1
B030 A MULTIPLE MODEL APPROACH TO HISTORY
MATCHING AND UNCERTAINTY ANALYSIS
USING TIME-LAPSE SEISMIC
KARL D STEPHEN*, JUAN SOLDO, COLIN MACBETH AND MIKE CHRISTIE
Heriot-Watt Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, Scotland, EH14 4AS, UK
Abstract
We present an automated multiple model history matching method, which integrates time-lapse
seismic with production data, and determines parameter uncertainty. For each simulation model,
we compare observed seismic attributes to synthetic impedance obtained via a petroelastic
modelling step and grid transformation. For seismic and production data, we then obtain a misfit
which is used to update our model parameters in a Bayesian framework and accounts for model
errors and data covariance. We show examples of the method's application to a UKCS field
study and discuss uncertainty of the reservoir parameters.
Introduction
Reservoir management requires tools such as simulation models to predict asset behaviour.
History matching is often employed to alter these models so that they compare favourably to
observed well rates and pressures. This well information is obtained at discrete locations, and
thus lacks the aerial coverage necessary to accurately constrain dynamic reservoir parameters
such as permeability and the precise location and effect of faults, among others. Time-lapse
seismic captures the effect of pressure and saturation on seismic impedance attributes giving
two-dimensional maps or three-dimensional volumes of the missing information. The process of
seismic history matching attempts to overlap the benefits of both types of information to improve
estimates of the reservoir model parameters.
We present an automated history matching method of including time-lapse seismic along with
production data, based on an integrated workflow (Figure 1). It improves on the classical
approach, where the engineer manually adjusts parameters in the simulation model. Our method
also improves on gradient-based methods, such as Steepest Descent, Gauss-Newton and
Levenberg-Marquardt algorithms (e.g. Lépine, et al., 1999, Dong and Oliver, 2003, Gosselin et
al., 2003, Mezghani, et al, 2004), which are good at finding local likelihood maxima but can fail
to find the global maximum. Our method is also faster than stochastic methods such as genetic
algorithms and simulated annealing, which often require more simulations and have slower
convergence rates, possibly ignoring previously determined misfit information when selecting
new parameters. Finally, multiple models are generated enabling uncertainty analysis in a
Bayesian framework. The posterior probability surface is resampled to obtain parameter
distributions.
We have applied our method to a UKCS turbidite reservoir, in which time-lapse seismic has
shown great promise. The original geological model was constructed by the field operator using
typical approaches where facies objects and static flow properties such as porosity and
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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permeabilities were distributed stochastically. The model was constrained to well logs and a
qualitative match to the baseline seismic survey was obtained. The operator upscaled and then
manually history matched the model to the well production data before we applied our method.
M=Σ…
Sample
from
distribution
Generate
Multiple
Models
Compare
Seismic &
Reservoir
Model
Evaluate
Misfit
Update
Probabilities
Figure 1. Schematic of the iterative automatic history matching process. The result of this process generates a
posterior probability density function (PPD), which can then be resampled.
Conventionally, geological models are constructed at a fine scale and upscaled (Christie, 1996)
while in our approach the models are generated at a scale appropriate for simulation. This speeds
up the generation process and removes the need for the upscaling process as part of history
matching.
We modified the model by combining geostatistical methods. We used simulated annealing to
update the three-dimensional model of Net To Gross (NTG). Permeabilities have been updated
using a pilot point method with Kriging (Deutsch and Journel, 1992). We also history matched
for fault transmissibility multipliers as well as rock physics parameters.
Method
The multi-dimensional global optimisation approach is broken into a number of components
(Figure 1) with an application of the stochastic Neighbourhood Algorithm (NA) (Sambridge,
1999a), which has been applied previously in history matching reservoirs to production data
(Subbey et al, 2002, Christie et al., 2002).
Sample from Distribution: The multiple model approach relies on sampling the parameter space
a number of times, ns (typically 10 to 64), and simultaneously running forward simulation to
obtain a misfit. Resampling takes place by dividing the parameter space into Voronoi cells. The
best nr models are selected and ns/nr new models are randomly located in each Voronoi cell in
this sub-sample. The process is repeated to build an approximate posterior probability
distribution (PPD), which may be resampled later as part of the uncertainty analysis.
