͩ 1
Ne − 1 Ne ͚ ͑y − y ͒͑Hy − Hy ͒
p
i i,j=1 p p
j p T ͪ , . . . . . . . (11) December 2008 SPE Reservoir Evaluation & Engineering where is a correlation function (matrix) discussed next and represents the flow path information extracted from the streamlines
(see Fig. 7). The operation ° in Eq. 11 denotes the Schur product
operator, which is an elementbyelement multiplication of the
matrices (Houtekamer and Mitchell 2001; Hamill et al. 2001).
We can think of the correlation function as a matrix with the
column j filled with ones at the grid locations i selected in Fig. 7.
For other gridblocks in the same column, the correlation function
is set equal to zero. A similar procedure is repeated for all other
producers j until matrix ij is completed. We can build the correlation function at each assimilation time. In fact, it is possible to
define different types of the correlation functions depending upon
physical considerations. In this work, we have investigated two
types of correlation functions: one based on streamline (denoted by
SL) trajectories and the other based on the time of flight and the
front location as given next. ij 0סGridblock i is crossed by a SL arriving at well j
1 Gridblock i is not crossed by any SL arriving at well j
ij 0סGridblock i is crossed by SLs arriving at well j and the
gridblock is behind the saturation front
1 Gridblock i is not crossed by any SL arriving at well j
or the gridblock is ahead of the water saturation front.
The saturation front along the streamline at any time t is given
by the following relationship (DattaGupta and King 2007;
Carter):
dfw
= , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)
dSw t
where dfw/dSw is the derivative of the fractional flow curve with
respect to water saturation at breakthrough and is the time of
flight along the streamlines. Although Eq. 12 is derived under
restrictive conditions (uniform saturation and injection conditions),
our experience indicates that it can be applied more generally for
the limited purpose of covariance localization. Generally, for cases
involving water injection, there is no observable difference between the two types of covariance localization.
The use of streamlinederived information can easily account
for changing field conditions. Because streamlines simply reflect
the underlying flow field, changing field conditions, for example
infill drilling, do not pose any problems. The streamlines are retraced at each pressure update to account for changes in the well
operating conditions. Consequently, in our approach the correlation matrix changes with flow dynamics and highlights the association between wells and the grid locations for the conditions
existing in the reservoir at the time of assimilation.
Application and Discussion
We demonstrate the application and advantages of the SL EnKF
using two examples. First, we use the PUNQS3 reservoir model
for quantitatively examining the advantages of the SL EnKF. The
PUNQS3 application demonstrates the generality of our approach
in reservoirs with threephase flow and highly compressible fluids
(free gas). Next, we illustrate the practical feasibility of our proposed approach using a field example.
The PUNQS3 Example. The PUNQS3 reservoir model was
developed in the European Union by a group of companies and
universities. Detailed description of the model can be found elsewhere (Floris et al. 2001; Barker et al. 2001; Carter). The PUNQS3 reservoir model consists of 19×28×5 gridlocks, of which 1,761
are active. The gridblock sizes are ⌬x=180 ft, ⌬y=180 ft, and ⌬z
ranges from 1.3 to 8.8 ft. The reservoir has a small gas cap in the
center of a dome shape structure. It has a fault to the east and south
and a strong aquifer zone to the west and north (see Fig. 10). The
field initially contains six production wells located around the
gas/oil contact. Because of the strong aquifer influence, no injection wells are present. All six producing wells were produced as
follows: an extended well testing during the first year, then a
shutin period lasting the following 3 years, and finally a 4year
production period. The well testing period consists of four time
December 2008 SPE Reservoir Evaluation & Engineering Fig. 10—Top surface map showing well locations [from Floris
et al. (2001)]. windows, each of which is 3 months long with constant flow rate.
The oil production rate is fixed at 150 sm3/day within the 4year
production period. All wells have a 2week shutin each year to
collect shutin pressure.
The reservoir properties and an input data file can be downloaded from reservoir project website (Carter). In our historymatching examples, we used a commercial streamline simulator
(Frontsim). Because of the differences in the simulator, we did not
use the 16.5 years of production data given in the PUNQS3 website. Instead, we regenerated our reference production data.
