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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

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

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

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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…

Generate MultipleModels

Compare Seismic & Reservoir Model

Evaluate Misfit

Update Probabilities

Sample from distribution

M=Σ…

Generate MultipleModels

Compare Seismic & Reservoir Model

Evaluate Misfit

Update Probabilities

Sample from distribution

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):

)P/Pexp(E rkeff

rk

rinfr

dry −+

κ=κ

1 (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

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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

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:

grf

rdry

rsat

)(

κφ−α

+κφ

α−+κ=κ

21 (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 r

sat+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:

cellp MVI ρρ == (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:

errori

modi

obsii,di,m

errori

modi

obsii xxxCCxxxJ +−++−= −− 11 (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, xi

error 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 :

∑=

)m(p)m|O(p)m(p)m|O(p)O|m(p

ii

(5)

where p(O|m) is the likelihood of the observed data obtained from

⎟⎠

⎞⎜⎝

⎛∝ ∑ 21 /Jexp/)O|m(p

ii (6)

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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.

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.

(a)

(b)

(c)

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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

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.

Producers

29 571 1113 1655 2196

Injectors

1 2

3 4

5 6

Producers

29 571 1113 1655 2196

Injectors Producers

29 571 1113 1655 219629 571 1113 1655 2196

Injectors

1 2

3 4

5 6

Figure 3. Plan view of permeability showing block multipliers. The same value was applied throughout each group of cells.

number of iterations

0 2000 4000 6000 8000

mis

fit

10

100

1000

10000

100000

1000000

Figure 4. Misfit value versus number of simulation runs when matching to production data only. A local minimum misfit was obtained.

number of iterations

0 500 1000 1500 2000 2500 3000

mis

fit

0.01

0.1

1

10

100

1000

10000

Figure 5. Misfit versus number of iterations matching to both production and seismic data. Weighting factors scale the misfit relative to Figure 4.

multiplier

0 1 2 3 4 5 6

prob

abilit

y

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

123456

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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Fault properties

1.0

0.01

0.0001

0.001

0.00001

0.1

1.0

0.01

0.0001

0.001

0.00001

0.1

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.

model number

0 40 80 120 160 200 240

mis

fit (n

orm

alis

ed to

min

imum

)

0

1

2

3

4

5

6

7

8

9total misfitInjector 1Injector 2seismic

Figure 8. Misfit variation when history matching for fault transmissibility multipliers. The total misfit is the sum of misfits for seismic plus two injectors and two producers.

Fault transmissibility multipliers (Figure 7) were altered by the operator when manually history matching the model. We therefore treated these parameters as uncertain in areas around the injectors and producers. When history matching, the misfit is the sum of misfits for seismic and well data (including a misfit for injection rate as simulation wells may switch to pressure control if the Bottom Hole Pressure limit is exceeded). Figure 8 shows the evolution of these misfits. The producer misfits are negligible compared to seismic and injector misfits. In this case, pressure around one of the injectors is still too high and the well is not matching on injection rate.

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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9th European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004

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.

Pressure2000 3000 4000 5000 6000

Sw

0.3

0.4

0.5

0.6

0.7

6.2 6.4 6.6 6.8 7.0

Pressure2000 3000 4000 5000 6000

Sw

0.3

0.4

0.5

0.6

0.7

6.8 6.9 7.0 7.1 7.2

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).

iteration

P k

0

10

20

30

40sandshale

model number

0 50 100 150 200 250 300 350 400

E k

0.2

0.4

0.6

0.8

1.0

1.2 sandshale

mis

fit (n

orm

alis

ed)

1

10

100

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

• 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.

(a)

(b)

(a)

(b)

(c)

8

• 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.

References

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