Critical Evaluation of the Ensemble Kalman Filter on History Matching of Geologic Facies
- Ning Liu (Chevron Corp.) | Dean S. Oliver (U. of Oklahoma)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- December 2005
- Document Type
- Journal Paper
- 470 - 477
- 2005. Society of Petroleum Engineers
- 5.6.9 Production Forecasting, 5.2 Reservoir Fluid Dynamics, 1.2.3 Rock properties, 5.1 Reservoir Characterisation, 3.3 Well & Reservoir Surveillance and Monitoring, 5.1.5 Geologic Modeling, 5.3.1 Flow in Porous Media, 5.5 Reservoir Simulation, 5.5.8 History Matching
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The objective of this paper is to compare the performance of the ensembleKalman filter (EnKF) to the performance of a gradient-based minimization methodfor the problem of estimation of facies boundaries in history matching. TheEnKF is a Monte Carlo method for data assimilation that uses an ensemble ofreservoir models to represent and update the covariance of variables. Inseveral published studies, it outperforms traditional history-matchingalgorithms in adaptability and efficiency.
Because of the approximate nature of the EnKF, the realizations from oneensemble tend to underestimate uncertainty, especially for problems that arehighly nonlinear. In this paper, the distributions of reservoir-modelrealizations from 20 independent ensembles are compared with the distributionsfrom 20 randomized-maximum-likelihood (RML) realizations for a 2D waterfloodmodel with one injector and four producers. RML is a gradient-based samplingmethod that generates one reservoir realization in each minimization of theobjective function. It is an approximate sampling method, but its samplingproperties are similar to the Markov-chain Monte Carlo (McMC) method on highlynonlinear problems and are relatively more efficient than McMC.
Despite the nonlinear relationship between the data (such as productionrates and facies observations) and the model variables, the EnKF was effectiveat history matching the production data. We find that the computational effortto generate 20 independent realizations was similar for the two methods,although the complexity of the code is substantially less for the EnKF.
Several questions regarding the use of the EnKF for history matching areaddressed in this paper. The most important is a comparison of the efficiencywith a gradient-based method for a history-matching problem with known faciesproperties but unknown boundary locations. Secondly, the EnKF and agradient-based method are unlikely to give identical estimates of modelvariables, so it is also important to know if one method generates betterrealizations. Finally, because there is often a desire to use thehistory-matched realizations to quantify uncertainty, it is important todetermine if one of the methods is more efficient at generating independentrealizations.
Gradient-based history matching can be performed in several ways (e.g.,assimilating data in batch or sequentially); a variety of minimizationalgorithms can be used (e.g., conjugate gradient or quasi-Newton); and severaldifferent methods for computing the gradient are available (e.g., adjoint orsensitivity equations). In this paper, we use what we believe is the mostefficient of the traditional gradient-based methods: an adjoint method tocompute the gradient of the squared data mismatch and the limited-memoryBroyden-Fletcher-Goldfarb-Shanno (LBFGS) method to compute the direction of thechange. The remaining choice is whether to incorporate all data at once orsequentially. Simultaneous, or batch, inversion of all data is clearly awell-established history-matching procedure. Although data from wells orsensors may arrive nearly continuously, the practice of updating reservoirmodels as the data arrive is not common. There are several reasons that makesequential assimilation of data difficult for large, nonlinear models: (1) thecovariance for all model variables must be updated as new data are assimilated,but the covariance matrix is very large; (2) the covariance may not be a goodmeasure of uncertainty for nonlinear problems; and (3) the sensitivity of adatum to changes in values of model variables is expensive to compute.
Bayesian updating in general is described by Woodbury. Modifying a methoddescribed by Tarantola, Oliver evaluated the possibility of using a sequentialassimilation approach for transient flow in porous media. He found that theresults from sequential assimilation could be almost as good as those frombatch assimilation if the order of the data was carefully selected. The problemwas quite small, however, and an extension to large models was impractical.Although a sequential method has the advantage of generating a sequence ofhistory-matched models that may all be useful at the time they are generated,our comparisons of efficiency will be based primarily on the effort required toassimilate all the data. If the intermediate predictions are needed (as theywould be for control of a reservoir), the comparison provided here willunderestimate the value of the sequential assimilation.
A secondary objective of history matching is often to assess the uncertaintyin the predictions of future reservoir performance or in the estimates ofreservoir properties such as permeability, porosity, or saturation. In general,uncertainty is estimated from an examination of a moderate number ofconditional simulations of the prediction or properties. Unless therealizations are generated fairly carefully and the sample is sufficientlylarge, however, the estimate of uncertainty could be quite poor. Two largecomparative studies of the ability of Monte Carlo methods to quantifyuncertainty in history matching have been carried out, one in groundwater andone in petroleum. Neither was conclusive, partly because of the small samplesize. Liu and Oliver used a smaller reservoir model (fewer variables), but amuch larger sample size. They found that the method that minimizes an objectivefunction containing a model mismatch part and a data mismatch part, with noiseadded to observations, created realizations that were distributed nearly thesame as realizations from McMC.
The EnKF is a Monte Carlo method for updating reservoir models. It solvesseveral problems with the application of the Kalman filter to large nonlinearproblems. It has been applied to reservoir flow problems with generally goodresults. There has been no examination, however, of the distribution of themembers of a single ensemble. The adequacy of the uncertainty estimate iscompletely unknown. In the first paper on the EnKF, Evensen described how theevolution of the probability density function for the model variables can beapproximated by the motion of "particles" or ensemble members in phase space.Any desired statistical quantities can be estimated from the ensemble ofpoints. When the size of the ensemble is relatively small, however, theapproximation of the covariance from the ensemble almost certainly containssubstantial errors. Houtekamer and Mitchell noted the tendency for a reductionin variance caused by "inbreeding." When the ensemble estimate is used in aKalman filter, van Leeuwen explained how nonlinearity in the covariance updaterelation causes growth in the error as additional data are assimilated. In thispaper, the comparison is made using history matching on a truncatedpluri-gaussian model for geologic facies. It provides a difficulthistory-matching problem with significant nonlinearities that make both theEnKF and the LBFGS method difficult to apply.
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