The initial ensemble members were generated using the
GSLIB. Values for porosity at well locations and anisotropy information used to generate the true model are given on the PUNQS3 website. The horizontal and vertical permeabilities were calculated using a deterministic relationship proposed elsewhere
(Barker et al. 2001):
log͑kh͒ = 9.02 + 0.77
kv = 0.31kh + 3.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)
We history matched the model using the standard EnKF and the
SL EnKF. Production data was assimilated for a period of 8 years
(2,936 days). Next, using the updated models, we performed flow
simulation for all the ensemble members, starting from time zero
and forecast the production response for the next 8.5 years. We
chose to start from the beginning rather than latest assimilation
time to examine the ability of the updated permeabilities to preserve matches at earlier times and also to avoid any potential
materialbalance problems arising from phase updating.
As mentioned earlier, there were no injection wells for this
example. The streamlines arriving at the producer at a given time
are indicative of the drainage area associated with the well.
Clearly, this drainage area will change with time. At early assimilation times, the streamlines cover regions around the well only as
shown in Fig. 11a. At later times, the streamlines cover the whole
reservoir model as shown in Fig. 11b. For watercutlike measurements, it is possible to further condition the flow path information
using the saturation front location derived from Eq. 12.
Our goal here is to demonstrate the advantages of the SL EnKF
compared to standard EnKF. For this purpose, we conduct the
history matching using different ensemble sizes: 30, 60, and 100
members. Each case was run using the standard EnKF and the SL
EnKF. Fig. 12 shows a comparison of the initial and updated
bottomhole pressure, gastooil ratio, and water cut for both the
standard EnKF and the SL EnKF using 100 members. Both meth1053 where M is the number of gridblocks, ktrue is the reference permeability, and kmean is the mean permeability from the ensemble. The
RMS for the permeability was computed for all the cases with
different numbers of members in the ensemble. Fig. 14a shows the
results for this calculation. Notice that the RMS error for the
natural logarithm of the permeability is less for the SL EnKF
compared to the standard EnKF, especially at later assimilation
steps. This trend is the same for all the cases with different ensemble sizes tested here. This seems to indicate a lessened, or at
least delayed, tendency for the filter to diverge in the presence of
the streamlinebased covariance localization. Fig. 14b shows a
comparison of the RMS for different number of members in the
ensemble. Again, the SL EnKF appears to show improved performance. The difference in the RMS error is particularly significant
for small ensemble sizes, indicating the potential advantage of the
SL EnKF for largescale field applications. Fig. 11—Flow path information from streamlines. (a) Zones affected by the streamlines arriving at each producer at early
time. (b) Zones selected by streamlines arriving at each producer at later time. ods are able to significantly decrease the spread between the observed and calculated production response as we increase the number of assimilation steps. However, the advantages of the SL EnKF
becomes more evident when we analyze the error in the static
variable viz. the permeability.
To compute the variance in permeability, we use the reference
permeability as the truth and each members from the ensemble as
one sample of the random variable. The variance at each gridblock
can now be computed as follows:
var͑ln͑k͒͒ = 1
Ne − 1 Ne ͚ ͑ln͑k ͒ − ln͑k
i 2
ref ͒͒ , . . . . . . . . . . . . . . . (14) i=1 where ln(k) is the natural logarithm of the permeability for the
ensemble member i at a given gridblock, kref is the reference
permeability for the corresponding gridblock, and Ne is the ensemble size. This operation is performed for every gridblock. Fig.
13 shows a comparison of the variance maps for the standard
EnKF and the SL EnKF. These maps were computed using the
updated permeability members at time 2,936 days. As expected,
the variance is small around the well position. This is especially
evident at Layers 4 and 3, where the wells are completed. A closer
look of the variance maps from both the techniques reveals that
overall the SL EnKF results in smaller variance compared to the
standard EnKF. This is simply a reflection of the lack of overshoot/
undershoot of the permeability for the SL EnKF. Thus, we are able
to better preserve the geologic continuity. Although we have
shown the maps from the 60 members ensemble only, similar
results were obtained with ensembles sizes of 30 and 100.
Additionally, one can also compute the deviation of the ensemble members from the ensemble mean. This quantity is an
indicator of the variability among the ensemble members. Our
experience shows a faster loss of variability over a sequence of
assimilation steps with the standard EnKF as compared to the SL
EnKF. Consequently, with SL EnKF, there is a lesser tendency for
the ensemble to ignore newer observations and filter divergence.