Generate Multiple Models: A set of models are created and converted into forward simulations
of the production process.
Compare Seismic and Reservoir Model: The observed seismic (Figure 2a) is compared to the
impedance attributes (Figure 2c) using a petroelastic model (e.g. MacBeth et al., 2003) and fluid
saturations and pressures from the simulations. For each simulation cell, we calculate the dry
bulk modulus using (MacBeth, 2004):
κ
r
dry
=
r
κ inf
1 + E kr exp( − Peff / Pkr )
(1)
where the superscript r identifies rock type (sand or shale), the constants κinf, Ek and Pk are
determined empirically, Peff is the difference between the overburden pressure and the pore
3
pressure. A similar equation is used for the shear modulus, µ r. We then use Gassmann’s
Equation (1951) to get the saturated bulk modulus:
κ
r
sat
=κ
r
dry
( 1 − α )2
+
φ α−φ
+
κg
κ rf
(2)
where φ is the porosity, κg is the bulk modulus of the mineral, α = (1-κrdry/κg) and κf r is the fluid
bulk modulus given by the saturation weighted harmonic average of the individual phase bulk
modulii. The P-wave modulii for sand and shale are obtained from M r=κ rsat+4µ r/3. (shale is
assumed to consist of dry frame only) and the value for each cell, Mcell, is obtained from the
harmonic mean of the sand and shale values, weighted by the respective fractional volumes
(Postma ,1955). Finally, the impedance is calculated using:
I = ρV p = ρM cell
(3)
where ρ is the bulk density of the cell obtained by averaging the densities of the rock frame and
the fluid densities. A map of impedance is obtained first by upscaling vertically using the volume
weighted arithmetic average and then downscaling to the seismic grid by interpolation. Predicted
impedances are then compared to the observed equivalent, obtained as integrated RMS of the
migrated stack after cross-equalisation and calibration to well seismic. Both data sets are
normalised prior to comparison.
Evaluate Misfit: A single misfit objective function is obtained for each model that we create
incorporating a comparison between observed and predicted production and seismic data. For
each variable being compared we use the following equation:
obs
mod
J i = xi − xi
error
+ xi
−1
−1
obs
mod
C m ,i + C d ,i x i − x i
error
+ xi
(4)
where the xi is the ith data vector under comparison, with superscript obs or mod for observed and
modelled respectively, and Cd,i is the data covariance matrix capturing data errors while Cm,i
captures model error covariance, xierror is the model error.
We obtain the data covariance by band pass filtering the observed seismic and production data to
remove high frequency noise. We can then assume the covariance matrix to be either: (i)
diagonal, (ii) an exponential model (iii) or it is obtained numerically. We find that random
sampling of the data by volume can speed up calculations by an order of magnitude. Otherwise,
calculation of the seismic misfit can take as long as the flow simulation.
We also account for the model error. Both observed and simulated data deviate from the true
answer by the observation error and the simulation error respectively and we can express the
misfit in terms of the difference of these errors. Model errors are then determined by fine scale
simulation of representative models and subsequent interpolation for other models.
Update probabilities: Our beliefs are updated from a prior PDF, p(m), using Bayes theorem to
give the probability of the model, m, given the new data, O :
p( m | O ) =
p( O | m ) p( m )
∑ p( O | mi ) p( mi )
(5)
where p(O|m) is the likelihood of the observed data obtained from
⎛
⎞
p( m | O ) ∝ 1 / exp⎜ ∑ J i / 2 ⎟
⎝ i
⎠
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Resample the PPD for uncertainty analysis: The PPD is then resampled using Markov Chain
Monte Carlo (MCMC) methods to determine probability distributions of the reservoir description
parameters (Sambridge, 1999b). If sufficient models are generated in the above process, this
MCMC resampling can continue without further reservoir simulation. Results are generated
equivalent to running an order of magnitude or more additional models.
(a)
(b)
(c)
Figure 2. Seismic maps (a) derived from RMS amplitudes, (b) the same upscaled to the simulation grid and (c)
predicted seismic derived by simulated annealing. The RMS values are normalised by subtracting the mean and
dividing the standard deviation. The simulation model contained cells measuring approximately 100 m x 100 m x 6
m while the seismic data was obtained for 12.5 m x 12.5 m x 25.0 m bins.
