1
B008 COMPARISON OF METHODS FOR DOWNSCALING
OF COARSE SCALE PERMEABILITY ESTIMATES
Alv-Arne Grimstad1 and Trond Mannseth1,2
1 RF-Rogaland Research
2Now with CIPR - Centre for Integrated Petroleum Research, University of Bergen
Abstract
Fine scale reservoir models with all available information integrated are important for predictions of future behavior. Traditionally, fine-scale realizations of the geostatistical model are
adjusted in some way to integrate production data. Since these data do not support high resolution of the permeability representation, strong regularization of the history matching problem
is needed.
In this paper, we study how integration of information may be approached from the other
direction. I.e., we study how information from well observations and a geostatistical model may
be integrated with a coarse scale permeability estimate obtained through a multiscale estimation
sequence (Adaptive multiscale estimation).
There are several possible solutions to this problem, depending on which aspects of the
geostatistical model it is important to match. We compare some methods with respect to how
the resulting permeabilities match the available information.
Introduction
A major motivation for building reservoir models is to be able to make predictions about the future
behavior of the reservoir under different scenarios. It is important to utilize as much information
as possible when building the model.
In addition, accurate simulation of some scenarios (e.g., investigation of placement of new wells)
require a detailed numerical gridding of the model, at least in parts of the reservoir. Thus, we
would like to have a fine scale reservoir model that is consistent with all available information.
Several types of information about the permeability in the reservoir may be available. Geostatistical
models try to quantify the spatial variability throughout the reservoir, but are based on assumptions
about this variability. Well logs and core samples may be detailed, but informs only about the close
vicinity of the wells. Production data, on the other hand, informs about the permeability in large
areas between the wells, but only about large scale variations. Integrating all the information at
the same scale of the reservoir model may not be advantageous, due to issues with computational
cost and numerical instability. Some up- or downscaling of information will be necessary if we want
to combine several information sources.
A possible solution to the data integration task is the scale splitting method[3]; Adaptive multiscale
estimation (AME)[2] followed by fine scale data integration. One aim of AME is to obtain a history
matched estimate where the resolution of the permeability is determined by the information content
in the available production data.
After the history matching, information from other data types is integrated in a downscaling step.
Several fine scale realizations may be generated from a single (or a few) history matched coarse
scale model(s). This method is in contrast to history matching several fine scale realizations of the
geostatistical model.
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August – 2 September 2004
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Since the time-consuming part of reservoir characterization is the conditioning to production history, the method with the fewest history matching runs should be the least expensive. However,
the quality of the resulting permeability realizations should also be discussed. The match to production history and well observations is relatively easy to assess. Adherence to the parameters of
the geostatistical model may also be quantified. But since the geostatistical model is only assumed,
it may not be necessary to match all aspects of the model. It is not obvious which parts of the
geostatistical model that are most important and need to be closely matched. The permeability
is not uniquely determined by the available information, so the choice of which aspects of the
geostatistical model to match will strongly influence the final permeability estimate(s).
In this paper, we start with a discussion of some possible methods for downscaling a history
matched (2D) coarse scale permeability estimate, i.e., integrating fine scale information such as well
observations and geostatistical model. Some examples with application of the discussed methods
are given, showing how the different ways of using the geostatistical model result in different types of
permeability estimates. The values of various “measurables” of the resulting permeability estimates
are calculated. These are the matches to the production history and well observations, and the
histogram of the permeability field.
Finally, we discuss and show how we may perform a final history matching step after the downscaling, using the permeability resolution obtained in AME, (i.e., using a limited number of parameters),
but with smooth basis functions.
Coarse scale history matched estimate
We use Adaptive multiscale estimation (AME)[2] to obtain a coarse scale estimate of the permeability. AME solves a series of history matching problems with increasing resolution of the permeability
representation. The permeability at each step is parameterized as k(x) = cTN ψ N (x), where ψ N
is a vector of piecewise constant basis functions. The parameters cN are estimated by finding the
minimum of the objective function
J(cN ) = (d − m(cN ))T D−1 (d − m(cN )),
(1)
where d is the measured production data, m is the corresponding data calculated with the reservoir
simulator, and D is the covariance matrix of the measurement errors.
Between the parameter estimation steps of AME, the parameterization is evaluated, investigating
which refinement(s) would allow the largest improvement of the match to the data. This iterative
procedure is terminated when the data is sufficiently well matched (data mismatch comparable
to magnitude of measurement errors), or when no new improvement is achieved. In this way we
obtain a coarse scale history matched permeability estimate. (More than one coarse scale estimate
can be obtained with a modification of the algorithm.[5])
Downscaling methods
In this section we briefly discuss the methods we have investigated for downscaling of the coarse
scale estimate. The references given for each method should be consulted for detailed explanations.