Another possible way to check for errors in the estimated permeability is to compare the mean permeability from the ensemble
with the reference permeability using the RMS measure,
RMSln͑k͒ = 1054 ͱ 1
M M ͚ ͑ln͑k true,i ͒ i=1 − ln͑kmean,i͒͒2, . . . . . . . . . . . (15) The Goldsmith Field Case. We now discuss the application of the
standard EnKF to a field case and compare the results with the SL
EnKF. The field case is from the Goldsmith San Andres Unit
(GSAU), a dolomite formation in west Texas. We matched 20
years of waterflood production history. The pilot area (see Fig. 15)
consists of nine inverted fivespot patterns covering approximately
320 acres with an average thickness of 100 ft. The area has more
than 50 years of production history before the initiation of the CO2
project in 1996. Because of practical difficulties describing the
correct boundary conditions for the pilot area, wells around the
pilot area were included in this study. The study area includes 11
injectors and 31 producers. Production history information from
only nine producers is used because only these have significant
watercut response. The detailed production rate and the well
schedule, including infill drilling, well conversions, and well shutin, can be found elsewhere (He and DattaGupta 2001). The study
area was discretized into 58×53×10 gridblocks. The initial 100
realizations of porosity and permeability were obtained using sequential Gaussian cosimulation conditioned to well and seismic
data. Fig. 16 shows some of the initial members; notice the PDF is
not Gaussian because of the presence of multiple geologic facies.
Fig. 17 shows some of the ensemble members after assimilation of watercut data. Notice that the permeability distribution
after a sequence of assimilation steps became totally Gaussian. The
standard EnKF was unable to preserve the initial density function
of the permeability. The multiGaussian distribution leads to loss
of flow channels and barriers because of its maximum entropy
character. Also, initially the ensemble members have permeabilities ranging from 0.005 md to 500 md. However, after the data
assimilation, the updated permeabilities range between 2×10−6 md
and 1×106 md, which clearly indicates very large and unrealistic
changes. Such overshooting/undershooting problems have also
been observed by others (Gu and Oliver 2005, 2006; Naevdal et al.
2005). Fig. 18 shows the initial and final spread of water cut at the
wells. As before, a significant reduction in spread is observed.
Fig. 19 shows the ensemble mean permeability before and after
updating. The loss of structure in the permeability field is quite
apparent here. For this field example, the wells that contain watercut information are located toward the center of the model. Injectors are located around the producers (see Fig. 15). This suggests
that most of the water arriving at producers probably travels across
the reservoir volume surrounded by the injectors and the producing
wells. Thus, the major changes from the standard EnKF should be
preferentially found in the area subscribed by the injectors and
wells with observations. Unfortunately, this is not the case here.
The standard EnKF has produced changes in all parts of the model
regardless of the wells and injector position (see Fig. 19).
We now repeat the same history matching, but this time by
conditioning the covariance matrix using the streamline path information. To select the gridblocks for a specific producer, we use
only those streamlines that have broken through at a given time
(see Eq. 12). Fig. 20 illustrates such regions, considering all the
streamlines arriving at producer P6 at three different times. The
watercut matches are shown in Fig. 18 and exhibit similar behavior as in the case of the standard EnKF.
December 2008 SPE Reservoir Evaluation & Engineering Fig. 12—(a) BHP spread at Well PRO15; (b) GOR at Well PRO1; and (c) water cut at producer PRO11. The red bold line is the
reference. The gray line around 3,000 days shows time up to which the information was assimilated.
December 2008 SPE Reservoir Evaluation & Engineering 1055 Fig. 13—Variance map of the natural logarithm of permeability at 2,936 days. (a) 60 members ensemble using the streamlineassisted EnKF; (b) 60 members ensemble using the standard EnKF. Fig. 14—RMS error from the mean of the ensemble. (a) RMS error for the natural logarithm of the permeability after each assimilation step. (b) RMS error for the natural logarithm of the permeability for the standard EnkF and the streamlineassisted EnKF for
different number of members in the ensemble. The real advantage of using the SL EnKF becomes evident
during analysis of the final permeability fields. Fig. 21 shows the
updated permeability for some of the members in the ensemble.
