Multiresolution Wavelet Analysis for Improved Reservoir Description
- Isha Sahni (Stanford U.) | Roland N. Horne (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- February 2005
- Document Type
- Journal Paper
- 53 - 69
- 2005. Society of Petroleum Engineers
- 5.6.5 Tracers, 5.5 Reservoir Simulation, 7.2.1 Risk, Uncertainty and Risk Assessment, 5.8.6 Naturally Fractured Reservoir, 4.3.4 Scale, 5.3.2 Multiphase Flow, 5.1 Reservoir Characterisation, 4.1.2 Separation and Treating, 5.1.5 Geologic Modeling, 5.5.8 History Matching, 5.6.4 Drillstem/Well Testing, 6.1.5 Human Resources, Competence and Training, 1.6 Drilling Operations, 7.6.2 Data Integration, 5.6.9 Production Forecasting, 4.1.5 Processing Equipment
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It is well documented that history matching is a problem with possiblynonunique solutions. In the past few years, several automated or semiautomatedhistory-matching algorithms have been proposed. Depending on the algorithmused, it is possible that the final estimated reservoir-property distributionthat allows for a good history match may not be geologically realistic.Therefore, there is a need to include other constraints to generate multiple,geologically realistic history-matched realizations. These constraints might,for example, include the variogram, a training image, the distribution ofnet-to-gross, pore volume, or other geostatistical information about thereservoir. This inclusion is particularly useful because it introducesuncertainty information in the reservoir description when we have limitedhistory from existing wells in the field and intend to drill infill wells orimplement a secondary-recovery process.
The algorithm proposed in this paper uses multiresolution wavelet analysisto integrate history data with the geostatistical information contained in thevariogram proposed for the reservoir. Wavelets allow the representation andmanipulation of property distributions at various resolutions at the same time.Using wavelets, information from different sources such as production historyand seismic surveys (that would be at different resolutions) can beincorporated directly at the appropriate resolution level. In the first step,we fix the wavelet coefficients sensitive to the history-match data. This hasthe effect of fixing the field history without fixing individual gridblockproperties. In the second step, the remaining free wavelet coefficients aremodified to integrate variogram information into the reservoir description.Generating multiple realizations of only the second set of waveletscoefficients results in multiple history-matched, variogram-constraineddescriptions of the reservoir. The computational investment is very modestbecause the history match is done only once.
In a number of example cases, different areal Gaussian fields with varyingamounts of available production-history data were studied to test thealgorithm. It was found that the wavelet coefficients constraining the historycan be decoupled from those constraining the variogram. The implication of thisobservation is that the history data and variogram can be integratedsequentially into the reservoir model—that is, after the initial history match,new information can be added to the model without disturbing the original matchto yield multiple history-matched and geostatistically constrainedrealizations.
Reservoir modeling is essential for forecasting the performance of areservoir, for reservoir management, for risk analysis, and for making keyeconomic decisions. The purpose of reservoir modeling is to develop a model ofthe reservoir that closely resembles the actual reservoir based on availableinformation. This model then can be used to forecast future performance andoptimize reservoir-management decisions. The more accurate the reservoir model,the better the predictions will be. Therein lies the importance of generating agood reservoir model. History matching is but one step in this direction.Merely achieving a good history match does not ensure sound predictions fromthe reservoir model; it is therefore essential that all sources of informationabout the reservoir be used appropriately to come up with a good model.
Early automated history-matching procedures were discussed by Jacquard andJain,1 adapted from variational analysis in electric networking. Since then,there have been several developments of concepts and algorithms along similarlines. In general, the objective is to determine the spatial distribution of aset of gridded reservoir properties such as permeabilities and porosities,given the response of the field in terms of fluid flow to an external impulsesuch as drainage and injection of fluids, as well as geostatistical data.Production history from existing wells is an important source of informationabout the reservoir, in terms of the average permeabilities, spatialdistribution of permeabilities, net-to-gross, etc. Production history could bein the form of the pressure or saturation distribution in the reservoir inresponse to injection or production impulses. A good reservoir model musttherefore, when run through a flow simulator, give the same response to thesame impulse as the real reservoir. Many studies have shown favorable resultsfrom integrating dynamic data into reservoir modeling using streamlinesimulators (e.g., Datta-Gupta et al.2).
However, not only does history matching alone not ensure sound productionforecast, it also does not guarantee physical consistency and might produceartifacts based on the algorithm used. The results thus obtained might give aperfect history match, but if they are aphysical, use of the model will lead tofurther error in prediction of future performance because the model may not beclose enough in a geological sense to the actual reservoir. This situationarises because there may be a number of different solutions to thehistory-matching problem. In other words, a number of different permeabilitydistributions may be found, all of which give the same response to a givenimpulse. As such, we need to integrate geostatistical data that will constrainthe problem and make the model more realistic. Landa and Horne3 and Landa4investigated the impact of different data on reservoir characterization anduncertainty. Integration of static and dynamic data into reservoir models hasbeen attempted in the Bayesian framework5-7 and with gradual deformation.8
Multiresolution wavelet analysis forms the basis for efficientrepresentation of the field as well as a reduction in the number of parametersto be estimated. As described in the following section, the griddedreservoir-property distribution can be transformed linearly to give a uniqueset of wavelet coefficients. It has been found9,10 that a specific subset ofthese wavelet coefficients is sufficient to determine the response of thereservoir to production. The conjecture is that the remaining set can bemodified subject to constraints based on geological, seismic, or othersubjective information about the spatial distribution of the permeabilities.This study showed that the sets of wavelets constraining the history match andthose constraining geostatistical parameters (variograms in particular) canindeed be decoupled and evaluated separately to yield a set of differentpermeability distributions stochastically.
Most history-matching algorithms involve flow simulation at each iterationwhile minimizing the objective function. The advantage of our new algorithm isthat instead of doing repeated history matches, it fixes a set of waveletcoefficients that constrain the history, thereby fixing the history up to sometolerance. The objective function endeavors to enforce a proposed variogram ofspatial distribution of the permeabilities. As such, the algorithm takesorders-of-magnitude less time to yield permeability distributions that areconstrained by both the history and the variogram of the field.
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