Kriging
Kriging (see, e.g., [1]) is a minimal-variance correction to an initial estimate, k0 , incorporating
well observations (point measurements) of permeability in a manner consistent with a covariance
model/function (the geostatistical model). The equations for the correction, ∆kKr , can be formulated as:
(2)
∆kKr (x) = CAT (ACAT + Σd )−1 (d − Ak0 (x)).
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In this formulation, Σd is the error covariance matrix of the permeability measurements. C is the
covariance matrix of the permeability field prior to incorporating the point measurements, and A
is an observation operator, such that Ak0 are the estimated permeabilities corresponding to the
observations d.
To use Kriging for downscaling, we set k0 = kAME , and use equation (2) with C = (CP−1 + G)−1 ,
where CP is the (prior) covariance matrix for the geostatistical model, G is the (fine scale) local basis
information matrix from the AME estimation problem. This has the result that the perturbation
∆kKr only slightly (according to the uncertainty of the estimated permeability) alters the mean
value of the permeability in the parameterization regions defined by AME.
The Kriged estimate integrates production data and well observation information, and the permeability perturbation is consistent with the geostatistical model. However, since the coarse scale
estimate, kAME , is piecewise constant, the downscaled estimate will only be piecewise smooth.
Multiple realization of the permeability may be obtained by drawing realizations from a distribution
with the updated covariance matrix after Kriging, and adding these to the Kriged permeability.
The realizations obtained in this manner will, like the Kriged estimate, be piecewise smooth.
Conditioning using Kriging
The equation for the Kriging update may also be used in another way. If k0 is an unconditional
realization from a distribution with covariance matrix C, then k0 + ∆kKr , with ∆kKr given by
equation (2), will also be a realization of the same distribution, but conditioned on d − Ak = e,
where e is a random vector with covariance matrix Σd . (See, e.g., section 3.3 of [6].)
To use this in downscaling, we use as “observations” the estimated mean values in the parameterization regions determined in AME, and the available well observations. The observation operator,
A, then consists of two parts: The first NAME rows extract the mean value of the permeability in
the parameterization regions, while the NW last rows extract the permeability estimates in the grid
blocks where the wells are located.
Some consideration should go into the construction of the measurement error covariance matrix,
Σd . For the first part of the observation vector, we use the covariance matrix of the estimated
region means. For the second part we use the covariance matrix of the permeability measurement
errors. Since these are different types of measurements, the optimal scaling between the two are
not obvious. However, for the cases in this paper, we have disregarded this issue. Regarding the
cross covariance: Even if the mean permeability of a parameterization region and the permeability
of a single block inside this region have a nonzero correlation, the observation error for these two
values should not be correlated. Thus, we use
ΣAME
0
,
(3)
Σd =
0
ΣW
where ΣAME is the estimation covariance matrix for the regional permeability means, and ΣW is
the covariance matrix of the permeability measurement errors.
This way, we can turn unconditional realizations of the prior geostatistical model into realizations
conditioned on the coarse scale estimate and the well observations.
Gradual deformation
This was proposed as a method for conditioning realizations of a geostatistical model to production
data (see, e.g., [4]). It consists of a series of perturbations of an initial realization. At each step,
k
(k+1)
= a1 k
(k)
+
n
ai ki ,
i=2
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August – 2 September 2004
(4)
4
where k(k) is the old estimate, k(k+1) is the new estimate, and {ki } are realizations of the geostatistical model. The coefficients {ai } that give the best match to the production
data are sought. To
ensure that the covariance properties of k(n) is the same as those of {ki }, ni=1 a2i = 1 is enforced.
The method was further developed[7] by noting that the mean value is also conserved if ni=1 ai = 1.
To have non-trivial solutions for the coefficients {ai }, we need n > 3. This development of the
method allows one to use as {ki } realizations that are conditioned on well observations, and have
k(n) also be conditioned on these observations.
In this paper, we use this method with realizations based on the Kriged coarse scale permeability.
This will give a better starting point for obtaining final realizations that match both types of data.
Other methods
Sequential simulation with Block Kriging[8] is another method for downscaling a coarse scale history
matched estimate. Due to limited time, we could not examine this method closely.
Final history matching step
Since the fluid flow in the reservoir is influenced by fine scale permeability variations, the downscaling will most likely cause the match to the production history to deteriorate. A final history
matching step may be needed to obtain a satisfactory match to all the data. However, since the
downscaled permeability is strongly influenced by the history matched permeability, the required
perturbations in a final history matching step will likely be small.