Results
Synthetic models
As an initial test, we used a small sector realization of the operator’s simulation model as a truth
case and matched to the production and synthetic seismic data. As unknowns, we defined six
permeability multipliers (Figure 3). Matching to production data alone (Figure 4) shows that a
local minimum misfit was obtained after 8000 simulations. The NA method is quasi-global such
that it depends on the parameter space being sufficiently sampled. Under-sampling of the
parameter space combined with the non-uniqueness of the well data resulted in the local
minimum. The additional constraint invoked by the seismic data resulted in a near perfect match
to the truth case (Figure 5), however. Analysis of the PPD quantifies the uncertainty of the
values obtained (Figure 6).
Matching observed baseline data
We found that the history matching process was dramatically improved when the baseline survey
was more closely matched, compared to the initial model provided by the operator. To do this,
we upscaled the observed RMS amplitudes (Figure 2a) onto the simulator grid (Figure 2b) and
normalised the data to give an estimated Net to Gross (NTG) map assuming a linear relationship.
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This map was then used as a constraint in simulated annealing together with a NTG distribution
from well data and a covariance model determined from the seismic. We retained the original
NTG values at the wells that were used in the operator’s model. A new three-dimensional model
of NTG was obtained with a much improved comparison (Figure 2c) and correlation between the
NTG map and RMS amplitudes. The misfit was reduced by one third.
1000000
29
571
1113
1655
2196
100000
1
3
4
5
6
10000
misfit
2
1000
100
10
Injectors
0
Producers
Figure 3. Plan view of permeability showing block
multipliers. The same value was applied throughout
each group of cells.
2000
4000
8000
Figure 4. Misfit value versus number of simulation
runs when matching to production data only. A local
minimum misfit was obtained.
0.18
10000
1
2
3
4
5
6
0.16
1000
0.14
100
0.12
probability
misfit
6000
number of iterations
10
1
0.10
0.08
0.06
0.04
0.1
0.02
0.01
0
500
1000
1500
2000
2500
3000
number of iterations
Figure 5. Misfit versus number of iterations matching
to both production and seismic data. Weighting factors
scale the misfit relative to Figure 4.
0.00
0
1
2
3
4
5
6
multiplier
Figure 6. Distributions of the permeability multipliers
obtained by MCMC sampling of the PPD. The legend
indicates the cell in Figure 3. Cell 2 had a value of 4.0
in the truth case while the rest had a value of 1.0.
Scaling issues
The differences in scales between the predicted and observed seismic result in a minimum
resolution that can be obtained in the history matching process. The best that we can hope to
achieve is a predicted seismic similar to the upscaled seismic (Figure 2b) used in the estimation
of our NTG map. We calculated the minimum from the upscaled data (Figure 2b) and normalise
subsequent misfits to this value.
Thresholding to eliminate noise
Time-lapse seismic data inevitably contains noise, which we would like to remove. Where
saturation effects dominated Equation 2 we can apply material balance to account for changes in
the seismic attributes (MacBeth et al., 2004) thereby identifying noise thresholds. Since the
reservoir is more affected by pressure, we applied different thresholds simultaneously to the
observed and predicted impedances. The misfit was reduced (data not shown) but the ranking of
initial models is unchanged producing very similar final parameter values.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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Fault properties
1.0
0.1
0.01
0.001
0.0001
0.00001
Figure 7. Cells in the simulation model (equivalent to Figure 1) that contain faults. Transmissibility multipliers are
coloured. Gray represents sealing faults. History matching was carried out by grouping the transmissibility
multipliers (indicated by rings) and applying a variable multiplier across each group.
Fault transmissibility multipliers (Figure 7)
were altered by the operator when manually
total misfit
8
history matching the model. We therefore
Injector 1
Injector
2
7
treated these parameters as uncertain in areas
seismic
around the injectors and producers. When
6
history matching, the misfit is the sum of
5
misfits for seismic and well data (including a
4
misfit for injection rate as simulation wells
3
may switch to pressure control if the Bottom
2
Hole Pressure limit is exceeded). Figure 8
1
shows the evolution of these misfits. The
0
producer misfits are negligible compared to
0
40
80
120
160
200
240
seismic and injector misfits. In this case,
model number
pressure around one of the injectors is still
Figure 8. Misfit variation when history matching for
too high and the well is not matching on
fault transmissibility multipliers. The total misfit is the
sum of misfits for seismic plus two injectors and two injection rate.
misfit (normalised to minimum)
9
producers.
