Automatic History Matching in a Bayesian Framework, Example Applications
- Fengjun Zhang (Chevron Corp.) | Jan-Arild Skjervheim (University of Bergen) | Albert C. Reynolds (U. of Tulsa) | Dean S. Oliver (U. of Oklahoma)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- June 2005
- Document Type
- Journal Paper
- 214 - 223
- 2005. Society of Petroleum Engineers
- 3.3.1 Production Logging, 5.8.7 Carbonate Reservoir, 4.3.4 Scale, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation, 3.3 Well & Reservoir Surveillance and Monitoring, 5.5.8 History Matching, 5.6.4 Drillstem/Well Testing, 3.3.6 Integrated Modeling, 5.1 Reservoir Characterisation, 5.2.1 Phase Behavior and PVT Measurements, 5.4.2 Gas Injection Methods, 5.1.1 Exploration, Development, Structural Geology, 5.5.3 Scaling Methods, 5.6.3 Pressure Transient Testing, 5.6.1 Open hole/cased hole log analysis, 7.6.2 Data Integration, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 5.1.5 Geologic Modeling
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The Bayesian framework allows one to integrate production and static datainto an a posteriori probability density function (pdf) for reservoir variables(model parameters). The problem of generating realizations of the reservoirvariables for the assessment of uncertainty in reservoir description orpredicted reservoir performance then becomes a problem of sampling this aposteriori pdf to obtain a suite of realizations. Generation of a realizationby the randomized-maximum-likelihood method requires the minimization of anobjective function that includes production-data misfit terms and a modelmisfit term that arises from a prior model constructed from static data.Minimization of this objective function with an optimization algorithm isequivalent to the automatic history matching of production data, with a priormodel constructed from static data providing regularization. Because of thecomputational cost of computing sensitivity coefficients and the need to solvematrix problems involving the covariance matrix for the prior model, thisapproach has not been applied to problems in which the number of data and thenumber of reservoir-model parameters are both large and the forward problem issolved by a conventional finite-difference simulator.
In this work, we illustrate that computational efficiency problems can beovercome by using a scaled limited-memory Broyden-Fletcher-Goldfarb-Shanno(LBFGS) algorithm to minimize the objective function and by using approximatecomputational stencils to approximate the multiplication of a vector by theprior covariance matrix or its inverse. Implementation of the LBFGS methodrequires only the gradient of the objective function, which can be obtainedfrom a single solution of the adjoint problem; individual sensitivitycoefficients are not needed. We apply the overall process to two examples. Thefirst is a true field example in which a realization of log permeabilities at26,019 gridblocks is generated by the automatic history matching of pressuredata, and the second is a pseudofield example that provides a very roughapproximation to a North Sea reservoir in which a realization of logpermeabilities at 9,750 gridblocks is computed by the automatic historymatching of gas/oil ratio (GOR) and pressure data.
The Bayes theorem provides a general framework for updating a pdf as newdata or information on the model becomes available. The Bayesian setting offersa distinct advantage. If one can generate a suite of realizations thatrepresent a correct sampling of the a posteriori pdf, then the suite of samplesprovides an assessment of the uncertainty in reservoir variables. Moreover, bypredicting future reservoir performance under proposed operating conditions foreach realization, one can characterize the uncertainty in future performancepredictions by constructing statistics for the set of outcomes. Liu and Oliverhave recently presented a comparison of methods for sampling the a posterioripdf. Their results indicate that the randomized-maximum-likelihood method isadequate for evaluating uncertainty with a relatively limited number ofsamples. In this work, we consider the case in which a prior geostatisticalmodel constructed from static data is available and is represented by amultivariate Gaussian pdf.Then, the a posteriori pdf conditional toproduction data is such that calculation of the maximum a posteriori estimateor generation of a realization by the randomized-maximum-likelihood method isequivalent to the minimization of an appropriate objective function.
History-matching problems of interest to us involve a few thousand to tensof thousands of reservoir variables and a few hundred to a few thousandproduction data. Thus, an optimization algorithm suitable for large-scaleproblems is needed. Our belief is that nongradient-based algorithms such assimulated annealing and the genetic algorithm are not competitive withgradient-based algorithms in terms of computational efficiency. Classicalgradient-based algorithms such as the Gauss-Newton and Levenberg-Marquardttypically converge fairly quickly and have been applied successfully toautomatic history matching for both single-phase- and multiphase-flow problems.No multiphase-flow example considered in these papers involved more than 1,500reservoir variables. For single-phase-flow problems, He et al. and Reynolds etal. have generated realizations of models involving up to 12,500 reservoirvariables by automatic history matching of pressure data. However, they used aprocedure based on their generalization of the method of Carter et al. tocalculate sensitivity coefficients; this method assumes that thepartial-differential equation solved by reservoir simulation is linear and doesnot apply for multiphase-flow problems.
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