This history matching step will need to use fine scale permeability representation, but there is no
need for as many degrees of freedom as the number of grid blocks. The parameterization obtained
in AME should be a good indication on the required resolution. The AME parameterization
could be used as it is for the final perturbations, but if the geostatistical model prescribes smooth
permeability variation, it could be better to use perturbations that are smooth. Both non-smooth
and smooth perturbations will most likely be able to produce final estimates that match production
data. The choice between the two approaches depends on how closely we want to follow the
geostatistical model.
In the following, we consider how to construct perturbations to the downscaled permeability that
have a smoothness consistent with the geostatistical model, and a resolution consistent with the
AME parameterization. To do this, we use equation (2) to define an expansion basis for the
perturbations: Let the perturbations to the downscaled permeability be of the form ∆k(p) = Bp,
where
(5)
B = CAT (ACAT + Σd )−1 ,
and minimize the objective function with respect to p. The columns of B are linear combinations of
the columns of C, and the dimension of p is (NAME + NW ). Examples of the type of perturbations
obtained in this way are shown in Figure 6 in the examples section.
Examples
The examples presented in this section are based on a synthetic horizontal 2D case, where the
reservoir is discretized into a 48 × 48 × 1 simulation grid. Initially, the reservoir contains 100% oil,
and it is produced using 3 water injection wells and 5 production wells.
The information available for reservoir characterization is:
Production history data: Observed time series of pressure in injection and production wells, oil
production rates and water production rates in production wells; 150 observations in each series,
2700 observations in all.
Well observations: The permeability at the position of the wells, 8 observations.
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Geostatistical model: Assumed type of model (multigaussian with Gaussian covariance function),
with assumed correlation function and mean value for the permeability.
In addition, we assume known the porosity of the reservoir (uniform), and relative permeability
and capillary pressure functions.
Reference permeability and coarse scale history match
The reference permeability is generated as a random realization of the geostatistical model. The
logarithm of the permeability is assumed to follow a multigaussian distribution with a Gaussian
correlation function. The mean value is loge (200) = 5.3, the variance is 0.3 and the range is 25 grid
blocks.
Synthetic production data is obtained by running the simulator with the reference permeability as
input, and adding uncorrelated Gaussian noise to the simulator output.
6
6
5.5
5.5
5
5
4.5
4.5
Figure 1: Left: Reference permeability. Right: Coarse scale estimate obtained by AME with parameterization
regions. The wells are shown as dots.
The reference permeability is shown in Figure 1, together with the coarse scale history matched
permeability obtained with AME. The coarse scale estimate uses 17 parameters and obtains an
objective function value of 2750, which is close to the expected value, (Nobs − Npar ). In the AME
permeability we recognize many of the coarse scale features of the reference permeability.
The match to the production history with the AME permeability is shown in Figure 2.
70
300
2500
3500
2000
3000
1500
2500
1000
2000
500
1500
60
250
50
40
200
30
20
150
10
100
0
50
100
150
0
0
50
100
150
0
0
50
100
150
1000
0
50
100
150
Figure 2: Production history from reference permeability and match obtained with the coarse scale estimate (AME).
From left to right: Pressure in injection wells, pressure in production wells, water production rate and oil production
rate.
Downscaling step
In this section we show the permeabilities obtained with the downscaling methods discussed above.
Figure 7 should be consulted for measures of the match to production history, well observations
and geostatistical model for each of the permeabilities.
The permeability estimate obtained when we adjust the AME permeability by Kriging against
the mismatch with observations in the well grid blocks is shown in the left plot of Figure 3. The
right plot in the same figure shows the smooth change from the AME permeability. The resulting
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August – 2 September 2004
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permeability now contains small scale variation, but the discontinuities from AME still dominates.
6
0.2
0.15
0.1
5.5
0.05
0
−0.05
5
−0.1
−0.15
4.5
−0.2
Figure 3: Left: Downscaled estimate obtained with Kriging (Kr). Right: perturbation from coarse scale estimate.
The first two plots in Figure 4 show realizations of the permeability obtained from the Kriged
permeability in Figure 3. The small scale variability of these realizations is consistent with the
geostatistical model, but the overall result is dominated by the discontinuities of the AME permeability. Still, the match to both production data and well observations is satisfactory.
The last two plots of Figure 4 show realizations of the permeability obtained with Conditioning
using Kriging. This method produce smooth permeabilities, as expected, but we note that a
strong conditioning on the observations (regional mean and wells) may cause the variance of the
permeability to increase somewhat.