Rock Physics parameters
We investigated the impact of the parameters in Equations (1-3). Figure 9 shows contour plots of
impedance as a function of saturation and pressure calculated for two values of Ek in Equation 1.
Reducing Ek increases the relative impact of saturation. Taking a realisation of the model, we
optimised for separate values of Ek and Pk in sand and shale. Figure 10a shows the change in
misfit during the history matching procedure. The pressure constant, Pk rapidly reaches the final
value (Figure 10b) while the Ek constants take longer (Figure 10c).
Discussion and conclusions
We have developed a seismic history matching method based on a multiple model approach
using a quasi-global stochastic method in a Bayesian framework. Our approach contrasts with
those of others, who have tended to use gradient based methods to determine new parameter
values (e.g. Dong and Oliver, 2003, Gosselin et al., 2003, Mezghani, et al, 2004). Similar to
others, however, we compare seismic impedance/attributes rather than forward modelling
seismic into the time domain. This reduces complexity of the modelling process as well as
simulation time. Mezghani et al. (2004) determine impedances by downscaling saturations and
pressures prior to application of the petro-elastic transform. This is advantageous in that
impedances reflect the fine grid geological model. However, each model generation requires an
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upscaling step prior to flow simulation adding uncontrollable errors as well as increasing the
demand on computer resources for downscaling, upscaling and application of the covariance in
the misfit calculations. To avoid this, we calculate impedances on the simulation grid. While we
have downscaled predicted seismic, we recognise that this reduces the resolution. In future we
will investigate comparison of seismic on the simulation grid.
Overall, we find that we must match the baseline survey to improve our chances of matching 4D
seismic, similar to other workers (Gosselin et al, 2003). The uncertainty of the rock physics is
important also. Given uncertainty issues in the upscaling and representivity of core derived rock
physics parameters, we would recommend that they be included as unknowns in the history
matching process. Their uncertainty can then be captured more effectively.
(a)
0.7
6.2
6.4
6.6
6.8
0.5
0.4
7.0
0.3
2000
3000
4000
5000
6000
Pressure
misfit (normalised)
Sw
0.6
(b)
(a)
10
1
40
Pk
0.7
0.6
Sw
100
0.3
(c)
sand
shale
30
20
0
1.2
iteration
1.0
Ek
0.4
sand
shale
10
6.8
6.9
7.0
7.1
7.2
0.5
(b)
0.8
0.6
0.4
2000
3000
4000
5000
6000
Pressure
Figure 9. Impedance as a function of saturation and
pressure for cells with NTG = 1.0 and φ = 0.3. The
contour plots were obtained using different parameters
in Equations 1-3 with (a) Ek=0.5 and (b) Ek=0.25. Also
shown are trajectories of pressure and saturation for
three cells within the model located close to injector 1
(solid circles), injector 2 (empty circles) and near
producer 1 (triangles).
0.2
0
50
100
150
200
250
300
350
400
model number
Figure 10. (a) Change in misfit (normalized to the minimum
obtained due to resolution limts) when optimizing for
parameters Ek and Pk in Equation 1, (b) Ek values and (c) Pk
values.
Conclusions
9th
•
A multi-model approach to history matching has been developed to include seismic
data systematically.
•
Base line seismic should be matched prior to history matching using time-lapse.
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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•
Reservoir description parameters such as fault transmissibility multipliers and rock
physics parameters in Gassmann’s equation are all important and simultaneous
matching is necessary.
Acknowledgements
The following organizations are thanked for funding of this work: Amerada Hess Corporation, BG Group, BP,
Chevron-Texaco, Kerr McGee Corporation, Shell, Total and The UK Department for Trade and Industry, Enterprise
Oil. We thank Schlumberger Geoquest and Hampson-Russel for use of their software. The EPSRC and IBM are
thanked for funding computer resources.
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