Notice how the SL EnKF tends to preserve the initial permeability
distribution. Both the spatial continuity and the bimodal nature of
the permeability distribution are maintained. Furthermore, the
strong overshooting problems encountered while using the standard EnKF does not seem to exist here. An analysis of the changes
proposed by the SL EnKF (see Fig. 19) reveals that the changes are
small and mostly localized in waterswept zones, which is physically reasonable. Thus, the updated ensemble members are able to
preserve and inherit the nonGaussian distribution by limiting and
localizing the changes to regions supported by the dynamic data.
Parallel Implementation of the Algorithm and
Scalability
A major advantage of the EnKF is that parallelization of the code
can be achieved with relatively little effort. Several authors have
1056 Fig. 15—Goldsmith study area; well distribution producer with
watercut information highlighted with circles; injectors are
darker uncircled dots. [from He and DattaGupta (2001)].
December 2008 SPE Reservoir Evaluation & Engineering Fig. 16—Permeability fields for the Goldsmith case (randomly selected members: 17 left, 58 center, and 97 right.). Permeability
maps generated using sequential Gaussian cosimulation conditioned to wells and seismic data. Below each map, the histogram
of the permeability is shown. Fig. 17—Updated permeability maps using the standard EnKF conditioned to water cut. Below each map is shown the histogram.
December 2008 SPE Reservoir Evaluation & Engineering 1057 Fig. 18—Water cut 100 members ensemble spread at Wells P1, P3, P7. Initial (top row), after standard EnKF (center row), and after
SL EnKF (bottom row). The gray line around 4,000 days shows the time up to which the information was assimilated. Fig. 19—(First) Initial mean from the 100 members; (second) mean from the 100 members after standard EnKF update; (third) mean
from the 100 members after SL EnKF update; (fourth) standard EnKF changes in permeability; and (fifth) SL EnKF changes in
permeability. Fig. 20—Regions selected by the streamlines arriving to producer P6 (black circle) at different times: (left) time 1,680 days, (center)
2,280 days, and (right) 3,960 days.
1058 December 2008 SPE Reservoir Evaluation & Engineering Fig. 21—Updated permeability map using the SL EnKF after
water cut assimilation; below each map is shown the corresponding histograms. Notice the SL EnKF is able to preserve
the bimodal prior density function. recognized the feasibility of parallel computation for the EnKF
analysis. A rather simplistic approach to parallelize any EnKF
implementation will require running each forward model in parallel. Further parallelization can be done by parallelizing the Kalman update equation as discussed by other authors (Keppenne
2000). In our parallel implementation, flow simulation for each
member is performed in a different CPU (node). Results from each
member are written by the simulator in binaries to the same network file system (NFS). After each forward run, information is
retrieved from the NFS by the master node with a minimal level of
interprocessor communication.
Most of the examples presented in this paper could be run
within acceptable CPU time. Synthetic cases with grid sizes of
50×50×1 and 100 ensemble members typically took from 2 to 3
hours to assimilate 25 to 40 observations in a single CPU computer. While running our field case example on a single CPU
computer, it would take from 20 to 25 hours to assimilate the
observation data for an ensemble of 100 members.
We used the Message Parsing Interface (MPI) to parallelize our
code. Tests were run in the 128 CPU sharedmemory supercomputer at Texas A&M University. Even from our rather unoptimized
parallel implementation, the results appear to indicate that the
EnKF scales quite well in terms of CPU time as shown in Fig. 22.
It is clear from our experiments that the application of the SL
EnKF for continuous data assimilation into highresolution reservoir
models is quite feasible with the current level of computing power.
Conclusions
We have presented a novel streamlineassisted EnKF for continuous model updating using the selective flow path information from
streamlines. The approach avoids much of the problems associated
with the standard EnKF related to parameter overshoots and loss of geologic realism during history matching. We demonstrate the
power and utility of our proposed approach using both synthetic
and field examples. Some specific conclusions from this paper are
summarized next.
The streamline trajectorybased covariance localization appears
to eliminate and/or minimize previously reported problems when
using the standard EnKF for reservoir history matching. Some of
the reported problems include overshooting of the reservoir parameters, the loss of variance, and the loss of geologic continuities
for nonlinear problems or nonGaussian distribution, particularly
for modest ensemble sizes (<100).
When comparing the standard EnKF and the streamlineassisted EnKF in terms of reservoir static parameters (for example,
permeability), the SL EnKF is able to better preserve the histograms and the spatial continuity. The standard EnKF tends to make
the updated parameters Gaussian and results in loss of connectivities in the extreme values. This adversely impacts the performance
forecasting because the flow channels and barriers are no longer
represented properly in the geologic model.