6
6
6
6
5.5
5.5
5.5
5.5
5
5
5
5
4.5
4.5
4.5
4.5
Figure 4: First two plots: Permeability realizations based on Kriged downscaled estimate (KrR). Last two plots:
Downscaled estimates obtained with Conditioning using Kriging (CK)
Two permeability realizations obtained when using Gradual Deformation on realizations based on
the Kriged estimate (Figure 3) are shown in the first two plots of Figure 5. The discontinuities of
the realizations based on Kriging are inherited.
6
6
6
6
5.5
5.5
5.5
5.5
5
5
5
5
4.5
4.5
4.5
4.5
Figure 5: First two plots: Downscaled estimates obtained with the Gradual Deformation method (GD) on KrR
realizations. Last two plots: Results after the CK permeabilities are subjected to a final HM step (CKHM).
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August – 2 September 2004
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Additional HM step
The Gradual Deformation method is a way to perform a final history match on the downscaled
estimates obtained with the Kriging method. This may be necessary since the match to production
data has deteriorated as a result of the downscaling. Each minimization problem solved in the
course of this method should be well posed due to the small number of parameters. However,
because of the randomness in choice of perturbation direction, the progress is slow.
The realizations from Conditioning using Kriging (last two plots of Figure 4) has also been subjected
to a final history matching step. This history matching follows the method outlined in the section
on “Final history matching step”. The resulting permeabilities are shown in last two plots of
Figure 5. The change between kCK and kCKHM can be expressed as a linear combination of smooth
basis functions, where each of the basis functions is a linear combination of the columns of the
covariance matrix. A few of the basis functions are shown in Figure 6. Note that these functions
have global support, but primarily change the value of only one “observation” of the A operator
(e.g., the mean value of the permeability in one of the parameterization regions of AME). Due to
the term Σd , some change is allowed in the value of the other observations also.
1
1
1
1
0.5
0.5
0.5
0.5
0
0
0
0
−0.5
−0.5
−0.5
−0.5
−1
−1
−1
−1
Figure 6: Some of the basis functions for the final history matching step. The parameterization regions of AME are
also shown.
Discussion
Figure 7 shows how the permeabilities downscaled with the different methods match the available
information. In the left plot we note that the AME permeability (AME) gives a good match to
the production history. As the permeability is Kriged (Kr), the match to well observations (middle
plot) is improved at the expense of the match to production history. As expected, the realizations
based on the Kriged permeability (KrR) show some variations in the match to both kinds of data.
2
2
10
6.5
10
6
1
10
5.5
1
10
5
0
10
4.5
0
10
−1
AME
Kr
KrR
GD
CK
CKHM
10
AME
Kr
KrR
GD
CK
CKHM
4
AME
Kr
KrR
GD
CK CKHM
Figure 7: Left: Mean square normalized error in match to production history. Middle: Mean square normalized
error in match to well observations. Right: Observed histograms.
The permeabilities obtained with Gradual Deformation (GD) further improve both the history
match and the match to well observations compared to the Kriging realizations.
9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August – 2 September 2004
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The realizations obtained with Conditioning using Kriging (CK) exhibit a dramatic improvement
in the match to production data through the final history matching step (CKHM). The match to
well observations is also improved, but not as much.
The right plot in Figure 7 shows the histograms of the permeabilities, with the prior model shown
as a reference to the left.
The results show that with a final history matching of the realizations obtained with Conditioning
using Kriging, we obtain fine scale permeabilities that match both production data and well data,
as well as the assumed histogram of the permeability distribution. The final history matching step
is performed with a fine scale permeability, but with only few degrees of freedom, comparable to
the resolution obtained in AME. Thus the computational work is reduced compared to minimizing
with full grid block parameterization. Due to the relatively good starting point, the history match
for each of the realizations converges with only a few iterations of the parameters.
Summary
The different areal support and associated length scale of available information make integration
of the information into a permeability estimate a difficult task. Which aspects to match for each
information type influences the integration method and the resulting estimates. The issue of what
constitutes a good match (of the different types of information) deserves more attention.
We have investigated some possibilities for downscaling a coarse scale history matched permeability estimate obtained with Adaptive multiscale estimation. The downscaling incorporates well
observations and assumptions about the geostatistical model into the estimate.
We have also shown how a final fine scale history matching step may be performed on the fine scale
realizations using only few degrees of freedom, making good use of the permeability resolution
obtained with Adaptive multiscale estimation.
Acknowledgements
The research presented in this paper has been financially supported by ENI SpA, Norsk Hydro
ASA, and The Research Council of Norway.
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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August – 2 September 2004