When comparing the standard EnKF and the SL EnKF in terms
of their ability to match and reproduce historical production data,
both approaches appear to perform equally well.
The results presented here suggest that the streamlinebased
covariance localization may significantly improve the quality of
results from the standard EnKF or its variants. The cost of localizing the covariance will be significantly less compared to the cost
of generating a large enough ensemble to achieve the same level
of accuracy.
The parallel implementation of our approach appears to scale
favorably with respect to model size and the number of members
in the ensemble, making the approach suitable for history matching
and uncertainty quantification using detailed geologic models.
Nomenclature
CD סdata observation covariance matrix
Cp סjoint model parameterdata state vector covariance
⌿
matrix
Cp Ht סmodel data crosscovariance matrix
⌿
dfw
dSw סfractional flow derivative at the saturation front
dk סobserved or calculated data
Dk סensemble observation at time k
H סmeasurement matrix
HCp Ht סcalculated data covariance matrix
⌿
kv /kh סvertical/horizontal permeability
K סKalman gain
ms סstatic variable vector at time k
k
md סdynamic variable vector at time k
k
Ne סnumber of members in the ensemble Fig. 22—(a) Total running time for the Goldsmith field case as a function of the number of processors. (b) Speedup factor for the
same dataset.
December 2008 SPE Reservoir Evaluation & Engineering 1059 yp סprior joint model parameterdata state vector, at time k
k
y p סmean of the prior joint mode parameter data state
vector
␦ij סKroneker delta
i סwhite random noise data observation
סstreamline base correlation function
סstreamline time of flight
סporosity
⌿k סensemble of joint modeldata state vectors
Acknowledgments
The authors would like to acknowledge financial support from
members of the Texas A&M Joint Industry Project, MCERI
(Model Calibration and Efficient Reservoir Imaging) and from the
U.S. Department of Energy. We also would like to thank the Texas
A&M University Supercomputing Facility for their support.
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Geophysics 61(4):12091227. DOI:10.1190/1.1444040. Elkin Arroyo is a reservoir engineer currently working for Occidental Oil & Gas. He has previously worked for the Colombian
National Oil Company Ecopetrol ICP and for Numerica Limited, an engineering consulting firm in Colombia. He won first
place in the SPE Golf Coast Region Student Paper Contest
(2006). He holds a BS degree in mechanical engineering from
the Universidad Industrial de Santander in Colombia and an
MS degree in petroleum engineering from Texas A&M University. Deepak Devegowda is Assistant Professor in the Petroleum
Engineering Department at the University of Oklahoma. His research is focused on problems related to reservoir characterization, geostatistics and unconventional oil, and gas recovery. He holds PhD and Masters degrees in petroleum engineering from Texas A&M University. Akhil DattaGupta is Professor
and holder of the LeSuer Chair in the Petroleum Engineering
Department at Texas A&M University. He holds a PhD degree in
petroleum engineering from the University of Texas at Austin
and has worked for BP Exploration/Research and the
Lawrence Berkeley National Laboratory. He is the recipient of
the 2003 SPE Lester C. Uren Award for significant technical contributions in petroleum reservoir characterization and streamlinebased flow simulation. He is an SPE Distinguished Member
(2001), Distinguished Lecturer (1999–2000), and Distinguished
Author (2000) and was selected as an Outstanding Technical
Editor in 1996. He also has received the Cedric K. Ferguson
Certificate (2000, 2006) and the AIME Rossitter W. Raymond
Award (1992). Jonggeun Choe is an associate professor in the
department of energy system engineering at Seoul National
University email: [email protected] Choe is also a member
of the Research Institute Of Engineering Science in the university. His main areas of research are well control and inverse
modeling for data integration and characterization. Before
joining the university, he was a research engineer at Texas
A&M University and developed conventional and SMD wellcontrol simulators. He was the recipient of the AIME Rossiter W.
Raymond Memorial Award in 2000. He holds BS and MS degrees in mineral and petroleum engineering from Seoul National University and a PhD degree in petroleum engineering
from Texas A&M University. He is one of technical editors of SPE
Drilling & Completion and a member of SPE, IAMG, and KSGE.
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